LMFDB - Embedded newform 363.4.a.v.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,4,Mod(1,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("363.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	363.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$21.4176933321$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 30x^{4} + 3x^{3} + 211x^{2} + 208x + 44 $ x^6 - x^5 - 30x^4 + 3x^3 + 211x^2 + 208x + 44
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.37704$ 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)

$f(q)$	$=$	\(q+3.37704 q^{2} -3.00000 q^{3} +3.40437 q^{4} +14.0063 q^{5} -10.1311 q^{6} +23.3415 q^{7} -15.5196 q^{8} +9.00000 q^{9} +47.2999 q^{10} -10.2131 q^{12} +37.7358 q^{13} +78.8251 q^{14} -42.0190 q^{15} -79.6452 q^{16} +8.60337 q^{17} +30.3933 q^{18} -88.5343 q^{19} +47.6828 q^{20} -70.0245 q^{21} +137.456 q^{23} +46.5588 q^{24} +71.1777 q^{25} +127.435 q^{26} -27.0000 q^{27} +79.4632 q^{28} +222.330 q^{29} -141.900 q^{30} +241.466 q^{31} -144.808 q^{32} +29.0539 q^{34} +326.929 q^{35} +30.6394 q^{36} +150.601 q^{37} -298.983 q^{38} -113.207 q^{39} -217.373 q^{40} -141.643 q^{41} -236.475 q^{42} +246.830 q^{43} +126.057 q^{45} +464.193 q^{46} +62.2452 q^{47} +238.936 q^{48} +201.826 q^{49} +240.370 q^{50} -25.8101 q^{51} +128.467 q^{52} -497.366 q^{53} -91.1800 q^{54} -362.251 q^{56} +265.603 q^{57} +750.816 q^{58} -586.139 q^{59} -143.048 q^{60} +646.573 q^{61} +815.440 q^{62} +210.074 q^{63} +148.140 q^{64} +528.540 q^{65} -963.421 q^{67} +29.2891 q^{68} -412.367 q^{69} +1104.05 q^{70} -149.817 q^{71} -139.676 q^{72} +432.996 q^{73} +508.584 q^{74} -213.533 q^{75} -301.404 q^{76} -382.305 q^{78} -1241.92 q^{79} -1115.54 q^{80} +81.0000 q^{81} -478.332 q^{82} +713.378 q^{83} -238.390 q^{84} +120.502 q^{85} +833.553 q^{86} -666.989 q^{87} +224.035 q^{89} +425.699 q^{90} +880.810 q^{91} +467.950 q^{92} -724.398 q^{93} +210.204 q^{94} -1240.04 q^{95} +434.424 q^{96} -1210.69 q^{97} +681.572 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{2} - 18 q^{3} + 17 q^{4} + 9 q^{5} - 15 q^{6} + q^{7} + 24 q^{8} + 54 q^{9} + 50 q^{10} - 51 q^{12} + 66 q^{13} - 42 q^{14} - 27 q^{15} - 71 q^{16} + 80 q^{17} + 45 q^{18} - 90 q^{19} + 455 q^{20}+ \cdots + 1405 q^{98}+O(q^{100}) $ 6 * q + 5 * q^2 - 18 * q^3 + 17 * q^4 + 9 * q^5 - 15 * q^6 + q^7 + 24 * q^8 + 54 * q^9 + 50 * q^10 - 51 * q^12 + 66 * q^13 - 42 * q^14 - 27 * q^15 - 71 * q^16 + 80 * q^17 + 45 * q^18 - 90 * q^19 + 455 * q^20 - 3 * q^21 - 196 * q^23 - 72 * q^24 + 351 * q^25 + 93 * q^26 - 162 * q^27 - 52 * q^28 + 579 * q^29 - 150 * q^30 + 129 * q^31 + 657 * q^32 + 530 * q^34 + 618 * q^35 + 153 * q^36 + 192 * q^37 - 769 * q^38 - 198 * q^39 + 1476 * q^40 + 990 * q^41 + 126 * q^42 + 208 * q^43 + 81 * q^45 + 493 * q^46 + 680 * q^47 + 213 * q^48 - 963 * q^49 + 1656 * q^50 - 240 * q^51 + 887 * q^52 + 197 * q^53 - 135 * q^54 - 429 * q^56 + 270 * q^57 + 395 * q^58 - 1229 * q^59 - 1365 * q^60 + 928 * q^61 + 2117 * q^62 + 9 * q^63 + 844 * q^64 + 2278 * q^65 - 264 * q^67 + 3279 * q^68 + 588 * q^69 + 665 * q^70 - 786 * q^71 + 216 * q^72 + 593 * q^73 + 1543 * q^74 - 1053 * q^75 - 434 * q^76 - 279 * q^78 - 421 * q^79 + 18 * q^80 + 486 * q^81 - 739 * q^82 + 2043 * q^83 + 156 * q^84 + 1900 * q^85 + 2683 * q^86 - 1737 * q^87 + 158 * q^89 + 450 * q^90 + 698 * q^91 - 3915 * q^92 - 387 * q^93 - 4934 * q^94 + 1038 * q^95 - 1971 * q^96 - 1711 * q^97 + 1405 * 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 $

$2$	3.37704	1.19396	0.596981	−	0.802255i	$-0.296367\pi$
$2$	3.37704	1.19396	0.596981	+	0.802255i	$0.296367\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	3.40437	0.425547
$5$	14.0063	1.25277	0.626383	−	0.779516i	$-0.284535\pi$
$5$	14.0063	1.25277	0.626383	+	0.779516i	$0.284535\pi$
$6$	−10.1311	−0.689335
$7$	23.3415	1.26032	0.630161	−	0.776464i	$-0.282989\pi$
$7$	23.3415	1.26032	0.630161	+	0.776464i	$0.282989\pi$
$8$	−15.5196	−0.685876
$9$	9.00000	0.333333
$10$	47.2999	1.49576
$11$	0	0
$11$	0	0
$12$	−10.2131	−0.245690
$13$	37.7358	0.805079	0.402539	−	0.915403i	$-0.368128\pi$
$13$	37.7358	0.805079	0.402539	+	0.915403i	$0.368128\pi$
$14$	78.8251	1.50478
$15$	−42.0190	−0.723284
$16$	−79.6452	−1.24446
$17$	8.60337	0.122743	0.0613713	−	0.998115i	$-0.480453\pi$
$17$	8.60337	0.122743	0.0613713	+	0.998115i	$0.480453\pi$
$18$	30.3933	0.397988
$19$	−88.5343	−1.06901	−0.534504	−	0.845166i	$-0.679502\pi$
$19$	−88.5343	−1.06901	−0.534504	+	0.845166i	$0.679502\pi$
$20$	47.6828	0.533110
$21$	−70.0245	−0.727648
$22$	0	0
$23$	137.456	1.24615	0.623076	−	0.782162i	$-0.285883\pi$
$23$	137.456	1.24615	0.623076	+	0.782162i	$0.285883\pi$
$24$	46.5588	0.395991
$25$	71.1777	0.569421
$26$	127.435	0.961234
$27$	−27.0000	−0.192450
$28$	79.4632	0.536326
$29$	222.330	1.42364	0.711821	−	0.702361i	$-0.247871\pi$
$29$	222.330	1.42364	0.711821	+	0.702361i	$0.247871\pi$
$30$	−141.900	−0.863575
$31$	241.466	1.39899	0.699493	−	0.714639i	$-0.253409\pi$
$31$	241.466	1.39899	0.699493	+	0.714639i	$0.253409\pi$
$32$	−144.808	−0.799959
$33$	0	0
$34$	29.0539	0.146550
$35$	326.929	1.57889
$36$	30.6394	0.141849
$37$	150.601	0.669151	0.334575	−	0.942369i	$-0.391407\pi$
$37$	150.601	0.669151	0.334575	+	0.942369i	$0.391407\pi$
$38$	−298.983	−1.27636
$39$	−113.207	−0.464812
$40$	−217.373	−0.859241
$41$	−141.643	−0.539533	−0.269767	−	0.962926i	$-0.586947\pi$
$41$	−141.643	−0.539533	−0.269767	+	0.962926i	$0.586947\pi$
$42$	−236.475	−0.868784
$43$	246.830	0.875376	0.437688	−	0.899127i	$-0.355797\pi$
$43$	246.830	0.875376	0.437688	+	0.899127i	$0.355797\pi$
$44$	0	0
$45$	126.057	0.417588
$46$	464.193	1.48786
$47$	62.2452	0.193179	0.0965893	−	0.995324i	$-0.469207\pi$
$47$	62.2452	0.193179	0.0965893	+	0.995324i	$0.469207\pi$
$48$	238.936	0.718487
$49$	201.826	0.588413
$50$	240.370	0.679868
$51$	−25.8101	−0.0708654
$52$	128.467	0.342599
$53$	−497.366	−1.28903	−0.644514	−	0.764592i	$-0.722940\pi$
$53$	−497.366	−1.28903	−0.644514	+	0.764592i	$0.722940\pi$
$54$	−91.1800	−0.229778
$55$	0	0
$56$	−362.251	−0.864425
$57$	265.603	0.617192
$58$	750.816	1.69977
$59$	−586.139	−1.29337	−0.646685	−	0.762757i	$-0.723845\pi$
$59$	−586.139	−1.29337	−0.646685	+	0.762757i	$0.723845\pi$
$60$	−143.048	−0.307791
$61$	646.573	1.35714	0.678568	−	0.734538i	$-0.262601\pi$
$61$	646.573	1.35714	0.678568	+	0.734538i	$0.262601\pi$
$62$	815.440	1.67034
$63$	210.074	0.420108
$64$	148.140	0.289336
$65$	528.540	1.00857
$66$	0	0
$67$	−963.421	−1.75673	−0.878363	−	0.477995i	$-0.841364\pi$
$67$	−963.421	−1.75673	−0.878363	+	0.477995i	$0.841364\pi$
$68$	29.2891	0.0522327
$69$	−412.367	−0.719466
$70$	1104.05	1.88513
$71$	−149.817	−0.250423	−0.125212	−	0.992130i	$-0.539961\pi$
$71$	−149.817	−0.250423	−0.125212	+	0.992130i	$0.539961\pi$
$72$	−139.676	−0.228625
$73$	432.996	0.694224	0.347112	−	0.937824i	$-0.387162\pi$
$73$	432.996	0.694224	0.347112	+	0.937824i	$0.387162\pi$
$74$	508.584	0.798941
$75$	−213.533	−0.328756
$76$	−301.404	−0.454913
$77$	0	0
$78$	−382.305	−0.554969
$79$	−1241.92	−1.76869	−0.884347	−	0.466829i	$-0.845396\pi$
$79$	−1241.92	−1.76869	−0.884347	+	0.466829i	$0.845396\pi$
$80$	−1115.54	−1.55901
$81$	81.0000	0.111111
$82$	−478.332	−0.644182
$83$	713.378	0.943415	0.471707	−	0.881755i	$-0.343638\pi$
$83$	713.378	0.943415	0.471707	+	0.881755i	$0.343638\pi$
$84$	−238.390	−0.309648
$85$	120.502	0.153768
$86$	833.553	1.04517
$87$	−666.989	−0.821940
$88$	0	0
$89$	224.035	0.266828	0.133414	−	0.991060i	$-0.457406\pi$
$89$	224.035	0.266828	0.133414	+	0.991060i	$0.457406\pi$
$90$	425.699	0.498585
$91$	880.810	1.01466
$92$	467.950	0.530295
$93$	−724.398	−0.807705
$94$	210.204	0.230648
$95$	−1240.04	−1.33922
$96$	434.424	0.461857
$97$	−1210.69	−1.26729	−0.633645	−	0.773624i	$-0.718442\pi$
$97$	−1210.69	−1.26729	−0.633645	+	0.773624i	$0.718442\pi$
$98$	681.572	0.702543
$99$	0	0
$100$	242.315	0.242315
$101$	1085.20	1.06913	0.534563	−	0.845128i	$-0.320476\pi$
$101$	1085.20	1.06913	0.534563	+	0.845128i	$0.320476\pi$
$102$	−87.1617	−0.0846107
$103$	−1054.83	−1.00908	−0.504540	−	0.863388i	$-0.668338\pi$
$103$	−1054.83	−1.00908	−0.504540	+	0.863388i	$0.668338\pi$
$104$	−585.644	−0.552184
$105$	−980.787	−0.911572
$106$	−1679.62	−1.53905
$107$	−538.523	−0.486551	−0.243275	−	0.969957i	$-0.578222\pi$
$107$	−538.523	−0.486551	−0.243275	+	0.969957i	$0.578222\pi$
$108$	−91.9181	−0.0818965
$109$	−125.822	−0.110565	−0.0552825	−	0.998471i	$-0.517606\pi$
$109$	−125.822	−0.110565	−0.0552825	+	0.998471i	$0.517606\pi$
$110$	0	0
$111$	−451.802	−0.386334
$112$	−1859.04	−1.56842
$113$	−1700.43	−1.41560	−0.707802	−	0.706411i	$-0.750313\pi$
$113$	−1700.43	−1.41560	−0.707802	+	0.706411i	$0.750313\pi$
$114$	896.950	0.736904
$115$	1925.25	1.56113
$116$	756.894	0.605826
$117$	339.622	0.268360
$118$	−1979.41	−1.54424
$119$	200.816	0.154695
$120$	652.118	0.496083
$121$	0	0
$122$	2183.50	1.62037
$123$	424.928	0.311500
$124$	822.041	0.595334
$125$	−753.854	−0.539414
$126$	709.426	0.501593
$127$	−1720.80	−1.20233	−0.601166	−	0.799124i	$-0.705297\pi$
$127$	−1720.80	−1.20233	−0.601166	+	0.799124i	$0.705297\pi$
$128$	1658.74	1.14541
$129$	−740.489	−0.505399
$130$	1784.90	1.20420
$131$	−676.499	−0.451191	−0.225595	−	0.974221i	$-0.572433\pi$
$131$	−676.499	−0.451191	−0.225595	+	0.974221i	$0.572433\pi$
$132$	0	0
$133$	−2066.52	−1.34730
$134$	−3253.51	−2.09746
$135$	−378.171	−0.241095
$136$	−133.521	−0.0841861
$137$	282.000	0.175860	0.0879302	−	0.996127i	$-0.471975\pi$
$137$	282.000	0.175860	0.0879302	+	0.996127i	$0.471975\pi$
$138$	−1392.58	−0.859015
$139$	−508.885	−0.310526	−0.155263	−	0.987873i	$-0.549622\pi$
$139$	−508.885	−0.310526	−0.155263	+	0.987873i	$0.549622\pi$
$140$	1112.99	0.671891
$141$	−186.736	−0.111532
$142$	−505.939	−0.298996
$143$	0	0
$144$	−716.807	−0.414819
$145$	3114.03	1.78349
$146$	1462.24	0.828877
$147$	−605.477	−0.339720
$148$	512.701	0.284755
$149$	−224.536	−0.123454	−0.0617271	−	0.998093i	$-0.519661\pi$
$149$	−224.536	−0.123454	−0.0617271	+	0.998093i	$0.519661\pi$
$150$	−721.109	−0.392522
$151$	2640.88	1.42326	0.711629	−	0.702556i	$-0.247958\pi$
$151$	2640.88	1.42326	0.711629	+	0.702556i	$0.247958\pi$
$152$	1374.02	0.733207
$153$	77.4303	0.0409142
$154$	0	0
$155$	3382.06	1.75260
$156$	−385.400	−0.197799
$157$	314.983	0.160117	0.0800587	−	0.996790i	$-0.474489\pi$
$157$	314.983	0.160117	0.0800587	+	0.996790i	$0.474489\pi$
$158$	−4194.01	−2.11176
$159$	1492.10	0.744221
$160$	−2028.23	−1.00216
$161$	3208.42	1.57055
$162$	273.540	0.132663
$163$	−1398.07	−0.671814	−0.335907	−	0.941895i	$-0.609043\pi$
$163$	−1398.07	−0.671814	−0.335907	+	0.941895i	$0.609043\pi$
$164$	−482.204	−0.229597
$165$	0	0
$166$	2409.10	1.12640
$167$	−1157.20	−0.536206	−0.268103	−	0.963390i	$-0.586397\pi$
$167$	−1157.20	−0.536206	−0.268103	+	0.963390i	$0.586397\pi$
$168$	1086.75	0.499076
$169$	−773.011	−0.351849
$170$	406.939	0.183593
$171$	−796.809	−0.356336
$172$	840.301	0.372513
$173$	709.698	0.311892	0.155946	−	0.987766i	$-0.450157\pi$
$173$	709.698	0.311892	0.155946	+	0.987766i	$0.450157\pi$
$174$	−2252.45	−0.981366
$175$	1661.39	0.717654
$176$	0	0
$177$	1758.42	0.746727
$178$	756.576	0.318583
$179$	−398.925	−0.166576	−0.0832878	−	0.996526i	$-0.526542\pi$
$179$	−398.925	−0.166576	−0.0832878	+	0.996526i	$0.526542\pi$
$180$	429.145	0.177703
$181$	339.146	0.139273	0.0696367	−	0.997572i	$-0.477816\pi$
$181$	339.146	0.139273	0.0696367	+	0.997572i	$0.477816\pi$
$182$	2974.53	1.21146
$183$	−1939.72	−0.783542
$184$	−2133.26	−0.854705
$185$	2109.36	0.838289
$186$	−2446.32	−0.964370
$187$	0	0
$188$	211.906	0.0822065
$189$	−630.221	−0.242549
$190$	−4187.67	−1.59897
$191$	−1499.60	−0.568100	−0.284050	−	0.958809i	$-0.591678\pi$
$191$	−1499.60	−0.568100	−0.284050	+	0.958809i	$0.591678\pi$
$192$	−444.419	−0.167048
$193$	−1269.54	−0.473489	−0.236745	−	0.971572i	$-0.576080\pi$
$193$	−1269.54	−0.473489	−0.236745	+	0.971572i	$0.576080\pi$
$194$	−4088.55	−1.51310
$195$	−1585.62	−0.582301
$196$	687.090	0.250397
$197$	1970.86	0.712782	0.356391	−	0.934337i	$-0.384007\pi$
$197$	1970.86	0.712782	0.356391	+	0.934337i	$0.384007\pi$
$198$	0	0
$199$	3706.78	1.32044	0.660218	−	0.751074i	$-0.270464\pi$
$199$	3706.78	1.32044	0.660218	+	0.751074i	$0.270464\pi$
$200$	−1104.65	−0.390552
$201$	2890.26	1.01425
$202$	3664.77	1.27650
$203$	5189.51	1.79425
$204$	−87.8672	−0.0301566
$205$	−1983.90	−0.675909
$206$	−3562.19	−1.20480
$207$	1237.10	0.415384
$208$	−3005.47	−1.00189
$209$	0	0
$210$	−3312.15	−1.08838
$211$	−2899.74	−0.946098	−0.473049	−	0.881036i	$-0.656847\pi$
$211$	−2899.74	−0.946098	−0.473049	+	0.881036i	$0.656847\pi$
$212$	−1693.22	−0.548542
$213$	449.452	0.144582
$214$	−1818.61	−0.580924
$215$	3457.18	1.09664
$216$	419.029	0.131997
$217$	5636.18	1.76317
$218$	−424.907	−0.132011
$219$	−1298.99	−0.400810
$220$	0	0
$221$	324.655	0.0988174
$222$	−1525.75	−0.461269
$223$	−6270.93	−1.88311	−0.941553	−	0.336865i	$-0.890633\pi$
$223$	−6270.93	−1.88311	−0.941553	+	0.336865i	$0.890633\pi$
$224$	−3380.04	−1.00821
$225$	640.599	0.189807
$226$	−5742.42	−1.69018
$227$	1500.61	0.438761	0.219380	−	0.975639i	$-0.429596\pi$
$227$	1500.61	0.438761	0.219380	+	0.975639i	$0.429596\pi$
$228$	904.211	0.262644
$229$	−2159.77	−0.623240	−0.311620	−	0.950207i	$-0.600872\pi$
$229$	−2159.77	−0.623240	−0.311620	+	0.950207i	$0.600872\pi$
$230$	6501.64	1.86394
$231$	0	0
$232$	−3450.47	−0.976441
$233$	2000.73	0.562541	0.281270	−	0.959629i	$-0.409244\pi$
$233$	2000.73	0.562541	0.281270	+	0.959629i	$0.409244\pi$
$234$	1146.92	0.320411
$235$	871.827	0.242007
$236$	−1995.44	−0.550389
$237$	3725.76	1.02116
$238$	678.161	0.184700
$239$	1765.46	0.477817	0.238908	−	0.971042i	$-0.423210\pi$
$239$	1765.46	0.477817	0.238908	+	0.971042i	$0.423210\pi$
$240$	3346.62	0.900096
$241$	−2787.34	−0.745014	−0.372507	−	0.928029i	$-0.621502\pi$
$241$	−2787.34	−0.745014	−0.372507	+	0.928029i	$0.621502\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	2201.18	0.577524
$245$	2826.84	0.737143
$246$	1435.00	0.371919
$247$	−3340.91	−0.860636
$248$	−3747.46	−0.959531
$249$	−2140.13	−0.544681
$250$	−2545.79	−0.644040
$251$	−1025.23	−0.257816	−0.128908	−	0.991657i	$-0.541147\pi$
$251$	−1025.23	−0.257816	−0.128908	+	0.991657i	$0.541147\pi$
$252$	715.169	0.178775
$253$	0	0
$254$	−5811.20	−1.43554
$255$	−361.505	−0.0887778
$256$	4416.50	1.07825
$257$	1693.79	0.411111	0.205555	−	0.978645i	$-0.434100\pi$
$257$	1693.79	0.411111	0.205555	+	0.978645i	$0.434100\pi$
$258$	−2500.66	−0.603427
$259$	3515.24	0.843346
$260$	1799.35	0.429196
$261$	2000.97	0.474547
$262$	−2284.56	−0.538705
$263$	−1139.53	−0.267172	−0.133586	−	0.991037i	$-0.542649\pi$
$263$	−1139.53	−0.267172	−0.133586	+	0.991037i	$0.542649\pi$
$264$	0	0
$265$	−6966.29	−1.61485
$266$	−6978.72	−1.60862
$267$	−672.106	−0.154053
$268$	−3279.84	−0.747569
$269$	−5330.08	−1.20811	−0.604053	−	0.796944i	$-0.706449\pi$
$269$	−5330.08	−1.20811	−0.604053	+	0.796944i	$0.706449\pi$
$270$	−1277.10	−0.287858
$271$	−1522.14	−0.341193	−0.170596	−	0.985341i	$-0.554569\pi$
$271$	−1522.14	−0.341193	−0.170596	+	0.985341i	$0.554569\pi$
$272$	−685.217	−0.152748
$273$	−2642.43	−0.585813
$274$	952.324	0.209971
$275$	0	0
$276$	−1403.85	−0.306166
$277$	2728.98	0.591943	0.295972	−	0.955197i	$-0.404357\pi$
$277$	2728.98	0.591943	0.295972	+	0.955197i	$0.404357\pi$
$278$	−1718.52	−0.370756
$279$	2173.19	0.466329
$280$	−5073.81	−1.08292
$281$	6880.19	1.46063	0.730316	−	0.683109i	$-0.239373\pi$
$281$	6880.19	1.46063	0.730316	+	0.683109i	$0.239373\pi$
$282$	−630.613	−0.133165
$283$	−871.549	−0.183068	−0.0915339	−	0.995802i	$-0.529177\pi$
$283$	−871.549	−0.183068	−0.0915339	+	0.995802i	$0.529177\pi$
$284$	−510.034	−0.106567
$285$	3720.12	0.773197
$286$	0	0
$287$	−3306.15	−0.679986
$288$	−1303.27	−0.266653
$289$	−4838.98	−0.984934
$290$	10516.2	2.12942
$291$	3632.08	0.731671
$292$	1474.08	0.295425
$293$	4560.87	0.909383	0.454691	−	0.890649i	$-0.349750\pi$
$293$	4560.87	0.909383	0.454691	+	0.890649i	$0.349750\pi$
$294$	−2044.72	−0.405613
$295$	−8209.67	−1.62029
$296$	−2337.26	−0.458954
$297$	0	0
$298$	−758.265	−0.147400
$299$	5186.99	1.00325
$300$	−726.946	−0.139901
$301$	5761.38	1.10326
$302$	8918.35	1.69932
$303$	−3255.61	−0.617261
$304$	7051.33	1.33033
$305$	9056.13	1.70017
$306$	261.485	0.0488500
$307$	−4841.69	−0.900097	−0.450048	−	0.893004i	$-0.648593\pi$
$307$	−4841.69	−0.900097	−0.450048	+	0.893004i	$0.648593\pi$
$308$	0	0
$309$	3164.48	0.582592
$310$	11421.3	2.09254
$311$	7270.61	1.32565	0.662827	−	0.748773i	$-0.269357\pi$
$311$	7270.61	1.32565	0.662827	+	0.748773i	$0.269357\pi$
$312$	1756.93	0.318804
$313$	2242.43	0.404952	0.202476	−	0.979287i	$-0.435101\pi$
$313$	2242.43	0.404952	0.202476	+	0.979287i	$0.435101\pi$
$314$	1063.71	0.191174
$315$	2942.36	0.526296
$316$	−4227.96	−0.752662
$317$	5439.43	0.963751	0.481875	−	0.876240i	$-0.339956\pi$
$317$	5439.43	0.963751	0.481875	+	0.876240i	$0.339956\pi$
$318$	5038.87	0.888572
$319$	0	0
$320$	2074.90	0.362470
$321$	1615.57	0.280910
$322$	10835.0	1.87518
$323$	−761.693	−0.131213
$324$	275.754	0.0472830
$325$	2685.94	0.458429
$326$	−4721.35	−0.802120
$327$	377.467	0.0638348
$328$	2198.24	0.370053
$329$	1452.90	0.243467
$330$	0	0
$331$	8447.66	1.40280	0.701398	−	0.712770i	$-0.252560\pi$
$331$	8447.66	1.40280	0.701398	+	0.712770i	$0.252560\pi$
$332$	2428.61	0.401467
$333$	1355.41	0.223050
$334$	−3907.89	−0.640210
$335$	−13494.0	−2.20076
$336$	5577.12	0.905526
$337$	−3116.36	−0.503736	−0.251868	−	0.967762i	$-0.581045\pi$
$337$	−3116.36	−0.503736	−0.251868	+	0.967762i	$0.581045\pi$
$338$	−2610.49	−0.420094
$339$	5101.30	0.817299
$340$	410.233	0.0654353
$341$	0	0
$342$	−2690.85	−0.425452
$343$	−3295.22	−0.518732
$344$	−3830.70	−0.600399
$345$	−5775.75	−0.901322
$346$	2396.67	0.372387
$347$	3813.71	0.590002	0.295001	−	0.955497i	$-0.404680\pi$
$347$	3813.71	0.590002	0.295001	+	0.955497i	$0.404680\pi$
$348$	−2270.68	−0.349774
$349$	−2006.69	−0.307781	−0.153890	−	0.988088i	$-0.549180\pi$
$349$	−2006.69	−0.307781	−0.153890	+	0.988088i	$0.549180\pi$
$350$	5610.59	0.856853
$351$	−1018.87	−0.154937
$352$	0	0
$353$	7986.75	1.20423	0.602113	−	0.798411i	$-0.294326\pi$
$353$	7986.75	1.20423	0.602113	+	0.798411i	$0.294326\pi$
$354$	5938.24	0.891565
$355$	−2098.39	−0.313722
$356$	762.700	0.113548
$357$	−602.447	−0.0893133
$358$	−1347.18	−0.198885
$359$	−514.645	−0.0756600	−0.0378300	−	0.999284i	$-0.512045\pi$
$359$	−514.645	−0.0756600	−0.0378300	+	0.999284i	$0.512045\pi$
$360$	−1956.36	−0.286414
$361$	979.319	0.142779
$362$	1145.31	0.166287
$363$	0	0
$364$	2998.61	0.431785
$365$	6064.69	0.869699
$366$	−6550.51	−0.935520
$367$	−2755.17	−0.391876	−0.195938	−	0.980616i	$-0.562775\pi$
$367$	−2755.17	−0.391876	−0.195938	+	0.980616i	$0.562775\pi$
$368$	−10947.7	−1.55078
$369$	−1274.78	−0.179844
$370$	7123.40	1.00089
$371$	−11609.3	−1.62459
$372$	−2466.12	−0.343716
$373$	191.959	0.0266468	0.0133234	−	0.999911i	$-0.495759\pi$
$373$	191.959	0.0266468	0.0133234	+	0.999911i	$0.495759\pi$
$374$	0	0
$375$	2261.56	0.311431
$376$	−966.020	−0.132497
$377$	8389.79	1.14614
$378$	−2128.28	−0.289595
$379$	1571.38	0.212972	0.106486	−	0.994314i	$-0.466040\pi$
$379$	1571.38	0.212972	0.106486	+	0.994314i	$0.466040\pi$
$380$	−4221.56	−0.569899
$381$	5162.40	0.694167
$382$	−5064.20	−0.678290
$383$	−14348.3	−1.91427	−0.957134	−	0.289645i	$-0.906463\pi$
$383$	−14348.3	−1.91427	−0.957134	+	0.289645i	$0.906463\pi$
$384$	−4976.21	−0.661306
$385$	0	0
$386$	−4287.28	−0.565328
$387$	2221.47	0.291792
$388$	−4121.65	−0.539291
$389$	7543.09	0.983162	0.491581	−	0.870832i	$-0.336419\pi$
$389$	7543.09	0.983162	0.491581	+	0.870832i	$0.336419\pi$
$390$	−5354.70	−0.695245
$391$	1182.58	0.152956
$392$	−3132.25	−0.403578
$393$	2029.50	0.260495
$394$	6655.67	0.851035
$395$	−17394.8	−2.21576
$396$	0	0
$397$	−3008.38	−0.380319	−0.190159	−	0.981753i	$-0.560900\pi$
$397$	−3008.38	−0.380319	−0.190159	+	0.981753i	$0.560900\pi$
$398$	12517.9	1.57655
$399$	6199.57	0.777861
$400$	−5668.96	−0.708620
$401$	13178.4	1.64115	0.820573	−	0.571541i	$-0.193654\pi$
$401$	13178.4	1.64115	0.820573	+	0.571541i	$0.193654\pi$
$402$	9760.52	1.21097
$403$	9111.91	1.12629
$404$	3694.44	0.454963
$405$	1134.51	0.139196
$406$	17525.2	2.14226
$407$	0	0
$408$	400.562	0.0486049
$409$	−10753.4	−1.30005	−0.650025	−	0.759912i	$-0.725242\pi$
$409$	−10753.4	−1.30005	−0.650025	+	0.759912i	$0.725242\pi$
$410$	−6699.69	−0.807009
$411$	−846.000	−0.101533
$412$	−3591.03	−0.429410
$413$	−13681.4	−1.63006
$414$	4177.73	0.495953
$415$	9991.82	1.18188
$416$	−5464.44	−0.644030
$417$	1526.65	0.179282
$418$	0	0
$419$	440.292	0.0513358	0.0256679	−	0.999671i	$-0.491829\pi$
$419$	440.292	0.0513358	0.0256679	+	0.999671i	$0.491829\pi$
$420$	−3338.97	−0.387916
$421$	10982.0	1.27133	0.635666	−	0.771964i	$-0.280725\pi$
$421$	10982.0	1.27133	0.635666	+	0.771964i	$0.280725\pi$
$422$	−9792.54	−1.12961
$423$	560.207	0.0643929
$424$	7718.93	0.884114
$425$	612.368	0.0698922
$426$	1517.82	0.172625
$427$	15092.0	1.71043
$428$	−1833.33	−0.207050
$429$	0	0
$430$	11675.0	1.30935
$431$	4312.39	0.481950	0.240975	−	0.970531i	$-0.422533\pi$
$431$	4312.39	0.481950	0.240975	+	0.970531i	$0.422533\pi$
$432$	2150.42	0.239496
$433$	10927.6	1.21281	0.606407	−	0.795154i	$-0.292610\pi$
$433$	10927.6	1.21281	0.606407	+	0.795154i	$0.292610\pi$
$434$	19033.6	2.10516
$435$	−9342.08	−1.02970
$436$	−428.346	−0.0470506
$437$	−12169.5	−1.33215
$438$	−4386.73	−0.478552
$439$	−5655.67	−0.614876	−0.307438	−	0.951568i	$-0.599472\pi$
$439$	−5655.67	−0.614876	−0.307438	+	0.951568i	$0.599472\pi$
$440$	0	0
$441$	1816.43	0.196138
$442$	1096.37	0.117984
$443$	2903.15	0.311361	0.155681	−	0.987807i	$-0.450243\pi$
$443$	2903.15	0.311361	0.155681	+	0.987807i	$0.450243\pi$
$444$	−1538.10	−0.164403
$445$	3137.92	0.334273
$446$	−21177.2	−2.24836
$447$	673.607	0.0712763
$448$	3457.81	0.364656
$449$	3477.50	0.365509	0.182754	−	0.983159i	$-0.441499\pi$
$449$	3477.50	0.365509	0.182754	+	0.983159i	$0.441499\pi$
$450$	2163.33	0.226623
$451$	0	0
$452$	−5788.91	−0.602406
$453$	−7922.65	−0.821718
$454$	5067.60	0.523864
$455$	12336.9	1.27113
$456$	−4122.05	−0.423317
$457$	3650.23	0.373634	0.186817	−	0.982395i	$-0.440183\pi$
$457$	3650.23	0.373634	0.186817	+	0.982395i	$0.440183\pi$
$458$	−7293.64	−0.744125
$459$	−232.291	−0.0236218
$460$	6554.27	0.664336
$461$	11721.5	1.18422	0.592111	−	0.805856i	$-0.298295\pi$
$461$	11721.5	1.18422	0.592111	+	0.805856i	$0.298295\pi$
$462$	0	0
$463$	−7337.56	−0.736512	−0.368256	−	0.929724i	$-0.620045\pi$
$463$	−7337.56	−0.736512	−0.368256	+	0.929724i	$0.620045\pi$
$464$	−17707.5	−1.77166
$465$	−10146.2	−1.01187
$466$	6756.53	0.671653
$467$	−10527.0	−1.04311	−0.521553	−	0.853219i	$-0.674647\pi$
$467$	−10527.0	−1.04311	−0.521553	+	0.853219i	$0.674647\pi$
$468$	1156.20	0.114200
$469$	−22487.7	−2.21404
$470$	2944.19	0.288948
$471$	−944.950	−0.0924438
$472$	9096.64	0.887091
$473$	0	0
$474$	12582.0	1.21922
$475$	−6301.66	−0.608716
$476$	683.651	0.0658300
$477$	−4476.30	−0.429676
$478$	5962.03	0.570495
$479$	−10530.9	−1.00453	−0.502265	−	0.864714i	$-0.667500\pi$
$479$	−10530.9	−1.00453	−0.502265	+	0.864714i	$0.667500\pi$
$480$	6084.69	0.578598
$481$	5683.03	0.538719
$482$	−9412.94	−0.889518
$483$	−9625.26	−0.906759
$484$	0	0
$485$	−16957.4	−1.58762
$486$	−820.620	−0.0765927
$487$	844.460	0.0785753	0.0392876	−	0.999228i	$-0.487491\pi$
$487$	844.460	0.0785753	0.0392876	+	0.999228i	$0.487491\pi$
$488$	−10034.6	−0.930826
$489$	4194.22	0.387872
$490$	9546.34	0.880122
$491$	−13213.6	−1.21451	−0.607254	−	0.794508i	$-0.707729\pi$
$491$	−13213.6	−1.21451	−0.607254	+	0.794508i	$0.707729\pi$
$492$	1446.61	0.132558
$493$	1912.79	0.174741
$494$	−11282.4	−1.02757
$495$	0	0
$496$	−19231.6	−1.74098
$497$	−3496.96	−0.315614
$498$	−7227.31	−0.650329
$499$	−16991.4	−1.52433	−0.762165	−	0.647383i	$-0.775864\pi$
$499$	−16991.4	−1.52433	−0.762165	+	0.647383i	$0.775864\pi$
$500$	−2566.40	−0.229546
$501$	3471.59	0.309579
$502$	−3462.24	−0.307823
$503$	2571.04	0.227906	0.113953	−	0.993486i	$-0.463649\pi$
$503$	2571.04	0.227906	0.113953	+	0.993486i	$0.463649\pi$
$504$	−3260.26	−0.288142
$505$	15199.7	1.33936
$506$	0	0
$507$	2319.03	0.203140
$508$	−5858.24	−0.511649
$509$	−20905.9	−1.82051	−0.910255	−	0.414048i	$-0.864115\pi$
$509$	−20905.9	−1.82051	−0.910255	+	0.414048i	$0.864115\pi$
$510$	−1220.82	−0.105997
$511$	10106.8	0.874946
$512$	1644.78	0.141972
$513$	2390.43	0.205731
$514$	5719.98	0.490851
$515$	−14774.3	−1.26414
$516$	−2520.90	−0.215071
$517$	0	0
$518$	11871.1	1.00692
$519$	−2129.09	−0.180071
$520$	−8202.73	−0.691757
$521$	736.058	0.0618949	0.0309475	−	0.999521i	$-0.490148\pi$
$521$	736.058	0.0618949	0.0309475	+	0.999521i	$0.490148\pi$
$522$	6757.34	0.566592
$523$	13165.7	1.10075	0.550377	−	0.834916i	$-0.314484\pi$
$523$	13165.7	1.10075	0.550377	+	0.834916i	$0.314484\pi$
$524$	−2303.06	−0.192003
$525$	−4984.18	−0.414338
$526$	−3848.22	−0.318993
$527$	2077.42	0.171715
$528$	0	0
$529$	6727.04	0.552892
$530$	−23525.4	−1.92807
$531$	−5275.25	−0.431123
$532$	−7035.22	−0.573337
$533$	−5344.99	−0.434367
$534$	−2269.73	−0.183934
$535$	−7542.73	−0.609534
$536$	14951.9	1.20490
$537$	1196.77	0.0961724
$538$	−17999.9	−1.44243
$539$	0	0
$540$	−1287.44	−0.102597
$541$	−21964.9	−1.74555	−0.872776	−	0.488121i	$-0.837682\pi$
$541$	−21964.9	−1.74555	−0.872776	+	0.488121i	$0.837682\pi$
$542$	−5140.31	−0.407371
$543$	−1017.44	−0.0804096
$544$	−1245.84	−0.0981890
$545$	−1762.31	−0.138512
$546$	−8923.58	−0.699439
$547$	−17257.9	−1.34899	−0.674493	−	0.738281i	$-0.735638\pi$
$547$	−17257.9	−1.34899	−0.674493	+	0.738281i	$0.735638\pi$
$548$	960.033	0.0748368
$549$	5819.16	0.452378
$550$	0	0
$551$	−19683.8	−1.52188
$552$	6399.77	0.493464
$553$	−28988.3	−2.22913
$554$	9215.85	0.706758
$555$	−6328.09	−0.483986
$556$	−1732.43	−0.132143
$557$	18174.4	1.38254	0.691269	−	0.722597i	$-0.257052\pi$
$557$	18174.4	1.38254	0.691269	+	0.722597i	$0.257052\pi$
$558$	7338.96	0.556779
$559$	9314.31	0.704747
$560$	−26038.3	−1.96486
$561$	0	0
$562$	23234.7	1.74394
$563$	22447.6	1.68038	0.840190	−	0.542292i	$-0.182443\pi$
$563$	22447.6	1.68038	0.840190	+	0.542292i	$0.182443\pi$
$564$	−635.718	−0.0474619
$565$	−23816.8	−1.77342
$566$	−2943.25	−0.218576
$567$	1890.66	0.140036
$568$	2325.11	0.171759
$569$	9383.01	0.691311	0.345656	−	0.938361i	$-0.387657\pi$
$569$	9383.01	0.691311	0.345656	+	0.938361i	$0.387657\pi$
$570$	12563.0	0.923168
$571$	7968.11	0.583984	0.291992	−	0.956421i	$-0.405682\pi$
$571$	7968.11	0.583984	0.291992	+	0.956421i	$0.405682\pi$
$572$	0	0
$573$	4498.79	0.327993
$574$	−11165.0	−0.811878
$575$	9783.77	0.709585
$576$	1333.26	0.0964452
$577$	−4929.58	−0.355669	−0.177835	−	0.984060i	$-0.556909\pi$
$577$	−4929.58	−0.355669	−0.177835	+	0.984060i	$0.556909\pi$
$578$	−16341.4	−1.17597
$579$	3808.62	0.273369
$580$	10601.3	0.758958
$581$	16651.3	1.18901
$582$	12265.7	0.873587
$583$	0	0
$584$	−6719.92	−0.476151
$585$	4756.86	0.336192
$586$	15402.2	1.08577
$587$	−2395.01	−0.168403	−0.0842016	−	0.996449i	$-0.526834\pi$
$587$	−2395.01	−0.168403	−0.0842016	+	0.996449i	$0.526834\pi$
$588$	−2061.27	−0.144567
$589$	−21378.0	−1.49553
$590$	−27724.3	−1.93456
$591$	−5912.58	−0.411525
$592$	−11994.6	−0.832729
$593$	−7389.73	−0.511737	−0.255868	−	0.966712i	$-0.582361\pi$
$593$	−7389.73	−0.511737	−0.255868	+	0.966712i	$0.582361\pi$
$594$	0	0
$595$	2812.69	0.193797
$596$	−764.403	−0.0525355
$597$	−11120.3	−0.762354
$598$	17516.7	1.19784
$599$	21832.8	1.48926	0.744629	−	0.667478i	$-0.232626\pi$
$599$	21832.8	1.48926	0.744629	+	0.667478i	$0.232626\pi$
$600$	3313.95	0.225485
$601$	−4785.00	−0.324766	−0.162383	−	0.986728i	$-0.551918\pi$
$601$	−4785.00	−0.324766	−0.162383	+	0.986728i	$0.551918\pi$
$602$	19456.4	1.31725
$603$	−8670.79	−0.585575
$604$	8990.55	0.605663
$605$	0	0
$606$	−10994.3	−0.736986
$607$	−16848.3	−1.12661	−0.563305	−	0.826249i	$-0.690471\pi$
$607$	−16848.3	−1.12661	−0.563305	+	0.826249i	$0.690471\pi$
$608$	12820.5	0.855163
$609$	−15568.5	−1.03591
$610$	30582.9	2.02994
$611$	2348.87	0.155524
$612$	263.602	0.0174109
$613$	3437.67	0.226503	0.113251	−	0.993566i	$-0.463873\pi$
$613$	3437.67	0.226503	0.113251	+	0.993566i	$0.463873\pi$
$614$	−16350.6	−1.07468
$615$	5951.69	0.390236
$616$	0	0
$617$	12049.1	0.786186	0.393093	−	0.919499i	$-0.371405\pi$
$617$	12049.1	0.786186	0.393093	+	0.919499i	$0.371405\pi$
$618$	10686.6	0.695593
$619$	22046.7	1.43156	0.715778	−	0.698328i	$-0.246072\pi$
$619$	22046.7	1.43156	0.715778	+	0.698328i	$0.246072\pi$
$620$	11513.8	0.745814
$621$	−3711.30	−0.239822
$622$	24553.1	1.58278
$623$	5229.32	0.336290
$624$	9016.42	0.578439
$625$	−19455.9	−1.24518
$626$	7572.78	0.483497
$627$	0	0
$628$	1072.32	0.0681374
$629$	1295.67	0.0821333
$630$	9936.46	0.628378
$631$	2847.78	0.179664	0.0898322	−	0.995957i	$-0.471367\pi$
$631$	2847.78	0.179664	0.0898322	+	0.995957i	$0.471367\pi$
$632$	19274.1	1.21310
$633$	8699.23	0.546230
$634$	18369.2	1.15068
$635$	−24102.1	−1.50624
$636$	5079.66	0.316701
$637$	7616.05	0.473719
$638$	0	0
$639$	−1348.36	−0.0834744
$640$	23232.9	1.43494
$641$	10179.3	0.627236	0.313618	−	0.949549i	$-0.398459\pi$
$641$	10179.3	0.627236	0.313618	+	0.949549i	$0.398459\pi$
$642$	5455.83	0.335396
$643$	−15705.5	−0.963240	−0.481620	−	0.876380i	$-0.659952\pi$
$643$	−15705.5	−0.963240	−0.481620	+	0.876380i	$0.659952\pi$
$644$	10922.7	0.668343
$645$	−10371.5	−0.633146
$646$	−2572.27	−0.156663
$647$	−3404.79	−0.206887	−0.103444	−	0.994635i	$-0.532986\pi$
$647$	−3404.79	−0.206887	−0.103444	+	0.994635i	$0.532986\pi$
$648$	−1257.09	−0.0762084
$649$	0	0
$650$	9070.53	0.547347
$651$	−16908.5	−1.01797
$652$	−4759.57	−0.285888
$653$	25658.3	1.53765	0.768826	−	0.639458i	$-0.220841\pi$
$653$	25658.3	1.53765	0.768826	+	0.639458i	$0.220841\pi$
$654$	1274.72	0.0762163
$655$	−9475.28	−0.565236
$656$	11281.2	0.671426
$657$	3896.96	0.231408
$658$	4906.48	0.290691
$659$	30937.0	1.82873	0.914367	−	0.404887i	$-0.132689\pi$
$659$	30937.0	1.82873	0.914367	+	0.404887i	$0.132689\pi$
$660$	0	0
$661$	−17637.6	−1.03786	−0.518928	−	0.854818i	$-0.673669\pi$
$661$	−17637.6	−1.03786	−0.518928	+	0.854818i	$0.673669\pi$
$662$	28528.1	1.67489
$663$	−973.964	−0.0570523
$664$	−11071.3	−0.647065
$665$	−28944.4	−1.68784
$666$	4577.25	0.266314
$667$	30560.5	1.77407
$668$	−3939.53	−0.228181
$669$	18812.8	1.08721
$670$	−45569.7	−2.62763
$671$	0	0
$672$	10140.1	0.582088
$673$	27408.3	1.56985	0.784927	−	0.619588i	$-0.212700\pi$
$673$	27408.3	1.56985	0.784927	+	0.619588i	$0.212700\pi$
$674$	−10524.1	−0.601442
$675$	−1921.80	−0.109585
$676$	−2631.62	−0.149728
$677$	1842.79	0.104615	0.0523074	−	0.998631i	$-0.483342\pi$
$677$	1842.79	0.104615	0.0523074	+	0.998631i	$0.483342\pi$
$678$	17227.3	0.975825
$679$	−28259.4	−1.59720
$680$	−1870.14	−0.105465
$681$	−4501.82	−0.253319
$682$	0	0
$683$	9024.46	0.505580	0.252790	−	0.967521i	$-0.418652\pi$
$683$	9024.46	0.505580	0.252790	+	0.967521i	$0.418652\pi$
$684$	−2712.63	−0.151638
$685$	3949.79	0.220312
$686$	−11128.1	−0.619347
$687$	6479.32	0.359828
$688$	−19658.8	−1.08937
$689$	−18768.5	−1.03777
$690$	−19504.9	−1.07614
$691$	7697.01	0.423745	0.211873	−	0.977297i	$-0.432044\pi$
$691$	7697.01	0.423745	0.211873	+	0.977297i	$0.432044\pi$
$692$	2416.08	0.132725
$693$	0	0
$694$	12879.0	0.704440
$695$	−7127.62	−0.389016
$696$	10351.4	0.563749
$697$	−1218.60	−0.0662237
$698$	−6776.65	−0.367478
$699$	−6002.18	−0.324783
$700$	5656.00	0.305395
$701$	−13348.5	−0.719212	−0.359606	−	0.933104i	$-0.617089\pi$
$701$	−13348.5	−0.719212	−0.359606	+	0.933104i	$0.617089\pi$
$702$	−3440.75	−0.184990
$703$	−13333.3	−0.715328
$704$	0	0
$705$	−2615.48	−0.139723
$706$	26971.5	1.43780
$707$	25330.3	1.34744
$708$	5986.31	0.317767
$709$	−24251.4	−1.28460	−0.642299	−	0.766455i	$-0.722019\pi$
$709$	−24251.4	−1.28460	−0.642299	+	0.766455i	$0.722019\pi$
$710$	−7086.35	−0.374572
$711$	−11177.3	−0.589565
$712$	−3476.94	−0.183011
$713$	33190.9	1.74335
$714$	−2034.48	−0.106637
$715$	0	0
$716$	−1358.09	−0.0708857
$717$	−5296.38	−0.275868
$718$	−1737.98	−0.0903352
$719$	−14231.5	−0.738170	−0.369085	−	0.929396i	$-0.620329\pi$
$719$	−14231.5	−0.738170	−0.369085	+	0.929396i	$0.620329\pi$
$720$	−10039.8	−0.519671
$721$	−24621.2	−1.27177
$722$	3307.19	0.170472
$723$	8362.02	0.430134
$724$	1154.58	0.0592674
$725$	15824.9	0.810652
$726$	0	0
$727$	−9582.44	−0.488849	−0.244424	−	0.969668i	$-0.578599\pi$
$727$	−9582.44	−0.488849	−0.244424	+	0.969668i	$0.578599\pi$
$728$	−13669.8	−0.695930
$729$	729.000	0.0370370
$730$	20480.7	1.03839
$731$	2123.57	0.107446
$732$	−6603.53	−0.333434
$733$	18811.0	0.947883	0.473942	−	0.880556i	$-0.342831\pi$
$733$	18811.0	0.947883	0.473942	+	0.880556i	$0.342831\pi$
$734$	−9304.30	−0.467885
$735$	−8480.52	−0.425590
$736$	−19904.7	−0.996870
$737$	0	0
$738$	−4304.99	−0.214727
$739$	14421.3	0.717854	0.358927	−	0.933366i	$-0.383143\pi$
$739$	14421.3	0.717854	0.358927	+	0.933366i	$0.383143\pi$
$740$	7181.06	0.356731
$741$	10022.7	0.496888
$742$	−39205.0	−1.93970
$743$	9349.25	0.461629	0.230815	−	0.972998i	$-0.425861\pi$
$743$	9349.25	0.461629	0.230815	+	0.972998i	$0.425861\pi$
$744$	11242.4	0.553985
$745$	−3144.92	−0.154659
$746$	648.253	0.0318153
$747$	6420.40	0.314472
$748$	0	0
$749$	−12569.9	−0.613211
$750$	7637.38	0.371837
$751$	−7699.82	−0.374128	−0.187064	−	0.982348i	$-0.559897\pi$
$751$	−7699.82	−0.374128	−0.187064	+	0.982348i	$0.559897\pi$
$752$	−4957.53	−0.240402
$753$	3075.69	0.148850
$754$	28332.6	1.36845
$755$	36989.1	1.78301
$756$	−2145.51	−0.103216
$757$	39855.1	1.91355	0.956776	−	0.290825i	$-0.0939297\pi$
$757$	39855.1	1.91355	0.956776	+	0.290825i	$0.0939297\pi$
$758$	5306.60	0.254280
$759$	0	0
$760$	19244.9	0.918536
$761$	29206.0	1.39122	0.695609	−	0.718421i	$-0.255135\pi$
$761$	29206.0	1.39122	0.695609	+	0.718421i	$0.255135\pi$
$762$	17433.6	0.828810
$763$	−2936.88	−0.139348
$764$	−5105.19	−0.241753
$765$	1084.52	0.0512559
$766$	−48454.8	−2.28556
$767$	−22118.4	−1.04126
$768$	−13249.5	−0.622526
$769$	19737.4	0.925553	0.462777	−	0.886475i	$-0.346853\pi$
$769$	19737.4	0.925553	0.462777	+	0.886475i	$0.346853\pi$
$770$	0	0
$771$	−5081.36	−0.237355
$772$	−4321.98	−0.201492
$773$	−36154.9	−1.68228	−0.841139	−	0.540820i	$-0.818114\pi$
$773$	−36154.9	−1.68228	−0.841139	+	0.540820i	$0.818114\pi$
$774$	7501.98	0.348389
$775$	17187.0	0.796613
$776$	18789.5	0.869204
$777$	−10545.7	−0.486906
$778$	25473.3	1.17386
$779$	12540.2	0.576765
$780$	−5398.05	−0.247796
$781$	0	0
$782$	3993.62	0.182623
$783$	−6002.90	−0.273980
$784$	−16074.4	−0.732254
$785$	4411.77	0.200589
$786$	6853.68	0.311021
$787$	10538.2	0.477316	0.238658	−	0.971104i	$-0.423293\pi$
$787$	10538.2	0.477316	0.238658	+	0.971104i	$0.423293\pi$
$788$	6709.55	0.303322
$789$	3418.58	0.154252
$790$	−58742.7	−2.64553
$791$	−39690.7	−1.78412
$792$	0	0
$793$	24398.9	1.09260
$794$	−10159.4	−0.454086
$795$	20898.9	0.932335
$796$	12619.3	0.561907
$797$	−27550.3	−1.22444	−0.612222	−	0.790686i	$-0.709724\pi$
$797$	−27550.3	−1.22444	−0.612222	+	0.790686i	$0.709724\pi$
$798$	20936.2	0.928737
$799$	535.518	0.0237112
$800$	−10307.1	−0.455514
$801$	2016.32	0.0889427
$802$	44504.1	1.95947
$803$	0	0
$804$	9839.53	0.431609
$805$	44938.2	1.96753
$806$	30771.2	1.34475
$807$	15990.2	0.697501
$808$	−16841.9	−0.733288
$809$	42469.4	1.84567	0.922834	−	0.385198i	$-0.125867\pi$
$809$	42469.4	1.84567	0.922834	+	0.385198i	$0.125867\pi$
$810$	3831.29	0.166195
$811$	−3094.04	−0.133966	−0.0669830	−	0.997754i	$-0.521337\pi$
$811$	−3094.04	−0.133966	−0.0669830	+	0.997754i	$0.521337\pi$
$812$	17667.0	0.763536
$813$	4566.41	0.196988
$814$	0	0
$815$	−19581.9	−0.841625
$816$	2055.65	0.0881890
$817$	−21852.9	−0.935784
$818$	−36314.6	−1.55221
$819$	7927.29	0.338220
$820$	−6753.92	−0.287631
$821$	−29731.6	−1.26387	−0.631937	−	0.775020i	$-0.717740\pi$
$821$	−29731.6	−1.26387	−0.631937	+	0.775020i	$0.717740\pi$
$822$	−2856.97	−0.121227
$823$	−29074.9	−1.23145	−0.615727	−	0.787959i	$-0.711138\pi$
$823$	−29074.9	−1.23145	−0.615727	+	0.787959i	$0.711138\pi$
$824$	16370.5	0.692103
$825$	0	0
$826$	−46202.5	−1.94623
$827$	−34713.8	−1.45963	−0.729817	−	0.683643i	$-0.760395\pi$
$827$	−34713.8	−1.45963	−0.729817	+	0.683643i	$0.760395\pi$
$828$	4211.55	0.176765
$829$	36539.1	1.53083	0.765413	−	0.643540i	$-0.222535\pi$
$829$	36539.1	1.53083	0.765413	+	0.643540i	$0.222535\pi$
$830$	33742.7	1.41112
$831$	−8186.93	−0.341758
$832$	5590.17	0.232938
$833$	1736.38	0.0722233
$834$	5155.57	0.214056
$835$	−16208.1	−0.671741
$836$	0	0
$837$	−6519.58	−0.269235
$838$	1486.88	0.0612930
$839$	4839.31	0.199132	0.0995658	−	0.995031i	$-0.468255\pi$
$839$	4839.31	0.199132	0.0995658	+	0.995031i	$0.468255\pi$
$840$	15221.4	0.625225
$841$	25041.5	1.02676
$842$	37086.7	1.51792
$843$	−20640.6	−0.843297
$844$	−9871.81	−0.402609
$845$	−10827.1	−0.440784
$846$	1891.84	0.0768827
$847$	0	0
$848$	39612.9	1.60414
$849$	2614.65	0.105694
$850$	2067.99	0.0834487
$851$	20700.9	0.833863
$852$	1530.10	0.0615264
$853$	−42549.0	−1.70791	−0.853957	−	0.520344i	$-0.825804\pi$
$853$	−42549.0	−1.70791	−0.853957	+	0.520344i	$0.825804\pi$
$854$	50966.2	2.04219
$855$	−11160.4	−0.446406
$856$	8357.65	0.333714
$857$	4754.94	0.189528	0.0947640	−	0.995500i	$-0.469790\pi$
$857$	4754.94	0.189528	0.0947640	+	0.995500i	$0.469790\pi$
$858$	0	0
$859$	−15180.5	−0.602972	−0.301486	−	0.953471i	$-0.597483\pi$
$859$	−15180.5	−0.602972	−0.301486	+	0.953471i	$0.597483\pi$
$860$	11769.5	0.466672
$861$	9918.45	0.392590
$862$	14563.1	0.575430
$863$	−32968.3	−1.30041	−0.650206	−	0.759758i	$-0.725317\pi$
$863$	−32968.3	−1.30041	−0.650206	+	0.759758i	$0.725317\pi$
$864$	3909.82	0.153952
$865$	9940.27	0.390728
$866$	36903.0	1.44805
$867$	14516.9	0.568652
$868$	19187.7	0.750313
$869$	0	0
$870$	−31548.6	−1.22942
$871$	−36355.4	−1.41430
$872$	1952.71	0.0758339
$873$	−10896.2	−0.422430
$874$	−41097.0	−1.59053
$875$	−17596.1	−0.679836
$876$	−4422.24	−0.170563
$877$	40465.2	1.55805	0.779027	−	0.626990i	$-0.215713\pi$
$877$	40465.2	1.55805	0.779027	+	0.626990i	$0.215713\pi$
$878$	−19099.4	−0.734138
$879$	−13682.6	−0.525032
$880$	0	0
$881$	−33385.1	−1.27670	−0.638350	−	0.769746i	$-0.720383\pi$
$881$	−33385.1	−1.27670	−0.638350	+	0.769746i	$0.720383\pi$
$882$	6134.15	0.234181
$883$	−12966.0	−0.494155	−0.247078	−	0.968996i	$-0.579470\pi$
$883$	−12966.0	−0.494155	−0.247078	+	0.968996i	$0.579470\pi$
$884$	1105.25	0.0420514
$885$	24629.0	0.935474
$886$	9804.06	0.371754
$887$	14228.6	0.538612	0.269306	−	0.963055i	$-0.413206\pi$
$887$	14228.6	0.538612	0.269306	+	0.963055i	$0.413206\pi$
$888$	7011.78	0.264977
$889$	−40166.0	−1.51533
$890$	10596.9	0.399110
$891$	0	0
$892$	−21348.6	−0.801349
$893$	−5510.83	−0.206509
$894$	2274.79	0.0851013
$895$	−5587.47	−0.208680
$896$	38717.4	1.44359
$897$	−15561.0	−0.579226
$898$	11743.7	0.436404
$899$	53685.1	1.99166
$900$	2180.84	0.0807718
$901$	−4279.03	−0.158219
$902$	0	0
$903$	−17284.1	−0.636965
$904$	26390.0	0.970929
$905$	4750.19	0.174477
$906$	−26755.1	−0.981101
$907$	21189.4	0.775726	0.387863	−	0.921717i	$-0.373213\pi$
$907$	21189.4	0.775726	0.387863	+	0.921717i	$0.373213\pi$
$908$	5108.62	0.186713
$909$	9766.83	0.356376
$910$	41662.2	1.51768
$911$	−28318.4	−1.02989	−0.514946	−	0.857223i	$-0.672188\pi$
$911$	−28318.4	−1.02989	−0.514946	+	0.857223i	$0.672188\pi$
$912$	−21154.0	−0.768069
$913$	0	0
$914$	12327.0	0.446105
$915$	−27168.4	−0.981595
$916$	−7352.68	−0.265218
$917$	−15790.5	−0.568646
$918$	−784.455	−0.0282036
$919$	−7151.26	−0.256690	−0.128345	−	0.991730i	$-0.540967\pi$
$919$	−7151.26	−0.256690	−0.128345	+	0.991730i	$0.540967\pi$
$920$	−29879.1	−1.07074
$921$	14525.1	0.519671
$922$	39584.1	1.41392
$923$	−5653.47	−0.201610
$924$	0	0
$925$	10719.4	0.381029
$926$	−24779.2	−0.879368
$927$	−9493.44	−0.336360
$928$	−32195.1	−1.13886
$929$	−33378.1	−1.17880	−0.589398	−	0.807843i	$-0.700635\pi$
$929$	−33378.1	−1.17880	−0.589398	+	0.807843i	$0.700635\pi$
$930$	−34264.0	−1.20813
$931$	−17868.5	−0.629018
$932$	6811.22	0.239387
$933$	−21811.8	−0.765367
$934$	−35550.0	−1.24543
$935$	0	0
$936$	−5270.80	−0.184061
$937$	10090.5	0.351808	0.175904	−	0.984407i	$-0.443715\pi$
$937$	10090.5	0.351808	0.175904	+	0.984407i	$0.443715\pi$
$938$	−75941.7	−2.64348
$939$	−6727.30	−0.233799
$940$	2968.03	0.102985
$941$	20807.3	0.720828	0.360414	−	0.932792i	$-0.382635\pi$
$941$	20807.3	0.720828	0.360414	+	0.932792i	$0.382635\pi$
$942$	−3191.13	−0.110374
$943$	−19469.6	−0.672340
$944$	46683.2	1.60954
$945$	−8827.08	−0.303857
$946$	0	0
$947$	35396.1	1.21459	0.607296	−	0.794476i	$-0.292254\pi$
$947$	35396.1	1.21459	0.607296	+	0.794476i	$0.292254\pi$
$948$	12683.9	0.434550
$949$	16339.4	0.558904
$950$	−21280.9	−0.726784
$951$	−16318.3	−0.556422
$952$	−3116.58	−0.106102
$953$	−31152.7	−1.05890	−0.529452	−	0.848340i	$-0.677603\pi$
$953$	−31152.7	−1.05890	−0.529452	+	0.848340i	$0.677603\pi$
$954$	−15116.6	−0.513017
$955$	−21003.9	−0.711696
$956$	6010.29	0.203333
$957$	0	0
$958$	−35563.3	−1.19937
$959$	6582.30	0.221641
$960$	−6224.69	−0.209272
$961$	28514.9	0.957164
$962$	19191.8	0.643210
$963$	−4846.70	−0.162184
$964$	−9489.14	−0.317038
$965$	−17781.6	−0.593171
$966$	−32504.9	−1.08264
$967$	−10002.0	−0.332620	−0.166310	−	0.986074i	$-0.553185\pi$
$967$	−10002.0	−0.332620	−0.166310	+	0.986074i	$0.553185\pi$
$968$	0	0
$969$	2285.08	0.0757557
$970$	−57265.7	−1.89556
$971$	20654.6	0.682633	0.341316	−	0.939948i	$-0.389127\pi$
$971$	20654.6	0.682633	0.341316	+	0.939948i	$0.389127\pi$
$972$	−827.263	−0.0272988
$973$	−11878.1	−0.391362
$974$	2851.77	0.0938159
$975$	−8057.83	−0.264674
$976$	−51496.5	−1.68890
$977$	−8005.35	−0.262143	−0.131072	−	0.991373i	$-0.541842\pi$
$977$	−8005.35	−0.262143	−0.131072	+	0.991373i	$0.541842\pi$
$978$	14164.0	0.463104
$979$	0	0
$980$	9623.62	0.313689
$981$	−1132.40	−0.0368550
$982$	−44622.9	−1.45008
$983$	−5988.91	−0.194320	−0.0971600	−	0.995269i	$-0.530976\pi$
$983$	−5988.91	−0.194320	−0.0971600	+	0.995269i	$0.530976\pi$
$984$	−6594.71	−0.213650
$985$	27604.6	0.892949
$986$	6459.55	0.208635
$987$	−4358.69	−0.140566
$988$	−11373.7	−0.366241
$989$	33928.1	1.09085
$990$	0	0
$991$	24763.7	0.793787	0.396894	−	0.917865i	$-0.370088\pi$
$991$	24763.7	0.793787	0.396894	+	0.917865i	$0.370088\pi$
$992$	−34966.2	−1.11913
$993$	−25343.0	−0.809905
$994$	−11809.4	−0.376831
$995$	51918.5	1.65420
$996$	−7285.82	−0.231787
$997$	−10142.2	−0.322174	−0.161087	−	0.986940i	$-0.551500\pi$
$997$	−10142.2	−0.322174	−0.161087	+	0.986940i	$0.551500\pi$
$998$	−57380.7	−1.81999
$999$	−4066.22	−0.128778

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.4.a.v.1.5		6
3.2	odd	2		1089.4.a.bi.1.2		6
11.5	even	5		33.4.e.c.25.1	yes	12
11.9	even	5		33.4.e.c.4.1	✓	12
11.10	odd	2		363.4.a.u.1.2		6
33.5	odd	10		99.4.f.d.91.3		12
33.20	odd	10		99.4.f.d.37.3		12
33.32	even	2		1089.4.a.bk.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.e.c.4.1	✓	12	11.9	even	5
33.4.e.c.25.1	yes	12	11.5	even	5
99.4.f.d.37.3		12	33.20	odd	10
99.4.f.d.91.3		12	33.5	odd	10
363.4.a.u.1.2		6	11.10	odd	2
363.4.a.v.1.5		6	1.1	even	1	trivial
1089.4.a.bi.1.2		6	3.2	odd	2
1089.4.a.bk.1.5		6	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q+3.37704 q^{2} -3.00000 q^{3} +3.40437 q^{4} +14.0063 q^{5} -10.1311 q^{6} +23.3415 q^{7} -15.5196 q^{8} +9.00000 q^{9} +47.2999 q^{10} -10.2131 q^{12} +37.7358 q^{13} +78.8251 q^{14} -42.0190 q^{15} -79.6452 q^{16} +8.60337 q^{17} +30.3933 q^{18} -88.5343 q^{19} +47.6828 q^{20} -70.0245 q^{21} +137.456 q^{23} +46.5588 q^{24} +71.1777 q^{25} +127.435 q^{26} -27.0000 q^{27} +79.4632 q^{28} +222.330 q^{29} -141.900 q^{30} +241.466 q^{31} -144.808 q^{32} +29.0539 q^{34} +326.929 q^{35} +30.6394 q^{36} +150.601 q^{37} -298.983 q^{38} -113.207 q^{39} -217.373 q^{40} -141.643 q^{41} -236.475 q^{42} +246.830 q^{43} +126.057 q^{45} +464.193 q^{46} +62.2452 q^{47} +238.936 q^{48} +201.826 q^{49} +240.370 q^{50} -25.8101 q^{51} +128.467 q^{52} -497.366 q^{53} -91.1800 q^{54} -362.251 q^{56} +265.603 q^{57} +750.816 q^{58} -586.139 q^{59} -143.048 q^{60} +646.573 q^{61} +815.440 q^{62} +210.074 q^{63} +148.140 q^{64} +528.540 q^{65} -963.421 q^{67} +29.2891 q^{68} -412.367 q^{69} +1104.05 q^{70} -149.817 q^{71} -139.676 q^{72} +432.996 q^{73} +508.584 q^{74} -213.533 q^{75} -301.404 q^{76} -382.305 q^{78} -1241.92 q^{79} -1115.54 q^{80} +81.0000 q^{81} -478.332 q^{82} +713.378 q^{83} -238.390 q^{84} +120.502 q^{85} +833.553 q^{86} -666.989 q^{87} +224.035 q^{89} +425.699 q^{90} +880.810 q^{91} +467.950 q^{92} -724.398 q^{93} +210.204 q^{94} -1240.04 q^{95} +434.424 q^{96} -1210.69 q^{97} +681.572 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{2} - 18 q^{3} + 17 q^{4} + 9 q^{5} - 15 q^{6} + q^{7} + 24 q^{8} + 54 q^{9} + 50 q^{10} - 51 q^{12} + 66 q^{13} - 42 q^{14} - 27 q^{15} - 71 q^{16} + 80 q^{17} + 45 q^{18} - 90 q^{19} + 455 q^{20}+ \cdots + 1405 q^{98}+O(q^{100}) \) 6 * q + 5 * q^2 - 18 * q^3 + 17 * q^4 + 9 * q^5 - 15 * q^6 + q^7 + 24 * q^8 + 54 * q^9 + 50 * q^10 - 51 * q^12 + 66 * q^13 - 42 * q^14 - 27 * q^15 - 71 * q^16 + 80 * q^17 + 45 * q^18 - 90 * q^19 + 455 * q^20 - 3 * q^21 - 196 * q^23 - 72 * q^24 + 351 * q^25 + 93 * q^26 - 162 * q^27 - 52 * q^28 + 579 * q^29 - 150 * q^30 + 129 * q^31 + 657 * q^32 + 530 * q^34 + 618 * q^35 + 153 * q^36 + 192 * q^37 - 769 * q^38 - 198 * q^39 + 1476 * q^40 + 990 * q^41 + 126 * q^42 + 208 * q^43 + 81 * q^45 + 493 * q^46 + 680 * q^47 + 213 * q^48 - 963 * q^49 + 1656 * q^50 - 240 * q^51 + 887 * q^52 + 197 * q^53 - 135 * q^54 - 429 * q^56 + 270 * q^57 + 395 * q^58 - 1229 * q^59 - 1365 * q^60 + 928 * q^61 + 2117 * q^62 + 9 * q^63 + 844 * q^64 + 2278 * q^65 - 264 * q^67 + 3279 * q^68 + 588 * q^69 + 665 * q^70 - 786 * q^71 + 216 * q^72 + 593 * q^73 + 1543 * q^74 - 1053 * q^75 - 434 * q^76 - 279 * q^78 - 421 * q^79 + 18 * q^80 + 486 * q^81 - 739 * q^82 + 2043 * q^83 + 156 * q^84 + 1900 * q^85 + 2683 * q^86 - 1737 * q^87 + 158 * q^89 + 450 * q^90 + 698 * q^91 - 3915 * q^92 - 387 * q^93 - 4934 * q^94 + 1038 * q^95 - 1971 * q^96 - 1711 * q^97 + 1405 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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