LMFDB - Embedded newform 63.10.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,10,Mod(1,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("63.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	63.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:	$32.4472576783$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 426x + 2016 $ x^3 - x^2 - 426*x + 2016
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.96128$ of defining polynomial
Character	$\chi$	$=$	63.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-41.8019 q^{2} +1235.40 q^{4} +1791.89 q^{5} +2401.00 q^{7} -30239.6 q^{8} -74904.4 q^{10} -17401.5 q^{11} -122541. q^{13} -100366. q^{14} +631550. q^{16} -331933. q^{17} +761707. q^{19} +2.21370e6 q^{20} +727418. q^{22} -1.23249e6 q^{23} +1.25774e6 q^{25} +5.12246e6 q^{26} +2.96620e6 q^{28} -634604. q^{29} -5.38069e6 q^{31} -1.09173e7 q^{32} +1.38754e7 q^{34} +4.30232e6 q^{35} -3.03611e6 q^{37} -3.18408e7 q^{38} -5.41861e7 q^{40} +7.37009e6 q^{41} -2.06990e7 q^{43} -2.14979e7 q^{44} +5.15203e7 q^{46} -2.03632e7 q^{47} +5.76480e6 q^{49} -5.25760e7 q^{50} -1.51388e8 q^{52} +5.97380e7 q^{53} -3.11816e7 q^{55} -7.26054e7 q^{56} +2.65277e7 q^{58} -6.03461e7 q^{59} -9.44357e6 q^{61} +2.24923e8 q^{62} +1.33012e8 q^{64} -2.19580e8 q^{65} -2.19187e8 q^{67} -4.10071e8 q^{68} -1.79846e8 q^{70} +5.58741e7 q^{71} +4.54332e8 q^{73} +1.26915e8 q^{74} +9.41015e8 q^{76} -4.17811e7 q^{77} +4.51057e7 q^{79} +1.13167e9 q^{80} -3.08084e8 q^{82} +3.34665e8 q^{83} -5.94786e8 q^{85} +8.65259e8 q^{86} +5.26216e8 q^{88} -6.51886e8 q^{89} -2.94221e8 q^{91} -1.52262e9 q^{92} +8.51220e8 q^{94} +1.36489e9 q^{95} -1.42804e9 q^{97} -2.40980e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 21 q^{2} + 1557 q^{4} - 1554 q^{5} + 7203 q^{7} - 14055 q^{8} - 97860 q^{10} + 3444 q^{11} - 19782 q^{13} - 50421 q^{14} + 482961 q^{16} - 1016694 q^{17} + 222852 q^{19} + 1922088 q^{20} - 2847048 q^{22}+ \cdots - 121060821 q^{98}+O(q^{100}) $ 3 * q - 21 * q^2 + 1557 * q^4 - 1554 * q^5 + 7203 * q^7 - 14055 * q^8 - 97860 * q^10 + 3444 * q^11 - 19782 * q^13 - 50421 * q^14 + 482961 * q^16 - 1016694 * q^17 + 222852 * q^19 + 1922088 * q^20 - 2847048 * q^22 - 1885632 * q^23 + 3073221 * q^25 + 8785056 * q^26 + 3738357 * q^28 - 4081818 * q^29 + 2869440 * q^31 - 25221951 * q^32 - 3981642 * q^34 - 3731154 * q^35 + 1395618 * q^37 - 43479870 * q^38 - 82859280 * q^40 + 14420658 * q^41 - 61631172 * q^43 - 97011984 * q^44 + 89747664 * q^46 + 10368960 * q^47 + 17294403 * q^49 - 73325055 * q^50 - 80908044 * q^52 - 67502610 * q^53 - 105823032 * q^55 - 33746055 * q^56 - 159163830 * q^58 + 42590100 * q^59 + 191746842 * q^61 + 46983468 * q^62 + 7852161 * q^64 - 364283220 * q^65 - 255175788 * q^67 - 743485806 * q^68 - 234961860 * q^70 - 296514504 * q^71 + 344213310 * q^73 + 690696462 * q^74 + 728839986 * q^76 + 8269044 * q^77 - 960412656 * q^79 + 1333333344 * q^80 + 562675302 * q^82 + 1100517180 * q^83 + 438179412 * q^85 + 880982256 * q^86 + 1206124800 * q^88 - 506816478 * q^89 - 47496582 * q^91 - 691123488 * q^92 + 1388004828 * q^94 + 2203071072 * q^95 - 647498250 * q^97 - 121060821 * q^98

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 $

$2$	−41.8019	−1.84740	−0.923701	−	0.383114i	$-0.874852\pi$
$2$	−41.8019	−1.84740	−0.923701	+	0.383114i	$0.874852\pi$
$3$	0	0
$3$	0	0
$4$	1235.40	2.41290
$5$	1791.89	1.28217	0.641086	−	0.767469i	$-0.278484\pi$
$5$	1791.89	1.28217	0.641086	+	0.767469i	$0.278484\pi$
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	−30239.6	−2.61019
$9$	0	0
$10$	−74904.4	−2.36869
$11$	−17401.5	−0.358361	−0.179180	−	0.983816i	$-0.557345\pi$
$11$	−17401.5	−0.358361	−0.179180	+	0.983816i	$0.557345\pi$
$12$	0	0
$13$	−122541.	−1.18997	−0.594986	−	0.803736i	$-0.702842\pi$
$13$	−122541.	−1.18997	−0.594986	+	0.803736i	$0.702842\pi$
$14$	−100366.	−0.698253
$15$	0	0
$16$	631550.	2.40917
$17$	−331933.	−0.963895	−0.481948	−	0.876200i	$-0.660070\pi$
$17$	−331933.	−0.963895	−0.481948	+	0.876200i	$0.660070\pi$
$18$	0	0
$19$	761707.	1.34090	0.670450	−	0.741954i	$-0.266101\pi$
$19$	761707.	1.34090	0.670450	+	0.741954i	$0.266101\pi$
$20$	2.21370e6	3.09375
$21$	0	0
$22$	727418.	0.662037
$23$	−1.23249e6	−0.918347	−0.459173	−	0.888347i	$-0.651854\pi$
$23$	−1.23249e6	−0.918347	−0.459173	+	0.888347i	$0.651854\pi$
$24$	0	0
$25$	1.25774e6	0.643963
$26$	5.12246e6	2.19836
$27$	0	0
$28$	2.96620e6	0.911989
$29$	−634604.	−0.166614	−0.0833071	−	0.996524i	$-0.526548\pi$
$29$	−634604.	−0.166614	−0.0833071	+	0.996524i	$0.526548\pi$
$30$	0	0
$31$	−5.38069e6	−1.04643	−0.523215	−	0.852201i	$-0.675268\pi$
$31$	−5.38069e6	−1.04643	−0.523215	+	0.852201i	$0.675268\pi$
$32$	−1.09173e7	−1.84052
$33$	0	0
$34$	1.38754e7	1.78070
$35$	4.30232e6	0.484615
$36$	0	0
$37$	−3.03611e6	−0.266324	−0.133162	−	0.991094i	$-0.542513\pi$
$37$	−3.03611e6	−0.266324	−0.133162	+	0.991094i	$0.542513\pi$
$38$	−3.18408e7	−2.47718
$39$	0	0
$40$	−5.41861e7	−3.34671
$41$	7.37009e6	0.407329	0.203665	−	0.979041i	$-0.434715\pi$
$41$	7.37009e6	0.407329	0.203665	+	0.979041i	$0.434715\pi$
$42$	0	0
$43$	−2.06990e7	−0.923297	−0.461649	−	0.887063i	$-0.652742\pi$
$43$	−2.06990e7	−0.923297	−0.461649	+	0.887063i	$0.652742\pi$
$44$	−2.14979e7	−0.864687
$45$	0	0
$46$	5.15203e7	1.69656
$47$	−2.03632e7	−0.608702	−0.304351	−	0.952560i	$-0.598440\pi$
$47$	−2.03632e7	−0.608702	−0.304351	+	0.952560i	$0.598440\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	−5.25760e7	−1.18966
$51$	0	0
$52$	−1.51388e8	−2.87128
$53$	5.97380e7	1.03994	0.519971	−	0.854184i	$-0.325943\pi$
$53$	5.97380e7	1.03994	0.519971	+	0.854184i	$0.325943\pi$
$54$	0	0
$55$	−3.11816e7	−0.459480
$56$	−7.26054e7	−0.986558
$57$	0	0
$58$	2.65277e7	0.307803
$59$	−6.03461e7	−0.648358	−0.324179	−	0.945996i	$-0.605088\pi$
$59$	−6.03461e7	−0.648358	−0.324179	+	0.945996i	$0.605088\pi$
$60$	0	0
$61$	−9.44357e6	−0.0873277	−0.0436639	−	0.999046i	$-0.513903\pi$
$61$	−9.44357e6	−0.0873277	−0.0436639	+	0.999046i	$0.513903\pi$
$62$	2.24923e8	1.93318
$63$	0	0
$64$	1.33012e8	0.991013
$65$	−2.19580e8	−1.52575
$66$	0	0
$67$	−2.19187e8	−1.32885	−0.664427	−	0.747353i	$-0.731324\pi$
$67$	−2.19187e8	−1.32885	−0.664427	+	0.747353i	$0.731324\pi$
$68$	−4.10071e8	−2.32578
$69$	0	0
$70$	−1.79846e8	−0.895279
$71$	5.58741e7	0.260944	0.130472	−	0.991452i	$-0.458351\pi$
$71$	5.58741e7	0.260944	0.130472	+	0.991452i	$0.458351\pi$
$72$	0	0
$73$	4.54332e8	1.87250	0.936248	−	0.351340i	$-0.114274\pi$
$73$	4.54332e8	1.87250	0.936248	+	0.351340i	$0.114274\pi$
$74$	1.26915e8	0.492007
$75$	0	0
$76$	9.41015e8	3.23545
$77$	−4.17811e7	−0.135448
$78$	0	0
$79$	4.51057e7	0.130289	0.0651447	−	0.997876i	$-0.479249\pi$
$79$	4.51057e7	0.130289	0.0651447	+	0.997876i	$0.479249\pi$
$80$	1.13167e9	3.08897
$81$	0	0
$82$	−3.08084e8	−0.752501
$83$	3.34665e8	0.774031	0.387016	−	0.922073i	$-0.373506\pi$
$83$	3.34665e8	0.774031	0.387016	+	0.922073i	$0.373506\pi$
$84$	0	0
$85$	−5.94786e8	−1.23588
$86$	8.65259e8	1.70570
$87$	0	0
$88$	5.26216e8	0.935389
$89$	−6.51886e8	−1.10133	−0.550664	−	0.834727i	$-0.685625\pi$
$89$	−6.51886e8	−1.10133	−0.550664	+	0.834727i	$0.685625\pi$
$90$	0	0
$91$	−2.94221e8	−0.449767
$92$	−1.52262e9	−2.21588
$93$	0	0
$94$	8.51220e8	1.12452
$95$	1.36489e9	1.71926
$96$	0	0
$97$	−1.42804e9	−1.63783	−0.818914	−	0.573916i	$-0.805424\pi$
$97$	−1.42804e9	−1.63783	−0.818914	+	0.573916i	$0.805424\pi$
$98$	−2.40980e8	−0.263915
$99$	0	0
$100$	1.55382e9	1.55382
$101$	−8.91532e8	−0.852493	−0.426247	−	0.904607i	$-0.640164\pi$
$101$	−8.91532e8	−0.852493	−0.426247	+	0.904607i	$0.640164\pi$
$102$	0	0
$103$	−7.12000e8	−0.623322	−0.311661	−	0.950193i	$-0.600885\pi$
$103$	−7.12000e8	−0.623322	−0.311661	+	0.950193i	$0.600885\pi$
$104$	3.70560e9	3.10605
$105$	0	0
$106$	−2.49716e9	−1.92119
$107$	−2.48598e9	−1.83346	−0.916729	−	0.399510i	$-0.869180\pi$
$107$	−2.48598e9	−1.83346	−0.916729	+	0.399510i	$0.869180\pi$
$108$	0	0
$109$	3.83455e8	0.260193	0.130096	−	0.991501i	$-0.458471\pi$
$109$	3.83455e8	0.260193	0.130096	+	0.991501i	$0.458471\pi$
$110$	1.30345e9	0.848844
$111$	0	0
$112$	1.51635e9	0.910581
$113$	−2.39582e9	−1.38230	−0.691148	−	0.722713i	$-0.742895\pi$
$113$	−2.39582e9	−1.38230	−0.691148	+	0.722713i	$0.742895\pi$
$114$	0	0
$115$	−2.20848e9	−1.17748
$116$	−7.83991e8	−0.402023
$117$	0	0
$118$	2.52258e9	1.19778
$119$	−7.96970e8	−0.364318
$120$	0	0
$121$	−2.05513e9	−0.871578
$122$	3.94760e8	0.161329
$123$	0	0
$124$	−6.64732e9	−2.52493
$125$	−1.24605e9	−0.456501
$126$	0	0
$127$	1.88261e9	0.642161	0.321080	−	0.947052i	$-0.395954\pi$
$127$	1.88261e9	0.642161	0.321080	+	0.947052i	$0.395954\pi$
$128$	2.95237e7	0.00972134
$129$	0	0
$130$	9.17888e9	2.81867
$131$	1.49241e9	0.442760	0.221380	−	0.975188i	$-0.428944\pi$
$131$	1.49241e9	0.442760	0.221380	+	0.975188i	$0.428944\pi$
$132$	0	0
$133$	1.82886e9	0.506813
$134$	9.16242e9	2.45493
$135$	0	0
$136$	1.00375e10	2.51595
$137$	3.21118e9	0.778792	0.389396	−	0.921070i	$-0.372684\pi$
$137$	3.21118e9	0.778792	0.389396	+	0.921070i	$0.372684\pi$
$138$	0	0
$139$	−3.03934e9	−0.690577	−0.345289	−	0.938497i	$-0.612219\pi$
$139$	−3.03934e9	−0.690577	−0.345289	+	0.938497i	$0.612219\pi$
$140$	5.31510e9	1.16933
$141$	0	0
$142$	−2.33565e9	−0.482070
$143$	2.13240e9	0.426439
$144$	0	0
$145$	−1.13714e9	−0.213628
$146$	−1.89920e10	−3.45925
$147$	0	0
$148$	−3.75082e9	−0.642611
$149$	3.64286e9	0.605487	0.302743	−	0.953072i	$-0.402098\pi$
$149$	3.64286e9	0.605487	0.302743	+	0.953072i	$0.402098\pi$
$150$	0	0
$151$	5.05862e9	0.791837	0.395919	−	0.918286i	$-0.370426\pi$
$151$	5.05862e9	0.791837	0.395919	+	0.918286i	$0.370426\pi$
$152$	−2.30337e10	−3.50000
$153$	0	0
$154$	1.74653e9	0.250226
$155$	−9.64159e9	−1.34170
$156$	0	0
$157$	−1.19687e10	−1.57216	−0.786082	−	0.618122i	$-0.787894\pi$
$157$	−1.19687e10	−1.57216	−0.786082	+	0.618122i	$0.787894\pi$
$158$	−1.88551e9	−0.240697
$159$	0	0
$160$	−1.95626e10	−2.35986
$161$	−2.95920e9	−0.347102
$162$	0	0
$163$	1.40960e10	1.56405	0.782027	−	0.623245i	$-0.214186\pi$
$163$	1.40960e10	1.56405	0.782027	+	0.623245i	$0.214186\pi$
$164$	9.10503e9	0.982843
$165$	0	0
$166$	−1.39896e10	−1.42995
$167$	−3.56028e9	−0.354210	−0.177105	−	0.984192i	$-0.556673\pi$
$167$	−3.56028e9	−0.354210	−0.177105	+	0.984192i	$0.556673\pi$
$168$	0	0
$169$	4.41184e9	0.416034
$170$	2.48632e10	2.28317
$171$	0	0
$172$	−2.55716e10	−2.22782
$173$	−2.64068e9	−0.224134	−0.112067	−	0.993701i	$-0.535747\pi$
$173$	−2.64068e9	−0.224134	−0.112067	+	0.993701i	$0.535747\pi$
$174$	0	0
$175$	3.01983e9	0.243395
$176$	−1.09899e10	−0.863353
$177$	0	0
$178$	2.72501e10	2.03459
$179$	−1.26973e9	−0.0924430	−0.0462215	−	0.998931i	$-0.514718\pi$
$179$	−1.26973e9	−0.0924430	−0.0462215	+	0.998931i	$0.514718\pi$
$180$	0	0
$181$	−2.17242e10	−1.50449	−0.752247	−	0.658881i	$-0.771030\pi$
$181$	−2.17242e10	−1.50449	−0.752247	+	0.658881i	$0.771030\pi$
$182$	1.22990e10	0.830901
$183$	0	0
$184$	3.72699e10	2.39706
$185$	−5.44037e9	−0.341473
$186$	0	0
$187$	5.77614e9	0.345422
$188$	−2.51567e10	−1.46874
$189$	0	0
$190$	−5.70552e10	−3.17617
$191$	1.07035e10	0.581937	0.290968	−	0.956733i	$-0.406023\pi$
$191$	1.07035e10	0.581937	0.290968	+	0.956733i	$0.406023\pi$
$192$	0	0
$193$	7.66063e9	0.397426	0.198713	−	0.980058i	$-0.436324\pi$
$193$	7.66063e9	0.397426	0.198713	+	0.980058i	$0.436324\pi$
$194$	5.96950e10	3.02573
$195$	0	0
$196$	7.12185e9	0.344699
$197$	4.26160e9	0.201592	0.100796	−	0.994907i	$-0.467861\pi$
$197$	4.26160e9	0.201592	0.100796	+	0.994907i	$0.467861\pi$
$198$	0	0
$199$	3.16291e10	1.42971	0.714856	−	0.699272i	$-0.246492\pi$
$199$	3.16291e10	1.42971	0.714856	+	0.699272i	$0.246492\pi$
$200$	−3.80336e10	−1.68086
$201$	0	0
$202$	3.72678e10	1.57490
$203$	−1.52368e9	−0.0629742
$204$	0	0
$205$	1.32064e10	0.522266
$206$	2.97630e10	1.15153
$207$	0	0
$208$	−7.73909e10	−2.86685
$209$	−1.32549e10	−0.480526
$210$	0	0
$211$	1.76561e10	0.613229	0.306615	−	0.951834i	$-0.400804\pi$
$211$	1.76561e10	0.613229	0.306615	+	0.951834i	$0.400804\pi$
$212$	7.38005e10	2.50927
$213$	0	0
$214$	1.03919e11	3.38713
$215$	−3.70903e10	−1.18383
$216$	0	0
$217$	−1.29190e10	−0.395513
$218$	−1.60292e10	−0.480681
$219$	0	0
$220$	−3.85219e10	−1.10868
$221$	4.06754e10	1.14701
$222$	0	0
$223$	−1.04686e10	−0.283476	−0.141738	−	0.989904i	$-0.545269\pi$
$223$	−1.04686e10	−0.283476	−0.141738	+	0.989904i	$0.545269\pi$
$224$	−2.62125e10	−0.695652
$225$	0	0
$226$	1.00150e11	2.55366
$227$	−1.95043e10	−0.487543	−0.243772	−	0.969833i	$-0.578385\pi$
$227$	−1.95043e10	−0.487543	−0.243772	+	0.969833i	$0.578385\pi$
$228$	0	0
$229$	−5.96135e10	−1.43247	−0.716234	−	0.697860i	$-0.754136\pi$
$229$	−5.96135e10	−1.43247	−0.716234	+	0.697860i	$0.754136\pi$
$230$	9.23187e10	2.17528
$231$	0	0
$232$	1.91902e10	0.434894
$233$	−8.42619e10	−1.87296	−0.936482	−	0.350716i	$-0.885938\pi$
$233$	−8.42619e10	−1.87296	−0.936482	+	0.350716i	$0.885938\pi$
$234$	0	0
$235$	−3.64885e10	−0.780460
$236$	−7.45517e10	−1.56442
$237$	0	0
$238$	3.33149e10	0.673042
$239$	7.28353e10	1.44395	0.721974	−	0.691920i	$-0.243235\pi$
$239$	7.28353e10	1.44395	0.721974	+	0.691920i	$0.243235\pi$
$240$	0	0
$241$	4.45555e10	0.850795	0.425397	−	0.905007i	$-0.360134\pi$
$241$	4.45555e10	0.850795	0.425397	+	0.905007i	$0.360134\pi$
$242$	8.59086e10	1.61015
$243$	0	0
$244$	−1.16666e10	−0.210713
$245$	1.03299e10	0.183167
$246$	0	0
$247$	−9.33404e10	−1.59564
$248$	1.62710e11	2.73138
$249$	0	0
$250$	5.20874e10	0.843340
$251$	9.55068e10	1.51881	0.759404	−	0.650620i	$-0.225491\pi$
$251$	9.55068e10	1.51881	0.759404	+	0.650620i	$0.225491\pi$
$252$	0	0
$253$	2.14472e10	0.329100
$254$	−7.86968e10	−1.18633
$255$	0	0
$256$	−6.93361e10	−1.00897
$257$	−9.12794e10	−1.30519	−0.652595	−	0.757707i	$-0.726320\pi$
$257$	−9.12794e10	−1.30519	−0.652595	+	0.757707i	$0.726320\pi$
$258$	0	0
$259$	−7.28970e9	−0.100661
$260$	−2.71270e11	−3.68147
$261$	0	0
$262$	−6.23858e10	−0.817956
$263$	−1.50976e11	−1.94584	−0.972921	−	0.231138i	$-0.925755\pi$
$263$	−1.50976e11	−1.94584	−0.972921	+	0.231138i	$0.925755\pi$
$264$	0	0
$265$	1.07044e11	1.33338
$266$	−7.64498e10	−0.936287
$267$	0	0
$268$	−2.70784e11	−3.20639
$269$	−2.39622e10	−0.279024	−0.139512	−	0.990220i	$-0.544553\pi$
$269$	−2.39622e10	−0.279024	−0.139512	+	0.990220i	$0.544553\pi$
$270$	0	0
$271$	−5.51248e10	−0.620848	−0.310424	−	0.950598i	$-0.600471\pi$
$271$	−5.51248e10	−0.620848	−0.310424	+	0.950598i	$0.600471\pi$
$272$	−2.09632e11	−2.32219
$273$	0	0
$274$	−1.34233e11	−1.43874
$275$	−2.18866e10	−0.230771
$276$	0	0
$277$	2.19632e10	0.224149	0.112074	−	0.993700i	$-0.464251\pi$
$277$	2.19632e10	0.224149	0.112074	+	0.993700i	$0.464251\pi$
$278$	1.27050e11	1.27577
$279$	0	0
$280$	−1.30101e11	−1.26494
$281$	1.62670e10	0.155643	0.0778213	−	0.996967i	$-0.475204\pi$
$281$	1.62670e10	0.155643	0.0778213	+	0.996967i	$0.475204\pi$
$282$	0	0
$283$	1.35590e10	0.125657	0.0628287	−	0.998024i	$-0.479988\pi$
$283$	1.35590e10	0.125657	0.0628287	+	0.998024i	$0.479988\pi$
$284$	6.90270e10	0.629632
$285$	0	0
$286$	−8.91387e10	−0.787805
$287$	1.76956e10	0.153956
$288$	0	0
$289$	−8.40858e9	−0.0709059
$290$	4.75346e10	0.394657
$291$	0	0
$292$	5.61284e11	4.51814
$293$	−5.41900e10	−0.429551	−0.214776	−	0.976663i	$-0.568902\pi$
$293$	−5.41900e10	−0.429551	−0.214776	+	0.976663i	$0.568902\pi$
$294$	0	0
$295$	−1.08133e11	−0.831306
$296$	9.18109e10	0.695155
$297$	0	0
$298$	−1.52279e11	−1.11858
$299$	1.51030e11	1.09281
$300$	0	0
$301$	−4.96983e10	−0.348974
$302$	−2.11460e11	−1.46284
$303$	0	0
$304$	4.81056e11	3.23046
$305$	−1.69218e10	−0.111969
$306$	0	0
$307$	7.32498e10	0.470634	0.235317	−	0.971919i	$-0.424387\pi$
$307$	7.32498e10	0.470634	0.235317	+	0.971919i	$0.424387\pi$
$308$	−5.16165e10	−0.326821
$309$	0	0
$310$	4.03037e11	2.47866
$311$	2.44502e11	1.48204	0.741021	−	0.671482i	$-0.234342\pi$
$311$	2.44502e11	1.48204	0.741021	+	0.671482i	$0.234342\pi$
$312$	0	0
$313$	−4.39895e10	−0.259060	−0.129530	−	0.991576i	$-0.541347\pi$
$313$	−4.39895e10	−0.259060	−0.129530	+	0.991576i	$0.541347\pi$
$314$	5.00314e11	2.90442
$315$	0	0
$316$	5.57237e10	0.314375
$317$	2.17653e11	1.21059	0.605297	−	0.795999i	$-0.293054\pi$
$317$	2.17653e11	1.21059	0.605297	+	0.795999i	$0.293054\pi$
$318$	0	0
$319$	1.10431e10	0.0597080
$320$	2.38342e11	1.27065
$321$	0	0
$322$	1.23700e11	0.641238
$323$	−2.52835e11	−1.29249
$324$	0	0
$325$	−1.54125e11	−0.766298
$326$	−5.89240e11	−2.88944
$327$	0	0
$328$	−2.22869e11	−1.06321
$329$	−4.88920e10	−0.230068
$330$	0	0
$331$	−3.66945e11	−1.68026	−0.840128	−	0.542389i	$-0.817520\pi$
$331$	−3.66945e11	−1.68026	−0.840128	+	0.542389i	$0.817520\pi$
$332$	4.13446e11	1.86766
$333$	0	0
$334$	1.48827e11	0.654368
$335$	−3.92758e11	−1.70382
$336$	0	0
$337$	1.99451e11	0.842367	0.421183	−	0.906976i	$-0.361615\pi$
$337$	1.99451e11	0.842367	0.421183	+	0.906976i	$0.361615\pi$
$338$	−1.84423e11	−0.768583
$339$	0	0
$340$	−7.34801e11	−2.98205
$341$	9.36322e10	0.374999
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	6.25931e11	2.40998
$345$	0	0
$346$	1.10385e11	0.414066
$347$	−6.63180e10	−0.245555	−0.122777	−	0.992434i	$-0.539180\pi$
$347$	−6.63180e10	−0.245555	−0.122777	+	0.992434i	$0.539180\pi$
$348$	0	0
$349$	−1.20301e11	−0.434065	−0.217033	−	0.976164i	$-0.569638\pi$
$349$	−1.20301e11	−0.434065	−0.217033	+	0.976164i	$0.569638\pi$
$350$	−1.26235e11	−0.449649
$351$	0	0
$352$	1.89978e11	0.659571
$353$	−2.43060e11	−0.833159	−0.416580	−	0.909099i	$-0.636771\pi$
$353$	−2.43060e11	−0.833159	−0.416580	+	0.909099i	$0.636771\pi$
$354$	0	0
$355$	1.00120e11	0.334575
$356$	−8.05341e11	−2.65739
$357$	0	0
$358$	5.30773e10	0.170779
$359$	−2.81654e11	−0.894933	−0.447466	−	0.894301i	$-0.647674\pi$
$359$	−2.81654e11	−0.894933	−0.447466	+	0.894301i	$0.647674\pi$
$360$	0	0
$361$	2.57510e11	0.798015
$362$	9.08115e11	2.77941
$363$	0	0
$364$	−3.63482e11	−1.08524
$365$	8.14113e11	2.40086
$366$	0	0
$367$	−2.41000e11	−0.693458	−0.346729	−	0.937965i	$-0.612708\pi$
$367$	−2.41000e11	−0.693458	−0.346729	+	0.937965i	$0.612708\pi$
$368$	−7.78376e11	−2.21246
$369$	0	0
$370$	2.27418e11	0.630837
$371$	1.43431e11	0.393061
$372$	0	0
$373$	−1.33921e11	−0.358227	−0.179114	−	0.983828i	$-0.557323\pi$
$373$	−1.33921e11	−0.358227	−0.179114	+	0.983828i	$0.557323\pi$
$374$	−2.41454e11	−0.638134
$375$	0	0
$376$	6.15775e11	1.58883
$377$	7.77651e10	0.198266
$378$	0	0
$379$	5.67201e11	1.41208	0.706042	−	0.708170i	$-0.250479\pi$
$379$	5.67201e11	1.41208	0.706042	+	0.708170i	$0.250479\pi$
$380$	1.68619e12	4.14841
$381$	0	0
$382$	−4.47427e11	−1.07507
$383$	1.03509e11	0.245800	0.122900	−	0.992419i	$-0.460781\pi$
$383$	1.03509e11	0.245800	0.122900	+	0.992419i	$0.460781\pi$
$384$	0	0
$385$	−7.48671e10	−0.173667
$386$	−3.20229e11	−0.734206
$387$	0	0
$388$	−1.76421e12	−3.95191
$389$	5.83600e11	1.29224	0.646119	−	0.763237i	$-0.276391\pi$
$389$	5.83600e11	1.29224	0.646119	+	0.763237i	$0.276391\pi$
$390$	0	0
$391$	4.09102e11	0.885190
$392$	−1.74326e11	−0.372884
$393$	0	0
$394$	−1.78143e11	−0.372422
$395$	8.08243e10	0.167053
$396$	0	0
$397$	−6.54542e10	−0.132245	−0.0661226	−	0.997812i	$-0.521063\pi$
$397$	−6.54542e10	−0.132245	−0.0661226	+	0.997812i	$0.521063\pi$
$398$	−1.32216e12	−2.64125
$399$	0	0
$400$	7.94326e11	1.55142
$401$	−4.06647e10	−0.0785359	−0.0392679	−	0.999229i	$-0.512503\pi$
$401$	−4.06647e10	−0.0785359	−0.0392679	+	0.999229i	$0.512503\pi$
$402$	0	0
$403$	6.59356e11	1.24522
$404$	−1.10140e12	−2.05698
$405$	0	0
$406$	6.36930e10	0.116339
$407$	5.28330e10	0.0954400
$408$	0	0
$409$	1.06163e12	1.87594	0.937970	−	0.346716i	$-0.112703\pi$
$409$	1.06163e12	1.87594	0.937970	+	0.346716i	$0.112703\pi$
$410$	−5.52052e11	−0.964835
$411$	0	0
$412$	−8.79607e11	−1.50401
$413$	−1.44891e11	−0.245056
$414$	0	0
$415$	5.99682e11	0.992441
$416$	1.33782e12	2.19017
$417$	0	0
$418$	5.54079e11	0.887726
$419$	1.19335e12	1.89149	0.945744	−	0.324912i	$-0.105335\pi$
$419$	1.19335e12	1.89149	0.945744	+	0.324912i	$0.105335\pi$
$420$	0	0
$421$	−5.88397e11	−0.912853	−0.456427	−	0.889761i	$-0.650871\pi$
$421$	−5.88397e11	−0.912853	−0.456427	+	0.889761i	$0.650871\pi$
$422$	−7.38058e11	−1.13288
$423$	0	0
$424$	−1.80646e12	−2.71444
$425$	−4.17485e11	−0.620713
$426$	0	0
$427$	−2.26740e10	−0.0330068
$428$	−3.07119e12	−4.42394
$429$	0	0
$430$	1.55045e12	2.18700
$431$	1.22932e12	1.71599	0.857997	−	0.513655i	$-0.171709\pi$
$431$	1.22932e12	1.71599	0.857997	+	0.513655i	$0.171709\pi$
$432$	0	0
$433$	−4.73819e10	−0.0647764	−0.0323882	−	0.999475i	$-0.510311\pi$
$433$	−4.73819e10	−0.0647764	−0.0323882	+	0.999475i	$0.510311\pi$
$434$	5.40041e11	0.730672
$435$	0	0
$436$	4.73722e11	0.627819
$437$	−9.38793e11	−1.23141
$438$	0	0
$439$	6.14236e11	0.789305	0.394652	−	0.918830i	$-0.370865\pi$
$439$	6.14236e11	0.789305	0.394652	+	0.918830i	$0.370865\pi$
$440$	9.42921e11	1.19933
$441$	0	0
$442$	−1.70031e12	−2.11899
$443$	8.19199e11	1.01058	0.505292	−	0.862948i	$-0.331385\pi$
$443$	8.19199e11	1.01058	0.505292	+	0.862948i	$0.331385\pi$
$444$	0	0
$445$	−1.16811e12	−1.41209
$446$	4.37607e11	0.523694
$447$	0	0
$448$	3.19361e11	0.374568
$449$	−4.81029e11	−0.558551	−0.279276	−	0.960211i	$-0.590094\pi$
$449$	−4.81029e11	−0.558551	−0.279276	+	0.960211i	$0.590094\pi$
$450$	0	0
$451$	−1.28251e11	−0.145971
$452$	−2.95980e12	−3.33534
$453$	0	0
$454$	8.15316e11	0.900689
$455$	−5.27212e11	−0.576679
$456$	0	0
$457$	1.15328e12	1.23684	0.618419	−	0.785849i	$-0.287774\pi$
$457$	1.15328e12	1.23684	0.618419	+	0.785849i	$0.287774\pi$
$458$	2.49196e12	2.64635
$459$	0	0
$460$	−2.72836e12	−2.84113
$461$	6.06986e11	0.625928	0.312964	−	0.949765i	$-0.398678\pi$
$461$	6.06986e11	0.625928	0.312964	+	0.949765i	$0.398678\pi$
$462$	0	0
$463$	−8.87758e11	−0.897801	−0.448900	−	0.893582i	$-0.648184\pi$
$463$	−8.87758e11	−0.897801	−0.448900	+	0.893582i	$0.648184\pi$
$464$	−4.00784e11	−0.401402
$465$	0	0
$466$	3.52231e12	3.46012
$467$	1.09779e12	1.06806	0.534029	−	0.845466i	$-0.320677\pi$
$467$	1.09779e12	1.06806	0.534029	+	0.845466i	$0.320677\pi$
$468$	0	0
$469$	−5.26267e11	−0.502260
$470$	1.52529e12	1.44182
$471$	0	0
$472$	1.82484e12	1.69234
$473$	3.60195e11	0.330874
$474$	0	0
$475$	9.58029e11	0.863490
$476$	−9.84579e11	−0.879062
$477$	0	0
$478$	−3.04466e12	−2.66755
$479$	3.12229e11	0.270996	0.135498	−	0.990778i	$-0.456737\pi$
$479$	3.12229e11	0.270996	0.135498	+	0.990778i	$0.456737\pi$
$480$	0	0
$481$	3.72048e11	0.316918
$482$	−1.86251e12	−1.57176
$483$	0	0
$484$	−2.53892e12	−2.10303
$485$	−2.55889e12	−2.09998
$486$	0	0
$487$	1.08377e12	0.873088	0.436544	−	0.899683i	$-0.356202\pi$
$487$	1.08377e12	0.873088	0.436544	+	0.899683i	$0.356202\pi$
$488$	2.85570e11	0.227942
$489$	0	0
$490$	−4.31809e11	−0.338384
$491$	−2.14702e12	−1.66713	−0.833567	−	0.552419i	$-0.813705\pi$
$491$	−2.14702e12	−1.66713	−0.833567	+	0.552419i	$0.813705\pi$
$492$	0	0
$493$	2.10646e11	0.160599
$494$	3.90181e12	2.94778
$495$	0	0
$496$	−3.39817e12	−2.52103
$497$	1.34154e11	0.0986277
$498$	0	0
$499$	−1.85705e12	−1.34082	−0.670411	−	0.741990i	$-0.733882\pi$
$499$	−1.85705e12	−1.34082	−0.670411	+	0.741990i	$0.733882\pi$
$500$	−1.53938e12	−1.10149
$501$	0	0
$502$	−3.99237e12	−2.80585
$503$	1.59040e12	1.10777	0.553885	−	0.832593i	$-0.313145\pi$
$503$	1.59040e12	1.10777	0.553885	+	0.832593i	$0.313145\pi$
$504$	0	0
$505$	−1.59753e12	−1.09304
$506$	−8.96533e11	−0.607979
$507$	0	0
$508$	2.32578e12	1.54947
$509$	−2.32864e12	−1.53771	−0.768853	−	0.639426i	$-0.779172\pi$
$509$	−2.32864e12	−1.53771	−0.768853	+	0.639426i	$0.779172\pi$
$510$	0	0
$511$	1.09085e12	0.707737
$512$	2.88327e12	1.85426
$513$	0	0
$514$	3.81566e12	2.41121
$515$	−1.27582e12	−0.799205
$516$	0	0
$517$	3.54350e11	0.218135
$518$	3.04724e11	0.185961
$519$	0	0
$520$	6.64003e12	3.98249
$521$	1.48366e12	0.882197	0.441099	−	0.897459i	$-0.354589\pi$
$521$	1.48366e12	0.882197	0.441099	+	0.897459i	$0.354589\pi$
$522$	0	0
$523$	−1.95584e12	−1.14308	−0.571540	−	0.820574i	$-0.693654\pi$
$523$	−1.95584e12	−1.14308	−0.571540	+	0.820574i	$0.693654\pi$
$524$	1.84373e12	1.06833
$525$	0	0
$526$	6.31110e12	3.59475
$527$	1.78603e12	1.00865
$528$	0	0
$529$	−2.82131e11	−0.156639
$530$	−4.47464e12	−2.46330
$531$	0	0
$532$	2.25938e12	1.22289
$533$	−9.03139e11	−0.484710
$534$	0	0
$535$	−4.45460e12	−2.35081
$536$	6.62812e12	3.46856
$537$	0	0
$538$	1.00167e12	0.515470
$539$	−1.00316e11	−0.0511944
$540$	0	0
$541$	−1.34802e12	−0.676563	−0.338281	−	0.941045i	$-0.609846\pi$
$541$	−1.34802e12	−0.676563	−0.338281	+	0.941045i	$0.609846\pi$
$542$	2.30433e12	1.14696
$543$	0	0
$544$	3.62381e12	1.77407
$545$	6.87109e11	0.333612
$546$	0	0
$547$	1.02503e12	0.489548	0.244774	−	0.969580i	$-0.421286\pi$
$547$	1.02503e12	0.489548	0.244774	+	0.969580i	$0.421286\pi$
$548$	3.96710e12	1.87914
$549$	0	0
$550$	9.14903e11	0.426327
$551$	−4.83382e11	−0.223413
$552$	0	0
$553$	1.08299e11	0.0492448
$554$	−9.18103e11	−0.414093
$555$	0	0
$556$	−3.75481e12	−1.66629
$557$	−1.75343e12	−0.771864	−0.385932	−	0.922527i	$-0.626120\pi$
$557$	−1.75343e12	−0.771864	−0.385932	+	0.922527i	$0.626120\pi$
$558$	0	0
$559$	2.53648e12	1.09870
$560$	2.71713e12	1.16752
$561$	0	0
$562$	−6.79991e11	−0.287535
$563$	2.17456e12	0.912185	0.456092	−	0.889932i	$-0.349249\pi$
$563$	2.17456e12	0.912185	0.456092	+	0.889932i	$0.349249\pi$
$564$	0	0
$565$	−4.29304e12	−1.77234
$566$	−5.66792e11	−0.232140
$567$	0	0
$568$	−1.68961e12	−0.681114
$569$	−3.13812e12	−1.25506	−0.627531	−	0.778592i	$-0.715934\pi$
$569$	−3.13812e12	−1.25506	−0.627531	+	0.778592i	$0.715934\pi$
$570$	0	0
$571$	2.05979e12	0.810887	0.405444	−	0.914120i	$-0.367117\pi$
$571$	2.05979e12	0.810887	0.405444	+	0.914120i	$0.367117\pi$
$572$	2.63438e12	1.02895
$573$	0	0
$574$	−7.39710e11	−0.284419
$575$	−1.55015e12	−0.591381
$576$	0	0
$577$	3.94380e12	1.48123	0.740617	−	0.671927i	$-0.234533\pi$
$577$	3.94380e12	1.48123	0.740617	+	0.671927i	$0.234533\pi$
$578$	3.51495e11	0.130992
$579$	0	0
$580$	−1.40483e12	−0.515462
$581$	8.03530e11	0.292556
$582$	0	0
$583$	−1.03953e12	−0.372675
$584$	−1.37389e13	−4.88757
$585$	0	0
$586$	2.26525e12	0.793555
$587$	−3.68239e12	−1.28014	−0.640071	−	0.768315i	$-0.721095\pi$
$587$	−3.68239e12	−1.28014	−0.640071	+	0.768315i	$0.721095\pi$
$588$	0	0
$589$	−4.09851e12	−1.40316
$590$	4.52019e12	1.53576
$591$	0	0
$592$	−1.91746e12	−0.641620
$593$	1.17114e12	0.388921	0.194460	−	0.980910i	$-0.437704\pi$
$593$	1.17114e12	0.388921	0.194460	+	0.980910i	$0.437704\pi$
$594$	0	0
$595$	−1.42808e12	−0.467118
$596$	4.50040e12	1.46098
$597$	0	0
$598$	−6.31336e12	−2.01886
$599$	−2.83757e12	−0.900586	−0.450293	−	0.892881i	$-0.648681\pi$
$599$	−2.83757e12	−0.900586	−0.450293	+	0.892881i	$0.648681\pi$
$600$	0	0
$601$	2.37066e12	0.741198	0.370599	−	0.928793i	$-0.379152\pi$
$601$	2.37066e12	0.741198	0.370599	+	0.928793i	$0.379152\pi$
$602$	2.07749e12	0.644695
$603$	0	0
$604$	6.24943e12	1.91062
$605$	−3.68257e12	−1.11751
$606$	0	0
$607$	4.40080e12	1.31578	0.657889	−	0.753115i	$-0.271450\pi$
$607$	4.40080e12	1.31578	0.657889	+	0.753115i	$0.271450\pi$
$608$	−8.31580e12	−2.46796
$609$	0	0
$610$	7.07366e11	0.206852
$611$	2.49533e12	0.724339
$612$	0	0
$613$	−1.87570e12	−0.536527	−0.268264	−	0.963346i	$-0.586450\pi$
$613$	−1.87570e12	−0.536527	−0.268264	+	0.963346i	$0.586450\pi$
$614$	−3.06198e12	−0.869451
$615$	0	0
$616$	1.26345e12	0.353544
$617$	3.18170e12	0.883845	0.441922	−	0.897053i	$-0.354297\pi$
$617$	3.18170e12	0.883845	0.441922	+	0.897053i	$0.354297\pi$
$618$	0	0
$619$	−1.16476e11	−0.0318880	−0.0159440	−	0.999873i	$-0.505075\pi$
$619$	−1.16476e11	−0.0318880	−0.0159440	+	0.999873i	$0.505075\pi$
$620$	−1.19113e13	−3.23739
$621$	0	0
$622$	−1.02207e13	−2.73793
$623$	−1.56518e12	−0.416263
$624$	0	0
$625$	−4.68931e12	−1.22927
$626$	1.83885e12	0.478587
$627$	0	0
$628$	−1.47861e13	−3.79347
$629$	1.00778e12	0.256708
$630$	0	0
$631$	−3.71283e12	−0.932338	−0.466169	−	0.884696i	$-0.654366\pi$
$631$	−3.71283e12	−0.932338	−0.466169	+	0.884696i	$0.654366\pi$
$632$	−1.36398e12	−0.340080
$633$	0	0
$634$	−9.09834e12	−2.23646
$635$	3.37343e12	0.823360
$636$	0	0
$637$	−7.06425e11	−0.169996
$638$	−4.61622e11	−0.110305
$639$	0	0
$640$	5.29032e10	0.0124644
$641$	−2.93764e11	−0.0687285	−0.0343642	−	0.999409i	$-0.510941\pi$
$641$	−2.93764e11	−0.0687285	−0.0343642	+	0.999409i	$0.510941\pi$
$642$	0	0
$643$	5.27980e12	1.21806	0.609029	−	0.793148i	$-0.291559\pi$
$643$	5.27980e12	1.21806	0.609029	+	0.793148i	$0.291559\pi$
$644$	−3.65580e12	−0.837522
$645$	0	0
$646$	1.05690e13	2.38775
$647$	2.62000e12	0.587804	0.293902	−	0.955836i	$-0.405046\pi$
$647$	2.62000e12	0.587804	0.293902	+	0.955836i	$0.405046\pi$
$648$	0	0
$649$	1.05011e12	0.232346
$650$	6.44272e12	1.41566
$651$	0	0
$652$	1.74142e13	3.77390
$653$	−3.15918e12	−0.679931	−0.339965	−	0.940438i	$-0.610415\pi$
$653$	−3.15918e12	−0.679931	−0.339965	+	0.940438i	$0.610415\pi$
$654$	0	0
$655$	2.67424e12	0.567694
$656$	4.65458e12	0.981326
$657$	0	0
$658$	2.04378e12	0.425028
$659$	−8.77292e11	−0.181201	−0.0906004	−	0.995887i	$-0.528879\pi$
$659$	−8.77292e11	−0.181201	−0.0906004	+	0.995887i	$0.528879\pi$
$660$	0	0
$661$	8.22108e12	1.67503	0.837514	−	0.546416i	$-0.184008\pi$
$661$	8.22108e12	1.67503	0.837514	+	0.546416i	$0.184008\pi$
$662$	1.53390e13	3.10411
$663$	0	0
$664$	−1.01201e13	−2.02037
$665$	3.27711e12	0.649821
$666$	0	0
$667$	7.82140e11	0.153010
$668$	−4.39838e12	−0.854671
$669$	0	0
$670$	1.64180e13	3.14764
$671$	1.64333e11	0.0312948
$672$	0	0
$673$	−7.47304e12	−1.40420	−0.702101	−	0.712077i	$-0.747754\pi$
$673$	−7.47304e12	−1.40420	−0.702101	+	0.712077i	$0.747754\pi$
$674$	−8.33743e12	−1.55619
$675$	0	0
$676$	5.45039e12	1.00385
$677$	−8.71309e12	−1.59413	−0.797064	−	0.603895i	$-0.793614\pi$
$677$	−8.71309e12	−1.59413	−0.797064	+	0.603895i	$0.793614\pi$
$678$	0	0
$679$	−3.42873e12	−0.619041
$680$	1.79861e13	3.22588
$681$	0	0
$682$	−3.91401e12	−0.692775
$683$	3.73903e12	0.657455	0.328727	−	0.944425i	$-0.393380\pi$
$683$	3.73903e12	0.657455	0.328727	+	0.944425i	$0.393380\pi$
$684$	0	0
$685$	5.75407e12	0.998545
$686$	−5.78593e11	−0.0997504
$687$	0	0
$688$	−1.30725e13	−2.22438
$689$	−7.32036e12	−1.23750
$690$	0	0
$691$	−6.26971e12	−1.04616	−0.523078	−	0.852285i	$-0.675216\pi$
$691$	−6.26971e12	−1.04616	−0.523078	+	0.852285i	$0.675216\pi$
$692$	−3.26230e12	−0.540812
$693$	0	0
$694$	2.77222e12	0.453639
$695$	−5.44616e12	−0.885439
$696$	0	0
$697$	−2.44637e12	−0.392623
$698$	5.02882e12	0.801893
$699$	0	0
$700$	3.73071e12	0.587287
$701$	5.13904e12	0.803805	0.401903	−	0.915682i	$-0.368349\pi$
$701$	5.13904e12	0.803805	0.401903	+	0.915682i	$0.368349\pi$
$702$	0	0
$703$	−2.31263e12	−0.357114
$704$	−2.31461e12	−0.355140
$705$	0	0
$706$	1.01604e13	1.53918
$707$	−2.14057e12	−0.322212
$708$	0	0
$709$	−2.95438e12	−0.439094	−0.219547	−	0.975602i	$-0.570458\pi$
$709$	−2.95438e12	−0.439094	−0.219547	+	0.975602i	$0.570458\pi$
$710$	−4.18522e12	−0.618096
$711$	0	0
$712$	1.97128e13	2.87467
$713$	6.63162e12	0.960986
$714$	0	0
$715$	3.82103e12	0.546768
$716$	−1.56863e12	−0.223055
$717$	0	0
$718$	1.17737e13	1.65330
$719$	1.21819e13	1.69994	0.849971	−	0.526829i	$-0.176619\pi$
$719$	1.21819e13	1.69994	0.849971	+	0.526829i	$0.176619\pi$
$720$	0	0
$721$	−1.70951e12	−0.235594
$722$	−1.07644e13	−1.47426
$723$	0	0
$724$	−2.68382e13	−3.63019
$725$	−7.98167e11	−0.107293
$726$	0	0
$727$	1.26291e13	1.67675	0.838376	−	0.545093i	$-0.183506\pi$
$727$	1.26291e13	1.67675	0.838376	+	0.545093i	$0.183506\pi$
$728$	8.89715e12	1.17398
$729$	0	0
$730$	−3.40315e13	−4.43536
$731$	6.87068e12	0.889962
$732$	0	0
$733$	9.16594e12	1.17276	0.586380	−	0.810036i	$-0.300553\pi$
$733$	9.16594e12	1.17276	0.586380	+	0.810036i	$0.300553\pi$
$734$	1.00743e13	1.28110
$735$	0	0
$736$	1.34554e13	1.69024
$737$	3.81418e12	0.476209
$738$	0	0
$739$	1.10658e13	1.36484	0.682420	−	0.730960i	$-0.260927\pi$
$739$	1.10658e13	1.36484	0.682420	+	0.730960i	$0.260927\pi$
$740$	−6.72105e12	−0.823938
$741$	0	0
$742$	−5.99569e12	−0.726142
$743$	−1.84642e12	−0.222270	−0.111135	−	0.993805i	$-0.535449\pi$
$743$	−1.84642e12	−0.222270	−0.111135	+	0.993805i	$0.535449\pi$
$744$	0	0
$745$	6.52760e12	0.776337
$746$	5.59816e12	0.661790
$747$	0	0
$748$	7.13586e12	0.833468
$749$	−5.96884e12	−0.692982
$750$	0	0
$751$	−9.19672e12	−1.05500	−0.527501	−	0.849555i	$-0.676871\pi$
$751$	−9.19672e12	−1.05500	−0.527501	+	0.849555i	$0.676871\pi$
$752$	−1.28604e13	−1.46647
$753$	0	0
$754$	−3.25073e12	−0.366277
$755$	9.06449e12	1.01527
$756$	0	0
$757$	8.81778e12	0.975950	0.487975	−	0.872858i	$-0.337736\pi$
$757$	8.81778e12	0.975950	0.487975	+	0.872858i	$0.337736\pi$
$758$	−2.37101e13	−2.60869
$759$	0	0
$760$	−4.12739e13	−4.48760
$761$	2.42043e12	0.261614	0.130807	−	0.991408i	$-0.458243\pi$
$761$	2.42043e12	0.261614	0.130807	+	0.991408i	$0.458243\pi$
$762$	0	0
$763$	9.20676e11	0.0983437
$764$	1.32231e13	1.40415
$765$	0	0
$766$	−4.32686e12	−0.454092
$767$	7.39488e12	0.771528
$768$	0	0
$769$	7.67945e12	0.791885	0.395942	−	0.918275i	$-0.370418\pi$
$769$	7.67945e12	0.791885	0.395942	+	0.918275i	$0.370418\pi$
$770$	3.12959e12	0.320833
$771$	0	0
$772$	9.46396e12	0.958948
$773$	−1.04693e13	−1.05466	−0.527328	−	0.849662i	$-0.676806\pi$
$773$	−1.04693e13	−1.05466	−0.527328	+	0.849662i	$0.676806\pi$
$774$	0	0
$775$	−6.76751e12	−0.673862
$776$	4.31835e13	4.27504
$777$	0	0
$778$	−2.43956e13	−2.38728
$779$	5.61385e12	0.546188
$780$	0	0
$781$	−9.72296e11	−0.0935123
$782$	−1.71013e13	−1.63530
$783$	0	0
$784$	3.64076e12	0.344167
$785$	−2.14465e13	−2.01578
$786$	0	0
$787$	−1.26490e13	−1.17535	−0.587677	−	0.809095i	$-0.699958\pi$
$787$	−1.26490e13	−1.17535	−0.587677	+	0.809095i	$0.699958\pi$
$788$	5.26479e12	0.486422
$789$	0	0
$790$	−3.37862e12	−0.308615
$791$	−5.75236e12	−0.522459
$792$	0	0
$793$	1.15723e12	0.103918
$794$	2.73611e12	0.244310
$795$	0	0
$796$	3.90747e13	3.44975
$797$	6.02337e11	0.0528782	0.0264391	−	0.999650i	$-0.491583\pi$
$797$	6.02337e11	0.0528782	0.0264391	+	0.999650i	$0.491583\pi$
$798$	0	0
$799$	6.75920e12	0.586725
$800$	−1.37311e13	−1.18523
$801$	0	0
$802$	1.69986e12	0.145087
$803$	−7.90608e12	−0.671029
$804$	0	0
$805$	−5.30256e12	−0.445045
$806$	−2.75624e13	−2.30043
$807$	0	0
$808$	2.69596e13	2.22517
$809$	7.72967e12	0.634443	0.317222	−	0.948351i	$-0.397250\pi$
$809$	7.72967e12	0.634443	0.317222	+	0.948351i	$0.397250\pi$
$810$	0	0
$811$	−1.20157e13	−0.975335	−0.487668	−	0.873029i	$-0.662152\pi$
$811$	−1.20157e13	−0.975335	−0.487668	+	0.873029i	$0.662152\pi$
$812$	−1.88236e12	−0.151950
$813$	0	0
$814$	−2.20852e12	−0.176316
$815$	2.52585e13	2.00538
$816$	0	0
$817$	−1.57666e13	−1.23805
$818$	−4.43783e13	−3.46562
$819$	0	0
$820$	1.63152e13	1.26017
$821$	−1.39952e13	−1.07507	−0.537533	−	0.843243i	$-0.680644\pi$
$821$	−1.39952e13	−1.07507	−0.537533	+	0.843243i	$0.680644\pi$
$822$	0	0
$823$	1.61982e13	1.23074	0.615371	−	0.788238i	$-0.289006\pi$
$823$	1.61982e13	1.23074	0.615371	+	0.788238i	$0.289006\pi$
$824$	2.15306e13	1.62699
$825$	0	0
$826$	6.05672e12	0.452718
$827$	1.30127e13	0.967374	0.483687	−	0.875241i	$-0.339297\pi$
$827$	1.30127e13	0.967374	0.483687	+	0.875241i	$0.339297\pi$
$828$	0	0
$829$	−1.31364e13	−0.966007	−0.483004	−	0.875618i	$-0.660454\pi$
$829$	−1.31364e13	−0.966007	−0.483004	+	0.875618i	$0.660454\pi$
$830$	−2.50679e13	−1.83344
$831$	0	0
$832$	−1.62994e13	−1.17928
$833$	−1.91353e12	−0.137699
$834$	0	0
$835$	−6.37963e12	−0.454157
$836$	−1.63751e13	−1.15946
$837$	0	0
$838$	−4.98842e13	−3.49434
$839$	5.22420e12	0.363991	0.181996	−	0.983299i	$-0.441744\pi$
$839$	5.22420e12	0.363991	0.181996	+	0.983299i	$0.441744\pi$
$840$	0	0
$841$	−1.41044e13	−0.972240
$842$	2.45961e13	1.68641
$843$	0	0
$844$	2.18124e13	1.47966
$845$	7.90552e12	0.533427
$846$	0	0
$847$	−4.93438e12	−0.329425
$848$	3.77275e13	2.50540
$849$	0	0
$850$	1.74517e13	1.14671
$851$	3.74196e12	0.244578
$852$	0	0
$853$	−5.65076e12	−0.365457	−0.182728	−	0.983163i	$-0.558493\pi$
$853$	−5.65076e12	−0.365457	−0.182728	+	0.983163i	$0.558493\pi$
$854$	9.47818e11	0.0609768
$855$	0	0
$856$	7.51752e13	4.78567
$857$	−1.42516e12	−0.0902503	−0.0451251	−	0.998981i	$-0.514369\pi$
$857$	−1.42516e12	−0.0902503	−0.0451251	+	0.998981i	$0.514369\pi$
$858$	0	0
$859$	2.77961e13	1.74186	0.870932	−	0.491404i	$-0.163516\pi$
$859$	2.77961e13	1.74186	0.870932	+	0.491404i	$0.163516\pi$
$860$	−4.58215e13	−2.85645
$861$	0	0
$862$	−5.13878e13	−3.17013
$863$	1.14117e13	0.700326	0.350163	−	0.936689i	$-0.386126\pi$
$863$	1.14117e13	0.700326	0.350163	+	0.936689i	$0.386126\pi$
$864$	0	0
$865$	−4.73180e12	−0.287378
$866$	1.98066e12	0.119668
$867$	0	0
$868$	−1.59602e13	−0.954333
$869$	−7.84908e11	−0.0466906
$870$	0	0
$871$	2.68594e13	1.58130
$872$	−1.15956e13	−0.679153
$873$	0	0
$874$	3.92434e13	2.27491
$875$	−2.99177e12	−0.172541
$876$	0	0
$877$	−1.21188e13	−0.691769	−0.345884	−	0.938277i	$-0.612421\pi$
$877$	−1.21188e13	−0.691769	−0.345884	+	0.938277i	$0.612421\pi$
$878$	−2.56763e13	−1.45816
$879$	0	0
$880$	−1.96927e13	−1.10697
$881$	4.60741e12	0.257671	0.128836	−	0.991666i	$-0.458876\pi$
$881$	4.60741e12	0.257671	0.128836	+	0.991666i	$0.458876\pi$
$882$	0	0
$883$	1.04709e13	0.579642	0.289821	−	0.957081i	$-0.406404\pi$
$883$	1.04709e13	0.579642	0.289821	+	0.957081i	$0.406404\pi$
$884$	5.02505e13	2.76761
$885$	0	0
$886$	−3.42441e13	−1.86696
$887$	3.68711e12	0.200000	0.0999999	−	0.994987i	$-0.468116\pi$
$887$	3.68711e12	0.200000	0.0999999	+	0.994987i	$0.468116\pi$
$888$	0	0
$889$	4.52015e12	0.242714
$890$	4.88291e13	2.60870
$891$	0	0
$892$	−1.29329e13	−0.683998
$893$	−1.55108e13	−0.816209
$894$	0	0
$895$	−2.27522e12	−0.118528
$896$	7.08864e10	0.00367432
$897$	0	0
$898$	2.01080e13	1.03187
$899$	3.41461e12	0.174350
$900$	0	0
$901$	−1.98290e13	−1.00240
$902$	5.36114e12	0.269667
$903$	0	0
$904$	7.24487e13	3.60805
$905$	−3.89274e13	−1.92902
$906$	0	0
$907$	−6.05209e11	−0.0296943	−0.0148471	−	0.999890i	$-0.504726\pi$
$907$	−6.05209e11	−0.0296943	−0.0148471	+	0.999890i	$0.504726\pi$
$908$	−2.40956e13	−1.17639
$909$	0	0
$910$	2.20385e13	1.06536
$911$	3.27450e12	0.157512	0.0787559	−	0.996894i	$-0.474905\pi$
$911$	3.27450e12	0.157512	0.0787559	+	0.996894i	$0.474905\pi$
$912$	0	0
$913$	−5.82368e12	−0.277383
$914$	−4.82094e13	−2.28494
$915$	0	0
$916$	−7.36467e13	−3.45640
$917$	3.58329e12	0.167348
$918$	0	0
$919$	2.21117e11	0.0102259	0.00511295	−	0.999987i	$-0.498372\pi$
$919$	2.21117e11	0.0102259	0.00511295	+	0.999987i	$0.498372\pi$
$920$	6.67836e13	3.07344
$921$	0	0
$922$	−2.53732e13	−1.15634
$923$	−6.84688e12	−0.310517
$924$	0	0
$925$	−3.81864e12	−0.171503
$926$	3.71100e13	1.65860
$927$	0	0
$928$	6.92817e12	0.306657
$929$	4.00147e12	0.176258	0.0881290	−	0.996109i	$-0.471911\pi$
$929$	4.00147e12	0.176258	0.0881290	+	0.996109i	$0.471911\pi$
$930$	0	0
$931$	4.39109e12	0.191557
$932$	−1.04097e14	−4.51927
$933$	0	0
$934$	−4.58899e13	−1.97313
$935$	1.03502e13	0.442890
$936$	0	0
$937$	−3.76138e10	−0.00159411	−0.000797056	−	1.00000i	$-0.500254\pi$
$937$	−3.76138e10	−0.00159411	−0.000797056		1.00000i	$0.500254\pi$
$938$	2.19990e13	0.927876
$939$	0	0
$940$	−4.50780e13	−1.88317
$941$	2.21494e12	0.0920894	0.0460447	−	0.998939i	$-0.485338\pi$
$941$	2.21494e12	0.0920894	0.0460447	+	0.998939i	$0.485338\pi$
$942$	0	0
$943$	−9.08353e12	−0.374069
$944$	−3.81116e13	−1.56201
$945$	0	0
$946$	−1.50568e13	−0.611257
$947$	−2.36266e13	−0.954610	−0.477305	−	0.878738i	$-0.658386\pi$
$947$	−2.36266e13	−0.954610	−0.477305	+	0.878738i	$0.658386\pi$
$948$	0	0
$949$	−5.56744e13	−2.22822
$950$	−4.00475e13	−1.59521
$951$	0	0
$952$	2.41001e13	0.950939
$953$	1.15037e13	0.451771	0.225885	−	0.974154i	$-0.427473\pi$
$953$	1.15037e13	0.451771	0.225885	+	0.974154i	$0.427473\pi$
$954$	0	0
$955$	1.91795e13	0.746142
$956$	8.99810e13	3.48410
$957$	0	0
$958$	−1.30518e13	−0.500639
$959$	7.71003e12	0.294356
$960$	0	0
$961$	2.51218e12	0.0950156
$962$	−1.55524e13	−0.585475
$963$	0	0
$964$	5.50440e13	2.05288
$965$	1.37270e13	0.509568
$966$	0	0
$967$	−1.36863e13	−0.503346	−0.251673	−	0.967812i	$-0.580981\pi$
$967$	−1.36863e13	−0.503346	−0.251673	+	0.967812i	$0.580981\pi$
$968$	6.21465e13	2.27498
$969$	0	0
$970$	1.06967e14	3.87950
$971$	−9.95259e12	−0.359294	−0.179647	−	0.983731i	$-0.557496\pi$
$971$	−9.95259e12	−0.359294	−0.179647	+	0.983731i	$0.557496\pi$
$972$	0	0
$973$	−7.29745e12	−0.261014
$974$	−4.53038e13	−1.61295
$975$	0	0
$976$	−5.96409e12	−0.210387
$977$	−1.51387e13	−0.531575	−0.265787	−	0.964032i	$-0.585632\pi$
$977$	−1.51387e13	−0.531575	−0.265787	+	0.964032i	$0.585632\pi$
$978$	0	0
$979$	1.13438e13	0.394673
$980$	1.27616e13	0.441964
$981$	0	0
$982$	8.97498e13	3.07987
$983$	−5.47141e13	−1.86899	−0.934497	−	0.355970i	$-0.884151\pi$
$983$	−5.47141e13	−1.86899	−0.934497	+	0.355970i	$0.884151\pi$
$984$	0	0
$985$	7.63631e12	0.258476
$986$	−8.80540e12	−0.296690
$987$	0	0
$988$	−1.15313e14	−3.85010
$989$	2.55112e13	0.847907
$990$	0	0
$991$	4.86384e13	1.60195	0.800974	−	0.598699i	$-0.204316\pi$
$991$	4.86384e13	1.60195	0.800974	+	0.598699i	$0.204316\pi$
$992$	5.87427e13	1.92598
$993$	0	0
$994$	−5.60789e12	−0.182205
$995$	5.66759e13	1.83314
$996$	0	0
$997$	2.77758e13	0.890304	0.445152	−	0.895455i	$-0.353150\pi$
$997$	2.77758e13	0.890304	0.445152	+	0.895455i	$0.353150\pi$
$998$	7.76283e13	2.47704
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.10.a.e.1.1		3
3.2	odd	2		7.10.a.b.1.3	✓	3
12.11	even	2		112.10.a.h.1.2		3
15.2	even	4		175.10.b.d.99.6		6
15.8	even	4		175.10.b.d.99.1		6
15.14	odd	2		175.10.a.d.1.1		3
21.2	odd	6		49.10.c.d.18.1		6
21.5	even	6		49.10.c.e.18.1		6
21.11	odd	6		49.10.c.d.30.1		6
21.17	even	6		49.10.c.e.30.1		6
21.20	even	2		49.10.a.c.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.a.b.1.3	✓	3	3.2	odd	2
49.10.a.c.1.3		3	21.20	even	2
49.10.c.d.18.1		6	21.2	odd	6
49.10.c.d.30.1		6	21.11	odd	6
49.10.c.e.18.1		6	21.5	even	6
49.10.c.e.30.1		6	21.17	even	6
63.10.a.e.1.1		3	1.1	even	1	trivial
112.10.a.h.1.2		3	12.11	even	2
175.10.a.d.1.1		3	15.14	odd	2
175.10.b.d.99.1		6	15.8	even	4
175.10.b.d.99.6		6	15.2	even	4

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:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	63.a (trivial)

\(f(q)\)	\(=\)	\(q-41.8019 q^{2} +1235.40 q^{4} +1791.89 q^{5} +2401.00 q^{7} -30239.6 q^{8} -74904.4 q^{10} -17401.5 q^{11} -122541. q^{13} -100366. q^{14} +631550. q^{16} -331933. q^{17} +761707. q^{19} +2.21370e6 q^{20} +727418. q^{22} -1.23249e6 q^{23} +1.25774e6 q^{25} +5.12246e6 q^{26} +2.96620e6 q^{28} -634604. q^{29} -5.38069e6 q^{31} -1.09173e7 q^{32} +1.38754e7 q^{34} +4.30232e6 q^{35} -3.03611e6 q^{37} -3.18408e7 q^{38} -5.41861e7 q^{40} +7.37009e6 q^{41} -2.06990e7 q^{43} -2.14979e7 q^{44} +5.15203e7 q^{46} -2.03632e7 q^{47} +5.76480e6 q^{49} -5.25760e7 q^{50} -1.51388e8 q^{52} +5.97380e7 q^{53} -3.11816e7 q^{55} -7.26054e7 q^{56} +2.65277e7 q^{58} -6.03461e7 q^{59} -9.44357e6 q^{61} +2.24923e8 q^{62} +1.33012e8 q^{64} -2.19580e8 q^{65} -2.19187e8 q^{67} -4.10071e8 q^{68} -1.79846e8 q^{70} +5.58741e7 q^{71} +4.54332e8 q^{73} +1.26915e8 q^{74} +9.41015e8 q^{76} -4.17811e7 q^{77} +4.51057e7 q^{79} +1.13167e9 q^{80} -3.08084e8 q^{82} +3.34665e8 q^{83} -5.94786e8 q^{85} +8.65259e8 q^{86} +5.26216e8 q^{88} -6.51886e8 q^{89} -2.94221e8 q^{91} -1.52262e9 q^{92} +8.51220e8 q^{94} +1.36489e9 q^{95} -1.42804e9 q^{97} -2.40980e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 21 q^{2} + 1557 q^{4} - 1554 q^{5} + 7203 q^{7} - 14055 q^{8} - 97860 q^{10} + 3444 q^{11} - 19782 q^{13} - 50421 q^{14} + 482961 q^{16} - 1016694 q^{17} + 222852 q^{19} + 1922088 q^{20} - 2847048 q^{22}+ \cdots - 121060821 q^{98}+O(q^{100}) \) 3 * q - 21 * q^2 + 1557 * q^4 - 1554 * q^5 + 7203 * q^7 - 14055 * q^8 - 97860 * q^10 + 3444 * q^11 - 19782 * q^13 - 50421 * q^14 + 482961 * q^16 - 1016694 * q^17 + 222852 * q^19 + 1922088 * q^20 - 2847048 * q^22 - 1885632 * q^23 + 3073221 * q^25 + 8785056 * q^26 + 3738357 * q^28 - 4081818 * q^29 + 2869440 * q^31 - 25221951 * q^32 - 3981642 * q^34 - 3731154 * q^35 + 1395618 * q^37 - 43479870 * q^38 - 82859280 * q^40 + 14420658 * q^41 - 61631172 * q^43 - 97011984 * q^44 + 89747664 * q^46 + 10368960 * q^47 + 17294403 * q^49 - 73325055 * q^50 - 80908044 * q^52 - 67502610 * q^53 - 105823032 * q^55 - 33746055 * q^56 - 159163830 * q^58 + 42590100 * q^59 + 191746842 * q^61 + 46983468 * q^62 + 7852161 * q^64 - 364283220 * q^65 - 255175788 * q^67 - 743485806 * q^68 - 234961860 * q^70 - 296514504 * q^71 + 344213310 * q^73 + 690696462 * q^74 + 728839986 * q^76 + 8269044 * q^77 - 960412656 * q^79 + 1333333344 * q^80 + 562675302 * q^82 + 1100517180 * q^83 + 438179412 * q^85 + 880982256 * q^86 + 1206124800 * q^88 - 506816478 * q^89 - 47496582 * q^91 - 691123488 * q^92 + 1388004828 * q^94 + 2203071072 * q^95 - 647498250 * q^97 - 121060821 * q^98

Introduction

L-functions

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