LMFDB - Embedded newform 357.2.a.h.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [357,2,Mod(1,357)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(357, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("357.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$357 = 3 \cdot 7 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	357.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,2,-4,6,-2,-2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.85065935216$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7232.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 2x^{3} - 5x^{2} + 4x + 4$ x^4 - 2x^3 - 5x^2 + 4*x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$2$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.06644$ of defining polynomial
Character	$\chi$	$=$	357.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.65222 q^{2} -1.00000 q^{3} +0.729840 q^{4} +3.48065 q^{5} -1.65222 q^{6} +1.00000 q^{7} -2.09859 q^{8} +1.00000 q^{9} +5.75081 q^{10} +3.57461 q^{11} -0.729840 q^{12} -3.48065 q^{13} +1.65222 q^{14} -3.48065 q^{15} -4.92701 q^{16} -1.00000 q^{17} +1.65222 q^{18} +4.17620 q^{19} +2.54032 q^{20} -1.00000 q^{21} +5.90604 q^{22} +7.92701 q^{23} +2.09859 q^{24} +7.11492 q^{25} -5.75081 q^{26} -1.00000 q^{27} +0.729840 q^{28} -3.30445 q^{29} -5.75081 q^{30} -9.59557 q^{31} -3.94335 q^{32} -3.57461 q^{33} -1.65222 q^{34} +3.48065 q^{35} +0.729840 q^{36} -7.05526 q^{37} +6.90002 q^{38} +3.48065 q^{39} -7.30445 q^{40} -6.94033 q^{41} -1.65222 q^{42} +4.96571 q^{43} +2.60889 q^{44} +3.48065 q^{45} +13.0972 q^{46} -9.59557 q^{47} +4.92701 q^{48} +1.00000 q^{49} +11.7554 q^{50} +1.00000 q^{51} -2.54032 q^{52} -3.59557 q^{53} -1.65222 q^{54} +12.4420 q^{55} -2.09859 q^{56} -4.17620 q^{57} -5.45968 q^{58} -9.40766 q^{59} -2.54032 q^{60} -12.1718 q^{61} -15.8540 q^{62} +1.00000 q^{63} +3.33873 q^{64} -12.1149 q^{65} -5.90604 q^{66} +8.54032 q^{67} -0.729840 q^{68} -7.92701 q^{69} +5.75081 q^{70} -2.06857 q^{71} -2.09859 q^{72} +10.4537 q^{73} -11.6569 q^{74} -7.11492 q^{75} +3.04796 q^{76} +3.57461 q^{77} +5.75081 q^{78} +0.162528 q^{79} -17.1492 q^{80} +1.00000 q^{81} -11.4670 q^{82} +8.54032 q^{83} -0.729840 q^{84} -3.48065 q^{85} +8.20447 q^{86} +3.30445 q^{87} -7.50162 q^{88} +0.961300 q^{89} +5.75081 q^{90} -3.48065 q^{91} +5.78546 q^{92} +9.59557 q^{93} -15.8540 q^{94} +14.5359 q^{95} +3.94335 q^{96} +9.72543 q^{97} +1.65222 q^{98} +3.57461 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{2} - 4 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 4 q^{9} + 4 q^{10} + 2 q^{11} - 6 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{15} + 6 q^{16} - 4 q^{17} + 2 q^{18} + 10 q^{19} + 4 q^{20}+ \cdots + 2 q^{99}+O(q^{100})$ 4 * q + 2 * q^2 - 4 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^6 + 4 * q^7 + 6 * q^8 + 4 * q^9 + 4 * q^10 + 2 * q^11 - 6 * q^12 + 2 * q^13 + 2 * q^14 + 2 * q^15 + 6 * q^16 - 4 * q^17 + 2 * q^18 + 10 * q^19 + 4 * q^20 - 4 * q^21 + 20 * q^22 + 6 * q^23 - 6 * q^24 + 10 * q^25 - 4 * q^26 - 4 * q^27 + 6 * q^28 - 4 * q^29 - 4 * q^30 - 4 * q^31 + 14 * q^32 - 2 * q^33 - 2 * q^34 - 2 * q^35 + 6 * q^36 - 16 * q^38 - 2 * q^39 - 20 * q^40 - 18 * q^41 - 2 * q^42 + 26 * q^43 - 8 * q^44 - 2 * q^45 - 20 * q^46 - 4 * q^47 - 6 * q^48 + 4 * q^49 + 10 * q^50 + 4 * q^51 - 4 * q^52 + 20 * q^53 - 2 * q^54 + 2 * q^55 + 6 * q^56 - 10 * q^57 - 28 * q^58 + 4 * q^59 - 4 * q^60 - 4 * q^61 - 12 * q^62 + 4 * q^63 - 2 * q^64 - 30 * q^65 - 20 * q^66 + 28 * q^67 - 6 * q^68 - 6 * q^69 + 4 * q^70 + 4 * q^71 + 6 * q^72 + 8 * q^73 - 24 * q^74 - 10 * q^75 + 8 * q^76 + 2 * q^77 + 4 * q^78 - 8 * q^79 - 44 * q^80 + 4 * q^81 - 28 * q^82 + 28 * q^83 - 6 * q^84 + 2 * q^85 - 20 * q^86 + 4 * q^87 + 8 * q^88 - 28 * q^89 + 4 * q^90 + 2 * q^91 - 16 * q^92 + 4 * q^93 - 12 * q^94 + 14 * q^95 - 14 * q^96 + 4 * q^97 + 2 * q^98 + 2 * q^99

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	1.65222	1.16830	0.584149	−	0.811646i	$-0.301428\pi$
$2$	1.65222	1.16830	0.584149	+	0.811646i	$0.301428\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.729840	0.364920
$5$	3.48065	1.55659	0.778297	−	0.627896i	$-0.216084\pi$
$5$	3.48065	1.55659	0.778297	+	0.627896i	$0.216084\pi$
$6$	−1.65222	−0.674517
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.09859	−0.741962
$9$	1.00000	0.333333
$10$	5.75081	1.81857
$11$	3.57461	1.07778	0.538892	−	0.842375i	$-0.318843\pi$
$11$	3.57461	1.07778	0.538892	+	0.842375i	$0.318843\pi$
$12$	−0.729840	−0.210687
$13$	−3.48065	−0.965359	−0.482679	−	0.875797i	$-0.660336\pi$
$13$	−3.48065	−0.965359	−0.482679	+	0.875797i	$0.660336\pi$
$14$	1.65222	0.441575
$15$	−3.48065	−0.898700
$16$	−4.92701	−1.23175
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.65222	0.389433
$19$	4.17620	0.958087	0.479043	−	0.877791i	$-0.340984\pi$
$19$	4.17620	0.958087	0.479043	+	0.877791i	$0.340984\pi$
$20$	2.54032	0.568033
$21$	−1.00000	−0.218218
$22$	5.90604	1.25917
$23$	7.92701	1.65290	0.826448	−	0.563013i	$-0.190358\pi$
$23$	7.92701	1.65290	0.826448	+	0.563013i	$0.190358\pi$
$24$	2.09859	0.428372
$25$	7.11492	1.42298
$26$	−5.75081	−1.12783
$27$	−1.00000	−0.192450
$28$	0.729840	0.137927
$29$	−3.30445	−0.613620	−0.306810	−	0.951771i	$-0.599262\pi$
$29$	−3.30445	−0.613620	−0.306810	+	0.951771i	$0.599262\pi$
$30$	−5.75081	−1.04995
$31$	−9.59557	−1.72342	−0.861708	−	0.507404i	$-0.830605\pi$
$31$	−9.59557	−1.72342	−0.861708	+	0.507404i	$0.830605\pi$
$32$	−3.94335	−0.697093
$33$	−3.57461	−0.622259
$34$	−1.65222	−0.283354
$35$	3.48065	0.588337
$36$	0.729840	0.121640
$37$	−7.05526	−1.15988	−0.579938	−	0.814660i	$-0.696923\pi$
$37$	−7.05526	−1.15988	−0.579938	+	0.814660i	$0.696923\pi$
$38$	6.90002	1.11933
$39$	3.48065	0.557350
$40$	−7.30445	−1.15493
$41$	−6.94033	−1.08390	−0.541949	−	0.840412i	$-0.682313\pi$
$41$	−6.94033	−1.08390	−0.541949	+	0.840412i	$0.682313\pi$
$42$	−1.65222	−0.254944
$43$	4.96571	0.757264	0.378632	−	0.925547i	$-0.376395\pi$
$43$	4.96571	0.757264	0.378632	+	0.925547i	$0.376395\pi$
$44$	2.60889	0.393305
$45$	3.48065	0.518865
$46$	13.0972	1.93108
$47$	−9.59557	−1.39966	−0.699829	−	0.714310i	$-0.746741\pi$
$47$	−9.59557	−1.39966	−0.699829	+	0.714310i	$0.746741\pi$
$48$	4.92701	0.711153
$49$	1.00000	0.142857
$50$	11.7554	1.66247
$51$	1.00000	0.140028
$52$	−2.54032	−0.352279
$53$	−3.59557	−0.493890	−0.246945	−	0.969029i	$-0.579427\pi$
$53$	−3.59557	−0.493890	−0.246945	+	0.969029i	$0.579427\pi$
$54$	−1.65222	−0.224839
$55$	12.4420	1.67767
$56$	−2.09859	−0.280435
$57$	−4.17620	−0.553152
$58$	−5.45968	−0.716891
$59$	−9.40766	−1.22477	−0.612387	−	0.790558i	$-0.709790\pi$
$59$	−9.40766	−1.22477	−0.612387	+	0.790558i	$0.709790\pi$
$60$	−2.54032	−0.327954
$61$	−12.1718	−1.55844	−0.779219	−	0.626752i	$-0.784384\pi$
$61$	−12.1718	−1.55844	−0.779219	+	0.626752i	$0.784384\pi$
$62$	−15.8540	−2.01346
$63$	1.00000	0.125988
$64$	3.33873	0.417341
$65$	−12.1149	−1.50267
$66$	−5.90604	−0.726984
$67$	8.54032	1.04337	0.521683	−	0.853139i	$-0.325304\pi$
$67$	8.54032	1.04337	0.521683	+	0.853139i	$0.325304\pi$
$68$	−0.729840	−0.0885062
$69$	−7.92701	−0.954300
$70$	5.75081	0.687353
$71$	−2.06857	−0.245494	−0.122747	−	0.992438i	$-0.539170\pi$
$71$	−2.06857	−0.245494	−0.122747	+	0.992438i	$0.539170\pi$
$72$	−2.09859	−0.247321
$73$	10.4537	1.22351	0.611754	−	0.791048i	$-0.290464\pi$
$73$	10.4537	1.22351	0.611754	+	0.791048i	$0.290464\pi$
$74$	−11.6569	−1.35508
$75$	−7.11492	−0.821561
$76$	3.04796	0.349625
$77$	3.57461	0.407364
$78$	5.75081	0.651151
$79$	0.162528	0.0182858	0.00914290	−	0.999958i	$-0.497090\pi$
$79$	0.162528	0.0182858	0.00914290	+	0.999958i	$0.497090\pi$
$80$	−17.1492	−1.91734
$81$	1.00000	0.111111
$82$	−11.4670	−1.26632
$83$	8.54032	0.937422	0.468711	−	0.883352i	$-0.344719\pi$
$83$	8.54032	0.937422	0.468711	+	0.883352i	$0.344719\pi$
$84$	−0.729840	−0.0796321
$85$	−3.48065	−0.377529
$86$	8.20447	0.884710
$87$	3.30445	0.354274
$88$	−7.50162	−0.799675
$89$	0.961300	0.101898	0.0509488	−	0.998701i	$-0.483775\pi$
$89$	0.961300	0.101898	0.0509488	+	0.998701i	$0.483775\pi$
$90$	5.75081	0.606189
$91$	−3.48065	−0.364871
$92$	5.78546	0.603175
$93$	9.59557	0.995015
$94$	−15.8540	−1.63522
$95$	14.5359	1.49135
$96$	3.94335	0.402467
$97$	9.72543	0.987467	0.493734	−	0.869613i	$-0.335632\pi$
$97$	9.72543	0.987467	0.493734	+	0.869613i	$0.335632\pi$
$98$	1.65222	0.166900
$99$	3.57461	0.359261
$100$	5.19276	0.519276
$101$	−10.7028	−1.06497	−0.532487	−	0.846438i	$-0.678742\pi$
$101$	−10.7028	−1.06497	−0.532487	+	0.846438i	$0.678742\pi$
$102$	1.65222	0.163594
$103$	19.3674	1.90832	0.954161	−	0.299294i	$-0.0967511\pi$
$103$	19.3674	1.90832	0.954161	+	0.299294i	$0.0967511\pi$
$104$	7.30445	0.716260
$105$	−3.48065	−0.339677
$106$	−5.94069	−0.577011
$107$	10.8464	1.04856	0.524279	−	0.851546i	$-0.324335\pi$
$107$	10.8464	1.04856	0.524279	+	0.851546i	$0.324335\pi$
$108$	−0.729840	−0.0702289
$109$	3.55364	0.340377	0.170188	−	0.985412i	$-0.445562\pi$
$109$	3.55364	0.340377	0.170188	+	0.985412i	$0.445562\pi$
$110$	20.5569	1.96002
$111$	7.05526	0.669655
$112$	−4.92701	−0.465559
$113$	−11.9597	−1.12507	−0.562536	−	0.826772i	$-0.690174\pi$
$113$	−11.9597	−1.12507	−0.562536	+	0.826772i	$0.690174\pi$
$114$	−6.90002	−0.646246
$115$	27.5912	2.57289
$116$	−2.41172	−0.223922
$117$	−3.48065	−0.321786
$118$	−15.5436	−1.43090
$119$	−1.00000	−0.0916698
$120$	7.30445	0.666802
$121$	1.77780	0.161618
$122$	−20.1105	−1.82072
$123$	6.94033	0.625789
$124$	−7.00324	−0.628909
$125$	7.36131	0.658416
$126$	1.65222	0.147192
$127$	−3.92701	−0.348466	−0.174233	−	0.984704i	$-0.555745\pi$
$127$	−3.92701	−0.348466	−0.174233	+	0.984704i	$0.555745\pi$
$128$	13.4030	1.18467
$129$	−4.96571	−0.437207
$130$	−20.0166	−1.75557
$131$	−16.7944	−1.46733	−0.733665	−	0.679511i	$-0.762192\pi$
$131$	−16.7944	−1.46733	−0.733665	+	0.679511i	$0.762192\pi$
$132$	−2.60889	−0.227075
$133$	4.17620	0.362123
$134$	14.1105	1.21896
$135$	−3.48065	−0.299567
$136$	2.09859	0.179952
$137$	5.17459	0.442095	0.221048	−	0.975263i	$-0.429052\pi$
$137$	5.17459	0.442095	0.221048	+	0.975263i	$0.429052\pi$
$138$	−13.0972	−1.11491
$139$	13.8807	1.17734	0.588671	−	0.808373i	$-0.299651\pi$
$139$	13.8807	1.17734	0.588671	+	0.808373i	$0.299651\pi$
$140$	2.54032	0.214696
$141$	9.59557	0.808093
$142$	−3.41774	−0.286811
$143$	−12.4420	−1.04045
$144$	−4.92701	−0.410584
$145$	−11.5016	−0.955157
$146$	17.2718	1.42942
$147$	−1.00000	−0.0824786
$148$	−5.14921	−0.423263
$149$	14.5823	1.19463	0.597313	−	0.802009i	$-0.296235\pi$
$149$	14.5823	1.19463	0.597313	+	0.802009i	$0.296235\pi$
$150$	−11.7554	−0.959828
$151$	9.88066	0.804077	0.402038	−	0.915623i	$-0.368302\pi$
$151$	9.88066	0.804077	0.402038	+	0.915623i	$0.368302\pi$
$152$	−8.76413	−0.710865
$153$	−1.00000	−0.0808452
$154$	5.90604	0.475923
$155$	−33.3988	−2.68266
$156$	2.54032	0.203388
$157$	−4.20888	−0.335905	−0.167953	−	0.985795i	$-0.553716\pi$
$157$	−4.20888	−0.335905	−0.167953	+	0.985795i	$0.553716\pi$
$158$	0.268532	0.0213633
$159$	3.59557	0.285148
$160$	−13.7254	−1.08509
$161$	7.92701	0.624736
$162$	1.65222	0.129811
$163$	0.540319	0.0423211	0.0211605	−	0.999776i	$-0.493264\pi$
$163$	0.540319	0.0423211	0.0211605	+	0.999776i	$0.493264\pi$
$164$	−5.06533	−0.395536
$165$	−12.4420	−0.968604
$166$	14.1105	1.09519
$167$	−4.08954	−0.316458	−0.158229	−	0.987402i	$-0.550578\pi$
$167$	−4.08954	−0.316458	−0.158229	+	0.987402i	$0.550578\pi$
$168$	2.09859	0.161909
$169$	−0.885076	−0.0680827
$170$	−5.75081	−0.441067
$171$	4.17620	0.319362
$172$	3.62418	0.276341
$173$	15.1702	1.15337	0.576684	−	0.816968i	$-0.304347\pi$
$173$	15.1702	1.15337	0.576684	+	0.816968i	$0.304347\pi$
$174$	5.45968	0.413897
$175$	7.11492	0.537838
$176$	−17.6121	−1.32756
$177$	9.40766	0.707123
$178$	1.58828	0.119047
$179$	11.9133	0.890445	0.445222	−	0.895420i	$-0.353125\pi$
$179$	11.9133	0.890445	0.445222	+	0.895420i	$0.353125\pi$
$180$	2.54032	0.189344
$181$	6.45366	0.479697	0.239848	−	0.970810i	$-0.422902\pi$
$181$	6.45366	0.479697	0.239848	+	0.970810i	$0.422902\pi$
$182$	−5.75081	−0.426278
$183$	12.1718	0.899765
$184$	−16.6355	−1.22639
$185$	−24.5569	−1.80546
$186$	15.8540	1.16247
$187$	−3.57461	−0.261401
$188$	−7.00324	−0.510764
$189$	−1.00000	−0.0727393
$190$	24.0166	1.74234
$191$	−23.0391	−1.66705	−0.833527	−	0.552479i	$-0.813682\pi$
$191$	−23.0391	−1.66705	−0.833527	+	0.552479i	$0.813682\pi$
$192$	−3.33873	−0.240952
$193$	−6.51494	−0.468955	−0.234478	−	0.972122i	$-0.575338\pi$
$193$	−6.51494	−0.468955	−0.234478	+	0.972122i	$0.575338\pi$
$194$	16.0686	1.15366
$195$	12.1149	0.867568
$196$	0.729840	0.0521315
$197$	0.622568	0.0443561	0.0221781	−	0.999754i	$-0.492940\pi$
$197$	0.622568	0.0443561	0.0221781	+	0.999754i	$0.492940\pi$
$198$	5.90604	0.419724
$199$	5.05526	0.358358	0.179179	−	0.983817i	$-0.442656\pi$
$199$	5.05526	0.358358	0.179179	+	0.983817i	$0.442656\pi$
$200$	−14.9313	−1.05580
$201$	−8.54032	−0.602388
$202$	−17.6835	−1.24421
$203$	−3.30445	−0.231927
$204$	0.729840	0.0510991
$205$	−24.1569	−1.68719
$206$	31.9992	2.22949
$207$	7.92701	0.550966
$208$	17.1492	1.18908
$209$	14.9283	1.03261
$210$	−5.75081	−0.396844
$211$	−10.6343	−0.732094	−0.366047	−	0.930596i	$-0.619289\pi$
$211$	−10.6343	−0.732094	−0.366047	+	0.930596i	$0.619289\pi$
$212$	−2.62420	−0.180230
$213$	2.06857	0.141736
$214$	17.9206	1.22503
$215$	17.2839	1.17875
$216$	2.09859	0.142791
$217$	−9.59557	−0.651390
$218$	5.87140	0.397661
$219$	−10.4537	−0.706393
$220$	9.08064	0.612217
$221$	3.48065	0.234134
$222$	11.6569	0.782357
$223$	−17.9109	−1.19940	−0.599701	−	0.800224i	$-0.704714\pi$
$223$	−17.9109	−1.19940	−0.599701	+	0.800224i	$0.704714\pi$
$224$	−3.94335	−0.263476
$225$	7.11492	0.474328
$226$	−19.7601	−1.31442
$227$	5.97903	0.396842	0.198421	−	0.980117i	$-0.436419\pi$
$227$	5.97903	0.396842	0.198421	+	0.980117i	$0.436419\pi$
$228$	−3.04796	−0.201856
$229$	5.27177	0.348368	0.174184	−	0.984713i	$-0.444271\pi$
$229$	5.27177	0.348368	0.174184	+	0.984713i	$0.444271\pi$
$230$	45.5867	3.00590
$231$	−3.57461	−0.235192
$232$	6.93467	0.455283
$233$	−25.5065	−1.67098	−0.835492	−	0.549502i	$-0.814817\pi$
$233$	−25.5065	−1.67098	−0.835492	+	0.549502i	$0.814817\pi$
$234$	−5.75081	−0.375942
$235$	−33.3988	−2.17870
$236$	−6.86609	−0.446945
$237$	−0.162528	−0.0105573
$238$	−1.65222	−0.107098
$239$	11.5629	0.747942	0.373971	−	0.927440i	$-0.377996\pi$
$239$	11.5629	0.747942	0.373971	+	0.927440i	$0.377996\pi$
$240$	17.1492	1.10698
$241$	24.7287	1.59291	0.796457	−	0.604696i	$-0.206705\pi$
$241$	24.7287	1.59291	0.796457	+	0.604696i	$0.206705\pi$
$242$	2.93733	0.188819
$243$	−1.00000	−0.0641500
$244$	−8.88346	−0.568706
$245$	3.48065	0.222371
$246$	11.4670	0.731108
$247$	−14.5359	−0.924898
$248$	20.1371	1.27871
$249$	−8.54032	−0.541221
$250$	12.1625	0.769226
$251$	−4.86734	−0.307224	−0.153612	−	0.988131i	$-0.549091\pi$
$251$	−4.86734	−0.307224	−0.153612	+	0.988131i	$0.549091\pi$
$252$	0.729840	0.0459756
$253$	28.3359	1.78147
$254$	−6.48830	−0.407112
$255$	3.48065	0.217967
$256$	15.4673	0.966708
$257$	26.7867	1.67091	0.835455	−	0.549559i	$-0.185204\pi$
$257$	26.7867	1.67091	0.835455	+	0.549559i	$0.185204\pi$
$258$	−8.20447	−0.510788
$259$	−7.05526	−0.438392
$260$	−8.84196	−0.548355
$261$	−3.30445	−0.204540
$262$	−27.7480	−1.71428
$263$	30.6867	1.89222	0.946112	−	0.323839i	$-0.104974\pi$
$263$	30.6867	1.89222	0.946112	+	0.323839i	$0.104974\pi$
$264$	7.50162	0.461693
$265$	−12.5149	−0.768786
$266$	6.90002	0.423067
$267$	−0.961300	−0.0588306
$268$	6.23307	0.380745
$269$	5.59116	0.340899	0.170450	−	0.985366i	$-0.445478\pi$
$269$	5.59116	0.340899	0.170450	+	0.985366i	$0.445478\pi$
$270$	−5.75081	−0.349983
$271$	8.88102	0.539484	0.269742	−	0.962933i	$-0.413062\pi$
$271$	8.88102	0.539484	0.269742	+	0.962933i	$0.413062\pi$
$272$	4.92701	0.298744
$273$	3.48065	0.210659
$274$	8.54958	0.516499
$275$	25.4330	1.53367
$276$	−5.78546	−0.348243
$277$	−1.50162	−0.0902235	−0.0451118	−	0.998982i	$-0.514364\pi$
$277$	−1.50162	−0.0902235	−0.0451118	+	0.998982i	$0.514364\pi$
$278$	22.9339	1.37549
$279$	−9.59557	−0.574472
$280$	−7.30445	−0.436524
$281$	2.35241	0.140333	0.0701665	−	0.997535i	$-0.477647\pi$
$281$	2.35241	0.140333	0.0701665	+	0.997535i	$0.477647\pi$
$282$	15.8540	0.944094
$283$	−17.4077	−1.03478	−0.517389	−	0.855750i	$-0.673096\pi$
$283$	−17.4077	−1.03478	−0.517389	+	0.855750i	$0.673096\pi$
$284$	−1.50973	−0.0895859
$285$	−14.5359	−0.861033
$286$	−20.5569	−1.21555
$287$	−6.94033	−0.409675
$288$	−3.94335	−0.232364
$289$	1.00000	0.0588235
$290$	−19.0032	−1.11591
$291$	−9.72543	−0.570115
$292$	7.62950	0.446483
$293$	−15.9314	−0.930724	−0.465362	−	0.885120i	$-0.654076\pi$
$293$	−15.9314	−0.930724	−0.465362	+	0.885120i	$0.654076\pi$
$294$	−1.65222	−0.0963596
$295$	−32.7448	−1.90648
$296$	14.8061	0.860585
$297$	−3.57461	−0.207420
$298$	24.0931	1.39568
$299$	−27.5912	−1.59564
$300$	−5.19276	−0.299804
$301$	4.96571	0.286219
$302$	16.3251	0.939401
$303$	10.7028	0.614862
$304$	−20.5762	−1.18013
$305$	−42.3657	−2.42586
$306$	−1.65222	−0.0944513
$307$	−11.5795	−0.660874	−0.330437	−	0.943828i	$-0.607196\pi$
$307$	−11.5795	−0.660874	−0.330437	+	0.943828i	$0.607196\pi$
$308$	2.60889	0.148655
$309$	−19.3674	−1.10177
$310$	−55.1823	−3.13415
$311$	−1.52825	−0.0866593	−0.0433296	−	0.999061i	$-0.513797\pi$
$311$	−1.52825	−0.0866593	−0.0433296	+	0.999061i	$0.513797\pi$
$312$	−7.30445	−0.413533
$313$	15.1565	0.856696	0.428348	−	0.903614i	$-0.359096\pi$
$313$	15.1565	0.856696	0.428348	+	0.903614i	$0.359096\pi$
$314$	−6.95401	−0.392437
$315$	3.48065	0.196112
$316$	0.118619	0.00667286
$317$	12.8746	0.723112	0.361556	−	0.932350i	$-0.382246\pi$
$317$	12.8746	0.723112	0.361556	+	0.932350i	$0.382246\pi$
$318$	5.94069	0.333137
$319$	−11.8121	−0.661350
$320$	11.6210	0.649631
$321$	−10.8464	−0.605386
$322$	13.0972	0.729878
$323$	−4.17620	−0.232370
$324$	0.729840	0.0405467
$325$	−24.7646	−1.37369
$326$	0.892728	0.0494436
$327$	−3.55364	−0.196517
$328$	14.5649	0.804211
$329$	−9.59557	−0.529021
$330$	−20.5569	−1.13162
$331$	24.2794	1.33452	0.667259	−	0.744826i	$-0.267467\pi$
$331$	24.2794	1.33452	0.667259	+	0.744826i	$0.267467\pi$
$332$	6.23307	0.342084
$333$	−7.05526	−0.386626
$334$	−6.75683	−0.369718
$335$	29.7259	1.62410
$336$	4.92701	0.268791
$337$	−4.78025	−0.260397	−0.130198	−	0.991488i	$-0.541561\pi$
$337$	−4.78025	−0.260397	−0.130198	+	0.991488i	$0.541561\pi$
$338$	−1.46234	−0.0795409
$339$	11.9597	0.649561
$340$	−2.54032	−0.137768
$341$	−34.3004	−1.85747
$342$	6.90002	0.373110
$343$	1.00000	0.0539949
$344$	−10.4210	−0.561862
$345$	−27.5912	−1.48546
$346$	25.0645	1.34748
$347$	22.6508	1.21596	0.607980	−	0.793952i	$-0.291980\pi$
$347$	22.6508	1.21596	0.607980	+	0.793952i	$0.291980\pi$
$348$	2.41172	0.129282
$349$	21.2573	1.13788	0.568938	−	0.822380i	$-0.307354\pi$
$349$	21.2573	1.13788	0.568938	+	0.822380i	$0.307354\pi$
$350$	11.7554	0.628355
$351$	3.48065	0.185783
$352$	−14.0959	−0.751315
$353$	3.55364	0.189141	0.0945705	−	0.995518i	$-0.469852\pi$
$353$	3.55364	0.189141	0.0945705	+	0.995518i	$0.469852\pi$
$354$	15.5436	0.826131
$355$	−7.19998	−0.382135
$356$	0.701596	0.0371845
$357$	1.00000	0.0529256
$358$	19.6835	1.04030
$359$	−16.8314	−0.888330	−0.444165	−	0.895945i	$-0.646500\pi$
$359$	−16.8314	−0.888330	−0.444165	+	0.895945i	$0.646500\pi$
$360$	−7.30445	−0.384978
$361$	−1.55932	−0.0820694
$362$	10.6629	0.560428
$363$	−1.77780	−0.0933105
$364$	−2.54032	−0.133149
$365$	36.3855	1.90451
$366$	20.1105	1.05119
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−39.0565	−2.03596
$369$	−6.94033	−0.361299
$370$	−40.5734	−2.10931
$371$	−3.59557	−0.186673
$372$	7.00324	0.363101
$373$	−1.14921	−0.0595039	−0.0297519	−	0.999557i	$-0.509472\pi$
$373$	−1.14921	−0.0595039	−0.0297519	+	0.999557i	$0.509472\pi$
$374$	−5.90604	−0.305394
$375$	−7.36131	−0.380137
$376$	20.1371	1.03849
$377$	11.5016	0.592364
$378$	−1.65222	−0.0849812
$379$	−35.0032	−1.79800	−0.898998	−	0.437953i	$-0.855704\pi$
$379$	−35.0032	−1.79800	−0.898998	+	0.437953i	$0.855704\pi$
$380$	10.6089	0.544225
$381$	3.92701	0.201187
$382$	−38.0658	−1.94762
$383$	−15.7867	−0.806663	−0.403332	−	0.915054i	$-0.632148\pi$
$383$	−15.7867	−0.806663	−0.403332	+	0.915054i	$0.632148\pi$
$384$	−13.4030	−0.683971
$385$	12.4420	0.634100
$386$	−10.7641	−0.547880
$387$	4.96571	0.252421
$388$	7.09801	0.360347
$389$	−8.65083	−0.438614	−0.219307	−	0.975656i	$-0.570380\pi$
$389$	−8.65083	−0.438614	−0.219307	+	0.975656i	$0.570380\pi$
$390$	20.0166	1.01358
$391$	−7.92701	−0.400886
$392$	−2.09859	−0.105995
$393$	16.7944	0.847163
$394$	1.02862	0.0518212
$395$	0.565702	0.0284636
$396$	2.60889	0.131102
$397$	−19.9985	−1.00369	−0.501847	−	0.864956i	$-0.667346\pi$
$397$	−19.9985	−1.00369	−0.501847	+	0.864956i	$0.667346\pi$
$398$	8.35241	0.418668
$399$	−4.17620	−0.209072
$400$	−35.0553	−1.75277
$401$	−35.7630	−1.78592	−0.892958	−	0.450139i	$-0.851374\pi$
$401$	−35.7630	−1.78592	−0.892958	+	0.450139i	$0.851374\pi$
$402$	−14.1105	−0.703768
$403$	33.3988	1.66371
$404$	−7.81137	−0.388630
$405$	3.48065	0.172955
$406$	−5.45968	−0.270959
$407$	−25.2198	−1.25010
$408$	−2.09859	−0.103896
$409$	15.3435	0.758688	0.379344	−	0.925256i	$-0.376150\pi$
$409$	15.3435	0.758688	0.379344	+	0.925256i	$0.376150\pi$
$410$	−39.9125	−1.97114
$411$	−5.17459	−0.255244
$412$	14.1351	0.696385
$413$	−9.40766	−0.462921
$414$	13.0972	0.643692
$415$	29.7259	1.45919
$416$	13.7254	0.672944
$417$	−13.8807	−0.679739
$418$	24.6648	1.20640
$419$	−1.24514	−0.0608289	−0.0304144	−	0.999537i	$-0.509683\pi$
$419$	−1.24514	−0.0608289	−0.0304144	+	0.999537i	$0.509683\pi$
$420$	−2.54032	−0.123955
$421$	36.3626	1.77220	0.886102	−	0.463491i	$-0.153403\pi$
$421$	36.3626	1.77220	0.886102	+	0.463491i	$0.153403\pi$
$422$	−17.5702	−0.855304
$423$	−9.59557	−0.466553
$424$	7.54562	0.366448
$425$	−7.11492	−0.345125
$426$	3.41774	0.165590
$427$	−12.1718	−0.589034
$428$	7.91612	0.382640
$429$	12.4420	0.600703
$430$	28.5569	1.37713
$431$	−14.6089	−0.703686	−0.351843	−	0.936059i	$-0.614445\pi$
$431$	−14.6089	−0.703686	−0.351843	+	0.936059i	$0.614445\pi$
$432$	4.92701	0.237051
$433$	−20.5525	−0.987688	−0.493844	−	0.869550i	$-0.664409\pi$
$433$	−20.5525	−0.987688	−0.493844	+	0.869550i	$0.664409\pi$
$434$	−15.8540	−0.761018
$435$	11.5016	0.551460
$436$	2.59359	0.124210
$437$	33.1048	1.58362
$438$	−17.2718	−0.825277
$439$	−14.5835	−0.696033	−0.348016	−	0.937488i	$-0.613145\pi$
$439$	−14.5835	−0.696033	−0.348016	+	0.937488i	$0.613145\pi$
$440$	−26.1105	−1.24477
$441$	1.00000	0.0476190
$442$	5.75081	0.273538
$443$	34.0258	1.61662	0.808308	−	0.588760i	$-0.200384\pi$
$443$	34.0258	1.61662	0.808308	+	0.588760i	$0.200384\pi$
$444$	5.14921	0.244371
$445$	3.34595	0.158613
$446$	−29.5928	−1.40126
$447$	−14.5823	−0.689717
$448$	3.33873	0.157740
$449$	−34.4956	−1.62795	−0.813974	−	0.580901i	$-0.802700\pi$
$449$	−34.4956	−1.62795	−0.813974	+	0.580901i	$0.802700\pi$
$450$	11.7554	0.554157
$451$	−24.8089	−1.16821
$452$	−8.72866	−0.410562
$453$	−9.88066	−0.464234
$454$	9.87869	0.463630
$455$	−12.1149	−0.567956
$456$	8.76413	0.410418
$457$	−3.00765	−0.140692	−0.0703460	−	0.997523i	$-0.522410\pi$
$457$	−3.00765	−0.140692	−0.0703460	+	0.997523i	$0.522410\pi$
$458$	8.71014	0.406998
$459$	1.00000	0.0466760
$460$	20.1371	0.938899
$461$	0.258453	0.0120374	0.00601868	−	0.999982i	$-0.498084\pi$
$461$	0.258453	0.0120374	0.00601868	+	0.999982i	$0.498084\pi$
$462$	−5.90604	−0.274774
$463$	20.5734	0.956128	0.478064	−	0.878325i	$-0.341339\pi$
$463$	20.5734	0.956128	0.478064	+	0.878325i	$0.341339\pi$
$464$	16.2810	0.755829
$465$	33.3988	1.54883
$466$	−42.1424	−1.95221
$467$	−11.9766	−0.554210	−0.277105	−	0.960840i	$-0.589375\pi$
$467$	−11.9766	−0.554210	−0.277105	+	0.960840i	$0.589375\pi$
$468$	−2.54032	−0.117426
$469$	8.54032	0.394355
$470$	−55.1823	−2.54537
$471$	4.20888	0.193935
$472$	19.7428	0.908736
$473$	17.7505	0.816167
$474$	−0.268532	−0.0123341
$475$	29.7134	1.36334
$476$	−0.729840	−0.0334522
$477$	−3.59557	−0.164630
$478$	19.1045	0.873819
$479$	−2.56129	−0.117028	−0.0585141	−	0.998287i	$-0.518636\pi$
$479$	−2.56129	−0.117028	−0.0585141	+	0.998287i	$0.518636\pi$
$480$	13.7254	0.626477
$481$	24.5569	1.11970
$482$	40.8573	1.86100
$483$	−7.92701	−0.360692
$484$	1.29751	0.0589778
$485$	33.8508	1.53709
$486$	−1.65222	−0.0749464
$487$	13.2451	0.600194	0.300097	−	0.953909i	$-0.402981\pi$
$487$	13.2451	0.600194	0.300097	+	0.953909i	$0.402981\pi$
$488$	25.5436	1.15630
$489$	−0.540319	−0.0244341
$490$	5.75081	0.259795
$491$	32.4302	1.46356	0.731778	−	0.681543i	$-0.238691\pi$
$491$	32.4302	1.46356	0.731778	+	0.681543i	$0.238691\pi$
$492$	5.06533	0.228363
$493$	3.30445	0.148825
$494$	−24.0166	−1.08056
$495$	12.4420	0.559224
$496$	47.2775	2.12282
$497$	−2.06857	−0.0927882
$498$	−14.1105	−0.632307
$499$	−36.1956	−1.62034	−0.810170	−	0.586195i	$-0.800625\pi$
$499$	−36.1956	−1.62034	−0.810170	+	0.586195i	$0.800625\pi$
$500$	5.37258	0.240269
$501$	4.08954	0.182707
$502$	−8.04194	−0.358929
$503$	−23.0743	−1.02883	−0.514415	−	0.857541i	$-0.671991\pi$
$503$	−23.0743	−1.02883	−0.514415	+	0.857541i	$0.671991\pi$
$504$	−2.09859	−0.0934785
$505$	−37.2529	−1.65773
$506$	46.8173	2.08128
$507$	0.885076	0.0393076
$508$	−2.86609	−0.127162
$509$	−21.1492	−0.937422	−0.468711	−	0.883351i	$-0.655281\pi$
$509$	−21.1492	−0.937422	−0.468711	+	0.883351i	$0.655281\pi$
$510$	5.75081	0.254650
$511$	10.4537	0.462443
$512$	−1.25058	−0.0552685
$513$	−4.17620	−0.184384
$514$	44.2576	1.95212
$515$	67.4110	2.97048
$516$	−3.62418	−0.159546
$517$	−34.3004	−1.50853
$518$	−11.6569	−0.512173
$519$	−15.1702	−0.665897
$520$	25.4242	1.11493
$521$	−42.6630	−1.86910	−0.934549	−	0.355834i	$-0.884197\pi$
$521$	−42.6630	−1.86910	−0.934549	+	0.355834i	$0.884197\pi$
$522$	−5.45968	−0.238964
$523$	8.31651	0.363656	0.181828	−	0.983330i	$-0.441799\pi$
$523$	8.31651	0.363656	0.181828	+	0.983330i	$0.441799\pi$
$524$	−12.2572	−0.535458
$525$	−7.11492	−0.310521
$526$	50.7013	2.21068
$527$	9.59557	0.417990
$528$	17.6121	0.766470
$529$	39.8375	1.73207
$530$	−20.6775	−0.898172
$531$	−9.40766	−0.408258
$532$	3.04796	0.132146
$533$	24.1569	1.04635
$534$	−1.58828	−0.0687317
$535$	37.7524	1.63218
$536$	−17.9226	−0.774138
$537$	−11.9133	−0.514098
$538$	9.23784	0.398272
$539$	3.57461	0.153969
$540$	−2.54032	−0.109318
$541$	40.4960	1.74106	0.870530	−	0.492115i	$-0.163776\pi$
$541$	40.4960	1.74106	0.870530	+	0.492115i	$0.163776\pi$
$542$	14.6734	0.630278
$543$	−6.45366	−0.276953
$544$	3.94335	0.169070
$545$	12.3690	0.529828
$546$	5.75081	0.246112
$547$	6.67621	0.285454	0.142727	−	0.989762i	$-0.454413\pi$
$547$	6.67621	0.285454	0.142727	+	0.989762i	$0.454413\pi$
$548$	3.77663	0.161330
$549$	−12.1718	−0.519479
$550$	42.0211	1.79178
$551$	−13.8000	−0.587901
$552$	16.6355	0.708055
$553$	0.162528	0.00691138
$554$	−2.48101	−0.105408
$555$	24.5569	1.04238
$556$	10.1307	0.429636
$557$	1.99803	0.0846592	0.0423296	−	0.999104i	$-0.486522\pi$
$557$	1.99803	0.0846592	0.0423296	+	0.999104i	$0.486522\pi$
$558$	−15.8540	−0.671155
$559$	−17.2839	−0.731032
$560$	−17.1492	−0.724686
$561$	3.57461	0.150920
$562$	3.88670	0.163951
$563$	29.9645	1.26285	0.631427	−	0.775435i	$-0.282469\pi$
$563$	29.9645	1.26285	0.631427	+	0.775435i	$0.282469\pi$
$564$	7.00324	0.294890
$565$	−41.6275	−1.75128
$566$	−28.7613	−1.20893
$567$	1.00000	0.0419961
$568$	4.34108	0.182148
$569$	−13.3556	−0.559898	−0.279949	−	0.960015i	$-0.590318\pi$
$569$	−13.3556	−0.559898	−0.279949	+	0.960015i	$0.590318\pi$
$570$	−24.0166	−1.00594
$571$	27.5347	1.15229	0.576146	−	0.817347i	$-0.304556\pi$
$571$	27.5347	1.15229	0.576146	+	0.817347i	$0.304556\pi$
$572$	−9.08064	−0.379681
$573$	23.0391	0.962474
$574$	−11.4670	−0.478622
$575$	56.4001	2.35205
$576$	3.33873	0.139114
$577$	29.0928	1.21115	0.605574	−	0.795789i	$-0.292944\pi$
$577$	29.0928	1.21115	0.605574	+	0.795789i	$0.292944\pi$
$578$	1.65222	0.0687234
$579$	6.51494	0.270752
$580$	−8.39435	−0.348556
$581$	8.54032	0.354312
$582$	−16.0686	−0.666064
$583$	−12.8528	−0.532307
$584$	−21.9379	−0.907797
$585$	−12.1149	−0.500890
$586$	−26.3223	−1.08736
$587$	−9.08837	−0.375117	−0.187559	−	0.982253i	$-0.560057\pi$
$587$	−9.08837	−0.375117	−0.187559	+	0.982253i	$0.560057\pi$
$588$	−0.729840	−0.0300981
$589$	−40.0731	−1.65118
$590$	−54.1017	−2.22733
$591$	−0.622568	−0.0256090
$592$	34.7613	1.42868
$593$	1.06730	0.0438289	0.0219145	−	0.999760i	$-0.493024\pi$
$593$	1.06730	0.0438289	0.0219145	+	0.999760i	$0.493024\pi$
$594$	−5.90604	−0.242328
$595$	−3.48065	−0.142693
$596$	10.6427	0.435943
$597$	−5.05526	−0.206898
$598$	−45.5867	−1.86418
$599$	13.2537	0.541531	0.270765	−	0.962645i	$-0.412723\pi$
$599$	13.2537	0.541531	0.270765	+	0.962645i	$0.412723\pi$
$600$	14.9313	0.609567
$601$	−18.6855	−0.762196	−0.381098	−	0.924535i	$-0.624454\pi$
$601$	−18.6855	−0.762196	−0.381098	+	0.924535i	$0.624454\pi$
$602$	8.20447	0.334389
$603$	8.54032	0.347789
$604$	7.21131	0.293424
$605$	6.18791	0.251574
$606$	17.6835	0.718343
$607$	36.2376	1.47084	0.735419	−	0.677613i	$-0.236985\pi$
$607$	36.2376	1.47084	0.735419	+	0.677613i	$0.236985\pi$
$608$	−16.4682	−0.667875
$609$	3.30445	0.133903
$610$	−69.9976	−2.83412
$611$	33.3988	1.35117
$612$	−0.729840	−0.0295021
$613$	−16.9835	−0.685958	−0.342979	−	0.939343i	$-0.611436\pi$
$613$	−16.9835	−0.685958	−0.342979	+	0.939343i	$0.611436\pi$
$614$	−19.1318	−0.772098
$615$	24.1569	0.974099
$616$	−7.50162	−0.302249
$617$	−18.6182	−0.749539	−0.374769	−	0.927118i	$-0.622278\pi$
$617$	−18.6182	−0.749539	−0.374769	+	0.927118i	$0.622278\pi$
$618$	−31.9992	−1.28720
$619$	42.3989	1.70415	0.852077	−	0.523417i	$-0.175343\pi$
$619$	42.3989	1.70415	0.852077	+	0.523417i	$0.175343\pi$
$620$	−24.3758	−0.978957
$621$	−7.92701	−0.318100
$622$	−2.52502	−0.101244
$623$	0.961300	0.0385137
$624$	−17.1492	−0.686518
$625$	−9.95247	−0.398099
$626$	25.0419	1.00088
$627$	−14.9283	−0.596178
$628$	−3.07181	−0.122579
$629$	7.05526	0.281311
$630$	5.75081	0.229118
$631$	−31.2134	−1.24258	−0.621292	−	0.783579i	$-0.713392\pi$
$631$	−31.2134	−1.24258	−0.621292	+	0.783579i	$0.713392\pi$
$632$	−0.341078	−0.0135674
$633$	10.6343	0.422674
$634$	21.2718	0.844810
$635$	−13.6686	−0.542420
$636$	2.62420	0.104056
$637$	−3.48065	−0.137908
$638$	−19.5162	−0.772654
$639$	−2.06857	−0.0818315
$640$	46.6513	1.84405
$641$	−14.6677	−0.579340	−0.289670	−	0.957127i	$-0.593546\pi$
$641$	−14.6677	−0.579340	−0.289670	+	0.957127i	$0.593546\pi$
$642$	−17.9206	−0.707271
$643$	−18.3823	−0.724927	−0.362464	−	0.931998i	$-0.618064\pi$
$643$	−18.3823	−0.724927	−0.362464	+	0.931998i	$0.618064\pi$
$644$	5.78546	0.227979
$645$	−17.2839	−0.680553
$646$	−6.90002	−0.271478
$647$	−24.4363	−0.960689	−0.480345	−	0.877080i	$-0.659488\pi$
$647$	−24.4363	−0.960689	−0.480345	+	0.877080i	$0.659488\pi$
$648$	−2.09859	−0.0824403
$649$	−33.6287	−1.32004
$650$	−40.9166	−1.60488
$651$	9.59557	0.376080
$652$	0.394347	0.0154438
$653$	0.595215	0.0232926	0.0116463	−	0.999932i	$-0.496293\pi$
$653$	0.595215	0.0232926	0.0116463	+	0.999932i	$0.496293\pi$
$654$	−5.87140	−0.229590
$655$	−58.4553	−2.28404
$656$	34.1951	1.33509
$657$	10.4537	0.407836
$658$	−15.8540	−0.618054
$659$	32.6493	1.27184	0.635918	−	0.771757i	$-0.280622\pi$
$659$	32.6493	1.27184	0.635918	+	0.771757i	$0.280622\pi$
$660$	−9.08064	−0.353463
$661$	4.89517	0.190400	0.0952000	−	0.995458i	$-0.469651\pi$
$661$	4.89517	0.190400	0.0952000	+	0.995458i	$0.469651\pi$
$662$	40.1150	1.55911
$663$	−3.48065	−0.135177
$664$	−17.9226	−0.695532
$665$	14.5359	0.563678
$666$	−11.6569	−0.451694
$667$	−26.1944	−1.01425
$668$	−2.98471	−0.115482
$669$	17.9109	0.692475
$670$	49.1137	1.89743
$671$	−43.5093	−1.67966
$672$	3.94335	0.152118
$673$	−2.77017	−0.106782	−0.0533911	−	0.998574i	$-0.517003\pi$
$673$	−2.77017	−0.106782	−0.0533911	+	0.998574i	$0.517003\pi$
$674$	−7.89803	−0.304221
$675$	−7.11492	−0.273854
$676$	−0.645964	−0.0248448
$677$	26.9315	1.03506	0.517531	−	0.855665i	$-0.326851\pi$
$677$	26.9315	1.03506	0.517531	+	0.855665i	$0.326851\pi$
$678$	19.7601	0.758881
$679$	9.72543	0.373228
$680$	7.30445	0.280113
$681$	−5.97903	−0.229117
$682$	−56.6719	−2.17008
$683$	19.8762	0.760543	0.380272	−	0.924875i	$-0.375830\pi$
$683$	19.8762	0.760543	0.380272	+	0.924875i	$0.375830\pi$
$684$	3.04796	0.116542
$685$	18.0109	0.688163
$686$	1.65222	0.0630822
$687$	−5.27177	−0.201131
$688$	−24.4661	−0.932763
$689$	12.5149	0.476781
$690$	−45.5867	−1.73546
$691$	−20.4915	−0.779534	−0.389767	−	0.920913i	$-0.627445\pi$
$691$	−20.4915	−0.779534	−0.389767	+	0.920913i	$0.627445\pi$
$692$	11.0718	0.420887
$693$	3.57461	0.135788
$694$	37.4242	1.42060
$695$	48.3137	1.83264
$696$	−6.93467	−0.262858
$697$	6.94033	0.262884
$698$	35.1218	1.32938
$699$	25.5065	0.964743
$700$	5.19276	0.196268
$701$	−1.19998	−0.0453225	−0.0226612	−	0.999743i	$-0.507214\pi$
$701$	−1.19998	−0.0453225	−0.0226612	+	0.999743i	$0.507214\pi$
$702$	5.75081	0.217050
$703$	−29.4642	−1.11126
$704$	11.9346	0.449804
$705$	33.3988	1.25787
$706$	5.87140	0.220973
$707$	−10.7028	−0.402522
$708$	6.86609	0.258044
$709$	37.0032	1.38969	0.694843	−	0.719162i	$-0.255474\pi$
$709$	37.0032	1.38969	0.694843	+	0.719162i	$0.255474\pi$
$710$	−11.8960	−0.446448
$711$	0.162528	0.00609527
$712$	−2.01737	−0.0756042
$713$	−76.0643	−2.84863
$714$	1.65222	0.0618329
$715$	−43.3061	−1.61956
$716$	8.69484	0.324941
$717$	−11.5629	−0.431824
$718$	−27.8093	−1.03783
$719$	−26.1347	−0.974660	−0.487330	−	0.873218i	$-0.662029\pi$
$719$	−26.1347	−0.974660	−0.487330	+	0.873218i	$0.662029\pi$
$720$	−17.1492	−0.639113
$721$	19.3674	0.721278
$722$	−2.57634	−0.0958815
$723$	−24.7287	−0.919669
$724$	4.71014	0.175051
$725$	−23.5109	−0.873172
$726$	−2.93733	−0.109014
$727$	21.4689	0.796239	0.398120	−	0.917334i	$-0.369663\pi$
$727$	21.4689	0.796239	0.398120	+	0.917334i	$0.369663\pi$
$728$	7.30445	0.270721
$729$	1.00000	0.0370370
$730$	60.1170	2.22503
$731$	−4.96571	−0.183664
$732$	8.88346	0.328342
$733$	50.1703	1.85308	0.926540	−	0.376195i	$-0.122768\pi$
$733$	50.1703	1.85308	0.926540	+	0.376195i	$0.122768\pi$
$734$	0	0
$735$	−3.48065	−0.128386
$736$	−31.2590	−1.15222
$737$	30.5283	1.12452
$738$	−11.4670	−0.422105
$739$	−12.4166	−0.456751	−0.228375	−	0.973573i	$-0.573341\pi$
$739$	−12.4166	−0.456751	−0.228375	+	0.973573i	$0.573341\pi$
$740$	−17.9226	−0.658848
$741$	14.5359	0.533990
$742$	−5.94069	−0.218090
$743$	−28.4363	−1.04323	−0.521613	−	0.853182i	$-0.674670\pi$
$743$	−28.4363	−1.04323	−0.521613	+	0.853182i	$0.674670\pi$
$744$	−20.1371	−0.738264
$745$	50.7557	1.85955
$746$	−1.89875	−0.0695183
$747$	8.54032	0.312474
$748$	−2.60889	−0.0953905
$749$	10.8464	0.396318
$750$	−12.1625	−0.444113
$751$	22.1359	0.807750	0.403875	−	0.914814i	$-0.367663\pi$
$751$	22.1359	0.807750	0.403875	+	0.914814i	$0.367663\pi$
$752$	47.2775	1.72403
$753$	4.86734	0.177376
$754$	19.0032	0.692057
$755$	34.3911	1.25162
$756$	−0.729840	−0.0265440
$757$	−22.3028	−0.810610	−0.405305	−	0.914181i	$-0.632835\pi$
$757$	−22.3028	−0.810610	−0.405305	+	0.914181i	$0.632835\pi$
$758$	−57.8332	−2.10059
$759$	−28.3359	−1.02853
$760$	−30.5049	−1.10653
$761$	−1.29518	−0.0469503	−0.0234752	−	0.999724i	$-0.507473\pi$
$761$	−1.29518	−0.0469503	−0.0234752	+	0.999724i	$0.507473\pi$
$762$	6.48830	0.235046
$763$	3.55364	0.128650
$764$	−16.8149	−0.608341
$765$	−3.48065	−0.125843
$766$	−26.0832	−0.942423
$767$	32.7448	1.18235
$768$	−15.4673	−0.558129
$769$	−6.42342	−0.231635	−0.115817	−	0.993271i	$-0.536949\pi$
$769$	−6.42342	−0.231635	−0.115817	+	0.993271i	$0.536949\pi$
$770$	20.5569	0.740818
$771$	−26.7867	−0.964700
$772$	−4.75486	−0.171131
$773$	13.7061	0.492974	0.246487	−	0.969146i	$-0.420724\pi$
$773$	13.7061	0.492974	0.246487	+	0.969146i	$0.420724\pi$
$774$	8.20447	0.294903
$775$	−68.2718	−2.45239
$776$	−20.4097	−0.732664
$777$	7.05526	0.253106
$778$	−14.2931	−0.512432
$779$	−28.9842	−1.03847
$780$	8.84196	0.316593
$781$	−7.39433	−0.264590
$782$	−13.0972	−0.468355
$783$	3.30445	0.118091
$784$	−4.92701	−0.175965
$785$	−14.6496	−0.522868
$786$	27.7480	0.989739
$787$	48.3328	1.72288	0.861439	−	0.507861i	$-0.169564\pi$
$787$	48.3328	1.72288	0.861439	+	0.507861i	$0.169564\pi$
$788$	0.454375	0.0161865
$789$	−30.6867	−1.09248
$790$	0.934666	0.0332539
$791$	−11.9597	−0.425238
$792$	−7.50162	−0.266558
$793$	42.3657	1.50445
$794$	−33.0419	−1.17261
$795$	12.5149	0.443859
$796$	3.68953	0.130772
$797$	−32.9105	−1.16575	−0.582875	−	0.812562i	$-0.698072\pi$
$797$	−32.9105	−1.16575	−0.582875	+	0.812562i	$0.698072\pi$
$798$	−6.90002	−0.244258
$799$	9.59557	0.339467
$800$	−28.0566	−0.991952
$801$	0.961300	0.0339659
$802$	−59.0884	−2.08648
$803$	37.3677	1.31868
$804$	−6.23307	−0.219823
$805$	27.5912	0.972461
$806$	55.1823	1.94371
$807$	−5.59116	−0.196818
$808$	22.4609	0.790170
$809$	28.1807	0.990781	0.495391	−	0.868670i	$-0.335025\pi$
$809$	28.1807	0.990781	0.495391	+	0.868670i	$0.335025\pi$
$810$	5.75081	0.202063
$811$	20.9347	0.735116	0.367558	−	0.930001i	$-0.380194\pi$
$811$	20.9347	0.735116	0.367558	+	0.930001i	$0.380194\pi$
$812$	−2.41172	−0.0846347
$813$	−8.88102	−0.311471
$814$	−41.6687	−1.46049
$815$	1.88066	0.0658767
$816$	−4.92701	−0.172480
$817$	20.7378	0.725525
$818$	25.3509	0.886373
$819$	−3.48065	−0.121624
$820$	−17.6307	−0.615689
$821$	−53.9896	−1.88425	−0.942125	−	0.335263i	$-0.891175\pi$
$821$	−53.9896	−1.88425	−0.942125	+	0.335263i	$0.891175\pi$
$822$	−8.54958	−0.298201
$823$	−25.9714	−0.905306	−0.452653	−	0.891687i	$-0.649522\pi$
$823$	−25.9714	−0.905306	−0.452653	+	0.891687i	$0.649522\pi$
$824$	−40.6441	−1.41590
$825$	−25.4330	−0.885465
$826$	−15.5436	−0.540830
$827$	7.78987	0.270880	0.135440	−	0.990786i	$-0.456755\pi$
$827$	7.78987	0.270880	0.135440	+	0.990786i	$0.456755\pi$
$828$	5.78546	0.201058
$829$	−8.09593	−0.281183	−0.140592	−	0.990068i	$-0.544900\pi$
$829$	−8.09593	−0.281183	−0.140592	+	0.990068i	$0.544900\pi$
$830$	49.1137	1.70476
$831$	1.50162	0.0520906
$832$	−11.6210	−0.402884
$833$	−1.00000	−0.0346479
$834$	−22.9339	−0.794138
$835$	−14.2343	−0.492597
$836$	10.8953	0.376821
$837$	9.59557	0.331672
$838$	−2.05724	−0.0710663
$839$	15.0743	0.520421	0.260211	−	0.965552i	$-0.416208\pi$
$839$	15.0743	0.520421	0.260211	+	0.965552i	$0.416208\pi$
$840$	7.30445	0.252027
$841$	−18.0806	−0.623470
$842$	60.0791	2.07046
$843$	−2.35241	−0.0810213
$844$	−7.76132	−0.267156
$845$	−3.08064	−0.105977
$846$	−15.8540	−0.545073
$847$	1.77780	0.0610860
$848$	17.7154	0.608351
$849$	17.4077	0.597429
$850$	−11.7554	−0.403208
$851$	−55.9271	−1.91716
$852$	1.50973	0.0517224
$853$	12.0581	0.412860	0.206430	−	0.978461i	$-0.433815\pi$
$853$	12.0581	0.412860	0.206430	+	0.978461i	$0.433815\pi$
$854$	−20.1105	−0.688168
$855$	14.5359	0.497117
$856$	−22.7621	−0.777991
$857$	40.3677	1.37893	0.689467	−	0.724317i	$-0.257845\pi$
$857$	40.3677	1.37893	0.689467	+	0.724317i	$0.257845\pi$
$858$	20.5569	0.701800
$859$	41.9110	1.42998	0.714992	−	0.699132i	$-0.246430\pi$
$859$	41.9110	1.42998	0.714992	+	0.699132i	$0.246430\pi$
$860$	12.6145	0.430151
$861$	6.94033	0.236526
$862$	−24.1371	−0.822114
$863$	−51.0258	−1.73694	−0.868470	−	0.495742i	$-0.834896\pi$
$863$	−51.0258	−1.73694	−0.868470	+	0.495742i	$0.834896\pi$
$864$	3.94335	0.134156
$865$	52.8021	1.79532
$866$	−33.9572	−1.15391
$867$	−1.00000	−0.0339618
$868$	−7.00324	−0.237705
$869$	0.580972	0.0197081
$870$	19.0032	0.644270
$871$	−29.7259	−1.00722
$872$	−7.45761	−0.252547
$873$	9.72543	0.329156
$874$	54.6966	1.85014
$875$	7.36131	0.248858
$876$	−7.62950	−0.257777
$877$	−29.8286	−1.00724	−0.503621	−	0.863925i	$-0.667999\pi$
$877$	−29.8286	−1.00724	−0.503621	+	0.863925i	$0.667999\pi$
$878$	−24.0952	−0.813174
$879$	15.9314	0.537354
$880$	−61.3017	−2.06648
$881$	−24.2477	−0.816924	−0.408462	−	0.912775i	$-0.633935\pi$
$881$	−24.2477	−0.816924	−0.408462	+	0.912775i	$0.633935\pi$
$882$	1.65222	0.0556332
$883$	17.7569	0.597568	0.298784	−	0.954321i	$-0.403419\pi$
$883$	17.7569	0.597568	0.298784	+	0.954321i	$0.403419\pi$
$884$	2.54032	0.0854402
$885$	32.7448	1.10070
$886$	56.2182	1.88869
$887$	6.57587	0.220796	0.110398	−	0.993887i	$-0.464787\pi$
$887$	6.57587	0.220796	0.110398	+	0.993887i	$0.464787\pi$
$888$	−14.8061	−0.496859
$889$	−3.92701	−0.131708
$890$	5.52825	0.185307
$891$	3.57461	0.119754
$892$	−13.0721	−0.437686
$893$	−40.0731	−1.34099
$894$	−24.0931	−0.805795
$895$	41.4662	1.38606
$896$	13.4030	0.447764
$897$	27.5912	0.921242
$898$	−56.9944	−1.90193
$899$	31.7081	1.05752
$900$	5.19276	0.173092
$901$	3.59557	0.119786
$902$	−40.9899	−1.36481
$903$	−4.96571	−0.165249
$904$	25.0984	0.834762
$905$	22.4629	0.746693
$906$	−16.3251	−0.542364
$907$	−38.4122	−1.27545	−0.637727	−	0.770262i	$-0.720125\pi$
$907$	−38.4122	−1.27545	−0.637727	+	0.770262i	$0.720125\pi$
$908$	4.36374	0.144816
$909$	−10.7028	−0.354991
$910$	−20.0166	−0.663542
$911$	40.0674	1.32749	0.663746	−	0.747958i	$-0.268965\pi$
$911$	40.0674	1.32749	0.663746	+	0.747958i	$0.268965\pi$
$912$	20.5762	0.681347
$913$	30.5283	1.01034
$914$	−4.96931	−0.164370
$915$	42.3657	1.40057
$916$	3.84755	0.127127
$917$	−16.7944	−0.554599
$918$	1.65222	0.0545315
$919$	−42.4940	−1.40175	−0.700873	−	0.713286i	$-0.747206\pi$
$919$	−42.4940	−1.40175	−0.700873	+	0.713286i	$0.747206\pi$
$920$	−57.9024	−1.90899
$921$	11.5795	0.381556
$922$	0.427022	0.0140632
$923$	7.19998	0.236990
$924$	−2.60889	−0.0858262
$925$	−50.1976	−1.65049
$926$	33.9919	1.11704
$927$	19.3674	0.636107
$928$	13.0306	0.427750
$929$	45.0001	1.47640	0.738202	−	0.674579i	$-0.235675\pi$
$929$	45.0001	1.47640	0.738202	+	0.674579i	$0.235675\pi$
$930$	55.1823	1.80950
$931$	4.17620	0.136870
$932$	−18.6157	−0.609776
$933$	1.52825	0.0500328
$934$	−19.7880	−0.647483
$935$	−12.4420	−0.406895
$936$	7.30445	0.238753
$937$	17.0267	0.556236	0.278118	−	0.960547i	$-0.410289\pi$
$937$	17.0267	0.556236	0.278118	+	0.960547i	$0.410289\pi$
$938$	14.1105	0.460725
$939$	−15.1565	−0.494614
$940$	−24.3758	−0.795052
$941$	28.2452	0.920766	0.460383	−	0.887720i	$-0.347712\pi$
$941$	28.2452	0.920766	0.460383	+	0.887720i	$0.347712\pi$
$942$	6.95401	0.226574
$943$	−55.0161	−1.79157
$944$	46.3517	1.50862
$945$	−3.48065	−0.113226
$946$	29.3277	0.953527
$947$	54.8065	1.78097	0.890486	−	0.455011i	$-0.150365\pi$
$947$	54.8065	1.78097	0.890486	+	0.455011i	$0.150365\pi$
$948$	−0.118619	−0.00385258
$949$	−36.3855	−1.18112
$950$	49.0931	1.59279
$951$	−12.8746	−0.417489
$952$	2.09859	0.0680156
$953$	32.4609	1.05151	0.525757	−	0.850635i	$-0.323782\pi$
$953$	32.4609	1.05151	0.525757	+	0.850635i	$0.323782\pi$
$954$	−5.94069	−0.192337
$955$	−80.1912	−2.59493
$956$	8.43907	0.272939
$957$	11.8121	0.381831
$958$	−4.23182	−0.136724
$959$	5.17459	0.167096
$960$	−11.6210	−0.375065
$961$	61.0750	1.97016
$962$	40.5734	1.30814
$963$	10.8464	0.349520
$964$	18.0480	0.581286
$965$	−22.6762	−0.729973
$966$	−13.0972	−0.421395
$967$	20.6279	0.663348	0.331674	−	0.943394i	$-0.392387\pi$
$967$	20.6279	0.663348	0.331674	+	0.943394i	$0.392387\pi$
$968$	−3.73087	−0.119915
$969$	4.17620	0.134159
$970$	55.9291	1.79577
$971$	31.6992	1.01728	0.508638	−	0.860980i	$-0.330149\pi$
$971$	31.6992	1.01728	0.508638	+	0.860980i	$0.330149\pi$
$972$	−0.729840	−0.0234096
$973$	13.8807	0.444994
$974$	21.8839	0.701206
$975$	24.7646	0.793101
$976$	59.9706	1.91961
$977$	−2.82093	−0.0902497	−0.0451248	−	0.998981i	$-0.514369\pi$
$977$	−2.82093	−0.0902497	−0.0451248	+	0.998981i	$0.514369\pi$
$978$	−0.892728	−0.0285463
$979$	3.43627	0.109824
$980$	2.54032	0.0811475
$981$	3.55364	0.113459
$982$	53.5820	1.70987
$983$	36.7557	1.17232	0.586162	−	0.810194i	$-0.300638\pi$
$983$	36.7557	1.17232	0.586162	+	0.810194i	$0.300638\pi$
$984$	−14.5649	−0.464312
$985$	2.16694	0.0690445
$986$	5.45968	0.173872
$987$	9.59557	0.305430
$988$	−10.6089	−0.337514
$989$	39.3633	1.25168
$990$	20.5569	0.653340
$991$	−49.0630	−1.55854	−0.779269	−	0.626690i	$-0.784409\pi$
$991$	−49.0630	−1.55854	−0.779269	+	0.626690i	$0.784409\pi$
$992$	37.8387	1.20138
$993$	−24.2794	−0.770484
$994$	−3.41774	−0.108404
$995$	17.5956	0.557817
$996$	−6.23307	−0.197502
$997$	−49.3343	−1.56243	−0.781217	−	0.624260i	$-0.785401\pi$
$997$	−49.3343	−1.56243	−0.781217	+	0.624260i	$0.785401\pi$
$998$	−59.8033	−1.89304
$999$	7.05526	0.223218

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	357.2.a.h.1.3	✓	4
3.2	odd	2		1071.2.a.j.1.2		4
4.3	odd	2		5712.2.a.bx.1.4		4
5.4	even	2		8925.2.a.bs.1.2		4
7.6	odd	2		2499.2.a.z.1.3		4
17.16	even	2		6069.2.a.s.1.3		4
21.20	even	2		7497.2.a.be.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.a.h.1.3	✓	4	1.1	even	1	trivial
1071.2.a.j.1.2		4	3.2	odd	2
2499.2.a.z.1.3		4	7.6	odd	2
5712.2.a.bx.1.4		4	4.3	odd	2
6069.2.a.s.1.3		4	17.16	even	2
7497.2.a.be.1.2		4	21.20	even	2
8925.2.a.bs.1.2		4	5.4	even	2

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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

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Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	357.2.a.h.1.3

Level	$357$
Weight	$2$
Character	357.1
Self dual	yes
Analytic conductor	$2.851$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$