LMFDB - Embedded newform 7225.2.a.bp.1.9

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 = [12,0,0,12,0,0,0] 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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 30x^{10} + 343x^{8} - 1860x^{6} + 4823x^{4} - 5230x^{2} + 1681 $ x^12 - 30x^10 + 343x^8 - 1860x^6 + 4823x^4 - 5230*x^2 + 1681
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.05077$ 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)

$f(q)$	$=$	\(q+1.80333 q^{2} -0.561698 q^{3} +1.25200 q^{4} -1.01293 q^{6} -3.11199 q^{7} -1.34889 q^{8} -2.68450 q^{9} -5.61195 q^{11} -0.703246 q^{12} +1.24880 q^{13} -5.61195 q^{14} -4.93650 q^{16} -4.84103 q^{18} +4.00000 q^{19} +1.74800 q^{21} -10.1202 q^{22} -2.32777 q^{23} +0.757669 q^{24} +2.25200 q^{26} +3.19297 q^{27} -3.89622 q^{28} +6.62488 q^{29} -4.55412 q^{31} -6.20435 q^{32} +3.15222 q^{33} -3.36099 q^{36} -1.90762 q^{37} +7.21332 q^{38} -0.701449 q^{39} -5.92343 q^{41} +3.15222 q^{42} +2.04316 q^{43} -7.02616 q^{44} -4.19774 q^{46} +4.85546 q^{47} +2.77282 q^{48} +2.68450 q^{49} +1.56350 q^{52} -9.11674 q^{53} +5.75798 q^{54} +4.19774 q^{56} -2.24679 q^{57} +11.9468 q^{58} +6.00000 q^{59} -5.65685 q^{61} -8.21258 q^{62} +8.35413 q^{63} -1.31550 q^{64} +5.68450 q^{66} +8.46212 q^{67} +1.30750 q^{69} +8.79676 q^{71} +3.62109 q^{72} +1.56845 q^{73} -3.44007 q^{74} +5.00800 q^{76} +17.4643 q^{77} -1.26494 q^{78} -4.91050 q^{79} +6.26000 q^{81} -10.6819 q^{82} +3.94658 q^{83} +2.18850 q^{84} +3.68450 q^{86} -3.72118 q^{87} +7.56991 q^{88} -10.6130 q^{89} -3.88626 q^{91} -2.91437 q^{92} +2.55804 q^{93} +8.75600 q^{94} +3.48497 q^{96} +12.3919 q^{97} +4.84103 q^{98} +15.0653 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{4} + 28 q^{9} + 4 q^{16} + 48 q^{19} + 24 q^{21} + 24 q^{26} + 68 q^{36} - 28 q^{49} + 72 q^{59} - 76 q^{64} + 8 q^{66} + 88 q^{69} + 48 q^{76} + 60 q^{81} - 40 q^{84} - 16 q^{86} - 16 q^{89}+ \cdots + 96 q^{94}+O(q^{100}) $ 12 * q + 12 * q^4 + 28 * q^9 + 4 * q^16 + 48 * q^19 + 24 * q^21 + 24 * q^26 + 68 * q^36 - 28 * q^49 + 72 * q^59 - 76 * q^64 + 8 * q^66 + 88 * q^69 + 48 * q^76 + 60 * q^81 - 40 * q^84 - 16 * q^86 - 16 * q^89 + 96 * q^94

Coefficient data

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

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

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

$2$	1.80333	1.27515	0.637574	−	0.770389i	$-0.279938\pi$
$2$	1.80333	1.27515	0.637574	+	0.770389i	$0.279938\pi$
$3$	−0.561698	−0.324296	−0.162148	−	0.986766i	$-0.551842\pi$
$3$	−0.561698	−0.324296	−0.162148	+	0.986766i	$0.551842\pi$
$4$	1.25200	0.626000
$5$	0	0
$5$	0	0
$6$	−1.01293	−0.413526
$7$	−3.11199	−1.17622	−0.588111	−	0.808780i	$-0.700128\pi$
$7$	−3.11199	−1.17622	−0.588111	+	0.808780i	$0.700128\pi$
$8$	−1.34889	−0.476905
$9$	−2.68450	−0.894832
$10$	0	0
$11$	−5.61195	−1.69207	−0.846033	−	0.533130i	$-0.821016\pi$
$11$	−5.61195	−1.69207	−0.846033	+	0.533130i	$0.821016\pi$
$12$	−0.703246	−0.203010
$13$	1.24880	0.346355	0.173178	−	0.984891i	$-0.444597\pi$
$13$	1.24880	0.346355	0.173178	+	0.984891i	$0.444597\pi$
$14$	−5.61195	−1.49986
$15$	0	0
$16$	−4.93650	−1.23412
$17$	0	0
$17$	0	0
$18$	−4.84103	−1.14104
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	1.74800	0.381445
$22$	−10.1202	−2.15763
$23$	−2.32777	−0.485373	−0.242687	−	0.970105i	$-0.578029\pi$
$23$	−2.32777	−0.485373	−0.242687	+	0.970105i	$0.578029\pi$
$24$	0.757669	0.154659
$25$	0	0
$26$	2.25200	0.441654
$27$	3.19297	0.614487
$28$	−3.89622	−0.736316
$29$	6.62488	1.23021	0.615104	−	0.788446i	$-0.289114\pi$
$29$	6.62488	1.23021	0.615104	+	0.788446i	$0.289114\pi$
$30$	0	0
$31$	−4.55412	−0.817944	−0.408972	−	0.912547i	$-0.634113\pi$
$31$	−4.55412	−0.817944	−0.408972	+	0.912547i	$0.634113\pi$
$32$	−6.20435	−1.09678
$33$	3.15222	0.548731
$34$	0	0
$35$	0	0
$36$	−3.36099	−0.560165
$37$	−1.90762	−0.313611	−0.156805	−	0.987630i	$-0.550120\pi$
$37$	−1.90762	−0.313611	−0.156805	+	0.987630i	$0.550120\pi$
$38$	7.21332	1.17016
$39$	−0.701449	−0.112322
$40$	0	0
$41$	−5.92343	−0.925084	−0.462542	−	0.886597i	$-0.653063\pi$
$41$	−5.92343	−0.925084	−0.462542	+	0.886597i	$0.653063\pi$
$42$	3.15222	0.486398
$43$	2.04316	0.311579	0.155790	−	0.987790i	$-0.450208\pi$
$43$	2.04316	0.311579	0.155790	+	0.987790i	$0.450208\pi$
$44$	−7.02616	−1.05923
$45$	0	0
$46$	−4.19774	−0.618922
$47$	4.85546	0.708242	0.354121	−	0.935200i	$-0.384780\pi$
$47$	4.85546	0.708242	0.354121	+	0.935200i	$0.384780\pi$
$48$	2.77282	0.400222
$49$	2.68450	0.383499
$50$	0	0
$51$	0	0
$52$	1.56350	0.216818
$53$	−9.11674	−1.25228	−0.626140	−	0.779710i	$-0.715366\pi$
$53$	−9.11674	−1.25228	−0.626140	+	0.779710i	$0.715366\pi$
$54$	5.75798	0.783562
$55$	0	0
$56$	4.19774	0.560946
$57$	−2.24679	−0.297595
$58$	11.9468	1.56870
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−5.65685	−0.724286	−0.362143	−	0.932123i	$-0.617955\pi$
$61$	−5.65685	−0.724286	−0.362143	+	0.932123i	$0.617955\pi$
$62$	−8.21258	−1.04300
$63$	8.35413	1.05252
$64$	−1.31550	−0.164438
$65$	0	0
$66$	5.68450	0.699713
$67$	8.46212	1.03381	0.516906	−	0.856042i	$-0.327084\pi$
$67$	8.46212	1.03381	0.516906	+	0.856042i	$0.327084\pi$
$68$	0	0
$69$	1.30750	0.157405
$70$	0	0
$71$	8.79676	1.04398	0.521992	−	0.852951i	$-0.325189\pi$
$71$	8.79676	1.04398	0.521992	+	0.852951i	$0.325189\pi$
$72$	3.62109	0.426750
$73$	1.56845	0.183573	0.0917864	−	0.995779i	$-0.470742\pi$
$73$	1.56845	0.183573	0.0917864	+	0.995779i	$0.470742\pi$
$74$	−3.44007	−0.399900
$75$	0	0
$76$	5.00800	0.574457
$77$	17.4643	1.99025
$78$	−1.26494	−0.143227
$79$	−4.91050	−0.552475	−0.276237	−	0.961089i	$-0.589088\pi$
$79$	−4.91050	−0.552475	−0.276237	+	0.961089i	$0.589088\pi$
$80$	0	0
$81$	6.26000	0.695556
$82$	−10.6819	−1.17962
$83$	3.94658	0.433194	0.216597	−	0.976261i	$-0.430504\pi$
$83$	3.94658	0.433194	0.216597	+	0.976261i	$0.430504\pi$
$84$	2.18850	0.238785
$85$	0	0
$86$	3.68450	0.397309
$87$	−3.72118	−0.398952
$88$	7.56991	0.806955
$89$	−10.6130	−1.12497	−0.562487	−	0.826806i	$-0.690155\pi$
$89$	−10.6130	−1.12497	−0.562487	+	0.826806i	$0.690155\pi$
$90$	0	0
$91$	−3.88626	−0.407391
$92$	−2.91437	−0.303844
$93$	2.55804	0.265256
$94$	8.75600	0.903113
$95$	0	0
$96$	3.48497	0.355683
$97$	12.3919	1.25821	0.629103	−	0.777322i	$-0.283422\pi$
$97$	12.3919	1.25821	0.629103	+	0.777322i	$0.283422\pi$
$98$	4.84103	0.489018
$99$	15.0653	1.51412
$100$	0	0
$101$	9.62099	0.957324	0.478662	−	0.877999i	$-0.341122\pi$
$101$	9.62099	0.957324	0.478662	+	0.877999i	$0.341122\pi$
$102$	0	0
$103$	−1.84298	−0.181594	−0.0907972	−	0.995869i	$-0.528942\pi$
$103$	−1.84298	−0.181594	−0.0907972	+	0.995869i	$0.528942\pi$
$104$	−1.68450	−0.165178
$105$	0	0
$106$	−16.4405	−1.59684
$107$	−16.9869	−1.64218	−0.821091	−	0.570798i	$-0.806634\pi$
$107$	−16.9869	−1.64218	−0.821091	+	0.570798i	$0.806634\pi$
$108$	3.99760	0.384669
$109$	0.446191	0.0427373	0.0213687	−	0.999772i	$-0.493198\pi$
$109$	0.446191	0.0427373	0.0213687	+	0.999772i	$0.493198\pi$
$110$	0	0
$111$	1.07151	0.101703
$112$	15.3623	1.45160
$113$	−11.3246	−1.06533	−0.532663	−	0.846327i	$-0.678809\pi$
$113$	−11.3246	−1.06533	−0.532663	+	0.846327i	$0.678809\pi$
$114$	−4.05171	−0.379477
$115$	0	0
$116$	8.29435	0.770111
$117$	−3.35240	−0.309929
$118$	10.8200	0.996060
$119$	0	0
$120$	0	0
$121$	20.4940	1.86309
$122$	−10.2012	−0.923571
$123$	3.32718	0.300001
$124$	−5.70176	−0.512033
$125$	0	0
$126$	15.0653	1.34212
$127$	−5.85000	−0.519104	−0.259552	−	0.965729i	$-0.583575\pi$
$127$	−5.85000	−0.519104	−0.259552	+	0.965729i	$0.583575\pi$
$128$	10.0364	0.887102
$129$	−1.14764	−0.101044
$130$	0	0
$131$	19.5632	1.70924	0.854620	−	0.519253i	$-0.173790\pi$
$131$	19.5632	1.70924	0.854620	+	0.519253i	$0.173790\pi$
$132$	3.94658	0.343506
$133$	−12.4480	−1.07938
$134$	15.2600	1.31826
$135$	0	0
$136$	0	0
$137$	13.8577	1.18394	0.591971	−	0.805959i	$-0.298350\pi$
$137$	13.8577	1.18394	0.591971	+	0.805959i	$0.298350\pi$
$138$	2.35786	0.200714
$139$	10.1997	0.865124	0.432562	−	0.901604i	$-0.357610\pi$
$139$	10.1997	0.865124	0.432562	+	0.901604i	$0.357610\pi$
$140$	0	0
$141$	−2.72730	−0.229680
$142$	15.8635	1.33123
$143$	−7.00821	−0.586056
$144$	13.2520	1.10433
$145$	0	0
$146$	2.82843	0.234082
$147$	−1.50788	−0.124367
$148$	−2.38834	−0.196320
$149$	15.9365	1.30557	0.652784	−	0.757544i	$-0.273601\pi$
$149$	15.9365	1.30557	0.652784	+	0.757544i	$0.273601\pi$
$150$	0	0
$151$	−3.36899	−0.274165	−0.137082	−	0.990560i	$-0.543772\pi$
$151$	−3.36899	−0.274165	−0.137082	+	0.990560i	$0.543772\pi$
$152$	−5.39556	−0.437638
$153$	0	0
$154$	31.4940	2.53786
$155$	0	0
$156$	−0.878214	−0.0703134
$157$	3.92136	0.312959	0.156479	−	0.987681i	$-0.449986\pi$
$157$	3.92136	0.312959	0.156479	+	0.987681i	$0.449986\pi$
$158$	−8.85526	−0.704486
$159$	5.12085	0.406110
$160$	0	0
$161$	7.24400	0.570907
$162$	11.2889	0.886936
$163$	14.7197	1.15293	0.576466	−	0.817121i	$-0.304431\pi$
$163$	14.7197	1.15293	0.576466	+	0.817121i	$0.304431\pi$
$164$	−7.41613	−0.579103
$165$	0	0
$166$	7.11699	0.552386
$167$	−0.420150	−0.0325122	−0.0162561	−	0.999868i	$-0.505175\pi$
$167$	−0.420150	−0.0325122	−0.0162561	+	0.999868i	$0.505175\pi$
$168$	−2.35786	−0.181913
$169$	−11.4405	−0.880038
$170$	0	0
$171$	−10.7380	−0.821154
$172$	2.55804	0.195049
$173$	−16.6024	−1.26226	−0.631128	−	0.775679i	$-0.717408\pi$
$173$	−16.6024	−1.26226	−0.631128	+	0.775679i	$0.717408\pi$
$174$	−6.71052	−0.508723
$175$	0	0
$176$	27.7034	2.08822
$177$	−3.37019	−0.253319
$178$	−19.1387	−1.43451
$179$	19.7935	1.47943	0.739717	−	0.672918i	$-0.234959\pi$
$179$	19.7935	1.47943	0.739717	+	0.672918i	$0.234959\pi$
$180$	0	0
$181$	9.36350	0.695983	0.347992	−	0.937498i	$-0.386864\pi$
$181$	9.36350	0.695983	0.347992	+	0.937498i	$0.386864\pi$
$182$	−7.00821	−0.519483
$183$	3.17744	0.234883
$184$	3.13991	0.231477
$185$	0	0
$186$	4.61299	0.338241
$187$	0	0
$188$	6.07904	0.443360
$189$	−9.93650	−0.722774
$190$	0	0
$191$	13.9365	1.00841	0.504205	−	0.863584i	$-0.331786\pi$
$191$	13.9365	1.00841	0.504205	+	0.863584i	$0.331786\pi$
$192$	0.738916	0.0533267
$193$	−4.17482	−0.300510	−0.150255	−	0.988647i	$-0.548009\pi$
$193$	−4.17482	−0.300510	−0.150255	+	0.988647i	$0.548009\pi$
$194$	22.3467	1.60440
$195$	0	0
$196$	3.36099	0.240071
$197$	17.6901	1.26037	0.630184	−	0.776446i	$-0.282979\pi$
$197$	17.6901	1.26037	0.630184	+	0.776446i	$0.282979\pi$
$198$	27.1676	1.93072
$199$	5.60063	0.397018	0.198509	−	0.980099i	$-0.436390\pi$
$199$	5.60063	0.397018	0.198509	+	0.980099i	$0.436390\pi$
$200$	0	0
$201$	−4.75316	−0.335262
$202$	17.3498	1.22073
$203$	−20.6166	−1.44700
$204$	0	0
$205$	0	0
$206$	−3.32351	−0.231560
$207$	6.24889	0.434328
$208$	−6.16470	−0.427445
$209$	−22.4478	−1.55275
$210$	0	0
$211$	0.923118	0.0635501	0.0317750	−	0.999495i	$-0.489884\pi$
$211$	0.923118	0.0635501	0.0317750	+	0.999495i	$0.489884\pi$
$212$	−11.4142	−0.783928
$213$	−4.94112	−0.338560
$214$	−30.6329	−2.09402
$215$	0	0
$216$	−4.30697	−0.293052
$217$	14.1724	0.962084
$218$	0.804630	0.0544964
$219$	−0.880993	−0.0595320
$220$	0	0
$221$	0	0
$222$	1.93228	0.129686
$223$	1.24880	0.0836259	0.0418129	−	0.999125i	$-0.486687\pi$
$223$	1.24880	0.0836259	0.0418129	+	0.999125i	$0.486687\pi$
$224$	19.3079	1.29006
$225$	0	0
$226$	−20.4219	−1.35845
$227$	11.5828	0.768775	0.384388	−	0.923172i	$-0.374413\pi$
$227$	11.5828	0.768775	0.384388	+	0.923172i	$0.374413\pi$
$228$	−2.81298	−0.186294
$229$	11.7480	0.776330	0.388165	−	0.921590i	$-0.373109\pi$
$229$	11.7480	0.776330	0.388165	+	0.921590i	$0.373109\pi$
$230$	0	0
$231$	−9.80969	−0.645430
$232$	−8.93623	−0.586692
$233$	−11.3042	−0.740561	−0.370280	−	0.928920i	$-0.620738\pi$
$233$	−11.3042	−0.740561	−0.370280	+	0.928920i	$0.620738\pi$
$234$	−6.04548	−0.395206
$235$	0	0
$236$	7.51200	0.488990
$237$	2.75822	0.179166
$238$	0	0
$239$	5.13501	0.332156	0.166078	−	0.986113i	$-0.446890\pi$
$239$	5.13501	0.332156	0.166078	+	0.986113i	$0.446890\pi$
$240$	0	0
$241$	−8.48528	−0.546585	−0.273293	−	0.961931i	$-0.588113\pi$
$241$	−8.48528	−0.546585	−0.273293	+	0.961931i	$0.588113\pi$
$242$	36.9574	2.37571
$243$	−13.0951	−0.840054
$244$	−7.08238	−0.453403
$245$	0	0
$246$	6.00000	0.382546
$247$	4.99520	0.317837
$248$	6.14301	0.390081
$249$	−2.21679	−0.140483
$250$	0	0
$251$	−19.8095	−1.25036	−0.625182	−	0.780479i	$-0.714975\pi$
$251$	−19.8095	−1.25036	−0.625182	+	0.780479i	$0.714975\pi$
$252$	10.4594	0.658879
$253$	13.0633	0.821284
$254$	−10.5495	−0.661934
$255$	0	0
$256$	20.7300	1.29562
$257$	−24.3982	−1.52192	−0.760958	−	0.648801i	$-0.775271\pi$
$257$	−24.3982	−1.52192	−0.760958	+	0.648801i	$0.775271\pi$
$258$	−2.06957	−0.128846
$259$	5.93650	0.368876
$260$	0	0
$261$	−17.7845	−1.10083
$262$	35.2788	2.17953
$263$	18.0585	1.11354	0.556768	−	0.830668i	$-0.312041\pi$
$263$	18.0585	1.11354	0.556768	+	0.830668i	$0.312041\pi$
$264$	−4.25200	−0.261693
$265$	0	0
$266$	−22.4478	−1.37636
$267$	5.96129	0.364825
$268$	10.5946	0.647167
$269$	−17.8601	−1.08895	−0.544475	−	0.838777i	$-0.683271\pi$
$269$	−17.8601	−1.08895	−0.544475	+	0.838777i	$0.683271\pi$
$270$	0	0
$271$	−15.8570	−0.963243	−0.481622	−	0.876379i	$-0.659952\pi$
$271$	−15.8570	−0.963243	−0.481622	+	0.876379i	$0.659952\pi$
$272$	0	0
$273$	2.18290	0.132115
$274$	24.9900	1.50970
$275$	0	0
$276$	1.63699	0.0985355
$277$	14.1784	0.851896	0.425948	−	0.904748i	$-0.359941\pi$
$277$	14.1784	0.851896	0.425948	+	0.904748i	$0.359941\pi$
$278$	18.3934	1.10316
$279$	12.2255	0.731922
$280$	0	0
$281$	−7.80949	−0.465875	−0.232937	−	0.972492i	$-0.574834\pi$
$281$	−7.80949	−0.465875	−0.232937	+	0.972492i	$0.574834\pi$
$282$	−4.91823	−0.292876
$283$	−20.9080	−1.24285	−0.621425	−	0.783474i	$-0.713446\pi$
$283$	−20.9080	−1.24285	−0.621425	+	0.783474i	$0.713446\pi$
$284$	11.0135	0.653534
$285$	0	0
$286$	−12.6381	−0.747307
$287$	18.4337	1.08810
$288$	16.6556	0.981438
$289$	0	0
$290$	0	0
$291$	−6.96050	−0.408032
$292$	1.96370	0.114917
$293$	4.79502	0.280128	0.140064	−	0.990142i	$-0.455269\pi$
$293$	4.79502	0.280128	0.140064	+	0.990142i	$0.455269\pi$
$294$	−2.71920	−0.158587
$295$	0	0
$296$	2.57317	0.149562
$297$	−17.9188	−1.03975
$298$	28.7388	1.66479
$299$	−2.90692	−0.168112
$300$	0	0
$301$	−6.35830	−0.366486
$302$	−6.07540	−0.349600
$303$	−5.40409	−0.310457
$304$	−19.7460	−1.13251
$305$	0	0
$306$	0	0
$307$	−24.9924	−1.42639	−0.713195	−	0.700966i	$-0.752753\pi$
$307$	−24.9924	−1.42639	−0.713195	+	0.700966i	$0.752753\pi$
$308$	21.8654	1.24589
$309$	1.03520	0.0588904
$310$	0	0
$311$	−4.03229	−0.228650	−0.114325	−	0.993443i	$-0.536471\pi$
$311$	−4.03229	−0.228650	−0.114325	+	0.993443i	$0.536471\pi$
$312$	0.946178	0.0535668
$313$	−22.6491	−1.28021	−0.640103	−	0.768289i	$-0.721108\pi$
$313$	−22.6491	−1.28021	−0.640103	+	0.768289i	$0.721108\pi$
$314$	7.07151	0.399068
$315$	0	0
$316$	−6.14795	−0.345849
$317$	−8.27315	−0.464666	−0.232333	−	0.972636i	$-0.574636\pi$
$317$	−8.27315	−0.464666	−0.232333	+	0.972636i	$0.574636\pi$
$318$	9.23459	0.517850
$319$	−37.1785	−2.08160
$320$	0	0
$321$	9.54148	0.532554
$322$	13.0633	0.727991
$323$	0	0
$324$	7.83752	0.435418
$325$	0	0
$326$	26.5444	1.47016
$327$	−0.250624	−0.0138596
$328$	7.99006	0.441177
$329$	−15.1102	−0.833050
$330$	0	0
$331$	−5.51200	−0.302967	−0.151484	−	0.988460i	$-0.548405\pi$
$331$	−5.51200	−0.302967	−0.151484	+	0.988460i	$0.548405\pi$
$332$	4.94112	0.271179
$333$	5.12099	0.280629
$334$	−0.757669	−0.0414578
$335$	0	0
$336$	−8.62899	−0.470750
$337$	−9.53810	−0.519573	−0.259787	−	0.965666i	$-0.583652\pi$
$337$	−9.53810	−0.519573	−0.259787	+	0.965666i	$0.583652\pi$
$338$	−20.6310	−1.12218
$339$	6.36099	0.345482
$340$	0	0
$341$	25.5575	1.38402
$342$	−19.3641	−1.04709
$343$	13.4298	0.725142
$344$	−2.75600	−0.148594
$345$	0	0
$346$	−29.9396	−1.60956
$347$	11.5471	0.619881	0.309940	−	0.950756i	$-0.399691\pi$
$347$	11.5471	0.619881	0.309940	+	0.950756i	$0.399691\pi$
$348$	−4.65892	−0.249744
$349$	−11.4325	−0.611967	−0.305984	−	0.952037i	$-0.598985\pi$
$349$	−11.4325	−0.611967	−0.305984	+	0.952037i	$0.598985\pi$
$350$	0	0
$351$	3.98738	0.212831
$352$	34.8185	1.85583
$353$	−31.8017	−1.69263	−0.846317	−	0.532680i	$-0.821185\pi$
$353$	−31.8017	−1.69263	−0.846317	+	0.532680i	$0.821185\pi$
$354$	−6.07756	−0.323019
$355$	0	0
$356$	−13.2875	−0.704234
$357$	0	0
$358$	35.6942	1.88650
$359$	25.1785	1.32887	0.664435	−	0.747346i	$-0.268672\pi$
$359$	25.1785	1.32887	0.664435	+	0.747346i	$0.268672\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	16.8855	0.887481
$363$	−11.5114	−0.604193
$364$	−4.86560	−0.255027
$365$	0	0
$366$	5.72998	0.299511
$367$	−6.08693	−0.317735	−0.158868	−	0.987300i	$-0.550784\pi$
$367$	−6.08693	−0.317735	−0.158868	+	0.987300i	$0.550784\pi$
$368$	11.4910	0.599011
$369$	15.9014	0.827795
$370$	0	0
$371$	28.3712	1.47296
$372$	3.20267	0.166050
$373$	−30.0732	−1.55713	−0.778566	−	0.627562i	$-0.784053\pi$
$373$	−30.0732	−1.55713	−0.778566	+	0.627562i	$0.784053\pi$
$374$	0	0
$375$	0	0
$376$	−6.54949	−0.337764
$377$	8.27315	0.426089
$378$	−17.9188	−0.921643
$379$	−1.01293	−0.0520306	−0.0260153	−	0.999662i	$-0.508282\pi$
$379$	−1.01293	−0.0520306	−0.0260153	+	0.999662i	$0.508282\pi$
$380$	0	0
$381$	3.28593	0.168343
$382$	25.1321	1.28587
$383$	−25.8660	−1.32169	−0.660846	−	0.750521i	$-0.729802\pi$
$383$	−25.8660	−1.32169	−0.660846	+	0.750521i	$0.729802\pi$
$384$	−5.63743	−0.287684
$385$	0	0
$386$	−7.52857	−0.383194
$387$	−5.48486	−0.278811
$388$	15.5147	0.787637
$389$	33.0695	1.67669	0.838345	−	0.545140i	$-0.183524\pi$
$389$	33.0695	1.67669	0.838345	+	0.545140i	$0.183524\pi$
$390$	0	0
$391$	0	0
$392$	−3.62109	−0.182893
$393$	−10.9886	−0.554301
$394$	31.9011	1.60715
$395$	0	0
$396$	18.8617	0.947836
$397$	−8.15201	−0.409138	−0.204569	−	0.978852i	$-0.565579\pi$
$397$	−8.15201	−0.409138	−0.204569	+	0.978852i	$0.565579\pi$
$398$	10.0998	0.506257
$399$	6.99200	0.350038
$400$	0	0
$401$	0.165449	0.00826215	0.00413108	−	0.999991i	$-0.498685\pi$
$401$	0.165449	0.00826215	0.00413108	+	0.999991i	$0.498685\pi$
$402$	−8.57151	−0.427508
$403$	−5.68719	−0.283299
$404$	12.0455	0.599285
$405$	0	0
$406$	−37.1785	−1.84514
$407$	10.7055	0.530650
$408$	0	0
$409$	13.3690	0.661054	0.330527	−	0.943797i	$-0.392774\pi$
$409$	13.3690	0.661054	0.330527	+	0.943797i	$0.392774\pi$
$410$	0	0
$411$	−7.78383	−0.383948
$412$	−2.30741	−0.113678
$413$	−18.6720	−0.918787
$414$	11.2688	0.553832
$415$	0	0
$416$	−7.74800	−0.379877
$417$	−5.72913	−0.280557
$418$	−40.4808	−1.97998
$419$	23.5732	1.15162	0.575812	−	0.817582i	$-0.304686\pi$
$419$	23.5732	1.15162	0.575812	+	0.817582i	$0.304686\pi$
$420$	0	0
$421$	−5.82751	−0.284015	−0.142008	−	0.989866i	$-0.545356\pi$
$421$	−5.82751	−0.284015	−0.142008	+	0.989866i	$0.545356\pi$
$422$	1.66469	0.0810357
$423$	−13.0345	−0.633757
$424$	12.2975	0.597219
$425$	0	0
$426$	−8.91047	−0.431714
$427$	17.6041	0.851921
$428$	−21.2676	−1.02801
$429$	3.93650	0.190056
$430$	0	0
$431$	−6.93636	−0.334112	−0.167056	−	0.985947i	$-0.553426\pi$
$431$	−6.93636	−0.334112	−0.167056	+	0.985947i	$0.553426\pi$
$432$	−15.7621	−0.758353
$433$	15.4464	0.742307	0.371153	−	0.928572i	$-0.378962\pi$
$433$	15.4464	0.742307	0.371153	+	0.928572i	$0.378962\pi$
$434$	25.5575	1.22680
$435$	0	0
$436$	0.558631	0.0267536
$437$	−9.31108	−0.445409
$438$	−1.58872	−0.0759121
$439$	1.47043	0.0701800	0.0350900	−	0.999384i	$-0.488828\pi$
$439$	1.47043	0.0701800	0.0350900	+	0.999384i	$0.488828\pi$
$440$	0	0
$441$	−7.20652	−0.343167
$442$	0	0
$443$	−12.7838	−0.607379	−0.303689	−	0.952771i	$-0.598218\pi$
$443$	−12.7838	−0.607379	−0.303689	+	0.952771i	$0.598218\pi$
$444$	1.34153	0.0636660
$445$	0	0
$446$	2.25200	0.106635
$447$	−8.95150	−0.423391
$448$	4.09384	0.193416
$449$	−9.73119	−0.459243	−0.229622	−	0.973280i	$-0.573749\pi$
$449$	−9.73119	−0.459243	−0.229622	+	0.973280i	$0.573749\pi$
$450$	0	0
$451$	33.2420	1.56530
$452$	−14.1784	−0.666894
$453$	1.89236	0.0889106
$454$	20.8876	0.980302
$455$	0	0
$456$	3.03068	0.141924
$457$	−36.2279	−1.69467	−0.847336	−	0.531058i	$-0.821795\pi$
$457$	−36.2279	−1.69467	−0.847336	+	0.531058i	$0.821795\pi$
$458$	21.1855	0.989935
$459$	0	0
$460$	0	0
$461$	−40.0595	−1.86576	−0.932878	−	0.360193i	$-0.882711\pi$
$461$	−40.0595	−1.86576	−0.932878	+	0.360193i	$0.882711\pi$
$462$	−17.6901	−0.823018
$463$	−15.3607	−0.713874	−0.356937	−	0.934128i	$-0.616179\pi$
$463$	−15.3607	−0.713874	−0.356937	+	0.934128i	$0.616179\pi$
$464$	−32.7037	−1.51823
$465$	0	0
$466$	−20.3851	−0.944324
$467$	19.4471	0.899903	0.449952	−	0.893053i	$-0.351441\pi$
$467$	19.4471	0.899903	0.449952	+	0.893053i	$0.351441\pi$
$468$	−4.19721	−0.194016
$469$	−26.3341	−1.21599
$470$	0	0
$471$	−2.20262	−0.101491
$472$	−8.09334	−0.372526
$473$	−11.4661	−0.527213
$474$	4.97398	0.228462
$475$	0	0
$476$	0	0
$477$	24.4739	1.12058
$478$	9.26012	0.423548
$479$	14.5547	0.665023	0.332511	−	0.943099i	$-0.392104\pi$
$479$	14.5547	0.665023	0.332511	+	0.943099i	$0.392104\pi$
$480$	0	0
$481$	−2.38224	−0.108621
$482$	−15.3018	−0.696976
$483$	−4.06894	−0.185143
$484$	25.6585	1.16629
$485$	0	0
$486$	−23.6149	−1.07119
$487$	29.6975	1.34572	0.672861	−	0.739768i	$-0.265065\pi$
$487$	29.6975	1.34572	0.672861	+	0.739768i	$0.265065\pi$
$488$	7.63048	0.345415
$489$	−8.26800	−0.373892
$490$	0	0
$491$	21.0715	0.950944	0.475472	−	0.879731i	$-0.342277\pi$
$491$	21.0715	0.950944	0.475472	+	0.879731i	$0.342277\pi$
$492$	4.16563	0.187801
$493$	0	0
$494$	9.00800	0.405289
$495$	0	0
$496$	22.4814	1.00944
$497$	−27.3754	−1.22796
$498$	−3.99760	−0.179137
$499$	−9.40840	−0.421178	−0.210589	−	0.977575i	$-0.567538\pi$
$499$	−9.40840	−0.421178	−0.210589	+	0.977575i	$0.567538\pi$
$500$	0	0
$501$	0.235997	0.0105436
$502$	−35.7231	−1.59440
$503$	29.3788	1.30993	0.654967	−	0.755658i	$-0.272683\pi$
$503$	29.3788	1.30993	0.654967	+	0.755658i	$0.272683\pi$
$504$	−11.2688	−0.501952
$505$	0	0
$506$	23.5575	1.04726
$507$	6.42610	0.285393
$508$	−7.32420	−0.324959
$509$	14.3930	0.637958	0.318979	−	0.947762i	$-0.396660\pi$
$509$	14.3930	0.637958	0.318979	+	0.947762i	$0.396660\pi$
$510$	0	0
$511$	−4.88099	−0.215922
$512$	17.3102	0.765009
$513$	12.7719	0.563892
$514$	−43.9980	−1.94067
$515$	0	0
$516$	−1.43685	−0.0632536
$517$	−27.2486	−1.19839
$518$	10.7055	0.470371
$519$	9.32552	0.409345
$520$	0	0
$521$	27.6698	1.21224	0.606118	−	0.795375i	$-0.292726\pi$
$521$	27.6698	1.21224	0.606118	+	0.795375i	$0.292726\pi$
$522$	−32.0712	−1.40372
$523$	37.4867	1.63918	0.819590	−	0.572950i	$-0.194201\pi$
$523$	37.4867	1.63918	0.819590	+	0.572950i	$0.194201\pi$
$524$	24.4931	1.06998
$525$	0	0
$526$	32.5655	1.41992
$527$	0	0
$528$	−15.5609	−0.677202
$529$	−17.5815	−0.764413
$530$	0	0
$531$	−16.1070	−0.698983
$532$	−15.5849	−0.675689
$533$	−7.39718	−0.320408
$534$	10.7502	0.465206
$535$	0	0
$536$	−11.4145	−0.493030
$537$	−11.1180	−0.479775
$538$	−32.2076	−1.38857
$539$	−15.0653	−0.648906
$540$	0	0
$541$	12.7279	0.547216	0.273608	−	0.961841i	$-0.411783\pi$
$541$	12.7279	0.547216	0.273608	+	0.961841i	$0.411783\pi$
$542$	−28.5954	−1.22828
$543$	−5.25946	−0.225705
$544$	0	0
$545$	0	0
$546$	3.93650	0.168466
$547$	28.7922	1.23106	0.615532	−	0.788112i	$-0.288941\pi$
$547$	28.7922	1.23106	0.615532	+	0.788112i	$0.288941\pi$
$548$	17.3498	0.741148
$549$	15.1858	0.648114
$550$	0	0
$551$	26.4995	1.12892
$552$	−1.76368	−0.0750671
$553$	15.2814	0.649833
$554$	25.5683	1.08629
$555$	0	0
$556$	12.7700	0.541568
$557$	−4.93477	−0.209093	−0.104546	−	0.994520i	$-0.533339\pi$
$557$	−4.93477	−0.209093	−0.104546	+	0.994520i	$0.533339\pi$
$558$	22.0466	0.933308
$559$	2.55150	0.107917
$560$	0	0
$561$	0	0
$562$	−14.0831	−0.594059
$563$	−3.97544	−0.167545	−0.0837724	−	0.996485i	$-0.526697\pi$
$563$	−3.97544	−0.167545	−0.0837724	+	0.996485i	$0.526697\pi$
$564$	−3.41458	−0.143780
$565$	0	0
$566$	−37.7040	−1.58482
$567$	−19.4811	−0.818128
$568$	−11.8659	−0.497881
$569$	−5.49600	−0.230404	−0.115202	−	0.993342i	$-0.536752\pi$
$569$	−5.49600	−0.230404	−0.115202	+	0.993342i	$0.536752\pi$
$570$	0	0
$571$	17.4475	0.730155	0.365077	−	0.930977i	$-0.381043\pi$
$571$	17.4475	0.730155	0.365077	+	0.930977i	$0.381043\pi$
$572$	−8.77428	−0.366871
$573$	−7.82810	−0.327024
$574$	33.2420	1.38749
$575$	0	0
$576$	3.53147	0.147144
$577$	−32.3418	−1.34641	−0.673203	−	0.739458i	$-0.735082\pi$
$577$	−32.3418	−1.34641	−0.673203	+	0.739458i	$0.735082\pi$
$578$	0	0
$579$	2.34499	0.0974543
$580$	0	0
$581$	−12.2817	−0.509532
$582$	−12.5521	−0.520301
$583$	51.1627	2.11894
$584$	−2.11566	−0.0875467
$585$	0	0
$586$	8.64701	0.357205
$587$	−28.6847	−1.18394	−0.591972	−	0.805959i	$-0.701650\pi$
$587$	−28.6847	−1.18394	−0.591972	+	0.805959i	$0.701650\pi$
$588$	−1.88786	−0.0778541
$589$	−18.2165	−0.750597
$590$	0	0
$591$	−9.93650	−0.408733
$592$	9.41695	0.387034
$593$	9.74614	0.400226	0.200113	−	0.979773i	$-0.435869\pi$
$593$	9.74614	0.400226	0.200113	+	0.979773i	$0.435869\pi$
$594$	−32.3135	−1.32584
$595$	0	0
$596$	19.9525	0.817286
$597$	−3.14586	−0.128752
$598$	−5.24214	−0.214367
$599$	8.07951	0.330120	0.165060	−	0.986284i	$-0.447218\pi$
$599$	8.07951	0.330120	0.165060	+	0.986284i	$0.447218\pi$
$600$	0	0
$601$	18.2165	0.743066	0.371533	−	0.928420i	$-0.378832\pi$
$601$	18.2165	0.743066	0.371533	+	0.928420i	$0.378832\pi$
$602$	−11.4661	−0.467324
$603$	−22.7165	−0.925089
$604$	−4.21798	−0.171627
$605$	0	0
$606$	−9.74536	−0.395878
$607$	−11.8506	−0.481001	−0.240501	−	0.970649i	$-0.577312\pi$
$607$	−11.8506	−0.481001	−0.240501	+	0.970649i	$0.577312\pi$
$608$	−24.8174	−1.00648
$609$	11.5803	0.469257
$610$	0	0
$611$	6.06350	0.245303
$612$	0	0
$613$	−23.1494	−0.934995	−0.467497	−	0.883994i	$-0.654844\pi$
$613$	−23.1494	−0.934995	−0.467497	+	0.883994i	$0.654844\pi$
$614$	−45.0695	−1.81886
$615$	0	0
$616$	−23.5575	−0.949158
$617$	10.4843	0.422081	0.211040	−	0.977477i	$-0.432315\pi$
$617$	10.4843	0.422081	0.211040	+	0.977477i	$0.432315\pi$
$618$	1.86681	0.0750940
$619$	23.2924	0.936202	0.468101	−	0.883675i	$-0.344938\pi$
$619$	23.2924	0.936202	0.468101	+	0.883675i	$0.344938\pi$
$620$	0	0
$621$	−7.43250	−0.298256
$622$	−7.27155	−0.291562
$623$	33.0275	1.32322
$624$	3.46270	0.138619
$625$	0	0
$626$	−40.8439	−1.63245
$627$	12.6089	0.503550
$628$	4.90954	0.195912
$629$	0	0
$630$	0	0
$631$	41.1310	1.63740	0.818699	−	0.574223i	$-0.194696\pi$
$631$	41.1310	1.63740	0.818699	+	0.574223i	$0.194696\pi$
$632$	6.62373	0.263478
$633$	−0.518514	−0.0206091
$634$	−14.9192	−0.592518
$635$	0	0
$636$	6.41131	0.254225
$637$	3.35240	0.132827
$638$	−67.0451	−2.65434
$639$	−23.6149	−0.934189
$640$	0	0
$641$	47.2807	1.86747	0.933737	−	0.357959i	$-0.116527\pi$
$641$	47.2807	1.86747	0.933737	+	0.357959i	$0.116527\pi$
$642$	17.2064	0.679084
$643$	7.06878	0.278765	0.139383	−	0.990239i	$-0.455488\pi$
$643$	7.06878	0.278765	0.139383	+	0.990239i	$0.455488\pi$
$644$	9.06949	0.357388
$645$	0	0
$646$	0	0
$647$	−26.9462	−1.05937	−0.529683	−	0.848196i	$-0.677689\pi$
$647$	−26.9462	−1.05937	−0.529683	+	0.848196i	$0.677689\pi$
$648$	−8.44406	−0.331714
$649$	−33.6717	−1.32173
$650$	0	0
$651$	−7.96060	−0.312000
$652$	18.4290	0.721736
$653$	28.2509	1.10554	0.552771	−	0.833333i	$-0.313570\pi$
$653$	28.2509	1.10554	0.552771	+	0.833333i	$0.313570\pi$
$654$	−0.451959	−0.0176730
$655$	0	0
$656$	29.2410	1.14167
$657$	−4.21049	−0.164267
$658$	−27.2486	−1.06226
$659$	14.1230	0.550153	0.275077	−	0.961422i	$-0.411297\pi$
$659$	14.1230	0.550153	0.275077	+	0.961422i	$0.411297\pi$
$660$	0	0
$661$	49.0355	1.90726	0.953629	−	0.300984i	$-0.0973150\pi$
$661$	49.0355	1.90726	0.953629	+	0.300984i	$0.0973150\pi$
$662$	−9.93996	−0.386328
$663$	0	0
$664$	−5.32351	−0.206592
$665$	0	0
$666$	9.23485	0.357843
$667$	−15.4212	−0.597111
$668$	−0.526028	−0.0203526
$669$	−0.701449	−0.0271196
$670$	0	0
$671$	31.7460	1.22554
$672$	−10.8452	−0.418363
$673$	−31.4528	−1.21242	−0.606209	−	0.795306i	$-0.707310\pi$
$673$	−31.4528	−1.21242	−0.606209	+	0.795306i	$0.707310\pi$
$674$	−17.2003	−0.662532
$675$	0	0
$676$	−14.3235	−0.550904
$677$	30.2796	1.16374	0.581870	−	0.813282i	$-0.302321\pi$
$677$	30.2796	1.16374	0.581870	+	0.813282i	$0.302321\pi$
$678$	11.4710	0.440540
$679$	−38.5635	−1.47993
$680$	0	0
$681$	−6.50602	−0.249311
$682$	46.0886	1.76482
$683$	46.1788	1.76698	0.883490	−	0.468449i	$-0.155187\pi$
$683$	46.1788	1.76698	0.883490	+	0.468449i	$0.155187\pi$
$684$	−13.4440	−0.514043
$685$	0	0
$686$	24.2184	0.924663
$687$	−6.59883	−0.251761
$688$	−10.0861	−0.384527
$689$	−11.3850	−0.433734
$690$	0	0
$691$	10.6458	0.404987	0.202494	−	0.979284i	$-0.435095\pi$
$691$	10.6458	0.404987	0.202494	+	0.979284i	$0.435095\pi$
$692$	−20.7862	−0.790172
$693$	−46.8830	−1.78094
$694$	20.8232	0.790439
$695$	0	0
$696$	5.01946	0.190262
$697$	0	0
$698$	−20.6166	−0.780349
$699$	6.34953	0.240161
$700$	0	0
$701$	39.4145	1.48866	0.744332	−	0.667810i	$-0.232768\pi$
$701$	39.4145	1.48866	0.744332	+	0.667810i	$0.232768\pi$
$702$	7.19057	0.271391
$703$	−7.63048	−0.287789
$704$	7.38255	0.278240
$705$	0	0
$706$	−57.3490	−2.15836
$707$	−29.9404	−1.12603
$708$	−4.21948	−0.158578
$709$	−15.1971	−0.570740	−0.285370	−	0.958417i	$-0.592116\pi$
$709$	−15.1971	−0.570740	−0.285370	+	0.958417i	$0.592116\pi$
$710$	0	0
$711$	13.1822	0.494372
$712$	14.3158	0.536506
$713$	10.6009	0.397008
$714$	0	0
$715$	0	0
$716$	24.7815	0.926126
$717$	−2.88432	−0.107717
$718$	45.4051	1.69450
$719$	6.85786	0.255755	0.127878	−	0.991790i	$-0.459184\pi$
$719$	6.85786	0.255755	0.127878	+	0.991790i	$0.459184\pi$
$720$	0	0
$721$	5.73535	0.213595
$722$	−5.40999	−0.201339
$723$	4.76617	0.177256
$724$	11.7231	0.435686
$725$	0	0
$726$	−20.7589	−0.770435
$727$	−38.5606	−1.43013	−0.715066	−	0.699057i	$-0.753603\pi$
$727$	−38.5606	−1.43013	−0.715066	+	0.699057i	$0.753603\pi$
$728$	5.24214	0.194287
$729$	−11.4245	−0.423129
$730$	0	0
$731$	0	0
$732$	3.97816	0.147037
$733$	−42.7425	−1.57873	−0.789366	−	0.613923i	$-0.789591\pi$
$733$	−42.7425	−1.57873	−0.789366	+	0.613923i	$0.789591\pi$
$734$	−10.9767	−0.405159
$735$	0	0
$736$	14.4423	0.532350
$737$	−47.4890	−1.74928
$738$	28.6755	1.05556
$739$	6.00000	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$740$	0	0
$741$	−2.80580	−0.103073
$742$	51.1627	1.87824
$743$	52.2906	1.91836	0.959178	−	0.282804i	$-0.0912645\pi$
$743$	52.2906	1.91836	0.959178	+	0.282804i	$0.0912645\pi$
$744$	−3.45051	−0.126502
$745$	0	0
$746$	−54.2320	−1.98557
$747$	−10.5946	−0.387635
$748$	0	0
$749$	52.8630	1.93157
$750$	0	0
$751$	−39.4604	−1.43993	−0.719966	−	0.694010i	$-0.755842\pi$
$751$	−39.4604	−1.43993	−0.719966	+	0.694010i	$0.755842\pi$
$752$	−23.9690	−0.874058
$753$	11.1269	0.405489
$754$	14.9192	0.543326
$755$	0	0
$756$	−12.4405	−0.452456
$757$	−8.61186	−0.313003	−0.156502	−	0.987678i	$-0.550022\pi$
$757$	−8.61186	−0.313003	−0.156502	+	0.987678i	$0.550022\pi$
$758$	−1.82664	−0.0663466
$759$	−7.33764	−0.266340
$760$	0	0
$761$	41.2400	1.49495	0.747474	−	0.664291i	$-0.231267\pi$
$761$	41.2400	1.49495	0.747474	+	0.664291i	$0.231267\pi$
$762$	5.92562	0.214663
$763$	−1.38854	−0.0502686
$764$	17.4485	0.631265
$765$	0	0
$766$	−46.6450	−1.68535
$767$	7.49280	0.270549
$768$	−11.6440	−0.420166
$769$	17.3510	0.625692	0.312846	−	0.949804i	$-0.398718\pi$
$769$	17.3510	0.625692	0.312846	+	0.949804i	$0.398718\pi$
$770$	0	0
$771$	13.7044	0.493552
$772$	−5.22687	−0.188119
$773$	20.0124	0.719796	0.359898	−	0.932992i	$-0.382812\pi$
$773$	20.0124	0.719796	0.359898	+	0.932992i	$0.382812\pi$
$774$	−9.89101	−0.355525
$775$	0	0
$776$	−16.7153	−0.600044
$777$	−3.33452	−0.119625
$778$	59.6352	2.13803
$779$	−23.6937	−0.848915
$780$	0	0
$781$	−49.3670	−1.76649
$782$	0	0
$783$	21.1530	0.755948
$784$	−13.2520	−0.473286
$785$	0	0
$786$	−19.8160	−0.706815
$787$	33.9131	1.20887	0.604437	−	0.796653i	$-0.293398\pi$
$787$	33.9131	1.20887	0.604437	+	0.796653i	$0.293398\pi$
$788$	22.1480	0.788990
$789$	−10.1434	−0.361116
$790$	0	0
$791$	35.2420	1.25306
$792$	−20.3214	−0.722089
$793$	−7.06428	−0.250860
$794$	−14.7008	−0.521711
$795$	0	0
$796$	7.01200	0.248534
$797$	0.730287	0.0258681	0.0129340	−	0.999916i	$-0.495883\pi$
$797$	0.730287	0.0258681	0.0129340	+	0.999916i	$0.495883\pi$
$798$	12.6089	0.446350
$799$	0	0
$800$	0	0
$801$	28.4905	1.00666
$802$	0.298360	0.0105355
$803$	−8.80204	−0.310617
$804$	−5.95095	−0.209874
$805$	0	0
$806$	−10.2559	−0.361248
$807$	10.0320	0.353142
$808$	−12.9777	−0.456553
$809$	−20.4333	−0.718395	−0.359198	−	0.933262i	$-0.616950\pi$
$809$	−20.4333	−0.718395	−0.359198	+	0.933262i	$0.616950\pi$
$810$	0	0
$811$	−2.51695	−0.0883820	−0.0441910	−	0.999023i	$-0.514071\pi$
$811$	−2.51695	−0.0883820	−0.0441910	+	0.999023i	$0.514071\pi$
$812$	−25.8119	−0.905822
$813$	8.90684	0.312376
$814$	19.3055	0.676657
$815$	0	0
$816$	0	0
$817$	8.17265	0.285925
$818$	24.1087	0.842941
$819$	10.4326	0.364546
$820$	0	0
$821$	15.2982	0.533912	0.266956	−	0.963709i	$-0.413982\pi$
$821$	15.2982	0.533912	0.266956	+	0.963709i	$0.413982\pi$
$822$	−14.0368	−0.489590
$823$	−14.3448	−0.500029	−0.250014	−	0.968242i	$-0.580435\pi$
$823$	−14.3448	−0.500029	−0.250014	+	0.968242i	$0.580435\pi$
$824$	2.48598	0.0866033
$825$	0	0
$826$	−33.6717	−1.17159
$827$	40.3144	1.40187	0.700934	−	0.713226i	$-0.252767\pi$
$827$	40.3144	1.40187	0.700934	+	0.713226i	$0.252767\pi$
$828$	7.82361	0.271889
$829$	37.1150	1.28906	0.644528	−	0.764581i	$-0.277054\pi$
$829$	37.1150	1.28906	0.644528	+	0.764581i	$0.277054\pi$
$830$	0	0
$831$	−7.96396	−0.276267
$832$	−1.64280	−0.0569540
$833$	0	0
$834$	−10.3315	−0.357751
$835$	0	0
$836$	−28.1047	−0.972020
$837$	−14.5412	−0.502616
$838$	42.5102	1.46849
$839$	−7.10180	−0.245182	−0.122591	−	0.992457i	$-0.539120\pi$
$839$	−7.10180	−0.245182	−0.122591	+	0.992457i	$0.539120\pi$
$840$	0	0
$841$	14.8890	0.513414
$842$	−10.5089	−0.362161
$843$	4.38657	0.151082
$844$	1.15574	0.0397824
$845$	0	0
$846$	−23.5054	−0.808134
$847$	−63.7771	−2.19141
$848$	45.0048	1.54547
$849$	11.7440	0.403052
$850$	0	0
$851$	4.44050	0.152218
$852$	−6.18629	−0.211939
$853$	38.3399	1.31273	0.656366	−	0.754442i	$-0.272093\pi$
$853$	38.3399	1.31273	0.656366	+	0.754442i	$0.272093\pi$
$854$	31.7460	1.08633
$855$	0	0
$856$	22.9134	0.783164
$857$	−1.73040	−0.0591094	−0.0295547	−	0.999563i	$-0.509409\pi$
$857$	−1.73040	−0.0591094	−0.0295547	+	0.999563i	$0.509409\pi$
$858$	7.09880	0.242349
$859$	30.9245	1.05513	0.527565	−	0.849515i	$-0.323105\pi$
$859$	30.9245	1.05513	0.527565	+	0.849515i	$0.323105\pi$
$860$	0	0
$861$	−10.3542	−0.352868
$862$	−12.5085	−0.426043
$863$	−40.2134	−1.36888	−0.684439	−	0.729070i	$-0.739953\pi$
$863$	−40.2134	−1.36888	−0.684439	+	0.729070i	$0.739953\pi$
$864$	−19.8103	−0.673960
$865$	0	0
$866$	27.8550	0.946550
$867$	0	0
$868$	17.7438	0.602265
$869$	27.5575	0.934824
$870$	0	0
$871$	10.5675	0.358066
$872$	−0.601863	−0.0203816
$873$	−33.2660	−1.12588
$874$	−16.7909	−0.567962
$875$	0	0
$876$	−1.10300	−0.0372670
$877$	−1.01124	−0.0341472	−0.0170736	−	0.999854i	$-0.505435\pi$
$877$	−1.01124	−0.0341472	−0.0170736	+	0.999854i	$0.505435\pi$
$878$	2.65168	0.0894898
$879$	−2.69336	−0.0908446
$880$	0	0
$881$	−19.5296	−0.657968	−0.328984	−	0.944336i	$-0.606706\pi$
$881$	−19.5296	−0.657968	−0.328984	+	0.944336i	$0.606706\pi$
$882$	−12.9957	−0.437589
$883$	−12.6125	−0.424445	−0.212223	−	0.977221i	$-0.568070\pi$
$883$	−12.6125	−0.424445	−0.212223	+	0.977221i	$0.568070\pi$
$884$	0	0
$885$	0	0
$886$	−23.0535	−0.774497
$887$	−37.1151	−1.24620	−0.623102	−	0.782141i	$-0.714128\pi$
$887$	−37.1151	−1.24620	−0.623102	+	0.782141i	$0.714128\pi$
$888$	−1.44534	−0.0485026
$889$	18.2052	0.610581
$890$	0	0
$891$	−35.1308	−1.17693
$892$	1.56350	0.0523498
$893$	19.4218	0.649927
$894$	−16.1425	−0.539886
$895$	0	0
$896$	−31.2332	−1.04343
$897$	1.63281	0.0545180
$898$	−17.5486	−0.585603
$899$	−30.1705	−1.00624
$900$	0	0
$901$	0	0
$902$	59.9463	1.99599
$903$	3.57145	0.118850
$904$	15.2756	0.508059
$905$	0	0
$906$	3.41254	0.113374
$907$	34.4437	1.14368	0.571842	−	0.820364i	$-0.306229\pi$
$907$	34.4437	1.14368	0.571842	+	0.820364i	$0.306229\pi$
$908$	14.5016	0.481254
$909$	−25.8275	−0.856644
$910$	0	0
$911$	−10.1968	−0.337835	−0.168918	−	0.985630i	$-0.554027\pi$
$911$	−10.1968	−0.337835	−0.168918	+	0.985630i	$0.554027\pi$
$912$	11.0913	0.367269
$913$	−22.1480	−0.732992
$914$	−65.3309	−2.16096
$915$	0	0
$916$	14.7085	0.485983
$917$	−60.8804	−2.01045
$918$	0	0
$919$	−50.5315	−1.66688	−0.833440	−	0.552611i	$-0.813632\pi$
$919$	−50.5315	−1.66688	−0.833440	+	0.552611i	$0.813632\pi$
$920$	0	0
$921$	14.0382	0.462573
$922$	−72.2405	−2.37911
$923$	10.9854	0.361589
$924$	−12.2817	−0.404039
$925$	0	0
$926$	−27.7005	−0.910295
$927$	4.94748	0.162496
$928$	−41.1031	−1.34927
$929$	28.3712	0.930830	0.465415	−	0.885093i	$-0.345905\pi$
$929$	28.3712	0.930830	0.465415	+	0.885093i	$0.345905\pi$
$930$	0	0
$931$	10.7380	0.351923
$932$	−14.1528	−0.463591
$933$	2.26493	0.0741504
$934$	35.0695	1.14751
$935$	0	0
$936$	4.52202	0.147807
$937$	47.5024	1.55183	0.775917	−	0.630835i	$-0.217287\pi$
$937$	47.5024	1.55183	0.775917	+	0.630835i	$0.217287\pi$
$938$	−47.4890	−1.55057
$939$	12.7220	0.415166
$940$	0	0
$941$	−46.0404	−1.50087	−0.750437	−	0.660942i	$-0.770157\pi$
$941$	−46.0404	−1.50087	−0.750437	+	0.660942i	$0.770157\pi$
$942$	−3.97205	−0.129416
$943$	13.7884	0.449011
$944$	−29.6190	−0.964016
$945$	0	0
$946$	−20.6772	−0.672274
$947$	−29.7383	−0.966366	−0.483183	−	0.875519i	$-0.660519\pi$
$947$	−29.7383	−0.966366	−0.483183	+	0.875519i	$0.660519\pi$
$948$	3.45329	0.112158
$949$	1.95868	0.0635814
$950$	0	0
$951$	4.64701	0.150690
$952$	0	0
$953$	−9.02109	−0.292222	−0.146111	−	0.989268i	$-0.546676\pi$
$953$	−9.02109	−0.292222	−0.146111	+	0.989268i	$0.546676\pi$
$954$	44.1344	1.42891
$955$	0	0
$956$	6.42903	0.207930
$957$	20.8831	0.675054
$958$	26.2470	0.848002
$959$	−43.1250	−1.39258
$960$	0	0
$961$	−10.2600	−0.330968
$962$	−4.29596	−0.138507
$963$	45.6011	1.46948
$964$	−10.6236	−0.342162
$965$	0	0
$966$	−7.33764	−0.236085
$967$	11.7541	0.377986	0.188993	−	0.981978i	$-0.439478\pi$
$967$	11.7541	0.377986	0.188993	+	0.981978i	$0.439478\pi$
$968$	−27.6441	−0.888516
$969$	0	0
$970$	0	0
$971$	19.9525	0.640306	0.320153	−	0.947366i	$-0.396266\pi$
$971$	19.9525	0.640306	0.320153	+	0.947366i	$0.396266\pi$
$972$	−16.3951	−0.525874
$973$	−31.7413	−1.01758
$974$	53.5544	1.71599
$975$	0	0
$976$	27.9250	0.893859
$977$	27.2005	0.870221	0.435110	−	0.900377i	$-0.356709\pi$
$977$	27.2005	0.870221	0.435110	+	0.900377i	$0.356709\pi$
$978$	−14.9099	−0.476767
$979$	59.5596	1.90353
$980$	0	0
$981$	−1.19780	−0.0382427
$982$	37.9989	1.21259
$983$	−7.12485	−0.227248	−0.113624	−	0.993524i	$-0.536246\pi$
$983$	−7.12485	−0.227248	−0.113624	+	0.993524i	$0.536246\pi$
$984$	−4.48800	−0.143072
$985$	0	0
$986$	0	0
$987$	8.48734	0.270155
$988$	6.25400	0.198966
$989$	−4.75601	−0.151232
$990$	0	0
$991$	56.9670	1.80962	0.904808	−	0.425820i	$-0.140014\pi$
$991$	56.9670	1.80962	0.904808	+	0.425820i	$0.140014\pi$
$992$	28.2554	0.897108
$993$	3.09608	0.0982511
$994$	−49.3670	−1.56583
$995$	0	0
$996$	−2.77542	−0.0879425
$997$	−5.04451	−0.159761	−0.0798806	−	0.996804i	$-0.525454\pi$
$997$	−5.04451	−0.159761	−0.0798806	+	0.996804i	$0.525454\pi$
$998$	−16.9665	−0.537064
$999$	−6.09097	−0.192710

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bp.1.9		12
5.2	odd	4		1445.2.b.f.579.10		12
5.3	odd	4		1445.2.b.f.579.3		12
5.4	even	2	inner	7225.2.a.bp.1.4		12
17.8	even	8		425.2.e.d.251.2		12
17.15	even	8		425.2.e.d.276.5		12
17.16	even	2	inner	7225.2.a.bp.1.10		12
85.8	odd	8		85.2.j.c.64.2	yes	12
85.32	odd	8		85.2.j.c.4.2	✓	12
85.33	odd	4		1445.2.b.f.579.4		12
85.42	odd	8		85.2.j.c.64.5	yes	12
85.49	even	8		425.2.e.d.276.2		12
85.59	even	8		425.2.e.d.251.5		12
85.67	odd	4		1445.2.b.f.579.9		12
85.83	odd	8		85.2.j.c.4.5	yes	12
85.84	even	2	inner	7225.2.a.bp.1.3		12
255.8	even	8		765.2.t.e.64.5		12
255.32	even	8		765.2.t.e.514.5		12
255.83	even	8		765.2.t.e.514.2		12
255.212	even	8		765.2.t.e.64.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.j.c.4.2	✓	12	85.32	odd	8
85.2.j.c.4.5	yes	12	85.83	odd	8
85.2.j.c.64.2	yes	12	85.8	odd	8
85.2.j.c.64.5	yes	12	85.42	odd	8
425.2.e.d.251.2		12	17.8	even	8
425.2.e.d.251.5		12	85.59	even	8
425.2.e.d.276.2		12	85.49	even	8
425.2.e.d.276.5		12	17.15	even	8
765.2.t.e.64.2		12	255.212	even	8
765.2.t.e.64.5		12	255.8	even	8
765.2.t.e.514.2		12	255.83	even	8
765.2.t.e.514.5		12	255.32	even	8
1445.2.b.f.579.3		12	5.3	odd	4
1445.2.b.f.579.4		12	85.33	odd	4
1445.2.b.f.579.9		12	85.67	odd	4
1445.2.b.f.579.10		12	5.2	odd	4
7225.2.a.bp.1.3		12	85.84	even	2	inner
7225.2.a.bp.1.4		12	5.4	even	2	inner
7225.2.a.bp.1.9		12	1.1	even	1	trivial
7225.2.a.bp.1.10		12	17.16	even	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.80333 q^{2} -0.561698 q^{3} +1.25200 q^{4} -1.01293 q^{6} -3.11199 q^{7} -1.34889 q^{8} -2.68450 q^{9} -5.61195 q^{11} -0.703246 q^{12} +1.24880 q^{13} -5.61195 q^{14} -4.93650 q^{16} -4.84103 q^{18} +4.00000 q^{19} +1.74800 q^{21} -10.1202 q^{22} -2.32777 q^{23} +0.757669 q^{24} +2.25200 q^{26} +3.19297 q^{27} -3.89622 q^{28} +6.62488 q^{29} -4.55412 q^{31} -6.20435 q^{32} +3.15222 q^{33} -3.36099 q^{36} -1.90762 q^{37} +7.21332 q^{38} -0.701449 q^{39} -5.92343 q^{41} +3.15222 q^{42} +2.04316 q^{43} -7.02616 q^{44} -4.19774 q^{46} +4.85546 q^{47} +2.77282 q^{48} +2.68450 q^{49} +1.56350 q^{52} -9.11674 q^{53} +5.75798 q^{54} +4.19774 q^{56} -2.24679 q^{57} +11.9468 q^{58} +6.00000 q^{59} -5.65685 q^{61} -8.21258 q^{62} +8.35413 q^{63} -1.31550 q^{64} +5.68450 q^{66} +8.46212 q^{67} +1.30750 q^{69} +8.79676 q^{71} +3.62109 q^{72} +1.56845 q^{73} -3.44007 q^{74} +5.00800 q^{76} +17.4643 q^{77} -1.26494 q^{78} -4.91050 q^{79} +6.26000 q^{81} -10.6819 q^{82} +3.94658 q^{83} +2.18850 q^{84} +3.68450 q^{86} -3.72118 q^{87} +7.56991 q^{88} -10.6130 q^{89} -3.88626 q^{91} -2.91437 q^{92} +2.55804 q^{93} +8.75600 q^{94} +3.48497 q^{96} +12.3919 q^{97} +4.84103 q^{98} +15.0653 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{4} + 28 q^{9} + 4 q^{16} + 48 q^{19} + 24 q^{21} + 24 q^{26} + 68 q^{36} - 28 q^{49} + 72 q^{59} - 76 q^{64} + 8 q^{66} + 88 q^{69} + 48 q^{76} + 60 q^{81} - 40 q^{84} - 16 q^{86} - 16 q^{89}+ \cdots + 96 q^{94}+O(q^{100}) \) 12 * q + 12 * q^4 + 28 * q^9 + 4 * q^16 + 48 * q^19 + 24 * q^21 + 24 * q^26 + 68 * q^36 - 28 * q^49 + 72 * q^59 - 76 * q^64 + 8 * q^66 + 88 * q^69 + 48 * q^76 + 60 * q^81 - 40 * q^84 - 16 * q^86 - 16 * q^89 + 96 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more