LMFDB - Embedded newform 169.10.a.f.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,10,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	169.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:	$87.0410563117$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 7679 x^{18} + 24599364 x^{16} - 42662336000 x^{14} + 43527566862400 x^{12} + \cdots + 25\!\cdots\!36 $ x^20 - 7679x^18 + 24599364x^16 - 42662336000x^14 + 43527566862400x^12 - 26625453181634304x^10 + 9550565219960082432x^8 - 1876223091685443895296x^6 + 172106193081772189679616x^4 - 4841822934658519419322368*x^2 + 25162285487879223389454336
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{25}\cdot 3^{10}\cdot 13^{12} $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$27.6771$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+27.6771 q^{2} -44.2389 q^{3} +254.024 q^{4} -2467.22 q^{5} -1224.41 q^{6} +3580.38 q^{7} -7140.04 q^{8} -17725.9 q^{9} -68285.5 q^{10} -79291.0 q^{11} -11237.7 q^{12} +99094.7 q^{14} +109147. q^{15} -327676. q^{16} +8421.95 q^{17} -490603. q^{18} -434724. q^{19} -626732. q^{20} -158392. q^{21} -2.19455e6 q^{22} +1.39043e6 q^{23} +315867. q^{24} +4.13403e6 q^{25} +1.65493e6 q^{27} +909503. q^{28} +2.73548e6 q^{29} +3.02087e6 q^{30} -1.77534e6 q^{31} -5.41344e6 q^{32} +3.50775e6 q^{33} +233095. q^{34} -8.83357e6 q^{35} -4.50281e6 q^{36} -1.57016e7 q^{37} -1.20319e7 q^{38} +1.76160e7 q^{40} +112436. q^{41} -4.38384e6 q^{42} -2.68150e7 q^{43} -2.01418e7 q^{44} +4.37337e7 q^{45} +3.84831e7 q^{46} +3.59357e7 q^{47} +1.44960e7 q^{48} -2.75345e7 q^{49} +1.14418e8 q^{50} -372578. q^{51} -3.83324e7 q^{53} +4.58037e7 q^{54} +1.95628e8 q^{55} -2.55641e7 q^{56} +1.92317e7 q^{57} +7.57103e7 q^{58} -646636. q^{59} +2.77259e7 q^{60} +4.42458e7 q^{61} -4.91363e7 q^{62} -6.34656e7 q^{63} +1.79417e7 q^{64} +9.70844e7 q^{66} +1.65124e8 q^{67} +2.13938e6 q^{68} -6.15110e7 q^{69} -2.44488e8 q^{70} +1.69848e8 q^{71} +1.26564e8 q^{72} -8.20084e7 q^{73} -4.34574e8 q^{74} -1.82885e8 q^{75} -1.10430e8 q^{76} -2.83892e8 q^{77} +5.10327e8 q^{79} +8.08448e8 q^{80} +2.75687e8 q^{81} +3.11191e6 q^{82} -4.20094e8 q^{83} -4.02354e7 q^{84} -2.07788e7 q^{85} -7.42162e8 q^{86} -1.21015e8 q^{87} +5.66141e8 q^{88} +3.97908e8 q^{89} +1.21042e9 q^{90} +3.53202e8 q^{92} +7.85390e7 q^{93} +9.94598e8 q^{94} +1.07256e9 q^{95} +2.39484e8 q^{96} -2.57864e8 q^{97} -7.62076e8 q^{98} +1.40551e9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 326 q^{3} + 5118 q^{4} + 129526 q^{9} + 88390 q^{10} + 427652 q^{12} + 473556 q^{14} + 1189618 q^{16} - 99312 q^{17} - 5073532 q^{22} + 6252378 q^{23} + 1529274 q^{25} + 18052718 q^{27} + 5424828 q^{29}+ \cdots + 9251202540 q^{95}+O(q^{100}) $ 20 * q + 326 * q^3 + 5118 * q^4 + 129526 * q^9 + 88390 * q^10 + 427652 * q^12 + 473556 * q^14 + 1189618 * q^16 - 99312 * q^17 - 5073532 * q^22 + 6252378 * q^23 + 1529274 * q^25 + 18052718 * q^27 + 5424828 * q^29 + 30045516 * q^30 - 15098160 * q^35 + 54078886 * q^36 + 39021096 * q^38 + 134360674 * q^40 + 121600068 * q^42 + 53242058 * q^43 + 82771076 * q^48 + 14055298 * q^49 + 10208934 * q^51 - 36429786 * q^53 - 131454160 * q^55 + 127272672 * q^56 + 536425792 * q^61 - 890759796 * q^62 - 343337066 * q^64 - 445758060 * q^66 + 336594090 * q^68 + 1083885582 * q^69 + 1083869262 * q^74 - 1218826882 * q^75 + 1470187374 * q^77 + 1556703616 * q^79 + 2139127516 * q^81 + 3719322754 * q^82 + 1288246146 * q^87 - 4109160928 * q^88 + 6489878934 * q^90 + 10148843820 * q^92 + 4894712828 * q^94 + 9251202540 * q^95

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 $

$2$	27.6771	1.22317	0.611584	−	0.791179i	$-0.290533\pi$
$2$	27.6771	1.22317	0.611584	+	0.791179i	$0.290533\pi$
$3$	−44.2389	−0.315325	−0.157663	−	0.987493i	$-0.550396\pi$
$3$	−44.2389	−0.315325	−0.157663	+	0.987493i	$0.550396\pi$
$4$	254.024	0.496141
$5$	−2467.22	−1.76540	−0.882698	−	0.469941i	$-0.844275\pi$
$5$	−2467.22	−1.76540	−0.882698	+	0.469941i	$0.844275\pi$
$6$	−1224.41	−0.385696
$7$	3580.38	0.563622	0.281811	−	0.959470i	$-0.409065\pi$
$7$	3580.38	0.563622	0.281811	+	0.959470i	$0.409065\pi$
$8$	−7140.04	−0.616305
$9$	−17725.9	−0.900570
$10$	−68285.5	−2.15938
$11$	−79291.0	−1.63289	−0.816445	−	0.577424i	$-0.804058\pi$
$11$	−79291.0	−1.63289	−0.816445	+	0.577424i	$0.804058\pi$
$12$	−11237.7	−0.156446
$13$	0	0
$13$	0	0
$14$	99094.7	0.689405
$15$	109147.	0.556674
$16$	−327676.	−1.24999
$17$	8421.95	0.0244564	0.0122282	−	0.999925i	$-0.496108\pi$
$17$	8421.95	0.0244564	0.0122282	+	0.999925i	$0.496108\pi$
$18$	−490603.	−1.10155
$19$	−434724.	−0.765284	−0.382642	−	0.923897i	$-0.624986\pi$
$19$	−434724.	−0.765284	−0.382642	+	0.923897i	$0.624986\pi$
$20$	−626732.	−0.875885
$21$	−158392.	−0.177724
$22$	−2.19455e6	−1.99730
$23$	1.39043e6	1.03603	0.518016	−	0.855371i	$-0.326671\pi$
$23$	1.39043e6	1.03603	0.518016	+	0.855371i	$0.326671\pi$
$24$	315867.	0.194336
$25$	4.13403e6	2.11662
$26$	0	0
$27$	1.65493e6	0.599297
$28$	909503.	0.279636
$29$	2.73548e6	0.718195	0.359098	−	0.933300i	$-0.383084\pi$
$29$	2.73548e6	0.718195	0.359098	+	0.933300i	$0.383084\pi$
$30$	3.02087e6	0.680905
$31$	−1.77534e6	−0.345266	−0.172633	−	0.984986i	$-0.555227\pi$
$31$	−1.77534e6	−0.345266	−0.172633	+	0.984986i	$0.555227\pi$
$32$	−5.41344e6	−0.912637
$33$	3.50775e6	0.514891
$34$	233095.	0.0299143
$35$	−8.83357e6	−0.995016
$36$	−4.50281e6	−0.446809
$37$	−1.57016e7	−1.37732	−0.688661	−	0.725084i	$-0.741801\pi$
$37$	−1.57016e7	−1.37732	−0.688661	+	0.725084i	$0.741801\pi$
$38$	−1.20319e7	−0.936071
$39$	0	0
$40$	1.76160e7	1.08802
$41$	112436.	0.00621411	0.00310706	−	0.999995i	$-0.499011\pi$
$41$	112436.	0.00621411	0.00310706	+	0.999995i	$0.499011\pi$
$42$	−4.38384e6	−0.217387
$43$	−2.68150e7	−1.19611	−0.598053	−	0.801457i	$-0.704059\pi$
$43$	−2.68150e7	−1.19611	−0.598053	+	0.801457i	$0.704059\pi$
$44$	−2.01418e7	−0.810143
$45$	4.37337e7	1.58986
$46$	3.84831e7	1.26724
$47$	3.59357e7	1.07420	0.537101	−	0.843518i	$-0.319519\pi$
$47$	3.59357e7	1.07420	0.537101	+	0.843518i	$0.319519\pi$
$48$	1.44960e7	0.394152
$49$	−2.75345e7	−0.682330
$50$	1.14418e8	2.58898
$51$	−372578.	−0.00771172
$52$	0	0
$53$	−3.83324e7	−0.667305	−0.333652	−	0.942696i	$-0.608281\pi$
$53$	−3.83324e7	−0.667305	−0.333652	+	0.942696i	$0.608281\pi$
$54$	4.58037e7	0.733042
$55$	1.95628e8	2.88270
$56$	−2.55641e7	−0.347363
$57$	1.92317e7	0.241313
$58$	7.57103e7	0.878474
$59$	−646636.	−0.00694745	−0.00347373	−	0.999994i	$-0.501106\pi$
$59$	−646636.	−0.00694745	−0.00347373	+	0.999994i	$0.501106\pi$
$60$	2.77259e7	0.276188
$61$	4.42458e7	0.409155	0.204577	−	0.978850i	$-0.434418\pi$
$61$	4.42458e7	0.409155	0.204577	+	0.978850i	$0.434418\pi$
$62$	−4.91363e7	−0.422318
$63$	−6.34656e7	−0.507581
$64$	1.79417e7	0.133676
$65$	0	0
$66$	9.70844e7	0.629798
$67$	1.65124e8	1.00109	0.500545	−	0.865711i	$-0.333133\pi$
$67$	1.65124e8	1.00109	0.500545	+	0.865711i	$0.333133\pi$
$68$	2.13938e6	0.0121338
$69$	−6.15110e7	−0.326687
$70$	−2.44488e8	−1.21707
$71$	1.69848e8	0.793228	0.396614	−	0.917985i	$-0.370185\pi$
$71$	1.69848e8	0.793228	0.396614	+	0.917985i	$0.370185\pi$
$72$	1.26564e8	0.555026
$73$	−8.20084e7	−0.337991	−0.168996	−	0.985617i	$-0.554052\pi$
$73$	−8.20084e7	−0.337991	−0.168996	+	0.985617i	$0.554052\pi$
$74$	−4.34574e8	−1.68470
$75$	−1.82885e8	−0.667424
$76$	−1.10430e8	−0.379688
$77$	−2.83892e8	−0.920333
$78$	0	0
$79$	5.10327e8	1.47410	0.737050	−	0.675839i	$-0.236218\pi$
$79$	5.10327e8	1.47410	0.737050	+	0.675839i	$0.236218\pi$
$80$	8.08448e8	2.20672
$81$	2.75687e8	0.711597
$82$	3.11191e6	0.00760090
$83$	−4.20094e8	−0.971618	−0.485809	−	0.874065i	$-0.661475\pi$
$83$	−4.20094e8	−0.971618	−0.485809	+	0.874065i	$0.661475\pi$
$84$	−4.02354e7	−0.0881762
$85$	−2.07788e7	−0.0431752
$86$	−7.42162e8	−1.46304
$87$	−1.21015e8	−0.226465
$88$	5.66141e8	1.00636
$89$	3.97908e8	0.672245	0.336123	−	0.941818i	$-0.390884\pi$
$89$	3.97908e8	0.672245	0.336123	+	0.941818i	$0.390884\pi$
$90$	1.21042e9	1.94467
$91$	0	0
$92$	3.53202e8	0.514018
$93$	7.85390e7	0.108871
$94$	9.94598e8	1.31393
$95$	1.07256e9	1.35103
$96$	2.39484e8	0.287777
$97$	−2.57864e8	−0.295746	−0.147873	−	0.989006i	$-0.547243\pi$
$97$	−2.57864e8	−0.295746	−0.147873	+	0.989006i	$0.547243\pi$
$98$	−7.62076e8	−0.834604
$99$	1.40551e9	1.47053
$100$	1.05014e9	1.05014
$101$	−1.65112e9	−1.57882	−0.789409	−	0.613868i	$-0.789613\pi$
$101$	−1.65112e9	−1.57882	−0.789409	+	0.613868i	$0.789613\pi$
$102$	−1.03119e7	−0.00943273
$103$	−9.79825e8	−0.857790	−0.428895	−	0.903354i	$-0.641097\pi$
$103$	−9.79825e8	−0.857790	−0.428895	+	0.903354i	$0.641097\pi$
$104$	0	0
$105$	3.90787e8	0.313754
$106$	−1.06093e9	−0.816226
$107$	−6.09542e8	−0.449549	−0.224774	−	0.974411i	$-0.572165\pi$
$107$	−6.09542e8	−0.449549	−0.224774	+	0.974411i	$0.572165\pi$
$108$	4.20392e8	0.297336
$109$	−5.84277e8	−0.396460	−0.198230	−	0.980156i	$-0.563519\pi$
$109$	−5.84277e8	−0.396460	−0.198230	+	0.980156i	$0.563519\pi$
$110$	5.41442e9	3.52602
$111$	6.94620e8	0.434304
$112$	−1.17321e9	−0.704519
$113$	3.38612e9	1.95366	0.976831	−	0.214013i	$-0.0686533\pi$
$113$	3.38612e9	1.95366	0.976831	+	0.214013i	$0.0686533\pi$
$114$	5.32279e8	0.295167
$115$	−3.43049e9	−1.82901
$116$	6.94878e8	0.356326
$117$	0	0
$118$	−1.78970e7	−0.00849790
$119$	3.01538e7	0.0137842
$120$	−7.79313e8	−0.343081
$121$	3.92911e9	1.66633
$122$	1.22460e9	0.500465
$123$	−4.97405e6	−0.00195946
$124$	−4.50979e8	−0.171300
$125$	−5.38075e9	−1.97128
$126$	−1.75654e9	−0.620857
$127$	−1.23693e9	−0.421920	−0.210960	−	0.977495i	$-0.567659\pi$
$127$	−1.23693e9	−0.421920	−0.210960	+	0.977495i	$0.567659\pi$
$128$	3.26825e9	1.07615
$129$	1.18627e9	0.377162
$130$	0	0
$131$	−3.86097e9	−1.14545	−0.572724	−	0.819748i	$-0.694114\pi$
$131$	−3.86097e9	−1.14545	−0.572724	+	0.819748i	$0.694114\pi$
$132$	8.91052e8	0.255458
$133$	−1.55648e9	−0.431331
$134$	4.57015e9	1.22450
$135$	−4.08307e9	−1.05800
$136$	−6.01330e7	−0.0150726
$137$	−5.58699e9	−1.35499	−0.677494	−	0.735529i	$-0.736934\pi$
$137$	−5.58699e9	−1.35499	−0.677494	+	0.735529i	$0.736934\pi$
$138$	−1.70245e9	−0.399593
$139$	3.62704e9	0.824111	0.412055	−	0.911159i	$-0.364811\pi$
$139$	3.62704e9	0.824111	0.412055	+	0.911159i	$0.364811\pi$
$140$	−2.24394e9	−0.493668
$141$	−1.58976e9	−0.338723
$142$	4.70091e9	0.970252
$143$	0	0
$144$	5.80836e9	1.12570
$145$	−6.74902e9	−1.26790
$146$	−2.26976e9	−0.413420
$147$	1.21809e9	0.215156
$148$	−3.98858e9	−0.683345
$149$	4.54812e8	0.0755951	0.0377975	−	0.999285i	$-0.487966\pi$
$149$	4.54812e8	0.0755951	0.0377975	+	0.999285i	$0.487966\pi$
$150$	−5.06173e9	−0.816372
$151$	−8.49413e9	−1.32960	−0.664802	−	0.747019i	$-0.731484\pi$
$151$	−8.49413e9	−1.32960	−0.664802	+	0.747019i	$0.731484\pi$
$152$	3.10395e9	0.471648
$153$	−1.49287e8	−0.0220247
$154$	−7.85732e9	−1.12572
$155$	4.38014e9	0.609531
$156$	0	0
$157$	8.94981e9	1.17562	0.587808	−	0.809000i	$-0.299991\pi$
$157$	8.94981e9	1.17562	0.587808	+	0.809000i	$0.299991\pi$
$158$	1.41244e10	1.80307
$159$	1.69578e9	0.210418
$160$	1.33561e10	1.61117
$161$	4.97826e9	0.583931
$162$	7.63023e9	0.870402
$163$	−3.72319e9	−0.413115	−0.206557	−	0.978435i	$-0.566226\pi$
$163$	−3.72319e9	−0.413115	−0.206557	+	0.978435i	$0.566226\pi$
$164$	2.85615e7	0.00308307
$165$	−8.65436e9	−0.908986
$166$	−1.16270e10	−1.18845
$167$	−3.39705e9	−0.337970	−0.168985	−	0.985619i	$-0.554049\pi$
$167$	−3.39705e9	−0.337970	−0.168985	+	0.985619i	$0.554049\pi$
$168$	1.13093e9	0.109532
$169$	0	0
$170$	−5.75097e8	−0.0528106
$171$	7.70588e9	0.689192
$172$	−6.81165e9	−0.593437
$173$	−9.36068e9	−0.794511	−0.397255	−	0.917708i	$-0.630037\pi$
$173$	−9.36068e9	−0.794511	−0.397255	+	0.917708i	$0.630037\pi$
$174$	−3.34934e9	−0.277005
$175$	1.48014e10	1.19297
$176$	2.59818e10	2.04109
$177$	2.86064e7	0.00219071
$178$	1.10130e10	0.822269
$179$	1.31097e10	0.954450	0.477225	−	0.878781i	$-0.341643\pi$
$179$	1.31097e10	0.954450	0.477225	+	0.878781i	$0.341643\pi$
$180$	1.11094e10	0.788795
$181$	−1.03171e10	−0.714506	−0.357253	−	0.934008i	$-0.616287\pi$
$181$	−1.03171e10	−0.714506	−0.357253	+	0.934008i	$0.616287\pi$
$182$	0	0
$183$	−1.95738e9	−0.129017
$184$	−9.92771e9	−0.638512
$185$	3.87391e10	2.43152
$186$	2.17374e9	0.133168
$187$	−6.67785e8	−0.0399346
$188$	9.12854e9	0.532955
$189$	5.92528e9	0.337777
$190$	2.96853e10	1.65254
$191$	2.12665e9	0.115624	0.0578118	−	0.998327i	$-0.481588\pi$
$191$	2.12665e9	0.115624	0.0578118	+	0.998327i	$0.481588\pi$
$192$	−7.93720e8	−0.0421514
$193$	2.86528e10	1.48648	0.743241	−	0.669024i	$-0.233288\pi$
$193$	2.86528e10	1.48648	0.743241	+	0.669024i	$0.233288\pi$
$194$	−7.13695e9	−0.361747
$195$	0	0
$196$	−6.99442e9	−0.338532
$197$	1.59594e10	0.754951	0.377476	−	0.926019i	$-0.376792\pi$
$197$	1.59594e10	0.754951	0.377476	+	0.926019i	$0.376792\pi$
$198$	3.89004e10	1.79871
$199$	−1.04597e10	−0.472802	−0.236401	−	0.971656i	$-0.575968\pi$
$199$	−1.04597e10	−0.472802	−0.236401	+	0.971656i	$0.575968\pi$
$200$	−2.95171e10	−1.30448
$201$	−7.30489e9	−0.315669
$202$	−4.56982e10	−1.93116
$203$	9.79406e9	0.404791
$204$	−9.46437e7	−0.00382610
$205$	−2.77404e8	−0.0109704
$206$	−2.71187e10	−1.04922
$207$	−2.46466e10	−0.933020
$208$	0	0
$209$	3.44697e10	1.24962
$210$	1.08159e10	0.383773
$211$	−2.31000e10	−0.802307	−0.401154	−	0.916011i	$-0.631391\pi$
$211$	−2.31000e10	−0.802307	−0.401154	+	0.916011i	$0.631391\pi$
$212$	−9.73734e9	−0.331077
$213$	−7.51389e9	−0.250125
$214$	−1.68704e10	−0.549874
$215$	6.61584e10	2.11160
$216$	−1.18163e10	−0.369350
$217$	−6.35639e9	−0.194600
$218$	−1.61711e10	−0.484937
$219$	3.62796e9	0.106577
$220$	4.96942e10	1.43022
$221$	0	0
$222$	1.92251e10	0.531227
$223$	1.27352e9	0.0344853	0.0172426	−	0.999851i	$-0.494511\pi$
$223$	1.27352e9	0.0344853	0.0172426	+	0.999851i	$0.494511\pi$
$224$	−1.93822e10	−0.514383
$225$	−7.32794e10	−1.90617
$226$	9.37181e10	2.38966
$227$	−1.78201e10	−0.445445	−0.222722	−	0.974882i	$-0.571494\pi$
$227$	−1.78201e10	−0.445445	−0.222722	+	0.974882i	$0.571494\pi$
$228$	4.88531e9	0.119725
$229$	−6.06264e10	−1.45681	−0.728403	−	0.685149i	$-0.759737\pi$
$229$	−6.06264e10	−1.45681	−0.728403	+	0.685149i	$0.759737\pi$
$230$	−9.49461e10	−2.23718
$231$	1.25591e10	0.290204
$232$	−1.95314e10	−0.442627
$233$	−1.65760e10	−0.368449	−0.184224	−	0.982884i	$-0.558977\pi$
$233$	−1.65760e10	−0.368449	−0.184224	+	0.982884i	$0.558977\pi$
$234$	0	0
$235$	−8.86612e10	−1.89639
$236$	−1.64261e8	−0.00344691
$237$	−2.25763e10	−0.464820
$238$	8.34571e8	0.0168604
$239$	−5.26394e10	−1.04357	−0.521784	−	0.853078i	$-0.674733\pi$
$239$	−5.26394e10	−1.04357	−0.521784	+	0.853078i	$0.674733\pi$
$240$	−3.57648e10	−0.695834
$241$	−2.09645e10	−0.400321	−0.200161	−	0.979763i	$-0.564146\pi$
$241$	−2.09645e10	−0.400321	−0.200161	+	0.979763i	$0.564146\pi$
$242$	1.08747e11	2.03820
$243$	−4.47701e10	−0.823682
$244$	1.12395e10	0.202998
$245$	6.79335e10	1.20458
$246$	−1.37668e8	−0.00239676
$247$	0	0
$248$	1.26760e10	0.212789
$249$	1.85845e10	0.306376
$250$	−1.48924e11	−2.41121
$251$	3.94060e10	0.626658	0.313329	−	0.949645i	$-0.398556\pi$
$251$	3.94060e10	0.626658	0.313329	+	0.949645i	$0.398556\pi$
$252$	−1.61218e10	−0.251832
$253$	−1.10248e11	−1.69173
$254$	−3.42348e10	−0.516079
$255$	9.19229e8	0.0136142
$256$	8.12698e10	1.18263
$257$	1.03941e11	1.48624	0.743120	−	0.669159i	$-0.233345\pi$
$257$	1.03941e11	1.48624	0.743120	+	0.669159i	$0.233345\pi$
$258$	3.28324e10	0.461333
$259$	−5.62176e10	−0.776289
$260$	0	0
$261$	−4.84889e10	−0.646785
$262$	−1.06861e11	−1.40108
$263$	1.03626e11	1.33557	0.667787	−	0.744352i	$-0.267242\pi$
$263$	1.03626e11	1.33557	0.667787	+	0.744352i	$0.267242\pi$
$264$	−2.50454e10	−0.317330
$265$	9.45742e10	1.17806
$266$	−4.30788e10	−0.527590
$267$	−1.76030e10	−0.211976
$268$	4.19454e10	0.496681
$269$	1.33651e11	1.55627	0.778136	−	0.628096i	$-0.216165\pi$
$269$	1.33651e11	1.55627	0.778136	+	0.628096i	$0.216165\pi$
$270$	−1.13008e11	−1.29411
$271$	−1.18011e11	−1.32911	−0.664557	−	0.747238i	$-0.731380\pi$
$271$	−1.18011e11	−1.32911	−0.664557	+	0.747238i	$0.731380\pi$
$272$	−2.75967e9	−0.0305701
$273$	0	0
$274$	−1.54632e11	−1.65738
$275$	−3.27791e11	−3.45621
$276$	−1.56253e10	−0.162083
$277$	−2.22522e10	−0.227099	−0.113549	−	0.993532i	$-0.536222\pi$
$277$	−2.22522e10	−0.227099	−0.113549	+	0.993532i	$0.536222\pi$
$278$	1.00386e11	1.00803
$279$	3.14695e10	0.310936
$280$	6.30720e10	0.613233
$281$	−1.75160e10	−0.167593	−0.0837966	−	0.996483i	$-0.526705\pi$
$281$	−1.75160e10	−0.167593	−0.0837966	+	0.996483i	$0.526705\pi$
$282$	−4.39999e10	−0.414315
$283$	9.41554e10	0.872582	0.436291	−	0.899806i	$-0.356292\pi$
$283$	9.41554e10	0.872582	0.436291	+	0.899806i	$0.356292\pi$
$284$	4.31455e10	0.393553
$285$	−4.74488e10	−0.426013
$286$	0	0
$287$	4.02565e8	0.00350241
$288$	9.59582e10	0.821894
$289$	−1.18517e11	−0.999402
$290$	−1.86794e11	−1.55085
$291$	1.14076e10	0.0932561
$292$	−2.08321e10	−0.167691
$293$	4.60347e10	0.364906	0.182453	−	0.983215i	$-0.441596\pi$
$293$	4.60347e10	0.364906	0.182453	+	0.983215i	$0.441596\pi$
$294$	3.37134e10	0.263172
$295$	1.59539e9	0.0122650
$296$	1.12110e11	0.848850
$297$	−1.31221e11	−0.978586
$298$	1.25879e10	0.0924655
$299$	0	0
$300$	−4.64571e10	−0.331136
$301$	−9.60079e10	−0.674152
$302$	−2.35093e11	−1.62633
$303$	7.30436e10	0.497841
$304$	1.42449e11	0.956593
$305$	−1.09164e11	−0.722320
$306$	−4.13183e9	−0.0269399
$307$	−8.99981e9	−0.0578243	−0.0289122	−	0.999582i	$-0.509204\pi$
$307$	−8.99981e9	−0.0578243	−0.0289122	+	0.999582i	$0.509204\pi$
$308$	−7.21154e10	−0.456615
$309$	4.33463e10	0.270483
$310$	1.21230e11	0.745559
$311$	7.38621e9	0.0447713	0.0223857	−	0.999749i	$-0.492874\pi$
$311$	7.38621e9	0.0447713	0.0223857	+	0.999749i	$0.492874\pi$
$312$	0	0
$313$	2.33361e11	1.37429	0.687145	−	0.726521i	$-0.258864\pi$
$313$	2.33361e11	1.37429	0.687145	+	0.726521i	$0.258864\pi$
$314$	2.47705e11	1.43798
$315$	1.56583e11	0.896082
$316$	1.29635e11	0.731361
$317$	−1.01774e10	−0.0566070	−0.0283035	−	0.999599i	$-0.509010\pi$
$317$	−1.01774e10	−0.0566070	−0.0283035	+	0.999599i	$0.509010\pi$
$318$	4.69344e10	0.257376
$319$	−2.16899e11	−1.17273
$320$	−4.42660e10	−0.235991
$321$	2.69655e10	0.141754
$322$	1.37784e11	0.714246
$323$	−3.66122e9	−0.0187161
$324$	7.00311e10	0.353052
$325$	0	0
$326$	−1.03047e11	−0.505309
$327$	2.58478e10	0.125014
$328$	−8.02799e8	−0.00382979
$329$	1.28664e11	0.605444
$330$	−2.39528e11	−1.11184
$331$	−3.78785e11	−1.73447	−0.867236	−	0.497897i	$-0.834106\pi$
$331$	−3.78785e11	−1.73447	−0.867236	+	0.497897i	$0.834106\pi$
$332$	−1.06714e11	−0.482059
$333$	2.78325e11	1.24037
$334$	−9.40206e10	−0.413394
$335$	−4.07396e11	−1.76732
$336$	5.19013e10	0.222153
$337$	−1.27459e11	−0.538314	−0.269157	−	0.963096i	$-0.586745\pi$
$337$	−1.27459e11	−0.538314	−0.269157	+	0.963096i	$0.586745\pi$
$338$	0	0
$339$	−1.49798e11	−0.616039
$340$	−5.27831e9	−0.0214210
$341$	1.40768e11	0.563781
$342$	2.13277e11	0.842997
$343$	−2.43065e11	−0.948199
$344$	1.91460e11	0.737166
$345$	1.51761e11	0.576732
$346$	−2.59077e11	−0.971820
$347$	−4.97009e10	−0.184027	−0.0920136	−	0.995758i	$-0.529330\pi$
$347$	−4.97009e10	−0.184027	−0.0920136	+	0.995758i	$0.529330\pi$
$348$	−3.07406e10	−0.112359
$349$	6.73686e10	0.243077	0.121538	−	0.992587i	$-0.461217\pi$
$349$	6.73686e10	0.243077	0.121538	+	0.992587i	$0.461217\pi$
$350$	4.09660e11	1.45921
$351$	0	0
$352$	4.29237e11	1.49024
$353$	5.64977e11	1.93662	0.968311	−	0.249749i	$-0.0803480\pi$
$353$	5.64977e11	1.93662	0.968311	+	0.249749i	$0.0803480\pi$
$354$	7.91744e8	0.00267960
$355$	−4.19052e11	−1.40036
$356$	1.01078e11	0.333528
$357$	−1.33397e9	−0.00434649
$358$	3.62838e11	1.16745
$359$	−5.75596e11	−1.82891	−0.914456	−	0.404686i	$-0.867381\pi$
$359$	−5.75596e11	−1.82891	−0.914456	+	0.404686i	$0.867381\pi$
$360$	−3.12260e11	−0.979840
$361$	−1.33703e11	−0.414341
$362$	−2.85549e11	−0.873961
$363$	−1.73820e11	−0.525435
$364$	0	0
$365$	2.02332e11	0.596688
$366$	−5.41748e10	−0.157809
$367$	6.01271e11	1.73011	0.865053	−	0.501680i	$-0.167285\pi$
$367$	6.01271e11	1.73011	0.865053	+	0.501680i	$0.167285\pi$
$368$	−4.55610e11	−1.29503
$369$	−1.99304e9	−0.00559624
$370$	1.07219e12	2.97415
$371$	−1.37244e11	−0.376108
$372$	1.99508e10	0.0540153
$373$	1.21541e11	0.325112	0.162556	−	0.986699i	$-0.448026\pi$
$373$	1.21541e11	0.325112	0.162556	+	0.986699i	$0.448026\pi$
$374$	−1.84824e10	−0.0488467
$375$	2.38039e11	0.621594
$376$	−2.56582e11	−0.662036
$377$	0	0
$378$	1.63995e11	0.413159
$379$	1.30959e11	0.326030	0.163015	−	0.986624i	$-0.447878\pi$
$379$	1.30959e11	0.326030	0.163015	+	0.986624i	$0.447878\pi$
$380$	2.72455e11	0.670300
$381$	5.47206e10	0.133042
$382$	5.88597e10	0.141427
$383$	6.95780e11	1.65226	0.826128	−	0.563483i	$-0.190539\pi$
$383$	6.95780e11	1.65226	0.826128	+	0.563483i	$0.190539\pi$
$384$	−1.44584e11	−0.339336
$385$	7.00423e11	1.62475
$386$	7.93029e11	1.81822
$387$	4.75320e11	1.07718
$388$	−6.55038e10	−0.146732
$389$	−1.49432e10	−0.0330880	−0.0165440	−	0.999863i	$-0.505266\pi$
$389$	−1.49432e10	−0.0330880	−0.0165440	+	0.999863i	$0.505266\pi$
$390$	0	0
$391$	1.17101e10	0.0253376
$392$	1.96597e11	0.420523
$393$	1.70805e11	0.361189
$394$	4.41711e11	0.923433
$395$	−1.25909e12	−2.60237
$396$	3.57032e11	0.729590
$397$	6.44574e11	1.30231	0.651157	−	0.758943i	$-0.274284\pi$
$397$	6.44574e11	1.30231	0.651157	+	0.758943i	$0.274284\pi$
$398$	−2.89494e11	−0.578316
$399$	6.88568e10	0.136009
$400$	−1.35462e12	−2.64575
$401$	8.71275e11	1.68270	0.841348	−	0.540494i	$-0.181763\pi$
$401$	8.71275e11	1.68270	0.841348	+	0.540494i	$0.181763\pi$
$402$	−2.02178e11	−0.386116
$403$	0	0
$404$	−4.19424e11	−0.783316
$405$	−6.80179e11	−1.25625
$406$	2.71072e11	0.495127
$407$	1.24499e12	2.24901
$408$	2.66022e9	0.00475277
$409$	−1.40219e11	−0.247772	−0.123886	−	0.992296i	$-0.539536\pi$
$409$	−1.40219e11	−0.247772	−0.123886	+	0.992296i	$0.539536\pi$
$410$	−7.67776e9	−0.0134186
$411$	2.47162e11	0.427261
$412$	−2.48899e11	−0.425584
$413$	−2.31520e9	−0.00391574
$414$	−6.82148e11	−1.14124
$415$	1.03646e12	1.71529
$416$	0	0
$417$	−1.60456e11	−0.259863
$418$	9.54023e11	1.52850
$419$	5.39944e11	0.855826	0.427913	−	0.903820i	$-0.359249\pi$
$419$	5.39944e11	0.855826	0.427913	+	0.903820i	$0.359249\pi$
$420$	9.92694e10	0.155666
$421$	8.59651e11	1.33368	0.666842	−	0.745199i	$-0.267646\pi$
$421$	8.59651e11	1.33368	0.666842	+	0.745199i	$0.267646\pi$
$422$	−6.39342e11	−0.981357
$423$	−6.36994e11	−0.967394
$424$	2.73694e11	0.411263
$425$	3.48166e10	0.0517649
$426$	−2.07963e11	−0.305945
$427$	1.58417e11	0.230609
$428$	−1.54838e11	−0.223039
$429$	0	0
$430$	1.83107e12	2.58284
$431$	−2.53427e11	−0.353758	−0.176879	−	0.984233i	$-0.556600\pi$
$431$	−2.53427e11	−0.353758	−0.176879	+	0.984233i	$0.556600\pi$
$432$	−5.42281e11	−0.749113
$433$	1.55645e11	0.212785	0.106392	−	0.994324i	$-0.466070\pi$
$433$	1.55645e11	0.212785	0.106392	+	0.994324i	$0.466070\pi$
$434$	−1.75927e11	−0.238028
$435$	2.98569e11	0.399800
$436$	−1.48420e11	−0.196700
$437$	−6.04453e11	−0.792859
$438$	1.00412e11	0.130362
$439$	8.15703e11	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$439$	8.15703e11	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$440$	−1.39679e12	−1.77662
$441$	4.88074e11	0.614486
$442$	0	0
$443$	−4.25413e11	−0.524800	−0.262400	−	0.964959i	$-0.584514\pi$
$443$	−4.25413e11	−0.524800	−0.262400	+	0.964959i	$0.584514\pi$
$444$	1.76450e11	0.215476
$445$	−9.81725e11	−1.18678
$446$	3.52474e10	0.0421813
$447$	−2.01204e10	−0.0238370
$448$	6.42381e10	0.0753427
$449$	−5.63347e11	−0.654135	−0.327068	−	0.945001i	$-0.606061\pi$
$449$	−5.63347e11	−0.654135	−0.327068	+	0.945001i	$0.606061\pi$
$450$	−2.02817e12	−2.33156
$451$	−8.91518e9	−0.0101470
$452$	8.60156e11	0.969291
$453$	3.75771e11	0.419258
$454$	−4.93209e11	−0.544854
$455$	0	0
$456$	−1.37315e11	−0.148722
$457$	8.80745e11	0.944556	0.472278	−	0.881450i	$-0.343432\pi$
$457$	8.80745e11	0.944556	0.472278	+	0.881450i	$0.343432\pi$
$458$	−1.67796e12	−1.78192
$459$	1.39377e10	0.0146567
$460$	−8.71426e11	−0.907445
$461$	5.49345e11	0.566488	0.283244	−	0.959048i	$-0.408589\pi$
$461$	5.49345e11	0.566488	0.283244	+	0.959048i	$0.408589\pi$
$462$	3.47599e11	0.354968
$463$	−4.06010e11	−0.410603	−0.205302	−	0.978699i	$-0.565818\pi$
$463$	−4.06010e11	−0.410603	−0.205302	+	0.978699i	$0.565818\pi$
$464$	−8.96352e11	−0.897734
$465$	−1.93773e11	−0.192200
$466$	−4.58775e11	−0.450675
$467$	1.33161e12	1.29554	0.647769	−	0.761837i	$-0.275702\pi$
$467$	1.33161e12	1.29554	0.647769	+	0.761837i	$0.275702\pi$
$468$	0	0
$469$	5.91206e11	0.564236
$470$	−2.45389e12	−2.31961
$471$	−3.95930e11	−0.370701
$472$	4.61700e9	0.00428175
$473$	2.12619e12	1.95311
$474$	−6.24847e11	−0.568554
$475$	−1.79716e12	−1.61982
$476$	7.65979e9	0.00683889
$477$	6.79476e11	0.600955
$478$	−1.45691e12	−1.27646
$479$	−9.28564e11	−0.805939	−0.402969	−	0.915214i	$-0.632022\pi$
$479$	−9.28564e11	−0.805939	−0.402969	+	0.915214i	$0.632022\pi$
$480$	−5.90860e11	−0.508041
$481$	0	0
$482$	−5.80238e11	−0.489660
$483$	−2.20233e11	−0.184128
$484$	9.98090e11	0.826733
$485$	6.36207e11	0.522108
$486$	−1.23911e12	−1.00750
$487$	−1.47674e12	−1.18966	−0.594829	−	0.803852i	$-0.702780\pi$
$487$	−1.47674e12	−1.18966	−0.594829	+	0.803852i	$0.702780\pi$
$488$	−3.15916e11	−0.252164
$489$	1.64710e11	0.130265
$490$	1.88020e12	1.47341
$491$	1.91948e12	1.49045	0.745225	−	0.666813i	$-0.232342\pi$
$491$	1.91948e12	1.49045	0.745225	+	0.666813i	$0.232342\pi$
$492$	−1.26353e9	−0.000972170 0
$493$	2.30381e10	0.0175645
$494$	0	0
$495$	−3.46769e12	−2.59607
$496$	5.81736e11	0.431577
$497$	6.08121e11	0.447081
$498$	5.14366e11	0.374749
$499$	9.40035e11	0.678721	0.339361	−	0.940656i	$-0.389789\pi$
$499$	9.40035e11	0.678721	0.339361	+	0.940656i	$0.389789\pi$
$500$	−1.36684e12	−0.978032
$501$	1.50282e11	0.106570
$502$	1.09065e12	0.766509
$503$	4.07654e11	0.283946	0.141973	−	0.989871i	$-0.454655\pi$
$503$	4.07654e11	0.283946	0.141973	+	0.989871i	$0.454655\pi$
$504$	4.53146e11	0.312825
$505$	4.07366e12	2.78724
$506$	−3.05136e12	−2.06927
$507$	0	0
$508$	−3.14211e11	−0.209332
$509$	−8.18527e11	−0.540509	−0.270255	−	0.962789i	$-0.587108\pi$
$509$	−8.18527e11	−0.540509	−0.270255	+	0.962789i	$0.587108\pi$
$510$	2.54416e10	0.0166525
$511$	−2.93621e11	−0.190499
$512$	5.75969e11	0.370411
$513$	−7.19437e11	−0.458632
$514$	2.87680e12	1.81792
$515$	2.41744e12	1.51434
$516$	3.01340e11	0.187125
$517$	−2.84938e12	−1.75405
$518$	−1.55594e12	−0.949532
$519$	4.14106e11	0.250529
$520$	0	0
$521$	−1.15230e11	−0.0685165	−0.0342582	−	0.999413i	$-0.510907\pi$
$521$	−1.15230e11	−0.0685165	−0.0342582	+	0.999413i	$0.510907\pi$
$522$	−1.34203e12	−0.791127
$523$	−7.61800e11	−0.445229	−0.222615	−	0.974907i	$-0.571459\pi$
$523$	−7.61800e11	−0.445229	−0.222615	+	0.974907i	$0.571459\pi$
$524$	−9.80779e11	−0.568304
$525$	−6.54797e11	−0.376175
$526$	2.86807e12	1.63363
$527$	−1.49518e10	−0.00844396
$528$	−1.14940e12	−0.643606
$529$	1.32139e11	0.0733638
$530$	2.61754e12	1.44096
$531$	1.14622e10	0.00625667
$532$	−3.95383e11	−0.214001
$533$	0	0
$534$	−4.87201e11	−0.259282
$535$	1.50387e12	0.793632
$536$	−1.17899e12	−0.616976
$537$	−5.79958e11	−0.300962
$538$	3.69907e12	1.90358
$539$	2.18324e12	1.11417
$540$	−1.03720e12	−0.524915
$541$	3.17517e12	1.59360	0.796800	−	0.604243i	$-0.206524\pi$
$541$	3.17517e12	1.59360	0.796800	+	0.604243i	$0.206524\pi$
$542$	−3.26622e12	−1.62573
$543$	4.56419e11	0.225302
$544$	−4.55917e10	−0.0223198
$545$	1.44154e12	0.699909
$546$	0	0
$547$	1.41950e12	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$547$	1.41950e12	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$548$	−1.41923e12	−0.672264
$549$	−7.84297e11	−0.368472
$550$	−9.07232e12	−4.22753
$551$	−1.18918e12	−0.549623
$552$	4.39191e11	0.201339
$553$	1.82717e12	0.830835
$554$	−6.15878e11	−0.277780
$555$	−1.71378e12	−0.766718
$556$	9.21355e11	0.408875
$557$	1.79642e11	0.0790786	0.0395393	−	0.999218i	$-0.487411\pi$
$557$	1.79642e11	0.0790786	0.0395393	+	0.999218i	$0.487411\pi$
$558$	8.70986e11	0.380327
$559$	0	0
$560$	2.89455e12	1.24376
$561$	2.95421e10	0.0125924
$562$	−4.84793e11	−0.204995
$563$	2.86519e12	1.20189	0.600946	−	0.799290i	$-0.294791\pi$
$563$	2.86519e12	1.20189	0.600946	+	0.799290i	$0.294791\pi$
$564$	−4.03836e11	−0.168054
$565$	−8.35429e12	−3.44899
$566$	2.60595e12	1.06732
$567$	9.87065e11	0.401072
$568$	−1.21272e12	−0.488870
$569$	−1.82502e12	−0.729897	−0.364948	−	0.931028i	$-0.618913\pi$
$569$	−1.82502e12	−0.729897	−0.364948	+	0.931028i	$0.618913\pi$
$570$	−1.31325e12	−0.521086
$571$	−2.25174e12	−0.886454	−0.443227	−	0.896409i	$-0.646166\pi$
$571$	−2.25174e12	−0.886454	−0.443227	+	0.896409i	$0.646166\pi$
$572$	0	0
$573$	−9.40808e10	−0.0364590
$574$	1.11418e10	0.00428404
$575$	5.74807e12	2.19289
$576$	−3.18033e11	−0.120385
$577$	−2.42445e12	−0.910587	−0.455293	−	0.890342i	$-0.650466\pi$
$577$	−2.42445e12	−0.910587	−0.455293	+	0.890342i	$0.650466\pi$
$578$	−3.28021e12	−1.22244
$579$	−1.26757e12	−0.468725
$580$	−1.71441e12	−0.629056
$581$	−1.50410e12	−0.547625
$582$	3.15731e11	0.114068
$583$	3.03941e12	1.08963
$584$	5.85543e11	0.208306
$585$	0	0
$586$	1.27411e12	0.446341
$587$	3.06448e12	1.06533	0.532666	−	0.846325i	$-0.321190\pi$
$587$	3.06448e12	1.06533	0.532666	+	0.846325i	$0.321190\pi$
$588$	3.09425e11	0.106748
$589$	7.71782e11	0.264226
$590$	4.41558e10	0.0150022
$591$	−7.06027e11	−0.238055
$592$	5.14503e12	1.72163
$593$	4.24429e12	1.40948	0.704740	−	0.709466i	$-0.251064\pi$
$593$	4.24429e12	1.40948	0.704740	+	0.709466i	$0.251064\pi$
$594$	−3.63182e12	−1.19698
$595$	−7.43959e10	−0.0243345
$596$	1.15533e11	0.0375058
$597$	4.62724e11	0.149086
$598$	0	0
$599$	−5.54247e12	−1.75907	−0.879535	−	0.475835i	$-0.842146\pi$
$599$	−5.54247e12	−1.75907	−0.879535	+	0.475835i	$0.842146\pi$
$600$	1.30580e12	0.411337
$601$	1.14070e12	0.356646	0.178323	−	0.983972i	$-0.442933\pi$
$601$	1.14070e12	0.356646	0.178323	+	0.983972i	$0.442933\pi$
$602$	−2.65722e12	−0.824601
$603$	−2.92697e12	−0.901551
$604$	−2.15771e12	−0.659671
$605$	−9.69397e12	−2.94173
$606$	2.02164e12	0.608943
$607$	−1.62034e11	−0.0484459	−0.0242229	−	0.999707i	$-0.507711\pi$
$607$	−1.62034e11	−0.0484459	−0.0242229	+	0.999707i	$0.507711\pi$
$608$	2.35335e12	0.698426
$609$	−4.33278e11	−0.127641
$610$	−3.02134e12	−0.883519
$611$	0	0
$612$	−3.79224e10	−0.0109273
$613$	−4.29927e12	−1.22977	−0.614883	−	0.788619i	$-0.710797\pi$
$613$	−4.29927e12	−1.22977	−0.614883	+	0.788619i	$0.710797\pi$
$614$	−2.49089e11	−0.0707289
$615$	1.22721e10	0.00345923
$616$	2.02700e12	0.567205
$617$	2.36564e12	0.657152	0.328576	−	0.944477i	$-0.393431\pi$
$617$	2.36564e12	0.657152	0.328576	+	0.944477i	$0.393431\pi$
$618$	1.19970e12	0.330846
$619$	−5.26560e11	−0.144158	−0.0720792	−	0.997399i	$-0.522963\pi$
$619$	−5.26560e11	−0.144158	−0.0720792	+	0.997399i	$0.522963\pi$
$620$	1.11266e12	0.302413
$621$	2.30106e12	0.620892
$622$	2.04429e11	0.0547629
$623$	1.42466e12	0.378892
$624$	0	0
$625$	5.20121e12	1.36347
$626$	6.45876e12	1.68099
$627$	−1.52490e12	−0.394038
$628$	2.27347e12	0.583271
$629$	−1.32238e11	−0.0336843
$630$	4.33377e12	1.09606
$631$	6.16163e12	1.54726	0.773630	−	0.633638i	$-0.218439\pi$
$631$	6.16163e12	1.54726	0.773630	+	0.633638i	$0.218439\pi$
$632$	−3.64375e12	−0.908494
$633$	1.02192e12	0.252988
$634$	−2.81681e11	−0.0692399
$635$	3.05178e12	0.744856
$636$	4.30769e11	0.104397
$637$	0	0
$638$	−6.00314e12	−1.43445
$639$	−3.01071e12	−0.714358
$640$	−8.06349e12	−1.89982
$641$	−4.86391e12	−1.13795	−0.568977	−	0.822354i	$-0.692661\pi$
$641$	−4.86391e12	−1.13795	−0.568977	+	0.822354i	$0.692661\pi$
$642$	7.46327e11	0.173389
$643$	5.33732e12	1.23133	0.615664	−	0.788009i	$-0.288888\pi$
$643$	5.33732e12	1.23133	0.615664	+	0.788009i	$0.288888\pi$
$644$	1.26460e12	0.289712
$645$	−2.92677e12	−0.665840
$646$	−1.01332e11	−0.0228929
$647$	−2.91982e12	−0.655069	−0.327535	−	0.944839i	$-0.606218\pi$
$647$	−2.91982e12	−0.655069	−0.327535	+	0.944839i	$0.606218\pi$
$648$	−1.96842e12	−0.438560
$649$	5.12724e10	0.0113444
$650$	0	0
$651$	2.81200e11	0.0613621
$652$	−9.45779e11	−0.204963
$653$	1.57692e12	0.339391	0.169696	−	0.985497i	$-0.445722\pi$
$653$	1.57692e12	0.339391	0.169696	+	0.985497i	$0.445722\pi$
$654$	7.15392e11	0.152913
$655$	9.52585e12	2.02217
$656$	−3.68427e10	−0.00776755
$657$	1.45367e12	0.304385
$658$	3.56104e12	0.740560
$659$	8.06220e12	1.66521	0.832606	−	0.553866i	$-0.186848\pi$
$659$	8.06220e12	1.66521	0.832606	+	0.553866i	$0.186848\pi$
$660$	−2.19842e12	−0.450985
$661$	−2.48474e12	−0.506262	−0.253131	−	0.967432i	$-0.581460\pi$
$661$	−2.48474e12	−0.506262	−0.253131	+	0.967432i	$0.581460\pi$
$662$	−1.04837e13	−2.12155
$663$	0	0
$664$	2.99949e12	0.598813
$665$	3.84017e12	0.761470
$666$	7.70323e12	1.51719
$667$	3.80349e12	0.744074
$668$	−8.62932e11	−0.167680
$669$	−5.63391e10	−0.0108741
$670$	−1.12756e13	−2.16173
$671$	−3.50829e12	−0.668104
$672$	8.57446e11	0.162198
$673$	3.04990e12	0.573083	0.286542	−	0.958068i	$-0.407494\pi$
$673$	3.04990e12	0.573083	0.286542	+	0.958068i	$0.407494\pi$
$674$	−3.52770e12	−0.658449
$675$	6.84152e12	1.26849
$676$	0	0
$677$	−9.40636e12	−1.72097	−0.860483	−	0.509479i	$-0.829838\pi$
$677$	−9.40636e12	−1.72097	−0.860483	+	0.509479i	$0.829838\pi$
$678$	−4.14599e12	−0.753519
$679$	−9.23253e11	−0.166689
$680$	1.48361e11	0.0266091
$681$	7.88341e11	0.140460
$682$	3.89607e12	0.689599
$683$	−1.50847e12	−0.265243	−0.132621	−	0.991167i	$-0.542339\pi$
$683$	−1.50847e12	−0.265243	−0.132621	+	0.991167i	$0.542339\pi$
$684$	1.95748e12	0.341936
$685$	1.37843e13	2.39209
$686$	−6.72735e12	−1.15981
$687$	2.68204e12	0.459368
$688$	8.78663e12	1.49511
$689$	0	0
$690$	4.20031e12	0.705440
$691$	−9.80911e12	−1.63674	−0.818368	−	0.574695i	$-0.805121\pi$
$691$	−9.80911e12	−1.63674	−0.818368	+	0.574695i	$0.805121\pi$
$692$	−2.37784e12	−0.394189
$693$	5.03225e12	0.828824
$694$	−1.37558e12	−0.225096
$695$	−8.94868e12	−1.45488
$696$	8.64049e11	0.139571
$697$	9.46932e8	0.000151975 0
$698$	1.86457e12	0.297324
$699$	7.33302e11	0.116181
$700$	3.75991e12	0.591883
$701$	−2.97437e11	−0.0465226	−0.0232613	−	0.999729i	$-0.507405\pi$
$701$	−2.97437e11	−0.0465226	−0.0232613	+	0.999729i	$0.507405\pi$
$702$	0	0
$703$	6.82585e12	1.05404
$704$	−1.42261e12	−0.218278
$705$	3.92227e12	0.597980
$706$	1.56370e13	2.36881
$707$	−5.91163e12	−0.889857
$708$	7.26672e9	0.00108690
$709$	−1.52039e12	−0.225969	−0.112984	−	0.993597i	$-0.536041\pi$
$709$	−1.52039e12	−0.225969	−0.112984	+	0.993597i	$0.536041\pi$
$710$	−1.15982e13	−1.71288
$711$	−9.04602e12	−1.32753
$712$	−2.84108e12	−0.414308
$713$	−2.46848e12	−0.357707
$714$	−3.69205e10	−0.00531649
$715$	0	0
$716$	3.33017e12	0.473542
$717$	2.32871e12	0.329063
$718$	−1.59309e13	−2.23707
$719$	4.71840e12	0.658438	0.329219	−	0.944254i	$-0.393215\pi$
$719$	4.71840e12	0.658438	0.329219	+	0.944254i	$0.393215\pi$
$720$	−1.43305e13	−1.98730
$721$	−3.50815e12	−0.483469
$722$	−3.70051e12	−0.506809
$723$	9.27448e11	0.126231
$724$	−2.62080e12	−0.354496
$725$	1.13086e13	1.52015
$726$	−4.81083e12	−0.642696
$727$	−4.23948e12	−0.562870	−0.281435	−	0.959580i	$-0.590810\pi$
$727$	−4.23948e12	−0.562870	−0.281435	+	0.959580i	$0.590810\pi$
$728$	0	0
$729$	−3.44577e12	−0.451869
$730$	5.59998e12	0.729850
$731$	−2.25834e11	−0.0292524
$732$	−4.97222e11	−0.0640105
$733$	2.32506e12	0.297485	0.148743	−	0.988876i	$-0.452477\pi$
$733$	2.32506e12	0.297485	0.148743	+	0.988876i	$0.452477\pi$
$734$	1.66415e13	2.11621
$735$	−3.00530e12	−0.379835
$736$	−7.52700e12	−0.945522
$737$	−1.30928e13	−1.63467
$738$	−5.51615e10	−0.00684515
$739$	−3.94677e12	−0.486790	−0.243395	−	0.969927i	$-0.578261\pi$
$739$	−3.94677e12	−0.486790	−0.243395	+	0.969927i	$0.578261\pi$
$740$	9.84067e12	1.20637
$741$	0	0
$742$	−3.79853e12	−0.460043
$743$	6.17403e12	0.743222	0.371611	−	0.928388i	$-0.378805\pi$
$743$	6.17403e12	0.743222	0.371611	+	0.928388i	$0.378805\pi$
$744$	−5.60771e11	−0.0670977
$745$	−1.12212e12	−0.133455
$746$	3.36391e12	0.397667
$747$	7.44656e12	0.875010
$748$	−1.69633e11	−0.0198132
$749$	−2.18239e12	−0.253376
$750$	6.58823e12	0.760314
$751$	−5.93809e12	−0.681188	−0.340594	−	0.940211i	$-0.610628\pi$
$751$	−5.93809e12	−0.681188	−0.340594	+	0.940211i	$0.610628\pi$
$752$	−1.17753e13	−1.34274
$753$	−1.74328e12	−0.197601
$754$	0	0
$755$	2.09568e13	2.34728
$756$	1.50516e12	0.167585
$757$	−5.64069e12	−0.624311	−0.312155	−	0.950031i	$-0.601051\pi$
$757$	−5.64069e12	−0.624311	−0.312155	+	0.950031i	$0.601051\pi$
$758$	3.62456e12	0.398790
$759$	4.87727e12	0.533444
$760$	−7.65810e12	−0.832645
$761$	3.67608e12	0.397332	0.198666	−	0.980067i	$-0.436339\pi$
$761$	3.67608e12	0.397332	0.198666	+	0.980067i	$0.436339\pi$
$762$	1.51451e12	0.162733
$763$	−2.09193e12	−0.223454
$764$	5.40221e11	0.0573656
$765$	3.68323e11	0.0388823
$766$	1.92572e13	2.02099
$767$	0	0
$768$	−3.59529e12	−0.372913
$769$	−8.01554e11	−0.0826541	−0.0413270	−	0.999146i	$-0.513159\pi$
$769$	−8.01554e11	−0.0826541	−0.0413270	+	0.999146i	$0.513159\pi$
$770$	1.93857e13	1.98734
$771$	−4.59824e12	−0.468649
$772$	7.27851e12	0.737504
$773$	9.66089e12	0.973216	0.486608	−	0.873620i	$-0.338234\pi$
$773$	9.66089e12	0.973216	0.486608	+	0.873620i	$0.338234\pi$
$774$	1.31555e13	1.31757
$775$	−7.33930e12	−0.730797
$776$	1.84116e12	0.182270
$777$	2.48700e12	0.244783
$778$	−4.13585e11	−0.0404722
$779$	−4.88787e10	−0.00475556
$780$	0	0
$781$	−1.34674e13	−1.29525
$782$	3.24103e11	0.0309922
$783$	4.52703e12	0.430413
$784$	9.02239e12	0.852902
$785$	−2.20811e13	−2.07543
$786$	4.72740e12	0.441795
$787$	1.24505e12	0.115691	0.0578456	−	0.998326i	$-0.481577\pi$
$787$	1.24505e12	0.115691	0.0578456	+	0.998326i	$0.481577\pi$
$788$	4.05408e12	0.374562
$789$	−4.58430e12	−0.421140
$790$	−3.48479e13	−3.18313
$791$	1.21236e13	1.10113
$792$	−1.00354e13	−0.906296
$793$	0	0
$794$	1.78400e13	1.59295
$795$	−4.18386e12	−0.371471
$796$	−2.65701e12	−0.234576
$797$	9.96620e12	0.874918	0.437459	−	0.899238i	$-0.355879\pi$
$797$	9.96620e12	0.874918	0.437459	+	0.899238i	$0.355879\pi$
$798$	1.90576e12	0.166362
$799$	3.02649e11	0.0262711
$800$	−2.23793e13	−1.93171
$801$	−7.05328e12	−0.605404
$802$	2.41144e13	2.05822
$803$	6.50253e12	0.551902
$804$	−1.85562e12	−0.156616
$805$	−1.22825e13	−1.03087
$806$	0	0
$807$	−5.91255e12	−0.490732
$808$	1.17890e13	0.973033
$809$	1.65385e13	1.35746	0.678732	−	0.734386i	$-0.262530\pi$
$809$	1.65385e13	1.35746	0.678732	+	0.734386i	$0.262530\pi$
$810$	−1.88254e13	−1.53660
$811$	−7.93499e12	−0.644099	−0.322049	−	0.946723i	$-0.604372\pi$
$811$	−7.93499e12	−0.644099	−0.322049	+	0.946723i	$0.604372\pi$
$812$	2.48793e12	0.200833
$813$	5.22069e12	0.419103
$814$	3.44578e13	2.75092
$815$	9.18590e12	0.729311
$816$	1.22085e11	0.00963953
$817$	1.16571e13	0.915360
$818$	−3.88087e12	−0.303067
$819$	0	0
$820$	−7.04674e10	−0.00544284
$821$	4.39935e12	0.337944	0.168972	−	0.985621i	$-0.445955\pi$
$821$	4.39935e12	0.337944	0.168972	+	0.985621i	$0.445955\pi$
$822$	6.84074e12	0.522613
$823$	−1.96888e13	−1.49596	−0.747979	−	0.663723i	$-0.768976\pi$
$823$	−1.96888e13	−1.49596	−0.747979	+	0.663723i	$0.768976\pi$
$824$	6.99598e12	0.528660
$825$	1.45011e13	1.08983
$826$	−6.40782e10	−0.00478961
$827$	3.94021e12	0.292917	0.146459	−	0.989217i	$-0.453213\pi$
$827$	3.94021e12	0.292917	0.146459	+	0.989217i	$0.453213\pi$
$828$	−6.26084e12	−0.462909
$829$	8.09989e12	0.595640	0.297820	−	0.954622i	$-0.403740\pi$
$829$	8.09989e12	0.595640	0.297820	+	0.954622i	$0.403740\pi$
$830$	2.86863e13	2.09809
$831$	9.84413e11	0.0716099
$832$	0	0
$833$	−2.31894e11	−0.0166873
$834$	−4.44097e12	−0.317856
$835$	8.38125e12	0.596650
$836$	8.75613e12	0.619989
$837$	−2.93806e12	−0.206917
$838$	1.49441e13	1.04682
$839$	1.62696e13	1.13357	0.566784	−	0.823866i	$-0.308187\pi$
$839$	1.62696e13	1.13357	0.566784	+	0.823866i	$0.308187\pi$
$840$	−2.79024e12	−0.193368
$841$	−7.02429e12	−0.484195
$842$	2.37927e13	1.63132
$843$	7.74888e11	0.0528463
$844$	−5.86795e12	−0.398057
$845$	0	0
$846$	−1.76302e13	−1.18329
$847$	1.40677e13	0.939180
$848$	1.25606e13	0.834121
$849$	−4.16533e12	−0.275147
$850$	9.63623e11	0.0633172
$851$	−2.18319e13	−1.42695
$852$	−1.90871e12	−0.124097
$853$	−2.11246e13	−1.36621	−0.683106	−	0.730320i	$-0.739371\pi$
$853$	−2.11246e13	−1.36621	−0.683106	+	0.730320i	$0.739371\pi$
$854$	4.38452e12	0.282073
$855$	−1.90121e13	−1.21670
$856$	4.35215e12	0.277059
$857$	−1.66234e13	−1.05270	−0.526350	−	0.850268i	$-0.676440\pi$
$857$	−1.66234e13	−1.05270	−0.526350	+	0.850268i	$0.676440\pi$
$858$	0	0
$859$	2.12853e13	1.33386	0.666930	−	0.745121i	$-0.267608\pi$
$859$	2.12853e13	1.33386	0.666930	+	0.745121i	$0.267608\pi$
$860$	1.68058e13	1.04765
$861$	−1.78090e10	−0.00110440
$862$	−7.01415e12	−0.432705
$863$	1.58665e13	0.973714	0.486857	−	0.873482i	$-0.338143\pi$
$863$	1.58665e13	0.973714	0.486857	+	0.873482i	$0.338143\pi$
$864$	−8.95886e12	−0.546941
$865$	2.30948e13	1.40263
$866$	4.30782e12	0.260272
$867$	5.24306e12	0.315136
$868$	−1.61468e12	−0.0965487
$869$	−4.04643e13	−2.40704
$870$	8.26354e12	0.489023
$871$	0	0
$872$	4.17176e12	0.244340
$873$	4.57088e12	0.266340
$874$	−1.67295e13	−0.969800
$875$	−1.92652e13	−1.11106
$876$	9.21589e11	0.0528772
$877$	−6.92981e12	−0.395570	−0.197785	−	0.980245i	$-0.563375\pi$
$877$	−6.92981e12	−0.395570	−0.197785	+	0.980245i	$0.563375\pi$
$878$	2.25763e13	1.28212
$879$	−2.03652e12	−0.115064
$880$	−6.41026e13	−3.60333
$881$	1.50068e13	0.839259	0.419630	−	0.907695i	$-0.362160\pi$
$881$	1.50068e13	0.839259	0.419630	+	0.907695i	$0.362160\pi$
$882$	1.35085e13	0.751620
$883$	−1.63884e13	−0.907221	−0.453611	−	0.891200i	$-0.649864\pi$
$883$	−1.63884e13	−0.907221	−0.453611	+	0.891200i	$0.649864\pi$
$884$	0	0
$885$	−7.05782e10	−0.00386746
$886$	−1.17742e13	−0.641919
$887$	−2.70000e13	−1.46456	−0.732281	−	0.681002i	$-0.761544\pi$
$887$	−2.70000e13	−1.46456	−0.732281	+	0.681002i	$0.761544\pi$
$888$	−4.95961e12	−0.267664
$889$	−4.42870e12	−0.237803
$890$	−2.71713e13	−1.45163
$891$	−2.18595e13	−1.16196
$892$	3.23504e11	0.0171095
$893$	−1.56221e13	−0.822069
$894$	−5.56874e11	−0.0291567
$895$	−3.23444e13	−1.68498
$896$	1.17016e13	0.606540
$897$	0	0
$898$	−1.55918e13	−0.800118
$899$	−4.85641e12	−0.247968
$900$	−1.86147e13	−0.945727
$901$	−3.22833e11	−0.0163199
$902$	−2.46747e11	−0.0124114
$903$	4.24728e12	0.212577
$904$	−2.41770e13	−1.20405
$905$	2.54546e13	1.26139
$906$	1.04003e13	0.512823
$907$	−1.12074e13	−0.549884	−0.274942	−	0.961461i	$-0.588659\pi$
$907$	−1.12074e13	−0.549884	−0.274942	+	0.961461i	$0.588659\pi$
$908$	−4.52673e12	−0.221003
$909$	2.92676e13	1.42184
$910$	0	0
$911$	1.06250e13	0.511089	0.255545	−	0.966797i	$-0.417745\pi$
$911$	1.06250e13	0.511089	0.255545	+	0.966797i	$0.417745\pi$
$912$	−6.30177e12	−0.301638
$913$	3.33097e13	1.58654
$914$	2.43765e13	1.15535
$915$	4.82929e12	0.227766
$916$	−1.54006e13	−0.722781
$917$	−1.38237e13	−0.645600
$918$	3.85756e11	0.0179276
$919$	4.03137e13	1.86437	0.932186	−	0.361980i	$-0.117899\pi$
$919$	4.03137e13	1.86437	0.932186	+	0.361980i	$0.117899\pi$
$920$	2.44938e13	1.12723
$921$	3.98141e11	0.0182335
$922$	1.52043e13	0.692910
$923$	0	0
$924$	3.19030e12	0.143982
$925$	−6.49107e13	−2.91527
$926$	−1.12372e13	−0.502237
$927$	1.73683e13	0.772500
$928$	−1.48084e13	−0.655452
$929$	−3.30900e13	−1.45756	−0.728779	−	0.684749i	$-0.759912\pi$
$929$	−3.30900e13	−1.45756	−0.728779	+	0.684749i	$0.759912\pi$
$930$	−5.36307e12	−0.235093
$931$	1.19699e13	0.522176
$932$	−4.21069e12	−0.182802
$933$	−3.26758e11	−0.0141175
$934$	3.68551e13	1.58466
$935$	1.64757e12	0.0705004
$936$	0	0
$937$	−9.13054e12	−0.386962	−0.193481	−	0.981104i	$-0.561978\pi$
$937$	−9.13054e12	−0.386962	−0.193481	+	0.981104i	$0.561978\pi$
$938$	1.63629e13	0.690156
$939$	−1.03236e13	−0.433348
$940$	−2.25221e13	−0.940877
$941$	2.61735e13	1.08820	0.544099	−	0.839021i	$-0.316872\pi$
$941$	2.61735e13	1.08820	0.544099	+	0.839021i	$0.316872\pi$
$942$	−1.09582e13	−0.453430
$943$	1.56335e11	0.00643802
$944$	2.11887e11	0.00868421
$945$	−1.46189e13	−0.596311
$946$	5.88468e13	2.38898
$947$	3.70875e12	0.149849	0.0749243	−	0.997189i	$-0.476128\pi$
$947$	3.70875e12	0.149849	0.0749243	+	0.997189i	$0.476128\pi$
$948$	−5.73492e12	−0.230616
$949$	0	0
$950$	−4.97403e13	−1.98131
$951$	4.50237e11	0.0178496
$952$	−2.15299e11	−0.00849525
$953$	−1.82708e13	−0.717528	−0.358764	−	0.933428i	$-0.616802\pi$
$953$	−1.82708e13	−0.717528	−0.358764	+	0.933428i	$0.616802\pi$
$954$	1.88060e13	0.735069
$955$	−5.24691e12	−0.204121
$956$	−1.33717e13	−0.517756
$957$	9.59537e12	0.369792
$958$	−2.57000e13	−0.985798
$959$	−2.00035e13	−0.763701
$960$	1.95828e12	0.0744139
$961$	−2.32878e13	−0.880791
$962$	0	0
$963$	1.08047e13	0.404850
$964$	−5.32550e12	−0.198616
$965$	−7.06927e13	−2.62423
$966$	−6.09542e12	−0.225220
$967$	−6.64147e12	−0.244256	−0.122128	−	0.992514i	$-0.538972\pi$
$967$	−6.64147e12	−0.244256	−0.122128	+	0.992514i	$0.538972\pi$
$968$	−2.80540e13	−1.02697
$969$	1.61968e11	0.00590165
$970$	1.76084e13	0.638626
$971$	2.72248e13	0.982828	0.491414	−	0.870926i	$-0.336480\pi$
$971$	2.72248e13	0.982828	0.491414	+	0.870926i	$0.336480\pi$
$972$	−1.13727e13	−0.408662
$973$	1.29862e13	0.464487
$974$	−4.08718e13	−1.45515
$975$	0	0
$976$	−1.44983e13	−0.511437
$977$	5.44450e12	0.191175	0.0955877	−	0.995421i	$-0.469527\pi$
$977$	5.44450e12	0.191175	0.0955877	+	0.995421i	$0.469527\pi$
$978$	4.55869e12	0.159337
$979$	−3.15505e13	−1.09770
$980$	1.72567e13	0.597642
$981$	1.03568e13	0.357040
$982$	5.31258e13	1.82307
$983$	−2.73557e13	−0.934454	−0.467227	−	0.884138i	$-0.654747\pi$
$983$	−2.73557e13	−0.934454	−0.467227	+	0.884138i	$0.654747\pi$
$984$	3.55149e10	0.00120763
$985$	−3.93753e13	−1.33279
$986$	6.37628e11	0.0214843
$987$	−5.69193e12	−0.190912
$988$	0	0
$989$	−3.72843e13	−1.23920
$990$	−9.59756e13	−3.17543
$991$	−9.85914e12	−0.324719	−0.162359	−	0.986732i	$-0.551910\pi$
$991$	−9.85914e12	−0.324719	−0.162359	+	0.986732i	$0.551910\pi$
$992$	9.61069e12	0.315103
$993$	1.67570e13	0.546923
$994$	1.68310e13	0.546855
$995$	2.58063e13	0.834683
$996$	4.72091e12	0.152005
$997$	−2.13556e13	−0.684516	−0.342258	−	0.939606i	$-0.611192\pi$
$997$	−2.13556e13	−0.684516	−0.342258	+	0.939606i	$0.611192\pi$
$998$	2.60175e13	0.830190
$999$	−2.59850e13	−0.825425

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.10.a.f.1.16		20
13.6	odd	12		13.10.e.a.10.3	yes	20
13.11	odd	12		13.10.e.a.4.3	✓	20
13.12	even	2	inner	169.10.a.f.1.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.e.a.4.3	✓	20	13.11	odd	12
13.10.e.a.10.3	yes	20	13.6	odd	12
169.10.a.f.1.5		20	13.12	even	2	inner
169.10.a.f.1.16		20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q+27.6771 q^{2} -44.2389 q^{3} +254.024 q^{4} -2467.22 q^{5} -1224.41 q^{6} +3580.38 q^{7} -7140.04 q^{8} -17725.9 q^{9} -68285.5 q^{10} -79291.0 q^{11} -11237.7 q^{12} +99094.7 q^{14} +109147. q^{15} -327676. q^{16} +8421.95 q^{17} -490603. q^{18} -434724. q^{19} -626732. q^{20} -158392. q^{21} -2.19455e6 q^{22} +1.39043e6 q^{23} +315867. q^{24} +4.13403e6 q^{25} +1.65493e6 q^{27} +909503. q^{28} +2.73548e6 q^{29} +3.02087e6 q^{30} -1.77534e6 q^{31} -5.41344e6 q^{32} +3.50775e6 q^{33} +233095. q^{34} -8.83357e6 q^{35} -4.50281e6 q^{36} -1.57016e7 q^{37} -1.20319e7 q^{38} +1.76160e7 q^{40} +112436. q^{41} -4.38384e6 q^{42} -2.68150e7 q^{43} -2.01418e7 q^{44} +4.37337e7 q^{45} +3.84831e7 q^{46} +3.59357e7 q^{47} +1.44960e7 q^{48} -2.75345e7 q^{49} +1.14418e8 q^{50} -372578. q^{51} -3.83324e7 q^{53} +4.58037e7 q^{54} +1.95628e8 q^{55} -2.55641e7 q^{56} +1.92317e7 q^{57} +7.57103e7 q^{58} -646636. q^{59} +2.77259e7 q^{60} +4.42458e7 q^{61} -4.91363e7 q^{62} -6.34656e7 q^{63} +1.79417e7 q^{64} +9.70844e7 q^{66} +1.65124e8 q^{67} +2.13938e6 q^{68} -6.15110e7 q^{69} -2.44488e8 q^{70} +1.69848e8 q^{71} +1.26564e8 q^{72} -8.20084e7 q^{73} -4.34574e8 q^{74} -1.82885e8 q^{75} -1.10430e8 q^{76} -2.83892e8 q^{77} +5.10327e8 q^{79} +8.08448e8 q^{80} +2.75687e8 q^{81} +3.11191e6 q^{82} -4.20094e8 q^{83} -4.02354e7 q^{84} -2.07788e7 q^{85} -7.42162e8 q^{86} -1.21015e8 q^{87} +5.66141e8 q^{88} +3.97908e8 q^{89} +1.21042e9 q^{90} +3.53202e8 q^{92} +7.85390e7 q^{93} +9.94598e8 q^{94} +1.07256e9 q^{95} +2.39484e8 q^{96} -2.57864e8 q^{97} -7.62076e8 q^{98} +1.40551e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 326 q^{3} + 5118 q^{4} + 129526 q^{9} + 88390 q^{10} + 427652 q^{12} + 473556 q^{14} + 1189618 q^{16} - 99312 q^{17} - 5073532 q^{22} + 6252378 q^{23} + 1529274 q^{25} + 18052718 q^{27} + 5424828 q^{29}+ \cdots + 9251202540 q^{95}+O(q^{100}) \) 20 * q + 326 * q^3 + 5118 * q^4 + 129526 * q^9 + 88390 * q^10 + 427652 * q^12 + 473556 * q^14 + 1189618 * q^16 - 99312 * q^17 - 5073532 * q^22 + 6252378 * q^23 + 1529274 * q^25 + 18052718 * q^27 + 5424828 * q^29 + 30045516 * q^30 - 15098160 * q^35 + 54078886 * q^36 + 39021096 * q^38 + 134360674 * q^40 + 121600068 * q^42 + 53242058 * q^43 + 82771076 * q^48 + 14055298 * q^49 + 10208934 * q^51 - 36429786 * q^53 - 131454160 * q^55 + 127272672 * q^56 + 536425792 * q^61 - 890759796 * q^62 - 343337066 * q^64 - 445758060 * q^66 + 336594090 * q^68 + 1083885582 * q^69 + 1083869262 * q^74 - 1218826882 * q^75 + 1470187374 * q^77 + 1556703616 * q^79 + 2139127516 * q^81 + 3719322754 * q^82 + 1288246146 * q^87 - 4109160928 * q^88 + 6489878934 * q^90 + 10148843820 * q^92 + 4894712828 * q^94 + 9251202540 * q^95

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