LMFDB - Embedded newform 138.8.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,8,Mod(1,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("138.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	138.a (trivial)

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:	yes
Analytic conductor:	$43.1091335168$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 684x - 5052 $ x^3 - x^2 - 684*x - 5052
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-20.3930$ of defining polynomial
Character	$\chi$	$=$	138.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +41.5271 q^{5} -216.000 q^{6} -286.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +41.5271 q^{5} -216.000 q^{6} -286.000 q^{7} +512.000 q^{8} +729.000 q^{9} +332.217 q^{10} +2799.34 q^{11} -1728.00 q^{12} -15814.5 q^{13} -2288.00 q^{14} -1121.23 q^{15} +4096.00 q^{16} +35309.0 q^{17} +5832.00 q^{18} -19488.5 q^{19} +2657.73 q^{20} +7722.00 q^{21} +22394.7 q^{22} -12167.0 q^{23} -13824.0 q^{24} -76400.5 q^{25} -126516. q^{26} -19683.0 q^{27} -18304.0 q^{28} -24130.2 q^{29} -8969.85 q^{30} -30949.4 q^{31} +32768.0 q^{32} -75582.1 q^{33} +282472. q^{34} -11876.7 q^{35} +46656.0 q^{36} -208533. q^{37} -155908. q^{38} +426993. q^{39} +21261.9 q^{40} -116195. q^{41} +61776.0 q^{42} -522487. q^{43} +179158. q^{44} +30273.2 q^{45} -97336.0 q^{46} -1.17457e6 q^{47} -110592. q^{48} -741747. q^{49} -611204. q^{50} -953343. q^{51} -1.01213e6 q^{52} +172185. q^{53} -157464. q^{54} +116248. q^{55} -146432. q^{56} +526190. q^{57} -193042. q^{58} -486816. q^{59} -71758.8 q^{60} +817298. q^{61} -247595. q^{62} -208494. q^{63} +262144. q^{64} -656732. q^{65} -604657. q^{66} -1.37196e6 q^{67} +2.25978e6 q^{68} +328509. q^{69} -95014.0 q^{70} -5.53333e6 q^{71} +373248. q^{72} -3.89660e6 q^{73} -1.66827e6 q^{74} +2.06281e6 q^{75} -1.24726e6 q^{76} -800610. q^{77} +3.41594e6 q^{78} +2.16899e6 q^{79} +170095. q^{80} +531441. q^{81} -929559. q^{82} +9.13811e6 q^{83} +494208. q^{84} +1.46628e6 q^{85} -4.17990e6 q^{86} +651515. q^{87} +1.43326e6 q^{88} +3.66653e6 q^{89} +242186. q^{90} +4.52296e6 q^{91} -778688. q^{92} +835634. q^{93} -9.39658e6 q^{94} -809301. q^{95} -884736. q^{96} -9.00621e6 q^{97} -5.93398e6 q^{98} +2.04072e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 24 q^{2} - 81 q^{3} + 192 q^{4} + 92 q^{5} - 648 q^{6} - 858 q^{7} + 1536 q^{8} + 2187 q^{9}+O(q^{10}) $ 3 * q + 24 * q^2 - 81 * q^3 + 192 * q^4 + 92 * q^5 - 648 * q^6 - 858 * q^7 + 1536 * q^8 + 2187 * q^9	\( 3 q + 24 q^{2} - 81 q^{3} + 192 q^{4} + 92 q^{5} - 648 q^{6} - 858 q^{7} + 1536 q^{8} + 2187 q^{9} + 736 q^{10} - 1820 q^{11} - 5184 q^{12} - 3698 q^{13} - 6864 q^{14} - 2484 q^{15} + 12288 q^{16} + 3164 q^{17} + 17496 q^{18} - 14234 q^{19} + 5888 q^{20} + 23166 q^{21} - 14560 q^{22} - 36501 q^{23} - 41472 q^{24} - 170119 q^{25} - 29584 q^{26} - 59049 q^{27} - 54912 q^{28} - 171914 q^{29} - 19872 q^{30} - 124972 q^{31} + 98304 q^{32} + 49140 q^{33} + 25312 q^{34} - 26312 q^{35} + 139968 q^{36} - 679074 q^{37} - 113872 q^{38} + 99846 q^{39} + 47104 q^{40} - 331362 q^{41} + 185328 q^{42} - 145922 q^{43} - 116480 q^{44} + 67068 q^{45} - 292008 q^{46} - 1824192 q^{47} - 331776 q^{48} - 2225241 q^{49} - 1360952 q^{50} - 85428 q^{51} - 236672 q^{52} - 3442448 q^{53} - 472392 q^{54} - 1548176 q^{55} - 439296 q^{56} + 384318 q^{57} - 1375312 q^{58} - 3147660 q^{59} - 158976 q^{60} - 3444530 q^{61} - 999776 q^{62} - 625482 q^{63} + 786432 q^{64} - 2527960 q^{65} + 393120 q^{66} - 921870 q^{67} + 202496 q^{68} + 985527 q^{69} - 210496 q^{70} + 2594520 q^{71} + 1119744 q^{72} - 540826 q^{73} - 5432592 q^{74} + 4593213 q^{75} - 910976 q^{76} + 520520 q^{77} + 798768 q^{78} + 316994 q^{79} + 376832 q^{80} + 1594323 q^{81} - 2650896 q^{82} + 4119316 q^{83} + 1482624 q^{84} - 382312 q^{85} - 1167376 q^{86} + 4641678 q^{87} - 931840 q^{88} + 1417176 q^{89} + 536544 q^{90} + 1057628 q^{91} - 2336064 q^{92} + 3374244 q^{93} - 14593536 q^{94} + 12543632 q^{95} - 2654208 q^{96} - 15089326 q^{97} - 17801928 q^{98} - 1326780 q^{99}+O(q^{100}) \) 3 * q + 24 * q^2 - 81 * q^3 + 192 * q^4 + 92 * q^5 - 648 * q^6 - 858 * q^7 + 1536 * q^8 + 2187 * q^9 + 736 * q^10 - 1820 * q^11 - 5184 * q^12 - 3698 * q^13 - 6864 * q^14 - 2484 * q^15 + 12288 * q^16 + 3164 * q^17 + 17496 * q^18 - 14234 * q^19 + 5888 * q^20 + 23166 * q^21 - 14560 * q^22 - 36501 * q^23 - 41472 * q^24 - 170119 * q^25 - 29584 * q^26 - 59049 * q^27 - 54912 * q^28 - 171914 * q^29 - 19872 * q^30 - 124972 * q^31 + 98304 * q^32 + 49140 * q^33 + 25312 * q^34 - 26312 * q^35 + 139968 * q^36 - 679074 * q^37 - 113872 * q^38 + 99846 * q^39 + 47104 * q^40 - 331362 * q^41 + 185328 * q^42 - 145922 * q^43 - 116480 * q^44 + 67068 * q^45 - 292008 * q^46 - 1824192 * q^47 - 331776 * q^48 - 2225241 * q^49 - 1360952 * q^50 - 85428 * q^51 - 236672 * q^52 - 3442448 * q^53 - 472392 * q^54 - 1548176 * q^55 - 439296 * q^56 + 384318 * q^57 - 1375312 * q^58 - 3147660 * q^59 - 158976 * q^60 - 3444530 * q^61 - 999776 * q^62 - 625482 * q^63 + 786432 * q^64 - 2527960 * q^65 + 393120 * q^66 - 921870 * q^67 + 202496 * q^68 + 985527 * q^69 - 210496 * q^70 + 2594520 * q^71 + 1119744 * q^72 - 540826 * q^73 - 5432592 * q^74 + 4593213 * q^75 - 910976 * q^76 + 520520 * q^77 + 798768 * q^78 + 316994 * q^79 + 376832 * q^80 + 1594323 * q^81 - 2650896 * q^82 + 4119316 * q^83 + 1482624 * q^84 - 382312 * q^85 - 1167376 * q^86 + 4641678 * q^87 - 931840 * q^88 + 1417176 * q^89 + 536544 * q^90 + 1057628 * q^91 - 2336064 * q^92 + 3374244 * q^93 - 14593536 * q^94 + 12543632 * q^95 - 2654208 * q^96 - 15089326 * q^97 - 17801928 * q^98 - 1326780 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	64.0000	0.500000
$5$	41.5271	0.148572	0.0742859	−	0.997237i	$-0.476332\pi$
$5$	41.5271	0.148572	0.0742859	+	0.997237i	$0.476332\pi$
$6$	−216.000	−0.408248
$7$	−286.000	−0.315154	−0.157577	−	0.987507i	$-0.550368\pi$
$7$	−286.000	−0.315154	−0.157577	+	0.987507i	$0.550368\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	332.217	0.105056
$11$	2799.34	0.634134	0.317067	−	0.948403i	$-0.397302\pi$
$11$	2799.34	0.634134	0.317067	+	0.948403i	$0.397302\pi$
$12$	−1728.00	−0.288675
$13$	−15814.5	−1.99643	−0.998217	−	0.0596884i	$-0.980989\pi$
$13$	−15814.5	−1.99643	−0.998217	+	0.0596884i	$0.980989\pi$
$14$	−2288.00	−0.222848
$15$	−1121.23	−0.0857780
$16$	4096.00	0.250000
$17$	35309.0	1.74307	0.871534	−	0.490336i	$-0.163126\pi$
$17$	35309.0	1.74307	0.871534	+	0.490336i	$0.163126\pi$
$18$	5832.00	0.235702
$19$	−19488.5	−0.651840	−0.325920	−	0.945397i	$-0.605674\pi$
$19$	−19488.5	−0.651840	−0.325920	+	0.945397i	$0.605674\pi$
$20$	2657.73	0.0742859
$21$	7722.00	0.181954
$22$	22394.7	0.448400
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	−13824.0	−0.204124
$25$	−76400.5	−0.977926
$26$	−126516.	−1.41169
$27$	−19683.0	−0.192450
$28$	−18304.0	−0.157577
$29$	−24130.2	−0.183725	−0.0918624	−	0.995772i	$-0.529282\pi$
$29$	−24130.2	−0.183725	−0.0918624	+	0.995772i	$0.529282\pi$
$30$	−8969.85	−0.0606542
$31$	−30949.4	−0.186589	−0.0932946	−	0.995639i	$-0.529740\pi$
$31$	−30949.4	−0.186589	−0.0932946	+	0.995639i	$0.529740\pi$
$32$	32768.0	0.176777
$33$	−75582.1	−0.366117
$34$	282472.	1.23253
$35$	−11876.7	−0.0468230
$36$	46656.0	0.166667
$37$	−208533.	−0.676814	−0.338407	−	0.941000i	$-0.609888\pi$
$37$	−208533.	−0.676814	−0.338407	+	0.941000i	$0.609888\pi$
$38$	−155908.	−0.460920
$39$	426993.	1.15264
$40$	21261.9	0.0525281
$41$	−116195.	−0.263296	−0.131648	−	0.991297i	$-0.542027\pi$
$41$	−116195.	−0.263296	−0.131648	+	0.991297i	$0.542027\pi$
$42$	61776.0	0.128661
$43$	−522487.	−1.00216	−0.501079	−	0.865402i	$-0.667063\pi$
$43$	−522487.	−1.00216	−0.501079	+	0.865402i	$0.667063\pi$
$44$	179158.	0.317067
$45$	30273.2	0.0495239
$46$	−97336.0	−0.147442
$47$	−1.17457e6	−1.65020	−0.825101	−	0.564985i	$-0.808882\pi$
$47$	−1.17457e6	−1.65020	−0.825101	+	0.564985i	$0.808882\pi$
$48$	−110592.	−0.144338
$49$	−741747.	−0.900678
$50$	−611204.	−0.691498
$51$	−953343.	−1.00636
$52$	−1.01213e6	−0.998217
$53$	172185.	0.158866	0.0794328	−	0.996840i	$-0.474689\pi$
$53$	172185.	0.158866	0.0794328	+	0.996840i	$0.474689\pi$
$54$	−157464.	−0.136083
$55$	116248.	0.0942144
$56$	−146432.	−0.111424
$57$	526190.	0.376340
$58$	−193042.	−0.129913
$59$	−486816.	−0.308591	−0.154295	−	0.988025i	$-0.549311\pi$
$59$	−486816.	−0.308591	−0.154295	+	0.988025i	$0.549311\pi$
$60$	−71758.8	−0.0428890
$61$	817298.	0.461027	0.230513	−	0.973069i	$-0.425959\pi$
$61$	817298.	0.461027	0.230513	+	0.973069i	$0.425959\pi$
$62$	−247595.	−0.131938
$63$	−208494.	−0.105051
$64$	262144.	0.125000
$65$	−656732.	−0.296614
$66$	−604657.	−0.258884
$67$	−1.37196e6	−0.557289	−0.278644	−	0.960394i	$-0.589885\pi$
$67$	−1.37196e6	−0.557289	−0.278644	+	0.960394i	$0.589885\pi$
$68$	2.25978e6	0.871534
$69$	328509.	0.120386
$70$	−95014.0	−0.0331089
$71$	−5.53333e6	−1.83477	−0.917386	−	0.397998i	$-0.869705\pi$
$71$	−5.53333e6	−1.83477	−0.917386	+	0.397998i	$0.869705\pi$
$72$	373248.	0.117851
$73$	−3.89660e6	−1.17235	−0.586174	−	0.810186i	$-0.699366\pi$
$73$	−3.89660e6	−1.17235	−0.586174	+	0.810186i	$0.699366\pi$
$74$	−1.66827e6	−0.478580
$75$	2.06281e6	0.564606
$76$	−1.24726e6	−0.325920
$77$	−800610.	−0.199850
$78$	3.41594e6	0.815041
$79$	2.16899e6	0.494952	0.247476	−	0.968894i	$-0.420399\pi$
$79$	2.16899e6	0.494952	0.247476	+	0.968894i	$0.420399\pi$
$80$	170095.	0.0371430
$81$	531441.	0.111111
$82$	−929559.	−0.186178
$83$	9.13811e6	1.75422	0.877108	−	0.480293i	$-0.159470\pi$
$83$	9.13811e6	1.75422	0.877108	+	0.480293i	$0.159470\pi$
$84$	494208.	0.0909771
$85$	1.46628e6	0.258971
$86$	−4.17990e6	−0.708632
$87$	651515.	0.106074
$88$	1.43326e6	0.224200
$89$	3.66653e6	0.551302	0.275651	−	0.961258i	$-0.411107\pi$
$89$	3.66653e6	0.551302	0.275651	+	0.961258i	$0.411107\pi$
$90$	242186.	0.0350187
$91$	4.52296e6	0.629184
$92$	−778688.	−0.104257
$93$	835634.	0.107727
$94$	−9.39658e6	−1.16687
$95$	−809301.	−0.0968450
$96$	−884736.	−0.102062
$97$	−9.00621e6	−1.00194	−0.500969	−	0.865465i	$-0.667023\pi$
$97$	−9.00621e6	−1.00194	−0.500969	+	0.865465i	$0.667023\pi$
$98$	−5.93398e6	−0.636875
$99$	2.04072e6	0.211378
$100$	−4.88963e6	−0.488963
$101$	−5.30515e6	−0.512358	−0.256179	−	0.966629i	$-0.582464\pi$
$101$	−5.30515e6	−0.512358	−0.256179	+	0.966629i	$0.582464\pi$
$102$	−7.62674e6	−0.711604
$103$	1.50152e7	1.35394	0.676972	−	0.736009i	$-0.263292\pi$
$103$	1.50152e7	1.35394	0.676972	+	0.736009i	$0.263292\pi$
$104$	−8.09705e6	−0.705846
$105$	320672.	0.0270333
$106$	1.37748e6	0.112335
$107$	3.37007e6	0.265947	0.132974	−	0.991120i	$-0.457547\pi$
$107$	3.37007e6	0.265947	0.132974	+	0.991120i	$0.457547\pi$
$108$	−1.25971e6	−0.0962250
$109$	−2.08819e7	−1.54447	−0.772233	−	0.635340i	$-0.780860\pi$
$109$	−2.08819e7	−1.54447	−0.772233	+	0.635340i	$0.780860\pi$
$110$	929986.	0.0666196
$111$	5.63040e6	0.390759
$112$	−1.17146e6	−0.0787885
$113$	1.92780e7	1.25686	0.628431	−	0.777866i	$-0.283698\pi$
$113$	1.92780e7	1.25686	0.628431	+	0.777866i	$0.283698\pi$
$114$	4.20952e6	0.266113
$115$	−505260.	−0.0309794
$116$	−1.54433e6	−0.0918624
$117$	−1.15288e7	−0.665478
$118$	−3.89453e6	−0.218207
$119$	−1.00984e7	−0.549335
$120$	−574070.	−0.0303271
$121$	−1.16509e7	−0.597874
$122$	6.53838e6	0.325995
$123$	3.13726e6	0.152014
$124$	−1.98076e6	−0.0932946
$125$	−6.41699e6	−0.293864
$126$	−1.66795e6	−0.0742825
$127$	−1.23381e7	−0.534483	−0.267242	−	0.963630i	$-0.586112\pi$
$127$	−1.23381e7	−0.534483	−0.267242	+	0.963630i	$0.586112\pi$
$128$	2.09715e6	0.0883883
$129$	1.41071e7	0.578596
$130$	−5.25386e6	−0.209738
$131$	4.17151e7	1.62123	0.810613	−	0.585582i	$-0.199134\pi$
$131$	4.17151e7	1.62123	0.810613	+	0.585582i	$0.199134\pi$
$132$	−4.83725e6	−0.183059
$133$	5.57371e6	0.205430
$134$	−1.09757e7	−0.394063
$135$	−817378.	−0.0285927
$136$	1.80782e7	0.616267
$137$	−2.65906e7	−0.883498	−0.441749	−	0.897139i	$-0.645642\pi$
$137$	−2.65906e7	−0.883498	−0.441749	+	0.897139i	$0.645642\pi$
$138$	2.62807e6	0.0851257
$139$	5.74851e7	1.81553	0.907765	−	0.419479i	$-0.137787\pi$
$139$	5.74851e7	1.81553	0.907765	+	0.419479i	$0.137787\pi$
$140$	−760112.	−0.0234115
$141$	3.17135e7	0.952745
$142$	−4.42666e7	−1.29738
$143$	−4.42702e7	−1.26601
$144$	2.98598e6	0.0833333
$145$	−1.00206e6	−0.0272963
$146$	−3.11728e7	−0.828975
$147$	2.00272e7	0.520007
$148$	−1.33461e7	−0.338407
$149$	5.04249e7	1.24880	0.624400	−	0.781104i	$-0.285343\pi$
$149$	5.04249e7	1.24880	0.624400	+	0.781104i	$0.285343\pi$
$150$	1.65025e7	0.399237
$151$	5.21519e6	0.123268	0.0616341	−	0.998099i	$-0.480369\pi$
$151$	5.21519e6	0.123268	0.0616341	+	0.998099i	$0.480369\pi$
$152$	−9.97811e6	−0.230460
$153$	2.57403e7	0.581022
$154$	−6.40488e6	−0.141315
$155$	−1.28524e6	−0.0277219
$156$	2.73275e7	0.576321
$157$	8.03316e7	1.65668	0.828338	−	0.560228i	$-0.189287\pi$
$157$	8.03316e7	1.65668	0.828338	+	0.560228i	$0.189287\pi$
$158$	1.73519e7	0.349984
$159$	−4.64900e6	−0.0917211
$160$	1.36076e6	0.0262640
$161$	3.47976e6	0.0657142
$162$	4.25153e6	0.0785674
$163$	1.09451e7	0.197954	0.0989768	−	0.995090i	$-0.468443\pi$
$163$	1.09451e7	0.197954	0.0989768	+	0.995090i	$0.468443\pi$
$164$	−7.43648e6	−0.131648
$165$	−3.13870e6	−0.0543947
$166$	7.31049e7	1.24042
$167$	−6.00144e7	−0.997121	−0.498560	−	0.866855i	$-0.666138\pi$
$167$	−6.00144e7	−0.997121	−0.498560	+	0.866855i	$0.666138\pi$
$168$	3.95366e6	0.0643306
$169$	1.87351e8	2.98575
$170$	1.17302e7	0.183120
$171$	−1.42071e7	−0.217280
$172$	−3.34392e7	−0.501079
$173$	1.68210e7	0.246996	0.123498	−	0.992345i	$-0.460589\pi$
$173$	1.68210e7	0.246996	0.123498	+	0.992345i	$0.460589\pi$
$174$	5.21212e6	0.0750053
$175$	2.18505e7	0.308197
$176$	1.14661e7	0.158533
$177$	1.31440e7	0.178165
$178$	2.93322e7	0.389830
$179$	7.54499e7	0.983271	0.491635	−	0.870801i	$-0.336399\pi$
$179$	7.54499e7	0.983271	0.491635	+	0.870801i	$0.336399\pi$
$180$	1.93749e6	0.0247620
$181$	−5.67489e7	−0.711348	−0.355674	−	0.934610i	$-0.615749\pi$
$181$	−5.67489e7	−0.711348	−0.355674	+	0.934610i	$0.615749\pi$
$182$	3.61837e7	0.444900
$183$	−2.20670e7	−0.266174
$184$	−6.22950e6	−0.0737210
$185$	−8.65978e6	−0.100556
$186$	6.68507e6	0.0761747
$187$	9.88418e7	1.10534
$188$	−7.51726e7	−0.825101
$189$	5.62934e6	0.0606514
$190$	−6.47441e6	−0.0684798
$191$	1.76379e7	0.183160	0.0915800	−	0.995798i	$-0.470808\pi$
$191$	1.76379e7	0.183160	0.0915800	+	0.995798i	$0.470808\pi$
$192$	−7.07789e6	−0.0721688
$193$	−9.35162e7	−0.936346	−0.468173	−	0.883637i	$-0.655088\pi$
$193$	−9.35162e7	−0.936346	−0.468173	+	0.883637i	$0.655088\pi$
$194$	−7.20497e7	−0.708477
$195$	1.77318e7	0.171250
$196$	−4.74718e7	−0.450339
$197$	−1.48989e8	−1.38842	−0.694211	−	0.719771i	$-0.744247\pi$
$197$	−1.48989e8	−1.38842	−0.694211	+	0.719771i	$0.744247\pi$
$198$	1.63257e7	0.149467
$199$	3.14201e7	0.282632	0.141316	−	0.989965i	$-0.454867\pi$
$199$	3.14201e7	0.282632	0.141316	+	0.989965i	$0.454867\pi$
$200$	−3.91171e7	−0.345749
$201$	3.70430e7	0.321751
$202$	−4.24412e7	−0.362291
$203$	6.90124e6	0.0579016
$204$	−6.10139e7	−0.503180
$205$	−4.82524e6	−0.0391183
$206$	1.20122e8	0.957383
$207$	−8.86974e6	−0.0695048
$208$	−6.47764e7	−0.499109
$209$	−5.45549e7	−0.413354
$210$	2.56538e6	0.0191154
$211$	2.20634e8	1.61691	0.808453	−	0.588561i	$-0.200305\pi$
$211$	2.20634e8	1.61691	0.808453	+	0.588561i	$0.200305\pi$
$212$	1.10198e7	0.0794328
$213$	1.49400e8	1.05931
$214$	2.69605e7	0.188053
$215$	−2.16974e7	−0.148892
$216$	−1.00777e7	−0.0680414
$217$	8.85153e6	0.0588043
$218$	−1.67056e8	−1.09210
$219$	1.05208e8	0.676855
$220$	7.43989e6	0.0471072
$221$	−5.58396e8	−3.47992
$222$	4.50432e7	0.276308
$223$	−1.90688e8	−1.15148	−0.575740	−	0.817633i	$-0.695286\pi$
$223$	−1.90688e8	−1.15148	−0.575740	+	0.817633i	$0.695286\pi$
$224$	−9.37165e6	−0.0557119
$225$	−5.56960e7	−0.325975
$226$	1.54224e8	0.888735
$227$	1.41698e8	0.804032	0.402016	−	0.915633i	$-0.368310\pi$
$227$	1.41698e8	0.804032	0.402016	+	0.915633i	$0.368310\pi$
$228$	3.36761e7	0.188170
$229$	1.05476e8	0.580402	0.290201	−	0.956966i	$-0.406278\pi$
$229$	1.05476e8	0.580402	0.290201	+	0.956966i	$0.406278\pi$
$230$	−4.04208e6	−0.0219057
$231$	2.16165e7	0.115383
$232$	−1.23547e7	−0.0649565
$233$	−3.53285e8	−1.82970	−0.914850	−	0.403794i	$-0.867691\pi$
$233$	−3.53285e8	−1.82970	−0.914850	+	0.403794i	$0.867691\pi$
$234$	−9.22304e7	−0.470564
$235$	−4.87766e7	−0.245174
$236$	−3.11562e7	−0.154295
$237$	−5.85628e7	−0.285761
$238$	−8.07870e7	−0.388438
$239$	−3.00197e8	−1.42238	−0.711188	−	0.703002i	$-0.751842\pi$
$239$	−3.00197e8	−1.42238	−0.711188	+	0.703002i	$0.751842\pi$
$240$	−4.59256e6	−0.0214445
$241$	−5.74470e7	−0.264367	−0.132184	−	0.991225i	$-0.542199\pi$
$241$	−5.74470e7	−0.264367	−0.132184	+	0.991225i	$0.542199\pi$
$242$	−9.32071e7	−0.422761
$243$	−1.43489e7	−0.0641500
$244$	5.23071e7	0.230513
$245$	−3.08026e7	−0.133815
$246$	2.50981e7	0.107490
$247$	3.08202e8	1.30136
$248$	−1.58461e7	−0.0659692
$249$	−2.46729e8	−1.01280
$250$	−5.13359e7	−0.207793
$251$	−3.12841e8	−1.24872	−0.624362	−	0.781135i	$-0.714641\pi$
$251$	−3.12841e8	−1.24872	−0.624362	+	0.781135i	$0.714641\pi$
$252$	−1.33436e7	−0.0525257
$253$	−3.40595e7	−0.132226
$254$	−9.87045e7	−0.377937
$255$	−3.95895e7	−0.149517
$256$	1.67772e7	0.0625000
$257$	3.89180e8	1.43016	0.715080	−	0.699043i	$-0.246390\pi$
$257$	3.89180e8	1.43016	0.715080	+	0.699043i	$0.246390\pi$
$258$	1.12857e8	0.409129
$259$	5.96406e7	0.213301
$260$	−4.20308e7	−0.148307
$261$	−1.75909e7	−0.0612416
$262$	3.33721e8	1.14638
$263$	2.02685e8	0.687032	0.343516	−	0.939147i	$-0.388382\pi$
$263$	2.02685e8	0.687032	0.343516	+	0.939147i	$0.388382\pi$
$264$	−3.86980e7	−0.129442
$265$	7.15035e6	0.0236030
$266$	4.45897e7	0.145261
$267$	−9.89962e7	−0.318295
$268$	−8.78056e7	−0.278644
$269$	8.11193e7	0.254092	0.127046	−	0.991897i	$-0.459450\pi$
$269$	8.11193e7	0.254092	0.127046	+	0.991897i	$0.459450\pi$
$270$	−6.53902e6	−0.0202181
$271$	5.50191e7	0.167927	0.0839636	−	0.996469i	$-0.473242\pi$
$271$	5.50191e7	0.167927	0.0839636	+	0.996469i	$0.473242\pi$
$272$	1.44626e8	0.435767
$273$	−1.22120e8	−0.363260
$274$	−2.12725e8	−0.624728
$275$	−2.13871e8	−0.620136
$276$	2.10246e7	0.0601929
$277$	−3.88158e8	−1.09731	−0.548655	−	0.836049i	$-0.684860\pi$
$277$	−3.88158e8	−1.09731	−0.548655	+	0.836049i	$0.684860\pi$
$278$	4.59881e8	1.28377
$279$	−2.25621e7	−0.0621964
$280$	−6.08089e6	−0.0165544
$281$	−3.47211e8	−0.933516	−0.466758	−	0.884385i	$-0.654578\pi$
$281$	−3.47211e8	−0.933516	−0.466758	+	0.884385i	$0.654578\pi$
$282$	2.53708e8	0.673692
$283$	−6.96096e8	−1.82565	−0.912824	−	0.408354i	$-0.866103\pi$
$283$	−6.96096e8	−1.82565	−0.912824	+	0.408354i	$0.866103\pi$
$284$	−3.54133e8	−0.917386
$285$	2.18511e7	0.0559135
$286$	−3.54162e8	−0.895201
$287$	3.32317e7	0.0829787
$288$	2.38879e7	0.0589256
$289$	8.36386e8	2.03828
$290$	−8.01645e6	−0.0193014
$291$	2.43168e8	0.578469
$292$	−2.49383e8	−0.586174
$293$	−4.78765e8	−1.11195	−0.555976	−	0.831198i	$-0.687655\pi$
$293$	−4.78765e8	−1.11195	−0.555976	+	0.831198i	$0.687655\pi$
$294$	1.60217e8	0.367700
$295$	−2.02161e7	−0.0458479
$296$	−1.06769e8	−0.239290
$297$	−5.50994e7	−0.122039
$298$	4.03399e8	0.883036
$299$	1.92416e8	0.416285
$300$	1.32020e8	0.282303
$301$	1.49431e8	0.315834
$302$	4.17216e7	0.0871638
$303$	1.43239e8	0.295810
$304$	−7.98249e7	−0.162960
$305$	3.39400e7	0.0684955
$306$	2.05922e8	0.410845
$307$	4.35793e7	0.0859599	0.0429799	−	0.999076i	$-0.486315\pi$
$307$	4.35793e7	0.0859599	0.0429799	+	0.999076i	$0.486315\pi$
$308$	−5.12391e7	−0.0999249
$309$	−4.05410e8	−0.781700
$310$	−1.02819e7	−0.0196023
$311$	−1.47610e8	−0.278262	−0.139131	−	0.990274i	$-0.544431\pi$
$311$	−1.47610e8	−0.278262	−0.139131	+	0.990274i	$0.544431\pi$
$312$	2.18620e8	0.407520
$313$	2.13743e8	0.393991	0.196995	−	0.980404i	$-0.436882\pi$
$313$	2.13743e8	0.393991	0.196995	+	0.980404i	$0.436882\pi$
$314$	6.42653e8	1.17145
$315$	−8.65815e6	−0.0156077
$316$	1.38816e8	0.247476
$317$	−3.58434e8	−0.631979	−0.315989	−	0.948763i	$-0.602336\pi$
$317$	−3.58434e8	−0.631979	−0.315989	+	0.948763i	$0.602336\pi$
$318$	−3.71920e7	−0.0648566
$319$	−6.75486e7	−0.116506
$320$	1.08861e7	0.0185715
$321$	−9.09918e7	−0.153545
$322$	2.78381e7	0.0464669
$323$	−6.88119e8	−1.13620
$324$	3.40122e7	0.0555556
$325$	1.20824e9	1.95237
$326$	8.75609e7	0.139974
$327$	5.63812e8	0.891697
$328$	−5.94918e7	−0.0930891
$329$	3.35928e8	0.520068
$330$	−2.51096e7	−0.0384629
$331$	−4.29241e8	−0.650584	−0.325292	−	0.945614i	$-0.605463\pi$
$331$	−4.29241e8	−0.650584	−0.325292	+	0.945614i	$0.605463\pi$
$332$	5.84839e8	0.877108
$333$	−1.52021e8	−0.225605
$334$	−4.80115e8	−0.705071
$335$	−5.69736e7	−0.0827974
$336$	3.16293e7	0.0454886
$337$	−8.32243e8	−1.18453	−0.592264	−	0.805744i	$-0.701766\pi$
$337$	−8.32243e8	−1.18453	−0.592264	+	0.805744i	$0.701766\pi$
$338$	1.49881e9	2.11124
$339$	−5.20506e8	−0.725649
$340$	9.38419e7	0.129485
$341$	−8.66378e7	−0.118322
$342$	−1.13657e8	−0.153640
$343$	4.47673e8	0.599006
$344$	−2.67513e8	−0.354316
$345$	1.36420e7	0.0178859
$346$	1.34568e8	0.174653
$347$	−5.48701e7	−0.0704990	−0.0352495	−	0.999379i	$-0.511223\pi$
$347$	−5.48701e7	−0.0704990	−0.0352495	+	0.999379i	$0.511223\pi$
$348$	4.16970e7	0.0530368
$349$	9.49652e7	0.119585	0.0597923	−	0.998211i	$-0.480956\pi$
$349$	9.49652e7	0.119585	0.0597923	+	0.998211i	$0.480956\pi$
$350$	1.74804e8	0.217929
$351$	3.11278e8	0.384214
$352$	9.17287e7	0.112100
$353$	4.67066e8	0.565155	0.282577	−	0.959245i	$-0.408811\pi$
$353$	4.67066e8	0.565155	0.282577	+	0.959245i	$0.408811\pi$
$354$	1.05152e8	0.125982
$355$	−2.29783e8	−0.272595
$356$	2.34658e8	0.275651
$357$	2.72656e8	0.317159
$358$	6.03599e8	0.695277
$359$	3.65088e8	0.416454	0.208227	−	0.978080i	$-0.433231\pi$
$359$	3.65088e8	0.416454	0.208227	+	0.978080i	$0.433231\pi$
$360$	1.54999e7	0.0175094
$361$	−5.14070e8	−0.575105
$362$	−4.53991e8	−0.502999
$363$	3.14574e8	0.345183
$364$	2.89469e8	0.314592
$365$	−1.61815e8	−0.174178
$366$	−1.76536e8	−0.188213
$367$	1.72693e9	1.82366	0.911831	−	0.410566i	$-0.134669\pi$
$367$	1.72693e9	1.82366	0.911831	+	0.410566i	$0.134669\pi$
$368$	−4.98360e7	−0.0521286
$369$	−8.47061e7	−0.0877652
$370$	−6.92783e7	−0.0711035
$371$	−4.92449e7	−0.0500672
$372$	5.34806e7	0.0538636
$373$	−8.10865e8	−0.809036	−0.404518	−	0.914530i	$-0.632561\pi$
$373$	−8.10865e8	−0.809036	−0.404518	+	0.914530i	$0.632561\pi$
$374$	7.90734e8	0.781592
$375$	1.73259e8	0.169663
$376$	−6.01381e8	−0.583435
$377$	3.81608e8	0.366795
$378$	4.50347e7	0.0428870
$379$	9.40245e8	0.887163	0.443582	−	0.896234i	$-0.353708\pi$
$379$	9.40245e8	0.887163	0.443582	+	0.896234i	$0.353708\pi$
$380$	−5.17952e7	−0.0484225
$381$	3.33128e8	0.308584
$382$	1.41103e8	0.129514
$383$	1.88821e9	1.71733	0.858666	−	0.512536i	$-0.171294\pi$
$383$	1.88821e9	1.71733	0.858666	+	0.512536i	$0.171294\pi$
$384$	−5.66231e7	−0.0510310
$385$	−3.32470e7	−0.0296920
$386$	−7.48130e8	−0.662096
$387$	−3.80893e8	−0.334052
$388$	−5.76398e8	−0.500969
$389$	1.32543e9	1.14165	0.570825	−	0.821072i	$-0.306623\pi$
$389$	1.32543e9	1.14165	0.570825	+	0.821072i	$0.306623\pi$
$390$	1.41854e8	0.121092
$391$	−4.29605e8	−0.363455
$392$	−3.79774e8	−0.318438
$393$	−1.12631e9	−0.936015
$394$	−1.19191e9	−0.981763
$395$	9.00720e7	0.0735360
$396$	1.30606e8	0.105689
$397$	−1.40217e9	−1.12469	−0.562345	−	0.826903i	$-0.690101\pi$
$397$	−1.40217e9	−1.12469	−0.562345	+	0.826903i	$0.690101\pi$
$398$	2.51360e8	0.199851
$399$	−1.50490e8	−0.118605
$400$	−3.12936e8	−0.244482
$401$	8.02654e8	0.621618	0.310809	−	0.950472i	$-0.399400\pi$
$401$	8.02654e8	0.621618	0.310809	+	0.950472i	$0.399400\pi$
$402$	2.96344e8	0.227512
$403$	4.89451e8	0.372513
$404$	−3.39530e8	−0.256179
$405$	2.20692e7	0.0165080
$406$	5.52099e7	0.0409426
$407$	−5.83755e8	−0.429191
$408$	−4.88112e8	−0.355802
$409$	−1.32301e9	−0.956162	−0.478081	−	0.878316i	$-0.658668\pi$
$409$	−1.32301e9	−0.956162	−0.478081	+	0.878316i	$0.658668\pi$
$410$	−3.86019e7	−0.0276608
$411$	7.17945e8	0.510088
$412$	9.60972e8	0.676972
$413$	1.39229e8	0.0972536
$414$	−7.09579e7	−0.0491473
$415$	3.79479e8	0.260627
$416$	−5.18211e8	−0.352923
$417$	−1.55210e9	−1.04820
$418$	−4.36439e8	−0.292285
$419$	2.02443e8	0.134448	0.0672240	−	0.997738i	$-0.478586\pi$
$419$	2.02443e8	0.134448	0.0672240	+	0.997738i	$0.478586\pi$
$420$	2.05230e7	0.0135166
$421$	4.26232e8	0.278393	0.139196	−	0.990265i	$-0.455548\pi$
$421$	4.26232e8	0.278393	0.139196	+	0.990265i	$0.455548\pi$
$422$	1.76508e9	1.14332
$423$	−8.56263e8	−0.550067
$424$	8.81588e7	0.0561675
$425$	−2.69762e9	−1.70459
$426$	1.19520e9	0.749043
$427$	−2.33747e8	−0.145294
$428$	2.15684e8	0.132974
$429$	1.19530e9	0.730929
$430$	−1.73579e8	−0.105283
$431$	−1.56063e9	−0.938922	−0.469461	−	0.882953i	$-0.655552\pi$
$431$	−1.56063e9	−0.938922	−0.469461	+	0.882953i	$0.655552\pi$
$432$	−8.06216e7	−0.0481125
$433$	1.84368e9	1.09139	0.545694	−	0.837985i	$-0.316266\pi$
$433$	1.84368e9	1.09139	0.545694	+	0.837985i	$0.316266\pi$
$434$	7.08122e7	0.0415809
$435$	2.70555e7	0.0157595
$436$	−1.33644e9	−0.772233
$437$	2.37117e8	0.135918
$438$	8.41666e8	0.478609
$439$	6.78955e8	0.383015	0.191507	−	0.981491i	$-0.438662\pi$
$439$	6.78955e8	0.383015	0.191507	+	0.981491i	$0.438662\pi$
$440$	5.95191e7	0.0333098
$441$	−5.40734e8	−0.300226
$442$	−4.46717e9	−2.46067
$443$	3.62712e8	0.198221	0.0991104	−	0.995076i	$-0.468400\pi$
$443$	3.62712e8	0.198221	0.0991104	+	0.995076i	$0.468400\pi$
$444$	3.60346e8	0.195379
$445$	1.52260e8	0.0819080
$446$	−1.52550e9	−0.814219
$447$	−1.36147e9	−0.720996
$448$	−7.49732e7	−0.0393943
$449$	2.19299e9	1.14334	0.571668	−	0.820485i	$-0.306297\pi$
$449$	2.19299e9	1.14334	0.571668	+	0.820485i	$0.306297\pi$
$450$	−4.45568e8	−0.230499
$451$	−3.25269e8	−0.166965
$452$	1.23379e9	0.628431
$453$	−1.40810e8	−0.0711689
$454$	1.13358e9	0.568537
$455$	1.87825e8	0.0934791
$456$	2.69409e8	0.133056
$457$	−3.68778e9	−1.80741	−0.903707	−	0.428151i	$-0.859165\pi$
$457$	−3.68778e9	−1.80741	−0.903707	+	0.428151i	$0.859165\pi$
$458$	8.43807e8	0.410406
$459$	−6.94987e8	−0.335453
$460$	−3.23366e7	−0.0154897
$461$	1.17428e9	0.558235	0.279118	−	0.960257i	$-0.409958\pi$
$461$	1.17428e9	0.558235	0.279118	+	0.960257i	$0.409958\pi$
$462$	1.72932e8	0.0815883
$463$	2.83931e9	1.32947	0.664736	−	0.747078i	$-0.268544\pi$
$463$	2.83931e9	1.32947	0.664736	+	0.747078i	$0.268544\pi$
$464$	−9.88373e7	−0.0459312
$465$	3.47014e7	0.0160052
$466$	−2.82628e9	−1.29379
$467$	2.39046e9	1.08611	0.543053	−	0.839698i	$-0.317268\pi$
$467$	2.39046e9	1.08611	0.543053	+	0.839698i	$0.317268\pi$
$468$	−7.37843e8	−0.332739
$469$	3.92381e8	0.175632
$470$	−3.90212e8	−0.173364
$471$	−2.16895e9	−0.956482
$472$	−2.49250e8	−0.109103
$473$	−1.46262e9	−0.635502
$474$	−4.68503e8	−0.202063
$475$	1.48893e9	0.637451
$476$	−6.46296e8	−0.274667
$477$	1.25523e8	0.0529552
$478$	−2.40158e9	−1.00577
$479$	2.67749e9	1.11315	0.556575	−	0.830798i	$-0.312115\pi$
$479$	2.67749e9	1.11315	0.556575	+	0.830798i	$0.312115\pi$
$480$	−3.67405e7	−0.0151635
$481$	3.29786e9	1.35122
$482$	−4.59576e8	−0.186936
$483$	−9.39536e7	−0.0379401
$484$	−7.45656e8	−0.298937
$485$	−3.74002e8	−0.148860
$486$	−1.14791e8	−0.0453609
$487$	−2.90561e9	−1.13995	−0.569975	−	0.821662i	$-0.693047\pi$
$487$	−2.90561e9	−1.13995	−0.569975	+	0.821662i	$0.693047\pi$
$488$	4.18456e8	0.162997
$489$	−2.95518e8	−0.114289
$490$	−2.46421e8	−0.0946217
$491$	3.95853e9	1.50921	0.754603	−	0.656181i	$-0.227829\pi$
$491$	3.95853e9	1.50921	0.754603	+	0.656181i	$0.227829\pi$
$492$	2.00785e8	0.0760069
$493$	−8.52013e8	−0.320245
$494$	2.46561e9	0.920197
$495$	8.47450e7	0.0314048
$496$	−1.26769e8	−0.0466473
$497$	1.58253e9	0.578236
$498$	−1.97383e9	−0.716156
$499$	3.21375e8	0.115787	0.0578936	−	0.998323i	$-0.481562\pi$
$499$	3.21375e8	0.115787	0.0578936	+	0.998323i	$0.481562\pi$
$500$	−4.10688e8	−0.146932
$501$	1.62039e9	0.575688
$502$	−2.50273e9	−0.882981
$503$	−1.86840e9	−0.654608	−0.327304	−	0.944919i	$-0.606140\pi$
$503$	−1.86840e9	−0.654608	−0.327304	+	0.944919i	$0.606140\pi$
$504$	−1.06749e8	−0.0371413
$505$	−2.20308e8	−0.0761219
$506$	−2.72476e8	−0.0934979
$507$	−5.05849e9	−1.72382
$508$	−7.89636e8	−0.267242
$509$	−2.91991e9	−0.981427	−0.490713	−	0.871321i	$-0.663264\pi$
$509$	−2.91991e9	−0.981427	−0.490713	+	0.871321i	$0.663264\pi$
$510$	−3.16716e8	−0.105724
$511$	1.11443e9	0.369470
$512$	1.34218e8	0.0441942
$513$	3.83592e8	0.125447
$514$	3.11344e9	1.01128
$515$	6.23537e8	0.201158
$516$	9.02857e8	0.289298
$517$	−3.28802e9	−1.04645
$518$	4.77124e8	0.150826
$519$	−4.54167e8	−0.142603
$520$	−3.36247e8	−0.104869
$521$	−3.38750e9	−1.04942	−0.524708	−	0.851282i	$-0.675825\pi$
$521$	−3.38750e9	−1.04942	−0.524708	+	0.851282i	$0.675825\pi$
$522$	−1.40727e8	−0.0433044
$523$	3.35239e9	1.02471	0.512353	−	0.858775i	$-0.328774\pi$
$523$	3.35239e9	1.02471	0.512353	+	0.858775i	$0.328774\pi$
$524$	2.66977e9	0.810613
$525$	−5.89965e8	−0.177938
$526$	1.62148e9	0.485805
$527$	−1.09279e9	−0.325237
$528$	−3.09584e8	−0.0915293
$529$	1.48036e8	0.0434783
$530$	5.72028e7	0.0166898
$531$	−3.54889e8	−0.102864
$532$	3.56718e8	0.102715
$533$	1.83757e9	0.525652
$534$	−7.91970e8	−0.225068
$535$	1.39949e8	0.0395123
$536$	−7.02444e8	−0.197031
$537$	−2.03715e9	−0.567692
$538$	6.48955e8	0.179670
$539$	−2.07640e9	−0.571150
$540$	−5.23122e7	−0.0142963
$541$	−2.43644e9	−0.661554	−0.330777	−	0.943709i	$-0.607311\pi$
$541$	−2.43644e9	−0.661554	−0.330777	+	0.943709i	$0.607311\pi$
$542$	4.40153e8	0.118742
$543$	1.53222e9	0.410697
$544$	1.15701e9	0.308134
$545$	−8.67166e8	−0.229464
$546$	−9.76959e8	−0.256863
$547$	−1.40940e9	−0.368195	−0.184098	−	0.982908i	$-0.558936\pi$
$547$	−1.40940e9	−0.368195	−0.184098	+	0.982908i	$0.558936\pi$
$548$	−1.70180e9	−0.441749
$549$	5.95810e8	0.153676
$550$	−1.71097e9	−0.438502
$551$	4.70261e8	0.119759
$552$	1.68197e8	0.0425628
$553$	−6.20332e8	−0.155986
$554$	−3.10527e9	−0.775916
$555$	2.33814e8	0.0580558
$556$	3.67905e9	0.907765
$557$	6.02997e8	0.147850	0.0739252	−	0.997264i	$-0.476447\pi$
$557$	6.02997e8	0.147850	0.0739252	+	0.997264i	$0.476447\pi$
$558$	−1.80497e8	−0.0439795
$559$	8.26289e9	2.00074
$560$	−4.86472e7	−0.0117058
$561$	−2.66873e9	−0.638167
$562$	−2.77769e9	−0.660095
$563$	3.01135e7	0.00711184	0.00355592	−	0.999994i	$-0.498868\pi$
$563$	3.01135e7	0.00711184	0.00355592	+	0.999994i	$0.498868\pi$
$564$	2.02966e9	0.476372
$565$	8.00559e8	0.186734
$566$	−5.56877e9	−1.29093
$567$	−1.51992e8	−0.0350171
$568$	−2.83306e9	−0.648690
$569$	1.61510e9	0.367541	0.183770	−	0.982969i	$-0.441170\pi$
$569$	1.61510e9	0.367541	0.183770	+	0.982969i	$0.441170\pi$
$570$	1.74809e8	0.0395368
$571$	6.13454e9	1.37897	0.689487	−	0.724298i	$-0.257836\pi$
$571$	6.13454e9	1.37897	0.689487	+	0.724298i	$0.257836\pi$
$572$	−2.83330e9	−0.633003
$573$	−4.76224e8	−0.105747
$574$	2.65854e8	0.0586748
$575$	9.29565e8	0.203912
$576$	1.91103e8	0.0416667
$577$	3.77193e9	0.817425	0.408713	−	0.912663i	$-0.365978\pi$
$577$	3.77193e9	0.817425	0.408713	+	0.912663i	$0.365978\pi$
$578$	6.69109e9	1.44128
$579$	2.52494e9	0.540599
$580$	−6.41316e7	−0.0136482
$581$	−2.61350e9	−0.552848
$582$	1.94534e9	0.409040
$583$	4.82004e8	0.100742
$584$	−1.99506e9	−0.414487
$585$	−4.78758e8	−0.0988713
$586$	−3.83012e9	−0.786269
$587$	2.11289e9	0.431165	0.215582	−	0.976486i	$-0.430835\pi$
$587$	2.11289e9	0.431165	0.215582	+	0.976486i	$0.430835\pi$
$588$	1.28174e9	0.260003
$589$	6.03158e8	0.121626
$590$	−1.61728e8	−0.0324193
$591$	4.02270e9	0.801606
$592$	−8.54153e8	−0.169204
$593$	−5.78227e9	−1.13869	−0.569347	−	0.822098i	$-0.692804\pi$
$593$	−5.78227e9	−1.13869	−0.569347	+	0.822098i	$0.692804\pi$
$594$	−4.40795e8	−0.0862947
$595$	−4.19356e8	−0.0816156
$596$	3.22719e9	0.624400
$597$	−8.48341e8	−0.163178
$598$	1.53932e9	0.294358
$599$	1.32859e9	0.252579	0.126289	−	0.991993i	$-0.459693\pi$
$599$	1.32859e9	0.252579	0.126289	+	0.991993i	$0.459693\pi$
$600$	1.05616e9	0.199618
$601$	8.68405e8	0.163178	0.0815890	−	0.996666i	$-0.474001\pi$
$601$	8.68405e8	0.163178	0.0815890	+	0.996666i	$0.474001\pi$
$602$	1.19545e9	0.223328
$603$	−1.00016e9	−0.185763
$604$	3.33772e8	0.0616341
$605$	−4.83827e8	−0.0888273
$606$	1.14591e9	0.209169
$607$	−7.36455e9	−1.33655	−0.668276	−	0.743913i	$-0.732967\pi$
$607$	−7.36455e9	−1.33655	−0.668276	+	0.743913i	$0.732967\pi$
$608$	−6.38599e8	−0.115230
$609$	−1.86333e8	−0.0334295
$610$	2.71520e8	0.0484337
$611$	1.85753e10	3.29452
$612$	1.64738e9	0.290511
$613$	−5.35310e9	−0.938629	−0.469314	−	0.883031i	$-0.655499\pi$
$613$	−5.35310e9	−0.938629	−0.469314	+	0.883031i	$0.655499\pi$
$614$	3.48634e8	0.0607828
$615$	1.30281e8	0.0225850
$616$	−4.09913e8	−0.0706576
$617$	−7.31088e9	−1.25306	−0.626530	−	0.779397i	$-0.715525\pi$
$617$	−7.31088e9	−1.25306	−0.626530	+	0.779397i	$0.715525\pi$
$618$	−3.24328e9	−0.552745
$619$	−8.84911e9	−1.49962	−0.749811	−	0.661652i	$-0.769856\pi$
$619$	−8.84911e9	−1.49962	−0.749811	+	0.661652i	$0.769856\pi$
$620$	−8.22553e7	−0.0138609
$621$	2.39483e8	0.0401286
$622$	−1.18088e9	−0.196761
$623$	−1.04863e9	−0.173745
$624$	1.74896e9	0.288160
$625$	5.70231e9	0.934266
$626$	1.70994e9	0.278594
$627$	1.47298e9	0.238650
$628$	5.14122e9	0.828338
$629$	−7.36310e9	−1.17973
$630$	−6.92652e7	−0.0110363
$631$	2.69893e9	0.427651	0.213825	−	0.976872i	$-0.431408\pi$
$631$	2.69893e9	0.427651	0.213825	+	0.976872i	$0.431408\pi$
$632$	1.11052e9	0.174992
$633$	−5.95713e9	−0.933521
$634$	−2.86748e9	−0.446876
$635$	−5.12364e8	−0.0794091
$636$	−2.97536e8	−0.0458606
$637$	1.17304e10	1.79814
$638$	−5.40388e8	−0.0823823
$639$	−4.03380e9	−0.611591
$640$	8.70886e7	0.0131320
$641$	−1.08090e10	−1.62099	−0.810496	−	0.585744i	$-0.800802\pi$
$641$	−1.08090e10	−1.62099	−0.810496	+	0.585744i	$0.800802\pi$
$642$	−7.27935e8	−0.108572
$643$	9.83463e9	1.45888	0.729440	−	0.684044i	$-0.239781\pi$
$643$	9.83463e9	1.45888	0.729440	+	0.684044i	$0.239781\pi$
$644$	2.22705e8	0.0328571
$645$	5.85829e8	0.0859630
$646$	−5.50496e9	−0.803415
$647$	−1.88758e9	−0.273993	−0.136996	−	0.990572i	$-0.543745\pi$
$647$	−1.88758e9	−0.273993	−0.136996	+	0.990572i	$0.543745\pi$
$648$	2.72098e8	0.0392837
$649$	−1.36276e9	−0.195688
$650$	9.66591e9	1.38053
$651$	−2.38991e8	−0.0339507
$652$	7.00487e8	0.0989768
$653$	1.03442e10	1.45379	0.726896	−	0.686747i	$-0.240962\pi$
$653$	1.03442e10	1.45379	0.726896	+	0.686747i	$0.240962\pi$
$654$	4.51050e9	0.630525
$655$	1.73231e9	0.240868
$656$	−4.75934e8	−0.0658239
$657$	−2.84062e9	−0.390782
$658$	2.68742e9	0.367744
$659$	−2.15223e9	−0.292947	−0.146474	−	0.989215i	$-0.546792\pi$
$659$	−2.15223e9	−0.292947	−0.146474	+	0.989215i	$0.546792\pi$
$660$	−2.00877e8	−0.0271974
$661$	−6.05491e9	−0.815459	−0.407730	−	0.913103i	$-0.633679\pi$
$661$	−6.05491e9	−0.815459	−0.407730	+	0.913103i	$0.633679\pi$
$662$	−3.43393e9	−0.460032
$663$	1.50767e10	2.00913
$664$	4.67871e9	0.620209
$665$	2.31460e8	0.0305211
$666$	−1.21617e9	−0.159527
$667$	2.93592e8	0.0383093
$668$	−3.84092e9	−0.498560
$669$	5.14858e9	0.664807
$670$	−4.55789e8	−0.0585466
$671$	2.28789e9	0.292352
$672$	2.53034e8	0.0321653
$673$	1.52328e10	1.92631	0.963155	−	0.268946i	$-0.0866754\pi$
$673$	1.52328e10	1.92631	0.963155	+	0.268946i	$0.0866754\pi$
$674$	−6.65794e9	−0.837588
$675$	1.50379e9	0.188202
$676$	1.19905e10	1.49287
$677$	5.66966e9	0.702257	0.351129	−	0.936327i	$-0.385798\pi$
$677$	5.66966e9	0.702257	0.351129	+	0.936327i	$0.385798\pi$
$678$	−4.16405e9	−0.513111
$679$	2.57578e9	0.315765
$680$	7.50735e8	0.0915599
$681$	−3.82585e9	−0.464208
$682$	−6.93103e8	−0.0836666
$683$	−3.90755e9	−0.469280	−0.234640	−	0.972082i	$-0.575391\pi$
$683$	−3.90755e9	−0.469280	−0.234640	+	0.972082i	$0.575391\pi$
$684$	−9.09256e8	−0.108640
$685$	−1.10423e9	−0.131263
$686$	3.58138e9	0.423561
$687$	−2.84785e9	−0.335095
$688$	−2.14011e9	−0.250539
$689$	−2.72303e9	−0.317165
$690$	1.09136e8	0.0126473
$691$	−6.48163e9	−0.747328	−0.373664	−	0.927564i	$-0.621899\pi$
$691$	−6.48163e9	−0.747328	−0.373664	+	0.927564i	$0.621899\pi$
$692$	1.07654e9	0.123498
$693$	−5.83645e8	−0.0666166
$694$	−4.38961e8	−0.0498503
$695$	2.38719e9	0.269737
$696$	3.33576e8	0.0375027
$697$	−4.10273e9	−0.458942
$698$	7.59722e8	0.0845591
$699$	9.53870e9	1.05638
$700$	1.39843e9	0.154099
$701$	−1.97166e9	−0.216182	−0.108091	−	0.994141i	$-0.534474\pi$
$701$	−1.97166e9	−0.216182	−0.108091	+	0.994141i	$0.534474\pi$
$702$	2.49022e9	0.271680
$703$	4.06400e9	0.441175
$704$	7.33829e8	0.0792667
$705$	1.31697e9	0.141551
$706$	3.73653e9	0.399625
$707$	1.51727e9	0.161472
$708$	8.41218e8	0.0890825
$709$	−6.03784e9	−0.636238	−0.318119	−	0.948051i	$-0.603051\pi$
$709$	−6.03784e9	−0.636238	−0.318119	+	0.948051i	$0.603051\pi$
$710$	−1.83826e9	−0.192754
$711$	1.58120e9	0.164984
$712$	1.87726e9	0.194915
$713$	3.76561e8	0.0389065
$714$	2.18125e9	0.224265
$715$	−1.83841e9	−0.188093
$716$	4.82879e9	0.491635
$717$	8.10533e9	0.821209
$718$	2.92071e9	0.294478
$719$	−1.84605e9	−0.185222	−0.0926108	−	0.995702i	$-0.529521\pi$
$719$	−1.84605e9	−0.185222	−0.0926108	+	0.995702i	$0.529521\pi$
$720$	1.23999e8	0.0123810
$721$	−4.29434e9	−0.426701
$722$	−4.11256e9	−0.406660
$723$	1.55107e9	0.152633
$724$	−3.63193e9	−0.355674
$725$	1.84356e9	0.179669
$726$	2.51659e9	0.244081
$727$	−1.33017e10	−1.28392	−0.641959	−	0.766739i	$-0.721878\pi$
$727$	−1.33017e10	−1.28392	−0.641959	+	0.766739i	$0.721878\pi$
$728$	2.31576e9	0.222450
$729$	3.87420e8	0.0370370
$730$	−1.29452e9	−0.123162
$731$	−1.84485e10	−1.74683
$732$	−1.41229e9	−0.133087
$733$	−1.73074e10	−1.62319	−0.811594	−	0.584222i	$-0.801400\pi$
$733$	−1.73074e10	−1.62319	−0.811594	+	0.584222i	$0.801400\pi$
$734$	1.38155e10	1.28952
$735$	8.31670e8	0.0772583
$736$	−3.98688e8	−0.0368605
$737$	−3.84058e9	−0.353396
$738$	−6.77649e8	−0.0620594
$739$	−2.48391e9	−0.226402	−0.113201	−	0.993572i	$-0.536110\pi$
$739$	−2.48391e9	−0.226402	−0.113201	+	0.993572i	$0.536110\pi$
$740$	−5.54226e8	−0.0502778
$741$	−8.32145e9	−0.751338
$742$	−3.93960e8	−0.0354028
$743$	8.62228e9	0.771189	0.385595	−	0.922668i	$-0.373996\pi$
$743$	8.62228e9	0.771189	0.385595	+	0.922668i	$0.373996\pi$
$744$	4.27845e8	0.0380873
$745$	2.09400e9	0.185537
$746$	−6.48692e9	−0.572075
$747$	6.66168e9	0.584739
$748$	6.32587e9	0.552669
$749$	−9.63839e8	−0.0838143
$750$	1.38607e9	0.119970
$751$	5.32898e9	0.459097	0.229548	−	0.973297i	$-0.426275\pi$
$751$	5.32898e9	0.459097	0.229548	+	0.973297i	$0.426275\pi$
$752$	−4.81105e9	−0.412551
$753$	8.44672e9	0.720951
$754$	3.05286e9	0.259363
$755$	2.16572e8	0.0183142
$756$	3.60278e8	0.0303257
$757$	−1.49684e9	−0.125412	−0.0627060	−	0.998032i	$-0.519973\pi$
$757$	−1.49684e9	−0.125412	−0.0627060	+	0.998032i	$0.519973\pi$
$758$	7.52196e9	0.627319
$759$	9.19607e8	0.0763407
$760$	−4.14362e8	−0.0342399
$761$	−1.83995e10	−1.51342	−0.756712	−	0.653749i	$-0.773195\pi$
$761$	−1.83995e10	−1.51342	−0.756712	+	0.653749i	$0.773195\pi$
$762$	2.66502e9	0.218202
$763$	5.97224e9	0.486744
$764$	1.12883e9	0.0915800
$765$	1.06892e9	0.0863235
$766$	1.51057e10	1.21434
$767$	7.69877e9	0.616081
$768$	−4.52985e8	−0.0360844
$769$	−2.02930e10	−1.60918	−0.804588	−	0.593834i	$-0.797614\pi$
$769$	−2.02930e10	−1.60918	−0.804588	+	0.593834i	$0.797614\pi$
$770$	−2.65976e8	−0.0209954
$771$	−1.05079e10	−0.825703
$772$	−5.98504e9	−0.468173
$773$	−5.58397e9	−0.434826	−0.217413	−	0.976080i	$-0.569762\pi$
$773$	−5.58397e9	−0.434826	−0.217413	+	0.976080i	$0.569762\pi$
$774$	−3.04714e9	−0.236211
$775$	2.36455e9	0.182470
$776$	−4.61118e9	−0.354239
$777$	−1.61029e9	−0.123149
$778$	1.06034e10	0.807268
$779$	2.26447e9	0.171627
$780$	1.13483e9	0.0856250
$781$	−1.54897e10	−1.16349
$782$	−3.43684e9	−0.257001
$783$	4.74955e8	0.0353579
$784$	−3.03820e9	−0.225169
$785$	3.33594e9	0.246135
$786$	−9.01046e9	−0.661863
$787$	1.63322e10	1.19435	0.597177	−	0.802110i	$-0.296289\pi$
$787$	1.63322e10	1.19435	0.597177	+	0.802110i	$0.296289\pi$
$788$	−9.53528e9	−0.694211
$789$	−5.47250e9	−0.396658
$790$	7.20576e8	0.0519978
$791$	−5.51351e9	−0.396105
$792$	1.04485e9	0.0747334
$793$	−1.29252e10	−0.920409
$794$	−1.12173e10	−0.795276
$795$	−1.93059e8	−0.0136272
$796$	2.01088e9	0.141316
$797$	−1.68627e10	−1.17984	−0.589918	−	0.807463i	$-0.700840\pi$
$797$	−1.68627e10	−1.17984	−0.589918	+	0.807463i	$0.700840\pi$
$798$	−1.20392e9	−0.0838664
$799$	−4.14730e10	−2.87641
$800$	−2.50349e9	−0.172875
$801$	2.67290e9	0.183767
$802$	6.42124e9	0.439550
$803$	−1.09079e10	−0.743425
$804$	2.37075e9	0.160875
$805$	1.44504e8	0.00976327
$806$	3.91561e9	0.263406
$807$	−2.19022e9	−0.146700
$808$	−2.71624e9	−0.181146
$809$	2.98182e9	0.197998	0.0989990	−	0.995088i	$-0.468436\pi$
$809$	2.98182e9	0.197998	0.0989990	+	0.995088i	$0.468436\pi$
$810$	1.76554e8	0.0116729
$811$	−6.37441e9	−0.419631	−0.209815	−	0.977741i	$-0.567286\pi$
$811$	−6.37441e9	−0.419631	−0.209815	+	0.977741i	$0.567286\pi$
$812$	4.41679e8	0.0289508
$813$	−1.48552e9	−0.0969528
$814$	−4.67004e9	−0.303484
$815$	4.54519e8	0.0294103
$816$	−3.90489e9	−0.251590
$817$	1.01825e10	0.653246
$818$	−1.05841e10	−0.676109
$819$	3.29724e9	0.209728
$820$	−3.08815e8	−0.0195592
$821$	4.63903e9	0.292568	0.146284	−	0.989243i	$-0.453269\pi$
$821$	4.63903e9	0.292568	0.146284	+	0.989243i	$0.453269\pi$
$822$	5.74356e9	0.360687
$823$	−4.65918e9	−0.291347	−0.145673	−	0.989333i	$-0.546535\pi$
$823$	−4.65918e9	−0.291347	−0.145673	+	0.989333i	$0.546535\pi$
$824$	7.68778e9	0.478691
$825$	5.77451e9	0.358036
$826$	1.11384e9	0.0687687
$827$	−2.91694e10	−1.79332	−0.896659	−	0.442721i	$-0.854013\pi$
$827$	−2.91694e10	−1.79332	−0.896659	+	0.442721i	$0.854013\pi$
$828$	−5.67664e8	−0.0347524
$829$	1.35411e10	0.825491	0.412746	−	0.910846i	$-0.364570\pi$
$829$	1.35411e10	0.825491	0.412746	+	0.910846i	$0.364570\pi$
$830$	3.03583e9	0.184291
$831$	1.04803e10	0.633533
$832$	−4.14569e9	−0.249554
$833$	−2.61903e10	−1.56994
$834$	−1.24168e10	−0.741187
$835$	−2.49222e9	−0.148144
$836$	−3.49151e9	−0.206677
$837$	6.09177e8	0.0359091
$838$	1.61955e9	0.0950691
$839$	−1.13991e10	−0.666354	−0.333177	−	0.942864i	$-0.608121\pi$
$839$	−1.13991e10	−0.666354	−0.333177	+	0.942864i	$0.608121\pi$
$840$	1.64184e8	0.00955771
$841$	−1.66676e10	−0.966245
$842$	3.40985e9	0.196854
$843$	9.37470e9	0.538966
$844$	1.41206e10	0.808453
$845$	7.78015e9	0.443598
$846$	−6.85011e9	−0.388956
$847$	3.33215e9	0.188423
$848$	7.05270e8	0.0397164
$849$	1.87946e10	1.05404
$850$	−2.15810e10	−1.20533
$851$	2.53723e9	0.141126
$852$	9.56159e9	0.529653
$853$	−6.63132e9	−0.365829	−0.182915	−	0.983129i	$-0.558553\pi$
$853$	−6.63132e9	−0.365829	−0.182915	+	0.983129i	$0.558553\pi$
$854$	−1.86998e9	−0.102739
$855$	−5.89980e8	−0.0322817
$856$	1.72547e9	0.0940265
$857$	−1.37475e10	−0.746092	−0.373046	−	0.927813i	$-0.621687\pi$
$857$	−1.37475e10	−0.746092	−0.373046	+	0.927813i	$0.621687\pi$
$858$	9.56237e9	0.516845
$859$	3.46757e10	1.86659	0.933295	−	0.359111i	$-0.116920\pi$
$859$	3.46757e10	1.86659	0.933295	+	0.359111i	$0.116920\pi$
$860$	−1.38863e9	−0.0744462
$861$	−8.97257e8	−0.0479078
$862$	−1.24850e10	−0.663918
$863$	−8.11718e9	−0.429900	−0.214950	−	0.976625i	$-0.568959\pi$
$863$	−8.11718e9	−0.429900	−0.214950	+	0.976625i	$0.568959\pi$
$864$	−6.44973e8	−0.0340207
$865$	6.98527e8	0.0366967
$866$	1.47495e10	0.771727
$867$	−2.25824e10	−1.17680
$868$	5.66498e8	0.0294022
$869$	6.07174e9	0.313866
$870$	2.16444e8	0.0111437
$871$	2.16970e10	1.11259
$872$	−1.06916e10	−0.546051
$873$	−6.56553e9	−0.333979
$874$	1.89693e9	0.0961085
$875$	1.83526e9	0.0926125
$876$	6.73333e9	0.338427
$877$	2.84041e10	1.42194	0.710972	−	0.703220i	$-0.248255\pi$
$877$	2.84041e10	1.42194	0.710972	+	0.703220i	$0.248255\pi$
$878$	5.43164e9	0.270832
$879$	1.29267e10	0.641986
$880$	4.76153e8	0.0235536
$881$	−6.66399e9	−0.328336	−0.164168	−	0.986432i	$-0.552494\pi$
$881$	−6.66399e9	−0.328336	−0.164168	+	0.986432i	$0.552494\pi$
$882$	−4.32587e9	−0.212292
$883$	−4.54101e9	−0.221968	−0.110984	−	0.993822i	$-0.535400\pi$
$883$	−4.54101e9	−0.221968	−0.110984	+	0.993822i	$0.535400\pi$
$884$	−3.57373e10	−1.73996
$885$	5.45833e8	0.0264703
$886$	2.90170e9	0.140163
$887$	−2.10722e10	−1.01386	−0.506928	−	0.861988i	$-0.669219\pi$
$887$	−2.10722e10	−1.01386	−0.506928	+	0.861988i	$0.669219\pi$
$888$	2.88277e9	0.138154
$889$	3.52869e9	0.168445
$890$	1.21808e9	0.0579177
$891$	1.48768e9	0.0704593
$892$	−1.22040e10	−0.575740
$893$	2.28907e10	1.07567
$894$	−1.08918e10	−0.509821
$895$	3.13321e9	0.146086
$896$	−5.99785e8	−0.0278559
$897$	−5.19522e9	−0.240342
$898$	1.75439e10	0.808460
$899$	7.46815e8	0.0342811
$900$	−3.56454e9	−0.162988
$901$	6.07968e9	0.276914
$902$	−2.60215e9	−0.118062
$903$	−4.03464e9	−0.182347
$904$	9.87033e9	0.444367
$905$	−2.35661e9	−0.105686
$906$	−1.12648e9	−0.0503240
$907$	−3.05980e10	−1.36166	−0.680828	−	0.732444i	$-0.738380\pi$
$907$	−3.05980e10	−1.36166	−0.680828	+	0.732444i	$0.738380\pi$
$908$	9.06868e9	0.402016
$909$	−3.86746e9	−0.170786
$910$	1.50260e9	0.0660997
$911$	4.12373e10	1.80707	0.903537	−	0.428509i	$-0.140961\pi$
$911$	4.12373e10	1.80707	0.903537	+	0.428509i	$0.140961\pi$
$912$	2.15527e9	0.0940850
$913$	2.55807e10	1.11241
$914$	−2.95022e10	−1.27803
$915$	−9.16380e8	−0.0395459
$916$	6.75045e9	0.290201
$917$	−1.19305e10	−0.510936
$918$	−5.55990e9	−0.237201
$919$	−2.66763e10	−1.13376	−0.566881	−	0.823800i	$-0.691850\pi$
$919$	−2.66763e10	−1.13376	−0.566881	+	0.823800i	$0.691850\pi$
$920$	−2.58693e8	−0.0109529
$921$	−1.17664e9	−0.0496289
$922$	9.39422e9	0.394732
$923$	8.75071e10	3.66300
$924$	1.38345e9	0.0576917
$925$	1.59321e10	0.661875
$926$	2.27145e10	0.940079
$927$	1.09461e10	0.451314
$928$	−7.90698e8	−0.0324783
$929$	3.72780e10	1.52545	0.762724	−	0.646724i	$-0.223862\pi$
$929$	3.72780e10	1.52545	0.762724	+	0.646724i	$0.223862\pi$
$930$	2.77612e8	0.0113174
$931$	1.44555e10	0.587098
$932$	−2.26103e10	−0.914850
$933$	3.98546e9	0.160654
$934$	1.91237e10	0.767993
$935$	4.10461e9	0.164222
$936$	−5.90275e9	−0.235282
$937$	1.23246e10	0.489422	0.244711	−	0.969596i	$-0.421307\pi$
$937$	1.23246e10	0.489422	0.244711	+	0.969596i	$0.421307\pi$
$938$	3.13905e9	0.124190
$939$	−5.77105e9	−0.227471
$940$	−3.12170e9	−0.122587
$941$	4.83093e10	1.89002	0.945011	−	0.327040i	$-0.106051\pi$
$941$	4.83093e10	1.89002	0.945011	+	0.327040i	$0.106051\pi$
$942$	−1.73516e10	−0.676335
$943$	1.41374e9	0.0549009
$944$	−1.99400e9	−0.0771477
$945$	2.33770e8	0.00901109
$946$	−1.17009e10	−0.449368
$947$	2.68092e10	1.02579	0.512896	−	0.858451i	$-0.328573\pi$
$947$	2.68092e10	1.02579	0.512896	+	0.858451i	$0.328573\pi$
$948$	−3.74802e9	−0.142880
$949$	6.16230e10	2.34051
$950$	1.19115e10	0.450746
$951$	9.67773e9	0.364873
$952$	−5.17037e9	−0.194219
$953$	2.13166e10	0.797799	0.398900	−	0.916995i	$-0.369392\pi$
$953$	2.13166e10	0.797799	0.398900	+	0.916995i	$0.369392\pi$
$954$	1.00418e9	0.0374450
$955$	7.32451e8	0.0272124
$956$	−1.92126e10	−0.711188
$957$	1.82381e9	0.0672648
$958$	2.14199e10	0.787115
$959$	7.60490e9	0.278438
$960$	−2.93924e8	−0.0107222
$961$	−2.65547e10	−0.965184
$962$	2.63829e10	0.955454
$963$	2.45678e9	0.0886491
$964$	−3.67661e9	−0.132184
$965$	−3.88345e9	−0.139115
$966$	−7.51629e8	−0.0268277
$967$	−2.42096e10	−0.860985	−0.430493	−	0.902594i	$-0.641660\pi$
$967$	−2.42096e10	−0.860985	−0.430493	+	0.902594i	$0.641660\pi$
$968$	−5.96525e9	−0.211381
$969$	1.85792e10	0.655986
$970$	−2.99201e9	−0.105260
$971$	−1.79110e10	−0.627845	−0.313923	−	0.949449i	$-0.601643\pi$
$971$	−1.79110e10	−0.627845	−0.313923	+	0.949449i	$0.601643\pi$
$972$	−9.18330e8	−0.0320750
$973$	−1.64407e10	−0.572172
$974$	−2.32449e10	−0.806067
$975$	−3.26225e10	−1.12720
$976$	3.34765e9	0.115257
$977$	−1.92033e10	−0.658786	−0.329393	−	0.944193i	$-0.606844\pi$
$977$	−1.92033e10	−0.658786	−0.329393	+	0.944193i	$0.606844\pi$
$978$	−2.36414e9	−0.0808142
$979$	1.02638e10	0.349599
$980$	−1.97137e9	−0.0669077
$981$	−1.52229e10	−0.514822
$982$	3.16682e10	1.06717
$983$	2.40161e9	0.0806426	0.0403213	−	0.999187i	$-0.487162\pi$
$983$	2.40161e9	0.0806426	0.0403213	+	0.999187i	$0.487162\pi$
$984$	1.60628e9	0.0537450
$985$	−6.18707e9	−0.206280
$986$	−6.81610e9	−0.226447
$987$	−9.07005e9	−0.300261
$988$	1.97249e10	0.650678
$989$	6.35710e9	0.208964
$990$	6.77960e8	0.0222065
$991$	−1.38836e10	−0.453153	−0.226576	−	0.973993i	$-0.572753\pi$
$991$	−1.38836e10	−0.453153	−0.226576	+	0.973993i	$0.572753\pi$
$992$	−1.01415e9	−0.0329846
$993$	1.15895e10	0.375615
$994$	1.26603e10	0.408875
$995$	1.30478e9	0.0419911
$996$	−1.57907e10	−0.506399
$997$	1.45305e10	0.464354	0.232177	−	0.972674i	$-0.425415\pi$
$997$	1.45305e10	0.464354	0.232177	+	0.972674i	$0.425415\pi$
$998$	2.57100e9	0.0818739
$999$	4.10456e9	0.130253

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.8.a.d.1.2	✓	3
3.2	odd	2		414.8.a.c.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.8.a.d.1.2	✓	3	1.1	even	1	trivial
414.8.a.c.1.2		3	3.2	odd	2

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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +41.5271 q^{5} -216.000 q^{6} -286.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} +41.5271 q^{5} -216.000 q^{6} -286.000 q^{7} +512.000 q^{8} +729.000 q^{9} +332.217 q^{10} +2799.34 q^{11} -1728.00 q^{12} -15814.5 q^{13} -2288.00 q^{14} -1121.23 q^{15} +4096.00 q^{16} +35309.0 q^{17} +5832.00 q^{18} -19488.5 q^{19} +2657.73 q^{20} +7722.00 q^{21} +22394.7 q^{22} -12167.0 q^{23} -13824.0 q^{24} -76400.5 q^{25} -126516. q^{26} -19683.0 q^{27} -18304.0 q^{28} -24130.2 q^{29} -8969.85 q^{30} -30949.4 q^{31} +32768.0 q^{32} -75582.1 q^{33} +282472. q^{34} -11876.7 q^{35} +46656.0 q^{36} -208533. q^{37} -155908. q^{38} +426993. q^{39} +21261.9 q^{40} -116195. q^{41} +61776.0 q^{42} -522487. q^{43} +179158. q^{44} +30273.2 q^{45} -97336.0 q^{46} -1.17457e6 q^{47} -110592. q^{48} -741747. q^{49} -611204. q^{50} -953343. q^{51} -1.01213e6 q^{52} +172185. q^{53} -157464. q^{54} +116248. q^{55} -146432. q^{56} +526190. q^{57} -193042. q^{58} -486816. q^{59} -71758.8 q^{60} +817298. q^{61} -247595. q^{62} -208494. q^{63} +262144. q^{64} -656732. q^{65} -604657. q^{66} -1.37196e6 q^{67} +2.25978e6 q^{68} +328509. q^{69} -95014.0 q^{70} -5.53333e6 q^{71} +373248. q^{72} -3.89660e6 q^{73} -1.66827e6 q^{74} +2.06281e6 q^{75} -1.24726e6 q^{76} -800610. q^{77} +3.41594e6 q^{78} +2.16899e6 q^{79} +170095. q^{80} +531441. q^{81} -929559. q^{82} +9.13811e6 q^{83} +494208. q^{84} +1.46628e6 q^{85} -4.17990e6 q^{86} +651515. q^{87} +1.43326e6 q^{88} +3.66653e6 q^{89} +242186. q^{90} +4.52296e6 q^{91} -778688. q^{92} +835634. q^{93} -9.39658e6 q^{94} -809301. q^{95} -884736. q^{96} -9.00621e6 q^{97} -5.93398e6 q^{98} +2.04072e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 24 q^{2} - 81 q^{3} + 192 q^{4} + 92 q^{5} - 648 q^{6} - 858 q^{7} + 1536 q^{8} + 2187 q^{9}+O(q^{10}) \) 3 * q + 24 * q^2 - 81 * q^3 + 192 * q^4 + 92 * q^5 - 648 * q^6 - 858 * q^7 + 1536 * q^8 + 2187 * q^9	\( 3 q + 24 q^{2} - 81 q^{3} + 192 q^{4} + 92 q^{5} - 648 q^{6} - 858 q^{7} + 1536 q^{8} + 2187 q^{9} + 736 q^{10} - 1820 q^{11} - 5184 q^{12} - 3698 q^{13} - 6864 q^{14} - 2484 q^{15} + 12288 q^{16} + 3164 q^{17} + 17496 q^{18} - 14234 q^{19} + 5888 q^{20} + 23166 q^{21} - 14560 q^{22} - 36501 q^{23} - 41472 q^{24} - 170119 q^{25} - 29584 q^{26} - 59049 q^{27} - 54912 q^{28} - 171914 q^{29} - 19872 q^{30} - 124972 q^{31} + 98304 q^{32} + 49140 q^{33} + 25312 q^{34} - 26312 q^{35} + 139968 q^{36} - 679074 q^{37} - 113872 q^{38} + 99846 q^{39} + 47104 q^{40} - 331362 q^{41} + 185328 q^{42} - 145922 q^{43} - 116480 q^{44} + 67068 q^{45} - 292008 q^{46} - 1824192 q^{47} - 331776 q^{48} - 2225241 q^{49} - 1360952 q^{50} - 85428 q^{51} - 236672 q^{52} - 3442448 q^{53} - 472392 q^{54} - 1548176 q^{55} - 439296 q^{56} + 384318 q^{57} - 1375312 q^{58} - 3147660 q^{59} - 158976 q^{60} - 3444530 q^{61} - 999776 q^{62} - 625482 q^{63} + 786432 q^{64} - 2527960 q^{65} + 393120 q^{66} - 921870 q^{67} + 202496 q^{68} + 985527 q^{69} - 210496 q^{70} + 2594520 q^{71} + 1119744 q^{72} - 540826 q^{73} - 5432592 q^{74} + 4593213 q^{75} - 910976 q^{76} + 520520 q^{77} + 798768 q^{78} + 316994 q^{79} + 376832 q^{80} + 1594323 q^{81} - 2650896 q^{82} + 4119316 q^{83} + 1482624 q^{84} - 382312 q^{85} - 1167376 q^{86} + 4641678 q^{87} - 931840 q^{88} + 1417176 q^{89} + 536544 q^{90} + 1057628 q^{91} - 2336064 q^{92} + 3374244 q^{93} - 14593536 q^{94} + 12543632 q^{95} - 2654208 q^{96} - 15089326 q^{97} - 17801928 q^{98} - 1326780 q^{99}+O(q^{100}) \) 3 * q + 24 * q^2 - 81 * q^3 + 192 * q^4 + 92 * q^5 - 648 * q^6 - 858 * q^7 + 1536 * q^8 + 2187 * q^9 + 736 * q^10 - 1820 * q^11 - 5184 * q^12 - 3698 * q^13 - 6864 * q^14 - 2484 * q^15 + 12288 * q^16 + 3164 * q^17 + 17496 * q^18 - 14234 * q^19 + 5888 * q^20 + 23166 * q^21 - 14560 * q^22 - 36501 * q^23 - 41472 * q^24 - 170119 * q^25 - 29584 * q^26 - 59049 * q^27 - 54912 * q^28 - 171914 * q^29 - 19872 * q^30 - 124972 * q^31 + 98304 * q^32 + 49140 * q^33 + 25312 * q^34 - 26312 * q^35 + 139968 * q^36 - 679074 * q^37 - 113872 * q^38 + 99846 * q^39 + 47104 * q^40 - 331362 * q^41 + 185328 * q^42 - 145922 * q^43 - 116480 * q^44 + 67068 * q^45 - 292008 * q^46 - 1824192 * q^47 - 331776 * q^48 - 2225241 * q^49 - 1360952 * q^50 - 85428 * q^51 - 236672 * q^52 - 3442448 * q^53 - 472392 * q^54 - 1548176 * q^55 - 439296 * q^56 + 384318 * q^57 - 1375312 * q^58 - 3147660 * q^59 - 158976 * q^60 - 3444530 * q^61 - 999776 * q^62 - 625482 * q^63 + 786432 * q^64 - 2527960 * q^65 + 393120 * q^66 - 921870 * q^67 + 202496 * q^68 + 985527 * q^69 - 210496 * q^70 + 2594520 * q^71 + 1119744 * q^72 - 540826 * q^73 - 5432592 * q^74 + 4593213 * q^75 - 910976 * q^76 + 520520 * q^77 + 798768 * q^78 + 316994 * q^79 + 376832 * q^80 + 1594323 * q^81 - 2650896 * q^82 + 4119316 * q^83 + 1482624 * q^84 - 382312 * q^85 - 1167376 * q^86 + 4641678 * q^87 - 931840 * q^88 + 1417176 * q^89 + 536544 * q^90 + 1057628 * q^91 - 2336064 * q^92 + 3374244 * q^93 - 14593536 * q^94 + 12543632 * q^95 - 2654208 * q^96 - 15089326 * q^97 - 17801928 * q^98 - 1326780 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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