LMFDB - Embedded newform 363.8.a.q.1.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [363,8,Mod(1,363)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("363.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	363.a (trivial)

Newform invariants

comment:select newform

sage:traces = [14,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$113.395764251$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 1447 x^{12} + 2312 x^{11} + 800984 x^{10} - 1116196 x^{9} - 213596799 x^{8} + \cdots - 84027524573916 $ x^14 - 2x^13 - 1447x^12 + 2312x^11 + 800984x^10 - 1116196x^9 - 213596799x^8 + 277042820x^7 + 28586595076x^6 - 29838573108x^5 - 1805972868375x^4 + 712345293048x^3 + 42617153430527x^2 + 38928829671186*x - 84027524573916
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4}\cdot 5\cdot 11^{8} $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$20.9545$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+22.5726 q^{2} -27.0000 q^{3} +381.522 q^{4} +369.687 q^{5} -609.460 q^{6} +757.126 q^{7} +5722.64 q^{8} +729.000 q^{9} +8344.78 q^{10} -10301.1 q^{12} -2036.16 q^{13} +17090.3 q^{14} -9981.54 q^{15} +80339.9 q^{16} +4701.64 q^{17} +16455.4 q^{18} -485.463 q^{19} +141043. q^{20} -20442.4 q^{21} +10368.5 q^{23} -154511. q^{24} +58543.3 q^{25} -45961.4 q^{26} -19683.0 q^{27} +288860. q^{28} -120912. q^{29} -225309. q^{30} +103372. q^{31} +1.08098e6 q^{32} +106128. q^{34} +279900. q^{35} +278129. q^{36} -446110. q^{37} -10958.2 q^{38} +54976.3 q^{39} +2.11558e6 q^{40} +134189. q^{41} -461438. q^{42} -412551. q^{43} +269502. q^{45} +234045. q^{46} -688723. q^{47} -2.16918e6 q^{48} -250303. q^{49} +1.32147e6 q^{50} -126944. q^{51} -776838. q^{52} -1.14687e6 q^{53} -444296. q^{54} +4.33276e6 q^{56} +13107.5 q^{57} -2.72929e6 q^{58} -1.18564e6 q^{59} -3.80817e6 q^{60} -2.51141e6 q^{61} +2.33338e6 q^{62} +551945. q^{63} +1.41170e7 q^{64} -752741. q^{65} +2.82253e6 q^{67} +1.79378e6 q^{68} -279951. q^{69} +6.31806e6 q^{70} +2.93679e6 q^{71} +4.17180e6 q^{72} +3.65685e6 q^{73} -1.00699e7 q^{74} -1.58067e6 q^{75} -185215. q^{76} +1.24096e6 q^{78} +107494. q^{79} +2.97006e7 q^{80} +531441. q^{81} +3.02899e6 q^{82} -516375. q^{83} -7.79922e6 q^{84} +1.73814e6 q^{85} -9.31233e6 q^{86} +3.26461e6 q^{87} -3.22551e6 q^{89} +6.08335e6 q^{90} -1.54163e6 q^{91} +3.95582e6 q^{92} -2.79105e6 q^{93} -1.55463e7 q^{94} -179469. q^{95} -2.91865e7 q^{96} +6.24150e6 q^{97} -5.64998e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 9 q^{2} - 378 q^{3} + 1129 q^{4} - 71 q^{5} - 243 q^{6} - 534 q^{7} + 3384 q^{8} + 10206 q^{9} - 676 q^{10} - 30483 q^{12} + 12320 q^{13} - 39924 q^{14} + 1917 q^{15} + 121041 q^{16} + 33141 q^{17}+ \cdots - 8477109 q^{98}+O(q^{100}) $ 14 * q + 9 * q^2 - 378 * q^3 + 1129 * q^4 - 71 * q^5 - 243 * q^6 - 534 * q^7 + 3384 * q^8 + 10206 * q^9 - 676 * q^10 - 30483 * q^12 + 12320 * q^13 - 39924 * q^14 + 1917 * q^15 + 121041 * q^16 + 33141 * q^17 + 6561 * q^18 - 45019 * q^19 + 84003 * q^20 + 14418 * q^21 + 69254 * q^23 - 91368 * q^24 + 79811 * q^25 - 371495 * q^26 - 275562 * q^27 - 7064 * q^28 + 411600 * q^29 + 18252 * q^30 + 412799 * q^31 + 1363161 * q^32 - 771854 * q^34 + 850847 * q^35 + 823041 * q^36 - 831248 * q^37 + 948531 * q^38 - 332640 * q^39 - 50612 * q^40 + 616814 * q^41 + 1077948 * q^42 - 99892 * q^43 - 51759 * q^45 - 250599 * q^46 - 1139331 * q^47 - 3268107 * q^48 + 2854014 * q^49 - 3832250 * q^50 - 894807 * q^51 - 3588929 * q^52 + 2582463 * q^53 - 177147 * q^54 - 3153909 * q^56 + 1215513 * q^57 - 5034575 * q^58 - 1606585 * q^59 - 2268081 * q^60 - 3071551 * q^61 + 1169699 * q^62 - 389286 * q^63 + 19996212 * q^64 + 2447897 * q^65 - 1617843 * q^67 + 9075543 * q^68 - 1869858 * q^69 + 781473 * q^70 - 571759 * q^71 + 2466936 * q^72 + 7998216 * q^73 - 2349001 * q^74 - 2154897 * q^75 - 3267874 * q^76 + 10030365 * q^78 - 422038 * q^79 + 5233418 * q^80 + 7440174 * q^81 - 26251615 * q^82 + 7152906 * q^83 + 190728 * q^84 + 11341160 * q^85 + 12629343 * q^86 - 11113200 * q^87 - 20120730 * q^89 - 492804 * q^90 + 14374304 * q^91 + 21153333 * q^92 - 11145573 * q^93 - 37707710 * q^94 + 18573421 * q^95 - 36805347 * q^96 + 21339043 * q^97 - 8477109 * 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 $

$2$	22.5726	1.99515	0.997577	−	0.0695758i	$-0.0221646\pi$
$2$	22.5726	1.99515	0.997577	+	0.0695758i	$0.0221646\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	381.522	2.98064
$5$	369.687	1.32263	0.661316	−	0.750108i	$-0.269998\pi$
$5$	369.687	1.32263	0.661316	+	0.750108i	$0.269998\pi$
$6$	−609.460	−1.15190
$7$	757.126	0.834306	0.417153	−	0.908836i	$-0.363028\pi$
$7$	757.126	0.834306	0.417153	+	0.908836i	$0.363028\pi$
$8$	5722.64	3.95167
$9$	729.000	0.333333
$10$	8344.78	2.63885
$11$	0	0
$11$	0	0
$12$	−10301.1	−1.72087
$13$	−2036.16	−0.257045	−0.128523	−	0.991707i	$-0.541024\pi$
$13$	−2036.16	−0.257045	−0.128523	+	0.991707i	$0.541024\pi$
$14$	17090.3	1.66457
$15$	−9981.54	−0.763622
$16$	80339.9	4.90356
$17$	4701.64	0.232102	0.116051	−	0.993243i	$-0.462976\pi$
$17$	4701.64	0.232102	0.116051	+	0.993243i	$0.462976\pi$
$18$	16455.4	0.665051
$19$	−485.463	−0.0162375	−0.00811874	−	0.999967i	$-0.502584\pi$
$19$	−485.463	−0.0162375	−0.00811874	+	0.999967i	$0.502584\pi$
$20$	141043.	3.94228
$21$	−20442.4	−0.481687
$22$	0	0
$23$	10368.5	0.177693	0.0888465	−	0.996045i	$-0.471682\pi$
$23$	10368.5	0.177693	0.0888465	+	0.996045i	$0.471682\pi$
$24$	−154511.	−2.28150
$25$	58543.3	0.749354
$26$	−45961.4	−0.512845
$27$	−19683.0	−0.192450
$28$	288860.	2.48676
$29$	−120912.	−0.920608	−0.460304	−	0.887761i	$-0.652260\pi$
$29$	−120912.	−0.920608	−0.460304	+	0.887761i	$0.652260\pi$
$30$	−225309.	−1.52354
$31$	103372.	0.623215	0.311608	−	0.950211i	$-0.399133\pi$
$31$	103372.	0.623215	0.311608	+	0.950211i	$0.399133\pi$
$32$	1.08098e6	5.83168
$33$	0	0
$34$	106128.	0.463079
$35$	279900.	1.10348
$36$	278129.	0.993546
$37$	−446110.	−1.44789	−0.723945	−	0.689857i	$-0.757673\pi$
$37$	−446110.	−1.44789	−0.723945	+	0.689857i	$0.757673\pi$
$38$	−10958.2	−0.0323963
$39$	54976.3	0.148405
$40$	2.11558e6	5.22661
$41$	134189.	0.304070	0.152035	−	0.988375i	$-0.451417\pi$
$41$	134189.	0.304070	0.152035	+	0.988375i	$0.451417\pi$
$42$	−461438.	−0.961039
$43$	−412551.	−0.791293	−0.395647	−	0.918403i	$-0.629480\pi$
$43$	−412551.	−0.791293	−0.395647	+	0.918403i	$0.629480\pi$
$44$	0	0
$45$	269502.	0.440877
$46$	234045.	0.354525
$47$	−688723.	−0.967615	−0.483807	−	0.875175i	$-0.660746\pi$
$47$	−688723.	−0.967615	−0.483807	+	0.875175i	$0.660746\pi$
$48$	−2.16918e6	−2.83107
$49$	−250303.	−0.303934
$50$	1.32147e6	1.49508
$51$	−126944.	−0.134004
$52$	−776838.	−0.766159
$53$	−1.14687e6	−1.05816	−0.529078	−	0.848573i	$-0.677462\pi$
$53$	−1.14687e6	−1.05816	−0.529078	+	0.848573i	$0.677462\pi$
$54$	−444296.	−0.383967
$55$	0	0
$56$	4.33276e6	3.29690
$57$	13107.5	0.00937471
$58$	−2.72929e6	−1.83675
$59$	−1.18564e6	−0.751573	−0.375787	−	0.926706i	$-0.622627\pi$
$59$	−1.18564e6	−0.751573	−0.375787	+	0.926706i	$0.622627\pi$
$60$	−3.80817e6	−2.27608
$61$	−2.51141e6	−1.41665	−0.708326	−	0.705886i	$-0.750549\pi$
$61$	−2.51141e6	−1.41665	−0.708326	+	0.705886i	$0.750549\pi$
$62$	2.33338e6	1.24341
$63$	551945.	0.278102
$64$	1.41170e7	6.73153
$65$	−752741.	−0.339976
$66$	0	0
$67$	2.82253e6	1.14651	0.573254	−	0.819377i	$-0.305681\pi$
$67$	2.82253e6	1.14651	0.573254	+	0.819377i	$0.305681\pi$
$68$	1.79378e6	0.691811
$69$	−279951.	−0.102591
$70$	6.31806e6	2.20161
$71$	2.93679e6	0.973797	0.486899	−	0.873458i	$-0.338128\pi$
$71$	2.93679e6	0.973797	0.486899	+	0.873458i	$0.338128\pi$
$72$	4.17180e6	1.31722
$73$	3.65685e6	1.10021	0.550107	−	0.835094i	$-0.314587\pi$
$73$	3.65685e6	1.10021	0.550107	+	0.835094i	$0.314587\pi$
$74$	−1.00699e7	−2.88876
$75$	−1.58067e6	−0.432640
$76$	−185215.	−0.0483980
$77$	0	0
$78$	1.24096e6	0.296091
$79$	107494.	0.0245294	0.0122647	−	0.999925i	$-0.496096\pi$
$79$	107494.	0.0245294	0.0122647	+	0.999925i	$0.496096\pi$
$80$	2.97006e7	6.48560
$81$	531441.	0.111111
$82$	3.02899e6	0.606666
$83$	−516375.	−0.0991270	−0.0495635	−	0.998771i	$-0.515783\pi$
$83$	−516375.	−0.0991270	−0.0495635	+	0.998771i	$0.515783\pi$
$84$	−7.79922e6	−1.43573
$85$	1.73814e6	0.306985
$86$	−9.31233e6	−1.57875
$87$	3.26461e6	0.531513
$88$	0	0
$89$	−3.22551e6	−0.484990	−0.242495	−	0.970153i	$-0.577966\pi$
$89$	−3.22551e6	−0.484990	−0.242495	+	0.970153i	$0.577966\pi$
$90$	6.08335e6	0.879618
$91$	−1.54163e6	−0.214454
$92$	3.95582e6	0.529639
$93$	−2.79105e6	−0.359814
$94$	−1.55463e7	−1.93054
$95$	−179469.	−0.0214762
$96$	−2.91865e7	−3.36692
$97$	6.24150e6	0.694366	0.347183	−	0.937797i	$-0.387138\pi$
$97$	6.24150e6	0.694366	0.347183	+	0.937797i	$0.387138\pi$
$98$	−5.64998e6	−0.606395
$99$	0	0
$100$	2.23355e7	2.23355
$101$	1.42558e7	1.37679	0.688395	−	0.725336i	$-0.258316\pi$
$101$	1.42558e7	1.37679	0.688395	+	0.725336i	$0.258316\pi$
$102$	−2.86546e6	−0.267359
$103$	1.45860e6	0.131525	0.0657623	−	0.997835i	$-0.479052\pi$
$103$	1.45860e6	0.131525	0.0657623	+	0.997835i	$0.479052\pi$
$104$	−1.16522e7	−1.01576
$105$	−7.55729e6	−0.637094
$106$	−2.58879e7	−2.11118
$107$	2.14587e6	0.169340	0.0846701	−	0.996409i	$-0.473016\pi$
$107$	2.14587e6	0.169340	0.0846701	+	0.996409i	$0.473016\pi$
$108$	−7.50949e6	−0.573624
$109$	−2.97884e6	−0.220320	−0.110160	−	0.993914i	$-0.535136\pi$
$109$	−2.97884e6	−0.220320	−0.110160	+	0.993914i	$0.535136\pi$
$110$	0	0
$111$	1.20450e7	0.835940
$112$	6.08275e7	4.09107
$113$	−1.42275e7	−0.927587	−0.463793	−	0.885943i	$-0.653512\pi$
$113$	−1.42275e7	−0.927587	−0.463793	+	0.885943i	$0.653512\pi$
$114$	295870.	0.0187040
$115$	3.83311e6	0.235022
$116$	−4.61304e7	−2.74400
$117$	−1.48436e6	−0.0856818
$118$	−2.67630e7	−1.49950
$119$	3.55974e6	0.193644
$120$	−5.71207e7	−3.01758
$121$	0	0
$122$	−5.66890e7	−2.82644
$123$	−3.62310e6	−0.175555
$124$	3.94387e7	1.85758
$125$	−7.23910e6	−0.331512
$126$	1.24588e7	0.554856
$127$	−1.42832e6	−0.0618745	−0.0309373	−	0.999521i	$-0.509849\pi$
$127$	−1.42832e6	−0.0618745	−0.0309373	+	0.999521i	$0.509849\pi$
$128$	1.80293e8	7.59876
$129$	1.11389e7	0.456853
$130$	−1.69913e7	−0.678305
$131$	3.05006e7	1.18538	0.592691	−	0.805430i	$-0.298065\pi$
$131$	3.05006e7	1.18538	0.592691	+	0.805430i	$0.298065\pi$
$132$	0	0
$133$	−367557.	−0.0135470
$134$	6.37119e7	2.28746
$135$	−7.27654e6	−0.254541
$136$	2.69058e7	0.917191
$137$	4.27546e7	1.42056	0.710282	−	0.703917i	$-0.248567\pi$
$137$	4.27546e7	1.42056	0.710282	+	0.703917i	$0.248567\pi$
$138$	−6.31921e6	−0.204685
$139$	−1.93259e7	−0.610362	−0.305181	−	0.952294i	$-0.598717\pi$
$139$	−1.93259e7	−0.610362	−0.305181	+	0.952294i	$0.598717\pi$
$140$	1.06788e8	3.28907
$141$	1.85955e7	0.558653
$142$	6.62909e7	1.94287
$143$	0	0
$144$	5.85678e7	1.63452
$145$	−4.46994e7	−1.21763
$146$	8.25445e7	2.19509
$147$	6.75818e6	0.175476
$148$	−1.70200e8	−4.31564
$149$	6.96560e7	1.72507	0.862535	−	0.505998i	$-0.168876\pi$
$149$	6.96560e7	1.72507	0.862535	+	0.505998i	$0.168876\pi$
$150$	−3.56798e7	−0.863183
$151$	−6.29202e7	−1.48720	−0.743602	−	0.668622i	$-0.766884\pi$
$151$	−6.29202e7	−1.48720	−0.743602	+	0.668622i	$0.766884\pi$
$152$	−2.77813e6	−0.0641652
$153$	3.42750e6	0.0773673
$154$	0	0
$155$	3.82154e7	0.824284
$156$	2.09746e7	0.442342
$157$	792275.	0.0163391	0.00816953	−	0.999967i	$-0.497400\pi$
$157$	792275.	0.0163391	0.00816953	+	0.999967i	$0.497400\pi$
$158$	2.42641e6	0.0489400
$159$	3.09656e7	0.610927
$160$	3.99625e8	7.71316
$161$	7.85030e6	0.148250
$162$	1.19960e7	0.221684
$163$	5.63784e7	1.01966	0.509831	−	0.860275i	$-0.329708\pi$
$163$	5.63784e7	1.01966	0.509831	+	0.860275i	$0.329708\pi$
$164$	5.11960e7	0.906321
$165$	0	0
$166$	−1.16559e7	−0.197774
$167$	−1.11325e8	−1.84963	−0.924815	−	0.380418i	$-0.875780\pi$
$167$	−1.11325e8	−1.84963	−0.924815	+	0.380418i	$0.875780\pi$
$168$	−1.16984e8	−1.90347
$169$	−5.86026e7	−0.933928
$170$	3.92342e7	0.612482
$171$	−353903.	−0.00541249
$172$	−1.57397e8	−2.35856
$173$	−5.32446e7	−0.781834	−0.390917	−	0.920426i	$-0.627842\pi$
$173$	−5.32446e7	−0.781834	−0.390917	+	0.920426i	$0.627842\pi$
$174$	7.36907e7	1.06045
$175$	4.43246e7	0.625190
$176$	0	0
$177$	3.20123e7	0.433921
$178$	−7.28080e7	−0.967629
$179$	7.17234e7	0.934706	0.467353	−	0.884071i	$-0.345208\pi$
$179$	7.17234e7	0.934706	0.467353	+	0.884071i	$0.345208\pi$
$180$	1.02821e8	1.31409
$181$	1.15264e8	1.44483	0.722416	−	0.691458i	$-0.243031\pi$
$181$	1.15264e8	1.44483	0.722416	+	0.691458i	$0.243031\pi$
$182$	−3.47986e7	−0.427870
$183$	6.78081e7	0.817904
$184$	5.93354e7	0.702185
$185$	−1.64921e8	−1.91503
$186$	−6.30012e7	−0.717883
$187$	0	0
$188$	−2.62763e8	−2.88411
$189$	−1.49025e7	−0.160562
$190$	−4.05108e6	−0.0428483
$191$	−1.37922e8	−1.43224	−0.716120	−	0.697977i	$-0.754084\pi$
$191$	−1.37922e8	−1.43224	−0.716120	+	0.697977i	$0.754084\pi$
$192$	−3.81160e8	−3.88645
$193$	8.80512e6	0.0881627	0.0440813	−	0.999028i	$-0.485964\pi$
$193$	8.80512e6	0.0881627	0.0440813	+	0.999028i	$0.485964\pi$
$194$	1.40887e8	1.38537
$195$	2.03240e7	0.196285
$196$	−9.54959e7	−0.905917
$197$	7.10547e7	0.662157	0.331079	−	0.943603i	$-0.392587\pi$
$197$	7.10547e7	0.662157	0.331079	+	0.943603i	$0.392587\pi$
$198$	0	0
$199$	−6.79775e7	−0.611476	−0.305738	−	0.952116i	$-0.598903\pi$
$199$	−6.79775e7	−0.611476	−0.305738	+	0.952116i	$0.598903\pi$
$200$	3.35022e8	2.96120
$201$	−7.62084e7	−0.661937
$202$	3.21791e8	2.74691
$203$	−9.15453e7	−0.768069
$204$	−4.84320e7	−0.399417
$205$	4.96079e7	0.402172
$206$	3.29244e7	0.262412
$207$	7.55867e6	0.0592310
$208$	−1.63585e8	−1.26044
$209$	0	0
$210$	−1.70587e8	−1.27110
$211$	3.71879e7	0.272529	0.136265	−	0.990672i	$-0.456490\pi$
$211$	3.71879e7	0.272529	0.136265	+	0.990672i	$0.456490\pi$
$212$	−4.37557e8	−3.15398
$213$	−7.92933e7	−0.562222
$214$	4.84378e7	0.337860
$215$	−1.52514e8	−1.04659
$216$	−1.12639e8	−0.760500
$217$	7.82658e7	0.519952
$218$	−6.72401e7	−0.439573
$219$	−9.87349e7	−0.635208
$220$	0	0
$221$	−9.57329e6	−0.0596607
$222$	2.71886e8	1.66783
$223$	−7.92903e7	−0.478799	−0.239400	−	0.970921i	$-0.576951\pi$
$223$	−7.92903e7	−0.478799	−0.239400	+	0.970921i	$0.576951\pi$
$224$	8.18440e8	4.86540
$225$	4.26780e7	0.249785
$226$	−3.21152e8	−1.85068
$227$	−3.38486e8	−1.92066	−0.960328	−	0.278871i	$-0.910040\pi$
$227$	−3.38486e8	−1.92066	−0.960328	+	0.278871i	$0.910040\pi$
$228$	5.00079e6	0.0279426
$229$	2.29754e8	1.26427	0.632133	−	0.774860i	$-0.282180\pi$
$229$	2.29754e8	1.26427	0.632133	+	0.774860i	$0.282180\pi$
$230$	8.65233e7	0.468906
$231$	0	0
$232$	−6.91933e8	−3.63794
$233$	1.72505e8	0.893422	0.446711	−	0.894678i	$-0.352595\pi$
$233$	1.72505e8	0.893422	0.446711	+	0.894678i	$0.352595\pi$
$234$	−3.35058e7	−0.170948
$235$	−2.54612e8	−1.27980
$236$	−4.52348e8	−2.24017
$237$	−2.90233e6	−0.0141621
$238$	8.03525e7	0.386349
$239$	−5.91779e7	−0.280393	−0.140196	−	0.990124i	$-0.544773\pi$
$239$	−5.91779e7	−0.280393	−0.140196	+	0.990124i	$0.544773\pi$
$240$	−8.01916e8	−3.74446
$241$	−3.11722e8	−1.43452	−0.717262	−	0.696804i	$-0.754605\pi$
$241$	−3.11722e8	−1.43452	−0.717262	+	0.696804i	$0.754605\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	−9.58157e8	−4.22252
$245$	−9.25337e7	−0.401993
$246$	−8.17828e7	−0.350259
$247$	988480.	0.00417377
$248$	5.91562e8	2.46274
$249$	1.39421e7	0.0572310
$250$	−1.63405e8	−0.661418
$251$	−4.46504e8	−1.78224	−0.891122	−	0.453764i	$-0.850081\pi$
$251$	−4.46504e8	−1.78224	−0.891122	+	0.453764i	$0.850081\pi$
$252$	2.10579e8	0.828921
$253$	0	0
$254$	−3.22408e7	−0.123449
$255$	−4.69297e7	−0.177238
$256$	2.26269e9	8.42916
$257$	−3.52039e8	−1.29368	−0.646838	−	0.762628i	$-0.723909\pi$
$257$	−3.52039e8	−1.29368	−0.646838	+	0.762628i	$0.723909\pi$
$258$	2.51433e8	0.911493
$259$	−3.37761e8	−1.20798
$260$	−2.87187e8	−1.01335
$261$	−8.81445e7	−0.306869
$262$	6.88477e8	2.36502
$263$	2.75435e8	0.933630	0.466815	−	0.884355i	$-0.345401\pi$
$263$	2.75435e8	0.933630	0.466815	+	0.884355i	$0.345401\pi$
$264$	0	0
$265$	−4.23984e8	−1.39955
$266$	−8.29671e6	−0.0270284
$267$	8.70887e7	0.280009
$268$	1.07686e9	3.41733
$269$	−3.09082e8	−0.968147	−0.484074	−	0.875027i	$-0.660843\pi$
$269$	−3.09082e8	−0.968147	−0.484074	+	0.875027i	$0.660843\pi$
$270$	−1.64250e8	−0.507847
$271$	3.18134e8	0.970995	0.485498	−	0.874238i	$-0.338638\pi$
$271$	3.18134e8	0.970995	0.485498	+	0.874238i	$0.338638\pi$
$272$	3.77730e8	1.13812
$273$	4.16240e7	0.123815
$274$	9.65082e8	2.83424
$275$	0	0
$276$	−1.06807e8	−0.305787
$277$	4.17374e8	1.17990	0.589951	−	0.807439i	$-0.299147\pi$
$277$	4.17374e8	1.17990	0.589951	+	0.807439i	$0.299147\pi$
$278$	−4.36235e8	−1.21777
$279$	7.53584e7	0.207738
$280$	1.60176e9	4.36059
$281$	−2.78776e7	−0.0749520	−0.0374760	−	0.999298i	$-0.511932\pi$
$281$	−2.78776e7	−0.0749520	−0.0374760	+	0.999298i	$0.511932\pi$
$282$	4.19749e8	1.11460
$283$	7.00496e7	0.183719	0.0918593	−	0.995772i	$-0.470719\pi$
$283$	7.00496e7	0.183719	0.0918593	+	0.995772i	$0.470719\pi$
$284$	1.12045e9	2.90254
$285$	4.84567e6	0.0123993
$286$	0	0
$287$	1.01598e8	0.253687
$288$	7.88036e8	1.94389
$289$	−3.88233e8	−0.946129
$290$	−1.00898e9	−2.42935
$291$	−1.68521e8	−0.400892
$292$	1.39517e9	3.27934
$293$	5.10340e7	0.118528	0.0592642	−	0.998242i	$-0.481125\pi$
$293$	5.10340e7	0.118528	0.0592642	+	0.998242i	$0.481125\pi$
$294$	1.52550e8	0.350103
$295$	−4.38316e8	−0.994054
$296$	−2.55292e9	−5.72159
$297$	0	0
$298$	1.57232e9	3.44178
$299$	−2.11120e7	−0.0456752
$300$	−6.03059e8	−1.28954
$301$	−3.12353e8	−0.660180
$302$	−1.42027e9	−2.96720
$303$	−3.84907e8	−0.794890
$304$	−3.90021e7	−0.0796214
$305$	−9.28435e8	−1.87371
$306$	7.73675e7	0.154360
$307$	6.44678e8	1.27162	0.635812	−	0.771844i	$-0.280665\pi$
$307$	6.44678e8	1.27162	0.635812	+	0.771844i	$0.280665\pi$
$308$	0	0
$309$	−3.93823e7	−0.0759357
$310$	8.62619e8	1.64457
$311$	−3.38991e8	−0.639037	−0.319519	−	0.947580i	$-0.603521\pi$
$311$	−3.38991e8	−0.639037	−0.319519	+	0.947580i	$0.603521\pi$
$312$	3.14609e8	0.586449
$313$	2.07682e8	0.382819	0.191410	−	0.981510i	$-0.438694\pi$
$313$	2.07682e8	0.382819	0.191410	+	0.981510i	$0.438694\pi$
$314$	1.78837e7	0.0325989
$315$	2.04047e8	0.367826
$316$	4.10111e7	0.0731133
$317$	−2.14341e8	−0.377918	−0.188959	−	0.981985i	$-0.560511\pi$
$317$	−2.14341e8	−0.377918	−0.188959	+	0.981985i	$0.560511\pi$
$318$	6.98973e8	1.21889
$319$	0	0
$320$	5.21889e9	8.90334
$321$	−5.79385e7	−0.0977687
$322$	1.77202e8	0.295782
$323$	−2.28247e6	−0.00376875
$324$	2.02756e8	0.331182
$325$	−1.19203e8	−0.192618
$326$	1.27261e9	2.03438
$327$	8.04287e7	0.127202
$328$	7.67914e8	1.20158
$329$	−5.21451e8	−0.807286
$330$	0	0
$331$	−6.23530e8	−0.945060	−0.472530	−	0.881315i	$-0.656659\pi$
$331$	−6.23530e8	−0.945060	−0.472530	+	0.881315i	$0.656659\pi$
$332$	−1.97008e8	−0.295461
$333$	−3.25214e8	−0.482630
$334$	−2.51289e9	−3.69029
$335$	1.04345e9	1.51641
$336$	−1.64234e9	−2.36198
$337$	7.47270e8	1.06359	0.531793	−	0.846874i	$-0.321518\pi$
$337$	7.47270e8	1.06359	0.531793	+	0.846874i	$0.321518\pi$
$338$	−1.32281e9	−1.86333
$339$	3.84143e8	0.535542
$340$	6.63136e8	0.915011
$341$	0	0
$342$	−7.98849e6	−0.0107988
$343$	−8.13037e8	−1.08788
$344$	−2.36088e9	−3.12693
$345$	−1.03494e8	−0.135690
$346$	−1.20187e9	−1.55988
$347$	−2.67533e8	−0.343736	−0.171868	−	0.985120i	$-0.554980\pi$
$347$	−2.67533e8	−0.343736	−0.171868	+	0.985120i	$0.554980\pi$
$348$	1.24552e9	1.58425
$349$	1.32034e8	0.166263	0.0831315	−	0.996539i	$-0.473508\pi$
$349$	1.32034e8	0.166263	0.0831315	+	0.996539i	$0.473508\pi$
$350$	1.00052e9	1.24735
$351$	4.00777e7	0.0494684
$352$	0	0
$353$	−1.32339e9	−1.60131	−0.800656	−	0.599124i	$-0.795516\pi$
$353$	−1.32339e9	−1.60131	−0.800656	+	0.599124i	$0.795516\pi$
$354$	7.22601e8	0.865739
$355$	1.08569e9	1.28797
$356$	−1.23060e9	−1.44558
$357$	−9.61129e7	−0.111800
$358$	1.61898e9	1.86488
$359$	−8.72846e8	−0.995650	−0.497825	−	0.867277i	$-0.665868\pi$
$359$	−8.72846e8	−0.995650	−0.497825	+	0.867277i	$0.665868\pi$
$360$	1.54226e9	1.74220
$361$	−8.93636e8	−0.999736
$362$	2.60180e9	2.88266
$363$	0	0
$364$	−5.88165e8	−0.639211
$365$	1.35189e9	1.45518
$366$	1.53060e9	1.63184
$367$	−1.05019e9	−1.10901	−0.554506	−	0.832180i	$-0.687093\pi$
$367$	−1.05019e9	−1.10901	−0.554506	+	0.832180i	$0.687093\pi$
$368$	8.33008e8	0.871329
$369$	9.78237e7	0.101357
$370$	−3.72269e9	−3.82077
$371$	−8.68328e8	−0.882826
$372$	−1.06485e9	−1.07247
$373$	9.52962e8	0.950812	0.475406	−	0.879767i	$-0.342301\pi$
$373$	9.52962e8	0.950812	0.475406	+	0.879767i	$0.342301\pi$
$374$	0	0
$375$	1.95456e8	0.191399
$376$	−3.94131e9	−3.82370
$377$	2.46195e8	0.236638
$378$	−3.36388e8	−0.320346
$379$	1.32631e9	1.25143	0.625716	−	0.780051i	$-0.284807\pi$
$379$	1.32631e9	1.25143	0.625716	+	0.780051i	$0.284807\pi$
$380$	−6.84714e7	−0.0640127
$381$	3.85646e7	0.0357233
$382$	−3.11325e9	−2.85754
$383$	1.63402e9	1.48615	0.743073	−	0.669210i	$-0.233367\pi$
$383$	1.63402e9	1.48615	0.743073	+	0.669210i	$0.233367\pi$
$384$	−4.86790e9	−4.38715
$385$	0	0
$386$	1.98754e8	0.175898
$387$	−3.00749e8	−0.263764
$388$	2.38127e9	2.06965
$389$	3.84635e8	0.331303	0.165651	−	0.986184i	$-0.447027\pi$
$389$	3.84635e8	0.331303	0.165651	+	0.986184i	$0.447027\pi$
$390$	4.58765e8	0.391620
$391$	4.87492e7	0.0412429
$392$	−1.43239e9	−1.20105
$393$	−8.23516e8	−0.684381
$394$	1.60389e9	1.32111
$395$	3.97389e7	0.0324434
$396$	0	0
$397$	−2.14022e8	−0.171669	−0.0858345	−	0.996309i	$-0.527356\pi$
$397$	−2.14022e8	−0.171669	−0.0858345	+	0.996309i	$0.527356\pi$
$398$	−1.53443e9	−1.21999
$399$	9.92403e6	0.00782137
$400$	4.70336e9	3.67450
$401$	1.37334e9	1.06359	0.531795	−	0.846873i	$-0.321518\pi$
$401$	1.37334e9	1.06359	0.531795	+	0.846873i	$0.321518\pi$
$402$	−1.72022e9	−1.32067
$403$	−2.10482e8	−0.160195
$404$	5.43890e9	4.10371
$405$	1.96467e8	0.146959
$406$	−2.06641e9	−1.53241
$407$	0	0
$408$	−7.26456e8	−0.529540
$409$	1.88430e9	1.36182	0.680908	−	0.732369i	$-0.261585\pi$
$409$	1.88430e9	1.36182	0.680908	+	0.732369i	$0.261585\pi$
$410$	1.11978e9	0.802395
$411$	−1.15437e9	−0.820163
$412$	5.56488e8	0.392027
$413$	−8.97680e8	−0.627042
$414$	1.70619e8	0.118175
$415$	−1.90897e8	−0.131108
$416$	−2.20105e9	−1.49901
$417$	5.21799e8	0.352393
$418$	0	0
$419$	−4.02965e8	−0.267620	−0.133810	−	0.991007i	$-0.542721\pi$
$419$	−4.02965e8	−0.267620	−0.133810	+	0.991007i	$0.542721\pi$
$420$	−2.88327e9	−1.89895
$421$	−2.30810e6	−0.00150754	−0.000753768	−	1.00000i	$-0.500240\pi$
$421$	−2.30810e6	−0.00150754	−0.000753768		1.00000i	$0.500240\pi$
$422$	8.39427e8	0.543738
$423$	−5.02079e8	−0.322538
$424$	−6.56314e9	−4.18149
$425$	2.75250e8	0.173926
$426$	−1.78985e9	−1.12172
$427$	−1.90145e9	−1.18192
$428$	8.18696e8	0.504742
$429$	0	0
$430$	−3.44265e9	−2.08811
$431$	1.32080e9	0.794631	0.397315	−	0.917682i	$-0.369942\pi$
$431$	1.32080e9	0.794631	0.397315	+	0.917682i	$0.369942\pi$
$432$	−1.58133e9	−0.943690
$433$	−3.17778e8	−0.188112	−0.0940559	−	0.995567i	$-0.529983\pi$
$433$	−3.17778e8	−0.188112	−0.0940559	+	0.995567i	$0.529983\pi$
$434$	1.76666e9	1.03738
$435$	1.20688e9	0.702996
$436$	−1.13649e9	−0.656695
$437$	−5.03355e6	−0.00288529
$438$	−2.22870e9	−1.26734
$439$	1.30278e9	0.734928	0.367464	−	0.930038i	$-0.380226\pi$
$439$	1.30278e9	0.734928	0.367464	+	0.930038i	$0.380226\pi$
$440$	0	0
$441$	−1.82471e8	−0.101311
$442$	−2.16094e8	−0.119032
$443$	−1.19448e9	−0.652779	−0.326389	−	0.945235i	$-0.605832\pi$
$443$	−1.19448e9	−0.652779	−0.326389	+	0.945235i	$0.605832\pi$
$444$	4.59541e9	2.49163
$445$	−1.19243e9	−0.641463
$446$	−1.78979e9	−0.955277
$447$	−1.88071e9	−0.995969
$448$	1.06884e10	5.61616
$449$	−7.10632e8	−0.370495	−0.185247	−	0.982692i	$-0.559309\pi$
$449$	−7.10632e8	−0.370495	−0.185247	+	0.982692i	$0.559309\pi$
$450$	9.63354e8	0.498359
$451$	0	0
$452$	−5.42810e9	−2.76480
$453$	1.69884e9	0.858638
$454$	−7.64050e9	−3.83201
$455$	−5.69920e8	−0.283644
$456$	7.50095e7	0.0370458
$457$	3.47868e8	0.170494	0.0852468	−	0.996360i	$-0.472832\pi$
$457$	3.47868e8	0.170494	0.0852468	+	0.996360i	$0.472832\pi$
$458$	5.18614e9	2.52241
$459$	−9.25425e7	−0.0446680
$460$	1.46242e9	0.700517
$461$	1.99406e9	0.947948	0.473974	−	0.880539i	$-0.342819\pi$
$461$	1.99406e9	0.947948	0.473974	+	0.880539i	$0.342819\pi$
$462$	0	0
$463$	−9.86608e7	−0.0461967	−0.0230984	−	0.999733i	$-0.507353\pi$
$463$	−9.86608e7	−0.0461967	−0.0230984	+	0.999733i	$0.507353\pi$
$464$	−9.71403e9	−4.51426
$465$	−1.03181e9	−0.475901
$466$	3.89389e9	1.78251
$467$	4.62965e7	0.0210348	0.0105174	−	0.999945i	$-0.496652\pi$
$467$	4.62965e7	0.0210348	0.0105174	+	0.999945i	$0.496652\pi$
$468$	−5.66315e8	−0.255386
$469$	2.13701e9	0.956539
$470$	−5.74725e9	−2.55339
$471$	−2.13914e7	−0.00943336
$472$	−6.78499e9	−2.96997
$473$	0	0
$474$	−6.55130e7	−0.0282555
$475$	−2.84206e7	−0.0121676
$476$	1.35812e9	0.577182
$477$	−8.36070e8	−0.352719
$478$	−1.33580e9	−0.559427
$479$	−2.66837e9	−1.10936	−0.554680	−	0.832064i	$-0.687159\pi$
$479$	−2.66837e9	−1.10936	−0.554680	+	0.832064i	$0.687159\pi$
$480$	−1.07899e10	−4.45320
$481$	9.08351e8	0.372174
$482$	−7.03637e9	−2.86209
$483$	−2.11958e8	−0.0855924
$484$	0	0
$485$	2.30740e9	0.918390
$486$	−3.23892e8	−0.127989
$487$	−3.49516e9	−1.37125	−0.685623	−	0.727957i	$-0.740470\pi$
$487$	−3.49516e9	−1.37125	−0.685623	+	0.727957i	$0.740470\pi$
$488$	−1.43719e10	−5.59815
$489$	−1.52222e9	−0.588702
$490$	−2.08872e9	−0.802038
$491$	1.11582e7	0.00425413	0.00212706	−	0.999998i	$-0.499323\pi$
$491$	1.11582e7	0.00425413	0.00212706	+	0.999998i	$0.499323\pi$
$492$	−1.38229e9	−0.523265
$493$	−5.68483e8	−0.213675
$494$	2.23125e7	0.00832731
$495$	0	0
$496$	8.30492e9	3.05597
$497$	2.22352e9	0.812445
$498$	3.14710e8	0.114185
$499$	−1.66807e9	−0.600984	−0.300492	−	0.953784i	$-0.597151\pi$
$499$	−1.66807e9	−0.600984	−0.300492	+	0.953784i	$0.597151\pi$
$500$	−2.76187e9	−0.988118
$501$	3.00577e9	1.06788
$502$	−1.00787e10	−3.55585
$503$	3.13985e9	1.10007	0.550035	−	0.835141i	$-0.314614\pi$
$503$	3.13985e9	1.10007	0.550035	+	0.835141i	$0.314614\pi$
$504$	3.15858e9	1.09897
$505$	5.27019e9	1.82099
$506$	0	0
$507$	1.58227e9	0.539203
$508$	−5.44934e8	−0.184425
$509$	−1.32992e9	−0.447005	−0.223503	−	0.974703i	$-0.571749\pi$
$509$	−1.32992e9	−0.447005	−0.223503	+	0.974703i	$0.571749\pi$
$510$	−1.05932e9	−0.353617
$511$	2.76870e9	0.917914
$512$	2.79972e10	9.21871
$513$	9.55537e6	0.00312490
$514$	−7.94644e9	−2.58108
$515$	5.39226e8	0.173959
$516$	4.24972e9	1.36171
$517$	0	0
$518$	−7.62415e9	−2.41011
$519$	1.43761e9	0.451392
$520$	−4.30766e9	−1.34348
$521$	1.06631e9	0.330332	0.165166	−	0.986266i	$-0.447184\pi$
$521$	1.06631e9	0.330332	0.165166	+	0.986266i	$0.447184\pi$
$522$	−1.98965e9	−0.612252
$523$	2.59234e9	0.792385	0.396193	−	0.918167i	$-0.370331\pi$
$523$	2.59234e9	0.792385	0.396193	+	0.918167i	$0.370331\pi$
$524$	1.16366e10	3.53319
$525$	−1.19677e9	−0.360954
$526$	6.21729e9	1.86273
$527$	4.86020e8	0.144649
$528$	0	0
$529$	−3.29732e9	−0.968425
$530$	−9.57041e9	−2.79232
$531$	−8.64332e8	−0.250524
$532$	−1.40231e8	−0.0403787
$533$	−2.73230e8	−0.0781598
$534$	1.96582e9	0.558661
$535$	7.93300e8	0.223975
$536$	1.61523e10	4.53063
$537$	−1.93653e9	−0.539653
$538$	−6.97679e9	−1.93160
$539$	0	0
$540$	−2.77616e9	−0.758693
$541$	1.68997e9	0.458869	0.229435	−	0.973324i	$-0.426312\pi$
$541$	1.68997e9	0.458869	0.229435	+	0.973324i	$0.426312\pi$
$542$	7.18110e9	1.93728
$543$	−3.11212e9	−0.834174
$544$	5.08239e9	1.35354
$545$	−1.10124e9	−0.291403
$546$	9.39561e8	0.247031
$547$	1.31553e9	0.343674	0.171837	−	0.985125i	$-0.445030\pi$
$547$	1.31553e9	0.343674	0.171837	+	0.985125i	$0.445030\pi$
$548$	1.63118e10	4.23419
$549$	−1.83082e9	−0.472217
$550$	0	0
$551$	5.86981e7	0.0149484
$552$	−1.60206e9	−0.405407
$553$	8.13862e7	0.0204650
$554$	9.42121e9	2.35409
$555$	4.45286e9	1.10564
$556$	−7.37324e9	−1.81927
$557$	3.52954e9	0.865416	0.432708	−	0.901534i	$-0.357558\pi$
$557$	3.52954e9	0.865416	0.432708	+	0.901534i	$0.357558\pi$
$558$	1.70103e9	0.414470
$559$	8.40018e8	0.203398
$560$	2.24871e10	5.41097
$561$	0	0
$562$	−6.29270e8	−0.149541
$563$	1.74244e9	0.411507	0.205754	−	0.978604i	$-0.434035\pi$
$563$	1.74244e9	0.411507	0.205754	+	0.978604i	$0.434035\pi$
$564$	7.09460e9	1.66514
$565$	−5.25972e9	−1.22686
$566$	1.58120e9	0.366547
$567$	4.02368e8	0.0927006
$568$	1.68062e10	3.84813
$569$	1.65109e9	0.375731	0.187865	−	0.982195i	$-0.439843\pi$
$569$	1.65109e9	0.375731	0.187865	+	0.982195i	$0.439843\pi$
$570$	1.09379e8	0.0247385
$571$	−1.50229e9	−0.337698	−0.168849	−	0.985642i	$-0.554005\pi$
$571$	−1.50229e9	−0.337698	−0.168849	+	0.985642i	$0.554005\pi$
$572$	0	0
$573$	3.72389e9	0.826905
$574$	2.29333e9	0.506145
$575$	6.07009e8	0.133155
$576$	1.02913e10	2.24384
$577$	−7.21698e9	−1.56401	−0.782007	−	0.623270i	$-0.785804\pi$
$577$	−7.21698e9	−1.56401	−0.782007	+	0.623270i	$0.785804\pi$
$578$	−8.76343e9	−1.88767
$579$	−2.37738e8	−0.0509007
$580$	−1.70538e10	−3.62930
$581$	−3.90961e8	−0.0827022
$582$	−3.80395e9	−0.799841
$583$	0	0
$584$	2.09268e10	4.34768
$585$	−5.48748e8	−0.113325
$586$	1.15197e9	0.236482
$587$	−8.76023e9	−1.78765	−0.893825	−	0.448416i	$-0.851988\pi$
$587$	−8.76023e9	−1.78765	−0.893825	+	0.448416i	$0.851988\pi$
$588$	2.57839e9	0.523032
$589$	−5.01834e7	−0.0101194
$590$	−9.89392e9	−1.98329
$591$	−1.91848e9	−0.382297
$592$	−3.58404e10	−7.09982
$593$	−5.37529e9	−1.05855	−0.529273	−	0.848451i	$-0.677535\pi$
$593$	−5.37529e9	−1.05855	−0.529273	+	0.848451i	$0.677535\pi$
$594$	0	0
$595$	1.31599e9	0.256119
$596$	2.65753e10	5.14181
$597$	1.83539e9	0.353036
$598$	−4.76553e8	−0.0911290
$599$	4.15321e9	0.789568	0.394784	−	0.918774i	$-0.370819\pi$
$599$	4.15321e9	0.789568	0.394784	+	0.918774i	$0.370819\pi$
$600$	−9.04559e9	−1.70965
$601$	−2.03055e9	−0.381552	−0.190776	−	0.981634i	$-0.561100\pi$
$601$	−2.03055e9	−0.381552	−0.190776	+	0.981634i	$0.561100\pi$
$602$	−7.05061e9	−1.31716
$603$	2.05763e9	0.382170
$604$	−2.40054e10	−4.43282
$605$	0	0
$606$	−8.68835e9	−1.58593
$607$	−1.06928e10	−1.94058	−0.970292	−	0.241935i	$-0.922218\pi$
$607$	−1.06928e10	−1.94058	−0.970292	+	0.241935i	$0.922218\pi$
$608$	−5.24777e8	−0.0946917
$609$	2.47172e9	0.443445
$610$	−2.09572e10	−3.73834
$611$	1.40235e9	0.248721
$612$	1.30766e9	0.230604
$613$	6.43717e9	1.12871	0.564356	−	0.825531i	$-0.309124\pi$
$613$	6.43717e9	1.12871	0.564356	+	0.825531i	$0.309124\pi$
$614$	1.45520e10	2.53708
$615$	−1.33941e9	−0.232194
$616$	0	0
$617$	2.64748e9	0.453768	0.226884	−	0.973922i	$-0.427146\pi$
$617$	2.64748e9	0.453768	0.226884	+	0.973922i	$0.427146\pi$
$618$	−8.88960e8	−0.151503
$619$	−6.80618e9	−1.15342	−0.576708	−	0.816950i	$-0.695663\pi$
$619$	−6.80618e9	−1.15342	−0.576708	+	0.816950i	$0.695663\pi$
$620$	1.45800e10	2.45689
$621$	−2.04084e8	−0.0341971
$622$	−7.65189e9	−1.27498
$623$	−2.44212e9	−0.404630
$624$	4.41679e9	0.727714
$625$	−7.24989e9	−1.18782
$626$	4.68792e9	0.763783
$627$	0	0
$628$	3.02270e8	0.0487008
$629$	−2.09745e9	−0.336058
$630$	4.60586e9	0.733870
$631$	6.80335e9	1.07800	0.539002	−	0.842305i	$-0.318802\pi$
$631$	6.80335e9	1.07800	0.539002	+	0.842305i	$0.318802\pi$
$632$	6.15146e8	0.0969323
$633$	−1.00407e9	−0.157345
$634$	−4.83823e9	−0.754004
$635$	−5.28030e8	−0.0818372
$636$	1.18140e10	1.82095
$637$	5.09656e8	0.0781249
$638$	0	0
$639$	2.14092e9	0.324599
$640$	6.66518e10	10.0504
$641$	−1.23182e9	−0.184733	−0.0923665	−	0.995725i	$-0.529443\pi$
$641$	−1.23182e9	−0.184733	−0.0923665	+	0.995725i	$0.529443\pi$
$642$	−1.30782e9	−0.195063
$643$	8.94158e9	1.32640	0.663202	−	0.748440i	$-0.269197\pi$
$643$	8.94158e9	1.32640	0.663202	+	0.748440i	$0.269197\pi$
$644$	2.99506e9	0.441880
$645$	4.11789e9	0.604249
$646$	−5.15213e7	−0.00751923
$647$	−6.39256e9	−0.927919	−0.463959	−	0.885857i	$-0.653572\pi$
$647$	−6.39256e9	−0.927919	−0.463959	+	0.885857i	$0.653572\pi$
$648$	3.04124e9	0.439075
$649$	0	0
$650$	−2.69073e9	−0.384303
$651$	−2.11318e9	−0.300194
$652$	2.15096e10	3.03924
$653$	1.19315e10	1.67687	0.838434	−	0.545004i	$-0.183472\pi$
$653$	1.19315e10	1.67687	0.838434	+	0.545004i	$0.183472\pi$
$654$	1.81548e9	0.253788
$655$	1.12757e10	1.56782
$656$	1.07807e10	1.49102
$657$	2.66584e9	0.366738
$658$	−1.17705e10	−1.61066
$659$	−8.83201e9	−1.20215	−0.601077	−	0.799191i	$-0.705262\pi$
$659$	−8.83201e9	−1.20215	−0.601077	+	0.799191i	$0.705262\pi$
$660$	0	0
$661$	−5.97465e9	−0.804651	−0.402325	−	0.915497i	$-0.631798\pi$
$661$	−5.97465e9	−0.804651	−0.402325	+	0.915497i	$0.631798\pi$
$662$	−1.40747e10	−1.88554
$663$	2.58479e8	0.0344451
$664$	−2.95503e9	−0.391717
$665$	−1.35881e8	−0.0179177
$666$	−7.34092e9	−0.962921
$667$	−1.25368e9	−0.163586
$668$	−4.24728e10	−5.51307
$669$	2.14084e9	0.276435
$670$	2.35534e10	3.02547
$671$	0	0
$672$	−2.20979e10	−2.80904
$673$	6.44511e9	0.815038	0.407519	−	0.913197i	$-0.366394\pi$
$673$	6.44511e9	0.815038	0.407519	+	0.913197i	$0.366394\pi$
$674$	1.68678e10	2.12202
$675$	−1.15231e9	−0.144213
$676$	−2.23581e10	−2.78370
$677$	4.29897e9	0.532480	0.266240	−	0.963907i	$-0.414219\pi$
$677$	4.29897e9	0.532480	0.266240	+	0.963907i	$0.414219\pi$
$678$	8.67110e9	1.06849
$679$	4.72561e9	0.579313
$680$	9.94671e9	1.21311
$681$	9.13911e9	1.10889
$682$	0	0
$683$	3.81095e8	0.0457678	0.0228839	−	0.999738i	$-0.492715\pi$
$683$	3.81095e8	0.0457678	0.0228839	+	0.999738i	$0.492715\pi$
$684$	−1.35021e8	−0.0161327
$685$	1.58058e10	1.87888
$686$	−1.83523e10	−2.17049
$687$	−6.20335e9	−0.729925
$688$	−3.31443e10	−3.88015
$689$	2.33522e9	0.271994
$690$	−2.33613e9	−0.270723
$691$	8.25120e9	0.951358	0.475679	−	0.879619i	$-0.342202\pi$
$691$	8.25120e9	0.951358	0.475679	+	0.879619i	$0.342202\pi$
$692$	−2.03140e10	−2.33036
$693$	0	0
$694$	−6.03892e9	−0.685806
$695$	−7.14452e9	−0.807284
$696$	1.86822e10	2.10037
$697$	6.30909e8	0.0705751
$698$	2.98034e9	0.331720
$699$	−4.65764e9	−0.515817
$700$	1.69108e10	1.86346
$701$	8.83850e9	0.969093	0.484547	−	0.874765i	$-0.338985\pi$
$701$	8.83850e9	0.969093	0.484547	+	0.874765i	$0.338985\pi$
$702$	9.04658e8	0.0986971
$703$	2.16570e8	0.0235101
$704$	0	0
$705$	6.87452e9	0.738891
$706$	−2.98723e10	−3.19486
$707$	1.07935e10	1.14866
$708$	1.22134e10	1.29336
$709$	1.15716e10	1.21936	0.609680	−	0.792648i	$-0.291298\pi$
$709$	1.15716e10	1.21936	0.609680	+	0.792648i	$0.291298\pi$
$710$	2.45069e10	2.56971
$711$	7.83628e7	0.00817648
$712$	−1.84584e10	−1.91652
$713$	1.07182e9	0.110741
$714$	−2.16952e9	−0.223059
$715$	0	0
$716$	2.73640e10	2.78602
$717$	1.59780e9	0.161885
$718$	−1.97024e10	−1.98648
$719$	3.22237e9	0.323313	0.161657	−	0.986847i	$-0.448316\pi$
$719$	3.22237e9	0.323313	0.161657	+	0.986847i	$0.448316\pi$
$720$	2.16517e10	2.16187
$721$	1.10435e9	0.109732
$722$	−2.01717e10	−1.99463
$723$	8.41649e9	0.828223
$724$	4.39756e10	4.30652
$725$	−7.07856e9	−0.689861
$726$	0	0
$727$	4.17821e9	0.403292	0.201646	−	0.979458i	$-0.435371\pi$
$727$	4.17821e9	0.403292	0.201646	+	0.979458i	$0.435371\pi$
$728$	−8.82218e9	−0.847454
$729$	3.87420e8	0.0370370
$730$	3.05156e10	2.90330
$731$	−1.93967e9	−0.183661
$732$	2.58702e10	2.43788
$733$	−5.44726e9	−0.510874	−0.255437	−	0.966826i	$-0.582219\pi$
$733$	−5.44726e9	−0.510874	−0.255437	+	0.966826i	$0.582219\pi$
$734$	−2.37055e10	−2.21265
$735$	2.49841e9	0.232091
$736$	1.12082e10	1.03625
$737$	0	0
$738$	2.20813e9	0.202222
$739$	−4.05591e9	−0.369686	−0.184843	−	0.982768i	$-0.559178\pi$
$739$	−4.05591e9	−0.369686	−0.184843	+	0.982768i	$0.559178\pi$
$740$	−6.29209e10	−5.70800
$741$	−2.66890e7	−0.00240973
$742$	−1.96004e10	−1.76137
$743$	2.80251e9	0.250661	0.125330	−	0.992115i	$-0.460001\pi$
$743$	2.80251e9	0.250661	0.125330	+	0.992115i	$0.460001\pi$
$744$	−1.59722e10	−1.42187
$745$	2.57509e10	2.28163
$746$	2.15108e10	1.89702
$747$	−3.76437e8	−0.0330423
$748$	0	0
$749$	1.62469e9	0.141282
$750$	4.41194e9	0.381870
$751$	−1.66390e10	−1.43346	−0.716732	−	0.697349i	$-0.754363\pi$
$751$	−1.66390e10	−1.43346	−0.716732	+	0.697349i	$0.754363\pi$
$752$	−5.53320e10	−4.74476
$753$	1.20556e10	1.02898
$754$	5.55726e9	0.472130
$755$	−2.32608e10	−1.96702
$756$	−5.68563e9	−0.478578
$757$	−7.88200e9	−0.660390	−0.330195	−	0.943913i	$-0.607115\pi$
$757$	−7.88200e9	−0.660390	−0.330195	+	0.943913i	$0.607115\pi$
$758$	2.99382e10	2.49680
$759$	0	0
$760$	−1.02704e9	−0.0848669
$761$	2.13478e10	1.75592	0.877962	−	0.478730i	$-0.158903\pi$
$761$	2.13478e10	1.75592	0.877962	+	0.478730i	$0.158903\pi$
$762$	8.70502e8	0.0712734
$763$	−2.25536e9	−0.183815
$764$	−5.26201e10	−4.26899
$765$	1.26710e9	0.102328
$766$	3.68841e10	2.96509
$767$	2.41415e9	0.193188
$768$	−6.10925e10	−4.86658
$769$	−1.93037e10	−1.53073	−0.765365	−	0.643596i	$-0.777442\pi$
$769$	−1.93037e10	−1.53073	−0.765365	+	0.643596i	$0.777442\pi$
$770$	0	0
$771$	9.50506e9	0.746904
$772$	3.35934e9	0.262781
$773$	1.04348e10	0.812562	0.406281	−	0.913748i	$-0.366825\pi$
$773$	1.04348e10	0.812562	0.406281	+	0.913748i	$0.366825\pi$
$774$	−6.78869e9	−0.526251
$775$	6.05175e9	0.467009
$776$	3.57179e10	2.74391
$777$	9.11956e9	0.697429
$778$	8.68221e9	0.661000
$779$	−6.51438e7	−0.00493733
$780$	7.75405e9	0.585056
$781$	0	0
$782$	1.10040e9	0.0822859
$783$	2.37990e9	0.177171
$784$	−2.01093e10	−1.49036
$785$	2.92893e8	0.0216105
$786$	−1.85889e10	−1.36544
$787$	−1.43512e10	−1.04949	−0.524744	−	0.851260i	$-0.675839\pi$
$787$	−1.43512e10	−1.04949	−0.524744	+	0.851260i	$0.675839\pi$
$788$	2.71089e10	1.97365
$789$	−7.43676e9	−0.539031
$790$	8.97011e8	0.0647296
$791$	−1.07720e10	−0.773891
$792$	0	0
$793$	5.11363e9	0.364144
$794$	−4.83103e9	−0.342506
$795$	1.14476e10	0.808031
$796$	−2.59349e10	−1.82259
$797$	−1.61771e10	−1.13187	−0.565935	−	0.824450i	$-0.691485\pi$
$797$	−1.61771e10	−1.13187	−0.565935	+	0.824450i	$0.691485\pi$
$798$	2.24011e8	0.0156048
$799$	−3.23813e9	−0.224585
$800$	6.32842e10	4.36999
$801$	−2.35139e9	−0.161663
$802$	3.09999e10	2.12202
$803$	0	0
$804$	−2.90751e10	−1.97299
$805$	2.90215e9	0.196081
$806$	−4.75113e9	−0.319613
$807$	8.34523e9	0.558960
$808$	8.15809e10	5.44062
$809$	7.11605e9	0.472519	0.236259	−	0.971690i	$-0.424078\pi$
$809$	7.11605e9	0.472519	0.236259	+	0.971690i	$0.424078\pi$
$810$	4.43476e9	0.293206
$811$	2.50874e10	1.65152	0.825759	−	0.564024i	$-0.190747\pi$
$811$	2.50874e10	1.65152	0.825759	+	0.564024i	$0.190747\pi$
$812$	−3.49265e10	−2.28933
$813$	−8.58961e9	−0.560604
$814$	0	0
$815$	2.08423e10	1.34864
$816$	−1.01987e10	−0.657097
$817$	2.00278e8	0.0128486
$818$	4.25335e10	2.71703
$819$	−1.12385e9	−0.0714848
$820$	1.89265e10	1.19873
$821$	1.43609e10	0.905691	0.452846	−	0.891589i	$-0.350409\pi$
$821$	1.43609e10	0.905691	0.452846	+	0.891589i	$0.350409\pi$
$822$	−2.60572e10	−1.63635
$823$	−1.84581e10	−1.15422	−0.577109	−	0.816667i	$-0.695819\pi$
$823$	−1.84581e10	−1.15422	−0.577109	+	0.816667i	$0.695819\pi$
$824$	8.34705e9	0.519742
$825$	0	0
$826$	−2.02630e10	−1.25104
$827$	1.56854e9	0.0964334	0.0482167	−	0.998837i	$-0.484646\pi$
$827$	1.56854e9	0.0964334	0.0482167	+	0.998837i	$0.484646\pi$
$828$	2.88380e9	0.176546
$829$	2.62637e10	1.60109	0.800545	−	0.599273i	$-0.204543\pi$
$829$	2.62637e10	1.60109	0.800545	+	0.599273i	$0.204543\pi$
$830$	−4.30904e9	−0.261581
$831$	−1.12691e10	−0.681217
$832$	−2.87446e10	−1.73031
$833$	−1.17683e9	−0.0705437
$834$	1.17783e10	0.703077
$835$	−4.11553e10	−2.44638
$836$	0	0
$837$	−2.03468e9	−0.119938
$838$	−9.09596e9	−0.533943
$839$	−9.69865e9	−0.566950	−0.283475	−	0.958980i	$-0.591487\pi$
$839$	−9.69865e9	−0.566950	−0.283475	+	0.958980i	$0.591487\pi$
$840$	−4.32476e10	−2.51759
$841$	−2.63027e9	−0.152480
$842$	−5.20998e7	−0.00300776
$843$	7.52695e8	0.0432736
$844$	1.41880e10	0.812311
$845$	−2.16646e10	−1.23524
$846$	−1.13332e10	−0.643513
$847$	0	0
$848$	−9.21397e10	−5.18873
$849$	−1.89134e9	−0.106070
$850$	6.21310e9	0.347010
$851$	−4.62551e9	−0.257280
$852$	−3.02521e10	−1.67578
$853$	2.21086e10	1.21966	0.609830	−	0.792532i	$-0.291238\pi$
$853$	2.21086e10	1.21966	0.609830	+	0.792532i	$0.291238\pi$
$854$	−4.29207e10	−2.35811
$855$	−1.30833e8	−0.00715873
$856$	1.22800e10	0.669178
$857$	1.69648e10	0.920696	0.460348	−	0.887739i	$-0.347725\pi$
$857$	1.69648e10	0.920696	0.460348	+	0.887739i	$0.347725\pi$
$858$	0	0
$859$	5.51307e9	0.296768	0.148384	−	0.988930i	$-0.452593\pi$
$859$	5.51307e9	0.296768	0.148384	+	0.988930i	$0.452593\pi$
$860$	−5.81875e10	−3.11950
$861$	−2.74315e9	−0.146466
$862$	2.98138e10	1.58541
$863$	−6.16691e9	−0.326610	−0.163305	−	0.986576i	$-0.552215\pi$
$863$	−6.16691e9	−0.326610	−0.163305	+	0.986576i	$0.552215\pi$
$864$	−2.12770e10	−1.12231
$865$	−1.96838e10	−1.03408
$866$	−7.17307e9	−0.375312
$867$	1.04823e10	0.546248
$868$	2.98601e10	1.54979
$869$	0	0
$870$	2.72425e10	1.40259
$871$	−5.74713e9	−0.294705
$872$	−1.70468e10	−0.870634
$873$	4.55006e9	0.231455
$874$	−1.13620e8	−0.00575659
$875$	−5.48092e9	−0.276583
$876$	−3.76695e10	−1.89333
$877$	9.33197e9	0.467170	0.233585	−	0.972336i	$-0.424954\pi$
$877$	9.33197e9	0.467170	0.233585	+	0.972336i	$0.424954\pi$
$878$	2.94071e10	1.46629
$879$	−1.37792e9	−0.0684324
$880$	0	0
$881$	8.00522e9	0.394419	0.197209	−	0.980361i	$-0.436812\pi$
$881$	8.00522e9	0.394419	0.197209	+	0.980361i	$0.436812\pi$
$882$	−4.11884e9	−0.202132
$883$	−1.44308e10	−0.705387	−0.352693	−	0.935739i	$-0.614734\pi$
$883$	−1.44308e10	−0.705387	−0.352693	+	0.935739i	$0.614734\pi$
$884$	−3.65242e9	−0.177827
$885$	1.18345e10	0.573918
$886$	−2.69625e10	−1.30239
$887$	1.05088e10	0.505615	0.252807	−	0.967517i	$-0.418646\pi$
$887$	1.05088e10	0.505615	0.252807	+	0.967517i	$0.418646\pi$
$888$	6.89289e10	3.30336
$889$	−1.08142e9	−0.0516223
$890$	−2.69162e10	−1.27982
$891$	0	0
$892$	−3.02510e10	−1.42713
$893$	3.34350e8	0.0157116
$894$	−4.24525e10	−1.98711
$895$	2.65152e10	1.23627
$896$	1.36504e11	6.33969
$897$	5.70024e8	0.0263706
$898$	−1.60408e10	−0.739194
$899$	−1.24989e10	−0.573737
$900$	1.62826e10	0.744517
$901$	−5.39219e9	−0.245600
$902$	0	0
$903$	8.43353e9	0.381155
$904$	−8.14189e10	−3.66552
$905$	4.26115e10	1.91098
$906$	3.83473e10	1.71311
$907$	8.15333e9	0.362835	0.181418	−	0.983406i	$-0.441931\pi$
$907$	8.15333e9	0.362835	0.181418	+	0.983406i	$0.441931\pi$
$908$	−1.29140e11	−5.72478
$909$	1.03925e10	0.458930
$910$	−1.28646e10	−0.565914
$911$	−4.19000e10	−1.83611	−0.918057	−	0.396448i	$-0.870243\pi$
$911$	−4.19000e10	−1.83611	−0.918057	+	0.396448i	$0.870243\pi$
$912$	1.05306e9	0.0459695
$913$	0	0
$914$	7.85228e9	0.340161
$915$	2.50677e10	1.08179
$916$	8.76560e10	3.76832
$917$	2.30928e10	0.988971
$918$	−2.08892e9	−0.0891195
$919$	4.24752e10	1.80522	0.902612	−	0.430455i	$-0.141647\pi$
$919$	4.24752e10	1.80522	0.902612	+	0.430455i	$0.141647\pi$
$920$	2.19355e10	0.928732
$921$	−1.74063e10	−0.734172
$922$	4.50110e10	1.89130
$923$	−5.97977e9	−0.250310
$924$	0	0
$925$	−2.61167e10	−1.08498
$926$	−2.22703e9	−0.0921696
$927$	1.06332e9	0.0438415
$928$	−1.30703e11	−5.36869
$929$	3.11681e10	1.27543	0.637713	−	0.770274i	$-0.279881\pi$
$929$	3.11681e10	1.27543	0.637713	+	0.770274i	$0.279881\pi$
$930$	−2.32907e10	−0.949495
$931$	1.21513e8	0.00493512
$932$	6.58145e10	2.66297
$933$	9.15275e9	0.368948
$934$	1.04503e9	0.0419677
$935$	0	0
$936$	−8.49445e9	−0.338587
$937$	−3.47809e10	−1.38119	−0.690594	−	0.723243i	$-0.742651\pi$
$937$	−3.47809e10	−1.38119	−0.690594	+	0.723243i	$0.742651\pi$
$938$	4.82379e10	1.90844
$939$	−5.60741e9	−0.221021
$940$	−9.71399e10	−3.81461
$941$	1.69704e10	0.663940	0.331970	−	0.943290i	$-0.392287\pi$
$941$	1.69704e10	0.663940	0.331970	+	0.943290i	$0.392287\pi$
$942$	−4.82859e8	−0.0188210
$943$	1.39134e9	0.0540311
$944$	−9.52543e10	−3.68538
$945$	−5.50926e9	−0.212365
$946$	0	0
$947$	3.94704e10	1.51024	0.755121	−	0.655585i	$-0.227578\pi$
$947$	3.94704e10	1.51024	0.755121	+	0.655585i	$0.227578\pi$
$948$	−1.10730e9	−0.0422120
$949$	−7.44592e9	−0.282805
$950$	−6.41526e8	−0.0242763
$951$	5.78720e9	0.218191
$952$	2.03711e10	0.765217
$953$	3.97393e10	1.48729	0.743643	−	0.668577i	$-0.233096\pi$
$953$	3.97393e10	1.48729	0.743643	+	0.668577i	$0.233096\pi$
$954$	−1.88723e10	−0.703728
$955$	−5.09879e10	−1.89433
$956$	−2.25777e10	−0.835750
$957$	0	0
$958$	−6.02321e10	−2.21334
$959$	3.23706e10	1.18518
$960$	−1.40910e11	−5.14034
$961$	−1.68268e10	−0.611603
$962$	2.05038e10	0.742544
$963$	1.56434e9	0.0564468
$964$	−1.18929e11	−4.27579
$965$	3.25514e9	0.116607
$966$	−4.78444e9	−0.170770
$967$	4.14224e10	1.47314	0.736568	−	0.676363i	$-0.236445\pi$
$967$	4.14224e10	1.47314	0.736568	+	0.676363i	$0.236445\pi$
$968$	0	0
$969$	6.16268e7	0.00217589
$970$	5.20840e10	1.83233
$971$	4.51848e10	1.58389	0.791945	−	0.610592i	$-0.209068\pi$
$971$	4.51848e10	1.58389	0.791945	+	0.610592i	$0.209068\pi$
$972$	−5.47442e9	−0.191208
$973$	−1.46321e10	−0.509228
$974$	−7.88947e10	−2.73584
$975$	3.21849e9	0.111208
$976$	−2.01766e11	−6.94664
$977$	2.53625e10	0.870082	0.435041	−	0.900411i	$-0.356734\pi$
$977$	2.53625e10	0.870082	0.435041	+	0.900411i	$0.356734\pi$
$978$	−3.43603e10	−1.17455
$979$	0	0
$980$	−3.53036e10	−1.19819
$981$	−2.17158e9	−0.0734401
$982$	2.51870e8	0.00848764
$983$	−6.71205e9	−0.225381	−0.112691	−	0.993630i	$-0.535947\pi$
$983$	−6.71205e9	−0.225381	−0.112691	+	0.993630i	$0.535947\pi$
$984$	−2.07337e10	−0.693735
$985$	2.62680e10	0.875790
$986$	−1.28321e10	−0.426314
$987$	1.40792e10	0.466087
$988$	3.77126e8	0.0124405
$989$	−4.27755e9	−0.140607
$990$	0	0
$991$	2.79748e10	0.913082	0.456541	−	0.889702i	$-0.349088\pi$
$991$	2.79748e10	0.913082	0.456541	+	0.889702i	$0.349088\pi$
$992$	1.11744e11	3.63439
$993$	1.68353e10	0.545631
$994$	5.01906e10	1.62095
$995$	−2.51304e10	−0.808757
$996$	5.31922e9	0.170585
$997$	−4.39464e10	−1.40440	−0.702200	−	0.711980i	$-0.747799\pi$
$997$	−4.39464e10	−1.40440	−0.702200	+	0.711980i	$0.747799\pi$
$998$	−3.76527e10	−1.19906
$999$	8.78078e9	0.278647

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.8.a.q.1.14		14
11.5	even	5		33.8.e.a.25.1	yes	28
11.9	even	5		33.8.e.a.4.1	✓	28
11.10	odd	2		363.8.a.p.1.1		14
33.5	odd	10		99.8.f.c.91.7		28
33.20	odd	10		99.8.f.c.37.7		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.a.4.1	✓	28	11.9	even	5
33.8.e.a.25.1	yes	28	11.5	even	5
99.8.f.c.37.7		28	33.20	odd	10
99.8.f.c.91.7		28	33.5	odd	10
363.8.a.p.1.1		14	11.10	odd	2
363.8.a.q.1.14		14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q+22.5726 q^{2} -27.0000 q^{3} +381.522 q^{4} +369.687 q^{5} -609.460 q^{6} +757.126 q^{7} +5722.64 q^{8} +729.000 q^{9} +8344.78 q^{10} -10301.1 q^{12} -2036.16 q^{13} +17090.3 q^{14} -9981.54 q^{15} +80339.9 q^{16} +4701.64 q^{17} +16455.4 q^{18} -485.463 q^{19} +141043. q^{20} -20442.4 q^{21} +10368.5 q^{23} -154511. q^{24} +58543.3 q^{25} -45961.4 q^{26} -19683.0 q^{27} +288860. q^{28} -120912. q^{29} -225309. q^{30} +103372. q^{31} +1.08098e6 q^{32} +106128. q^{34} +279900. q^{35} +278129. q^{36} -446110. q^{37} -10958.2 q^{38} +54976.3 q^{39} +2.11558e6 q^{40} +134189. q^{41} -461438. q^{42} -412551. q^{43} +269502. q^{45} +234045. q^{46} -688723. q^{47} -2.16918e6 q^{48} -250303. q^{49} +1.32147e6 q^{50} -126944. q^{51} -776838. q^{52} -1.14687e6 q^{53} -444296. q^{54} +4.33276e6 q^{56} +13107.5 q^{57} -2.72929e6 q^{58} -1.18564e6 q^{59} -3.80817e6 q^{60} -2.51141e6 q^{61} +2.33338e6 q^{62} +551945. q^{63} +1.41170e7 q^{64} -752741. q^{65} +2.82253e6 q^{67} +1.79378e6 q^{68} -279951. q^{69} +6.31806e6 q^{70} +2.93679e6 q^{71} +4.17180e6 q^{72} +3.65685e6 q^{73} -1.00699e7 q^{74} -1.58067e6 q^{75} -185215. q^{76} +1.24096e6 q^{78} +107494. q^{79} +2.97006e7 q^{80} +531441. q^{81} +3.02899e6 q^{82} -516375. q^{83} -7.79922e6 q^{84} +1.73814e6 q^{85} -9.31233e6 q^{86} +3.26461e6 q^{87} -3.22551e6 q^{89} +6.08335e6 q^{90} -1.54163e6 q^{91} +3.95582e6 q^{92} -2.79105e6 q^{93} -1.55463e7 q^{94} -179469. q^{95} -2.91865e7 q^{96} +6.24150e6 q^{97} -5.64998e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 9 q^{2} - 378 q^{3} + 1129 q^{4} - 71 q^{5} - 243 q^{6} - 534 q^{7} + 3384 q^{8} + 10206 q^{9} - 676 q^{10} - 30483 q^{12} + 12320 q^{13} - 39924 q^{14} + 1917 q^{15} + 121041 q^{16} + 33141 q^{17}+ \cdots - 8477109 q^{98}+O(q^{100}) \) 14 * q + 9 * q^2 - 378 * q^3 + 1129 * q^4 - 71 * q^5 - 243 * q^6 - 534 * q^7 + 3384 * q^8 + 10206 * q^9 - 676 * q^10 - 30483 * q^12 + 12320 * q^13 - 39924 * q^14 + 1917 * q^15 + 121041 * q^16 + 33141 * q^17 + 6561 * q^18 - 45019 * q^19 + 84003 * q^20 + 14418 * q^21 + 69254 * q^23 - 91368 * q^24 + 79811 * q^25 - 371495 * q^26 - 275562 * q^27 - 7064 * q^28 + 411600 * q^29 + 18252 * q^30 + 412799 * q^31 + 1363161 * q^32 - 771854 * q^34 + 850847 * q^35 + 823041 * q^36 - 831248 * q^37 + 948531 * q^38 - 332640 * q^39 - 50612 * q^40 + 616814 * q^41 + 1077948 * q^42 - 99892 * q^43 - 51759 * q^45 - 250599 * q^46 - 1139331 * q^47 - 3268107 * q^48 + 2854014 * q^49 - 3832250 * q^50 - 894807 * q^51 - 3588929 * q^52 + 2582463 * q^53 - 177147 * q^54 - 3153909 * q^56 + 1215513 * q^57 - 5034575 * q^58 - 1606585 * q^59 - 2268081 * q^60 - 3071551 * q^61 + 1169699 * q^62 - 389286 * q^63 + 19996212 * q^64 + 2447897 * q^65 - 1617843 * q^67 + 9075543 * q^68 - 1869858 * q^69 + 781473 * q^70 - 571759 * q^71 + 2466936 * q^72 + 7998216 * q^73 - 2349001 * q^74 - 2154897 * q^75 - 3267874 * q^76 + 10030365 * q^78 - 422038 * q^79 + 5233418 * q^80 + 7440174 * q^81 - 26251615 * q^82 + 7152906 * q^83 + 190728 * q^84 + 11341160 * q^85 + 12629343 * q^86 - 11113200 * q^87 - 20120730 * q^89 - 492804 * q^90 + 14374304 * q^91 + 21153333 * q^92 - 11145573 * q^93 - 37707710 * q^94 + 18573421 * q^95 - 36805347 * q^96 + 21339043 * q^97 - 8477109 * q^98

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