LMFDB - Embedded newform 69.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,6,Mod(1,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("69.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	69.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:	$11.0664835671$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{29}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 7 $ x^2 - x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.19258$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.7703 q^{2} -9.00000 q^{3} +84.0000 q^{4} +41.6148 q^{5} +96.9330 q^{6} +0.236813 q^{7} -560.057 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-10.7703 q^{2} -9.00000 q^{3} +84.0000 q^{4} +41.6148 q^{5} +96.9330 q^{6} +0.236813 q^{7} -560.057 q^{8} +81.0000 q^{9} -448.205 q^{10} -421.598 q^{11} -756.000 q^{12} +254.502 q^{13} -2.55055 q^{14} -374.534 q^{15} +3344.00 q^{16} -975.007 q^{17} -872.397 q^{18} +2039.79 q^{19} +3495.65 q^{20} -2.13132 q^{21} +4540.75 q^{22} +529.000 q^{23} +5040.51 q^{24} -1393.21 q^{25} -2741.07 q^{26} -729.000 q^{27} +19.8923 q^{28} +2671.55 q^{29} +4033.85 q^{30} +9039.14 q^{31} -18094.2 q^{32} +3794.38 q^{33} +10501.1 q^{34} +9.85493 q^{35} +6804.00 q^{36} -12665.3 q^{37} -21969.2 q^{38} -2290.52 q^{39} -23306.7 q^{40} +10146.4 q^{41} +22.9550 q^{42} +19523.2 q^{43} -35414.2 q^{44} +3370.80 q^{45} -5697.50 q^{46} +27679.7 q^{47} -30096.0 q^{48} -16806.9 q^{49} +15005.3 q^{50} +8775.06 q^{51} +21378.2 q^{52} +10852.8 q^{53} +7851.57 q^{54} -17544.7 q^{55} -132.629 q^{56} -18358.1 q^{57} -28773.4 q^{58} +11907.7 q^{59} -31460.8 q^{60} +39861.9 q^{61} -97354.5 q^{62} +19.1818 q^{63} +87872.0 q^{64} +10591.1 q^{65} -40866.7 q^{66} -28550.5 q^{67} -81900.6 q^{68} -4761.00 q^{69} -106.141 q^{70} +52179.8 q^{71} -45364.6 q^{72} +56918.0 q^{73} +136410. q^{74} +12538.8 q^{75} +171342. q^{76} -99.8398 q^{77} +24669.7 q^{78} +23178.3 q^{79} +139160. q^{80} +6561.00 q^{81} -109281. q^{82} -18344.6 q^{83} -179.031 q^{84} -40574.8 q^{85} -210272. q^{86} -24043.9 q^{87} +236119. q^{88} +47362.4 q^{89} -36304.6 q^{90} +60.2694 q^{91} +44436.0 q^{92} -81352.3 q^{93} -298120. q^{94} +84885.4 q^{95} +162847. q^{96} -140379. q^{97} +181016. q^{98} -34149.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} + 168 q^{4} + 94 q^{5} - 118 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q - 18 * q^3 + 168 * q^4 + 94 * q^5 - 118 * q^7 + 162 * q^9	\( 2 q - 18 q^{3} + 168 q^{4} + 94 q^{5} - 118 q^{7} + 162 q^{9} + 116 q^{10} + 320 q^{11} - 1512 q^{12} - 288 q^{13} - 1276 q^{14} - 846 q^{15} + 6688 q^{16} - 1810 q^{17} + 730 q^{19} + 7896 q^{20} + 1062 q^{21} + 12528 q^{22} + 1058 q^{23} - 1774 q^{25} - 8584 q^{26} - 1458 q^{27} - 9912 q^{28} + 8208 q^{29} - 1044 q^{30} + 1772 q^{31} - 2880 q^{33} + 1508 q^{34} - 6184 q^{35} + 13608 q^{36} - 23112 q^{37} - 36076 q^{38} + 2592 q^{39} + 6032 q^{40} + 5516 q^{41} + 11484 q^{42} + 10322 q^{43} + 26880 q^{44} + 7614 q^{45} + 42952 q^{47} - 60192 q^{48} - 19634 q^{49} + 10904 q^{50} + 16290 q^{51} - 24192 q^{52} - 25350 q^{53} + 21304 q^{55} - 66352 q^{56} - 6570 q^{57} + 30856 q^{58} + 18344 q^{59} - 71064 q^{60} + 37224 q^{61} - 175624 q^{62} - 9558 q^{63} + 175744 q^{64} - 17828 q^{65} - 112752 q^{66} - 7482 q^{67} - 152040 q^{68} - 9522 q^{69} - 66816 q^{70} + 126848 q^{71} + 137660 q^{73} + 23896 q^{74} + 15966 q^{75} + 61320 q^{76} - 87784 q^{77} + 77256 q^{78} + 62286 q^{79} + 314336 q^{80} + 13122 q^{81} - 159152 q^{82} + 83120 q^{83} + 89208 q^{84} - 84316 q^{85} - 309372 q^{86} - 73872 q^{87} + 651456 q^{88} + 69770 q^{89} + 9396 q^{90} + 64204 q^{91} + 88872 q^{92} - 15948 q^{93} - 133632 q^{94} + 16272 q^{95} - 170104 q^{97} + 150568 q^{98} + 25920 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 + 168 * q^4 + 94 * q^5 - 118 * q^7 + 162 * q^9 + 116 * q^10 + 320 * q^11 - 1512 * q^12 - 288 * q^13 - 1276 * q^14 - 846 * q^15 + 6688 * q^16 - 1810 * q^17 + 730 * q^19 + 7896 * q^20 + 1062 * q^21 + 12528 * q^22 + 1058 * q^23 - 1774 * q^25 - 8584 * q^26 - 1458 * q^27 - 9912 * q^28 + 8208 * q^29 - 1044 * q^30 + 1772 * q^31 - 2880 * q^33 + 1508 * q^34 - 6184 * q^35 + 13608 * q^36 - 23112 * q^37 - 36076 * q^38 + 2592 * q^39 + 6032 * q^40 + 5516 * q^41 + 11484 * q^42 + 10322 * q^43 + 26880 * q^44 + 7614 * q^45 + 42952 * q^47 - 60192 * q^48 - 19634 * q^49 + 10904 * q^50 + 16290 * q^51 - 24192 * q^52 - 25350 * q^53 + 21304 * q^55 - 66352 * q^56 - 6570 * q^57 + 30856 * q^58 + 18344 * q^59 - 71064 * q^60 + 37224 * q^61 - 175624 * q^62 - 9558 * q^63 + 175744 * q^64 - 17828 * q^65 - 112752 * q^66 - 7482 * q^67 - 152040 * q^68 - 9522 * q^69 - 66816 * q^70 + 126848 * q^71 + 137660 * q^73 + 23896 * q^74 + 15966 * q^75 + 61320 * q^76 - 87784 * q^77 + 77256 * q^78 + 62286 * q^79 + 314336 * q^80 + 13122 * q^81 - 159152 * q^82 + 83120 * q^83 + 89208 * q^84 - 84316 * q^85 - 309372 * q^86 - 73872 * q^87 + 651456 * q^88 + 69770 * q^89 + 9396 * q^90 + 64204 * q^91 + 88872 * q^92 - 15948 * q^93 - 133632 * q^94 + 16272 * q^95 - 170104 * q^97 + 150568 * q^98 + 25920 * 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 $

$2$	−10.7703	−1.90394	−0.951972	−	0.306186i	$-0.900947\pi$
$2$	−10.7703	−1.90394	−0.951972	+	0.306186i	$0.900947\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	84.0000	2.62500
$5$	41.6148	0.744429	0.372214	−	0.928147i	$-0.378599\pi$
$5$	41.6148	0.744429	0.372214	+	0.928147i	$0.378599\pi$
$6$	96.9330	1.09924
$7$	0.236813	0.00182667	0.000913335	−	1.00000i	$-0.499709\pi$
$7$	0.236813	0.00182667	0.000913335		1.00000i	$0.499709\pi$
$8$	−560.057	−3.09391
$9$	81.0000	0.333333
$10$	−448.205	−1.41735
$11$	−421.598	−1.05055	−0.525275	−	0.850933i	$-0.676037\pi$
$11$	−421.598	−1.05055	−0.525275	+	0.850933i	$0.676037\pi$
$12$	−756.000	−1.51554
$13$	254.502	0.417670	0.208835	−	0.977951i	$-0.433033\pi$
$13$	254.502	0.417670	0.208835	+	0.977951i	$0.433033\pi$
$14$	−2.55055	−0.00347788
$15$	−374.534	−0.429796
$16$	3344.00	3.26562
$17$	−975.007	−0.818249	−0.409125	−	0.912479i	$-0.634166\pi$
$17$	−975.007	−0.818249	−0.409125	+	0.912479i	$0.634166\pi$
$18$	−872.397	−0.634648
$19$	2039.79	1.29629	0.648143	−	0.761519i	$-0.275546\pi$
$19$	2039.79	1.29629	0.648143	+	0.761519i	$0.275546\pi$
$20$	3495.65	1.95413
$21$	−2.13132	−0.00105463
$22$	4540.75	2.00019
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	5040.51	1.78627
$25$	−1393.21	−0.445826
$26$	−2741.07	−0.795220
$27$	−729.000	−0.192450
$28$	19.8923	0.00479501
$29$	2671.55	0.589885	0.294943	−	0.955515i	$-0.404699\pi$
$29$	2671.55	0.589885	0.294943	+	0.955515i	$0.404699\pi$
$30$	4033.85	0.818308
$31$	9039.14	1.68936	0.844681	−	0.535270i	$-0.179790\pi$
$31$	9039.14	1.68936	0.844681	+	0.535270i	$0.179790\pi$
$32$	−18094.2	−3.12366
$33$	3794.38	0.606535
$34$	10501.1	1.55790
$35$	9.85493	0.00135983
$36$	6804.00	0.875000
$37$	−12665.3	−1.52094	−0.760471	−	0.649372i	$-0.775032\pi$
$37$	−12665.3	−1.52094	−0.760471	+	0.649372i	$0.775032\pi$
$38$	−21969.2	−2.46805
$39$	−2290.52	−0.241142
$40$	−23306.7	−2.30319
$41$	10146.4	0.942658	0.471329	−	0.881957i	$-0.343774\pi$
$41$	10146.4	0.942658	0.471329	+	0.881957i	$0.343774\pi$
$42$	22.9550	0.00200795
$43$	19523.2	1.61020	0.805102	−	0.593137i	$-0.202111\pi$
$43$	19523.2	1.61020	0.805102	+	0.593137i	$0.202111\pi$
$44$	−35414.2	−2.75769
$45$	3370.80	0.248143
$46$	−5697.50	−0.397000
$47$	27679.7	1.82775	0.913875	−	0.405995i	$-0.133075\pi$
$47$	27679.7	1.82775	0.913875	+	0.405995i	$0.133075\pi$
$48$	−30096.0	−1.88541
$49$	−16806.9	−0.999997
$50$	15005.3	0.848827
$51$	8775.06	0.472416
$52$	21378.2	1.09638
$53$	10852.8	0.530703	0.265351	−	0.964152i	$-0.414512\pi$
$53$	10852.8	0.530703	0.265351	+	0.964152i	$0.414512\pi$
$54$	7851.57	0.366414
$55$	−17544.7	−0.782059
$56$	−132.629	−0.00565155
$57$	−18358.1	−0.748411
$58$	−28773.4	−1.12311
$59$	11907.7	0.445345	0.222672	−	0.974893i	$-0.428522\pi$
$59$	11907.7	0.445345	0.222672	+	0.974893i	$0.428522\pi$
$60$	−31460.8	−1.12821
$61$	39861.9	1.37162	0.685809	−	0.727782i	$-0.259449\pi$
$61$	39861.9	1.37162	0.685809	+	0.727782i	$0.259449\pi$
$62$	−97354.5	−3.21645
$63$	19.1818	0.000608890 0
$64$	87872.0	2.68164
$65$	10591.1	0.310925
$66$	−40866.7	−1.15481
$67$	−28550.5	−0.777009	−0.388504	−	0.921447i	$-0.627008\pi$
$67$	−28550.5	−0.777009	−0.388504	+	0.921447i	$0.627008\pi$
$68$	−81900.6	−2.14790
$69$	−4761.00	−0.120386
$70$	−106.141	−0.00258903
$71$	52179.8	1.22845	0.614223	−	0.789132i	$-0.289469\pi$
$71$	52179.8	1.22845	0.614223	+	0.789132i	$0.289469\pi$
$72$	−45364.6	−1.03130
$73$	56918.0	1.25009	0.625047	−	0.780587i	$-0.285080\pi$
$73$	56918.0	1.25009	0.625047	+	0.780587i	$0.285080\pi$
$74$	136410.	2.89579
$75$	12538.8	0.257398
$76$	171342.	3.40275
$77$	−99.8398	−0.00191901
$78$	24669.7	0.459120
$79$	23178.3	0.417844	0.208922	−	0.977932i	$-0.433004\pi$
$79$	23178.3	0.417844	0.208922	+	0.977932i	$0.433004\pi$
$80$	139160.	2.43103
$81$	6561.00	0.111111
$82$	−109281.	−1.79477
$83$	−18344.6	−0.292289	−0.146144	−	0.989263i	$-0.546686\pi$
$83$	−18344.6	−0.292289	−0.146144	+	0.989263i	$0.546686\pi$
$84$	−179.031	−0.00276840
$85$	−40574.8	−0.609128
$86$	−210272.	−3.06574
$87$	−24043.9	−0.340571
$88$	236119.	3.25030
$89$	47362.4	0.633810	0.316905	−	0.948457i	$-0.397356\pi$
$89$	47362.4	0.633810	0.316905	+	0.948457i	$0.397356\pi$
$90$	−36304.6	−0.472450
$91$	60.2694	0.000762945 0
$92$	44436.0	0.547350
$93$	−81352.3	−0.975354
$94$	−298120.	−3.47993
$95$	84885.4	0.964992
$96$	162847.	1.80344
$97$	−140379.	−1.51486	−0.757432	−	0.652915i	$-0.773546\pi$
$97$	−140379.	−1.51486	−0.757432	+	0.652915i	$0.773546\pi$
$98$	181016.	1.90394
$99$	−34149.4	−0.350183
$100$	−117029.	−1.17029
$101$	87080.3	0.849408	0.424704	−	0.905332i	$-0.360378\pi$
$101$	87080.3	0.849408	0.424704	+	0.905332i	$0.360378\pi$
$102$	−94510.3	−0.899454
$103$	17988.9	0.167075	0.0835375	−	0.996505i	$-0.473378\pi$
$103$	17988.9	0.167075	0.0835375	+	0.996505i	$0.473378\pi$
$104$	−142536.	−1.29223
$105$	−88.6944	−0.000785096 0
$106$	−116888.	−1.01043
$107$	−127485.	−1.07646	−0.538231	−	0.842797i	$-0.680907\pi$
$107$	−127485.	−1.07646	−0.538231	+	0.842797i	$0.680907\pi$
$108$	−61236.0	−0.505181
$109$	−136418.	−1.09978	−0.549890	−	0.835237i	$-0.685330\pi$
$109$	−136418.	−1.09978	−0.549890	+	0.835237i	$0.685330\pi$
$110$	188962.	1.48900
$111$	113988.	0.878116
$112$	791.902	0.00596522
$113$	−104012.	−0.766279	−0.383139	−	0.923691i	$-0.625157\pi$
$113$	−104012.	−0.766279	−0.383139	+	0.923691i	$0.625157\pi$
$114$	197723.	1.42493
$115$	22014.2	0.155224
$116$	224410.	1.54845
$117$	20614.7	0.139223
$118$	−128249.	−0.847912
$119$	−230.894	−0.00149467
$120$	209760.	1.32975
$121$	16693.7	0.103655
$122$	−429325.	−2.61148
$123$	−91318.0	−0.544244
$124$	759288.	4.43458
$125$	−188024.	−1.07631
$126$	−206.595	−0.00115929
$127$	203763.	1.12103	0.560513	−	0.828145i	$-0.310604\pi$
$127$	203763.	1.12103	0.560513	+	0.828145i	$0.310604\pi$
$128$	−367397.	−1.98203
$129$	−175709.	−0.929651
$130$	−114069.	−0.591984
$131$	60674.1	0.308905	0.154453	−	0.988000i	$-0.450639\pi$
$131$	60674.1	0.308905	0.154453	+	0.988000i	$0.450639\pi$
$132$	318728.	1.59215
$133$	483.048	0.00236789
$134$	307498.	1.47938
$135$	−30337.2	−0.143265
$136$	546060.	2.53159
$137$	94446.5	0.429917	0.214958	−	0.976623i	$-0.431038\pi$
$137$	94446.5	0.429917	0.214958	+	0.976623i	$0.431038\pi$
$138$	51277.5	0.229208
$139$	403508.	1.77139	0.885697	−	0.464263i	$-0.153681\pi$
$139$	403508.	1.77139	0.885697	+	0.464263i	$0.153681\pi$
$140$	827.814	0.00356954
$141$	−249117.	−1.05525
$142$	−561993.	−2.33889
$143$	−107298.	−0.438783
$144$	270864.	1.08854
$145$	111176.	0.439128
$146$	−613026.	−2.38011
$147$	151262.	0.577348
$148$	−1.06389e6	−3.99247
$149$	384528.	1.41893	0.709467	−	0.704738i	$-0.248936\pi$
$149$	384528.	1.41893	0.709467	+	0.704738i	$0.248936\pi$
$150$	−135048.	−0.490070
$151$	−442508.	−1.57935	−0.789675	−	0.613525i	$-0.789751\pi$
$151$	−442508.	−1.57935	−0.789675	+	0.613525i	$0.789751\pi$
$152$	−1.14240e6	−4.01059
$153$	−78975.6	−0.272750
$154$	1075.31	0.00365368
$155$	376162.	1.25761
$156$	−192404.	−0.632997
$157$	240724.	0.779416	0.389708	−	0.920938i	$-0.372576\pi$
$157$	240724.	0.779416	0.389708	+	0.920938i	$0.372576\pi$
$158$	−249638.	−0.795552
$159$	−97675.1	−0.306402
$160$	−752985.	−2.32534
$161$	125.274	0.000380887 0
$162$	−70664.1	−0.211549
$163$	−178188.	−0.525301	−0.262651	−	0.964891i	$-0.584597\pi$
$163$	−178188.	−0.525301	−0.262651	+	0.964891i	$0.584597\pi$
$164$	852301.	2.47448
$165$	157903.	0.451522
$166$	197577.	0.556502
$167$	25525.2	0.0708236	0.0354118	−	0.999373i	$-0.488726\pi$
$167$	25525.2	0.0708236	0.0354118	+	0.999373i	$0.488726\pi$
$168$	1193.66	0.00326292
$169$	−306522.	−0.825552
$170$	437004.	1.15975
$171$	165223.	0.432095
$172$	1.63995e6	4.22678
$173$	701865.	1.78295	0.891474	−	0.453072i	$-0.149672\pi$
$173$	701865.	1.78295	0.891474	+	0.453072i	$0.149672\pi$
$174$	258961.	0.648427
$175$	−329.929	−0.000814377 0
$176$	−1.40982e6	−3.43070
$177$	−107169.	−0.257120
$178$	−510109.	−1.20674
$179$	−292917.	−0.683302	−0.341651	−	0.939827i	$-0.610986\pi$
$179$	−292917.	−0.683302	−0.341651	+	0.939827i	$0.610986\pi$
$180$	283147.	0.651375
$181$	−38932.7	−0.0883321	−0.0441660	−	0.999024i	$-0.514063\pi$
$181$	−38932.7	−0.0883321	−0.0441660	+	0.999024i	$0.514063\pi$
$182$	−649.121	−0.00145260
$183$	−358757.	−0.791904
$184$	−296270.	−0.645124
$185$	−527066.	−1.13223
$186$	876191.	1.85702
$187$	411061.	0.859611
$188$	2.32510e6	4.79784
$189$	−172.637	−0.000351543 0
$190$	−914243.	−1.83729
$191$	37244.8	0.0738724	0.0369362	−	0.999318i	$-0.488240\pi$
$191$	37244.8	0.0738724	0.0369362	+	0.999318i	$0.488240\pi$
$192$	−790848.	−1.54825
$193$	−42454.3	−0.0820406	−0.0410203	−	0.999158i	$-0.513061\pi$
$193$	−42454.3	−0.0820406	−0.0410203	+	0.999158i	$0.513061\pi$
$194$	1.51193e6	2.88421
$195$	−95319.6	−0.179513
$196$	−1.41178e6	−2.62499
$197$	87335.1	0.160333	0.0801666	−	0.996781i	$-0.474455\pi$
$197$	87335.1	0.160333	0.0801666	+	0.996781i	$0.474455\pi$
$198$	367801.	0.666729
$199$	1.08779e6	1.94721	0.973604	−	0.228245i	$-0.0732989\pi$
$199$	1.08779e6	1.94721	0.973604	+	0.228245i	$0.0732989\pi$
$200$	780275.	1.37934
$201$	256954.	0.448606
$202$	−937883.	−1.61722
$203$	632.657	0.00107753
$204$	737105.	1.24009
$205$	422243.	0.701742
$206$	−193746.	−0.318101
$207$	42849.0	0.0695048
$208$	851055.	1.36395
$209$	−859969.	−1.36181
$210$	955.267	0.00149478
$211$	−1.02777e6	−1.58923	−0.794617	−	0.607111i	$-0.792328\pi$
$211$	−1.02777e6	−1.58923	−0.794617	+	0.607111i	$0.792328\pi$
$212$	911634.	1.39310
$213$	−469618.	−0.709244
$214$	1.37305e6	2.04952
$215$	812456.	1.19868
$216$	408282.	0.595423
$217$	2140.58	0.00308591
$218$	1.46927e6	2.09392
$219$	−512262.	−0.721742
$220$	−1.47376e6	−2.05291
$221$	−248141.	−0.341758
$222$	−1.22769e6	−1.67188
$223$	−209627.	−0.282284	−0.141142	−	0.989989i	$-0.545077\pi$
$223$	−209627.	−0.282284	−0.141142	+	0.989989i	$0.545077\pi$
$224$	−4284.93	−0.00570589
$225$	−112850.	−0.148609
$226$	1.12024e6	1.45895
$227$	−347103.	−0.447088	−0.223544	−	0.974694i	$-0.571763\pi$
$227$	−347103.	−0.447088	−0.223544	+	0.974694i	$0.571763\pi$
$228$	−1.54208e6	−1.96458
$229$	−474094.	−0.597414	−0.298707	−	0.954345i	$-0.596555\pi$
$229$	−474094.	−0.597414	−0.298707	+	0.954345i	$0.596555\pi$
$230$	−237101.	−0.295538
$231$	898.558	0.00110794
$232$	−1.49622e6	−1.82505
$233$	67426.2	0.0813652	0.0406826	−	0.999172i	$-0.487047\pi$
$233$	67426.2	0.0813652	0.0406826	+	0.999172i	$0.487047\pi$
$234$	−222027.	−0.265073
$235$	1.15189e6	1.36063
$236$	1.00024e6	1.16903
$237$	−208605.	−0.241243
$238$	2486.81	0.00284577
$239$	−631686.	−0.715331	−0.357665	−	0.933850i	$-0.616427\pi$
$239$	−631686.	−0.715331	−0.357665	+	0.933850i	$0.616427\pi$
$240$	−1.25244e6	−1.40355
$241$	−582036.	−0.645516	−0.322758	−	0.946482i	$-0.604610\pi$
$241$	−582036.	−0.645516	−0.322758	+	0.946482i	$0.604610\pi$
$242$	−179797.	−0.197353
$243$	−59049.0	−0.0641500
$244$	3.34840e6	3.60050
$245$	−699418.	−0.744426
$246$	983525.	1.03621
$247$	519130.	0.541419
$248$	−5.06243e6	−5.22673
$249$	165101.	0.168753
$250$	2.02508e6	2.04924
$251$	−1.28835e6	−1.29077	−0.645386	−	0.763856i	$-0.723304\pi$
$251$	−1.28835e6	−1.29077	−0.645386	+	0.763856i	$0.723304\pi$
$252$	1611.27	0.00159834
$253$	−223025.	−0.219055
$254$	−2.19459e6	−2.13437
$255$	365173.	0.351680
$256$	1.14509e6	1.09204
$257$	−204908.	−0.193520	−0.0967601	−	0.995308i	$-0.530848\pi$
$257$	−204908.	−0.193520	−0.0967601	+	0.995308i	$0.530848\pi$
$258$	1.89245e6	1.77000
$259$	−2999.32	−0.00277826
$260$	889650.	0.816179
$261$	216395.	0.196628
$262$	−653480.	−0.588138
$263$	418862.	0.373407	0.186703	−	0.982416i	$-0.440220\pi$
$263$	418862.	0.373407	0.186703	+	0.982416i	$0.440220\pi$
$264$	−2.12507e6	−1.87656
$265$	451637.	0.395071
$266$	−5202.58	−0.00450832
$267$	−426262.	−0.365930
$268$	−2.39824e6	−2.03965
$269$	−1.48662e6	−1.25262	−0.626308	−	0.779576i	$-0.715435\pi$
$269$	−1.48662e6	−1.25262	−0.626308	+	0.779576i	$0.715435\pi$
$270$	326742.	0.272769
$271$	−658640.	−0.544785	−0.272392	−	0.962186i	$-0.587815\pi$
$271$	−658640.	−0.544785	−0.272392	+	0.962186i	$0.587815\pi$
$272$	−3.26042e6	−2.67209
$273$	−542.425	−0.000440487 0
$274$	−1.01722e6	−0.818537
$275$	587372.	0.468362
$276$	−399924.	−0.316013
$277$	236526.	0.185216	0.0926082	−	0.995703i	$-0.470480\pi$
$277$	236526.	0.185216	0.0926082	+	0.995703i	$0.470480\pi$
$278$	−4.34592e6	−3.37263
$279$	732170.	0.563121
$280$	−5519.32	−0.00420718
$281$	−1.66253e6	−1.25604	−0.628020	−	0.778197i	$-0.716134\pi$
$281$	−1.66253e6	−1.25604	−0.628020	+	0.778197i	$0.716134\pi$
$282$	2.68308e6	2.00914
$283$	966812.	0.717589	0.358795	−	0.933417i	$-0.383188\pi$
$283$	966812.	0.717589	0.358795	+	0.933417i	$0.383188\pi$
$284$	4.38310e6	3.22467
$285$	−763968.	−0.557139
$286$	1.15563e6	0.835418
$287$	2402.81	0.00172193
$288$	−1.46563e6	−1.04122
$289$	−469218.	−0.330469
$290$	−1.19740e6	−0.836074
$291$	1.26341e6	0.874607
$292$	4.78111e6	3.28150
$293$	−82436.0	−0.0560981	−0.0280490	−	0.999607i	$-0.508929\pi$
$293$	−82436.0	−0.0560981	−0.0280490	+	0.999607i	$0.508929\pi$
$294$	−1.62915e6	−1.09924
$295$	495535.	0.331528
$296$	7.09332e6	4.70565
$297$	307345.	0.202178
$298$	−4.14149e6	−2.70157
$299$	134632.	0.0870902
$300$	1.05326e6	0.675669
$301$	4623.35	0.00294131
$302$	4.76595e6	3.00699
$303$	−783722.	−0.490406
$304$	6.82105e6	4.23318
$305$	1.65884e6	1.02107
$306$	850593.	0.519300
$307$	471104.	0.285280	0.142640	−	0.989775i	$-0.454441\pi$
$307$	471104.	0.285280	0.142640	+	0.989775i	$0.454441\pi$
$308$	−8386.54	−0.00503740
$309$	−161900.	−0.0964608
$310$	−4.05139e6	−2.39442
$311$	974141.	0.571112	0.285556	−	0.958362i	$-0.407822\pi$
$311$	974141.	0.571112	0.285556	+	0.958362i	$0.407822\pi$
$312$	1.28282e6	0.746071
$313$	203897.	0.117639	0.0588194	−	0.998269i	$-0.481266\pi$
$313$	203897.	0.117639	0.0588194	+	0.998269i	$0.481266\pi$
$314$	−2.59267e6	−1.48396
$315$	798.249	0.000453275 0
$316$	1.94698e6	1.09684
$317$	−1.11600e6	−0.623757	−0.311879	−	0.950122i	$-0.600958\pi$
$317$	−1.11600e6	−0.623757	−0.311879	+	0.950122i	$0.600958\pi$
$318$	1.05199e6	0.583371
$319$	−1.12632e6	−0.619704
$320$	3.65678e6	1.99629
$321$	1.14736e6	0.621496
$322$	−1349.24	−0.000725187 0
$323$	−1.98881e6	−1.06068
$324$	551124.	0.291667
$325$	−354574.	−0.186208
$326$	1.91914e6	1.00014
$327$	1.22776e6	0.634958
$328$	−5.68259e6	−2.91650
$329$	6554.91	0.00333870
$330$	−1.70066e6	−0.859673
$331$	2.08998e6	1.04851	0.524254	−	0.851562i	$-0.324344\pi$
$331$	2.08998e6	1.04851	0.524254	+	0.851562i	$0.324344\pi$
$332$	−1.54094e6	−0.767258
$333$	−1.02589e6	−0.506981
$334$	−274915.	−0.134844
$335$	−1.18812e6	−0.578428
$336$	−7127.12	−0.00344402
$337$	901699.	0.432501	0.216250	−	0.976338i	$-0.430617\pi$
$337$	901699.	0.432501	0.216250	+	0.976338i	$0.430617\pi$
$338$	3.30134e6	1.57180
$339$	936107.	0.442411
$340$	−3.40828e6	−1.59896
$341$	−3.81088e6	−1.77476
$342$	−1.77950e6	−0.822685
$343$	−7960.21	−0.00365333
$344$	−1.09341e7	−4.98182
$345$	−198128.	−0.0896187
$346$	−7.55932e6	−3.39463
$347$	−1.53207e6	−0.683053	−0.341526	−	0.939872i	$-0.610944\pi$
$347$	−1.53207e6	−0.683053	−0.341526	+	0.939872i	$0.610944\pi$
$348$	−2.01969e6	−0.893998
$349$	1.28353e6	0.564084	0.282042	−	0.959402i	$-0.408988\pi$
$349$	1.28353e6	0.564084	0.282042	+	0.959402i	$0.408988\pi$
$350$	3553.44	0.00155053
$351$	−185532.	−0.0803806
$352$	7.62846e6	3.28156
$353$	−4.45978e6	−1.90492	−0.952460	−	0.304663i	$-0.901456\pi$
$353$	−4.45978e6	−1.90492	−0.952460	+	0.304663i	$0.901456\pi$
$354$	1.15425e6	0.489542
$355$	2.17145e6	0.914491
$356$	3.97844e6	1.66375
$357$	2078.05	0.000862949 0
$358$	3.15482e6	1.30097
$359$	4.67506e6	1.91448	0.957240	−	0.289296i	$-0.0934213\pi$
$359$	4.67506e6	1.91448	0.957240	+	0.289296i	$0.0934213\pi$
$360$	−1.88784e6	−0.767731
$361$	1.68463e6	0.680356
$362$	419318.	0.168179
$363$	−150243.	−0.0598451
$364$	5062.63	0.00200273
$365$	2.36863e6	0.930606
$366$	3.86393e6	1.50774
$367$	2.43229e6	0.942650	0.471325	−	0.881960i	$-0.343776\pi$
$367$	2.43229e6	0.942650	0.471325	+	0.881960i	$0.343776\pi$
$368$	1.76898e6	0.680930
$369$	821862.	0.314219
$370$	5.67668e6	2.15571
$371$	2570.08	0.000969419 0
$372$	−6.83359e6	−2.56030
$373$	1.37458e6	0.511561	0.255780	−	0.966735i	$-0.417668\pi$
$373$	1.37458e6	0.511561	0.255780	+	0.966735i	$0.417668\pi$
$374$	−4.42726e6	−1.63665
$375$	1.69222e6	0.621410
$376$	−1.55022e7	−5.65489
$377$	679914.	0.246377
$378$	1859.35	0.000669318 0
$379$	2.01997e6	0.722350	0.361175	−	0.932498i	$-0.382376\pi$
$379$	2.01997e6	0.722350	0.361175	+	0.932498i	$0.382376\pi$
$380$	7.13037e6	2.53310
$381$	−1.83387e6	−0.647225
$382$	−401139.	−0.140649
$383$	2.00834e6	0.699584	0.349792	−	0.936827i	$-0.386252\pi$
$383$	2.00834e6	0.699584	0.349792	+	0.936827i	$0.386252\pi$
$384$	3.30658e6	1.14433
$385$	−4154.82	−0.00142857
$386$	457247.	0.156201
$387$	1.58138e6	0.536734
$388$	−1.17919e7	−3.97652
$389$	3.25055e6	1.08914	0.544569	−	0.838716i	$-0.316693\pi$
$389$	3.25055e6	1.08914	0.544569	+	0.838716i	$0.316693\pi$
$390$	1.02662e6	0.341782
$391$	−515779.	−0.170617
$392$	9.41285e6	3.09390
$393$	−546067.	−0.178346
$394$	−940628.	−0.305265
$395$	964563.	0.311055
$396$	−2.86855e6	−0.919231
$397$	5.43707e6	1.73137	0.865683	−	0.500592i	$-0.166884\pi$
$397$	5.43707e6	1.73137	0.865683	+	0.500592i	$0.166884\pi$
$398$	−1.17159e7	−3.70737
$399$	−4347.43	−0.00136710
$400$	−4.65888e6	−1.45590
$401$	3.27208e6	1.01616	0.508081	−	0.861309i	$-0.330355\pi$
$401$	3.27208e6	1.01616	0.508081	+	0.861309i	$0.330355\pi$
$402$	−2.76748e6	−0.854121
$403$	2.30048e6	0.705596
$404$	7.31474e6	2.22970
$405$	273035.	0.0827143
$406$	−6813.92	−0.00205155
$407$	5.33968e6	1.59783
$408$	−4.91454e6	−1.46161
$409$	−1.56299e6	−0.462007	−0.231003	−	0.972953i	$-0.574201\pi$
$409$	−1.56299e6	−0.462007	−0.231003	+	0.972953i	$0.574201\pi$
$410$	−4.54769e6	−1.33608
$411$	−850019.	−0.248213
$412$	1.51107e6	0.438572
$413$	2819.89	0.000813498 0
$414$	−461498.	−0.132333
$415$	−763406.	−0.217588
$416$	−4.60500e6	−1.30466
$417$	−3.63157e6	−1.02272
$418$	9.26215e6	2.59281
$419$	−4.76111e6	−1.32487	−0.662434	−	0.749120i	$-0.730477\pi$
$419$	−4.76111e6	−1.32487	−0.662434	+	0.749120i	$0.730477\pi$
$420$	−7450.33	−0.00206088
$421$	−4.56989e6	−1.25661	−0.628305	−	0.777967i	$-0.716251\pi$
$421$	−4.56989e6	−1.25661	−0.628305	+	0.777967i	$0.716251\pi$
$422$	1.10694e7	3.02581
$423$	2.24206e6	0.609250
$424$	−6.07818e6	−1.64195
$425$	1.35839e6	0.364796
$426$	5.05794e6	1.35036
$427$	9439.80	0.00250549
$428$	−1.07087e7	−2.82571
$429$	965678.	0.253331
$430$	−8.75042e6	−2.28222
$431$	−3.28891e6	−0.852824	−0.426412	−	0.904529i	$-0.640223\pi$
$431$	−3.28891e6	−0.852824	−0.426412	+	0.904529i	$0.640223\pi$
$432$	−2.43778e6	−0.628470
$433$	2.11805e6	0.542895	0.271447	−	0.962453i	$-0.412498\pi$
$433$	2.11805e6	0.542895	0.271447	+	0.962453i	$0.412498\pi$
$434$	−23054.8	−0.00587540
$435$	−1.00058e6	−0.253531
$436$	−1.14591e7	−2.88692
$437$	1.07905e6	0.270294
$438$	5.51723e6	1.37416
$439$	758443.	0.187829	0.0939143	−	0.995580i	$-0.470062\pi$
$439$	758443.	0.187829	0.0939143	+	0.995580i	$0.470062\pi$
$440$	9.82605e6	2.41962
$441$	−1.36136e6	−0.333332
$442$	2.67257e6	0.650688
$443$	650377.	0.157455	0.0787274	−	0.996896i	$-0.474914\pi$
$443$	650377.	0.157455	0.0787274	+	0.996896i	$0.474914\pi$
$444$	9.57500e6	2.30506
$445$	1.97098e6	0.471826
$446$	2.25776e6	0.537452
$447$	−3.46075e6	−0.819222
$448$	20809.2	0.00489847
$449$	−7.58111e6	−1.77467	−0.887333	−	0.461129i	$-0.847445\pi$
$449$	−7.58111e6	−1.77467	−0.887333	+	0.461129i	$0.847445\pi$
$450$	1.21543e6	0.282942
$451$	−4.27772e6	−0.990309
$452$	−8.73699e6	−2.01148
$453$	3.98257e6	0.911838
$454$	3.73841e6	0.851231
$455$	2508.10	0.000567958 0
$456$	1.02816e7	2.31551
$457$	−8.04537e6	−1.80200	−0.901002	−	0.433815i	$-0.857167\pi$
$457$	−8.04537e6	−1.80200	−0.901002	+	0.433815i	$0.857167\pi$
$458$	5.10614e6	1.13744
$459$	710780.	0.157472
$460$	1.84920e6	0.407463
$461$	5.63578e6	1.23510	0.617550	−	0.786532i	$-0.288125\pi$
$461$	5.63578e6	1.23510	0.617550	+	0.786532i	$0.288125\pi$
$462$	−9677.77	−0.00210945
$463$	1.39156e6	0.301682	0.150841	−	0.988558i	$-0.451802\pi$
$463$	1.39156e6	0.301682	0.150841	+	0.988558i	$0.451802\pi$
$464$	8.93365e6	1.92634
$465$	−3.38546e6	−0.726082
$466$	−726202.	−0.154915
$467$	4.05063e6	0.859469	0.429734	−	0.902955i	$-0.358607\pi$
$467$	4.05063e6	0.859469	0.429734	+	0.902955i	$0.358607\pi$
$468$	1.73163e6	0.365461
$469$	−6761.12	−0.00141934
$470$	−1.24062e7	−2.59056
$471$	−2.16651e6	−0.449996
$472$	−6.66897e6	−1.37786
$473$	−8.23095e6	−1.69160
$474$	2.24675e6	0.459312
$475$	−2.84184e6	−0.577917
$476$	−19395.1	−0.00392351
$477$	879076.	0.176901
$478$	6.80347e6	1.36195
$479$	6.55529e6	1.30543	0.652714	−	0.757605i	$-0.273630\pi$
$479$	6.55529e6	1.30543	0.652714	+	0.757605i	$0.273630\pi$
$480$	6.77687e6	1.34254
$481$	−3.22336e6	−0.635252
$482$	6.26872e6	1.22903
$483$	−1127.47	−0.000219905 0
$484$	1.40227e6	0.272094
$485$	−5.84186e6	−1.12771
$486$	635977.	0.122138
$487$	−1.98924e6	−0.380071	−0.190035	−	0.981777i	$-0.560860\pi$
$487$	−1.98924e6	−0.380071	−0.190035	+	0.981777i	$0.560860\pi$
$488$	−2.23249e7	−4.24366
$489$	1.60369e6	0.303283
$490$	7.53296e6	1.41735
$491$	7.99205e6	1.49608	0.748040	−	0.663654i	$-0.230995\pi$
$491$	7.99205e6	1.49608	0.748040	+	0.663654i	$0.230995\pi$
$492$	−7.67071e6	−1.42864
$493$	−2.60478e6	−0.482673
$494$	−5.59120e6	−1.03083
$495$	−1.42112e6	−0.260686
$496$	3.02269e7	5.51682
$497$	12356.8	0.00224397
$498$	−1.77819e6	−0.321296
$499$	−1.00308e7	−1.80338	−0.901688	−	0.432387i	$-0.857671\pi$
$499$	−1.00308e7	−1.80338	−0.901688	+	0.432387i	$0.857671\pi$
$500$	−1.57940e7	−2.82533
$501$	−229727.	−0.0408900
$502$	1.38760e7	2.45756
$503$	−4.82601e6	−0.850488	−0.425244	−	0.905079i	$-0.639812\pi$
$503$	−4.82601e6	−0.850488	−0.425244	+	0.905079i	$0.639812\pi$
$504$	−10742.9	−0.00188385
$505$	3.62383e6	0.632324
$506$	2.40206e6	0.417068
$507$	2.75869e6	0.476633
$508$	1.71161e7	2.94270
$509$	138289.	0.0236589	0.0118294	−	0.999930i	$-0.496234\pi$
$509$	138289.	0.0236589	0.0118294	+	0.999930i	$0.496234\pi$
$510$	−3.93303e6	−0.669579
$511$	13478.9	0.00228351
$512$	−576256.	−0.0971494
$513$	−1.48700e6	−0.249470
$514$	2.20693e6	0.368451
$515$	748605.	0.124375
$516$	−1.47596e7	−2.44033
$517$	−1.16697e7	−1.92014
$518$	32303.6	0.00528965
$519$	−6.31679e6	−1.02939
$520$	−5.93160e6	−0.961975
$521$	4.62831e6	0.747012	0.373506	−	0.927628i	$-0.378155\pi$
$521$	4.62831e6	0.747012	0.373506	+	0.927628i	$0.378155\pi$
$522$	−2.33065e6	−0.374369
$523$	−1.54914e6	−0.247649	−0.123825	−	0.992304i	$-0.539516\pi$
$523$	−1.54914e6	−0.247649	−0.123825	+	0.992304i	$0.539516\pi$
$524$	5.09662e6	0.810876
$525$	2969.36	0.000470181 0
$526$	−4.51129e6	−0.710945
$527$	−8.81323e6	−1.38232
$528$	1.26884e7	1.98072
$529$	279841.	0.0434783
$530$	−4.86428e6	−0.752192
$531$	964521.	0.148448
$532$	40576.0	0.00621570
$533$	2.58229e6	0.393720
$534$	4.59098e6	0.696710
$535$	−5.30526e6	−0.801349
$536$	1.59899e7	2.40399
$537$	2.63626e6	0.394505
$538$	1.60113e7	2.38491
$539$	7.08577e6	1.05055
$540$	−2.54833e6	−0.376072
$541$	−6.52702e6	−0.958787	−0.479393	−	0.877600i	$-0.659143\pi$
$541$	−6.52702e6	−0.958787	−0.479393	+	0.877600i	$0.659143\pi$
$542$	7.09377e6	1.03724
$543$	350395.	0.0509985
$544$	1.76419e7	2.55593
$545$	−5.67701e6	−0.818707
$546$	5842.09	0.000838662 0
$547$	1.06849e7	1.52686	0.763432	−	0.645888i	$-0.223513\pi$
$547$	1.06849e7	1.52686	0.763432	+	0.645888i	$0.223513\pi$
$548$	7.93351e6	1.12853
$549$	3.22881e6	0.457206
$550$	−6.32619e6	−0.891735
$551$	5.44938e6	0.764660
$552$	2.66643e6	0.372463
$553$	5488.93	0.000763264 0
$554$	−2.54746e6	−0.352642
$555$	4.74360e6	0.653695
$556$	3.38947e7	4.64991
$557$	−3.07804e6	−0.420374	−0.210187	−	0.977661i	$-0.567407\pi$
$557$	−3.07804e6	−0.420374	−0.210187	+	0.977661i	$0.567407\pi$
$558$	−7.88572e6	−1.07215
$559$	4.96871e6	0.672533
$560$	32954.9	0.00444068
$561$	−3.69955e6	−0.496297
$562$	1.79060e7	2.39143
$563$	1.10366e7	1.46745	0.733724	−	0.679448i	$-0.237781\pi$
$563$	1.10366e7	1.46745	0.733724	+	0.679448i	$0.237781\pi$
$564$	−2.09259e7	−2.77004
$565$	−4.32844e6	−0.570440
$566$	−1.04129e7	−1.36625
$567$	1553.73	0.000202963 0
$568$	−2.92237e7	−3.80070
$569$	−9.26308e6	−1.19943	−0.599715	−	0.800214i	$-0.704719\pi$
$569$	−9.26308e6	−1.19943	−0.599715	+	0.800214i	$0.704719\pi$
$570$	8.22819e6	1.06076
$571$	−1.31913e7	−1.69315	−0.846576	−	0.532267i	$-0.821340\pi$
$571$	−1.31913e7	−1.69315	−0.846576	+	0.532267i	$0.821340\pi$
$572$	−9.01300e6	−1.15181
$573$	−335203.	−0.0426503
$574$	−25879.0	−0.00327845
$575$	−737006.	−0.0929611
$576$	7.11763e6	0.893880
$577$	−1.29741e6	−0.162232	−0.0811160	−	0.996705i	$-0.525848\pi$
$577$	−1.29741e6	−0.162232	−0.0811160	+	0.996705i	$0.525848\pi$
$578$	5.05363e6	0.629193
$579$	382089.	0.0473661
$580$	9.33878e6	1.15271
$581$	−4344.23	−0.000533916 0
$582$	−1.36074e7	−1.66520
$583$	−4.57551e6	−0.557530
$584$	−3.18773e7	−3.86767
$585$	857876.	0.103642
$586$	887863.	0.106808
$587$	4.51203e6	0.540477	0.270238	−	0.962793i	$-0.412897\pi$
$587$	4.51203e6	0.540477	0.270238	+	0.962793i	$0.412897\pi$
$588$	1.27060e7	1.51554
$589$	1.84379e7	2.18990
$590$	−5.33708e6	−0.631210
$591$	−786016.	−0.0925684
$592$	−4.23529e7	−4.96683
$593$	9.27288e6	1.08287	0.541437	−	0.840741i	$-0.317880\pi$
$593$	9.27288e6	1.08287	0.541437	+	0.840741i	$0.317880\pi$
$594$	−3.31020e6	−0.384936
$595$	−9608.63	−0.00111268
$596$	3.23004e7	3.72470
$597$	−9.79011e6	−1.12422
$598$	−1.45003e6	−0.165815
$599$	−6.32225e6	−0.719954	−0.359977	−	0.932961i	$-0.617215\pi$
$599$	−6.32225e6	−0.719954	−0.359977	+	0.932961i	$0.617215\pi$
$600$	−7.02247e6	−0.796365
$601$	3.18785e6	0.360008	0.180004	−	0.983666i	$-0.442389\pi$
$601$	3.18785e6	0.360008	0.180004	+	0.983666i	$0.442389\pi$
$602$	−49795.0	−0.00560009
$603$	−2.31259e6	−0.259003
$604$	−3.71706e7	−4.14579
$605$	694706.	0.0771636
$606$	8.44095e6	0.933705
$607$	−5.70410e6	−0.628369	−0.314185	−	0.949362i	$-0.601731\pi$
$607$	−5.70410e6	−0.628369	−0.314185	+	0.949362i	$0.601731\pi$
$608$	−3.69082e7	−4.04915
$609$	−5693.91	−0.000622110 0
$610$	−1.78663e7	−1.94406
$611$	7.04455e6	0.763396
$612$	−6.63395e6	−0.715968
$613$	1.17835e7	1.26655	0.633276	−	0.773926i	$-0.281710\pi$
$613$	1.17835e7	1.26655	0.633276	+	0.773926i	$0.281710\pi$
$614$	−5.07394e6	−0.543156
$615$	−3.80018e6	−0.405151
$616$	55916.0	0.00593723
$617$	1.41628e6	0.149774	0.0748868	−	0.997192i	$-0.476140\pi$
$617$	1.41628e6	0.149774	0.0748868	+	0.997192i	$0.476140\pi$
$618$	1.74372e6	0.183656
$619$	−4.37752e6	−0.459200	−0.229600	−	0.973285i	$-0.573742\pi$
$619$	−4.37752e6	−0.459200	−0.229600	+	0.973285i	$0.573742\pi$
$620$	3.15976e7	3.30123
$621$	−385641.	−0.0401286
$622$	−1.04918e7	−1.08736
$623$	11216.0	0.00115776
$624$	−7.65950e6	−0.787479
$625$	−3.47084e6	−0.355414
$626$	−2.19604e6	−0.223978
$627$	7.73972e6	0.786243
$628$	2.02208e7	2.04597
$629$	1.23488e7	1.24451
$630$	−8597.41	−0.000863011 0
$631$	−1.61975e7	−1.61948	−0.809739	−	0.586790i	$-0.800391\pi$
$631$	−1.61975e7	−1.61948	−0.809739	+	0.586790i	$0.800391\pi$
$632$	−1.29812e7	−1.29277
$633$	9.24989e6	0.917545
$634$	1.20197e7	1.18760
$635$	8.47956e6	0.834525
$636$	−8.20471e6	−0.804304
$637$	−4.27740e6	−0.417668
$638$	1.21308e7	1.17988
$639$	4.22656e6	0.409482
$640$	−1.52892e7	−1.47548
$641$	−1.56439e7	−1.50383	−0.751915	−	0.659260i	$-0.770870\pi$
$641$	−1.56439e7	−1.50383	−0.751915	+	0.659260i	$0.770870\pi$
$642$	−1.23575e7	−1.18329
$643$	−1.57570e7	−1.50295	−0.751476	−	0.659760i	$-0.770658\pi$
$643$	−1.57570e7	−1.50295	−0.751476	+	0.659760i	$0.770658\pi$
$644$	10523.0	0.000999829 0
$645$	−7.31211e6	−0.692059
$646$	2.14201e7	2.01948
$647$	−660291.	−0.0620119	−0.0310059	−	0.999519i	$-0.509871\pi$
$647$	−660291.	−0.0620119	−0.0310059	+	0.999519i	$0.509871\pi$
$648$	−3.67453e6	−0.343768
$649$	−5.02024e6	−0.467857
$650$	3.81888e6	0.354529
$651$	−19265.3	−0.00178165
$652$	−1.49678e7	−1.37892
$653$	1.24310e7	1.14084	0.570419	−	0.821354i	$-0.306781\pi$
$653$	1.24310e7	1.14084	0.570419	+	0.821354i	$0.306781\pi$
$654$	−1.32234e7	−1.20892
$655$	2.52494e6	0.229958
$656$	3.39297e7	3.07837
$657$	4.61036e6	0.416698
$658$	−70598.6	−0.00635669
$659$	−13723.9	−0.00123101	−0.000615507	−	1.00000i	$-0.500196\pi$
$659$	−13723.9	−0.00123101	−0.000615507		1.00000i	$0.500196\pi$
$660$	1.32638e7	1.18525
$661$	6.54194e6	0.582375	0.291187	−	0.956666i	$-0.405950\pi$
$661$	6.54194e6	0.582375	0.291187	+	0.956666i	$0.405950\pi$
$662$	−2.25098e7	−1.99630
$663$	2.23327e6	0.197314
$664$	1.02740e7	0.904315
$665$	20101.9	0.00176272
$666$	1.10492e7	0.965262
$667$	1.41325e6	0.123000
$668$	2.14412e6	0.185912
$669$	1.88665e6	0.162977
$670$	1.27965e7	1.10129
$671$	−1.68057e7	−1.44095
$672$	38564.4	0.00329430
$673$	1.54266e7	1.31290	0.656451	−	0.754369i	$-0.272057\pi$
$673$	1.54266e7	1.31290	0.656451	+	0.754369i	$0.272057\pi$
$674$	−9.71159e6	−0.823456
$675$	1.01565e6	0.0857992
$676$	−2.57478e7	−2.16707
$677$	5.30462e6	0.444818	0.222409	−	0.974953i	$-0.428608\pi$
$677$	5.30462e6	0.444818	0.222409	+	0.974953i	$0.428608\pi$
$678$	−1.00822e7	−0.842326
$679$	−33243.6	−0.00276716
$680$	2.27242e7	1.88459
$681$	3.12392e6	0.258127
$682$	4.10444e7	3.37904
$683$	1.52230e7	1.24867	0.624337	−	0.781155i	$-0.285369\pi$
$683$	1.52230e7	1.24867	0.624337	+	0.781155i	$0.285369\pi$
$684$	1.38787e7	1.13425
$685$	3.93038e6	0.320043
$686$	85734.1	0.00695574
$687$	4.26684e6	0.344917
$688$	6.52857e7	5.25832
$689$	2.76206e6	0.221659
$690$	2.13391e6	0.170629
$691$	1.04711e7	0.834248	0.417124	−	0.908850i	$-0.363038\pi$
$691$	1.04711e7	0.834248	0.417124	+	0.908850i	$0.363038\pi$
$692$	5.89567e7	4.68024
$693$	−8087.02	−0.000639669 0
$694$	1.65009e7	1.30049
$695$	1.67919e7	1.31868
$696$	1.34660e7	1.05369
$697$	−9.89286e6	−0.771329
$698$	−1.38241e7	−1.07398
$699$	−606836.	−0.0469762
$700$	−27714.0	−0.00213774
$701$	−2.28874e7	−1.75914	−0.879572	−	0.475765i	$-0.842171\pi$
$701$	−2.28874e7	−1.75914	−0.879572	+	0.475765i	$0.842171\pi$
$702$	1.99824e6	0.153040
$703$	−2.58346e7	−1.97158
$704$	−3.70466e7	−2.81720
$705$	−1.03670e7	−0.785560
$706$	4.80333e7	3.62686
$707$	20621.7	0.00155159
$708$	−9.00219e6	−0.674940
$709$	1.85127e7	1.38310	0.691550	−	0.722329i	$-0.256928\pi$
$709$	1.85127e7	1.38310	0.691550	+	0.722329i	$0.256928\pi$
$710$	−2.33873e7	−1.74114
$711$	1.87745e6	0.139281
$712$	−2.65257e7	−1.96095
$713$	4.78170e6	0.352256
$714$	−22381.3	−0.00164301
$715$	−4.46517e6	−0.326643
$716$	−2.46051e7	−1.79367
$717$	5.68518e6	0.412996
$718$	−5.03519e7	−3.64506
$719$	1.25679e7	0.906652	0.453326	−	0.891345i	$-0.350237\pi$
$719$	1.25679e7	0.906652	0.453326	+	0.891345i	$0.350237\pi$
$720$	1.12720e7	0.810342
$721$	4260.00	0.000305191 0
$722$	−1.81440e7	−1.29536
$723$	5.23832e6	0.372689
$724$	−3.27035e6	−0.231872
$725$	−3.72201e6	−0.262986
$726$	1.61817e6	0.113942
$727$	3.64050e6	0.255462	0.127731	−	0.991809i	$-0.459231\pi$
$727$	3.64050e6	0.255462	0.127731	+	0.991809i	$0.459231\pi$
$728$	−33754.3	−0.00236048
$729$	531441.	0.0370370
$730$	−2.55110e7	−1.77182
$731$	−1.90353e7	−1.31755
$732$	−3.01356e7	−2.07875
$733$	−4.38735e6	−0.301608	−0.150804	−	0.988564i	$-0.548186\pi$
$733$	−4.38735e6	−0.301608	−0.150804	+	0.988564i	$0.548186\pi$
$734$	−2.61966e7	−1.79475
$735$	6.29476e6	0.429795
$736$	−9.57181e6	−0.651327
$737$	1.20368e7	0.816287
$738$	−8.85173e6	−0.598256
$739$	1.95857e7	1.31925	0.659625	−	0.751595i	$-0.270715\pi$
$739$	1.95857e7	1.31925	0.659625	+	0.751595i	$0.270715\pi$
$740$	−4.42736e7	−2.97211
$741$	−4.67217e6	−0.312589
$742$	−27680.6	−0.00184572
$743$	−2.94833e7	−1.95931	−0.979657	−	0.200678i	$-0.935685\pi$
$743$	−2.94833e7	−1.95931	−0.979657	+	0.200678i	$0.935685\pi$
$744$	4.55619e7	3.01766
$745$	1.60021e7	1.05630
$746$	−1.48047e7	−0.973983
$747$	−1.48591e6	−0.0974296
$748$	3.45291e7	2.25648
$749$	−30190.0	−0.00196634
$750$	−1.82258e7	−1.18313
$751$	1.27147e7	0.822634	0.411317	−	0.911492i	$-0.365069\pi$
$751$	1.27147e7	0.822634	0.411317	+	0.911492i	$0.365069\pi$
$752$	9.25609e7	5.96875
$753$	1.15952e7	0.745228
$754$	−7.32290e6	−0.469089
$755$	−1.84149e7	−1.17571
$756$	−14501.5	−0.000922800 0
$757$	1.63980e7	1.04004	0.520022	−	0.854153i	$-0.325924\pi$
$757$	1.63980e7	1.04004	0.520022	+	0.854153i	$0.325924\pi$
$758$	−2.17558e7	−1.37531
$759$	2.00723e6	0.126471
$760$	−4.75407e7	−2.98560
$761$	8.50863e6	0.532596	0.266298	−	0.963891i	$-0.414199\pi$
$761$	8.50863e6	0.532596	0.266298	+	0.963891i	$0.414199\pi$
$762$	1.97513e7	1.23228
$763$	−32305.5	−0.00200893
$764$	3.12856e6	0.193915
$765$	−3.28656e6	−0.203043
$766$	−2.16305e7	−1.33197
$767$	3.03053e6	0.186007
$768$	−1.03058e7	−0.630490
$769$	1.70141e7	1.03751	0.518755	−	0.854923i	$-0.326396\pi$
$769$	1.70141e7	1.03751	0.518755	+	0.854923i	$0.326396\pi$
$770$	44748.7	0.00271991
$771$	1.84417e6	0.111729
$772$	−3.56616e6	−0.215356
$773$	3.63916e6	0.219055	0.109527	−	0.993984i	$-0.465066\pi$
$773$	3.63916e6	0.219055	0.109527	+	0.993984i	$0.465066\pi$
$774$	−1.70320e7	−1.02191
$775$	−1.25934e7	−0.753161
$776$	7.86204e7	4.68685
$777$	26993.8	0.00160403
$778$	−3.50095e7	−2.07366
$779$	2.06966e7	1.22195
$780$	−8.00685e6	−0.471221
$781$	−2.19989e7	−1.29054
$782$	5.55511e6	0.324845
$783$	−1.94756e6	−0.113524
$784$	−5.62024e7	−3.26561
$785$	1.00177e7	0.580220
$786$	5.88132e6	0.339561
$787$	−5.30127e6	−0.305101	−0.152550	−	0.988296i	$-0.548749\pi$
$787$	−5.30127e6	−0.305101	−0.152550	+	0.988296i	$0.548749\pi$
$788$	7.33615e6	0.420875
$789$	−3.76976e6	−0.215586
$790$	−1.03887e7	−0.592232
$791$	−24631.3	−0.00139974
$792$	1.91256e7	1.08343
$793$	1.01449e7	0.572883
$794$	−5.85591e7	−3.29642
$795$	−4.06473e6	−0.228094
$796$	9.13743e7	5.11142
$797$	1.41955e7	0.791598	0.395799	−	0.918337i	$-0.370468\pi$
$797$	1.41955e7	0.791598	0.395799	+	0.918337i	$0.370468\pi$
$798$	46823.2	0.00260288
$799$	−2.69879e7	−1.49555
$800$	2.52089e7	1.39261
$801$	3.83636e6	0.211270
$802$	−3.52414e7	−1.93472
$803$	−2.39965e7	−1.31329
$804$	2.15841e7	1.17759
$805$	5213.26	0.000283543 0
$806$	−2.47769e7	−1.34341
$807$	1.33795e7	0.723198
$808$	−4.87699e7	−2.62799
$809$	−1.58542e7	−0.851675	−0.425838	−	0.904800i	$-0.640021\pi$
$809$	−1.58542e7	−0.851675	−0.425838	+	0.904800i	$0.640021\pi$
$810$	−2.94068e6	−0.157483
$811$	−1.40129e7	−0.748127	−0.374063	−	0.927403i	$-0.622036\pi$
$811$	−1.40129e7	−0.748127	−0.374063	+	0.927403i	$0.622036\pi$
$812$	53143.1	0.00282851
$813$	5.92776e6	0.314532
$814$	−5.75101e7	−3.04217
$815$	−7.41524e6	−0.391049
$816$	2.93438e7	1.54273
$817$	3.98232e7	2.08728
$818$	1.68339e7	0.879634
$819$	4881.82	0.000254315 0
$820$	3.54684e7	1.84207
$821$	1.16279e7	0.602065	0.301033	−	0.953614i	$-0.402669\pi$
$821$	1.16279e7	0.602065	0.301033	+	0.953614i	$0.402669\pi$
$822$	9.15498e6	0.472583
$823$	−2.76237e7	−1.42161	−0.710807	−	0.703387i	$-0.751670\pi$
$823$	−2.76237e7	−1.42161	−0.710807	+	0.703387i	$0.751670\pi$
$824$	−1.00748e7	−0.516914
$825$	−5.28635e6	−0.270409
$826$	−30371.1	−0.00154885
$827$	−2.78146e7	−1.41420	−0.707098	−	0.707116i	$-0.749996\pi$
$827$	−2.78146e7	−1.41420	−0.707098	+	0.707116i	$0.749996\pi$
$828$	3.59932e6	0.182450
$829$	−1.23659e7	−0.624943	−0.312471	−	0.949927i	$-0.601157\pi$
$829$	−1.23659e7	−0.624943	−0.312471	+	0.949927i	$0.601157\pi$
$830$	8.22214e6	0.414276
$831$	−2.12873e6	−0.106935
$832$	2.23636e7	1.12004
$833$	1.63869e7	0.818246
$834$	3.91132e7	1.94719
$835$	1.06223e6	0.0527231
$836$	−7.22374e7	−3.57476
$837$	−6.58953e6	−0.325118
$838$	5.12787e7	2.52247
$839$	−3.49502e7	−1.71413	−0.857067	−	0.515205i	$-0.827716\pi$
$839$	−3.49502e7	−1.71413	−0.857067	+	0.515205i	$0.827716\pi$
$840$	49673.9	0.00242901
$841$	−1.33740e7	−0.652035
$842$	4.92192e7	2.39251
$843$	1.49628e7	0.725175
$844$	−8.63323e7	−4.17174
$845$	−1.27558e7	−0.614565
$846$	−2.41477e7	−1.15998
$847$	3953.28	0.000189343 0
$848$	3.62917e7	1.73308
$849$	−8.70131e6	−0.414300
$850$	−1.46303e7	−0.694552
$851$	−6.69997e6	−0.317138
$852$	−3.94479e7	−1.86177
$853$	−9.99950e6	−0.470550	−0.235275	−	0.971929i	$-0.575599\pi$
$853$	−9.99950e6	−0.470550	−0.235275	+	0.971929i	$0.575599\pi$
$854$	−101670.	−0.00477032
$855$	6.87571e6	0.321664
$856$	7.13987e7	3.33047
$857$	9.71240e6	0.451725	0.225863	−	0.974159i	$-0.427480\pi$
$857$	9.71240e6	0.451725	0.225863	+	0.974159i	$0.427480\pi$
$858$	−1.04007e7	−0.482329
$859$	3.69393e7	1.70807	0.854035	−	0.520216i	$-0.174149\pi$
$859$	3.69393e7	1.70807	0.854035	+	0.520216i	$0.174149\pi$
$860$	6.82463e7	3.14654
$861$	−21625.3	−0.000994155 0
$862$	3.54227e7	1.62373
$863$	−3.31284e7	−1.51416	−0.757082	−	0.653319i	$-0.773376\pi$
$863$	−3.31284e7	−1.51416	−0.757082	+	0.653319i	$0.773376\pi$
$864$	1.31906e7	0.601148
$865$	2.92080e7	1.32728
$866$	−2.28121e7	−1.03364
$867$	4.22296e6	0.190796
$868$	179809.	0.00810051
$869$	−9.77194e6	−0.438966
$870$	1.07766e7	0.482708
$871$	−7.26615e6	−0.324533
$872$	7.64019e7	3.40262
$873$	−1.13707e7	−0.504954
$874$	−1.16217e7	−0.514625
$875$	−44526.6	−0.00196607
$876$	−4.30300e7	−1.89457
$877$	1.38007e7	0.605902	0.302951	−	0.953006i	$-0.402028\pi$
$877$	1.38007e7	0.605902	0.302951	+	0.953006i	$0.402028\pi$
$878$	−8.16869e6	−0.357615
$879$	741924.	0.0323882
$880$	−5.86696e7	−2.55391
$881$	−7.74868e6	−0.336347	−0.168174	−	0.985757i	$-0.553787\pi$
$881$	−7.74868e6	−0.336347	−0.168174	+	0.985757i	$0.553787\pi$
$882$	1.46623e7	0.634646
$883$	3.79838e7	1.63944	0.819722	−	0.572761i	$-0.194128\pi$
$883$	3.79838e7	1.63944	0.819722	+	0.572761i	$0.194128\pi$
$884$	−2.08439e7	−0.897115
$885$	−4.45982e6	−0.191408
$886$	−7.00477e6	−0.299785
$887$	−1.51721e7	−0.647495	−0.323747	−	0.946144i	$-0.604943\pi$
$887$	−1.51721e7	−0.647495	−0.323747	+	0.946144i	$0.604943\pi$
$888$	−6.38398e7	−2.71681
$889$	48253.7	0.00204775
$890$	−2.12281e7	−0.898330
$891$	−2.76610e6	−0.116728
$892$	−1.76087e7	−0.740995
$893$	5.64607e7	2.36929
$894$	3.72734e7	1.55975
$895$	−1.21897e7	−0.508670
$896$	−87004.5	−0.00362052
$897$	−1.21168e6	−0.0502815
$898$	8.16510e7	3.37886
$899$	2.41485e7	0.996530
$900$	−9.47937e6	−0.390098
$901$	−1.05815e7	−0.434247
$902$	4.60724e7	1.88549
$903$	−41610.2	−0.00169817
$904$	5.82526e7	2.37080
$905$	−1.62018e6	−0.0657569
$906$	−4.28936e7	−1.73609
$907$	8.48315e6	0.342404	0.171202	−	0.985236i	$-0.445235\pi$
$907$	8.48315e6	0.342404	0.171202	+	0.985236i	$0.445235\pi$
$908$	−2.91566e7	−1.17361
$909$	7.05350e6	0.283136
$910$	−27013.1	−0.00108136
$911$	2.96302e7	1.18288	0.591438	−	0.806351i	$-0.298561\pi$
$911$	2.96302e7	1.18288	0.591438	+	0.806351i	$0.298561\pi$
$912$	−6.13894e7	−2.44403
$913$	7.73403e6	0.307064
$914$	8.66513e7	3.43091
$915$	−1.49296e7	−0.589516
$916$	−3.98239e7	−1.56821
$917$	14368.4	0.000564268 0
$918$	−7.65534e6	−0.299818
$919$	−1.27080e7	−0.496352	−0.248176	−	0.968715i	$-0.579831\pi$
$919$	−1.27080e7	−0.496352	−0.248176	+	0.968715i	$0.579831\pi$
$920$	−1.23292e7	−0.480249
$921$	−4.23993e6	−0.164706
$922$	−6.06992e7	−2.35156
$923$	1.32799e7	0.513085
$924$	75478.9	0.00290834
$925$	1.76454e7	0.678075
$926$	−1.49875e7	−0.574385
$927$	1.45710e6	0.0556916
$928$	−4.83394e7	−1.84260
$929$	−4.50533e7	−1.71272	−0.856362	−	0.516376i	$-0.827281\pi$
$929$	−4.50533e7	−1.71272	−0.856362	+	0.516376i	$0.827281\pi$
$930$	3.64625e7	1.38242
$931$	−3.42826e7	−1.29628
$932$	5.66380e6	0.213584
$933$	−8.76727e6	−0.329731
$934$	−4.36266e7	−1.63638
$935$	1.71062e7	0.639919
$936$	−1.15454e7	−0.430744
$937$	−2.49632e7	−0.928864	−0.464432	−	0.885609i	$-0.653741\pi$
$937$	−2.49632e7	−0.928864	−0.464432	+	0.885609i	$0.653741\pi$
$938$	72819.4	0.00270234
$939$	−1.83508e6	−0.0679188
$940$	9.67585e7	3.57165
$941$	−3.36141e7	−1.23751	−0.618754	−	0.785585i	$-0.712362\pi$
$941$	−3.36141e7	−1.23751	−0.618754	+	0.785585i	$0.712362\pi$
$942$	2.33340e7	0.856767
$943$	5.36747e6	0.196558
$944$	3.98192e7	1.45433
$945$	−7184.24	−0.000261699 0
$946$	8.86501e7	3.22071
$947$	−5.33926e6	−0.193467	−0.0967333	−	0.995310i	$-0.530839\pi$
$947$	−5.33926e6	−0.193467	−0.0967333	+	0.995310i	$0.530839\pi$
$948$	−1.75228e7	−0.633262
$949$	1.44858e7	0.522127
$950$	3.06076e7	1.10032
$951$	1.00440e7	0.360126
$952$	129314.	0.00462438
$953$	−3.69391e7	−1.31751	−0.658755	−	0.752358i	$-0.728916\pi$
$953$	−3.69391e7	−1.31751	−0.658755	+	0.752358i	$0.728916\pi$
$954$	−9.46793e6	−0.336809
$955$	1.54994e6	0.0549928
$956$	−5.30617e7	−1.87774
$957$	1.01369e7	0.357786
$958$	−7.06026e7	−2.48546
$959$	22366.2	0.000785317 0
$960$	−3.29110e7	−1.15256
$961$	5.30769e7	1.85395
$962$	3.47166e7	1.20948
$963$	−1.03263e7	−0.358821
$964$	−4.88910e7	−1.69448
$965$	−1.76673e6	−0.0610734
$966$	12143.2	0.000418687 0
$967$	−4.47688e7	−1.53960	−0.769802	−	0.638283i	$-0.779645\pi$
$967$	−4.47688e7	−1.53960	−0.769802	+	0.638283i	$0.779645\pi$
$968$	−9.34943e6	−0.320698
$969$	1.78993e7	0.612386
$970$	6.29187e7	2.14709
$971$	−4.07416e7	−1.38672	−0.693361	−	0.720590i	$-0.743871\pi$
$971$	−4.07416e7	−1.38672	−0.693361	+	0.720590i	$0.743871\pi$
$972$	−4.96012e6	−0.168394
$973$	95555.9	0.00323575
$974$	2.14247e7	0.723633
$975$	3.19116e6	0.107507
$976$	1.33298e8	4.47919
$977$	1.07576e7	0.360561	0.180281	−	0.983615i	$-0.442299\pi$
$977$	1.07576e7	0.360561	0.180281	+	0.983615i	$0.442299\pi$
$978$	−1.72722e7	−0.577433
$979$	−1.99679e7	−0.665849
$980$	−5.87511e7	−1.95412
$981$	−1.10499e7	−0.366593
$982$	−8.60770e7	−2.84845
$983$	−2.44766e7	−0.807917	−0.403958	−	0.914777i	$-0.632366\pi$
$983$	−2.44766e7	−0.807917	−0.403958	+	0.914777i	$0.632366\pi$
$984$	5.11433e7	1.68384
$985$	3.63444e6	0.119357
$986$	2.80543e7	0.918982
$987$	−58994.2	−0.00192760
$988$	4.36069e7	1.42123
$989$	1.03278e7	0.335751
$990$	1.53060e7	0.496332
$991$	−2.09086e6	−0.0676301	−0.0338151	−	0.999428i	$-0.510766\pi$
$991$	−2.09086e6	−0.0676301	−0.0338151	+	0.999428i	$0.510766\pi$
$992$	−1.63556e8	−5.27699
$993$	−1.88098e7	−0.605357
$994$	−133087.	−0.00427239
$995$	4.52682e7	1.44956
$996$	1.38685e7	0.442977
$997$	1.49624e7	0.476721	0.238360	−	0.971177i	$-0.423390\pi$
$997$	1.49624e7	0.476721	0.238360	+	0.971177i	$0.423390\pi$
$998$	1.08036e8	3.43353
$999$	9.23304e6	0.292705

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.a.a.1.1	✓	2
3.2	odd	2		207.6.a.a.1.2		2
4.3	odd	2		1104.6.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.a.1.1	✓	2	1.1	even	1	trivial
207.6.a.a.1.2		2	3.2	odd	2
1104.6.a.h.1.1		2	4.3	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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q-10.7703 q^{2} -9.00000 q^{3} +84.0000 q^{4} +41.6148 q^{5} +96.9330 q^{6} +0.236813 q^{7} -560.057 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-10.7703 q^{2} -9.00000 q^{3} +84.0000 q^{4} +41.6148 q^{5} +96.9330 q^{6} +0.236813 q^{7} -560.057 q^{8} +81.0000 q^{9} -448.205 q^{10} -421.598 q^{11} -756.000 q^{12} +254.502 q^{13} -2.55055 q^{14} -374.534 q^{15} +3344.00 q^{16} -975.007 q^{17} -872.397 q^{18} +2039.79 q^{19} +3495.65 q^{20} -2.13132 q^{21} +4540.75 q^{22} +529.000 q^{23} +5040.51 q^{24} -1393.21 q^{25} -2741.07 q^{26} -729.000 q^{27} +19.8923 q^{28} +2671.55 q^{29} +4033.85 q^{30} +9039.14 q^{31} -18094.2 q^{32} +3794.38 q^{33} +10501.1 q^{34} +9.85493 q^{35} +6804.00 q^{36} -12665.3 q^{37} -21969.2 q^{38} -2290.52 q^{39} -23306.7 q^{40} +10146.4 q^{41} +22.9550 q^{42} +19523.2 q^{43} -35414.2 q^{44} +3370.80 q^{45} -5697.50 q^{46} +27679.7 q^{47} -30096.0 q^{48} -16806.9 q^{49} +15005.3 q^{50} +8775.06 q^{51} +21378.2 q^{52} +10852.8 q^{53} +7851.57 q^{54} -17544.7 q^{55} -132.629 q^{56} -18358.1 q^{57} -28773.4 q^{58} +11907.7 q^{59} -31460.8 q^{60} +39861.9 q^{61} -97354.5 q^{62} +19.1818 q^{63} +87872.0 q^{64} +10591.1 q^{65} -40866.7 q^{66} -28550.5 q^{67} -81900.6 q^{68} -4761.00 q^{69} -106.141 q^{70} +52179.8 q^{71} -45364.6 q^{72} +56918.0 q^{73} +136410. q^{74} +12538.8 q^{75} +171342. q^{76} -99.8398 q^{77} +24669.7 q^{78} +23178.3 q^{79} +139160. q^{80} +6561.00 q^{81} -109281. q^{82} -18344.6 q^{83} -179.031 q^{84} -40574.8 q^{85} -210272. q^{86} -24043.9 q^{87} +236119. q^{88} +47362.4 q^{89} -36304.6 q^{90} +60.2694 q^{91} +44436.0 q^{92} -81352.3 q^{93} -298120. q^{94} +84885.4 q^{95} +162847. q^{96} -140379. q^{97} +181016. q^{98} -34149.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} + 168 q^{4} + 94 q^{5} - 118 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q - 18 * q^3 + 168 * q^4 + 94 * q^5 - 118 * q^7 + 162 * q^9	\( 2 q - 18 q^{3} + 168 q^{4} + 94 q^{5} - 118 q^{7} + 162 q^{9} + 116 q^{10} + 320 q^{11} - 1512 q^{12} - 288 q^{13} - 1276 q^{14} - 846 q^{15} + 6688 q^{16} - 1810 q^{17} + 730 q^{19} + 7896 q^{20} + 1062 q^{21} + 12528 q^{22} + 1058 q^{23} - 1774 q^{25} - 8584 q^{26} - 1458 q^{27} - 9912 q^{28} + 8208 q^{29} - 1044 q^{30} + 1772 q^{31} - 2880 q^{33} + 1508 q^{34} - 6184 q^{35} + 13608 q^{36} - 23112 q^{37} - 36076 q^{38} + 2592 q^{39} + 6032 q^{40} + 5516 q^{41} + 11484 q^{42} + 10322 q^{43} + 26880 q^{44} + 7614 q^{45} + 42952 q^{47} - 60192 q^{48} - 19634 q^{49} + 10904 q^{50} + 16290 q^{51} - 24192 q^{52} - 25350 q^{53} + 21304 q^{55} - 66352 q^{56} - 6570 q^{57} + 30856 q^{58} + 18344 q^{59} - 71064 q^{60} + 37224 q^{61} - 175624 q^{62} - 9558 q^{63} + 175744 q^{64} - 17828 q^{65} - 112752 q^{66} - 7482 q^{67} - 152040 q^{68} - 9522 q^{69} - 66816 q^{70} + 126848 q^{71} + 137660 q^{73} + 23896 q^{74} + 15966 q^{75} + 61320 q^{76} - 87784 q^{77} + 77256 q^{78} + 62286 q^{79} + 314336 q^{80} + 13122 q^{81} - 159152 q^{82} + 83120 q^{83} + 89208 q^{84} - 84316 q^{85} - 309372 q^{86} - 73872 q^{87} + 651456 q^{88} + 69770 q^{89} + 9396 q^{90} + 64204 q^{91} + 88872 q^{92} - 15948 q^{93} - 133632 q^{94} + 16272 q^{95} - 170104 q^{97} + 150568 q^{98} + 25920 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 + 168 * q^4 + 94 * q^5 - 118 * q^7 + 162 * q^9 + 116 * q^10 + 320 * q^11 - 1512 * q^12 - 288 * q^13 - 1276 * q^14 - 846 * q^15 + 6688 * q^16 - 1810 * q^17 + 730 * q^19 + 7896 * q^20 + 1062 * q^21 + 12528 * q^22 + 1058 * q^23 - 1774 * q^25 - 8584 * q^26 - 1458 * q^27 - 9912 * q^28 + 8208 * q^29 - 1044 * q^30 + 1772 * q^31 - 2880 * q^33 + 1508 * q^34 - 6184 * q^35 + 13608 * q^36 - 23112 * q^37 - 36076 * q^38 + 2592 * q^39 + 6032 * q^40 + 5516 * q^41 + 11484 * q^42 + 10322 * q^43 + 26880 * q^44 + 7614 * q^45 + 42952 * q^47 - 60192 * q^48 - 19634 * q^49 + 10904 * q^50 + 16290 * q^51 - 24192 * q^52 - 25350 * q^53 + 21304 * q^55 - 66352 * q^56 - 6570 * q^57 + 30856 * q^58 + 18344 * q^59 - 71064 * q^60 + 37224 * q^61 - 175624 * q^62 - 9558 * q^63 + 175744 * q^64 - 17828 * q^65 - 112752 * q^66 - 7482 * q^67 - 152040 * q^68 - 9522 * q^69 - 66816 * q^70 + 126848 * q^71 + 137660 * q^73 + 23896 * q^74 + 15966 * q^75 + 61320 * q^76 - 87784 * q^77 + 77256 * q^78 + 62286 * q^79 + 314336 * q^80 + 13122 * q^81 - 159152 * q^82 + 83120 * q^83 + 89208 * q^84 - 84316 * q^85 - 309372 * q^86 - 73872 * q^87 + 651456 * q^88 + 69770 * q^89 + 9396 * q^90 + 64204 * q^91 + 88872 * q^92 - 15948 * q^93 - 133632 * q^94 + 16272 * q^95 - 170104 * q^97 + 150568 * q^98 + 25920 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more