LMFDB - Embedded newform 42.6.e.b.25.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [42,6,Mod(25,42)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(42, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("42.25");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 42 = 2 \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	42.e (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.73612043215$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			25.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	42.25
Dual form			42.6.e.b.37.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 + 3.46410i) q^{2} +(4.50000 - 7.79423i) q^{3} +(-8.00000 + 13.8564i) q^{4} +(-43.0000 - 74.4782i) q^{5} +36.0000 q^{6} +(24.5000 - 127.306i) q^{7} -64.0000 q^{8} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})$	\(q+(2.00000 + 3.46410i) q^{2} +(4.50000 - 7.79423i) q^{3} +(-8.00000 + 13.8564i) q^{4} +(-43.0000 - 74.4782i) q^{5} +36.0000 q^{6} +(24.5000 - 127.306i) q^{7} -64.0000 q^{8} +(-40.5000 - 70.1481i) q^{9} +(172.000 - 297.913i) q^{10} +(-17.0000 + 29.4449i) q^{11} +(72.0000 + 124.708i) q^{12} -3.00000 q^{13} +(490.000 - 169.741i) q^{14} -774.000 q^{15} +(-128.000 - 221.703i) q^{16} +(952.000 - 1648.91i) q^{17} +(162.000 - 280.592i) q^{18} +(744.500 + 1289.51i) q^{19} +1376.00 q^{20} +(-882.000 - 763.834i) q^{21} -136.000 q^{22} +(112.000 + 193.990i) q^{23} +(-288.000 + 498.831i) q^{24} +(-2135.50 + 3698.79i) q^{25} +(-6.00000 - 10.3923i) q^{26} -729.000 q^{27} +(1568.00 + 1357.93i) q^{28} -6508.00 q^{29} +(-1548.00 - 2681.21i) q^{30} +(-865.500 + 1499.09i) q^{31} +(512.000 - 886.810i) q^{32} +(153.000 + 265.004i) q^{33} +7616.00 q^{34} +(-10535.0 + 3649.43i) q^{35} +1296.00 q^{36} +(3816.50 + 6610.37i) q^{37} +(-2978.00 + 5158.05i) q^{38} +(-13.5000 + 23.3827i) q^{39} +(2752.00 + 4766.60i) q^{40} +15414.0 q^{41} +(882.000 - 4583.01i) q^{42} +18491.0 q^{43} +(-272.000 - 471.118i) q^{44} +(-3483.00 + 6032.73i) q^{45} +(-448.000 + 775.959i) q^{46} +(-9231.00 - 15988.6i) q^{47} -2304.00 q^{48} +(-15606.5 - 6237.98i) q^{49} -17084.0 q^{50} +(-8568.00 - 14840.2i) q^{51} +(24.0000 - 41.5692i) q^{52} +(9978.00 - 17282.4i) q^{53} +(-1458.00 - 2525.33i) q^{54} +2924.00 q^{55} +(-1568.00 + 8147.57i) q^{56} +13401.0 q^{57} +(-13016.0 - 22544.4i) q^{58} +(15914.0 - 27563.9i) q^{59} +(6192.00 - 10724.9i) q^{60} +(28827.0 + 49929.8i) q^{61} -6924.00 q^{62} +(-9922.50 + 3437.25i) q^{63} +4096.00 q^{64} +(129.000 + 223.435i) q^{65} +(-612.000 + 1060.02i) q^{66} +(30281.5 - 52449.1i) q^{67} +(15232.0 + 26382.6i) q^{68} +2016.00 q^{69} +(-33712.0 - 29195.4i) q^{70} -44834.0 q^{71} +(2592.00 + 4489.48i) q^{72} +(-10410.5 + 18031.5i) q^{73} +(-15266.0 + 26441.5i) q^{74} +(19219.5 + 33289.2i) q^{75} -23824.0 q^{76} +(3332.00 + 2885.60i) q^{77} -108.000 q^{78} +(15265.5 + 26440.6i) q^{79} +(-11008.0 + 19066.4i) q^{80} +(-3280.50 + 5681.99i) q^{81} +(30828.0 + 53395.7i) q^{82} +110602. q^{83} +(17640.0 - 6110.68i) q^{84} -163744. q^{85} +(36982.0 + 64054.7i) q^{86} +(-29286.0 + 50724.8i) q^{87} +(1088.00 - 1884.47i) q^{88} +(29496.0 + 51088.6i) q^{89} -27864.0 q^{90} +(-73.5000 + 381.917i) q^{91} -3584.00 q^{92} +(7789.50 + 13491.8i) q^{93} +(36924.0 - 63954.2i) q^{94} +(64027.0 - 110898. i) q^{95} +(-4608.00 - 7981.29i) q^{96} -119846. q^{97} +(-9604.00 - 66538.5i) q^{98} +2754.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 9 q^{3} - 16 q^{4} - 86 q^{5} + 72 q^{6} + 49 q^{7} - 128 q^{8} - 81 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 + 9 * q^3 - 16 * q^4 - 86 * q^5 + 72 * q^6 + 49 * q^7 - 128 * q^8 - 81 * q^9	\( 2 q + 4 q^{2} + 9 q^{3} - 16 q^{4} - 86 q^{5} + 72 q^{6} + 49 q^{7} - 128 q^{8} - 81 q^{9} + 344 q^{10} - 34 q^{11} + 144 q^{12} - 6 q^{13} + 980 q^{14} - 1548 q^{15} - 256 q^{16} + 1904 q^{17} + 324 q^{18} + 1489 q^{19} + 2752 q^{20} - 1764 q^{21} - 272 q^{22} + 224 q^{23} - 576 q^{24} - 4271 q^{25} - 12 q^{26} - 1458 q^{27} + 3136 q^{28} - 13016 q^{29} - 3096 q^{30} - 1731 q^{31} + 1024 q^{32} + 306 q^{33} + 15232 q^{34} - 21070 q^{35} + 2592 q^{36} + 7633 q^{37} - 5956 q^{38} - 27 q^{39} + 5504 q^{40} + 30828 q^{41} + 1764 q^{42} + 36982 q^{43} - 544 q^{44} - 6966 q^{45} - 896 q^{46} - 18462 q^{47} - 4608 q^{48} - 31213 q^{49} - 34168 q^{50} - 17136 q^{51} + 48 q^{52} + 19956 q^{53} - 2916 q^{54} + 5848 q^{55} - 3136 q^{56} + 26802 q^{57} - 26032 q^{58} + 31828 q^{59} + 12384 q^{60} + 57654 q^{61} - 13848 q^{62} - 19845 q^{63} + 8192 q^{64} + 258 q^{65} - 1224 q^{66} + 60563 q^{67} + 30464 q^{68} + 4032 q^{69} - 67424 q^{70} - 89668 q^{71} + 5184 q^{72} - 20821 q^{73} - 30532 q^{74} + 38439 q^{75} - 47648 q^{76} + 6664 q^{77} - 216 q^{78} + 30531 q^{79} - 22016 q^{80} - 6561 q^{81} + 61656 q^{82} + 221204 q^{83} + 35280 q^{84} - 327488 q^{85} + 73964 q^{86} - 58572 q^{87} + 2176 q^{88} + 58992 q^{89} - 55728 q^{90} - 147 q^{91} - 7168 q^{92} + 15579 q^{93} + 73848 q^{94} + 128054 q^{95} - 9216 q^{96} - 239692 q^{97} - 19208 q^{98} + 5508 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 9 * q^3 - 16 * q^4 - 86 * q^5 + 72 * q^6 + 49 * q^7 - 128 * q^8 - 81 * q^9 + 344 * q^10 - 34 * q^11 + 144 * q^12 - 6 * q^13 + 980 * q^14 - 1548 * q^15 - 256 * q^16 + 1904 * q^17 + 324 * q^18 + 1489 * q^19 + 2752 * q^20 - 1764 * q^21 - 272 * q^22 + 224 * q^23 - 576 * q^24 - 4271 * q^25 - 12 * q^26 - 1458 * q^27 + 3136 * q^28 - 13016 * q^29 - 3096 * q^30 - 1731 * q^31 + 1024 * q^32 + 306 * q^33 + 15232 * q^34 - 21070 * q^35 + 2592 * q^36 + 7633 * q^37 - 5956 * q^38 - 27 * q^39 + 5504 * q^40 + 30828 * q^41 + 1764 * q^42 + 36982 * q^43 - 544 * q^44 - 6966 * q^45 - 896 * q^46 - 18462 * q^47 - 4608 * q^48 - 31213 * q^49 - 34168 * q^50 - 17136 * q^51 + 48 * q^52 + 19956 * q^53 - 2916 * q^54 + 5848 * q^55 - 3136 * q^56 + 26802 * q^57 - 26032 * q^58 + 31828 * q^59 + 12384 * q^60 + 57654 * q^61 - 13848 * q^62 - 19845 * q^63 + 8192 * q^64 + 258 * q^65 - 1224 * q^66 + 60563 * q^67 + 30464 * q^68 + 4032 * q^69 - 67424 * q^70 - 89668 * q^71 + 5184 * q^72 - 20821 * q^73 - 30532 * q^74 + 38439 * q^75 - 47648 * q^76 + 6664 * q^77 - 216 * q^78 + 30531 * q^79 - 22016 * q^80 - 6561 * q^81 + 61656 * q^82 + 221204 * q^83 + 35280 * q^84 - 327488 * q^85 + 73964 * q^86 - 58572 * q^87 + 2176 * q^88 + 58992 * q^89 - 55728 * q^90 - 147 * q^91 - 7168 * q^92 + 15579 * q^93 + 73848 * q^94 + 128054 * q^95 - 9216 * q^96 - 239692 * q^97 - 19208 * q^98 + 5508 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/42\mathbb{Z}\right)^\times$.

$n$	$29$	$31$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	2.00000	+	3.46410i	0.353553	+	0.612372i
$2$	2.00000	+	3.46410i	0.353553	+	0.612372i
$3$	4.50000	−	7.79423i	0.288675	−	0.500000i
$3$	4.50000	−	7.79423i	0.288675	−	0.500000i
$4$	−8.00000	+	13.8564i	−0.250000	+	0.433013i
$5$	−43.0000	−	74.4782i	−0.769207	−	1.33231i	−0.937993	−	0.346654i	$-0.887318\pi$
$5$	−43.0000	−	74.4782i	−0.769207	−	1.33231i	0.168786	−	0.985653i	$-0.446015\pi$
$6$	36.0000			0.408248
$7$	24.5000	−	127.306i	0.188982	−	0.981981i
$7$	24.5000	−	127.306i	0.188982	−	0.981981i
$8$	−64.0000			−0.353553
$9$	−40.5000	−	70.1481i	−0.166667	−	0.288675i
$10$	172.000	−	297.913i	0.543912	−	0.942083i
$11$	−17.0000	+	29.4449i	−0.0423611	+	0.0733716i	−0.886429	−	0.462865i	$-0.846821\pi$
$11$	−17.0000	+	29.4449i	−0.0423611	+	0.0733716i	0.844067	+	0.536237i	$0.180155\pi$
$12$	72.0000	+	124.708i	0.144338	+	0.250000i
$13$	−3.00000			−0.00492337			−0.00246169	−	0.999997i	$-0.500784\pi$
$13$	−3.00000			−0.00492337			−0.00246169	+	0.999997i	$0.500784\pi$
$14$	490.000	−	169.741i	0.668153	−	0.231455i
$15$	−774.000			−0.888204
$16$	−128.000	−	221.703i	−0.125000	−	0.216506i
$17$	952.000	−	1648.91i	0.798941	−	1.38381i	−0.121366	−	0.992608i	$-0.538727\pi$
$17$	952.000	−	1648.91i	0.798941	−	1.38381i	0.920306	−	0.391198i	$-0.127939\pi$
$18$	162.000	−	280.592i	0.117851	−	0.204124i
$19$	744.500	+	1289.51i	0.473130	+	0.819486i	0.999527	−	0.0307534i	$-0.00979067\pi$
$19$	744.500	+	1289.51i	0.473130	+	0.819486i	−0.526397	+	0.850239i	$0.676457\pi$
$20$	1376.00			0.769207
$21$	−882.000	−	763.834i	−0.436436	−	0.377964i
$22$	−136.000			−0.0599076
$23$	112.000	+	193.990i	0.0441467	+	0.0764644i	0.887254	−	0.461280i	$-0.152610\pi$
$23$	112.000	+	193.990i	0.0441467	+	0.0764644i	−0.843108	+	0.537745i	$0.819276\pi$
$24$	−288.000	+	498.831i	−0.102062	+	0.176777i
$25$	−2135.50	+	3698.79i	−0.683360	+	1.18361i
$26$	−6.00000	−	10.3923i	−0.00174068	−	0.00301494i
$27$	−729.000			−0.192450
$28$	1568.00	+	1357.93i	0.377964	+	0.327327i
$29$	−6508.00			−1.43699			−0.718493	−	0.695534i	$-0.755168\pi$
$29$	−6508.00			−1.43699			−0.718493	+	0.695534i	$0.755168\pi$
$30$	−1548.00	−	2681.21i	−0.314028	−	0.543912i
$31$	−865.500	+	1499.09i	−0.161757	+	0.280171i	−0.935499	−	0.353330i	$-0.885049\pi$
$31$	−865.500	+	1499.09i	−0.161757	+	0.280171i	0.773742	+	0.633501i	$0.218383\pi$
$32$	512.000	−	886.810i	0.0883883	−	0.153093i
$33$	153.000	+	265.004i	0.0244572	+	0.0423611i
$34$	7616.00			1.12987
$35$	−10535.0	+	3649.43i	−1.45367	+	0.503564i
$36$	1296.00			0.166667
$37$	3816.50	+	6610.37i	0.458312	+	0.793819i	0.998872	−	0.0474862i	$-0.0151210\pi$
$37$	3816.50	+	6610.37i	0.458312	+	0.793819i	−0.540560	+	0.841305i	$0.681788\pi$
$38$	−2978.00	+	5158.05i	−0.334554	+	0.579464i
$39$	−13.5000	+	23.3827i	−0.00142126	+	0.00246169i
$40$	2752.00	+	4766.60i	0.271956	+	0.471041i
$41$	15414.0			1.43204			0.716021	−	0.698079i	$-0.245961\pi$
$41$	15414.0			1.43204			0.716021	+	0.698079i	$0.245961\pi$
$42$	882.000	−	4583.01i	0.0771517	−	0.400892i
$43$	18491.0			1.52507			0.762534	−	0.646948i	$-0.223955\pi$
$43$	18491.0			1.52507			0.762534	+	0.646948i	$0.223955\pi$
$44$	−272.000	−	471.118i	−0.0211805	−	0.0366858i
$45$	−3483.00	+	6032.73i	−0.256402	+	0.444102i
$46$	−448.000	+	775.959i	−0.0312164	+	0.0540685i
$47$	−9231.00	−	15988.6i	−0.609543	−	1.05576i	−0.991316	−	0.131503i	$-0.958020\pi$
$47$	−9231.00	−	15988.6i	−0.609543	−	1.05576i	0.381773	−	0.924256i	$-0.375314\pi$
$48$	−2304.00			−0.144338
$49$	−15606.5	−	6237.98i	−0.928571	−	0.371154i
$50$	−17084.0			−0.966417
$51$	−8568.00	−	14840.2i	−0.461269	−	0.798941i
$52$	24.0000	−	41.5692i	0.00123084	−	0.00213188i
$53$	9978.00	−	17282.4i	0.487926	−	0.845112i	−0.511978	−	0.858999i	$-0.671087\pi$
$53$	9978.00	−	17282.4i	0.487926	−	0.845112i	0.999904	+	0.0138864i	$0.00442031\pi$
$54$	−1458.00	−	2525.33i	−0.0680414	−	0.117851i
$55$	2924.00			0.130338
$56$	−1568.00	+	8147.57i	−0.0668153	+	0.347183i
$57$	13401.0			0.546324
$58$	−13016.0	−	22544.4i	−0.508051	−	0.879971i
$59$	15914.0	−	27563.9i	0.595181	−	1.03088i	−0.398340	−	0.917238i	$-0.630414\pi$
$59$	15914.0	−	27563.9i	0.595181	−	1.03088i	0.993521	−	0.113646i	$-0.0362531\pi$
$60$	6192.00	−	10724.9i	0.222051	−	0.384604i
$61$	28827.0	+	49929.8i	0.991916	+	1.71805i	0.605854	+	0.795576i	$0.292832\pi$
$61$	28827.0	+	49929.8i	0.991916	+	1.71805i	0.386062	+	0.922473i	$0.373835\pi$
$62$	−6924.00			−0.228759
$63$	−9922.50	+	3437.25i	−0.314970	+	0.109109i
$64$	4096.00			0.125000
$65$	129.000	+	223.435i	0.00378710	+	0.00655944i
$66$	−612.000	+	1060.02i	−0.0172938	+	0.0299538i
$67$	30281.5	−	52449.1i	0.824120	−	1.42742i	−0.0784702	−	0.996916i	$-0.525004\pi$
$67$	30281.5	−	52449.1i	0.824120	−	1.42742i	0.902590	−	0.430501i	$-0.141663\pi$
$68$	15232.0	+	26382.6i	0.399470	+	0.691903i
$69$	2016.00			0.0509762
$70$	−33712.0	−	29195.4i	−0.822317	−	0.712148i
$71$	−44834.0			−1.05551			−0.527754	−	0.849397i	$-0.676966\pi$
$71$	−44834.0			−1.05551			−0.527754	+	0.849397i	$0.676966\pi$
$72$	2592.00	+	4489.48i	0.0589256	+	0.102062i
$73$	−10410.5	+	18031.5i	−0.228646	+	0.396027i	−0.957407	−	0.288741i	$-0.906763\pi$
$73$	−10410.5	+	18031.5i	−0.228646	+	0.396027i	0.728761	+	0.684768i	$0.240097\pi$
$74$	−15266.0	+	26441.5i	−0.324075	+	0.561315i
$75$	19219.5	+	33289.2i	0.394538	+	0.683360i
$76$	−23824.0			−0.473130
$77$	3332.00	+	2885.60i	0.0640440	+	0.0554637i
$78$	−108.000			−0.00200996
$79$	15265.5	+	26440.6i	0.275197	+	0.476655i	0.970185	−	0.242367i	$-0.0779237\pi$
$79$	15265.5	+	26440.6i	0.275197	+	0.476655i	−0.694988	+	0.719021i	$0.744590\pi$
$80$	−11008.0	+	19066.4i	−0.192302	+	0.333077i
$81$	−3280.50	+	5681.99i	−0.0555556	+	0.0962250i
$82$	30828.0	+	53395.7i	0.506303	+	0.876943i
$83$	110602.			1.76225			0.881125	−	0.472883i	$-0.156787\pi$
$83$	110602.			1.76225			0.881125	+	0.472883i	$0.156787\pi$
$84$	17640.0	−	6110.68i	0.272772	−	0.0944911i
$85$	−163744.			−2.45820
$86$	36982.0	+	64054.7i	0.539193	+	0.933910i
$87$	−29286.0	+	50724.8i	−0.414822	+	0.718493i
$88$	1088.00	−	1884.47i	0.0149769	−	0.0259408i
$89$	29496.0	+	51088.6i	0.394719	+	0.683673i	0.993065	−	0.117564i	$-0.0375086\pi$
$89$	29496.0	+	51088.6i	0.394719	+	0.683673i	−0.598346	+	0.801238i	$0.704175\pi$
$90$	−27864.0			−0.362608
$91$	−73.5000	+	381.917i	−0.000930430	+	0.00483466i
$92$	−3584.00			−0.0441467
$93$	7789.50	+	13491.8i	0.0933904	+	0.161757i
$94$	36924.0	−	63954.2i	0.431012	−	0.746534i
$95$	64027.0	−	110898.i	0.727871	−	1.26071i
$96$	−4608.00	−	7981.29i	−0.0510310	−	0.0883883i
$97$	−119846.			−1.29328			−0.646642	−	0.762793i	$-0.723827\pi$
$97$	−119846.			−1.29328			−0.646642	+	0.762793i	$0.723827\pi$
$98$	−9604.00	−	66538.5i	−0.101015	−	0.699854i
$99$	2754.00			0.0282407
$100$	−34168.0	−	59180.7i	−0.341680	−	0.591807i
$101$	−50005.0	+	86611.2i	−0.487764	+	0.844833i	−0.999901	−	0.0140714i	$-0.995521\pi$
$101$	−50005.0	+	86611.2i	−0.487764	+	0.844833i	0.512137	+	0.858904i	$0.328854\pi$
$102$	34272.0	−	59360.8i	0.326166	−	0.564937i
$103$	−60845.5	−	105387.i	−0.565113	−	0.978805i	−0.997039	−	0.0768956i	$-0.975499\pi$
$103$	−60845.5	−	105387.i	−0.565113	−	0.978805i	0.431926	−	0.901909i	$-0.357834\pi$
$104$	192.000			0.00174068
$105$	−18963.0	+	98534.6i	−0.167855	+	0.872199i
$106$	79824.0			0.690031
$107$	24324.0	+	42130.4i	0.205388	+	0.355743i	0.950256	−	0.311469i	$-0.100821\pi$
$107$	24324.0	+	42130.4i	0.205388	+	0.355743i	−0.744868	+	0.667212i	$0.767488\pi$
$108$	5832.00	−	10101.3i	0.0481125	−	0.0833333i
$109$	76037.5	−	131701.i	0.613002	−	1.06175i	−0.377730	−	0.925916i	$-0.623295\pi$
$109$	76037.5	−	131701.i	0.613002	−	1.06175i	0.990732	−	0.135834i	$-0.0433714\pi$
$110$	5848.00	+	10129.0i	0.0460814	+	0.0798153i
$111$	68697.0			0.529213
$112$	−31360.0	+	10863.4i	−0.236228	+	0.0818317i
$113$	−60886.0			−0.448561			−0.224280	−	0.974525i	$-0.572003\pi$
$113$	−60886.0			−0.448561			−0.224280	+	0.974525i	$0.572003\pi$
$114$	26802.0	+	46422.4i	0.193155	+	0.334554i
$115$	9632.00	−	16683.1i	0.0679160	−	0.117634i
$116$	52064.0	−	90177.5i	0.359247	−	0.622233i
$117$	121.500	+	210.444i	0.000820562	+	0.00142126i
$118$	127312.			0.841714
$119$	−186592.	−	161593.i	−1.20789	−	1.04606i
$120$	49536.0			0.314028
$121$	79947.5	+	138473.i	0.496411	+	0.859809i
$122$	−115308.	+	199719.i	−0.701390	+	1.21484i
$123$	69363.0	−	120140.i	0.413395	−	0.716021i
$124$	−13848.0	−	23985.4i	−0.0808785	−	0.140086i
$125$	98556.0			0.564167
$126$	−31752.0	−	27498.0i	−0.178174	−	0.154303i
$127$	−151965.			−0.836054			−0.418027	−	0.908435i	$-0.637278\pi$
$127$	−151965.			−0.836054			−0.418027	+	0.908435i	$0.637278\pi$
$128$	8192.00	+	14189.0i	0.0441942	+	0.0765466i
$129$	83209.5	−	144123.i	0.440249	−	0.762534i
$130$	−516.000	+	893.738i	−0.00267788	+	0.00463823i
$131$	−117751.	−	203951.i	−0.599496	−	1.03836i	−0.992895	−	0.118990i	$-0.962034\pi$
$131$	−117751.	−	203951.i	−0.599496	−	1.03836i	0.393399	−	0.919368i	$-0.371299\pi$
$132$	−4896.00			−0.0244572
$133$	182402.	−	63186.1i	0.894132	−	0.309736i
$134$	242252.			1.16548
$135$	31347.0	+	54294.6i	0.148034	+	0.256402i
$136$	−60928.0	+	105530.i	−0.282468	+	0.489249i
$137$	−162754.	+	281898.i	−0.740850	+	1.28319i	0.211259	+	0.977430i	$0.432244\pi$
$137$	−162754.	+	281898.i	−0.740850	+	1.28319i	−0.952109	+	0.305760i	$0.901090\pi$
$138$	4032.00	+	6983.63i	0.0180228	+	0.0312164i
$139$	3211.00			0.0140962			0.00704812	−	0.999975i	$-0.497756\pi$
$139$	3211.00			0.0140962			0.00704812	+	0.999975i	$0.497756\pi$
$140$	33712.0	−	175173.i	0.145367	−	0.755347i
$141$	−166158.			−0.703839
$142$	−89668.0	−	155310.i	−0.373179	−	0.646364i
$143$	51.0000	−	88.3346i	0.000208560	−	0.000361236i
$144$	−10368.0	+	17957.9i	−0.0416667	+	0.0721688i
$145$	279844.	+	484704.i	1.10534	+	1.91451i
$146$	−83284.0			−0.323355
$147$	−118850.	+	93569.7i	−0.453632	+	0.357143i
$148$	−122128.			−0.458312
$149$	75942.0	+	131535.i	0.280231	+	0.485375i	0.971442	−	0.237279i	$-0.0762556\pi$
$149$	75942.0	+	131535.i	0.280231	+	0.485375i	−0.691210	+	0.722654i	$0.742922\pi$
$150$	−76878.0	+	133157.i	−0.278981	+	0.483208i
$151$	−38324.0	+	66379.1i	−0.136782	+	0.236913i	−0.926277	−	0.376844i	$-0.877009\pi$
$151$	−38324.0	+	66379.1i	−0.136782	+	0.236913i	0.789495	+	0.613757i	$0.210343\pi$
$152$	−47648.0	−	82528.8i	−0.167277	−	0.289732i
$153$	−154224.			−0.532627
$154$	−3332.00	+	17313.6i	−0.0113215	+	0.0588281i
$155$	148866.			0.497698
$156$	−216.000	−	374.123i	−0.000710628	−	0.00123084i
$157$	194855.	−	337499.i	0.630903	−	1.09276i	−0.356465	−	0.934309i	$-0.616018\pi$
$157$	194855.	−	337499.i	0.630903	−	1.09276i	0.987368	−	0.158447i	$-0.0506487\pi$
$158$	−61062.0	+	105762.i	−0.194593	+	0.337046i
$159$	−89802.0	−	155542.i	−0.281704	−	0.487926i
$160$	−88064.0			−0.271956
$161$	27440.0	−	9505.49i	0.0834295	−	0.0289008i
$162$	−26244.0			−0.0785674
$163$	56186.0	+	97317.0i	0.165638	+	0.286893i	0.936882	−	0.349647i	$-0.113698\pi$
$163$	56186.0	+	97317.0i	0.165638	+	0.286893i	−0.771244	+	0.636540i	$0.780365\pi$
$164$	−123312.	+	213583.i	−0.358010	+	0.620092i
$165$	13158.0	−	22790.3i	0.0376253	−	0.0651689i
$166$	221204.	+	383137.i	0.623050	+	1.07915i
$167$	52550.0			0.145808			0.0729040	−	0.997339i	$-0.476773\pi$
$167$	52550.0			0.145808			0.0729040	+	0.997339i	$0.476773\pi$
$168$	56448.0	+	48885.4i	0.154303	+	0.133631i
$169$	−371284.			−0.999976
$170$	−327488.	−	567226.i	−0.869107	−	1.50534i
$171$	60304.5	−	104450.i	0.157710	−	0.273162i
$172$	−147928.	+	256219.i	−0.381267	+	0.660374i
$173$	67628.0	+	117135.i	0.171795	+	0.297558i	0.939048	−	0.343787i	$-0.111710\pi$
$173$	67628.0	+	117135.i	0.171795	+	0.297558i	−0.767252	+	0.641345i	$0.778377\pi$
$174$	−234288.			−0.586647
$175$	418558.	+	362482.i	1.03314	+	0.894728i
$176$	8704.00			0.0211805
$177$	−143226.	−	248075.i	−0.343628	−	0.595181i
$178$	−117984.	+	204354.i	−0.279109	+	0.483430i
$179$	−125319.	+	217059.i	−0.292337	+	0.506343i	−0.974362	−	0.224986i	$-0.927766\pi$
$179$	−125319.	+	217059.i	−0.292337	+	0.506343i	0.682025	+	0.731329i	$0.261100\pi$
$180$	−55728.0	−	96523.7i	−0.128201	−	0.222051i
$181$	199233.			0.452027			0.226014	−	0.974124i	$-0.427431\pi$
$181$	199233.			0.452027			0.226014	+	0.974124i	$0.427431\pi$
$182$	−1470.00	+	509.223i	−0.00328957	+	0.00113954i
$183$	518886.			1.14537
$184$	−7168.00	−	12415.3i	−0.0156082	−	0.0270342i
$185$	328219.	−	568492.i	0.705074	−	1.22122i
$186$	−31158.0	+	53967.2i	−0.0660370	+	0.114379i
$187$	32368.0	+	56063.0i	0.0676880	+	0.117239i
$188$	295392.			0.609543
$189$	−17860.5	+	92805.9i	−0.0363696	+	0.188982i
$190$	512216.			1.02936
$191$	119385.	+	206781.i	0.236792	+	0.410135i	0.959792	−	0.280713i	$-0.0905708\pi$
$191$	119385.	+	206781.i	0.236792	+	0.410135i	−0.723000	+	0.690848i	$0.757237\pi$
$192$	18432.0	−	31925.2i	0.0360844	−	0.0625000i
$193$	−42845.5	+	74210.6i	−0.0827965	+	0.143408i	−0.904450	−	0.426579i	$-0.859718\pi$
$193$	−42845.5	+	74210.6i	−0.0827965	+	0.143408i	0.821654	+	0.569987i	$0.193052\pi$
$194$	−239692.	−	415159.i	−0.457245	−	0.791972i
$195$	2322.00			0.00437296
$196$	211288.	−	166346.i	0.392857	−	0.309295i
$197$	−71408.0			−0.131094			−0.0655468	−	0.997849i	$-0.520879\pi$
$197$	−71408.0			−0.131094			−0.0655468	+	0.997849i	$0.520879\pi$
$198$	5508.00	+	9540.14i	0.00998461	+	0.0172938i
$199$	355676.	−	616049.i	0.636681	−	1.10276i	−0.349475	−	0.936946i	$-0.613640\pi$
$199$	355676.	−	616049.i	0.636681	−	1.10276i	0.986156	−	0.165818i	$-0.0530265\pi$
$200$	136672.	−	236723.i	0.241604	−	0.418471i
$201$	−272534.	−	472042.i	−0.475806	−	0.824120i
$202$	−400040.			−0.689803
$203$	−159446.	+	828506.i	−0.271565	+	1.41109i
$204$	274176.			0.461269
$205$	−662802.	−	1.14801e6i	−1.10154	−	1.90792i
$206$	243382.	−	421550.i	0.399595	−	0.692119i
$207$	9072.00	−	15713.2i	0.0147156	−	0.0254881i
$208$	384.000	+	665.108i	0.000615422	+	0.00106594i
$209$	−50626.0			−0.0801693
$210$	−379260.	+	131380.i	−0.593456	+	0.205579i
$211$	−260260.			−0.402440			−0.201220	−	0.979546i	$-0.564491\pi$
$211$	−260260.			−0.402440			−0.201220	+	0.979546i	$0.564491\pi$
$212$	159648.	+	276518.i	0.243963	+	0.422556i
$213$	−201753.	+	349446.i	−0.304699	+	0.527754i
$214$	−97296.0	+	168522.i	−0.145231	+	0.251548i
$215$	−795113.	−	1.37718e6i	−1.17309	−	2.03186i
$216$	46656.0			0.0680414
$217$	169638.	+	146911.i	0.244553	+	0.211790i
$218$	608300.			0.866915
$219$	93694.5	+	162284.i	0.132009	+	0.228646i
$220$	−23392.0	+	40516.1i	−0.0325845	+	0.0564380i
$221$	−2856.00	+	4946.74i	−0.00393349	+	0.00681300i
$222$	137394.	+	237973.i	0.187105	+	0.324075i
$223$	105656.			0.142276			0.0711381	−	0.997466i	$-0.477337\pi$
$223$	105656.			0.142276			0.0711381	+	0.997466i	$0.477337\pi$
$224$	−100352.	−	86907.4i	−0.133631	−	0.115728i
$225$	345951.			0.455573
$226$	−121772.	−	210915.i	−0.158590	−	0.274686i
$227$	−327375.	+	567030.i	−0.421678	+	0.730368i	−0.996104	−	0.0881890i	$-0.971892\pi$
$227$	−327375.	+	567030.i	−0.421678	+	0.730368i	0.574426	+	0.818557i	$0.305225\pi$
$228$	−107208.	+	185690.i	−0.136581	+	0.236565i
$229$	−278856.	−	482994.i	−0.351392	−	0.608629i	0.635101	−	0.772429i	$-0.280958\pi$
$229$	−278856.	−	482994.i	−0.351392	−	0.608629i	−0.986494	+	0.163800i	$0.947625\pi$
$230$	77056.0			0.0960477
$231$	37485.0	−	12985.2i	0.0462197	−	0.0160110i
$232$	416512.			0.508051
$233$	623797.	+	1.08045e6i	0.752755	+	1.30381i	0.946483	+	0.322754i	$0.104609\pi$
$233$	623797.	+	1.08045e6i	0.752755	+	1.30381i	−0.193728	+	0.981055i	$0.562058\pi$
$234$	−486.000	+	841.777i	−0.000580225	+	0.00100498i
$235$	−793866.	+	1.37502e6i	−0.937729	+	1.62420i
$236$	254624.	+	441022.i	0.297591	+	0.515442i
$237$	274779.			0.317770
$238$	186592.	−	969560.i	0.213526	−	1.10951i
$239$	−496926.			−0.562726			−0.281363	−	0.959601i	$-0.590786\pi$
$239$	−496926.			−0.562726			−0.281363	+	0.959601i	$0.590786\pi$
$240$	99072.0	+	171598.i	0.111026	+	0.192302i
$241$	138809.	−	240424.i	0.153948	−	0.266646i	−0.778727	−	0.627363i	$-0.784134\pi$
$241$	138809.	−	240424.i	0.153948	−	0.266646i	0.932676	+	0.360716i	$0.117468\pi$
$242$	−319790.	+	553893.i	−0.351016	+	0.607977i
$243$	29524.5	+	51137.9i	0.0320750	+	0.0555556i
$244$	−922464.			−0.991916
$245$	206486.	+	1.43058e6i	0.219774	+	1.52264i
$246$	554904.			0.584629
$247$	−2233.50	−	3868.54i	−0.00232940	−	0.00403463i
$248$	55392.0	−	95941.8i	0.0571897	−	0.0990555i
$249$	497709.	−	862057.i	0.508718	−	0.881125i
$250$	197112.	+	341408.i	0.199463	+	0.345481i
$251$	−308328.			−0.308908			−0.154454	−	0.988000i	$-0.549362\pi$
$251$	−308328.			−0.308908			−0.154454	+	0.988000i	$0.549362\pi$
$252$	31752.0	−	164988.i	0.0314970	−	0.163663i
$253$	−7616.00			−0.00748041
$254$	−303930.	−	526422.i	−0.295590	−	0.511976i
$255$	−736848.	+	1.27626e6i	−0.709623	+	1.22910i
$256$	−32768.0	+	56755.8i	−0.0312500	+	0.0541266i
$257$	−204381.	−	353998.i	−0.193022	−	0.334325i	0.753228	−	0.657759i	$-0.228496\pi$
$257$	−204381.	−	353998.i	−0.193022	−	0.334325i	−0.946250	+	0.323435i	$0.895162\pi$
$258$	665676.			0.622606
$259$	935042.	−	323908.i	0.866128	−	0.300035i
$260$	−4128.00			−0.00378710
$261$	263574.	+	456524.i	0.239498	+	0.414822i
$262$	471004.	−	815803.i	0.423908	−	0.734230i
$263$	−544062.	+	942343.i	−0.485019	+	0.840078i	−0.999852	−	0.0172127i	$-0.994521\pi$
$263$	−544062.	+	942343.i	−0.485019	+	0.840078i	0.514833	+	0.857291i	$0.327854\pi$
$264$	−9792.00	−	16960.2i	−0.00864692	−	0.0149769i
$265$	−1.71622e6			−1.50126
$266$	583688.	+	505489.i	0.505797	+	0.438033i
$267$	530928.			0.455782
$268$	484504.	+	839186.i	0.412060	+	0.713709i
$269$	334145.	−	578756.i	0.281549	−	0.487657i	−0.690217	−	0.723602i	$-0.742485\pi$
$269$	334145.	−	578756.i	0.281549	−	0.487657i	0.971766	+	0.235945i	$0.0758184\pi$
$270$	−125388.	+	217178.i	−0.104676	+	0.181304i
$271$	415332.	+	719376.i	0.343536	+	0.595022i	0.985087	−	0.172059i	$-0.0550419\pi$
$271$	415332.	+	719376.i	0.343536	+	0.595022i	−0.641551	+	0.767081i	$0.721709\pi$
$272$	−487424.			−0.399470
$273$	2646.00	+	2291.50i	0.00214874	+	0.00186086i
$274$	−1.30203e6			−1.04772
$275$	−72607.0	−	125759.i	−0.0578958	−	0.100278i
$276$	−16128.0	+	27934.5i	−0.0127441	+	0.0220734i
$277$	−462537.	+	801137.i	−0.362198	+	0.627346i	−0.988322	−	0.152377i	$-0.951307\pi$
$277$	−462537.	+	801137.i	−0.362198	+	0.627346i	0.626124	+	0.779724i	$0.284640\pi$
$278$	6422.00	+	11123.2i	0.00498377	+	0.00863215i
$279$	140211.			0.107838
$280$	674240.	−	233564.i	0.513948	−	0.178037i
$281$	1.33635e6			1.00961			0.504805	−	0.863233i	$-0.331564\pi$
$281$	1.33635e6			1.00961			0.504805	+	0.863233i	$0.331564\pi$
$282$	−332316.	−	575588.i	−0.248845	−	0.431012i
$283$	496478.	−	859926.i	0.368497	−	0.638256i	−0.620833	−	0.783942i	$-0.713206\pi$
$283$	496478.	−	859926.i	0.368497	−	0.638256i	0.989331	+	0.145686i	$0.0465390\pi$
$284$	358672.	−	621238.i	0.263877	−	0.457048i
$285$	−576243.	−	998082.i	−0.420236	−	0.727871i
$286$	408.000			0.000294948
$287$	377643.	−	1.96229e6i	0.270630	−	1.40624i
$288$	−82944.0			−0.0589256
$289$	−1.10268e6	−	1.90990e6i	−0.776613	−	1.34513i
$290$	−1.11938e6	+	1.93882e6i	−0.781594	+	1.35376i
$291$	−539307.	+	934107.i	−0.373339	+	0.646642i
$292$	−166568.	−	288504.i	−0.114323	−	0.198014i
$293$	563544.			0.383494			0.191747	−	0.981444i	$-0.438585\pi$
$293$	563544.			0.383494			0.191747	+	0.981444i	$0.438585\pi$
$294$	−561834.	−	224567.i	−0.379088	−	0.151523i
$295$	−2.73721e6			−1.83127
$296$	−244256.	−	423064.i	−0.162038	−	0.280657i
$297$	12393.0	−	21465.3i	0.00815240	−	0.0141204i
$298$	−303768.	+	526142.i	−0.198153	+	0.343212i
$299$	−336.000	−	581.969i	−0.000217351	−	0.000376463i
$300$	−615024.			−0.394538
$301$	453029.	−	2.35401e6i	0.288211	−	1.49759i
$302$	−306592.			−0.193439
$303$	450045.	+	779501.i	0.281611	+	0.487764i
$304$	190592.	−	330115.i	0.118283	−	0.204871i
$305$	2.47912e6	−	4.29397e6i	1.52598	−	2.64307i
$306$	−308448.	−	534248.i	−0.188312	−	0.326166i
$307$	2.82703e6			1.71193			0.855963	−	0.517037i	$-0.172965\pi$
$307$	2.82703e6			1.71193			0.855963	+	0.517037i	$0.172965\pi$
$308$	−66640.0	+	23084.8i	−0.0400275	+	0.0138659i
$309$	−1.09522e6			−0.652536
$310$	297732.	+	515687.i	0.175963	+	0.304777i
$311$	563657.	−	976283.i	0.330456	−	0.572367i	−0.652145	−	0.758094i	$-0.726131\pi$
$311$	563657.	−	976283.i	0.330456	−	0.572367i	0.982601	+	0.185727i	$0.0594641\pi$
$312$	864.000	−	1496.49i	0.000502490	−	0.000870338i
$313$	1.18006e6	+	2.04393e6i	0.680840	+	1.17925i	0.974725	+	0.223408i	$0.0717183\pi$
$313$	1.18006e6	+	2.04393e6i	0.680840	+	1.17925i	−0.293885	+	0.955841i	$0.594948\pi$
$314$	1.55884e6			0.892231
$315$	682668.	+	591208.i	0.387644	+	0.335710i
$316$	−488496.			−0.275197
$317$	1.11210e6	+	1.92621e6i	0.621578	+	1.07660i	0.989192	+	0.146626i	$0.0468415\pi$
$317$	1.11210e6	+	1.92621e6i	0.621578	+	1.07660i	−0.367614	+	0.929979i	$0.619825\pi$
$318$	359208.	−	622167.i	0.199195	−	0.345016i
$319$	110636.	−	191627.i	0.0608723	−	0.105434i
$320$	−176128.	−	305063.i	−0.0961509	−	0.166538i
$321$	437832.			0.237162
$322$	87808.0	+	76044.0i	0.0471948	+	0.0408719i
$323$	2.83506e6			1.51201
$324$	−52488.0	−	90911.9i	−0.0277778	−	0.0481125i
$325$	6406.50	−	11096.4i	0.00336444	−	0.00582738i
$326$	−224744.	+	389268.i	−0.117124	+	0.202864i
$327$	−684338.	−	1.18531e6i	−0.353917	−	0.613002i
$328$	−986496.			−0.506303
$329$	−2.26160e6	+	783439.i	−1.15193	+	0.399039i
$330$	105264.			0.0532102
$331$	1.85152e6	+	3.20693e6i	0.928877	+	1.60886i	0.785204	+	0.619237i	$0.212558\pi$
$331$	1.85152e6	+	3.20693e6i	0.928877	+	1.60886i	0.143673	+	0.989625i	$0.454109\pi$
$332$	−884816.	+	1.53255e6i	−0.440563	+	0.763077i
$333$	309136.	−	535440.i	0.152771	−	0.264606i
$334$	105100.	+	182039.i	0.0515509	+	0.0892888i
$335$	−5.20842e6			−2.53568
$336$	−56448.0	+	293312.i	−0.0272772	+	0.141737i
$337$	1.21432e6			0.582452			0.291226	−	0.956654i	$-0.405937\pi$
$337$	1.21432e6			0.582452			0.291226	+	0.956654i	$0.405937\pi$
$338$	−742568.	−	1.28617e6i	−0.353545	−	0.612358i
$339$	−273987.	+	474559.i	−0.129488	+	0.224280i
$340$	1.30995e6	−	2.26890e6i	0.614551	−	1.06443i
$341$	−29427.0	−	50969.1i	−0.0137044	−	0.0237367i
$342$	482436.			0.223036
$343$	−1.17649e6	+	1.83397e6i	−0.539949	+	0.841698i
$344$	−1.18342e6			−0.539193
$345$	−86688.0	−	150148.i	−0.0392113	−	0.0679160i
$346$	−270512.	+	468541.i	−0.121478	+	0.210405i
$347$	488952.	−	846890.i	0.217993	−	0.377575i	−0.736201	−	0.676763i	$-0.763382\pi$
$347$	488952.	−	846890.i	0.217993	−	0.377575i	0.954194	+	0.299188i	$0.0967157\pi$
$348$	−468576.	−	811597.i	−0.207411	−	0.359247i
$349$	511282.			0.224697			0.112348	−	0.993669i	$-0.464163\pi$
$349$	511282.			0.224697			0.112348	+	0.993669i	$0.464163\pi$
$350$	−418558.	+	2.17489e6i	−0.182636	+	0.949003i
$351$	2187.00			0.000947504
$352$	17408.0	+	30151.5i	0.00748845	+	0.0129704i
$353$	−1.51376e6	+	2.62191e6i	−0.646577	+	1.11990i	0.337357	+	0.941377i	$0.390467\pi$
$353$	−1.51376e6	+	2.62191e6i	−0.646577	+	1.11990i	−0.983935	+	0.178528i	$0.942866\pi$
$354$	572904.	−	992299.i	0.242982	−	0.420857i
$355$	1.92786e6	+	3.33915e6i	0.811905	+	1.40626i
$356$	−943872.			−0.394719
$357$	−2.09916e6	+	727170.i	−0.871716	+	0.301971i
$358$	−1.00255e6			−0.413427
$359$	−2.29728e6	−	3.97900e6i	−0.940757	−	1.62944i	−0.764032	−	0.645179i	$-0.776783\pi$
$359$	−2.29728e6	−	3.97900e6i	−0.940757	−	1.62944i	−0.176725	−	0.984260i	$-0.556550\pi$
$360$	222912.	−	386095.i	0.0906520	−	0.157014i
$361$	129489.	−	224282.i	0.0522956	−	0.0905786i
$362$	398466.	+	690163.i	0.159816	+	0.276809i
$363$	1.43906e6			0.573206
$364$	−4704.00	−	4073.78i	−0.00186086	−	0.00161155i
$365$	1.79061e6			0.703506
$366$	1.03777e6	+	1.79747e6i	0.404948	+	0.701390i
$367$	556365.	−	963653.i	0.215623	−	0.373470i	−0.737842	−	0.674973i	$-0.764155\pi$
$367$	556365.	−	963653.i	0.215623	−	0.373470i	0.953465	+	0.301503i	$0.0974885\pi$
$368$	28672.0	−	49661.4i	0.0110367	−	0.0191161i
$369$	−624267.	−	1.08126e6i	−0.238674	−	0.413395i
$370$	2.62575e6			0.997125
$371$	−1.95569e6	−	1.69368e6i	−0.737675	−	0.638845i
$372$	−249264.			−0.0933904
$373$	1.22947e6	+	2.12951e6i	0.457559	+	0.792516i	0.998831	−	0.0483315i	$-0.0153904\pi$
$373$	1.22947e6	+	2.12951e6i	0.457559	+	0.792516i	−0.541272	+	0.840848i	$0.682057\pi$
$374$	−129472.	+	224252.i	−0.0478627	+	0.0829006i
$375$	443502.	−	768168.i	0.162861	−	0.282084i
$376$	590784.	+	1.02327e6i	0.215506	+	0.373267i
$377$	19524.0			0.00707482
$378$	−357210.	+	123741.i	−0.128586	+	0.0445435i
$379$	−4.10130e6			−1.46664			−0.733320	−	0.679884i	$-0.762030\pi$
$379$	−4.10130e6			−1.46664			−0.733320	+	0.679884i	$0.762030\pi$
$380$	1.02443e6	+	1.77437e6i	0.363935	+	0.630354i
$381$	−683842.	+	1.18445e6i	−0.241348	+	0.418027i
$382$	−477540.	+	827124.i	−0.167437	+	0.290009i
$383$	1.29707e6	+	2.24658e6i	0.451820	+	0.782575i	0.998499	−	0.0547673i	$-0.0174417\pi$
$383$	1.29707e6	+	2.24658e6i	0.451820	+	0.782575i	−0.546679	+	0.837342i	$0.684108\pi$
$384$	147456.			0.0510310
$385$	71638.0	−	372242.i	0.0246315	−	0.127989i
$386$	−342764.			−0.117092
$387$	−748886.	−	1.29711e6i	−0.254178	−	0.440249i
$388$	958768.	−	1.66063e6i	0.323321	−	0.560009i
$389$	−1.11703e6	+	1.93476e6i	−0.374276	+	0.648265i	−0.990218	−	0.139526i	$-0.955442\pi$
$389$	−1.11703e6	+	1.93476e6i	−0.374276	+	0.648265i	0.615942	+	0.787791i	$0.288775\pi$
$390$	4644.00	+	8043.64i	0.00154608	+	0.00267788i
$391$	426496.			0.141082
$392$	998816.	+	399231.i	0.328300	+	0.131223i
$393$	−2.11952e6			−0.692238
$394$	−142816.	−	247365.i	−0.0463486	−	0.0802781i
$395$	1.31283e6	−	2.27389e6i	0.423367	−	0.733293i
$396$	−22032.0	+	38160.5i	−0.00706018	+	0.0122286i
$397$	1.03200e6	+	1.78747e6i	0.328627	+	0.569198i	0.982240	−	0.187631i	$-0.0600809\pi$
$397$	1.03200e6	+	1.78747e6i	0.328627	+	0.569198i	−0.653613	+	0.756829i	$0.726748\pi$
$398$	2.84541e6			0.900403
$399$	328324.	−	1.70602e6i	0.103245	−	0.536479i
$400$	1.09338e6			0.341680
$401$	−576414.	−	998378.i	−0.179008	−	0.310052i	0.762533	−	0.646950i	$-0.223956\pi$
$401$	−576414.	−	998378.i	−0.179008	−	0.310052i	−0.941541	+	0.336898i	$0.890622\pi$
$402$	1.09013e6	−	1.88817e6i	0.336446	−	0.582741i
$403$	2596.50	−	4497.27i	0.000796390	−	0.00137939i
$404$	−800080.	−	1.38578e6i	−0.243882	−	0.422416i
$405$	564246.			0.170935
$406$	−3.18892e6	+	1.10467e6i	−0.960127	+	0.332598i
$407$	−259522.			−0.0776583
$408$	548352.	+	949774.i	0.163083	+	0.282468i
$409$	−2.96706e6	+	5.13910e6i	−0.877038	+	1.51907i	−0.0224623	+	0.999748i	$0.507151\pi$
$409$	−2.96706e6	+	5.13910e6i	−0.877038	+	1.51907i	−0.854576	+	0.519327i	$0.826183\pi$
$410$	2.65121e6	−	4.59203e6i	0.778904	−	1.34910i
$411$	1.46479e6	+	2.53708e6i	0.427730	+	0.740850i
$412$	1.94706e6			0.565113
$413$	−3.11914e6	−	2.70126e6i	−0.899830	−	0.779275i
$414$	72576.0			0.0208110
$415$	−4.75589e6	−	8.23744e6i	−1.35554	−	2.34786i
$416$	−1536.00	+	2660.43i	−0.000435169	+	0.000753735i
$417$	14449.5	−	25027.3i	0.00406923	−	0.00704812i
$418$	−101252.	−	175374.i	−0.0283441	−	0.0490934i
$419$	771666.			0.214731			0.107365	−	0.994220i	$-0.465759\pi$
$419$	771666.			0.214731			0.107365	+	0.994220i	$0.465759\pi$
$420$	−1.21363e6	−	1.05104e6i	−0.335710	−	0.290733i
$421$	−2.87542e6			−0.790671			−0.395336	−	0.918537i	$-0.629372\pi$
$421$	−2.87542e6			−0.790671			−0.395336	+	0.918537i	$0.629372\pi$
$422$	−520520.	−	901567.i	−0.142284	−	0.246443i
$423$	−747711.	+	1.29507e6i	−0.203181	+	0.351920i
$424$	−638592.	+	1.10607e6i	−0.172508	+	0.298792i
$425$	4.06599e6	+	7.04250e6i	1.09193	+	1.89128i
$426$	−1.61402e6			−0.430909
$427$	7.06261e6	−	2.44656e6i	1.87454	−	0.649361i
$428$	−778368.			−0.205388
$429$	−459.000	−	795.011i	−0.000120412	−	0.000208560i
$430$	3.18045e6	−	5.50870e6i	0.829503	−	1.43674i
$431$	68931.0	−	119392.i	0.0178740	−	0.0309587i	−0.856950	−	0.515399i	$-0.827644\pi$
$431$	68931.0	−	119392.i	0.0178740	−	0.0309587i	0.874824	+	0.484441i	$0.160977\pi$
$432$	93312.0	+	161621.i	0.0240563	+	0.0416667i
$433$	1.56526e6			0.401204			0.200602	−	0.979673i	$-0.435710\pi$
$433$	1.56526e6			0.401204			0.200602	+	0.979673i	$0.435710\pi$
$434$	−169638.	+	881465.i	−0.0432314	+	0.224637i
$435$	5.03719e6			1.27634
$436$	1.21660e6	+	2.10721e6i	0.306501	+	0.530875i
$437$	−166768.	+	288851.i	−0.0417743	+	0.0723552i
$438$	−374778.	+	649135.i	−0.0933445	+	0.161677i
$439$	−2.44079e6	−	4.22757e6i	−0.604462	−	1.04696i	−0.992136	−	0.125162i	$-0.960055\pi$
$439$	−2.44079e6	−	4.22757e6i	−0.604462	−	1.04696i	0.387675	−	0.921796i	$-0.373278\pi$
$440$	−187136.			−0.0460814
$441$	194481.	+	1.34740e6i	0.0476190	+	0.329914i
$442$	−22848.0			−0.00556279
$443$	650758.	+	1.12715e6i	0.157547	+	0.272879i	0.933984	−	0.357316i	$-0.116308\pi$
$443$	650758.	+	1.12715e6i	0.157547	+	0.272879i	−0.776437	+	0.630195i	$0.782975\pi$
$444$	−549576.	+	951894.i	−0.132303	+	0.229156i
$445$	2.53666e6	−	4.39362e6i	0.607242	−	1.05177i
$446$	211312.	+	366003.i	0.0503022	+	0.0871260i
$447$	1.36696e6			0.323583
$448$	100352.	−	521444.i	0.0236228	−	0.122748i
$449$	−3.13141e6			−0.733034			−0.366517	−	0.930411i	$-0.619450\pi$
$449$	−3.13141e6			−0.733034			−0.366517	+	0.930411i	$0.619450\pi$
$450$	691902.	+	1.19841e6i	0.161069	+	0.278981i
$451$	−262038.	+	453863.i	−0.0606629	+	0.105071i
$452$	487088.	−	843661.i	0.112140	−	0.194233i
$453$	344916.	+	597412.i	0.0789710	+	0.136782i
$454$	−2.61900e6			−0.596343
$455$	31605.0	−	10948.3i	0.00715694	−	0.00247924i
$456$	−857664.			−0.193155
$457$	−3.24634e6	−	5.62283e6i	−0.727116	−	1.25940i	−0.958097	−	0.286444i	$-0.907527\pi$
$457$	−3.24634e6	−	5.62283e6i	−0.727116	−	1.25940i	0.230981	−	0.972958i	$-0.425807\pi$
$458$	1.11543e6	−	1.93197e6i	0.248472	−	0.430366i
$459$	−694008.	+	1.20206e6i	−0.153756	+	0.266314i
$460$	154112.	+	266930.i	0.0339580	+	0.0588170i
$461$	5.34717e6			1.17185			0.585925	−	0.810365i	$-0.300731\pi$
$461$	5.34717e6			1.17185			0.585925	+	0.810365i	$0.300731\pi$
$462$	119952.	+	103881.i	0.0261458	+	0.0226430i
$463$	3.37285e6			0.731215			0.365607	−	0.930769i	$-0.380861\pi$
$463$	3.37285e6			0.731215			0.365607	+	0.930769i	$0.380861\pi$
$464$	833024.	+	1.44284e6i	0.179623	+	0.311117i
$465$	669897.	−	1.16030e6i	0.143673	−	0.248849i
$466$	−2.49519e6	+	4.32179e6i	−0.532278	+	0.921932i
$467$	−1.11726e6	−	1.93515e6i	−0.237062	−	0.410604i	0.722808	−	0.691049i	$-0.242851\pi$
$467$	−1.11726e6	−	1.93515e6i	−0.237062	−	0.410604i	−0.959870	+	0.280445i	$0.909518\pi$
$468$	−3888.00			−0.000820562
$469$	−5.93517e6	−	5.14001e6i	−1.24595	−	1.07903i
$470$	−6.35093e6			−1.32615
$471$	−1.75369e6	−	3.03749e6i	−0.364252	−	0.630903i
$472$	−1.01850e6	+	1.76409e6i	−0.210428	+	0.364473i
$473$	−314347.	+	544465.i	−0.0646036	+	0.111897i
$474$	549558.	+	951862.i	0.112349	+	0.194593i
$475$	−6.35952e6			−1.29327
$476$	3.73184e6	−	1.29275e6i	0.754928	−	0.261515i
$477$	−1.61644e6			−0.325284
$478$	−993852.	−	1.72140e6i	−0.198954	−	0.344598i
$479$	1.26068e6	−	2.18357e6i	0.251054	−	0.434838i	−0.712762	−	0.701406i	$-0.752556\pi$
$479$	1.26068e6	−	2.18357e6i	0.251054	−	0.434838i	0.963816	+	0.266568i	$0.0858896\pi$
$480$	−396288.	+	686391.i	−0.0785069	+	0.135978i
$481$	−11449.5	−	19831.1i	−0.00225644	−	0.00390827i
$482$	1.11047e6			0.217716
$483$	49392.0	−	256648.i	0.00963360	−	0.0500577i
$484$	−2.55832e6			−0.496411
$485$	5.15338e6	+	8.92591e6i	0.994804	+	1.72305i
$486$	−118098.	+	204552.i	−0.0226805	+	0.0392837i
$487$	−958358.	+	1.65993e6i	−0.183107	+	0.317151i	−0.942937	−	0.332971i	$-0.891949\pi$
$487$	−958358.	+	1.65993e6i	−0.183107	+	0.317151i	0.759830	+	0.650122i	$0.225282\pi$
$488$	−1.84493e6	−	3.19551e6i	−0.350695	−	0.607422i
$489$	1.01135e6			0.191262
$490$	−4.54269e6	+	3.57644e6i	−0.854718	+	0.672916i
$491$	5.82875e6			1.09112			0.545559	−	0.838073i	$-0.316317\pi$
$491$	5.82875e6			1.09112			0.545559	+	0.838073i	$0.316317\pi$
$492$	1.10981e6	+	1.92224e6i	0.206697	+	0.358010i
$493$	−6.19562e6	+	1.07311e7i	−1.14807	+	1.98851i
$494$	8934.00	−	15474.1i	0.00164713	−	0.00285292i
$495$	−118422.	−	205113.i	−0.0217230	−	0.0376253i
$496$	443136.			0.0808785
$497$	−1.09843e6	+	5.70763e6i	−0.199472	+	1.03649i
$498$	3.98167e6			0.719436
$499$	5.00243e6	+	8.66446e6i	0.899352	+	1.55772i	0.828324	+	0.560249i	$0.189295\pi$
$499$	5.00243e6	+	8.66446e6i	0.899352	+	1.55772i	0.0710277	+	0.997474i	$0.477372\pi$
$500$	−788448.	+	1.36563e6i	−0.141042	+	0.244292i
$501$	236475.	−	409587.i	0.0420912	−	0.0729040i
$502$	−616656.	−	1.06808e6i	−0.109215	−	0.189167i
$503$	1.13666e6			0.200313			0.100157	−	0.994972i	$-0.468066\pi$
$503$	1.13666e6			0.200313			0.100157	+	0.994972i	$0.468066\pi$
$504$	635040.	−	219984.i	0.111359	−	0.0385758i
$505$	8.60086e6			1.50077
$506$	−15232.0	−	26382.6i	−0.00264473	−	0.00458080i
$507$	−1.67078e6	+	2.89387e6i	−0.288668	+	0.499988i
$508$	1.21572e6	−	2.10569e6i	0.209013	−	0.362022i
$509$	−2.00968e6	−	3.48088e6i	−0.343822	−	0.595517i	0.641317	−	0.767276i	$-0.278388\pi$
$509$	−2.00968e6	−	3.48088e6i	−0.343822	−	0.595517i	−0.985139	+	0.171759i	$0.945055\pi$
$510$	−5.89478e6			−1.00356
$511$	2.04046e6	+	1.76709e6i	0.345681	+	0.299368i
$512$	−262144.			−0.0441942
$513$	−542740.	−	940054.i	−0.0910540	−	0.157710i
$514$	817524.	−	1.41599e6i	0.136487	−	0.236403i
$515$	−5.23271e6	+	9.06332e6i	−0.869378	+	1.50581i
$516$	1.33135e6	+	2.30597e6i	0.220125	+	0.381267i
$517$	627708.			0.103284
$518$	2.99214e6	+	2.59127e6i	0.489956	+	0.424314i
$519$	1.21730e6			0.198372
$520$	−8256.00	−	14299.8i	−0.00133894	−	0.00231911i
$521$	2.76014e6	−	4.78070e6i	0.445488	−	0.771609i	−0.552598	−	0.833448i	$-0.686363\pi$
$521$	2.76014e6	−	4.78070e6i	0.445488	−	0.771609i	0.998086	+	0.0618395i	$0.0196967\pi$
$522$	−1.05430e6	+	1.82609e6i	−0.169350	+	0.293324i
$523$	−4.47236e6	−	7.74636e6i	−0.714961	−	1.23835i	−0.962974	−	0.269593i	$-0.913111\pi$
$523$	−4.47236e6	−	7.74636e6i	−0.714961	−	1.23835i	0.248013	−	0.968757i	$-0.420222\pi$
$524$	3.76803e6			0.599496
$525$	4.70878e6	−	1.63117e6i	0.745607	−	0.258286i
$526$	−4.35250e6			−0.685921
$527$	1.64791e6	+	2.85427e6i	0.258468	+	0.447680i
$528$	39168.0	−	67841.0i	0.00611430	−	0.0105903i
$529$	3.19308e6	−	5.53058e6i	0.496102	−	0.859274i
$530$	−3.43243e6	−	5.94515e6i	−0.530777	−	0.919333i
$531$	−2.57807e6			−0.396788
$532$	−583688.	+	3.03293e6i	−0.0894132	+	0.464605i
$533$	−46242.0			−0.00705048
$534$	1.06186e6	+	1.83919e6i	0.161143	+	0.279109i
$535$	2.09186e6	−	3.62321e6i	0.315972	−	0.547280i
$536$	−1.93802e6	+	3.35674e6i	−0.291370	+	0.504668i
$537$	1.12787e6	+	1.95353e6i	0.168781	+	0.292337i
$538$	2.67316e6			0.398171
$539$	448987.	−	353486.i	0.0665674	−	0.0524083i
$540$	−1.00310e6			−0.148034
$541$	−424528.	−	735305.i	−0.0623611	−	0.108013i	0.833159	−	0.553033i	$-0.186530\pi$
$541$	−424528.	−	735305.i	−0.0623611	−	0.108013i	−0.895520	+	0.445021i	$0.853196\pi$
$542$	−1.66133e6	+	2.87750e6i	−0.242917	+	0.420744i
$543$	896548.	−	1.55287e6i	0.130489	−	0.226014i
$544$	−974848.	−	1.68849e6i	−0.141234	−	0.244625i
$545$	−1.30784e7			−1.88610
$546$	−2646.00	+	13749.0i	−0.000379847	+	0.00197374i
$547$	8.61340e6			1.23085			0.615426	−	0.788194i	$-0.288984\pi$
$547$	8.61340e6			1.23085			0.615426	+	0.788194i	$0.288984\pi$
$548$	−2.60406e6	−	4.51037e6i	−0.370425	−	0.641595i
$549$	2.33499e6	−	4.04432e6i	0.330639	−	0.572683i
$550$	290428.	−	503036.i	0.0409385	−	0.0709075i
$551$	−4.84521e6	−	8.39214e6i	−0.679882	−	1.17759i
$552$	−129024.			−0.0180228
$553$	3.74005e6	−	1.29559e6i	0.520073	−	0.180159i
$554$	−3.70029e6			−0.512226
$555$	−2.95397e6	−	5.11643e6i	−0.407074	−	0.705074i
$556$	−25688.0	+	44492.9i	−0.00352406	+	0.00610385i
$557$	−3.89940e6	+	6.75395e6i	−0.532549	+	0.922401i	0.466729	+	0.884400i	$0.345432\pi$
$557$	−3.89940e6	+	6.75395e6i	−0.532549	+	0.922401i	−0.999278	+	0.0380009i	$0.987901\pi$
$558$	280422.	+	485705.i	0.0381265	+	0.0660370i
$559$	−55473.0			−0.00750848
$560$	2.15757e6	+	1.86851e6i	0.290733	+	0.251782i
$561$	582624.			0.0781594
$562$	2.67270e6	+	4.62925e6i	0.356951	+	0.618258i
$563$	−536357.	+	928998.i	−0.0713153	+	0.123522i	−0.899478	−	0.436966i	$-0.856053\pi$
$563$	−536357.	+	928998.i	−0.0713153	+	0.123522i	0.828163	+	0.560488i	$0.189386\pi$
$564$	1.32926e6	−	2.30235e6i	0.175960	−	0.304771i
$565$	2.61810e6	+	4.53468e6i	0.345036	+	0.597620i
$566$	3.97183e6			0.521134
$567$	642978.	+	556835.i	0.0839921	+	0.0727393i
$568$	2.86938e6			0.373179
$569$	507621.	+	879225.i	0.0657293	+	0.113846i	0.897017	−	0.441996i	$-0.145729\pi$
$569$	507621.	+	879225.i	0.0657293	+	0.113846i	−0.831288	+	0.555842i	$0.812396\pi$
$570$	2.30497e6	−	3.99233e6i	0.297152	−	0.514682i
$571$	3.39047e6	−	5.87246e6i	0.435180	−	0.753754i	−0.562130	−	0.827049i	$-0.690018\pi$
$571$	3.39047e6	−	5.87246e6i	0.435180	−	0.753754i	0.997310	+	0.0732945i	$0.0233513\pi$
$572$	816.000	+	1413.35i	0.000104280	+	0.000180618i
$573$	2.14893e6			0.273423
$574$	7.55286e6	−	2.61639e6i	0.956823	−	0.331453i
$575$	−956704.			−0.120672
$576$	−165888.	−	287326.i	−0.0208333	−	0.0360844i
$577$	1.65268e6	−	2.86253e6i	0.206657	−	0.357941i	−0.744002	−	0.668177i	$-0.767075\pi$
$577$	1.65268e6	−	2.86253e6i	0.206657	−	0.357941i	0.950659	+	0.310236i	$0.100408\pi$
$578$	4.41072e6	−	7.63959e6i	0.549148	−	0.951153i
$579$	385610.	+	667895.i	0.0478026	+	0.0827965i
$580$	−8.95501e6			−1.10534
$581$	2.70975e6	−	1.40803e7i	0.333034	−	1.73050i
$582$	−4.31446e6			−0.527981
$583$	339252.	+	587602.i	0.0413381	+	0.0715998i
$584$	666272.	−	1.15402e6i	0.0808387	−	0.140017i
$585$	10449.0	−	18098.2i	0.00126237	−	0.00218648i
$586$	1.12709e6	+	1.95217e6i	0.135586	+	0.234841i
$587$	−1.19833e7			−1.43543			−0.717715	−	0.696337i	$-0.754812\pi$
$587$	−1.19833e7			−1.43543			−0.717715	+	0.696337i	$0.754812\pi$
$588$	−345744.	−	2.39538e6i	−0.0412393	−	0.285714i
$589$	−2.57746e6			−0.306128
$590$	−5.47442e6	−	9.48197e6i	−0.647452	−	1.12142i
$591$	−321336.	+	556570.i	−0.0378434	+	0.0655468i
$592$	977024.	−	1.69226e6i	0.114578	−	0.198455i
$593$	2.65510e6	+	4.59877e6i	0.310059	+	0.537038i	0.978375	−	0.206840i	$-0.0663179\pi$
$593$	2.65510e6	+	4.59877e6i	0.310059	+	0.537038i	−0.668316	+	0.743877i	$0.732985\pi$
$594$	99144.0			0.0115292
$595$	−4.01173e6	+	2.08456e7i	−0.464557	+	2.41391i
$596$	−2.43014e6			−0.280231
$597$	−3.20108e6	−	5.54444e6i	−0.367588	−	0.636681i
$598$	1344.00	−	2327.88i	0.000153690	−	0.000266199i
$599$	−1.21118e6	+	2.09782e6i	−0.137924	+	0.238892i	−0.926711	−	0.375775i	$-0.877376\pi$
$599$	−1.21118e6	+	2.09782e6i	−0.137924	+	0.238892i	0.788786	+	0.614667i	$0.210710\pi$
$600$	−1.23005e6	−	2.13051e6i	−0.139490	−	0.241604i
$601$	−7.10659e6			−0.802556			−0.401278	−	0.915956i	$-0.631434\pi$
$601$	−7.10659e6			−0.802556			−0.401278	+	0.915956i	$0.631434\pi$
$602$	9.06059e6	−	3.13868e6i	1.01898	−	0.352985i
$603$	−4.90560e6			−0.549413
$604$	−613184.	−	1.06207e6i	−0.0683909	−	0.118457i
$605$	6.87548e6	−	1.19087e7i	0.763686	−	1.32274i
$606$	−1.80018e6	+	3.11800e6i	−0.199129	+	0.344901i
$607$	−8.95970e6	−	1.55187e7i	−0.987011	−	1.70955i	−0.632636	−	0.774449i	$-0.718027\pi$
$607$	−8.95970e6	−	1.55187e7i	−0.987011	−	1.70955i	−0.354374	−	0.935104i	$-0.615306\pi$
$608$	1.52474e6			0.167277
$609$	5.74006e6	+	4.97103e6i	0.627152	+	0.543130i
$610$	1.98330e7			2.15806
$611$	27693.0	+	47965.7i	0.00300101	+	0.00519790i
$612$	1.23379e6	−	2.13699e6i	0.133157	−	0.230634i
$613$	−6.98950e6	+	1.21062e7i	−0.751269	+	1.30124i	0.195940	+	0.980616i	$0.437224\pi$
$613$	−6.98950e6	+	1.21062e7i	−0.751269	+	1.30124i	−0.947208	+	0.320619i	$0.896109\pi$
$614$	5.65407e6	+	9.79313e6i	0.605257	+	1.04834i
$615$	−1.19304e7			−1.27195
$616$	−213248.	−	184678.i	−0.0226430	−	0.0196094i
$617$	−5.25594e6			−0.555824			−0.277912	−	0.960606i	$-0.589642\pi$
$617$	−5.25594e6			−0.555824			−0.277912	+	0.960606i	$0.589642\pi$
$618$	−2.19044e6	−	3.79395e6i	−0.230706	−	0.399595i
$619$	−721506.	+	1.24969e6i	−0.0756857	+	0.131091i	−0.901384	−	0.433020i	$-0.857448\pi$
$619$	−721506.	+	1.24969e6i	−0.0756857	+	0.131091i	0.825699	+	0.564112i	$0.190781\pi$
$620$	−1.19093e6	+	2.06275e6i	−0.124425	+	0.215510i
$621$	−81648.0	−	141418.i	−0.00849604	−	0.0147156i
$622$	4.50926e6			0.467336
$623$	7.22652e6	−	2.50334e6i	0.745949	−	0.258404i
$624$	6912.00			0.000710628
$625$	2.43553e6	+	4.21846e6i	0.249398	+	0.431970i
$626$	−4.72026e6	+	8.17573e6i	−0.481426	+	0.833855i
$627$	−227817.	+	394591.i	−0.0231429	+	0.0400846i
$628$	3.11768e6	+	5.39998e6i	0.315451	+	0.546378i
$629$	1.45332e7			1.46466
$630$	−682668.	+	3.54725e6i	−0.0685264	+	0.356074i
$631$	1.51723e7			1.51697			0.758487	−	0.651688i	$-0.225939\pi$
$631$	1.51723e7			1.51697			0.758487	+	0.651688i	$0.225939\pi$
$632$	−976992.	−	1.69220e6i	−0.0972967	−	0.168523i
$633$	−1.17117e6	+	2.02853e6i	−0.116174	+	0.201220i
$634$	−4.44840e6	+	7.70485e6i	−0.439522	+	0.761275i
$635$	6.53450e6	+	1.13181e7i	0.643099	+	1.11388i
$636$	2.87366e6			0.281704
$637$	46819.5	+	18713.9i	0.00457170	+	0.00182733i
$638$	885088.			0.0860864
$639$	1.81578e6	+	3.14502e6i	0.175918	+	0.304699i
$640$	704512.	−	1.22025e6i	0.0679890	−	0.117760i
$641$	72424.0	−	125442.i	0.00696205	−	0.0120586i	−0.862523	−	0.506017i	$-0.831117\pi$
$641$	72424.0	−	125442.i	0.00696205	−	0.0120586i	0.869485	+	0.493959i	$0.164451\pi$
$642$	875664.	+	1.51669e6i	0.0838494	+	0.145231i
$643$	−27469.0			−0.00262009			−0.00131004	−	0.999999i	$-0.500417\pi$
$643$	−27469.0			−0.00262009			−0.00131004	+	0.999999i	$0.500417\pi$
$644$	−87808.0	+	456264.i	−0.00834295	+	0.0433512i
$645$	−1.43120e7			−1.35457
$646$	5.67011e6	+	9.82092e6i	0.534577	+	0.925915i
$647$	−4.27892e6	+	7.41130e6i	−0.401858	+	0.696039i	−0.993950	−	0.109831i	$-0.964969\pi$
$647$	−4.27892e6	+	7.41130e6i	−0.401858	+	0.696039i	0.592092	+	0.805870i	$0.298302\pi$
$648$	209952.	−	363648.i	0.0196419	−	0.0340207i
$649$	541076.	+	937171.i	0.0504251	+	0.0873388i
$650$	51252.0			0.00475803
$651$	1.90843e6	−	661099.i	0.176491	−	0.0611384i
$652$	−1.79795e6			−0.165638
$653$	−6.25152e6	−	1.08279e7i	−0.573723	−	0.993718i	−0.996179	−	0.0873343i	$-0.972165\pi$
$653$	−6.25152e6	−	1.08279e7i	−0.573723	−	0.993718i	0.422456	−	0.906384i	$-0.361168\pi$
$654$	2.73735e6	−	4.74123e6i	0.250257	−	0.433458i
$655$	−1.01266e7	+	1.75398e7i	−0.922274	+	1.59742i
$656$	−1.97299e6	−	3.41732e6i	−0.179005	−	0.310046i
$657$	1.68650e6			0.152431
$658$	−7.23710e6	−	6.26752e6i	−0.651629	−	0.564327i
$659$	1.80471e7			1.61880			0.809400	−	0.587258i	$-0.199793\pi$
$659$	1.80471e7			1.61880			0.809400	+	0.587258i	$0.199793\pi$
$660$	210528.	+	364645.i	0.0188127	+	0.0325845i
$661$	−1.67072e6	+	2.89377e6i	−0.148731	+	0.257609i	−0.930759	−	0.365634i	$-0.880852\pi$
$661$	−1.67072e6	+	2.89377e6i	−0.148731	+	0.257609i	0.782028	+	0.623243i	$0.214185\pi$
$662$	−7.40608e6	+	1.28277e7i	−0.656815	+	1.13764i
$663$	25704.0	+	44520.6i	0.00227100	+	0.00393349i
$664$	−7.07853e6			−0.623050
$665$	−1.25493e7	−	1.08680e7i	−1.10044	−	0.953006i
$666$	2.47309e6			0.216050
$667$	−728896.	−	1.26248e6i	−0.0634382	−	0.109878i
$668$	−420400.	+	728154.i	−0.0364520	+	0.0631367i
$669$	475452.	−	823507.i	0.0410716	−	0.0711381i
$670$	−1.04168e7	−	1.80425e7i	−0.896497	−	1.55278i
$671$	−1.96024e6			−0.168075
$672$	−1.12896e6	+	391083.i	−0.0964396	+	0.0334077i
$673$	−8.47066e6			−0.720907			−0.360454	−	0.932777i	$-0.617378\pi$
$673$	−8.47066e6			−0.720907			−0.360454	+	0.932777i	$0.617378\pi$
$674$	2.42865e6	+	4.20655e6i	0.205928	+	0.356678i
$675$	1.55678e6	−	2.69642e6i	0.131513	−	0.227787i
$676$	2.97027e6	−	5.14466e6i	0.249994	−	0.433002i
$677$	−7.32764e6	−	1.26918e7i	−0.614458	−	1.06427i	−0.990479	−	0.137662i	$-0.956041\pi$
$677$	−7.32764e6	−	1.26918e7i	−0.614458	−	1.06427i	0.376021	−	0.926611i	$-0.377292\pi$
$678$	−2.19190e6			−0.183124
$679$	−2.93623e6	+	1.52571e7i	−0.244408	+	1.26998i
$680$	1.04796e7			0.869107
$681$	2.94638e6	+	5.10327e6i	0.243456	+	0.421678i
$682$	117708.	−	203876.i	0.00969047	−	0.0167844i
$683$	−9.88081e6	+	1.71141e7i	−0.810477	+	1.40379i	0.102054	+	0.994779i	$0.467459\pi$
$683$	−9.88081e6	+	1.71141e7i	−0.810477	+	1.40379i	−0.912531	+	0.409008i	$0.865875\pi$
$684$	964872.	+	1.67121e6i	0.0788550	+	0.136581i
$685$	2.79937e7			2.27947
$686$	−8.70603e6	−	407548.i	−0.706333	−	0.0330650i
$687$	−5.01942e6			−0.405753
$688$	−2.36685e6	−	4.09950e6i	−0.190634	−	0.330187i
$689$	−29934.0	+	51847.2i	−0.00240224	+	0.00416080i
$690$	346752.	−	600592.i	0.0277266	−	0.0480238i
$691$	−6.79162e6	−	1.17634e7i	−0.541101	−	0.937214i	−0.998841	−	0.0481282i	$-0.984674\pi$
$691$	−6.79162e6	−	1.17634e7i	−0.541101	−	0.937214i	0.457740	−	0.889086i	$-0.348659\pi$
$692$	−2.16410e6			−0.171795
$693$	67473.0	−	350600.i	0.00533700	−	0.0277318i
$694$	3.91162e6			0.308289
$695$	−138073.	−	239149.i	−0.0108429	−	0.0187805i
$696$	1.87430e6	−	3.24639e6i	0.146662	−	0.254026i
$697$	1.46741e7	−	2.54163e7i	1.14412	−	1.98167i
$698$	1.02256e6	+	1.77113e6i	0.0794423	+	0.137598i
$699$	1.12283e7			0.869206
$700$	−8.37116e6	+	2.89985e6i	−0.645715	+	0.223682i
$701$	−1.29915e7			−0.998538			−0.499269	−	0.866447i	$-0.666398\pi$
$701$	−1.29915e7			−0.998538			−0.499269	+	0.866447i	$0.666398\pi$
$702$	4374.00	+	7575.99i	0.000334993	+	0.000580225i
$703$	−5.68277e6	+	9.84284e6i	−0.433682	+	0.751160i
$704$	−69632.0	+	120606.i	−0.00529514	+	0.00917145i
$705$	7.14479e6	+	1.23751e7i	0.541398	+	0.937729i
$706$	−1.21101e7			−0.914399
$707$	9.80098e6	+	8.48790e6i	0.737430	+	0.638633i
$708$	4.58323e6			0.343628
$709$	−954367.	−	1.65301e6i	−0.0713017	−	0.123498i	0.828170	−	0.560476i	$-0.189382\pi$
$709$	−954367.	−	1.65301e6i	−0.0713017	−	0.123498i	−0.899472	+	0.436978i	$0.856049\pi$
$710$	−7.71145e6	+	1.33566e7i	−0.574103	+	0.994376i
$711$	1.23651e6	−	2.14169e6i	0.0917323	−	0.158885i
$712$	−1.88774e6	−	3.26967e6i	−0.139554	−	0.241715i
$713$	−387744.			−0.0285641
$714$	−6.71731e6	−	5.81736e6i	−0.493117	−	0.427052i
$715$	−8772.00			−0.000641702
$716$	−2.00510e6	−	3.47294e6i	−0.146169	−	0.253172i
$717$	−2.23617e6	+	3.87315e6i	−0.162445	+	0.281363i
$718$	9.18911e6	−	1.59160e7i	0.665216	−	1.15219i
$719$	5.55999e6	+	9.63018e6i	0.401099	+	0.694724i	0.993859	−	0.110655i	$-0.0352950\pi$
$719$	5.55999e6	+	9.63018e6i	0.401099	+	0.694724i	−0.592760	+	0.805379i	$0.701962\pi$
$720$	1.78330e6			0.128201
$721$	−1.49071e7	+	5.16399e6i	−1.06796	+	0.369953i
$722$	1.03591e6			0.0739571
$723$	−1.24928e6	−	2.16382e6i	−0.0888821	−	0.153948i
$724$	−1.59386e6	+	2.76065e6i	−0.113007	+	0.195734i
$725$	1.38978e7	−	2.40718e7i	0.981979	−	1.70084i
$726$	2.87811e6	+	4.98503e6i	0.202659	+	0.351016i
$727$	8.37406e6			0.587624			0.293812	−	0.955863i	$-0.405076\pi$
$727$	8.37406e6			0.587624			0.293812	+	0.955863i	$0.405076\pi$
$728$	4704.00	−	24442.7i	0.000328957	−	0.00170931i
$729$	531441.			0.0370370
$730$	3.58121e6	+	6.20284e6i	0.248727	+	0.430808i
$731$	1.76034e7	−	3.04900e7i	1.21844	−	2.11040i
$732$	−4.15109e6	+	7.18990e6i	−0.286341	+	0.495958i
$733$	−2.32224e6	−	4.02224e6i	−0.159642	−	0.276508i	0.775098	−	0.631841i	$-0.217701\pi$
$733$	−2.32224e6	−	4.02224e6i	−0.159642	−	0.276508i	−0.934740	+	0.355334i	$0.884367\pi$
$734$	4.45092e6			0.304937
$735$	1.20794e7	+	4.82820e6i	0.824761	+	0.329660i
$736$	229376.			0.0156082
$737$	1.02957e6	+	1.78327e6i	0.0698212	+	0.120934i
$738$	2.49707e6	−	4.32505e6i	0.168768	−	0.292314i
$739$	5.53116e6	−	9.58026e6i	0.372568	−	0.645307i	−0.617392	−	0.786656i	$-0.711811\pi$
$739$	5.53116e6	−	9.58026e6i	0.372568	−	0.645307i	0.989960	+	0.141349i	$0.0451440\pi$
$740$	5.25150e6	+	9.09587e6i	0.352537	+	0.610612i
$741$	−40203.0			−0.00268976
$742$	1.95569e6	−	1.01621e7i	0.130404	−	0.677597i
$743$	1.97245e6			0.131079			0.0655395	−	0.997850i	$-0.479123\pi$
$743$	1.97245e6			0.131079			0.0655395	+	0.997850i	$0.479123\pi$
$744$	−498528.	−	863476.i	−0.0330185	−	0.0571897i
$745$	6.53101e6	−	1.13120e7i	0.431112	−	0.746707i
$746$	−4.91790e6	+	8.51805e6i	−0.323543	+	0.560393i
$747$	−4.47938e6	−	7.75852e6i	−0.293708	−	0.508718i
$748$	−1.03578e6			−0.0676880
$749$	5.95938e6	−	2.06439e6i	0.388147	−	0.134458i
$750$	3.54802e6			0.230320
$751$	−7.51232e6	−	1.30117e7i	−0.486042	−	0.841850i	0.513829	−	0.857893i	$-0.328227\pi$
$751$	−7.51232e6	−	1.30117e7i	−0.486042	−	0.841850i	−0.999871	+	0.0160425i	$0.994893\pi$
$752$	−2.36314e6	+	4.09307e6i	−0.152386	+	0.263940i
$753$	−1.38748e6	+	2.40318e6i	−0.0891740	+	0.154454i
$754$	39048.0	+	67633.1i	0.00250133	+	0.00433243i
$755$	6.59173e6			0.420854
$756$	−1.14307e6	−	989929.i	−0.0727393	−	0.0629941i
$757$	−4.72426e6			−0.299636			−0.149818	−	0.988714i	$-0.547869\pi$
$757$	−4.72426e6			−0.299636			−0.149818	+	0.988714i	$0.547869\pi$
$758$	−8.20260e6	−	1.42073e7i	−0.518535	−	0.898130i
$759$	−34272.0	+	59360.8i	−0.00215941	+	0.00374021i
$760$	−4.09773e6	+	7.09747e6i	−0.257341	+	0.445728i
$761$	4.28918e6	+	7.42907e6i	0.268480	+	0.465021i	0.968470	−	0.249132i	$-0.0801453\pi$
$761$	4.28918e6	+	7.42907e6i	0.268480	+	0.465021i	−0.699989	+	0.714153i	$0.746812\pi$
$762$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	42.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 + 3.46410i) q^{2} +(4.50000 - 7.79423i) q^{3} +(-8.00000 + 13.8564i) q^{4} +(-43.0000 - 74.4782i) q^{5} +36.0000 q^{6} +(24.5000 - 127.306i) q^{7} -64.0000 q^{8} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})\)	\(q+(2.00000 + 3.46410i) q^{2} +(4.50000 - 7.79423i) q^{3} +(-8.00000 + 13.8564i) q^{4} +(-43.0000 - 74.4782i) q^{5} +36.0000 q^{6} +(24.5000 - 127.306i) q^{7} -64.0000 q^{8} +(-40.5000 - 70.1481i) q^{9} +(172.000 - 297.913i) q^{10} +(-17.0000 + 29.4449i) q^{11} +(72.0000 + 124.708i) q^{12} -3.00000 q^{13} +(490.000 - 169.741i) q^{14} -774.000 q^{15} +(-128.000 - 221.703i) q^{16} +(952.000 - 1648.91i) q^{17} +(162.000 - 280.592i) q^{18} +(744.500 + 1289.51i) q^{19} +1376.00 q^{20} +(-882.000 - 763.834i) q^{21} -136.000 q^{22} +(112.000 + 193.990i) q^{23} +(-288.000 + 498.831i) q^{24} +(-2135.50 + 3698.79i) q^{25} +(-6.00000 - 10.3923i) q^{26} -729.000 q^{27} +(1568.00 + 1357.93i) q^{28} -6508.00 q^{29} +(-1548.00 - 2681.21i) q^{30} +(-865.500 + 1499.09i) q^{31} +(512.000 - 886.810i) q^{32} +(153.000 + 265.004i) q^{33} +7616.00 q^{34} +(-10535.0 + 3649.43i) q^{35} +1296.00 q^{36} +(3816.50 + 6610.37i) q^{37} +(-2978.00 + 5158.05i) q^{38} +(-13.5000 + 23.3827i) q^{39} +(2752.00 + 4766.60i) q^{40} +15414.0 q^{41} +(882.000 - 4583.01i) q^{42} +18491.0 q^{43} +(-272.000 - 471.118i) q^{44} +(-3483.00 + 6032.73i) q^{45} +(-448.000 + 775.959i) q^{46} +(-9231.00 - 15988.6i) q^{47} -2304.00 q^{48} +(-15606.5 - 6237.98i) q^{49} -17084.0 q^{50} +(-8568.00 - 14840.2i) q^{51} +(24.0000 - 41.5692i) q^{52} +(9978.00 - 17282.4i) q^{53} +(-1458.00 - 2525.33i) q^{54} +2924.00 q^{55} +(-1568.00 + 8147.57i) q^{56} +13401.0 q^{57} +(-13016.0 - 22544.4i) q^{58} +(15914.0 - 27563.9i) q^{59} +(6192.00 - 10724.9i) q^{60} +(28827.0 + 49929.8i) q^{61} -6924.00 q^{62} +(-9922.50 + 3437.25i) q^{63} +4096.00 q^{64} +(129.000 + 223.435i) q^{65} +(-612.000 + 1060.02i) q^{66} +(30281.5 - 52449.1i) q^{67} +(15232.0 + 26382.6i) q^{68} +2016.00 q^{69} +(-33712.0 - 29195.4i) q^{70} -44834.0 q^{71} +(2592.00 + 4489.48i) q^{72} +(-10410.5 + 18031.5i) q^{73} +(-15266.0 + 26441.5i) q^{74} +(19219.5 + 33289.2i) q^{75} -23824.0 q^{76} +(3332.00 + 2885.60i) q^{77} -108.000 q^{78} +(15265.5 + 26440.6i) q^{79} +(-11008.0 + 19066.4i) q^{80} +(-3280.50 + 5681.99i) q^{81} +(30828.0 + 53395.7i) q^{82} +110602. q^{83} +(17640.0 - 6110.68i) q^{84} -163744. q^{85} +(36982.0 + 64054.7i) q^{86} +(-29286.0 + 50724.8i) q^{87} +(1088.00 - 1884.47i) q^{88} +(29496.0 + 51088.6i) q^{89} -27864.0 q^{90} +(-73.5000 + 381.917i) q^{91} -3584.00 q^{92} +(7789.50 + 13491.8i) q^{93} +(36924.0 - 63954.2i) q^{94} +(64027.0 - 110898. i) q^{95} +(-4608.00 - 7981.29i) q^{96} -119846. q^{97} +(-9604.00 - 66538.5i) q^{98} +2754.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 9 q^{3} - 16 q^{4} - 86 q^{5} + 72 q^{6} + 49 q^{7} - 128 q^{8} - 81 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 + 9 * q^3 - 16 * q^4 - 86 * q^5 + 72 * q^6 + 49 * q^7 - 128 * q^8 - 81 * q^9	\( 2 q + 4 q^{2} + 9 q^{3} - 16 q^{4} - 86 q^{5} + 72 q^{6} + 49 q^{7} - 128 q^{8} - 81 q^{9} + 344 q^{10} - 34 q^{11} + 144 q^{12} - 6 q^{13} + 980 q^{14} - 1548 q^{15} - 256 q^{16} + 1904 q^{17} + 324 q^{18} + 1489 q^{19} + 2752 q^{20} - 1764 q^{21} - 272 q^{22} + 224 q^{23} - 576 q^{24} - 4271 q^{25} - 12 q^{26} - 1458 q^{27} + 3136 q^{28} - 13016 q^{29} - 3096 q^{30} - 1731 q^{31} + 1024 q^{32} + 306 q^{33} + 15232 q^{34} - 21070 q^{35} + 2592 q^{36} + 7633 q^{37} - 5956 q^{38} - 27 q^{39} + 5504 q^{40} + 30828 q^{41} + 1764 q^{42} + 36982 q^{43} - 544 q^{44} - 6966 q^{45} - 896 q^{46} - 18462 q^{47} - 4608 q^{48} - 31213 q^{49} - 34168 q^{50} - 17136 q^{51} + 48 q^{52} + 19956 q^{53} - 2916 q^{54} + 5848 q^{55} - 3136 q^{56} + 26802 q^{57} - 26032 q^{58} + 31828 q^{59} + 12384 q^{60} + 57654 q^{61} - 13848 q^{62} - 19845 q^{63} + 8192 q^{64} + 258 q^{65} - 1224 q^{66} + 60563 q^{67} + 30464 q^{68} + 4032 q^{69} - 67424 q^{70} - 89668 q^{71} + 5184 q^{72} - 20821 q^{73} - 30532 q^{74} + 38439 q^{75} - 47648 q^{76} + 6664 q^{77} - 216 q^{78} + 30531 q^{79} - 22016 q^{80} - 6561 q^{81} + 61656 q^{82} + 221204 q^{83} + 35280 q^{84} - 327488 q^{85} + 73964 q^{86} - 58572 q^{87} + 2176 q^{88} + 58992 q^{89} - 55728 q^{90} - 147 q^{91} - 7168 q^{92} + 15579 q^{93} + 73848 q^{94} + 128054 q^{95} - 9216 q^{96} - 239692 q^{97} - 19208 q^{98} + 5508 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 9 * q^3 - 16 * q^4 - 86 * q^5 + 72 * q^6 + 49 * q^7 - 128 * q^8 - 81 * q^9 + 344 * q^10 - 34 * q^11 + 144 * q^12 - 6 * q^13 + 980 * q^14 - 1548 * q^15 - 256 * q^16 + 1904 * q^17 + 324 * q^18 + 1489 * q^19 + 2752 * q^20 - 1764 * q^21 - 272 * q^22 + 224 * q^23 - 576 * q^24 - 4271 * q^25 - 12 * q^26 - 1458 * q^27 + 3136 * q^28 - 13016 * q^29 - 3096 * q^30 - 1731 * q^31 + 1024 * q^32 + 306 * q^33 + 15232 * q^34 - 21070 * q^35 + 2592 * q^36 + 7633 * q^37 - 5956 * q^38 - 27 * q^39 + 5504 * q^40 + 30828 * q^41 + 1764 * q^42 + 36982 * q^43 - 544 * q^44 - 6966 * q^45 - 896 * q^46 - 18462 * q^47 - 4608 * q^48 - 31213 * q^49 - 34168 * q^50 - 17136 * q^51 + 48 * q^52 + 19956 * q^53 - 2916 * q^54 + 5848 * q^55 - 3136 * q^56 + 26802 * q^57 - 26032 * q^58 + 31828 * q^59 + 12384 * q^60 + 57654 * q^61 - 13848 * q^62 - 19845 * q^63 + 8192 * q^64 + 258 * q^65 - 1224 * q^66 + 60563 * q^67 + 30464 * q^68 + 4032 * q^69 - 67424 * q^70 - 89668 * q^71 + 5184 * q^72 - 20821 * q^73 - 30532 * q^74 + 38439 * q^75 - 47648 * q^76 + 6664 * q^77 - 216 * q^78 + 30531 * q^79 - 22016 * q^80 - 6561 * q^81 + 61656 * q^82 + 221204 * q^83 + 35280 * q^84 - 327488 * q^85 + 73964 * q^86 - 58572 * q^87 + 2176 * q^88 + 58992 * q^89 - 55728 * q^90 - 147 * q^91 - 7168 * q^92 + 15579 * q^93 + 73848 * q^94 + 128054 * q^95 - 9216 * q^96 - 239692 * q^97 - 19208 * q^98 + 5508 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more