LMFDB - Embedded newform 7225.2.a.bh.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7225,2,Mod(1,7225)] 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("7225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,3,-3,3,0,-9,-6] 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:	$57.6919154604$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.1397493.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 3x^{4} + 10x^{3} + 3x^{2} - 6x + 1 $ x^6 - 3x^5 - 3x^4 + 10x^3 + 3x^2 - 6*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1445)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.05432$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.40162 q^{2} -3.14741 q^{3} +3.76778 q^{4} -7.55888 q^{6} +2.45595 q^{7} +4.24555 q^{8} +6.90618 q^{9} -2.67294 q^{11} -11.8588 q^{12} +2.57085 q^{13} +5.89825 q^{14} +2.66063 q^{16} +16.5860 q^{18} -6.25456 q^{19} -7.72986 q^{21} -6.41940 q^{22} -3.11473 q^{23} -13.3625 q^{24} +6.17420 q^{26} -12.2943 q^{27} +9.25348 q^{28} +0.710678 q^{29} -3.99271 q^{31} -2.10127 q^{32} +8.41285 q^{33} +26.0210 q^{36} -3.63287 q^{37} -15.0211 q^{38} -8.09151 q^{39} -8.29741 q^{41} -18.5642 q^{42} +12.7653 q^{43} -10.0711 q^{44} -7.48040 q^{46} -2.78467 q^{47} -8.37409 q^{48} -0.968329 q^{49} +9.68640 q^{52} -0.165984 q^{53} -29.5263 q^{54} +10.4268 q^{56} +19.6857 q^{57} +1.70678 q^{58} -12.2972 q^{59} +0.680871 q^{61} -9.58898 q^{62} +16.9612 q^{63} -10.3677 q^{64} +20.2045 q^{66} +1.50354 q^{67} +9.80333 q^{69} +2.51376 q^{71} +29.3205 q^{72} +8.24985 q^{73} -8.72478 q^{74} -23.5658 q^{76} -6.56461 q^{77} -19.4327 q^{78} -4.47987 q^{79} +17.9768 q^{81} -19.9272 q^{82} -12.1257 q^{83} -29.1245 q^{84} +30.6575 q^{86} -2.23680 q^{87} -11.3481 q^{88} +1.16707 q^{89} +6.31386 q^{91} -11.7356 q^{92} +12.5667 q^{93} -6.68772 q^{94} +6.61356 q^{96} +3.12613 q^{97} -2.32556 q^{98} -18.4598 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 9 q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9} - 6 q^{11} - 6 q^{12} + 9 q^{13} + 18 q^{14} - 3 q^{16} + 15 q^{18} - 21 q^{19} - 12 q^{21} + 3 q^{22} - 18 q^{23} - 6 q^{26} - 12 q^{27}+ \cdots - 18 q^{99}+O(q^{100}) $ 6 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 9 * q^6 - 6 * q^7 + 6 * q^8 + 3 * q^9 - 6 * q^11 - 6 * q^12 + 9 * q^13 + 18 * q^14 - 3 * q^16 + 15 * q^18 - 21 * q^19 - 12 * q^21 + 3 * q^22 - 18 * q^23 - 6 * q^26 - 12 * q^27 + 15 * q^28 - 18 * q^29 + 6 * q^31 + 24 * q^33 + 18 * q^36 + 9 * q^37 - 15 * q^38 - 9 * q^39 - 3 * q^41 - 6 * q^42 + 39 * q^43 - 6 * q^44 - 27 * q^46 - 3 * q^47 - 21 * q^48 + 6 * q^49 - 3 * q^52 + 24 * q^53 - 33 * q^54 - 9 * q^56 + 12 * q^57 - 33 * q^58 - 12 * q^59 - 27 * q^61 + 12 * q^63 - 12 * q^64 + 21 * q^66 + 30 * q^67 + 18 * q^71 + 30 * q^72 + 9 * q^73 - 27 * q^74 - 45 * q^76 + 21 * q^77 - 36 * q^78 - 3 * q^79 + 6 * q^81 - 15 * q^82 + 21 * q^83 - 27 * q^84 + 36 * q^86 - 6 * q^87 - 12 * q^88 + 6 * q^89 - 36 * q^91 - 9 * q^92 + 30 * q^93 + 3 * q^94 - 27 * q^96 + 18 * q^97 - 42 * q^98 - 18 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	2.40162	1.69820	0.849101	−	0.528230i	$-0.177144\pi$
$2$	2.40162	1.69820	0.849101	+	0.528230i	$0.177144\pi$
$3$	−3.14741	−1.81716	−0.908579	−	0.417714i	$-0.862831\pi$
$3$	−3.14741	−1.81716	−0.908579	+	0.417714i	$0.862831\pi$
$4$	3.76778	1.88389
$5$	0	0
$5$	0	0
$6$	−7.55888	−3.08590
$7$	2.45595	0.928260	0.464130	−	0.885767i	$-0.346367\pi$
$7$	2.45595	0.928260	0.464130	+	0.885767i	$0.346367\pi$
$8$	4.24555	1.50103
$9$	6.90618	2.30206
$10$	0	0
$11$	−2.67294	−0.805923	−0.402962	−	0.915217i	$-0.632019\pi$
$11$	−2.67294	−0.805923	−0.402962	+	0.915217i	$0.632019\pi$
$12$	−11.8588	−3.42333
$13$	2.57085	0.713025	0.356512	−	0.934291i	$-0.383966\pi$
$13$	2.57085	0.713025	0.356512	+	0.934291i	$0.383966\pi$
$14$	5.89825	1.57637
$15$	0	0
$16$	2.66063	0.665158
$17$	0	0
$17$	0	0
$18$	16.5860	3.90936
$19$	−6.25456	−1.43490	−0.717448	−	0.696613i	$-0.754690\pi$
$19$	−6.25456	−1.43490	−0.717448	+	0.696613i	$0.754690\pi$
$20$	0	0
$21$	−7.72986	−1.68679
$22$	−6.41940	−1.36862
$23$	−3.11473	−0.649466	−0.324733	−	0.945806i	$-0.605275\pi$
$23$	−3.11473	−0.649466	−0.324733	+	0.945806i	$0.605275\pi$
$24$	−13.3625	−2.72760
$25$	0	0
$26$	6.17420	1.21086
$27$	−12.2943	−2.36605
$28$	9.25348	1.74874
$29$	0.710678	0.131970	0.0659848	−	0.997821i	$-0.478981\pi$
$29$	0.710678	0.131970	0.0659848	+	0.997821i	$0.478981\pi$
$30$	0	0
$31$	−3.99271	−0.717112	−0.358556	−	0.933508i	$-0.616731\pi$
$31$	−3.99271	−0.717112	−0.358556	+	0.933508i	$0.616731\pi$
$32$	−2.10127	−0.371456
$33$	8.41285	1.46449
$34$	0	0
$35$	0	0
$36$	26.0210	4.33683
$37$	−3.63287	−0.597240	−0.298620	−	0.954372i	$-0.596526\pi$
$37$	−3.63287	−0.597240	−0.298620	+	0.954372i	$0.596526\pi$
$38$	−15.0211	−2.43674
$39$	−8.09151	−1.29568
$40$	0	0
$41$	−8.29741	−1.29584	−0.647919	−	0.761709i	$-0.724360\pi$
$41$	−8.29741	−1.29584	−0.647919	+	0.761709i	$0.724360\pi$
$42$	−18.5642	−2.86452
$43$	12.7653	1.94669	0.973347	−	0.229338i	$-0.0736563\pi$
$43$	12.7653	1.94669	0.973347	+	0.229338i	$0.0736563\pi$
$44$	−10.0711	−1.51827
$45$	0	0
$46$	−7.48040	−1.10293
$47$	−2.78467	−0.406185	−0.203093	−	0.979160i	$-0.565099\pi$
$47$	−2.78467	−0.406185	−0.203093	+	0.979160i	$0.565099\pi$
$48$	−8.37409	−1.20870
$49$	−0.968329	−0.138333
$50$	0	0
$51$	0	0
$52$	9.68640	1.34326
$53$	−0.165984	−0.0227996	−0.0113998	−	0.999935i	$-0.503629\pi$
$53$	−0.165984	−0.0227996	−0.0113998	+	0.999935i	$0.503629\pi$
$54$	−29.5263	−4.01803
$55$	0	0
$56$	10.4268	1.39334
$57$	19.6857	2.60743
$58$	1.70678	0.224111
$59$	−12.2972	−1.60095	−0.800476	−	0.599364i	$-0.795420\pi$
$59$	−12.2972	−1.60095	−0.800476	+	0.599364i	$0.795420\pi$
$60$	0	0
$61$	0.680871	0.0871767	0.0435883	−	0.999050i	$-0.486121\pi$
$61$	0.680871	0.0871767	0.0435883	+	0.999050i	$0.486121\pi$
$62$	−9.58898	−1.21780
$63$	16.9612	2.13691
$64$	−10.3677	−1.29596
$65$	0	0
$66$	20.2045	2.48700
$67$	1.50354	0.183687	0.0918433	−	0.995773i	$-0.470724\pi$
$67$	1.50354	0.183687	0.0918433	+	0.995773i	$0.470724\pi$
$68$	0	0
$69$	9.80333	1.18018
$70$	0	0
$71$	2.51376	0.298328	0.149164	−	0.988812i	$-0.452342\pi$
$71$	2.51376	0.298328	0.149164	+	0.988812i	$0.452342\pi$
$72$	29.3205	3.45546
$73$	8.24985	0.965572	0.482786	−	0.875738i	$-0.339625\pi$
$73$	8.24985	0.965572	0.482786	+	0.875738i	$0.339625\pi$
$74$	−8.72478	−1.01423
$75$	0	0
$76$	−23.5658	−2.70319
$77$	−6.56461	−0.748107
$78$	−19.4327	−2.20032
$79$	−4.47987	−0.504025	−0.252012	−	0.967724i	$-0.581092\pi$
$79$	−4.47987	−0.504025	−0.252012	+	0.967724i	$0.581092\pi$
$80$	0	0
$81$	17.9768	1.99742
$82$	−19.9272	−2.20060
$83$	−12.1257	−1.33097	−0.665484	−	0.746412i	$-0.731775\pi$
$83$	−12.1257	−1.33097	−0.665484	+	0.746412i	$0.731775\pi$
$84$	−29.1245	−3.17774
$85$	0	0
$86$	30.6575	3.30588
$87$	−2.23680	−0.239810
$88$	−11.3481	−1.20971
$89$	1.16707	0.123709	0.0618546	−	0.998085i	$-0.480298\pi$
$89$	1.16707	0.123709	0.0618546	+	0.998085i	$0.480298\pi$
$90$	0	0
$91$	6.31386	0.661873
$92$	−11.7356	−1.22352
$93$	12.5667	1.30310
$94$	−6.68772	−0.689785
$95$	0	0
$96$	6.61356	0.674993
$97$	3.12613	0.317410	0.158705	−	0.987326i	$-0.449268\pi$
$97$	3.12613	0.317410	0.158705	+	0.987326i	$0.449268\pi$
$98$	−2.32556	−0.234917
$99$	−18.4598	−1.85528
$100$	0	0
$101$	0.963333	0.0958553	0.0479276	−	0.998851i	$-0.484738\pi$
$101$	0.963333	0.0958553	0.0479276	+	0.998851i	$0.484738\pi$
$102$	0	0
$103$	13.1842	1.29908	0.649539	−	0.760329i	$-0.274962\pi$
$103$	13.1842	1.29908	0.649539	+	0.760329i	$0.274962\pi$
$104$	10.9147	1.07027
$105$	0	0
$106$	−0.398630	−0.0387184
$107$	1.20156	0.116159	0.0580797	−	0.998312i	$-0.481502\pi$
$107$	1.20156	0.116159	0.0580797	+	0.998312i	$0.481502\pi$
$108$	−46.3224	−4.45738
$109$	18.8394	1.80449	0.902243	−	0.431227i	$-0.141919\pi$
$109$	18.8394	1.80449	0.902243	+	0.431227i	$0.141919\pi$
$110$	0	0
$111$	11.4341	1.08528
$112$	6.53436	0.617439
$113$	−11.1118	−1.04531	−0.522656	−	0.852544i	$-0.675059\pi$
$113$	−11.1118	−1.04531	−0.522656	+	0.852544i	$0.675059\pi$
$114$	47.2775	4.42794
$115$	0	0
$116$	2.67768	0.248617
$117$	17.7547	1.64143
$118$	−29.5331	−2.71874
$119$	0	0
$120$	0	0
$121$	−3.85537	−0.350488
$122$	1.63520	0.148044
$123$	26.1153	2.35474
$124$	−15.0437	−1.35096
$125$	0	0
$126$	40.7344	3.62891
$127$	−1.62949	−0.144594	−0.0722970	−	0.997383i	$-0.523033\pi$
$127$	−1.62949	−0.144594	−0.0722970	+	0.997383i	$0.523033\pi$
$128$	−20.6968	−1.82935
$129$	−40.1777	−3.53745
$130$	0	0
$131$	−14.1257	−1.23417	−0.617083	−	0.786898i	$-0.711686\pi$
$131$	−14.1257	−1.23417	−0.617083	+	0.786898i	$0.711686\pi$
$132$	31.6978	2.75894
$133$	−15.3609	−1.33196
$134$	3.61093	0.311937
$135$	0	0
$136$	0	0
$137$	−12.0396	−1.02861	−0.514305	−	0.857607i	$-0.671950\pi$
$137$	−12.0396	−1.02861	−0.514305	+	0.857607i	$0.671950\pi$
$138$	23.5439	2.00419
$139$	12.7907	1.08489	0.542447	−	0.840090i	$-0.317498\pi$
$139$	12.7907	1.08489	0.542447	+	0.840090i	$0.317498\pi$
$140$	0	0
$141$	8.76448	0.738103
$142$	6.03709	0.506622
$143$	−6.87173	−0.574643
$144$	18.3748	1.53123
$145$	0	0
$146$	19.8130	1.63974
$147$	3.04773	0.251372
$148$	−13.6879	−1.12514
$149$	5.52099	0.452297	0.226149	−	0.974093i	$-0.427386\pi$
$149$	5.52099	0.452297	0.226149	+	0.974093i	$0.427386\pi$
$150$	0	0
$151$	−20.1224	−1.63754	−0.818769	−	0.574123i	$-0.805343\pi$
$151$	−20.1224	−1.63754	−0.818769	+	0.574123i	$0.805343\pi$
$152$	−26.5540	−2.15382
$153$	0	0
$154$	−15.7657	−1.27044
$155$	0	0
$156$	−30.4871	−2.44092
$157$	−6.77751	−0.540904	−0.270452	−	0.962733i	$-0.587173\pi$
$157$	−6.77751	−0.540904	−0.270452	+	0.962733i	$0.587173\pi$
$158$	−10.7589	−0.855936
$159$	0.522418	0.0414305
$160$	0	0
$161$	−7.64961	−0.602874
$162$	43.1734	3.39202
$163$	15.3503	1.20233	0.601165	−	0.799125i	$-0.294704\pi$
$163$	15.3503	1.20233	0.601165	+	0.799125i	$0.294704\pi$
$164$	−31.2629	−2.44122
$165$	0	0
$166$	−29.1213	−2.26025
$167$	−25.4830	−1.97193	−0.985965	−	0.166950i	$-0.946608\pi$
$167$	−25.4830	−1.97193	−0.985965	+	0.166950i	$0.946608\pi$
$168$	−32.8175	−2.53193
$169$	−6.39074	−0.491596
$170$	0	0
$171$	−43.1951	−3.30321
$172$	48.0970	3.66736
$173$	4.03225	0.306566	0.153283	−	0.988182i	$-0.451015\pi$
$173$	4.03225	0.306566	0.153283	+	0.988182i	$0.451015\pi$
$174$	−5.37193	−0.407245
$175$	0	0
$176$	−7.11172	−0.536066
$177$	38.7042	2.90918
$178$	2.80286	0.210083
$179$	−21.2384	−1.58743	−0.793717	−	0.608287i	$-0.791857\pi$
$179$	−21.2384	−1.58743	−0.793717	+	0.608287i	$0.791857\pi$
$180$	0	0
$181$	11.0451	0.820973	0.410486	−	0.911867i	$-0.365359\pi$
$181$	11.0451	0.820973	0.410486	+	0.911867i	$0.365359\pi$
$182$	15.1635	1.12399
$183$	−2.14298	−0.158414
$184$	−13.2237	−0.974867
$185$	0	0
$186$	30.1804	2.21294
$187$	0	0
$188$	−10.4920	−0.765210
$189$	−30.1942	−2.19631
$190$	0	0
$191$	−11.9621	−0.865544	−0.432772	−	0.901503i	$-0.642464\pi$
$191$	−11.9621	−0.865544	−0.432772	+	0.901503i	$0.642464\pi$
$192$	32.6314	2.35497
$193$	0.567470	0.0408474	0.0204237	−	0.999791i	$-0.493498\pi$
$193$	0.567470	0.0408474	0.0204237	+	0.999791i	$0.493498\pi$
$194$	7.50778	0.539027
$195$	0	0
$196$	−3.64846	−0.260604
$197$	8.93316	0.636461	0.318231	−	0.948013i	$-0.396911\pi$
$197$	8.93316	0.636461	0.318231	+	0.948013i	$0.396911\pi$
$198$	−44.3335	−3.15065
$199$	−16.6697	−1.18168	−0.590840	−	0.806789i	$-0.701204\pi$
$199$	−16.6697	−1.18168	−0.590840	+	0.806789i	$0.701204\pi$
$200$	0	0
$201$	−4.73225	−0.333787
$202$	2.31356	0.162782
$203$	1.74539	0.122502
$204$	0	0
$205$	0	0
$206$	31.6634	2.20610
$207$	−21.5109	−1.49511
$208$	6.84007	0.474274
$209$	16.7181	1.15642
$210$	0	0
$211$	19.6063	1.34975	0.674876	−	0.737931i	$-0.264197\pi$
$211$	19.6063	1.34975	0.674876	+	0.737931i	$0.264197\pi$
$212$	−0.625391	−0.0429520
$213$	−7.91182	−0.542109
$214$	2.88570	0.197262
$215$	0	0
$216$	−52.1962	−3.55150
$217$	−9.80588	−0.665666
$218$	45.2451	3.06438
$219$	−25.9657	−1.75460
$220$	0	0
$221$	0	0
$222$	27.4604	1.84302
$223$	−17.7907	−1.19135	−0.595677	−	0.803224i	$-0.703116\pi$
$223$	−17.7907	−1.19135	−0.595677	+	0.803224i	$0.703116\pi$
$224$	−5.16061	−0.344808
$225$	0	0
$226$	−26.6864	−1.77515
$227$	2.74184	0.181982	0.0909911	−	0.995852i	$-0.470997\pi$
$227$	2.74184	0.181982	0.0909911	+	0.995852i	$0.470997\pi$
$228$	74.1713	4.91212
$229$	28.4081	1.87726	0.938630	−	0.344925i	$-0.112096\pi$
$229$	28.4081	1.87726	0.938630	+	0.344925i	$0.112096\pi$
$230$	0	0
$231$	20.6615	1.35943
$232$	3.01722	0.198090
$233$	−14.7458	−0.966032	−0.483016	−	0.875611i	$-0.660459\pi$
$233$	−14.7458	−0.966032	−0.483016	+	0.875611i	$0.660459\pi$
$234$	42.6401	2.78747
$235$	0	0
$236$	−46.3330	−3.01602
$237$	14.1000	0.915892
$238$	0	0
$239$	8.93034	0.577656	0.288828	−	0.957381i	$-0.406734\pi$
$239$	8.93034	0.577656	0.288828	+	0.957381i	$0.406734\pi$
$240$	0	0
$241$	−25.4005	−1.63619	−0.818095	−	0.575083i	$-0.804970\pi$
$241$	−25.4005	−1.63619	−0.818095	+	0.575083i	$0.804970\pi$
$242$	−9.25913	−0.595199
$243$	−19.6972	−1.26358
$244$	2.56538	0.164231
$245$	0	0
$246$	62.7192	3.99883
$247$	−16.0795	−1.02312
$248$	−16.9512	−1.07640
$249$	38.1645	2.41858
$250$	0	0
$251$	−0.459279	−0.0289894	−0.0144947	−	0.999895i	$-0.504614\pi$
$251$	−0.459279	−0.0289894	−0.0144947	+	0.999895i	$0.504614\pi$
$252$	63.9062	4.02571
$253$	8.32550	0.523420
$254$	−3.91342	−0.245550
$255$	0	0
$256$	−28.9704	−1.81065
$257$	−18.6397	−1.16271	−0.581357	−	0.813648i	$-0.697478\pi$
$257$	−18.6397	−1.16271	−0.581357	+	0.813648i	$0.697478\pi$
$258$	−96.4916	−6.00730
$259$	−8.92213	−0.554394
$260$	0	0
$261$	4.90807	0.303802
$262$	−33.9245	−2.09586
$263$	16.5539	1.02076	0.510379	−	0.859950i	$-0.329505\pi$
$263$	16.5539	1.02076	0.510379	+	0.859950i	$0.329505\pi$
$264$	35.7172	2.19824
$265$	0	0
$266$	−36.8910	−2.26193
$267$	−3.67325	−0.224799
$268$	5.66501	0.346046
$269$	−7.70877	−0.470012	−0.235006	−	0.971994i	$-0.575511\pi$
$269$	−7.70877	−0.470012	−0.235006	+	0.971994i	$0.575511\pi$
$270$	0	0
$271$	−21.0971	−1.28156	−0.640780	−	0.767725i	$-0.721389\pi$
$271$	−21.0971	−1.28156	−0.640780	+	0.767725i	$0.721389\pi$
$272$	0	0
$273$	−19.8723	−1.20273
$274$	−28.9145	−1.74679
$275$	0	0
$276$	36.9368	2.22334
$277$	6.30787	0.379003	0.189501	−	0.981880i	$-0.439313\pi$
$277$	6.30787	0.379003	0.189501	+	0.981880i	$0.439313\pi$
$278$	30.7184	1.84237
$279$	−27.5744	−1.65083
$280$	0	0
$281$	−23.6264	−1.40943	−0.704716	−	0.709490i	$-0.748925\pi$
$281$	−23.6264	−1.40943	−0.704716	+	0.709490i	$0.748925\pi$
$282$	21.0490	1.25345
$283$	−19.3219	−1.14857	−0.574283	−	0.818657i	$-0.694719\pi$
$283$	−19.3219	−1.14857	−0.574283	+	0.818657i	$0.694719\pi$
$284$	9.47130	0.562018
$285$	0	0
$286$	−16.5033	−0.975861
$287$	−20.3780	−1.20288
$288$	−14.5118	−0.855113
$289$	0	0
$290$	0	0
$291$	−9.83920	−0.576784
$292$	31.0837	1.81903
$293$	11.1159	0.649397	0.324698	−	0.945818i	$-0.394737\pi$
$293$	11.1159	0.649397	0.324698	+	0.945818i	$0.394737\pi$
$294$	7.31949	0.426881
$295$	0	0
$296$	−15.4235	−0.896474
$297$	32.8621	1.90685
$298$	13.2593	0.768093
$299$	−8.00750	−0.463086
$300$	0	0
$301$	31.3509	1.80704
$302$	−48.3264	−2.78087
$303$	−3.03200	−0.174184
$304$	−16.6411	−0.954431
$305$	0	0
$306$	0	0
$307$	8.08367	0.461359	0.230680	−	0.973030i	$-0.425905\pi$
$307$	8.08367	0.461359	0.230680	+	0.973030i	$0.425905\pi$
$308$	−24.7340	−1.40935
$309$	−41.4960	−2.36063
$310$	0	0
$311$	13.5612	0.768988	0.384494	−	0.923128i	$-0.374376\pi$
$311$	13.5612	0.768988	0.384494	+	0.923128i	$0.374376\pi$
$312$	−34.3529	−1.94485
$313$	−9.12199	−0.515605	−0.257803	−	0.966198i	$-0.582998\pi$
$313$	−9.12199	−0.515605	−0.257803	+	0.966198i	$0.582998\pi$
$314$	−16.2770	−0.918565
$315$	0	0
$316$	−16.8792	−0.949528
$317$	33.7435	1.89522	0.947611	−	0.319426i	$-0.103490\pi$
$317$	33.7435	1.89522	0.947611	+	0.319426i	$0.103490\pi$
$318$	1.25465	0.0703573
$319$	−1.89960	−0.106357
$320$	0	0
$321$	−3.78181	−0.211080
$322$	−18.3715	−1.02380
$323$	0	0
$324$	67.7326	3.76292
$325$	0	0
$326$	36.8656	2.04180
$327$	−59.2953	−3.27904
$328$	−35.2271	−1.94509
$329$	−6.83899	−0.377046
$330$	0	0
$331$	−21.2983	−1.17066	−0.585329	−	0.810796i	$-0.699035\pi$
$331$	−21.2983	−1.17066	−0.585329	+	0.810796i	$0.699035\pi$
$332$	−45.6870	−2.50740
$333$	−25.0892	−1.37488
$334$	−61.2004	−3.34874
$335$	0	0
$336$	−20.5663	−1.12198
$337$	−22.4073	−1.22061	−0.610303	−	0.792168i	$-0.708952\pi$
$337$	−22.4073	−1.22061	−0.610303	+	0.792168i	$0.708952\pi$
$338$	−15.3481	−0.834829
$339$	34.9734	1.89949
$340$	0	0
$341$	10.6723	0.577937
$342$	−103.738	−5.60953
$343$	−19.5698	−1.05667
$344$	54.1958	2.92204
$345$	0	0
$346$	9.68393	0.520611
$347$	−18.0764	−0.970393	−0.485197	−	0.874405i	$-0.661252\pi$
$347$	−18.0764	−0.970393	−0.485197	+	0.874405i	$0.661252\pi$
$348$	−8.42776	−0.451775
$349$	−9.00587	−0.482073	−0.241037	−	0.970516i	$-0.577487\pi$
$349$	−9.00587	−0.482073	−0.241037	+	0.970516i	$0.577487\pi$
$350$	0	0
$351$	−31.6069	−1.68705
$352$	5.61658	0.299365
$353$	27.4982	1.46358	0.731791	−	0.681530i	$-0.238685\pi$
$353$	27.4982	1.46358	0.731791	+	0.681530i	$0.238685\pi$
$354$	92.9527	4.94038
$355$	0	0
$356$	4.39727	0.233055
$357$	0	0
$358$	−51.0066	−2.69578
$359$	24.2551	1.28013	0.640066	−	0.768320i	$-0.278907\pi$
$359$	24.2551	1.28013	0.640066	+	0.768320i	$0.278907\pi$
$360$	0	0
$361$	20.1195	1.05892
$362$	26.5261	1.39418
$363$	12.1344	0.636891
$364$	23.7893	1.24690
$365$	0	0
$366$	−5.14663	−0.269019
$367$	−25.1930	−1.31506	−0.657532	−	0.753427i	$-0.728399\pi$
$367$	−25.1930	−1.31506	−0.657532	+	0.753427i	$0.728399\pi$
$368$	−8.28715	−0.431997
$369$	−57.3034	−2.98310
$370$	0	0
$371$	−0.407647	−0.0211640
$372$	47.3486	2.45491
$373$	12.3570	0.639823	0.319911	−	0.947447i	$-0.396347\pi$
$373$	12.3570	0.639823	0.319911	+	0.947447i	$0.396347\pi$
$374$	0	0
$375$	0	0
$376$	−11.8224	−0.609696
$377$	1.82705	0.0940976
$378$	−72.5151	−3.72977
$379$	1.73645	0.0891957	0.0445978	−	0.999005i	$-0.485799\pi$
$379$	1.73645	0.0891957	0.0445978	+	0.999005i	$0.485799\pi$
$380$	0	0
$381$	5.12867	0.262750
$382$	−28.7283	−1.46987
$383$	−17.5969	−0.899158	−0.449579	−	0.893241i	$-0.648426\pi$
$383$	−17.5969	−0.899158	−0.449579	+	0.893241i	$0.648426\pi$
$384$	65.1412	3.32422
$385$	0	0
$386$	1.36285	0.0693671
$387$	88.1596	4.48140
$388$	11.7786	0.597967
$389$	−11.0817	−0.561866	−0.280933	−	0.959727i	$-0.590644\pi$
$389$	−11.0817	−0.561866	−0.280933	+	0.959727i	$0.590644\pi$
$390$	0	0
$391$	0	0
$392$	−4.11109	−0.207641
$393$	44.4593	2.24267
$394$	21.4541	1.08084
$395$	0	0
$396$	−69.5527	−3.49515
$397$	−8.03226	−0.403128	−0.201564	−	0.979475i	$-0.564602\pi$
$397$	−8.03226	−0.403128	−0.201564	+	0.979475i	$0.564602\pi$
$398$	−40.0342	−2.00673
$399$	48.3469	2.42037
$400$	0	0
$401$	10.9609	0.547360	0.273680	−	0.961821i	$-0.411759\pi$
$401$	10.9609	0.547360	0.273680	+	0.961821i	$0.411759\pi$
$402$	−11.3651	−0.566838
$403$	−10.2646	−0.511319
$404$	3.62963	0.180581
$405$	0	0
$406$	4.19176	0.208034
$407$	9.71046	0.481330
$408$	0	0
$409$	8.43948	0.417306	0.208653	−	0.977990i	$-0.433092\pi$
$409$	8.43948	0.417306	0.208653	+	0.977990i	$0.433092\pi$
$410$	0	0
$411$	37.8935	1.86915
$412$	49.6752	2.44732
$413$	−30.2011	−1.48610
$414$	−51.6610	−2.53900
$415$	0	0
$416$	−5.40205	−0.264857
$417$	−40.2576	−1.97142
$418$	40.1505	1.96383
$419$	14.4828	0.707533	0.353767	−	0.935334i	$-0.384901\pi$
$419$	14.4828	0.707533	0.353767	+	0.935334i	$0.384901\pi$
$420$	0	0
$421$	1.51690	0.0739292	0.0369646	−	0.999317i	$-0.488231\pi$
$421$	1.51690	0.0739292	0.0369646	+	0.999317i	$0.488231\pi$
$422$	47.0869	2.29215
$423$	−19.2314	−0.935063
$424$	−0.704692	−0.0342229
$425$	0	0
$426$	−19.0012	−0.920611
$427$	1.67218	0.0809226
$428$	4.52723	0.218832
$429$	21.6282	1.04422
$430$	0	0
$431$	19.7875	0.953129	0.476564	−	0.879140i	$-0.341882\pi$
$431$	19.7875	0.953129	0.476564	+	0.879140i	$0.341882\pi$
$432$	−32.7107	−1.57379
$433$	−9.49722	−0.456407	−0.228204	−	0.973613i	$-0.573285\pi$
$433$	−9.49722	−0.456407	−0.228204	+	0.973613i	$0.573285\pi$
$434$	−23.5500	−1.13044
$435$	0	0
$436$	70.9828	3.39946
$437$	19.4813	0.931916
$438$	−62.3597	−2.97966
$439$	−20.0395	−0.956433	−0.478217	−	0.878242i	$-0.658717\pi$
$439$	−20.0395	−0.956433	−0.478217	+	0.878242i	$0.658717\pi$
$440$	0	0
$441$	−6.68745	−0.318450
$442$	0	0
$443$	13.1946	0.626894	0.313447	−	0.949606i	$-0.398516\pi$
$443$	13.1946	0.626894	0.313447	+	0.949606i	$0.398516\pi$
$444$	43.0813	2.04455
$445$	0	0
$446$	−42.7265	−2.02316
$447$	−17.3768	−0.821895
$448$	−25.4626	−1.20299
$449$	16.1247	0.760973	0.380486	−	0.924786i	$-0.375757\pi$
$449$	16.1247	0.760973	0.380486	+	0.924786i	$0.375757\pi$
$450$	0	0
$451$	22.1785	1.04435
$452$	−41.8669	−1.96925
$453$	63.3334	2.97566
$454$	6.58486	0.309043
$455$	0	0
$456$	83.5764	3.91382
$457$	8.78267	0.410836	0.205418	−	0.978674i	$-0.434145\pi$
$457$	8.78267	0.410836	0.205418	+	0.978674i	$0.434145\pi$
$458$	68.2255	3.18797
$459$	0	0
$460$	0	0
$461$	18.3746	0.855789	0.427895	−	0.903829i	$-0.359255\pi$
$461$	18.3746	0.855789	0.427895	+	0.903829i	$0.359255\pi$
$462$	49.6211	2.30858
$463$	29.1662	1.35547	0.677735	−	0.735306i	$-0.262962\pi$
$463$	29.1662	1.35547	0.677735	+	0.735306i	$0.262962\pi$
$464$	1.89085	0.0877806
$465$	0	0
$466$	−35.4139	−1.64052
$467$	8.21883	0.380322	0.190161	−	0.981753i	$-0.439099\pi$
$467$	8.21883	0.380322	0.190161	+	0.981753i	$0.439099\pi$
$468$	66.8960	3.09227
$469$	3.69261	0.170509
$470$	0	0
$471$	21.3316	0.982908
$472$	−52.2082	−2.40307
$473$	−34.1210	−1.56889
$474$	33.8628	1.55537
$475$	0	0
$476$	0	0
$477$	−1.14631	−0.0524861
$478$	21.4473	0.980977
$479$	36.3883	1.66262	0.831312	−	0.555807i	$-0.187590\pi$
$479$	36.3883	1.66262	0.831312	+	0.555807i	$0.187590\pi$
$480$	0	0
$481$	−9.33955	−0.425847
$482$	−61.0024	−2.77858
$483$	24.0765	1.09552
$484$	−14.5262	−0.660281
$485$	0	0
$486$	−47.3052	−2.14581
$487$	−33.0821	−1.49909	−0.749547	−	0.661951i	$-0.769729\pi$
$487$	−33.0821	−1.49909	−0.749547	+	0.661951i	$0.769729\pi$
$488$	2.89067	0.130855
$489$	−48.3137	−2.18482
$490$	0	0
$491$	12.1162	0.546797	0.273398	−	0.961901i	$-0.411852\pi$
$491$	12.1162	0.546797	0.273398	+	0.961901i	$0.411852\pi$
$492$	98.3970	4.43608
$493$	0	0
$494$	−38.6169	−1.73746
$495$	0	0
$496$	−10.6231	−0.476992
$497$	6.17365	0.276926
$498$	91.6567	4.10723
$499$	−4.93983	−0.221137	−0.110568	−	0.993869i	$-0.535267\pi$
$499$	−4.93983	−0.221137	−0.110568	+	0.993869i	$0.535267\pi$
$500$	0	0
$501$	80.2053	3.58331
$502$	−1.10301	−0.0492299
$503$	−22.7414	−1.01399	−0.506995	−	0.861949i	$-0.669244\pi$
$503$	−22.7414	−1.01399	−0.506995	+	0.861949i	$0.669244\pi$
$504$	72.0096	3.20756
$505$	0	0
$506$	19.9947	0.888873
$507$	20.1143	0.893306
$508$	−6.13957	−0.272399
$509$	−28.2780	−1.25340	−0.626701	−	0.779260i	$-0.715595\pi$
$509$	−28.2780	−1.25340	−0.626701	+	0.779260i	$0.715595\pi$
$510$	0	0
$511$	20.2612	0.896303
$512$	−28.1824	−1.24550
$513$	76.8957	3.39503
$514$	−44.7656	−1.97452
$515$	0	0
$516$	−151.381	−6.66417
$517$	7.44326	0.327354
$518$	−21.4276	−0.941474
$519$	−12.6911	−0.557079
$520$	0	0
$521$	−15.0564	−0.659632	−0.329816	−	0.944045i	$-0.606987\pi$
$521$	−15.0564	−0.659632	−0.329816	+	0.944045i	$0.606987\pi$
$522$	11.7873	0.515917
$523$	−15.3033	−0.669165	−0.334583	−	0.942366i	$-0.608595\pi$
$523$	−15.3033	−0.669165	−0.334583	+	0.942366i	$0.608595\pi$
$524$	−53.2225	−2.32504
$525$	0	0
$526$	39.7562	1.73345
$527$	0	0
$528$	22.3835	0.974116
$529$	−13.2984	−0.578193
$530$	0	0
$531$	−84.9263	−3.68549
$532$	−57.8764	−2.50926
$533$	−21.3314	−0.923965
$534$	−8.82175	−0.381755
$535$	0	0
$536$	6.38335	0.275719
$537$	66.8460	2.88462
$538$	−18.5136	−0.798176
$539$	2.58829	0.111486
$540$	0	0
$541$	−5.89759	−0.253557	−0.126779	−	0.991931i	$-0.540464\pi$
$541$	−5.89759	−0.253557	−0.126779	+	0.991931i	$0.540464\pi$
$542$	−50.6673	−2.17635
$543$	−34.7633	−1.49184
$544$	0	0
$545$	0	0
$546$	−47.7257	−2.04247
$547$	13.6997	0.585758	0.292879	−	0.956149i	$-0.405387\pi$
$547$	13.6997	0.585758	0.292879	+	0.956149i	$0.405387\pi$
$548$	−45.3625	−1.93779
$549$	4.70222	0.200686
$550$	0	0
$551$	−4.44498	−0.189363
$552$	41.6205	1.77149
$553$	−11.0023	−0.467866
$554$	15.1491	0.643624
$555$	0	0
$556$	48.1926	2.04382
$557$	−26.3831	−1.11789	−0.558943	−	0.829206i	$-0.688793\pi$
$557$	−26.3831	−1.11789	−0.558943	+	0.829206i	$0.688793\pi$
$558$	−66.2232	−2.80345
$559$	32.8177	1.38804
$560$	0	0
$561$	0	0
$562$	−56.7416	−2.39350
$563$	22.3909	0.943666	0.471833	−	0.881688i	$-0.343593\pi$
$563$	22.3909	0.943666	0.471833	+	0.881688i	$0.343593\pi$
$564$	33.0227	1.39051
$565$	0	0
$566$	−46.4038	−1.95050
$567$	44.1500	1.85412
$568$	10.6723	0.447799
$569$	1.33840	0.0561085	0.0280542	−	0.999606i	$-0.491069\pi$
$569$	1.33840	0.0561085	0.0280542	+	0.999606i	$0.491069\pi$
$570$	0	0
$571$	−29.2226	−1.22293	−0.611464	−	0.791272i	$-0.709419\pi$
$571$	−29.2226	−1.22293	−0.611464	+	0.791272i	$0.709419\pi$
$572$	−25.8912	−1.08257
$573$	37.6495	1.57283
$574$	−48.9402	−2.04273
$575$	0	0
$576$	−71.6013	−2.98339
$577$	−14.0956	−0.586809	−0.293405	−	0.955988i	$-0.594788\pi$
$577$	−14.0956	−0.586809	−0.293405	+	0.955988i	$0.594788\pi$
$578$	0	0
$579$	−1.78606	−0.0742261
$580$	0	0
$581$	−29.7800	−1.23548
$582$	−23.6300	−0.979497
$583$	0.443665	0.0183747
$584$	35.0252	1.44935
$585$	0	0
$586$	26.6961	1.10281
$587$	2.14440	0.0885089	0.0442545	−	0.999020i	$-0.485909\pi$
$587$	2.14440	0.0885089	0.0442545	+	0.999020i	$0.485909\pi$
$588$	11.4832	0.473558
$589$	24.9727	1.02898
$590$	0	0
$591$	−28.1163	−1.15655
$592$	−9.66572	−0.397259
$593$	−20.0106	−0.821736	−0.410868	−	0.911695i	$-0.634774\pi$
$593$	−20.0106	−0.821736	−0.410868	+	0.911695i	$0.634774\pi$
$594$	78.9223	3.23822
$595$	0	0
$596$	20.8019	0.852079
$597$	52.4662	2.14730
$598$	−19.2310	−0.786413
$599$	18.2512	0.745724	0.372862	−	0.927887i	$-0.378377\pi$
$599$	18.2512	0.745724	0.372862	+	0.927887i	$0.378377\pi$
$600$	0	0
$601$	11.2285	0.458020	0.229010	−	0.973424i	$-0.426451\pi$
$601$	11.2285	0.458020	0.229010	+	0.973424i	$0.426451\pi$
$602$	75.2931	3.06872
$603$	10.3837	0.422857
$604$	−75.8169	−3.08495
$605$	0	0
$606$	−7.28172	−0.295800
$607$	−25.4461	−1.03282	−0.516412	−	0.856340i	$-0.672733\pi$
$607$	−25.4461	−1.03282	−0.516412	+	0.856340i	$0.672733\pi$
$608$	13.1425	0.533000
$609$	−5.49345	−0.222606
$610$	0	0
$611$	−7.15895	−0.289620
$612$	0	0
$613$	38.4941	1.55476	0.777380	−	0.629031i	$-0.216548\pi$
$613$	38.4941	1.55476	0.777380	+	0.629031i	$0.216548\pi$
$614$	19.4139	0.783482
$615$	0	0
$616$	−27.8704	−1.12293
$617$	23.0835	0.929305	0.464653	−	0.885493i	$-0.346179\pi$
$617$	23.0835	0.929305	0.464653	+	0.885493i	$0.346179\pi$
$618$	−99.6578	−4.00882
$619$	−7.27540	−0.292423	−0.146211	−	0.989253i	$-0.546708\pi$
$619$	−7.27540	−0.292423	−0.146211	+	0.989253i	$0.546708\pi$
$620$	0	0
$621$	38.2936	1.53667
$622$	32.5690	1.30590
$623$	2.86626	0.114834
$624$	−21.5285	−0.861830
$625$	0	0
$626$	−21.9076	−0.875602
$627$	−52.6187	−2.10139
$628$	−25.5362	−1.01901
$629$	0	0
$630$	0	0
$631$	33.2465	1.32352	0.661761	−	0.749715i	$-0.269809\pi$
$631$	33.2465	1.32352	0.661761	+	0.749715i	$0.269809\pi$
$632$	−19.0195	−0.756555
$633$	−61.7090	−2.45271
$634$	81.0390	3.21847
$635$	0	0
$636$	1.96836	0.0780505
$637$	−2.48943	−0.0986347
$638$	−4.56213	−0.180616
$639$	17.3605	0.686769
$640$	0	0
$641$	21.4007	0.845276	0.422638	−	0.906299i	$-0.361104\pi$
$641$	21.4007	0.845276	0.422638	+	0.906299i	$0.361104\pi$
$642$	−9.08247	−0.358457
$643$	12.3392	0.486610	0.243305	−	0.969950i	$-0.421768\pi$
$643$	12.3392	0.486610	0.243305	+	0.969950i	$0.421768\pi$
$644$	−28.8221	−1.13575
$645$	0	0
$646$	0	0
$647$	2.48180	0.0975696	0.0487848	−	0.998809i	$-0.484465\pi$
$647$	2.48180	0.0975696	0.0487848	+	0.998809i	$0.484465\pi$
$648$	76.3212	2.99818
$649$	32.8696	1.29024
$650$	0	0
$651$	30.8631	1.20962
$652$	57.8367	2.26506
$653$	24.2220	0.947882	0.473941	−	0.880557i	$-0.342831\pi$
$653$	24.2220	0.947882	0.473941	+	0.880557i	$0.342831\pi$
$654$	−142.405	−5.56847
$655$	0	0
$656$	−22.0763	−0.861936
$657$	56.9750	2.22281
$658$	−16.4247	−0.640300
$659$	−0.298916	−0.0116441	−0.00582205	−	0.999983i	$-0.501853\pi$
$659$	−0.298916	−0.0116441	−0.00582205	+	0.999983i	$0.501853\pi$
$660$	0	0
$661$	45.7248	1.77849	0.889244	−	0.457434i	$-0.151231\pi$
$661$	45.7248	1.77849	0.889244	+	0.457434i	$0.151231\pi$
$662$	−51.1504	−1.98802
$663$	0	0
$664$	−51.4802	−1.99782
$665$	0	0
$666$	−60.2549	−2.33483
$667$	−2.21357	−0.0857099
$668$	−96.0143	−3.71491
$669$	55.9946	2.16488
$670$	0	0
$671$	−1.81993	−0.0702577
$672$	16.2425	0.626570
$673$	24.6536	0.950325	0.475162	−	0.879898i	$-0.342389\pi$
$673$	24.6536	0.950325	0.475162	+	0.879898i	$0.342389\pi$
$674$	−53.8139	−2.07284
$675$	0	0
$676$	−24.0789	−0.926113
$677$	−34.2463	−1.31619	−0.658097	−	0.752933i	$-0.728638\pi$
$677$	−34.2463	−1.31619	−0.658097	+	0.752933i	$0.728638\pi$
$678$	83.9929	3.22573
$679$	7.67760	0.294639
$680$	0	0
$681$	−8.62969	−0.330690
$682$	25.6308	0.981454
$683$	46.3391	1.77312	0.886558	−	0.462618i	$-0.153090\pi$
$683$	46.3391	1.77312	0.886558	+	0.462618i	$0.153090\pi$
$684$	−162.750	−6.22290
$685$	0	0
$686$	−46.9992	−1.79444
$687$	−89.4119	−3.41128
$688$	33.9638	1.29486
$689$	−0.426719	−0.0162567
$690$	0	0
$691$	25.0441	0.952724	0.476362	−	0.879249i	$-0.341955\pi$
$691$	25.0441	0.952724	0.476362	+	0.879249i	$0.341955\pi$
$692$	15.1926	0.577537
$693$	−45.3364	−1.72219
$694$	−43.4127	−1.64792
$695$	0	0
$696$	−9.49642	−0.359961
$697$	0	0
$698$	−21.6287	−0.818658
$699$	46.4112	1.75543
$700$	0	0
$701$	−45.8247	−1.73078	−0.865388	−	0.501102i	$-0.832928\pi$
$701$	−45.8247	−1.73078	−0.865388	+	0.501102i	$0.832928\pi$
$702$	−75.9077	−2.86495
$703$	22.7220	0.856977
$704$	27.7123	1.04445
$705$	0	0
$706$	66.0402	2.48546
$707$	2.36589	0.0889786
$708$	145.829	5.48059
$709$	19.9724	0.750078	0.375039	−	0.927009i	$-0.377629\pi$
$709$	19.9724	0.750078	0.375039	+	0.927009i	$0.377629\pi$
$710$	0	0
$711$	−30.9388	−1.16029
$712$	4.95486	0.185691
$713$	12.4362	0.465740
$714$	0	0
$715$	0	0
$716$	−80.0218	−2.99055
$717$	−28.1074	−1.04969
$718$	58.2514	2.17392
$719$	−33.6934	−1.25655	−0.628277	−	0.777990i	$-0.716239\pi$
$719$	−33.6934	−1.25655	−0.628277	+	0.777990i	$0.716239\pi$
$720$	0	0
$721$	32.3797	1.20588
$722$	48.3195	1.79827
$723$	79.9457	2.97321
$724$	41.6154	1.54662
$725$	0	0
$726$	29.1423	1.08157
$727$	6.28209	0.232990	0.116495	−	0.993191i	$-0.462834\pi$
$727$	6.28209	0.232990	0.116495	+	0.993191i	$0.462834\pi$
$728$	26.8058	0.993489
$729$	8.06486	0.298699
$730$	0	0
$731$	0	0
$732$	−8.07429	−0.298434
$733$	−10.1710	−0.375675	−0.187837	−	0.982200i	$-0.560148\pi$
$733$	−10.1710	−0.375675	−0.187837	+	0.982200i	$0.560148\pi$
$734$	−60.5040	−2.23324
$735$	0	0
$736$	6.54489	0.241248
$737$	−4.01888	−0.148037
$738$	−137.621	−5.06590
$739$	−14.2869	−0.525552	−0.262776	−	0.964857i	$-0.584638\pi$
$739$	−14.2869	−0.525552	−0.262776	+	0.964857i	$0.584638\pi$
$740$	0	0
$741$	50.6088	1.85916
$742$	−0.979014	−0.0359407
$743$	−21.9933	−0.806855	−0.403428	−	0.915012i	$-0.632181\pi$
$743$	−21.9933	−0.806855	−0.403428	+	0.915012i	$0.632181\pi$
$744$	53.3525	1.95600
$745$	0	0
$746$	29.6769	1.08655
$747$	−83.7422	−3.06397
$748$	0	0
$749$	2.95097	0.107826
$750$	0	0
$751$	−11.9963	−0.437751	−0.218876	−	0.975753i	$-0.570239\pi$
$751$	−11.9963	−0.437751	−0.218876	+	0.975753i	$0.570239\pi$
$752$	−7.40897	−0.270177
$753$	1.44554	0.0526784
$754$	4.38787	0.159797
$755$	0	0
$756$	−113.765	−4.13761
$757$	41.6081	1.51227	0.756135	−	0.654415i	$-0.227085\pi$
$757$	41.6081	1.51227	0.756135	+	0.654415i	$0.227085\pi$
$758$	4.17031	0.151472
$759$	−26.2038	−0.951136
$760$	0	0
$761$	50.6689	1.83675	0.918374	−	0.395715i	$-0.129503\pi$
$761$	50.6689	1.83675	0.918374	+	0.395715i	$0.129503\pi$
$762$	12.3171	0.446203
$763$	46.2685	1.67503
$764$	−45.0705	−1.63059
$765$	0	0
$766$	−42.2610	−1.52695
$767$	−31.6141	−1.14152
$768$	91.1817	3.29024
$769$	36.4245	1.31350	0.656750	−	0.754108i	$-0.271931\pi$
$769$	36.4245	1.31350	0.656750	+	0.754108i	$0.271931\pi$
$770$	0	0
$771$	58.6668	2.11283
$772$	2.13810	0.0769521
$773$	−40.5949	−1.46010	−0.730049	−	0.683395i	$-0.760503\pi$
$773$	−40.5949	−1.46010	−0.730049	+	0.683395i	$0.760503\pi$
$774$	211.726	7.61033
$775$	0	0
$776$	13.2721	0.476442
$777$	28.0816	1.00742
$778$	−26.6141	−0.954163
$779$	51.8967	1.85939
$780$	0	0
$781$	−6.71913	−0.240430
$782$	0	0
$783$	−8.73732	−0.312246
$784$	−2.57637	−0.0920131
$785$	0	0
$786$	106.774	3.80851
$787$	−9.86307	−0.351580	−0.175790	−	0.984428i	$-0.556248\pi$
$787$	−9.86307	−0.351580	−0.175790	+	0.984428i	$0.556248\pi$
$788$	33.6582	1.19902
$789$	−52.1019	−1.85488
$790$	0	0
$791$	−27.2900	−0.970321
$792$	−78.3721	−2.78483
$793$	1.75042	0.0621591
$794$	−19.2904	−0.684592
$795$	0	0
$796$	−62.8076	−2.22616
$797$	39.4185	1.39628	0.698138	−	0.715964i	$-0.254012\pi$
$797$	39.4185	1.39628	0.698138	+	0.715964i	$0.254012\pi$
$798$	116.111	4.11028
$799$	0	0
$800$	0	0
$801$	8.06000	0.284786
$802$	26.3239	0.929528
$803$	−22.0514	−0.778177
$804$	−17.8301	−0.628819
$805$	0	0
$806$	−24.6518	−0.868322
$807$	24.2627	0.854086
$808$	4.08988	0.143881
$809$	9.42731	0.331447	0.165723	−	0.986172i	$-0.447004\pi$
$809$	9.42731	0.331447	0.165723	+	0.986172i	$0.447004\pi$
$810$	0	0
$811$	−22.2187	−0.780203	−0.390101	−	0.920772i	$-0.627560\pi$
$811$	−22.2187	−0.780203	−0.390101	+	0.920772i	$0.627560\pi$
$812$	6.57624	0.230781
$813$	66.4013	2.32880
$814$	23.3208	0.817395
$815$	0	0
$816$	0	0
$817$	−79.8415	−2.79330
$818$	20.2684	0.708670
$819$	43.6047	1.52367
$820$	0	0
$821$	14.1242	0.492938	0.246469	−	0.969151i	$-0.420730\pi$
$821$	14.1242	0.492938	0.246469	+	0.969151i	$0.420730\pi$
$822$	91.0058	3.17419
$823$	47.6084	1.65953	0.829763	−	0.558116i	$-0.188476\pi$
$823$	47.6084	1.65953	0.829763	+	0.558116i	$0.188476\pi$
$824$	55.9741	1.94995
$825$	0	0
$826$	−72.5317	−2.52370
$827$	−4.85003	−0.168652	−0.0843261	−	0.996438i	$-0.526874\pi$
$827$	−4.85003	−0.168652	−0.0843261	+	0.996438i	$0.526874\pi$
$828$	−81.0484	−2.81663
$829$	19.9101	0.691508	0.345754	−	0.938325i	$-0.387623\pi$
$829$	19.9101	0.691508	0.345754	+	0.938325i	$0.387623\pi$
$830$	0	0
$831$	−19.8534	−0.688708
$832$	−26.6538	−0.924055
$833$	0	0
$834$	−96.6835	−3.34787
$835$	0	0
$836$	62.9902	2.17856
$837$	49.0877	1.69672
$838$	34.7823	1.20153
$839$	−1.98935	−0.0686799	−0.0343400	−	0.999410i	$-0.510933\pi$
$839$	−1.98935	−0.0686799	−0.0343400	+	0.999410i	$0.510933\pi$
$840$	0	0
$841$	−28.4949	−0.982584
$842$	3.64302	0.125547
$843$	74.3618	2.56116
$844$	73.8722	2.54279
$845$	0	0
$846$	−46.1866	−1.58793
$847$	−9.46857	−0.325344
$848$	−0.441621	−0.0151653
$849$	60.8138	2.08712
$850$	0	0
$851$	11.3154	0.387887
$852$	−29.8100	−1.02127
$853$	−26.0104	−0.890579	−0.445289	−	0.895387i	$-0.646899\pi$
$853$	−26.0104	−0.890579	−0.445289	+	0.895387i	$0.646899\pi$
$854$	4.01595	0.137423
$855$	0	0
$856$	5.10129	0.174359
$857$	18.9964	0.648904	0.324452	−	0.945902i	$-0.394820\pi$
$857$	18.9964	0.648904	0.324452	+	0.945902i	$0.394820\pi$
$858$	51.9426	1.77329
$859$	15.6479	0.533899	0.266950	−	0.963710i	$-0.413984\pi$
$859$	15.6479	0.533899	0.266950	+	0.963710i	$0.413984\pi$
$860$	0	0
$861$	64.1379	2.18581
$862$	47.5220	1.61861
$863$	44.1746	1.50372	0.751860	−	0.659322i	$-0.229157\pi$
$863$	44.1746	1.50372	0.751860	+	0.659322i	$0.229157\pi$
$864$	25.8337	0.878881
$865$	0	0
$866$	−22.8087	−0.775072
$867$	0	0
$868$	−36.9464	−1.25404
$869$	11.9744	0.406205
$870$	0	0
$871$	3.86537	0.130973
$872$	79.9836	2.70859
$873$	21.5896	0.730697
$874$	46.7867	1.58258
$875$	0	0
$876$	−97.8330	−3.30547
$877$	−9.87744	−0.333538	−0.166769	−	0.985996i	$-0.553333\pi$
$877$	−9.87744	−0.333538	−0.166769	+	0.985996i	$0.553333\pi$
$878$	−48.1273	−1.62422
$879$	−34.9862	−1.18006
$880$	0	0
$881$	51.6330	1.73956	0.869780	−	0.493439i	$-0.164260\pi$
$881$	51.6330	1.73956	0.869780	+	0.493439i	$0.164260\pi$
$882$	−16.0607	−0.540793
$883$	17.8503	0.600710	0.300355	−	0.953827i	$-0.402895\pi$
$883$	17.8503	0.600710	0.300355	+	0.953827i	$0.402895\pi$
$884$	0	0
$885$	0	0
$886$	31.6884	1.06459
$887$	57.3894	1.92695	0.963474	−	0.267802i	$-0.0862974\pi$
$887$	57.3894	1.92695	0.963474	+	0.267802i	$0.0862974\pi$
$888$	48.5441	1.62903
$889$	−4.00194	−0.134221
$890$	0	0
$891$	−48.0509	−1.60977
$892$	−67.0315	−2.24438
$893$	17.4169	0.582833
$894$	−41.7325	−1.39574
$895$	0	0
$896$	−50.8302	−1.69812
$897$	25.2029	0.841499
$898$	38.7255	1.29229
$899$	−2.83753	−0.0946370
$900$	0	0
$901$	0	0
$902$	53.2644	1.77351
$903$	−98.6742	−3.28367
$904$	−47.1757	−1.56904
$905$	0	0
$906$	152.103	5.05328
$907$	7.21531	0.239580	0.119790	−	0.992799i	$-0.461778\pi$
$907$	7.21531	0.239580	0.119790	+	0.992799i	$0.461778\pi$
$908$	10.3307	0.342835
$909$	6.65295	0.220665
$910$	0	0
$911$	−47.1365	−1.56170	−0.780852	−	0.624717i	$-0.785214\pi$
$911$	−47.1365	−1.56170	−0.780852	+	0.624717i	$0.785214\pi$
$912$	52.3763	1.73435
$913$	32.4113	1.07266
$914$	21.0927	0.697683
$915$	0	0
$916$	107.036	3.53656
$917$	−34.6919	−1.14563
$918$	0	0
$919$	−3.10828	−0.102533	−0.0512663	−	0.998685i	$-0.516326\pi$
$919$	−3.10828	−0.102533	−0.0512663	+	0.998685i	$0.516326\pi$
$920$	0	0
$921$	−25.4426	−0.838363
$922$	44.1288	1.45330
$923$	6.46249	0.212715
$924$	77.8481	2.56101
$925$	0	0
$926$	70.0463	2.30186
$927$	91.0524	2.99055
$928$	−1.49333	−0.0490209
$929$	−18.2447	−0.598588	−0.299294	−	0.954161i	$-0.596751\pi$
$929$	−18.2447	−0.598588	−0.299294	+	0.954161i	$0.596751\pi$
$930$	0	0
$931$	6.05647	0.198493
$932$	−55.5592	−1.81990
$933$	−42.6828	−1.39737
$934$	19.7385	0.645864
$935$	0	0
$936$	75.3786	2.46383
$937$	14.3214	0.467860	0.233930	−	0.972253i	$-0.424841\pi$
$937$	14.3214	0.467860	0.233930	+	0.972253i	$0.424841\pi$
$938$	8.86825	0.289559
$939$	28.7106	0.936936
$940$	0	0
$941$	−29.4874	−0.961261	−0.480631	−	0.876923i	$-0.659592\pi$
$941$	−29.4874	−0.961261	−0.480631	+	0.876923i	$0.659592\pi$
$942$	51.2304	1.66918
$943$	25.8442	0.841603
$944$	−32.7182	−1.06489
$945$	0	0
$946$	−81.9457	−2.66429
$947$	−10.2467	−0.332973	−0.166486	−	0.986044i	$-0.553242\pi$
$947$	−10.2467	−0.332973	−0.166486	+	0.986044i	$0.553242\pi$
$948$	53.1257	1.72544
$949$	21.2091	0.688477
$950$	0	0
$951$	−106.204	−3.44392
$952$	0	0
$953$	30.5425	0.989370	0.494685	−	0.869072i	$-0.335283\pi$
$953$	30.5425	0.989370	0.494685	+	0.869072i	$0.335283\pi$
$954$	−2.75301	−0.0891320
$955$	0	0
$956$	33.6476	1.08824
$957$	5.97883	0.193268
$958$	87.3909	2.82347
$959$	−29.5686	−0.954818
$960$	0	0
$961$	−15.0583	−0.485751
$962$	−22.4301	−0.723175
$963$	8.29821	0.267406
$964$	−95.7036	−3.08241
$965$	0	0
$966$	57.8225	1.86041
$967$	50.0287	1.60881	0.804407	−	0.594078i	$-0.202483\pi$
$967$	50.0287	1.60881	0.804407	+	0.594078i	$0.202483\pi$
$968$	−16.3681	−0.526092
$969$	0	0
$970$	0	0
$971$	0.00176375	5.66015e−5 0	2.83007e−5	−	1.00000i	$-0.499991\pi$
$971$	0.00176375	5.66015e−5 0	2.83007e−5		1.00000i	$0.499991\pi$
$972$	−74.2148	−2.38044
$973$	31.4133	1.00706
$974$	−79.4507	−2.54577
$975$	0	0
$976$	1.81155	0.0579862
$977$	15.5052	0.496055	0.248028	−	0.968753i	$-0.420218\pi$
$977$	15.5052	0.496055	0.248028	+	0.968753i	$0.420218\pi$
$978$	−116.031	−3.71027
$979$	−3.11952	−0.0997002
$980$	0	0
$981$	130.108	4.15404
$982$	29.0985	0.928571
$983$	−16.7282	−0.533547	−0.266774	−	0.963759i	$-0.585958\pi$
$983$	−16.7282	−0.533547	−0.266774	+	0.963759i	$0.585958\pi$
$984$	110.874	3.53453
$985$	0	0
$986$	0	0
$987$	21.5251	0.685151
$988$	−60.5842	−1.92744
$989$	−39.7606	−1.26431
$990$	0	0
$991$	−2.57414	−0.0817701	−0.0408851	−	0.999164i	$-0.513018\pi$
$991$	−2.57414	−0.0817701	−0.0408851	+	0.999164i	$0.513018\pi$
$992$	8.38976	0.266375
$993$	67.0343	2.12727
$994$	14.8268	0.470277
$995$	0	0
$996$	143.796	4.55634
$997$	−16.8644	−0.534101	−0.267050	−	0.963683i	$-0.586049\pi$
$997$	−16.8644	−0.534101	−0.267050	+	0.963683i	$0.586049\pi$
$998$	−11.8636	−0.375535
$999$	44.6637	1.41310

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bh.1.6		6
5.4	even	2		1445.2.a.m.1.1	yes	6
17.16	even	2		7225.2.a.bi.1.6		6
85.4	even	4		1445.2.d.h.866.11		12
85.64	even	4		1445.2.d.h.866.12		12
85.84	even	2		1445.2.a.l.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1445.2.a.l.1.1	✓	6	85.84	even	2
1445.2.a.m.1.1	yes	6	5.4	even	2
1445.2.d.h.866.11		12	85.4	even	4
1445.2.d.h.866.12		12	85.64	even	4
7225.2.a.bh.1.6		6	1.1	even	1	trivial
7225.2.a.bi.1.6		6	17.16	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:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.40162 q^{2} -3.14741 q^{3} +3.76778 q^{4} -7.55888 q^{6} +2.45595 q^{7} +4.24555 q^{8} +6.90618 q^{9} -2.67294 q^{11} -11.8588 q^{12} +2.57085 q^{13} +5.89825 q^{14} +2.66063 q^{16} +16.5860 q^{18} -6.25456 q^{19} -7.72986 q^{21} -6.41940 q^{22} -3.11473 q^{23} -13.3625 q^{24} +6.17420 q^{26} -12.2943 q^{27} +9.25348 q^{28} +0.710678 q^{29} -3.99271 q^{31} -2.10127 q^{32} +8.41285 q^{33} +26.0210 q^{36} -3.63287 q^{37} -15.0211 q^{38} -8.09151 q^{39} -8.29741 q^{41} -18.5642 q^{42} +12.7653 q^{43} -10.0711 q^{44} -7.48040 q^{46} -2.78467 q^{47} -8.37409 q^{48} -0.968329 q^{49} +9.68640 q^{52} -0.165984 q^{53} -29.5263 q^{54} +10.4268 q^{56} +19.6857 q^{57} +1.70678 q^{58} -12.2972 q^{59} +0.680871 q^{61} -9.58898 q^{62} +16.9612 q^{63} -10.3677 q^{64} +20.2045 q^{66} +1.50354 q^{67} +9.80333 q^{69} +2.51376 q^{71} +29.3205 q^{72} +8.24985 q^{73} -8.72478 q^{74} -23.5658 q^{76} -6.56461 q^{77} -19.4327 q^{78} -4.47987 q^{79} +17.9768 q^{81} -19.9272 q^{82} -12.1257 q^{83} -29.1245 q^{84} +30.6575 q^{86} -2.23680 q^{87} -11.3481 q^{88} +1.16707 q^{89} +6.31386 q^{91} -11.7356 q^{92} +12.5667 q^{93} -6.68772 q^{94} +6.61356 q^{96} +3.12613 q^{97} -2.32556 q^{98} -18.4598 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 9 q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9} - 6 q^{11} - 6 q^{12} + 9 q^{13} + 18 q^{14} - 3 q^{16} + 15 q^{18} - 21 q^{19} - 12 q^{21} + 3 q^{22} - 18 q^{23} - 6 q^{26} - 12 q^{27}+ \cdots - 18 q^{99}+O(q^{100}) \) 6 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 9 * q^6 - 6 * q^7 + 6 * q^8 + 3 * q^9 - 6 * q^11 - 6 * q^12 + 9 * q^13 + 18 * q^14 - 3 * q^16 + 15 * q^18 - 21 * q^19 - 12 * q^21 + 3 * q^22 - 18 * q^23 - 6 * q^26 - 12 * q^27 + 15 * q^28 - 18 * q^29 + 6 * q^31 + 24 * q^33 + 18 * q^36 + 9 * q^37 - 15 * q^38 - 9 * q^39 - 3 * q^41 - 6 * q^42 + 39 * q^43 - 6 * q^44 - 27 * q^46 - 3 * q^47 - 21 * q^48 + 6 * q^49 - 3 * q^52 + 24 * q^53 - 33 * q^54 - 9 * q^56 + 12 * q^57 - 33 * q^58 - 12 * q^59 - 27 * q^61 + 12 * q^63 - 12 * q^64 + 21 * q^66 + 30 * q^67 + 18 * q^71 + 30 * q^72 + 9 * q^73 - 27 * q^74 - 45 * q^76 + 21 * q^77 - 36 * q^78 - 3 * q^79 + 6 * q^81 - 15 * q^82 + 21 * q^83 - 27 * q^84 + 36 * q^86 - 6 * q^87 - 12 * q^88 + 6 * q^89 - 36 * q^91 - 9 * q^92 + 30 * q^93 + 3 * q^94 - 27 * q^96 + 18 * q^97 - 42 * q^98 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more