LMFDB - Embedded newform 363.4.a.j.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [363,4,Mod(1,363)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [2,-1,6,1,16,-3,-2] 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:	$21.4176933321$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ 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)

$f(q)$	$=$	\(q-3.37228 q^{2} +3.00000 q^{3} +3.37228 q^{4} -3.48913 q^{5} -10.1168 q^{6} +4.74456 q^{7} +15.6060 q^{8} +9.00000 q^{9} +11.7663 q^{10} +10.1168 q^{12} +15.0217 q^{13} -16.0000 q^{14} -10.4674 q^{15} -79.6060 q^{16} -73.1684 q^{17} -30.3505 q^{18} +78.7011 q^{19} -11.7663 q^{20} +14.2337 q^{21} +112.000 q^{23} +46.8179 q^{24} -112.826 q^{25} -50.6576 q^{26} +27.0000 q^{27} +16.0000 q^{28} -243.125 q^{29} +35.2989 q^{30} +278.717 q^{31} +143.606 q^{32} +246.745 q^{34} -16.5544 q^{35} +30.3505 q^{36} +102.380 q^{37} -265.402 q^{38} +45.0652 q^{39} -54.4512 q^{40} +241.255 q^{41} -48.0000 q^{42} +280.016 q^{43} -31.4021 q^{45} -377.696 q^{46} -169.870 q^{47} -238.818 q^{48} -320.489 q^{49} +380.481 q^{50} -219.505 q^{51} +50.6576 q^{52} -409.652 q^{53} -91.0516 q^{54} +74.0435 q^{56} +236.103 q^{57} +819.886 q^{58} +196.000 q^{59} -35.2989 q^{60} +701.359 q^{61} -939.913 q^{62} +42.7011 q^{63} +152.568 q^{64} -52.4128 q^{65} +900.587 q^{67} -246.745 q^{68} +336.000 q^{69} +55.8260 q^{70} +756.500 q^{71} +140.454 q^{72} +1019.81 q^{73} -345.255 q^{74} -338.478 q^{75} +265.402 q^{76} -151.973 q^{78} +327.549 q^{79} +277.755 q^{80} +81.0000 q^{81} -813.581 q^{82} +756.619 q^{83} +48.0000 q^{84} +255.294 q^{85} -944.293 q^{86} -729.375 q^{87} +508.978 q^{89} +105.897 q^{90} +71.2716 q^{91} +377.696 q^{92} +836.152 q^{93} +572.848 q^{94} -274.598 q^{95} +430.818 q^{96} +614.358 q^{97} +1080.78 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 6 q^{3} + q^{4} + 16 q^{5} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 18 q^{9} + 58 q^{10} + 3 q^{12} + 76 q^{13} - 32 q^{14} + 48 q^{15} - 119 q^{16} + 26 q^{17} - 9 q^{18} + 54 q^{19} - 58 q^{20}+ \cdots + 375 q^{98}+O(q^{100}) $ 2 * q - q^2 + 6 * q^3 + q^4 + 16 * q^5 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 18 * q^9 + 58 * q^10 + 3 * q^12 + 76 * q^13 - 32 * q^14 + 48 * q^15 - 119 * q^16 + 26 * q^17 - 9 * q^18 + 54 * q^19 - 58 * q^20 - 6 * q^21 + 224 * q^23 - 27 * q^24 + 142 * q^25 + 94 * q^26 + 54 * q^27 + 32 * q^28 - 222 * q^29 + 174 * q^30 - 40 * q^31 + 247 * q^32 + 482 * q^34 - 148 * q^35 + 9 * q^36 - 48 * q^37 - 324 * q^38 + 228 * q^39 - 534 * q^40 + 494 * q^41 - 96 * q^42 + 66 * q^43 + 144 * q^45 - 112 * q^46 - 64 * q^47 - 357 * q^48 - 618 * q^49 + 985 * q^50 + 78 * q^51 - 94 * q^52 - 84 * q^53 - 27 * q^54 + 240 * q^56 + 162 * q^57 + 870 * q^58 + 392 * q^59 - 174 * q^60 + 1104 * q^61 - 1696 * q^62 - 18 * q^63 + 713 * q^64 + 1136 * q^65 + 928 * q^67 - 482 * q^68 + 672 * q^69 - 256 * q^70 + 456 * q^71 - 81 * q^72 + 592 * q^73 - 702 * q^74 + 426 * q^75 + 324 * q^76 + 282 * q^78 + 230 * q^79 - 490 * q^80 + 162 * q^81 - 214 * q^82 - 348 * q^83 + 96 * q^84 + 2188 * q^85 - 1452 * q^86 - 666 * q^87 + 972 * q^89 + 522 * q^90 - 340 * q^91 + 112 * q^92 - 120 * q^93 + 824 * q^94 - 756 * q^95 + 741 * q^96 - 1184 * q^97 + 375 * 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 $

$2$	−3.37228	−1.19228	−0.596141	−	0.802880i	$-0.703300\pi$
$2$	−3.37228	−1.19228	−0.596141	+	0.802880i	$0.703300\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	3.37228	0.421535
$5$	−3.48913	−0.312077	−0.156038	−	0.987751i	$-0.549872\pi$
$5$	−3.48913	−0.312077	−0.156038	+	0.987751i	$0.549872\pi$
$6$	−10.1168	−0.688364
$7$	4.74456	0.256182	0.128091	−	0.991762i	$-0.459115\pi$
$7$	4.74456	0.256182	0.128091	+	0.991762i	$0.459115\pi$
$8$	15.6060	0.689693
$9$	9.00000	0.333333
$10$	11.7663	0.372083
$11$	0	0
$11$	0	0
$12$	10.1168	0.243373
$13$	15.0217	0.320483	0.160242	−	0.987078i	$-0.448773\pi$
$13$	15.0217	0.320483	0.160242	+	0.987078i	$0.448773\pi$
$14$	−16.0000	−0.305441
$15$	−10.4674	−0.180178
$16$	−79.6060	−1.24384
$17$	−73.1684	−1.04388	−0.521940	−	0.852982i	$-0.674791\pi$
$17$	−73.1684	−1.04388	−0.521940	+	0.852982i	$0.674791\pi$
$18$	−30.3505	−0.397427
$19$	78.7011	0.950277	0.475138	−	0.879911i	$-0.342398\pi$
$19$	78.7011	0.950277	0.475138	+	0.879911i	$0.342398\pi$
$20$	−11.7663	−0.131551
$21$	14.2337	0.147907
$22$	0	0
$23$	112.000	1.01537	0.507687	−	0.861541i	$-0.330501\pi$
$23$	112.000	1.01537	0.507687	+	0.861541i	$0.330501\pi$
$24$	46.8179	0.398194
$25$	−112.826	−0.902608
$26$	−50.6576	−0.382106
$27$	27.0000	0.192450
$28$	16.0000	0.107990
$29$	−243.125	−1.55680	−0.778399	−	0.627769i	$-0.783968\pi$
$29$	−243.125	−1.55680	−0.778399	+	0.627769i	$0.783968\pi$
$30$	35.2989	0.214822
$31$	278.717	1.61481	0.807405	−	0.589998i	$-0.200871\pi$
$31$	278.717	1.61481	0.807405	+	0.589998i	$0.200871\pi$
$32$	143.606	0.793318
$33$	0	0
$34$	246.745	1.24460
$35$	−16.5544	−0.0799486
$36$	30.3505	0.140512
$37$	102.380	0.454898	0.227449	−	0.973790i	$-0.426961\pi$
$37$	102.380	0.454898	0.227449	+	0.973790i	$0.426961\pi$
$38$	−265.402	−1.13300
$39$	45.0652	0.185031
$40$	−54.4512	−0.215237
$41$	241.255	0.918970	0.459485	−	0.888186i	$-0.348034\pi$
$41$	241.255	0.918970	0.459485	+	0.888186i	$0.348034\pi$
$42$	−48.0000	−0.176347
$43$	280.016	0.993071	0.496536	−	0.868016i	$-0.334605\pi$
$43$	280.016	0.993071	0.496536	+	0.868016i	$0.334605\pi$
$44$	0	0
$45$	−31.4021	−0.104026
$46$	−377.696	−1.21061
$47$	−169.870	−0.527192	−0.263596	−	0.964633i	$-0.584909\pi$
$47$	−169.870	−0.527192	−0.263596	+	0.964633i	$0.584909\pi$
$48$	−238.818	−0.718133
$49$	−320.489	−0.934371
$50$	380.481	1.07616
$51$	−219.505	−0.602684
$52$	50.6576	0.135095
$53$	−409.652	−1.06170	−0.530849	−	0.847466i	$-0.678127\pi$
$53$	−409.652	−1.06170	−0.530849	+	0.847466i	$0.678127\pi$
$54$	−91.0516	−0.229455
$55$	0	0
$56$	74.0435	0.176687
$57$	236.103	0.548643
$58$	819.886	1.85614
$59$	196.000	0.432492	0.216246	−	0.976339i	$-0.430619\pi$
$59$	196.000	0.432492	0.216246	+	0.976339i	$0.430619\pi$
$60$	−35.2989	−0.0759512
$61$	701.359	1.47213	0.736064	−	0.676912i	$-0.236682\pi$
$61$	701.359	1.47213	0.736064	+	0.676912i	$0.236682\pi$
$62$	−939.913	−1.92531
$63$	42.7011	0.0853941
$64$	152.568	0.297984
$65$	−52.4128	−0.100015
$66$	0	0
$67$	900.587	1.64215	0.821076	−	0.570819i	$-0.193374\pi$
$67$	900.587	1.64215	0.821076	+	0.570819i	$0.193374\pi$
$68$	−246.745	−0.440032
$69$	336.000	0.586227
$70$	55.8260	0.0953212
$71$	756.500	1.26451	0.632254	−	0.774762i	$-0.282130\pi$
$71$	756.500	1.26451	0.632254	+	0.774762i	$0.282130\pi$
$72$	140.454	0.229898
$73$	1019.81	1.63507	0.817536	−	0.575877i	$-0.195339\pi$
$73$	1019.81	1.63507	0.817536	+	0.575877i	$0.195339\pi$
$74$	−345.255	−0.542367
$75$	−338.478	−0.521121
$76$	265.402	0.400575
$77$	0	0
$78$	−151.973	−0.220609
$79$	327.549	0.466483	0.233241	−	0.972419i	$-0.425067\pi$
$79$	327.549	0.466483	0.233241	+	0.972419i	$0.425067\pi$
$80$	277.755	0.388175
$81$	81.0000	0.111111
$82$	−813.581	−1.09567
$83$	756.619	1.00060	0.500300	−	0.865852i	$-0.333223\pi$
$83$	756.619	1.00060	0.500300	+	0.865852i	$0.333223\pi$
$84$	48.0000	0.0623480
$85$	255.294	0.325771
$86$	−944.293	−1.18402
$87$	−729.375	−0.898818
$88$	0	0
$89$	508.978	0.606198	0.303099	−	0.952959i	$-0.401979\pi$
$89$	508.978	0.606198	0.303099	+	0.952959i	$0.401979\pi$
$90$	105.897	0.124028
$91$	71.2716	0.0821022
$92$	377.696	0.428016
$93$	836.152	0.932311
$94$	572.848	0.628561
$95$	−274.598	−0.296559
$96$	430.818	0.458023
$97$	614.358	0.643079	0.321539	−	0.946896i	$-0.395800\pi$
$97$	614.358	0.643079	0.321539	+	0.946896i	$0.395800\pi$
$98$	1080.78	1.11403
$99$	0	0
$100$	−380.481	−0.380481
$101$	1015.92	1.00087	0.500434	−	0.865775i	$-0.333174\pi$
$101$	1015.92	1.00087	0.500434	+	0.865775i	$0.333174\pi$
$102$	740.234	0.718569
$103$	1102.16	1.05436	0.527181	−	0.849753i	$-0.323249\pi$
$103$	1102.16	1.05436	0.527181	+	0.849753i	$0.323249\pi$
$104$	234.429	0.221035
$105$	−49.6631	−0.0461583
$106$	1381.46	1.26584
$107$	−1377.58	−1.24463	−0.622315	−	0.782767i	$-0.713808\pi$
$107$	−1377.58	−1.24463	−0.622315	+	0.782767i	$0.713808\pi$
$108$	91.0516	0.0811245
$109$	−320.217	−0.281388	−0.140694	−	0.990053i	$-0.544933\pi$
$109$	−320.217	−0.281388	−0.140694	+	0.990053i	$0.544933\pi$
$110$	0	0
$111$	307.141	0.262636
$112$	−377.696	−0.318651
$113$	−1629.45	−1.35651	−0.678254	−	0.734828i	$-0.737263\pi$
$113$	−1629.45	−1.35651	−0.678254	+	0.734828i	$0.737263\pi$
$114$	−796.206	−0.654136
$115$	−390.782	−0.316875
$116$	−819.886	−0.656245
$117$	135.196	0.106828
$118$	−660.967	−0.515652
$119$	−347.152	−0.267423
$120$	−163.354	−0.124267
$121$	0	0
$122$	−2365.18	−1.75519
$123$	723.766	0.530568
$124$	939.913	0.680699
$125$	829.805	0.593760
$126$	−144.000	−0.101814
$127$	−2291.26	−1.60091	−0.800457	−	0.599390i	$-0.795410\pi$
$127$	−2291.26	−1.60091	−0.800457	+	0.599390i	$0.795410\pi$
$128$	−1663.35	−1.14860
$129$	840.049	0.573350
$130$	176.751	0.119247
$131$	1147.41	0.765267	0.382633	−	0.923900i	$-0.375017\pi$
$131$	1147.41	0.765267	0.382633	+	0.923900i	$0.375017\pi$
$132$	0	0
$133$	373.402	0.243444
$134$	−3037.03	−1.95791
$135$	−94.2064	−0.0600592
$136$	−1141.86	−0.719956
$137$	1268.60	0.791121	0.395561	−	0.918440i	$-0.370550\pi$
$137$	1268.60	0.791121	0.395561	+	0.918440i	$0.370550\pi$
$138$	−1133.09	−0.698947
$139$	486.288	0.296737	0.148368	−	0.988932i	$-0.452598\pi$
$139$	486.288	0.296737	0.148368	+	0.988932i	$0.452598\pi$
$140$	−55.8260	−0.0337011
$141$	−509.609	−0.304374
$142$	−2551.13	−1.50765
$143$	0	0
$144$	−716.454	−0.414614
$145$	848.293	0.485841
$146$	−3439.10	−1.94947
$147$	−961.467	−0.539459
$148$	345.255	0.191756
$149$	−2354.11	−1.29434	−0.647169	−	0.762346i	$-0.724047\pi$
$149$	−2354.11	−1.29434	−0.647169	+	0.762346i	$0.724047\pi$
$150$	1141.44	0.621323
$151$	570.070	0.307229	0.153615	−	0.988131i	$-0.450909\pi$
$151$	570.070	0.307229	0.153615	+	0.988131i	$0.450909\pi$
$152$	1228.21	0.655399
$153$	−658.516	−0.347960
$154$	0	0
$155$	−972.479	−0.503945
$156$	151.973	0.0779971
$157$	−2072.67	−1.05361	−0.526807	−	0.849985i	$-0.676611\pi$
$157$	−2072.67	−1.05361	−0.526807	+	0.849985i	$0.676611\pi$
$158$	−1104.59	−0.556179
$159$	−1228.96	−0.612972
$160$	−501.059	−0.247576
$161$	531.391	0.260121
$162$	−273.155	−0.132476
$163$	2676.51	1.28614	0.643069	−	0.765808i	$-0.277661\pi$
$163$	2676.51	1.28614	0.643069	+	0.765808i	$0.277661\pi$
$164$	813.581	0.387378
$165$	0	0
$166$	−2551.53	−1.19300
$167$	1188.12	0.550536	0.275268	−	0.961368i	$-0.411233\pi$
$167$	1188.12	0.550536	0.275268	+	0.961368i	$0.411233\pi$
$168$	222.130	0.102010
$169$	−1971.35	−0.897290
$170$	−860.923	−0.388410
$171$	708.310	0.316759
$172$	944.293	0.418615
$173$	−807.147	−0.354718	−0.177359	−	0.984146i	$-0.556755\pi$
$173$	−807.147	−0.354718	−0.177359	+	0.984146i	$0.556755\pi$
$174$	2459.66	1.07164
$175$	−535.310	−0.231232
$176$	0	0
$177$	588.000	0.249699
$178$	−1716.42	−0.722758
$179$	−1950.39	−0.814408	−0.407204	−	0.913337i	$-0.633496\pi$
$179$	−1950.39	−0.814408	−0.407204	+	0.913337i	$0.633496\pi$
$180$	−105.897	−0.0438505
$181$	1061.61	0.435959	0.217980	−	0.975953i	$-0.430053\pi$
$181$	1061.61	0.435959	0.217980	+	0.975953i	$0.430053\pi$
$182$	−240.348	−0.0978889
$183$	2104.08	0.849933
$184$	1747.87	0.700297
$185$	−357.218	−0.141963
$186$	−2819.74	−1.11158
$187$	0	0
$188$	−572.848	−0.222230
$189$	128.103	0.0493023
$190$	926.021	0.353582
$191$	2136.41	0.809348	0.404674	−	0.914461i	$-0.367385\pi$
$191$	2136.41	0.809348	0.404674	+	0.914461i	$0.367385\pi$
$192$	457.704	0.172041
$193$	−3947.76	−1.47236	−0.736181	−	0.676784i	$-0.763373\pi$
$193$	−3947.76	−1.47236	−0.736181	+	0.676784i	$0.763373\pi$
$194$	−2071.79	−0.766731
$195$	−157.238	−0.0577439
$196$	−1080.78	−0.393870
$197$	−923.886	−0.334133	−0.167066	−	0.985946i	$-0.553429\pi$
$197$	−923.886	−0.334133	−0.167066	+	0.985946i	$0.553429\pi$
$198$	0	0
$199$	−476.152	−0.169616	−0.0848078	−	0.996397i	$-0.527028\pi$
$199$	−476.152	−0.169616	−0.0848078	+	0.996397i	$0.527028\pi$
$200$	−1760.76	−0.622522
$201$	2701.76	0.948097
$202$	−3425.96	−1.19332
$203$	−1153.52	−0.398824
$204$	−740.234	−0.254053
$205$	−841.770	−0.286789
$206$	−3716.80	−1.25710
$207$	1008.00	0.338458
$208$	−1195.82	−0.398631
$209$	0	0
$210$	167.478	0.0550337
$211$	4918.24	1.60467	0.802336	−	0.596872i	$-0.203590\pi$
$211$	4918.24	1.60467	0.802336	+	0.596872i	$0.203590\pi$
$212$	−1381.46	−0.447543
$213$	2269.50	0.730064
$214$	4645.57	1.48395
$215$	−977.012	−0.309915
$216$	421.361	0.132731
$217$	1322.39	0.413686
$218$	1079.86	0.335494
$219$	3059.44	0.944010
$220$	0	0
$221$	−1099.12	−0.334546
$222$	−1035.77	−0.313136
$223$	2100.29	0.630700	0.315350	−	0.948975i	$-0.397878\pi$
$223$	2100.29	0.630700	0.315350	+	0.948975i	$0.397878\pi$
$224$	681.348	0.203234
$225$	−1015.43	−0.300869
$226$	5494.95	1.61734
$227$	2257.16	0.659970	0.329985	−	0.943986i	$-0.392956\pi$
$227$	2257.16	0.659970	0.329985	+	0.943986i	$0.392956\pi$
$228$	796.206	0.231272
$229$	−5311.07	−1.53260	−0.766301	−	0.642482i	$-0.777905\pi$
$229$	−5311.07	−1.53260	−0.766301	+	0.642482i	$0.777905\pi$
$230$	1317.83	0.377804
$231$	0	0
$232$	−3794.20	−1.07371
$233$	−2466.27	−0.693435	−0.346718	−	0.937970i	$-0.612704\pi$
$233$	−2466.27	−0.693435	−0.346718	+	0.937970i	$0.612704\pi$
$234$	−455.918	−0.127369
$235$	592.696	0.164524
$236$	660.967	0.182311
$237$	982.646	0.269324
$238$	1170.70	0.318844
$239$	−1429.40	−0.386863	−0.193432	−	0.981114i	$-0.561962\pi$
$239$	−1429.40	−0.386863	−0.193432	+	0.981114i	$0.561962\pi$
$240$	833.266	0.224113
$241$	978.989	0.261669	0.130835	−	0.991404i	$-0.458234\pi$
$241$	978.989	0.261669	0.130835	+	0.991404i	$0.458234\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	2365.18	0.620553
$245$	1118.23	0.291595
$246$	−2440.74	−0.632586
$247$	1182.23	0.304548
$248$	4349.65	1.11372
$249$	2269.86	0.577696
$250$	−2798.33	−0.707929
$251$	−6530.63	−1.64227	−0.821135	−	0.570734i	$-0.806659\pi$
$251$	−6530.63	−1.64227	−0.821135	+	0.570734i	$0.806659\pi$
$252$	144.000	0.0359966
$253$	0	0
$254$	7726.76	1.90874
$255$	765.882	0.188084
$256$	4388.74	1.07147
$257$	8130.26	1.97335	0.986676	−	0.162696i	$-0.0520188\pi$
$257$	8130.26	1.97335	0.986676	+	0.162696i	$0.0520188\pi$
$258$	−2832.88	−0.683595
$259$	485.750	0.116537
$260$	−176.751	−0.0421600
$261$	−2188.12	−0.518933
$262$	−3869.40	−0.912414
$263$	4549.42	1.06665	0.533326	−	0.845910i	$-0.320942\pi$
$263$	4549.42	1.06665	0.533326	+	0.845910i	$0.320942\pi$
$264$	0	0
$265$	1429.33	0.331332
$266$	−1259.22	−0.290254
$267$	1526.93	0.349988
$268$	3037.03	0.692225
$269$	−29.1522	−0.00660760	−0.00330380	−	0.999995i	$-0.501052\pi$
$269$	−29.1522	−0.00660760	−0.00330380	+	0.999995i	$0.501052\pi$
$270$	317.690	0.0716075
$271$	−7711.22	−1.72850	−0.864250	−	0.503063i	$-0.832206\pi$
$271$	−7711.22	−1.72850	−0.864250	+	0.503063i	$0.832206\pi$
$272$	5824.64	1.29842
$273$	213.815	0.0474017
$274$	−4278.07	−0.943239
$275$	0	0
$276$	1133.09	0.247115
$277$	−1127.52	−0.244571	−0.122286	−	0.992495i	$-0.539022\pi$
$277$	−1127.52	−0.244571	−0.122286	+	0.992495i	$0.539022\pi$
$278$	−1639.90	−0.353794
$279$	2508.46	0.538270
$280$	−258.347	−0.0551400
$281$	1872.47	0.397517	0.198758	−	0.980049i	$-0.436309\pi$
$281$	1872.47	0.397517	0.198758	+	0.980049i	$0.436309\pi$
$282$	1718.54	0.362900
$283$	−2124.48	−0.446245	−0.223123	−	0.974790i	$-0.571625\pi$
$283$	−2124.48	−0.446245	−0.223123	+	0.974790i	$0.571625\pi$
$284$	2551.13	0.533034
$285$	−823.794	−0.171219
$286$	0	0
$287$	1144.65	0.235424
$288$	1292.45	0.264439
$289$	440.621	0.0896846
$290$	−2860.68	−0.579259
$291$	1843.07	0.371282
$292$	3439.10	0.689241
$293$	3324.19	0.662802	0.331401	−	0.943490i	$-0.392479\pi$
$293$	3324.19	0.662802	0.331401	+	0.943490i	$0.392479\pi$
$294$	3242.34	0.643187
$295$	−683.869	−0.134971
$296$	1597.75	0.313740
$297$	0	0
$298$	7938.73	1.54322
$299$	1682.44	0.325411
$300$	−1141.44	−0.219671
$301$	1328.55	0.254407
$302$	−1922.44	−0.366304
$303$	3047.75	0.577851
$304$	−6265.07	−1.18200
$305$	−2447.13	−0.459417
$306$	2220.70	0.414866
$307$	1698.94	0.315843	0.157921	−	0.987452i	$-0.449521\pi$
$307$	1698.94	0.315843	0.157921	+	0.987452i	$0.449521\pi$
$308$	0	0
$309$	3306.49	0.608736
$310$	3279.47	0.600844
$311$	6928.83	1.26334	0.631668	−	0.775239i	$-0.282370\pi$
$311$	6928.83	1.26334	0.631668	+	0.775239i	$0.282370\pi$
$312$	703.287	0.127615
$313$	−3560.75	−0.643020	−0.321510	−	0.946906i	$-0.604190\pi$
$313$	−3560.75	−0.643020	−0.321510	+	0.946906i	$0.604190\pi$
$314$	6989.64	1.25620
$315$	−148.989	−0.0266495
$316$	1104.59	0.196639
$317$	332.750	0.0589561	0.0294780	−	0.999565i	$-0.490615\pi$
$317$	332.750	0.0589561	0.0294780	+	0.999565i	$0.490615\pi$
$318$	4144.39	0.730835
$319$	0	0
$320$	−532.329	−0.0929940
$321$	−4132.73	−0.718587
$322$	−1792.00	−0.310137
$323$	−5758.43	−0.991975
$324$	273.155	0.0468372
$325$	−1694.84	−0.289271
$326$	−9025.94	−1.53344
$327$	−960.652	−0.162459
$328$	3765.02	0.633807
$329$	−805.957	−0.135057
$330$	0	0
$331$	−541.445	−0.0899108	−0.0449554	−	0.998989i	$-0.514315\pi$
$331$	−541.445	−0.0899108	−0.0449554	+	0.998989i	$0.514315\pi$
$332$	2551.53	0.421788
$333$	921.423	0.151633
$334$	−4006.67	−0.656393
$335$	−3142.26	−0.512478
$336$	−1133.09	−0.183973
$337$	−816.531	−0.131986	−0.0659930	−	0.997820i	$-0.521022\pi$
$337$	−816.531	−0.131986	−0.0659930	+	0.997820i	$0.521022\pi$
$338$	6647.94	1.06982
$339$	−4888.34	−0.783180
$340$	860.923	0.137324
$341$	0	0
$342$	−2388.62	−0.377666
$343$	−3147.97	−0.495552
$344$	4369.92	0.684914
$345$	−1172.35	−0.182948
$346$	2721.93	0.422924
$347$	−6260.53	−0.968539	−0.484269	−	0.874919i	$-0.660914\pi$
$347$	−6260.53	−0.968539	−0.484269	+	0.874919i	$0.660914\pi$
$348$	−2459.66	−0.378884
$349$	12768.5	1.95840	0.979198	−	0.202906i	$-0.0650386\pi$
$349$	12768.5	1.95840	0.979198	+	0.202906i	$0.0650386\pi$
$350$	1805.22	0.275694
$351$	405.587	0.0616771
$352$	0	0
$353$	−2649.28	−0.399453	−0.199727	−	0.979852i	$-0.564005\pi$
$353$	−2649.28	−0.399453	−0.199727	+	0.979852i	$0.564005\pi$
$354$	−1982.90	−0.297712
$355$	−2639.52	−0.394623
$356$	1716.42	0.255534
$357$	−1041.46	−0.154397
$358$	6577.27	0.971004
$359$	3203.91	0.471020	0.235510	−	0.971872i	$-0.424324\pi$
$359$	3203.91	0.471020	0.235510	+	0.971872i	$0.424324\pi$
$360$	−490.061	−0.0717457
$361$	−665.143	−0.0969737
$362$	−3580.04	−0.519786
$363$	0	0
$364$	240.348	0.0346089
$365$	−3558.26	−0.510268
$366$	−7095.54	−1.01336
$367$	−8429.40	−1.19894	−0.599470	−	0.800397i	$-0.704622\pi$
$367$	−8429.40	−1.19894	−0.599470	+	0.800397i	$0.704622\pi$
$368$	−8915.87	−1.26297
$369$	2171.30	0.306323
$370$	1204.64	0.169260
$371$	−1943.62	−0.271988
$372$	2819.74	0.393002
$373$	9388.53	1.30327	0.651635	−	0.758533i	$-0.274083\pi$
$373$	9388.53	1.30327	0.651635	+	0.758533i	$0.274083\pi$
$374$	0	0
$375$	2489.41	0.342807
$376$	−2650.98	−0.363600
$377$	−3652.16	−0.498928
$378$	−432.000	−0.0587822
$379$	−14264.5	−1.93329	−0.966647	−	0.256112i	$-0.917558\pi$
$379$	−14264.5	−1.93329	−0.966647	+	0.256112i	$0.917558\pi$
$380$	−926.021	−0.125010
$381$	−6873.77	−0.924288
$382$	−7204.58	−0.964970
$383$	13462.2	1.79605	0.898026	−	0.439942i	$-0.145001\pi$
$383$	13462.2	1.79605	0.898026	+	0.439942i	$0.145001\pi$
$384$	−4990.05	−0.663144
$385$	0	0
$386$	13313.0	1.75547
$387$	2520.15	0.331024
$388$	2071.79	0.271080
$389$	−941.881	−0.122764	−0.0613821	−	0.998114i	$-0.519551\pi$
$389$	−941.881	−0.122764	−0.0613821	+	0.998114i	$0.519551\pi$
$390$	530.252	0.0688470
$391$	−8194.87	−1.05993
$392$	−5001.54	−0.644429
$393$	3442.24	0.441827
$394$	3115.60	0.398380
$395$	−1142.86	−0.145578
$396$	0	0
$397$	−847.839	−0.107183	−0.0535917	−	0.998563i	$-0.517067\pi$
$397$	−847.839	−0.107183	−0.0535917	+	0.998563i	$0.517067\pi$
$398$	1605.72	0.202230
$399$	1120.21	0.140553
$400$	8981.62	1.12270
$401$	12203.6	1.51975	0.759875	−	0.650069i	$-0.225260\pi$
$401$	12203.6	1.51975	0.759875	+	0.650069i	$0.225260\pi$
$402$	−9111.10	−1.13040
$403$	4186.82	0.517520
$404$	3425.96	0.421901
$405$	−282.619	−0.0346752
$406$	3890.00	0.475511
$407$	0	0
$408$	−3425.59	−0.415667
$409$	−8759.53	−1.05900	−0.529500	−	0.848310i	$-0.677620\pi$
$409$	−8759.53	−1.05900	−0.529500	+	0.848310i	$0.677620\pi$
$410$	2838.69	0.341934
$411$	3805.79	0.456754
$412$	3716.80	0.444451
$413$	929.934	0.110797
$414$	−3399.26	−0.403537
$415$	−2639.94	−0.312264
$416$	2157.21	0.254245
$417$	1458.86	0.171321
$418$	0	0
$419$	−11188.4	−1.30451	−0.652256	−	0.757999i	$-0.726177\pi$
$419$	−11188.4	−1.30451	−0.652256	+	0.757999i	$0.726177\pi$
$420$	−167.478	−0.0194574
$421$	−14082.3	−1.63023	−0.815116	−	0.579298i	$-0.803327\pi$
$421$	−14082.3	−1.63023	−0.815116	+	0.579298i	$0.803327\pi$
$422$	−16585.7	−1.91322
$423$	−1528.83	−0.175731
$424$	−6393.02	−0.732246
$425$	8255.30	0.942214
$426$	−7653.39	−0.870441
$427$	3327.64	0.377133
$428$	−4645.57	−0.524655
$429$	0	0
$430$	3294.76	0.369505
$431$	5616.05	0.627647	0.313823	−	0.949481i	$-0.398390\pi$
$431$	5616.05	0.627647	0.313823	+	0.949481i	$0.398390\pi$
$432$	−2149.36	−0.239378
$433$	7195.75	0.798627	0.399314	−	0.916814i	$-0.369248\pi$
$433$	7195.75	0.798627	0.399314	+	0.916814i	$0.369248\pi$
$434$	−4459.48	−0.493230
$435$	2544.88	0.280500
$436$	−1079.86	−0.118615
$437$	8814.52	0.964887
$438$	−10317.3	−1.12553
$439$	−101.959	−0.0110848	−0.00554240	−	0.999985i	$-0.501764\pi$
$439$	−101.959	−0.0110848	−0.00554240	+	0.999985i	$0.501764\pi$
$440$	0	0
$441$	−2884.40	−0.311457
$442$	3706.53	0.398873
$443$	4953.74	0.531285	0.265642	−	0.964072i	$-0.414416\pi$
$443$	4953.74	0.531285	0.265642	+	0.964072i	$0.414416\pi$
$444$	1035.77	0.110710
$445$	−1775.89	−0.189180
$446$	−7082.78	−0.751972
$447$	−7062.34	−0.747287
$448$	723.869	0.0763383
$449$	−11602.0	−1.21945	−0.609723	−	0.792615i	$-0.708719\pi$
$449$	−11602.0	−1.21945	−0.609723	+	0.792615i	$0.708719\pi$
$450$	3424.33	0.358721
$451$	0	0
$452$	−5494.95	−0.571816
$453$	1710.21	0.177379
$454$	−7611.79	−0.786870
$455$	−248.676	−0.0256222
$456$	3684.62	0.378395
$457$	3530.68	0.361397	0.180698	−	0.983539i	$-0.442164\pi$
$457$	3530.68	0.361397	0.180698	+	0.983539i	$0.442164\pi$
$458$	17910.4	1.82729
$459$	−1975.55	−0.200895
$460$	−1317.83	−0.133574
$461$	−11566.3	−1.16854	−0.584271	−	0.811559i	$-0.698619\pi$
$461$	−11566.3	−1.16854	−0.584271	+	0.811559i	$0.698619\pi$
$462$	0	0
$463$	10888.5	1.09294	0.546470	−	0.837479i	$-0.315971\pi$
$463$	10888.5	1.09294	0.546470	+	0.837479i	$0.315971\pi$
$464$	19354.2	1.93641
$465$	−2917.44	−0.290953
$466$	8316.94	0.826770
$467$	10688.0	1.05906	0.529529	−	0.848292i	$-0.322369\pi$
$467$	10688.0	1.05906	0.529529	+	0.848292i	$0.322369\pi$
$468$	455.918	0.0450317
$469$	4272.89	0.420690
$470$	−1998.74	−0.196159
$471$	−6218.02	−0.608304
$472$	3058.77	0.298287
$473$	0	0
$474$	−3313.76	−0.321110
$475$	−8879.53	−0.857728
$476$	−1170.70	−0.112728
$477$	−3686.87	−0.353900
$478$	4820.35	0.461250
$479$	−2341.90	−0.223391	−0.111696	−	0.993742i	$-0.535628\pi$
$479$	−2341.90	−0.223391	−0.111696	+	0.993742i	$0.535628\pi$
$480$	−1503.18	−0.142938
$481$	1537.93	0.145787
$482$	−3301.43	−0.311983
$483$	1594.17	0.150181
$484$	0	0
$485$	−2143.57	−0.200690
$486$	−819.464	−0.0764849
$487$	6748.91	0.627972	0.313986	−	0.949428i	$-0.398335\pi$
$487$	6748.91	0.627972	0.313986	+	0.949428i	$0.398335\pi$
$488$	10945.4	1.01532
$489$	8029.53	0.742552
$490$	−3770.98	−0.347664
$491$	−7361.40	−0.676609	−0.338305	−	0.941037i	$-0.609853\pi$
$491$	−7361.40	−0.676609	−0.338305	+	0.941037i	$0.609853\pi$
$492$	2440.74	0.223653
$493$	17789.1	1.62511
$494$	−3986.80	−0.363107
$495$	0	0
$496$	−22187.6	−2.00857
$497$	3589.26	0.323944
$498$	−7654.60	−0.688777
$499$	10381.7	0.931359	0.465680	−	0.884953i	$-0.345810\pi$
$499$	10381.7	0.931359	0.465680	+	0.884953i	$0.345810\pi$
$500$	2798.33	0.250291
$501$	3564.36	0.317852
$502$	22023.1	1.95805
$503$	−19149.0	−1.69744	−0.848721	−	0.528840i	$-0.822627\pi$
$503$	−19149.0	−1.69744	−0.848721	+	0.528840i	$0.822627\pi$
$504$	666.391	0.0588957
$505$	−3544.67	−0.312348
$506$	0	0
$507$	−5914.04	−0.518051
$508$	−7726.76	−0.674841
$509$	16073.2	1.39967	0.699836	−	0.714303i	$-0.253256\pi$
$509$	16073.2	1.39967	0.699836	+	0.714303i	$0.253256\pi$
$510$	−2582.77	−0.224249
$511$	4838.58	0.418877
$512$	−1493.27	−0.128894
$513$	2124.93	0.182881
$514$	−27417.5	−2.35279
$515$	−3845.58	−0.329042
$516$	2832.88	0.241687
$517$	0	0
$518$	−1638.09	−0.138945
$519$	−2421.44	−0.204797
$520$	−817.952	−0.0689799
$521$	−18955.3	−1.59395	−0.796975	−	0.604012i	$-0.793568\pi$
$521$	−18955.3	−1.59395	−0.796975	+	0.604012i	$0.793568\pi$
$522$	7378.97	0.618714
$523$	4442.19	0.371402	0.185701	−	0.982606i	$-0.440544\pi$
$523$	4442.19	0.371402	0.185701	+	0.982606i	$0.440544\pi$
$524$	3869.40	0.322587
$525$	−1605.93	−0.133502
$526$	−15341.9	−1.27175
$527$	−20393.3	−1.68567
$528$	0	0
$529$	377.000	0.0309855
$530$	−4820.09	−0.395041
$531$	1764.00	0.144164
$532$	1259.22	0.102620
$533$	3624.08	0.294515
$534$	−5149.25	−0.417285
$535$	4806.54	0.388420
$536$	14054.5	1.13258
$537$	−5851.17	−0.470199
$538$	98.3096	0.00787812
$539$	0	0
$540$	−317.690	−0.0253171
$541$	−2180.90	−0.173316	−0.0866580	−	0.996238i	$-0.527619\pi$
$541$	−2180.90	−0.173316	−0.0866580	+	0.996238i	$0.527619\pi$
$542$	26004.4	2.06086
$543$	3184.82	0.251701
$544$	−10507.4	−0.828129
$545$	1117.28	0.0878146
$546$	−721.044	−0.0565162
$547$	−8225.04	−0.642920	−0.321460	−	0.946923i	$-0.604174\pi$
$547$	−8225.04	−0.642920	−0.321460	+	0.946923i	$0.604174\pi$
$548$	4278.07	0.333485
$549$	6312.23	0.490709
$550$	0	0
$551$	−19134.2	−1.47939
$552$	5243.61	0.404316
$553$	1554.08	0.119505
$554$	3802.32	0.291598
$555$	−1071.65	−0.0819625
$556$	1639.90	0.125085
$557$	25181.9	1.91561	0.957804	−	0.287423i	$-0.0927986\pi$
$557$	25181.9	1.91561	0.957804	+	0.287423i	$0.0927986\pi$
$558$	−8459.22	−0.641769
$559$	4206.33	0.318263
$560$	1317.83	0.0994435
$561$	0	0
$562$	−6314.50	−0.473952
$563$	4504.50	0.337197	0.168599	−	0.985685i	$-0.446076\pi$
$563$	4504.50	0.337197	0.168599	+	0.985685i	$0.446076\pi$
$564$	−1718.54	−0.128304
$565$	5685.34	0.423335
$566$	7164.36	0.532050
$567$	384.310	0.0284647
$568$	11805.9	0.872122
$569$	13447.0	0.990732	0.495366	−	0.868684i	$-0.335034\pi$
$569$	13447.0	0.990732	0.495366	+	0.868684i	$0.335034\pi$
$570$	2778.06	0.204141
$571$	2605.52	0.190959	0.0954795	−	0.995431i	$-0.469562\pi$
$571$	2605.52	0.190959	0.0954795	+	0.995431i	$0.469562\pi$
$572$	0	0
$573$	6409.24	0.467277
$574$	−3860.09	−0.280691
$575$	−12636.5	−0.916485
$576$	1373.11	0.0993281
$577$	6339.65	0.457406	0.228703	−	0.973496i	$-0.426552\pi$
$577$	6339.65	0.457406	0.228703	+	0.973496i	$0.426552\pi$
$578$	−1485.90	−0.106929
$579$	−11843.3	−0.850069
$580$	2860.68	0.204799
$581$	3589.83	0.256336
$582$	−6215.37	−0.442672
$583$	0	0
$584$	15915.2	1.12770
$585$	−471.715	−0.0333385
$586$	−11210.1	−0.790247
$587$	−13370.6	−0.940140	−0.470070	−	0.882629i	$-0.655771\pi$
$587$	−13370.6	−0.940140	−0.470070	+	0.882629i	$0.655771\pi$
$588$	−3242.34	−0.227401
$589$	21935.3	1.53452
$590$	2306.20	0.160923
$591$	−2771.66	−0.192912
$592$	−8150.09	−0.565822
$593$	−14319.3	−0.991608	−0.495804	−	0.868434i	$-0.665127\pi$
$593$	−14319.3	−0.991608	−0.495804	+	0.868434i	$0.665127\pi$
$594$	0	0
$595$	1211.26	0.0834567
$596$	−7938.73	−0.545609
$597$	−1428.46	−0.0979276
$598$	−5673.65	−0.387981
$599$	−5788.63	−0.394853	−0.197427	−	0.980318i	$-0.563258\pi$
$599$	−5788.63	−0.394853	−0.197427	+	0.980318i	$0.563258\pi$
$600$	−5282.28	−0.359413
$601$	−23968.1	−1.62675	−0.813375	−	0.581739i	$-0.802372\pi$
$601$	−23968.1	−1.62675	−0.813375	+	0.581739i	$0.802372\pi$
$602$	−4480.26	−0.303325
$603$	8105.28	0.547384
$604$	1922.44	0.129508
$605$	0	0
$606$	−10277.9	−0.688961
$607$	23526.6	1.57317	0.786585	−	0.617482i	$-0.211847\pi$
$607$	23526.6	1.57317	0.786585	+	0.617482i	$0.211847\pi$
$608$	11301.9	0.753872
$609$	−3460.56	−0.230261
$610$	8252.40	0.547754
$611$	−2551.74	−0.168956
$612$	−2220.70	−0.146677
$613$	−1228.07	−0.0809159	−0.0404579	−	0.999181i	$-0.512882\pi$
$613$	−1228.07	−0.0809159	−0.0404579	+	0.999181i	$0.512882\pi$
$614$	−5729.31	−0.376573
$615$	−2525.31	−0.165578
$616$	0	0
$617$	−9844.90	−0.642368	−0.321184	−	0.947017i	$-0.604081\pi$
$617$	−9844.90	−0.642368	−0.321184	+	0.947017i	$0.604081\pi$
$618$	−11150.4	−0.725785
$619$	−6551.68	−0.425419	−0.212709	−	0.977115i	$-0.568229\pi$
$619$	−6551.68	−0.425419	−0.212709	+	0.977115i	$0.568229\pi$
$620$	−3279.47	−0.212430
$621$	3024.00	0.195409
$622$	−23365.9	−1.50625
$623$	2414.88	0.155297
$624$	−3587.46	−0.230150
$625$	11208.0	0.717309
$626$	12007.8	0.766661
$627$	0	0
$628$	−6989.64	−0.444135
$629$	−7491.01	−0.474859
$630$	502.434	0.0317737
$631$	−26440.5	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$631$	−26440.5	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$632$	5111.72	0.321730
$633$	14754.7	0.926458
$634$	−1122.13	−0.0702923
$635$	7994.48	0.499608
$636$	−4144.39	−0.258389
$637$	−4814.31	−0.299450
$638$	0	0
$639$	6808.50	0.421502
$640$	5803.64	0.358451
$641$	−27927.2	−1.72084	−0.860421	−	0.509584i	$-0.829799\pi$
$641$	−27927.2	−1.72084	−0.860421	+	0.509584i	$0.829799\pi$
$642$	13936.7	0.856758
$643$	−16737.7	−1.02655	−0.513274	−	0.858225i	$-0.671568\pi$
$643$	−16737.7	−1.02655	−0.513274	+	0.858225i	$0.671568\pi$
$644$	1792.00	0.109650
$645$	−2931.03	−0.178929
$646$	19419.1	1.18271
$647$	7818.70	0.475092	0.237546	−	0.971376i	$-0.423657\pi$
$647$	7818.70	0.475092	0.237546	+	0.971376i	$0.423657\pi$
$648$	1264.08	0.0766325
$649$	0	0
$650$	5715.49	0.344892
$651$	3967.17	0.238842
$652$	9025.94	0.542152
$653$	19747.6	1.18344	0.591719	−	0.806144i	$-0.298450\pi$
$653$	19747.6	1.18344	0.591719	+	0.806144i	$0.298450\pi$
$654$	3239.59	0.193697
$655$	−4003.47	−0.238822
$656$	−19205.4	−1.14305
$657$	9178.33	0.545024
$658$	2717.91	0.161026
$659$	−7867.72	−0.465072	−0.232536	−	0.972588i	$-0.574702\pi$
$659$	−7867.72	−0.465072	−0.232536	+	0.972588i	$0.574702\pi$
$660$	0	0
$661$	4227.41	0.248755	0.124378	−	0.992235i	$-0.460307\pi$
$661$	4227.41	0.248755	0.124378	+	0.992235i	$0.460307\pi$
$662$	1825.90	0.107199
$663$	−3297.35	−0.193150
$664$	11807.8	0.690106
$665$	−1302.85	−0.0759733
$666$	−3107.30	−0.180789
$667$	−27230.0	−1.58073
$668$	4006.67	0.232070
$669$	6300.88	0.364135
$670$	10596.6	0.611018
$671$	0	0
$672$	2044.04	0.117337
$673$	−29397.6	−1.68379	−0.841897	−	0.539638i	$-0.818561\pi$
$673$	−29397.6	−1.68379	−0.841897	+	0.539638i	$0.818561\pi$
$674$	2753.57	0.157364
$675$	−3046.30	−0.173707
$676$	−6647.94	−0.378239
$677$	−5737.14	−0.325696	−0.162848	−	0.986651i	$-0.552068\pi$
$677$	−5737.14	−0.325696	−0.162848	+	0.986651i	$0.552068\pi$
$678$	16484.8	0.933771
$679$	2914.86	0.164745
$680$	3984.11	0.224682
$681$	6771.49	0.381034
$682$	0	0
$683$	32097.6	1.79821	0.899107	−	0.437729i	$-0.144217\pi$
$683$	32097.6	1.79821	0.899107	+	0.437729i	$0.144217\pi$
$684$	2388.62	0.133525
$685$	−4426.30	−0.246891
$686$	10615.8	0.590837
$687$	−15933.2	−0.884848
$688$	−22291.0	−1.23523
$689$	−6153.69	−0.340257
$690$	3953.48	0.218125
$691$	−16456.2	−0.905965	−0.452983	−	0.891519i	$-0.649640\pi$
$691$	−16456.2	−0.905965	−0.452983	+	0.891519i	$0.649640\pi$
$692$	−2721.93	−0.149526
$693$	0	0
$694$	21112.3	1.15477
$695$	−1696.72	−0.0926047
$696$	−11382.6	−0.619909
$697$	−17652.3	−0.959294
$698$	−43058.9	−2.33496
$699$	−7398.80	−0.400355
$700$	−1805.22	−0.0974725
$701$	−27238.1	−1.46758	−0.733788	−	0.679379i	$-0.762249\pi$
$701$	−27238.1	−1.46758	−0.733788	+	0.679379i	$0.762249\pi$
$702$	−1367.75	−0.0735364
$703$	8057.44	0.432279
$704$	0	0
$705$	1778.09	0.0949882
$706$	8934.12	0.476261
$707$	4820.09	0.256405
$708$	1982.90	0.105257
$709$	28761.4	1.52349	0.761747	−	0.647875i	$-0.224342\pi$
$709$	28761.4	1.52349	0.761747	+	0.647875i	$0.224342\pi$
$710$	8901.21	0.470502
$711$	2947.94	0.155494
$712$	7943.10	0.418090
$713$	31216.3	1.63964
$714$	3512.09	0.184085
$715$	0	0
$716$	−6577.27	−0.343302
$717$	−4288.21	−0.223356
$718$	−10804.5	−0.561588
$719$	−27272.0	−1.41456	−0.707282	−	0.706931i	$-0.750079\pi$
$719$	−27272.0	−1.41456	−0.707282	+	0.706931i	$0.750079\pi$
$720$	2499.80	0.129392
$721$	5229.28	0.270109
$722$	2243.05	0.115620
$723$	2936.97	0.151075
$724$	3580.04	0.183772
$725$	27430.8	1.40518
$726$	0	0
$727$	3979.75	0.203027	0.101514	−	0.994834i	$-0.467631\pi$
$727$	3979.75	0.203027	0.101514	+	0.994834i	$0.467631\pi$
$728$	1112.26	0.0566253
$729$	729.000	0.0370370
$730$	11999.5	0.608383
$731$	−20488.3	−1.03665
$732$	7095.54	0.358277
$733$	−9342.48	−0.470767	−0.235384	−	0.971903i	$-0.575635\pi$
$733$	−9342.48	−0.470767	−0.235384	+	0.971903i	$0.575635\pi$
$734$	28426.3	1.42947
$735$	3354.68	0.168353
$736$	16083.9	0.805515
$737$	0	0
$738$	−7322.23	−0.365224
$739$	28928.0	1.43997	0.719983	−	0.693992i	$-0.244150\pi$
$739$	28928.0	1.43997	0.719983	+	0.693992i	$0.244150\pi$
$740$	−1204.64	−0.0598425
$741$	3546.68	0.175831
$742$	6554.43	0.324287
$743$	−4857.04	−0.239822	−0.119911	−	0.992785i	$-0.538261\pi$
$743$	−4857.04	−0.239822	−0.119911	+	0.992785i	$0.538261\pi$
$744$	13049.0	0.643008
$745$	8213.80	0.403933
$746$	−31660.8	−1.55386
$747$	6809.57	0.333533
$748$	0	0
$749$	−6536.00	−0.318852
$750$	−8395.00	−0.408723
$751$	14355.4	0.697517	0.348759	−	0.937213i	$-0.386603\pi$
$751$	14355.4	0.697517	0.348759	+	0.937213i	$0.386603\pi$
$752$	13522.6	0.655744
$753$	−19591.9	−0.948165
$754$	12316.1	0.594863
$755$	−1989.05	−0.0958792
$756$	432.000	0.0207827
$757$	−17714.9	−0.850538	−0.425269	−	0.905067i	$-0.639821\pi$
$757$	−17714.9	−0.850538	−0.425269	+	0.905067i	$0.639821\pi$
$758$	48103.9	2.30503
$759$	0	0
$760$	−4285.37	−0.204535
$761$	7945.82	0.378497	0.189248	−	0.981929i	$-0.439395\pi$
$761$	7945.82	0.378497	0.189248	+	0.981929i	$0.439395\pi$
$762$	23180.3	1.10201
$763$	−1519.29	−0.0720866
$764$	7204.58	0.341168
$765$	2297.64	0.108590
$766$	−45398.4	−2.14140
$767$	2944.26	0.138606
$768$	13166.2	0.618613
$769$	−27308.1	−1.28057	−0.640284	−	0.768139i	$-0.721183\pi$
$769$	−27308.1	−1.28057	−0.640284	+	0.768139i	$0.721183\pi$
$770$	0	0
$771$	24390.8	1.13932
$772$	−13313.0	−0.620653
$773$	−18872.6	−0.878136	−0.439068	−	0.898454i	$-0.644691\pi$
$773$	−18872.6	−0.878136	−0.439068	+	0.898454i	$0.644691\pi$
$774$	−8498.64	−0.394674
$775$	−31446.6	−1.45754
$776$	9587.65	0.443527
$777$	1457.25	0.0672826
$778$	3176.29	0.146369
$779$	18987.1	0.873276
$780$	−530.252	−0.0243411
$781$	0	0
$782$	27635.4	1.26373
$783$	−6564.37	−0.299606
$784$	25512.8	1.16221
$785$	7231.82	0.328808
$786$	−11608.2	−0.526782
$787$	14512.1	0.657307	0.328654	−	0.944450i	$-0.393405\pi$
$787$	14512.1	0.657307	0.328654	+	0.944450i	$0.393405\pi$
$788$	−3115.60	−0.140849
$789$	13648.3	0.615832
$790$	3854.04	0.173570
$791$	−7731.01	−0.347513
$792$	0	0
$793$	10535.6	0.471792
$794$	2859.15	0.127793
$795$	4287.98	0.191294
$796$	−1605.72	−0.0714989
$797$	29108.9	1.29371	0.646856	−	0.762612i	$-0.276083\pi$
$797$	29108.9	1.29371	0.646856	+	0.762612i	$0.276083\pi$
$798$	−3777.65	−0.167578
$799$	12429.1	0.550325
$800$	−16202.5	−0.716056
$801$	4580.80	0.202066
$802$	−41154.0	−1.81197
$803$	0	0
$804$	9111.10	0.399656
$805$	−1854.09	−0.0811777
$806$	−14119.1	−0.617029
$807$	−87.4567	−0.00381490
$808$	15854.4	0.690291
$809$	3000.83	0.130413	0.0652063	−	0.997872i	$-0.479229\pi$
$809$	3000.83	0.130413	0.0652063	+	0.997872i	$0.479229\pi$
$810$	953.071	0.0413426
$811$	−6239.39	−0.270154	−0.135077	−	0.990835i	$-0.543128\pi$
$811$	−6239.39	−0.270154	−0.135077	+	0.990835i	$0.543128\pi$
$812$	−3890.00	−0.168118
$813$	−23133.7	−0.997950
$814$	0	0
$815$	−9338.68	−0.401374
$816$	17473.9	0.749645
$817$	22037.6	0.943693
$818$	29539.6	1.26263
$819$	641.445	0.0273674
$820$	−2838.69	−0.120892
$821$	−14922.4	−0.634342	−0.317171	−	0.948368i	$-0.602733\pi$
$821$	−14922.4	−0.634342	−0.317171	+	0.948368i	$0.602733\pi$
$822$	−12834.2	−0.544580
$823$	−25737.8	−1.09011	−0.545057	−	0.838399i	$-0.683492\pi$
$823$	−25737.8	−1.09011	−0.545057	+	0.838399i	$0.683492\pi$
$824$	17200.3	0.727186
$825$	0	0
$826$	−3136.00	−0.132101
$827$	−27043.4	−1.13711	−0.568555	−	0.822645i	$-0.692497\pi$
$827$	−27043.4	−1.13711	−0.568555	+	0.822645i	$0.692497\pi$
$828$	3399.26	0.142672
$829$	−9795.41	−0.410384	−0.205192	−	0.978722i	$-0.565782\pi$
$829$	−9795.41	−0.410384	−0.205192	+	0.978722i	$0.565782\pi$
$830$	8902.62	0.372306
$831$	−3382.56	−0.141203
$832$	2291.84	0.0954990
$833$	23449.7	0.975370
$834$	−4919.70	−0.204263
$835$	−4145.50	−0.171809
$836$	0	0
$837$	7525.37	0.310770
$838$	37730.5	1.55535
$839$	28875.5	1.18819	0.594095	−	0.804395i	$-0.297510\pi$
$839$	28875.5	1.18819	0.594095	+	0.804395i	$0.297510\pi$
$840$	−775.041	−0.0318351
$841$	34720.7	1.42362
$842$	47489.3	1.94369
$843$	5617.41	0.229506
$844$	16585.7	0.676426
$845$	6878.28	0.280024
$846$	5155.63	0.209520
$847$	0	0
$848$	32610.7	1.32059
$849$	−6373.45	−0.257640
$850$	−27839.2	−1.12338
$851$	11466.6	0.461892
$852$	7653.39	0.307747
$853$	47157.1	1.89288	0.946441	−	0.322878i	$-0.104650\pi$
$853$	47157.1	1.89288	0.946441	+	0.322878i	$0.104650\pi$
$854$	−11221.7	−0.449649
$855$	−2471.38	−0.0988531
$856$	−21498.4	−0.858412
$857$	−5021.31	−0.200145	−0.100073	−	0.994980i	$-0.531908\pi$
$857$	−5021.31	−0.200145	−0.100073	+	0.994980i	$0.531908\pi$
$858$	0	0
$859$	−22921.1	−0.910428	−0.455214	−	0.890382i	$-0.650437\pi$
$859$	−22921.1	−0.910428	−0.455214	+	0.890382i	$0.650437\pi$
$860$	−3294.76	−0.130640
$861$	3433.95	0.135922
$862$	−18938.9	−0.748332
$863$	−19488.1	−0.768693	−0.384347	−	0.923189i	$-0.625573\pi$
$863$	−19488.1	−0.768693	−0.384347	+	0.923189i	$0.625573\pi$
$864$	3877.36	0.152674
$865$	2816.24	0.110699
$866$	−24266.1	−0.952188
$867$	1321.86	0.0517794
$868$	4459.48	0.174383
$869$	0	0
$870$	−8582.05	−0.334435
$871$	13528.4	0.526282
$872$	−4997.30	−0.194071
$873$	5529.22	0.214360
$874$	−29725.0	−1.15042
$875$	3937.06	0.152111
$876$	10317.3	0.397933
$877$	8455.67	0.325573	0.162787	−	0.986661i	$-0.447952\pi$
$877$	8455.67	0.325573	0.162787	+	0.986661i	$0.447952\pi$
$878$	343.834	0.0132162
$879$	9972.56	0.382669
$880$	0	0
$881$	−11291.2	−0.431794	−0.215897	−	0.976416i	$-0.569268\pi$
$881$	−11291.2	−0.431794	−0.215897	+	0.976416i	$0.569268\pi$
$882$	9727.02	0.371344
$883$	31818.1	1.21264	0.606322	−	0.795219i	$-0.292644\pi$
$883$	31818.1	1.21264	0.606322	+	0.795219i	$0.292644\pi$
$884$	−3706.53	−0.141023
$885$	−2051.61	−0.0779254
$886$	−16705.4	−0.633441
$887$	−17481.1	−0.661732	−0.330866	−	0.943678i	$-0.607341\pi$
$887$	−17481.1	−0.661732	−0.330866	+	0.943678i	$0.607341\pi$
$888$	4793.24	0.181138
$889$	−10871.0	−0.410126
$890$	5988.80	0.225556
$891$	0	0
$892$	7082.78	0.265862
$893$	−13368.9	−0.500978
$894$	23816.2	0.890976
$895$	6805.16	0.254158
$896$	−7891.87	−0.294251
$897$	5047.31	0.187876
$898$	39125.1	1.45392
$899$	−67763.1	−2.51393
$900$	−3424.33	−0.126827
$901$	29973.6	1.10829
$902$	0	0
$903$	3985.66	0.146882
$904$	−25429.1	−0.935574
$905$	−3704.08	−0.136053
$906$	−5767.31	−0.211486
$907$	10607.4	0.388326	0.194163	−	0.980969i	$-0.437801\pi$
$907$	10607.4	0.388326	0.194163	+	0.980969i	$0.437801\pi$
$908$	7611.79	0.278201
$909$	9143.26	0.333623
$910$	838.604	0.0305489
$911$	−41249.2	−1.50016	−0.750080	−	0.661347i	$-0.769985\pi$
$911$	−41249.2	−1.50016	−0.750080	+	0.661347i	$0.769985\pi$
$912$	−18795.2	−0.682425
$913$	0	0
$914$	−11906.5	−0.430887
$915$	−7341.38	−0.265244
$916$	−17910.4	−0.646045
$917$	5443.97	0.196048
$918$	6662.10	0.239523
$919$	13858.1	0.497429	0.248714	−	0.968577i	$-0.419992\pi$
$919$	13858.1	0.497429	0.248714	+	0.968577i	$0.419992\pi$
$920$	−6098.53	−0.218546
$921$	5096.83	0.182352
$922$	39004.9	1.39323
$923$	11363.9	0.405253
$924$	0	0
$925$	−11551.2	−0.410595
$926$	−36719.0	−1.30309
$927$	9919.47	0.351454
$928$	−34914.2	−1.23504
$929$	20893.7	0.737890	0.368945	−	0.929451i	$-0.379719\pi$
$929$	20893.7	0.737890	0.368945	+	0.929451i	$0.379719\pi$
$930$	9838.42	0.346897
$931$	−25222.8	−0.887911
$932$	−8316.94	−0.292307
$933$	20786.5	0.729388
$934$	−36042.9	−1.26270
$935$	0	0
$936$	2109.86	0.0736784
$937$	−3203.52	−0.111691	−0.0558454	−	0.998439i	$-0.517785\pi$
$937$	−3203.52	−0.111691	−0.0558454	+	0.998439i	$0.517785\pi$
$938$	−14409.4	−0.501581
$939$	−10682.2	−0.371248
$940$	1998.74	0.0693528
$941$	−19951.6	−0.691182	−0.345591	−	0.938385i	$-0.612322\pi$
$941$	−19951.6	−0.691182	−0.345591	+	0.938385i	$0.612322\pi$
$942$	20968.9	0.725270
$943$	27020.6	0.933099
$944$	−15602.8	−0.537952
$945$	−446.968	−0.0153861
$946$	0	0
$947$	−38216.7	−1.31138	−0.655689	−	0.755031i	$-0.727622\pi$
$947$	−38216.7	−1.31138	−0.655689	+	0.755031i	$0.727622\pi$
$948$	3313.76	0.113529
$949$	15319.4	0.524014
$950$	29944.3	1.02265
$951$	998.249	0.0340383
$952$	−5417.65	−0.184440
$953$	47661.4	1.62004	0.810022	−	0.586399i	$-0.199455\pi$
$953$	47661.4	1.62004	0.810022	+	0.586399i	$0.199455\pi$
$954$	12433.2	0.421948
$955$	−7454.21	−0.252579
$956$	−4820.35	−0.163077
$957$	0	0
$958$	7897.56	0.266345
$959$	6018.94	0.202671
$960$	−1596.99	−0.0536901
$961$	47892.3	1.60761
$962$	−5186.34	−0.173819
$963$	−12398.2	−0.414876
$964$	3301.43	0.110303
$965$	13774.2	0.459490
$966$	−5376.00	−0.179058
$967$	−18933.2	−0.629628	−0.314814	−	0.949153i	$-0.601942\pi$
$967$	−18933.2	−0.629628	−0.314814	+	0.949153i	$0.601942\pi$
$968$	0	0
$969$	−17275.3	−0.572717
$970$	7228.73	0.239279
$971$	−40660.3	−1.34382	−0.671911	−	0.740632i	$-0.734526\pi$
$971$	−40660.3	−1.34382	−0.671911	+	0.740632i	$0.734526\pi$
$972$	819.464	0.0270415
$973$	2307.23	0.0760188
$974$	−22759.2	−0.748720
$975$	−5084.53	−0.167011
$976$	−55832.3	−1.83110
$977$	−22502.8	−0.736876	−0.368438	−	0.929652i	$-0.620107\pi$
$977$	−22502.8	−0.736876	−0.368438	+	0.929652i	$0.620107\pi$
$978$	−27077.8	−0.885331
$979$	0	0
$980$	3770.98	0.122918
$981$	−2881.96	−0.0937959
$982$	24824.7	0.806709
$983$	−4435.20	−0.143907	−0.0719536	−	0.997408i	$-0.522923\pi$
$983$	−4435.20	−0.143907	−0.0719536	+	0.997408i	$0.522923\pi$
$984$	11295.1	0.365929
$985$	3223.55	0.104275
$986$	−59989.8	−1.93759
$987$	−2417.87	−0.0779753
$988$	3986.80	0.128378
$989$	31361.8	1.00834
$990$	0	0
$991$	7362.76	0.236010	0.118005	−	0.993013i	$-0.462350\pi$
$991$	7362.76	0.236010	0.118005	+	0.993013i	$0.462350\pi$
$992$	40025.5	1.28106
$993$	−1624.33	−0.0519101
$994$	−12104.0	−0.386233
$995$	1661.35	0.0529331
$996$	7654.60	0.243519
$997$	53480.1	1.69883	0.849413	−	0.527728i	$-0.176956\pi$
$997$	53480.1	1.69883	0.849413	+	0.527728i	$0.176956\pi$
$998$	−35010.0	−1.11044
$999$	2764.27	0.0875452

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.4.a.j.1.1		2
3.2	odd	2		1089.4.a.t.1.2		2
11.10	odd	2		33.4.a.d.1.2	✓	2
33.32	even	2		99.4.a.e.1.1		2
44.43	even	2		528.4.a.o.1.1		2
55.32	even	4		825.4.c.i.199.4		4
55.43	even	4		825.4.c.i.199.1		4
55.54	odd	2		825.4.a.k.1.1		2
77.76	even	2		1617.4.a.j.1.2		2
88.21	odd	2		2112.4.a.ba.1.2		2
88.43	even	2		2112.4.a.bh.1.2		2
132.131	odd	2		1584.4.a.x.1.2		2
165.164	even	2		2475.4.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.d.1.2	✓	2	11.10	odd	2
99.4.a.e.1.1		2	33.32	even	2
363.4.a.j.1.1		2	1.1	even	1	trivial
528.4.a.o.1.1		2	44.43	even	2
825.4.a.k.1.1		2	55.54	odd	2
825.4.c.i.199.1		4	55.43	even	4
825.4.c.i.199.4		4	55.32	even	4
1089.4.a.t.1.2		2	3.2	odd	2
1584.4.a.x.1.2		2	132.131	odd	2
1617.4.a.j.1.2		2	77.76	even	2
2112.4.a.ba.1.2		2	88.21	odd	2
2112.4.a.bh.1.2		2	88.43	even	2
2475.4.a.o.1.2		2	165.164	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-3.37228 q^{2} +3.00000 q^{3} +3.37228 q^{4} -3.48913 q^{5} -10.1168 q^{6} +4.74456 q^{7} +15.6060 q^{8} +9.00000 q^{9} +11.7663 q^{10} +10.1168 q^{12} +15.0217 q^{13} -16.0000 q^{14} -10.4674 q^{15} -79.6060 q^{16} -73.1684 q^{17} -30.3505 q^{18} +78.7011 q^{19} -11.7663 q^{20} +14.2337 q^{21} +112.000 q^{23} +46.8179 q^{24} -112.826 q^{25} -50.6576 q^{26} +27.0000 q^{27} +16.0000 q^{28} -243.125 q^{29} +35.2989 q^{30} +278.717 q^{31} +143.606 q^{32} +246.745 q^{34} -16.5544 q^{35} +30.3505 q^{36} +102.380 q^{37} -265.402 q^{38} +45.0652 q^{39} -54.4512 q^{40} +241.255 q^{41} -48.0000 q^{42} +280.016 q^{43} -31.4021 q^{45} -377.696 q^{46} -169.870 q^{47} -238.818 q^{48} -320.489 q^{49} +380.481 q^{50} -219.505 q^{51} +50.6576 q^{52} -409.652 q^{53} -91.0516 q^{54} +74.0435 q^{56} +236.103 q^{57} +819.886 q^{58} +196.000 q^{59} -35.2989 q^{60} +701.359 q^{61} -939.913 q^{62} +42.7011 q^{63} +152.568 q^{64} -52.4128 q^{65} +900.587 q^{67} -246.745 q^{68} +336.000 q^{69} +55.8260 q^{70} +756.500 q^{71} +140.454 q^{72} +1019.81 q^{73} -345.255 q^{74} -338.478 q^{75} +265.402 q^{76} -151.973 q^{78} +327.549 q^{79} +277.755 q^{80} +81.0000 q^{81} -813.581 q^{82} +756.619 q^{83} +48.0000 q^{84} +255.294 q^{85} -944.293 q^{86} -729.375 q^{87} +508.978 q^{89} +105.897 q^{90} +71.2716 q^{91} +377.696 q^{92} +836.152 q^{93} +572.848 q^{94} -274.598 q^{95} +430.818 q^{96} +614.358 q^{97} +1080.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 6 q^{3} + q^{4} + 16 q^{5} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 18 q^{9} + 58 q^{10} + 3 q^{12} + 76 q^{13} - 32 q^{14} + 48 q^{15} - 119 q^{16} + 26 q^{17} - 9 q^{18} + 54 q^{19} - 58 q^{20}+ \cdots + 375 q^{98}+O(q^{100}) \) 2 * q - q^2 + 6 * q^3 + q^4 + 16 * q^5 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 18 * q^9 + 58 * q^10 + 3 * q^12 + 76 * q^13 - 32 * q^14 + 48 * q^15 - 119 * q^16 + 26 * q^17 - 9 * q^18 + 54 * q^19 - 58 * q^20 - 6 * q^21 + 224 * q^23 - 27 * q^24 + 142 * q^25 + 94 * q^26 + 54 * q^27 + 32 * q^28 - 222 * q^29 + 174 * q^30 - 40 * q^31 + 247 * q^32 + 482 * q^34 - 148 * q^35 + 9 * q^36 - 48 * q^37 - 324 * q^38 + 228 * q^39 - 534 * q^40 + 494 * q^41 - 96 * q^42 + 66 * q^43 + 144 * q^45 - 112 * q^46 - 64 * q^47 - 357 * q^48 - 618 * q^49 + 985 * q^50 + 78 * q^51 - 94 * q^52 - 84 * q^53 - 27 * q^54 + 240 * q^56 + 162 * q^57 + 870 * q^58 + 392 * q^59 - 174 * q^60 + 1104 * q^61 - 1696 * q^62 - 18 * q^63 + 713 * q^64 + 1136 * q^65 + 928 * q^67 - 482 * q^68 + 672 * q^69 - 256 * q^70 + 456 * q^71 - 81 * q^72 + 592 * q^73 - 702 * q^74 + 426 * q^75 + 324 * q^76 + 282 * q^78 + 230 * q^79 - 490 * q^80 + 162 * q^81 - 214 * q^82 - 348 * q^83 + 96 * q^84 + 2188 * q^85 - 1452 * q^86 - 666 * q^87 + 972 * q^89 + 522 * q^90 - 340 * q^91 + 112 * q^92 - 120 * q^93 + 824 * q^94 - 756 * q^95 + 741 * q^96 - 1184 * q^97 + 375 * q^98

Introduction

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