LMFDB - Embedded newform 925.2.a.m.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [925,2,Mod(1,925)] 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("925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 925 = 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,5,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.38616218697$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 $ x^9 - 4x^8 - 6x^7 + 30x^6 + 15x^5 - 70x^4 - 22x^3 + 44x^2 + 4x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.15179$ of defining polynomial
Character	$\chi$	$=$	925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.15179 q^{2} +1.93821 q^{3} +2.63020 q^{4} +4.17062 q^{6} +4.19034 q^{7} +1.35605 q^{8} +0.756662 q^{9} +0.890817 q^{11} +5.09787 q^{12} -4.38953 q^{13} +9.01674 q^{14} -2.34246 q^{16} -3.99376 q^{17} +1.62818 q^{18} -2.99524 q^{19} +8.12177 q^{21} +1.91685 q^{22} +3.43248 q^{23} +2.62831 q^{24} -9.44533 q^{26} -4.34806 q^{27} +11.0214 q^{28} +0.0350663 q^{29} +5.73019 q^{31} -7.75258 q^{32} +1.72659 q^{33} -8.59373 q^{34} +1.99017 q^{36} -1.00000 q^{37} -6.44512 q^{38} -8.50783 q^{39} -0.0119268 q^{41} +17.4763 q^{42} -12.8668 q^{43} +2.34302 q^{44} +7.38598 q^{46} +7.16099 q^{47} -4.54019 q^{48} +10.5590 q^{49} -7.74075 q^{51} -11.5453 q^{52} +8.09263 q^{53} -9.35611 q^{54} +5.68230 q^{56} -5.80540 q^{57} +0.0754552 q^{58} -3.75697 q^{59} +13.4979 q^{61} +12.3302 q^{62} +3.17067 q^{63} -11.9970 q^{64} +3.71526 q^{66} +6.95341 q^{67} -10.5044 q^{68} +6.65288 q^{69} -10.7163 q^{71} +1.02607 q^{72} +11.1230 q^{73} -2.15179 q^{74} -7.87806 q^{76} +3.73283 q^{77} -18.3071 q^{78} -7.52510 q^{79} -10.6974 q^{81} -0.0256639 q^{82} +9.42416 q^{83} +21.3618 q^{84} -27.6867 q^{86} +0.0679658 q^{87} +1.20799 q^{88} -11.2080 q^{89} -18.3936 q^{91} +9.02810 q^{92} +11.1063 q^{93} +15.4089 q^{94} -15.0261 q^{96} -3.40068 q^{97} +22.7207 q^{98} +0.674048 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 8 q^{3} + 11 q^{4} + 2 q^{6} + 8 q^{7} + 15 q^{8} + 13 q^{9} + 16 q^{12} + 6 q^{13} - 4 q^{14} + 11 q^{16} + 18 q^{17} - 3 q^{18} - 4 q^{19} + 4 q^{21} + 6 q^{22} + 16 q^{23} + 6 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) $ 9 * q + 5 * q^2 + 8 * q^3 + 11 * q^4 + 2 * q^6 + 8 * q^7 + 15 * q^8 + 13 * q^9 + 16 * q^12 + 6 * q^13 - 4 * q^14 + 11 * q^16 + 18 * q^17 - 3 * q^18 - 4 * q^19 + 4 * q^21 + 6 * q^22 + 16 * q^23 + 6 * q^24 - 6 * q^26 + 32 * q^27 - 20 * q^28 - 2 * q^29 - 6 * q^31 + 35 * q^32 + 6 * q^33 + 6 * q^34 + 11 * q^36 - 9 * q^37 + 4 * q^38 - 16 * q^39 + 2 * q^41 + 4 * q^42 + 14 * q^43 + 4 * q^44 + 16 * q^46 + 34 * q^47 + 6 * q^48 - 7 * q^49 + 8 * q^51 + 16 * q^52 - 4 * q^53 - 18 * q^54 - 24 * q^56 - 4 * q^57 - 22 * q^58 + 10 * q^59 + 8 * q^61 + 26 * q^62 + 18 * q^63 + 15 * q^64 - 34 * q^66 + 16 * q^67 + 36 * q^68 - 10 * q^69 + 8 * q^71 + 9 * q^72 + 8 * q^73 - 5 * q^74 - 48 * q^76 + 14 * q^77 - 76 * q^78 - 26 * q^79 + 13 * q^81 - 4 * q^82 + 70 * q^83 - 24 * q^84 - 16 * q^86 - 12 * q^87 - 36 * q^88 - 8 * q^89 - 18 * q^91 + 16 * q^92 + 2 * q^93 - 26 * q^94 + 52 * q^96 - 2 * q^97 + 21 * q^98 + 8 * 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 $

$2$	2.15179	1.52154	0.760772	−	0.649019i	$-0.224820\pi$
$2$	2.15179	1.52154	0.760772	+	0.649019i	$0.224820\pi$
$3$	1.93821	1.11903	0.559513	−	0.828821i	$-0.310988\pi$
$3$	1.93821	1.11903	0.559513	+	0.828821i	$0.310988\pi$
$4$	2.63020	1.31510
$5$	0	0
$5$	0	0
$6$	4.17062	1.70265
$7$	4.19034	1.58380	0.791901	−	0.610650i	$-0.209092\pi$
$7$	4.19034	1.58380	0.791901	+	0.610650i	$0.209092\pi$
$8$	1.35605	0.479435
$9$	0.756662	0.252221
$10$	0	0
$11$	0.890817	0.268592	0.134296	−	0.990941i	$-0.457123\pi$
$11$	0.890817	0.268592	0.134296	+	0.990941i	$0.457123\pi$
$12$	5.09787	1.47163
$13$	−4.38953	−1.21744	−0.608718	−	0.793387i	$-0.708316\pi$
$13$	−4.38953	−1.21744	−0.608718	+	0.793387i	$0.708316\pi$
$14$	9.01674	2.40982
$15$	0	0
$16$	−2.34246	−0.585616
$17$	−3.99376	−0.968629	−0.484314	−	0.874894i	$-0.660931\pi$
$17$	−3.99376	−0.968629	−0.484314	+	0.874894i	$0.660931\pi$
$18$	1.62818	0.383765
$19$	−2.99524	−0.687155	−0.343577	−	0.939124i	$-0.611639\pi$
$19$	−2.99524	−0.687155	−0.343577	+	0.939124i	$0.611639\pi$
$20$	0	0
$21$	8.12177	1.77232
$22$	1.91685	0.408674
$23$	3.43248	0.715722	0.357861	−	0.933775i	$-0.383506\pi$
$23$	3.43248	0.715722	0.357861	+	0.933775i	$0.383506\pi$
$24$	2.62831	0.536501
$25$	0	0
$26$	−9.44533	−1.85238
$27$	−4.34806	−0.836785
$28$	11.0214	2.08285
$29$	0.0350663	0.00651164	0.00325582	−	0.999995i	$-0.498964\pi$
$29$	0.0350663	0.00651164	0.00325582	+	0.999995i	$0.498964\pi$
$30$	0	0
$31$	5.73019	1.02917	0.514587	−	0.857438i	$-0.327945\pi$
$31$	5.73019	1.02917	0.514587	+	0.857438i	$0.327945\pi$
$32$	−7.75258	−1.37048
$33$	1.72659	0.300561
$34$	−8.59373	−1.47381
$35$	0	0
$36$	1.99017	0.331695
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−6.44512	−1.04554
$39$	−8.50783	−1.36234
$40$	0	0
$41$	−0.0119268	−0.00186265	−0.000931325	−	1.00000i	$-0.500296\pi$
$41$	−0.0119268	−0.00186265	−0.000931325		1.00000i	$0.500296\pi$
$42$	17.4763	2.69666
$43$	−12.8668	−1.96217	−0.981087	−	0.193567i	$-0.937994\pi$
$43$	−12.8668	−1.96217	−0.981087	+	0.193567i	$0.937994\pi$
$44$	2.34302	0.353224
$45$	0	0
$46$	7.38598	1.08900
$47$	7.16099	1.04454	0.522269	−	0.852781i	$-0.325086\pi$
$47$	7.16099	1.04454	0.522269	+	0.852781i	$0.325086\pi$
$48$	−4.54019	−0.655320
$49$	10.5590	1.50843
$50$	0	0
$51$	−7.74075	−1.08392
$52$	−11.5453	−1.60105
$53$	8.09263	1.11161	0.555804	−	0.831313i	$-0.312410\pi$
$53$	8.09263	1.11161	0.555804	+	0.831313i	$0.312410\pi$
$54$	−9.35611	−1.27321
$55$	0	0
$56$	5.68230	0.759330
$57$	−5.80540	−0.768944
$58$	0.0754552	0.00990775
$59$	−3.75697	−0.489115	−0.244558	−	0.969635i	$-0.578643\pi$
$59$	−3.75697	−0.489115	−0.244558	+	0.969635i	$0.578643\pi$
$60$	0	0
$61$	13.4979	1.72823	0.864116	−	0.503293i	$-0.167878\pi$
$61$	13.4979	1.72823	0.864116	+	0.503293i	$0.167878\pi$
$62$	12.3302	1.56593
$63$	3.17067	0.399467
$64$	−11.9970	−1.49962
$65$	0	0
$66$	3.71526	0.457317
$67$	6.95341	0.849494	0.424747	−	0.905312i	$-0.360363\pi$
$67$	6.95341	0.849494	0.424747	+	0.905312i	$0.360363\pi$
$68$	−10.5044	−1.27384
$69$	6.65288	0.800912
$70$	0	0
$71$	−10.7163	−1.27179	−0.635894	−	0.771776i	$-0.719369\pi$
$71$	−10.7163	−1.27179	−0.635894	+	0.771776i	$0.719369\pi$
$72$	1.02607	0.120923
$73$	11.1230	1.30184	0.650922	−	0.759145i	$-0.274383\pi$
$73$	11.1230	1.30184	0.650922	+	0.759145i	$0.274383\pi$
$74$	−2.15179	−0.250140
$75$	0	0
$76$	−7.87806	−0.903675
$77$	3.73283	0.425396
$78$	−18.3071	−2.07287
$79$	−7.52510	−0.846640	−0.423320	−	0.905980i	$-0.639135\pi$
$79$	−7.52510	−0.846640	−0.423320	+	0.905980i	$0.639135\pi$
$80$	0	0
$81$	−10.6974	−1.18861
$82$	−0.0256639	−0.00283411
$83$	9.42416	1.03444	0.517218	−	0.855854i	$-0.326968\pi$
$83$	9.42416	1.03444	0.517218	+	0.855854i	$0.326968\pi$
$84$	21.3618	2.33077
$85$	0	0
$86$	−27.6867	−2.98553
$87$	0.0679658	0.00728670
$88$	1.20799	0.128772
$89$	−11.2080	−1.18805	−0.594023	−	0.804448i	$-0.702461\pi$
$89$	−11.2080	−1.18805	−0.594023	+	0.804448i	$0.702461\pi$
$90$	0	0
$91$	−18.3936	−1.92818
$92$	9.02810	0.941245
$93$	11.1063	1.15167
$94$	15.4089	1.58931
$95$	0	0
$96$	−15.0261	−1.53360
$97$	−3.40068	−0.345286	−0.172643	−	0.984984i	$-0.555231\pi$
$97$	−3.40068	−0.345286	−0.172643	+	0.984984i	$0.555231\pi$
$98$	22.7207	2.29514
$99$	0.674048	0.0677443
$100$	0	0
$101$	7.39980	0.736307	0.368154	−	0.929765i	$-0.379990\pi$
$101$	7.39980	0.736307	0.368154	+	0.929765i	$0.379990\pi$
$102$	−16.6565	−1.64923
$103$	2.88273	0.284044	0.142022	−	0.989863i	$-0.454640\pi$
$103$	2.88273	0.284044	0.142022	+	0.989863i	$0.454640\pi$
$104$	−5.95240	−0.583681
$105$	0	0
$106$	17.4136	1.69136
$107$	19.5051	1.88563	0.942816	−	0.333313i	$-0.108167\pi$
$107$	19.5051	1.88563	0.942816	+	0.333313i	$0.108167\pi$
$108$	−11.4363	−1.10045
$109$	3.34203	0.320108	0.160054	−	0.987108i	$-0.448833\pi$
$109$	3.34203	0.320108	0.160054	+	0.987108i	$0.448833\pi$
$110$	0	0
$111$	−1.93821	−0.183967
$112$	−9.81573	−0.927499
$113$	3.88547	0.365514	0.182757	−	0.983158i	$-0.441498\pi$
$113$	3.88547	0.365514	0.182757	+	0.983158i	$0.441498\pi$
$114$	−12.4920	−1.16998
$115$	0	0
$116$	0.0922311	0.00856344
$117$	−3.32139	−0.307062
$118$	−8.08420	−0.744211
$119$	−16.7352	−1.53412
$120$	0	0
$121$	−10.2064	−0.927859
$122$	29.0447	2.62958
$123$	−0.0231166	−0.00208436
$124$	15.0715	1.35346
$125$	0	0
$126$	6.82262	0.607807
$127$	−18.7816	−1.66660	−0.833298	−	0.552824i	$-0.813550\pi$
$127$	−18.7816	−1.66660	−0.833298	+	0.552824i	$0.813550\pi$
$128$	−10.3098	−0.911269
$129$	−24.9386	−2.19572
$130$	0	0
$131$	0.375087	0.0327715	0.0163858	−	0.999866i	$-0.494784\pi$
$131$	0.375087	0.0327715	0.0163858	+	0.999866i	$0.494784\pi$
$132$	4.54127	0.395267
$133$	−12.5511	−1.08832
$134$	14.9623	1.29254
$135$	0	0
$136$	−5.41573	−0.464395
$137$	−13.7889	−1.17806	−0.589032	−	0.808110i	$-0.700491\pi$
$137$	−13.7889	−1.17806	−0.589032	+	0.808110i	$0.700491\pi$
$138$	14.3156	1.21862
$139$	−10.6223	−0.900975	−0.450487	−	0.892783i	$-0.648750\pi$
$139$	−10.6223	−0.900975	−0.450487	+	0.892783i	$0.648750\pi$
$140$	0	0
$141$	13.8795	1.16887
$142$	−23.0592	−1.93508
$143$	−3.91027	−0.326993
$144$	−1.77245	−0.147704
$145$	0	0
$146$	23.9342	1.98081
$147$	20.4655	1.68797
$148$	−2.63020	−0.216201
$149$	8.30509	0.680380	0.340190	−	0.940357i	$-0.389509\pi$
$149$	8.30509	0.680380	0.340190	+	0.940357i	$0.389509\pi$
$150$	0	0
$151$	−20.7165	−1.68589	−0.842944	−	0.538001i	$-0.819180\pi$
$151$	−20.7165	−1.68589	−0.842944	+	0.538001i	$0.819180\pi$
$152$	−4.06168	−0.329446
$153$	−3.02193	−0.244308
$154$	8.03227	0.647258
$155$	0	0
$156$	−22.3773	−1.79161
$157$	5.00991	0.399834	0.199917	−	0.979813i	$-0.435933\pi$
$157$	5.00991	0.399834	0.199917	+	0.979813i	$0.435933\pi$
$158$	−16.1924	−1.28820
$159$	15.6852	1.24392
$160$	0	0
$161$	14.3833	1.13356
$162$	−23.0187	−1.80852
$163$	−5.93480	−0.464849	−0.232425	−	0.972614i	$-0.574666\pi$
$163$	−5.93480	−0.464849	−0.232425	+	0.972614i	$0.574666\pi$
$164$	−0.0313698	−0.00244957
$165$	0	0
$166$	20.2788	1.57394
$167$	−0.513319	−0.0397218	−0.0198609	−	0.999803i	$-0.506322\pi$
$167$	−0.513319	−0.0397218	−0.0198609	+	0.999803i	$0.506322\pi$
$168$	11.0135	0.849710
$169$	6.26794	0.482149
$170$	0	0
$171$	−2.26638	−0.173315
$172$	−33.8423	−2.58045
$173$	−0.177838	−0.0135208	−0.00676040	−	0.999977i	$-0.502152\pi$
$173$	−0.177838	−0.0135208	−0.00676040	+	0.999977i	$0.502152\pi$
$174$	0.146248	0.0110870
$175$	0	0
$176$	−2.08671	−0.157291
$177$	−7.28179	−0.547333
$178$	−24.1173	−1.80767
$179$	24.1056	1.80174	0.900869	−	0.434090i	$-0.142930\pi$
$179$	24.1056	1.80174	0.900869	+	0.434090i	$0.142930\pi$
$180$	0	0
$181$	−17.7038	−1.31591	−0.657955	−	0.753057i	$-0.728578\pi$
$181$	−17.7038	−1.31591	−0.657955	+	0.753057i	$0.728578\pi$
$182$	−39.5792	−2.93381
$183$	26.1618	1.93394
$184$	4.65461	0.343142
$185$	0	0
$186$	23.8985	1.75232
$187$	−3.55771	−0.260166
$188$	18.8348	1.37367
$189$	−18.2199	−1.32530
$190$	0	0
$191$	−3.05809	−0.221276	−0.110638	−	0.993861i	$-0.535289\pi$
$191$	−3.05809	−0.221276	−0.110638	+	0.993861i	$0.535289\pi$
$192$	−23.2527	−1.67812
$193$	11.2448	0.809420	0.404710	−	0.914445i	$-0.367372\pi$
$193$	11.2448	0.809420	0.404710	+	0.914445i	$0.367372\pi$
$194$	−7.31754	−0.525369
$195$	0	0
$196$	27.7722	1.98373
$197$	24.6046	1.75301	0.876503	−	0.481397i	$-0.159870\pi$
$197$	24.6046	1.75301	0.876503	+	0.481397i	$0.159870\pi$
$198$	1.45041	0.103076
$199$	10.9154	0.773771	0.386885	−	0.922128i	$-0.373551\pi$
$199$	10.9154	0.773771	0.386885	+	0.922128i	$0.373551\pi$
$200$	0	0
$201$	13.4772	0.950606
$202$	15.9228	1.12032
$203$	0.146940	0.0103131
$204$	−20.3597	−1.42546
$205$	0	0
$206$	6.20303	0.432186
$207$	2.59723	0.180520
$208$	10.2823	0.712950
$209$	−2.66821	−0.184564
$210$	0	0
$211$	5.92367	0.407802	0.203901	−	0.978992i	$-0.434638\pi$
$211$	5.92367	0.407802	0.203901	+	0.978992i	$0.434638\pi$
$212$	21.2852	1.46187
$213$	−20.7704	−1.42317
$214$	41.9709	2.86907
$215$	0	0
$216$	−5.89618	−0.401184
$217$	24.0115	1.63001
$218$	7.19134	0.487059
$219$	21.5586	1.45680
$220$	0	0
$221$	17.5307	1.17924
$222$	−4.17062	−0.279914
$223$	−12.2403	−0.819670	−0.409835	−	0.912160i	$-0.634414\pi$
$223$	−12.2403	−0.819670	−0.409835	+	0.912160i	$0.634414\pi$
$224$	−32.4860	−2.17056
$225$	0	0
$226$	8.36071	0.556146
$227$	−4.20072	−0.278812	−0.139406	−	0.990235i	$-0.544519\pi$
$227$	−4.20072	−0.278812	−0.139406	+	0.990235i	$0.544519\pi$
$228$	−15.2693	−1.01124
$229$	2.37350	0.156846	0.0784228	−	0.996920i	$-0.475012\pi$
$229$	2.37350	0.156846	0.0784228	+	0.996920i	$0.475012\pi$
$230$	0	0
$231$	7.23502	0.476029
$232$	0.0475515	0.00312191
$233$	21.0060	1.37615	0.688076	−	0.725639i	$-0.258456\pi$
$233$	21.0060	1.37615	0.688076	+	0.725639i	$0.258456\pi$
$234$	−7.14693	−0.467209
$235$	0	0
$236$	−9.88156	−0.643234
$237$	−14.5852	−0.947413
$238$	−36.0107	−2.33423
$239$	−11.1938	−0.724067	−0.362034	−	0.932165i	$-0.617917\pi$
$239$	−11.1938	−0.724067	−0.362034	+	0.932165i	$0.617917\pi$
$240$	0	0
$241$	3.09667	0.199474	0.0997372	−	0.995014i	$-0.468200\pi$
$241$	3.09667	0.199474	0.0997372	+	0.995014i	$0.468200\pi$
$242$	−21.9621	−1.41178
$243$	−7.68973	−0.493296
$244$	35.5022	2.27279
$245$	0	0
$246$	−0.0497421	−0.00317144
$247$	13.1477	0.836566
$248$	7.77041	0.493422
$249$	18.2660	1.15756
$250$	0	0
$251$	−2.10677	−0.132978	−0.0664891	−	0.997787i	$-0.521180\pi$
$251$	−2.10677	−0.132978	−0.0664891	+	0.997787i	$0.521180\pi$
$252$	8.33949	0.525339
$253$	3.05772	0.192237
$254$	−40.4140	−2.53580
$255$	0	0
$256$	1.80941	0.113088
$257$	−5.91601	−0.369030	−0.184515	−	0.982830i	$-0.559071\pi$
$257$	−5.91601	−0.369030	−0.184515	+	0.982830i	$0.559071\pi$
$258$	−53.6627	−3.34089
$259$	−4.19034	−0.260375
$260$	0	0
$261$	0.0265333	0.00164237
$262$	0.807108	0.0498633
$263$	−5.57668	−0.343873	−0.171936	−	0.985108i	$-0.555002\pi$
$263$	−5.57668	−0.343873	−0.171936	+	0.985108i	$0.555002\pi$
$264$	2.34134	0.144100
$265$	0	0
$266$	−27.0073	−1.65592
$267$	−21.7235	−1.32946
$268$	18.2888	1.11717
$269$	−9.27624	−0.565583	−0.282791	−	0.959181i	$-0.591260\pi$
$269$	−9.27624	−0.565583	−0.282791	+	0.959181i	$0.591260\pi$
$270$	0	0
$271$	26.4067	1.60409	0.802047	−	0.597261i	$-0.203744\pi$
$271$	26.4067	1.60409	0.802047	+	0.597261i	$0.203744\pi$
$272$	9.35524	0.567244
$273$	−35.6507	−2.15768
$274$	−29.6708	−1.79248
$275$	0	0
$276$	17.4984	1.05328
$277$	−0.119318	−0.00716913	−0.00358456	−	0.999994i	$-0.501141\pi$
$277$	−0.119318	−0.00716913	−0.00358456	+	0.999994i	$0.501141\pi$
$278$	−22.8570	−1.37087
$279$	4.33582	0.259579
$280$	0	0
$281$	−9.48414	−0.565776	−0.282888	−	0.959153i	$-0.591293\pi$
$281$	−9.48414	−0.565776	−0.282888	+	0.959153i	$0.591293\pi$
$282$	29.8658	1.77848
$283$	15.8066	0.939604	0.469802	−	0.882772i	$-0.344325\pi$
$283$	15.8066	0.939604	0.469802	+	0.882772i	$0.344325\pi$
$284$	−28.1859	−1.67253
$285$	0	0
$286$	−8.41407	−0.497534
$287$	−0.0499773	−0.00295007
$288$	−5.86608	−0.345662
$289$	−1.04989	−0.0617580
$290$	0	0
$291$	−6.59123	−0.386385
$292$	29.2555	1.71205
$293$	21.7705	1.27185	0.635924	−	0.771752i	$-0.280619\pi$
$293$	21.7705	1.27185	0.635924	+	0.771752i	$0.280619\pi$
$294$	44.0375	2.56832
$295$	0	0
$296$	−1.35605	−0.0788186
$297$	−3.87333	−0.224753
$298$	17.8708	1.03523
$299$	−15.0670	−0.871346
$300$	0	0
$301$	−53.9165	−3.10769
$302$	−44.5776	−2.56515
$303$	14.3424	0.823947
$304$	7.01623	0.402409
$305$	0	0
$306$	−6.50255	−0.371726
$307$	14.3387	0.818352	0.409176	−	0.912455i	$-0.365816\pi$
$307$	14.3387	0.818352	0.409176	+	0.912455i	$0.365816\pi$
$308$	9.81808	0.559437
$309$	5.58734	0.317853
$310$	0	0
$311$	−14.9445	−0.847422	−0.423711	−	0.905797i	$-0.639273\pi$
$311$	−14.9445	−0.847422	−0.423711	+	0.905797i	$0.639273\pi$
$312$	−11.5370	−0.653155
$313$	−12.8158	−0.724393	−0.362197	−	0.932102i	$-0.617973\pi$
$313$	−12.8158	−0.724393	−0.362197	+	0.932102i	$0.617973\pi$
$314$	10.7803	0.608365
$315$	0	0
$316$	−19.7925	−1.11341
$317$	−14.0723	−0.790378	−0.395189	−	0.918600i	$-0.629321\pi$
$317$	−14.0723	−0.790378	−0.395189	+	0.918600i	$0.629321\pi$
$318$	33.7513	1.89268
$319$	0.0312376	0.00174897
$320$	0	0
$321$	37.8051	2.11007
$322$	30.9498	1.72476
$323$	11.9623	0.665598
$324$	−28.1364	−1.56313
$325$	0	0
$326$	−12.7704	−0.707289
$327$	6.47755	0.358210
$328$	−0.0161733	−0.000893020 0
$329$	30.0070	1.65434
$330$	0	0
$331$	14.7404	0.810203	0.405102	−	0.914272i	$-0.367236\pi$
$331$	14.7404	0.810203	0.405102	+	0.914272i	$0.367236\pi$
$332$	24.7874	1.36038
$333$	−0.756662	−0.0414648
$334$	−1.10455	−0.0604385
$335$	0	0
$336$	−19.0250	−1.03790
$337$	−26.2754	−1.43131	−0.715657	−	0.698452i	$-0.753873\pi$
$337$	−26.2754	−1.43131	−0.715657	+	0.698452i	$0.753873\pi$
$338$	13.4873	0.733612
$339$	7.53086	0.409020
$340$	0	0
$341$	5.10456	0.276427
$342$	−4.87678	−0.263706
$343$	14.9134	0.805246
$344$	−17.4480	−0.940735
$345$	0	0
$346$	−0.382670	−0.0205725
$347$	13.1067	0.703604	0.351802	−	0.936074i	$-0.385569\pi$
$347$	13.1067	0.703604	0.351802	+	0.936074i	$0.385569\pi$
$348$	0.178763	0.00958272
$349$	32.6773	1.74917	0.874587	−	0.484868i	$-0.161133\pi$
$349$	32.6773	1.74917	0.874587	+	0.484868i	$0.161133\pi$
$350$	0	0
$351$	19.0859	1.01873
$352$	−6.90613	−0.368098
$353$	10.4045	0.553776	0.276888	−	0.960902i	$-0.410697\pi$
$353$	10.4045	0.553776	0.276888	+	0.960902i	$0.410697\pi$
$354$	−15.6689	−0.832792
$355$	0	0
$356$	−29.4792	−1.56240
$357$	−32.4364	−1.71672
$358$	51.8702	2.74143
$359$	−1.71765	−0.0906538	−0.0453269	−	0.998972i	$-0.514433\pi$
$359$	−1.71765	−0.0906538	−0.0453269	+	0.998972i	$0.514433\pi$
$360$	0	0
$361$	−10.0286	−0.527819
$362$	−38.0948	−2.00222
$363$	−19.7822	−1.03830
$364$	−48.3788	−2.53574
$365$	0	0
$366$	56.2947	2.94257
$367$	8.41206	0.439106	0.219553	−	0.975601i	$-0.429540\pi$
$367$	8.41206	0.439106	0.219553	+	0.975601i	$0.429540\pi$
$368$	−8.04047	−0.419138
$369$	−0.00902454	−0.000469799 0
$370$	0	0
$371$	33.9109	1.76057
$372$	29.2118	1.51456
$373$	13.3481	0.691138	0.345569	−	0.938393i	$-0.387686\pi$
$373$	13.3481	0.691138	0.345569	+	0.938393i	$0.387686\pi$
$374$	−7.65544	−0.395853
$375$	0	0
$376$	9.71064	0.500788
$377$	−0.153924	−0.00792750
$378$	−39.2053	−2.01650
$379$	−18.0051	−0.924861	−0.462430	−	0.886656i	$-0.653022\pi$
$379$	−18.0051	−0.924861	−0.462430	+	0.886656i	$0.653022\pi$
$380$	0	0
$381$	−36.4027	−1.86497
$382$	−6.58036	−0.336681
$383$	−9.11385	−0.465696	−0.232848	−	0.972513i	$-0.574804\pi$
$383$	−9.11385	−0.465696	−0.232848	+	0.972513i	$0.574804\pi$
$384$	−19.9826	−1.01973
$385$	0	0
$386$	24.1965	1.23157
$387$	−9.73585	−0.494901
$388$	−8.94444	−0.454085
$389$	12.6152	0.639617	0.319809	−	0.947482i	$-0.396381\pi$
$389$	12.6152	0.639617	0.319809	+	0.947482i	$0.396381\pi$
$390$	0	0
$391$	−13.7085	−0.693269
$392$	14.3185	0.723192
$393$	0.726998	0.0366722
$394$	52.9439	2.66728
$395$	0	0
$396$	1.77288	0.0890904
$397$	−24.4702	−1.22813	−0.614063	−	0.789257i	$-0.710466\pi$
$397$	−24.4702	−1.22813	−0.614063	+	0.789257i	$0.710466\pi$
$398$	23.4876	1.17733
$399$	−24.3266	−1.21785
$400$	0	0
$401$	−6.71024	−0.335094	−0.167547	−	0.985864i	$-0.553585\pi$
$401$	−6.71024	−0.335094	−0.167547	+	0.985864i	$0.553585\pi$
$402$	29.0000	1.44639
$403$	−25.1528	−1.25295
$404$	19.4629	0.968316
$405$	0	0
$406$	0.316183	0.0156919
$407$	−0.890817	−0.0441562
$408$	−10.4968	−0.519670
$409$	−0.917573	−0.0453710	−0.0226855	−	0.999743i	$-0.507222\pi$
$409$	−0.917573	−0.0453710	−0.0226855	+	0.999743i	$0.507222\pi$
$410$	0	0
$411$	−26.7258	−1.31829
$412$	7.58215	0.373546
$413$	−15.7430	−0.774662
$414$	5.58869	0.274669
$415$	0	0
$416$	34.0302	1.66847
$417$	−20.5883	−1.00822
$418$	−5.74142	−0.280822
$419$	9.21985	0.450419	0.225210	−	0.974310i	$-0.427693\pi$
$419$	9.21985	0.450419	0.225210	+	0.974310i	$0.427693\pi$
$420$	0	0
$421$	3.15954	0.153987	0.0769933	−	0.997032i	$-0.475468\pi$
$421$	3.15954	0.153987	0.0769933	+	0.997032i	$0.475468\pi$
$422$	12.7465	0.620489
$423$	5.41845	0.263454
$424$	10.9740	0.532944
$425$	0	0
$426$	−44.6935	−2.16541
$427$	56.5609	2.73718
$428$	51.3023	2.47979
$429$	−7.57892	−0.365914
$430$	0	0
$431$	−10.0895	−0.485996	−0.242998	−	0.970027i	$-0.578131\pi$
$431$	−10.0895	−0.485996	−0.242998	+	0.970027i	$0.578131\pi$
$432$	10.1852	0.490035
$433$	18.1592	0.872673	0.436337	−	0.899784i	$-0.356276\pi$
$433$	18.1592	0.872673	0.436337	+	0.899784i	$0.356276\pi$
$434$	51.6676	2.48013
$435$	0	0
$436$	8.79018	0.420973
$437$	−10.2811	−0.491812
$438$	46.3896	2.21658
$439$	−25.8313	−1.23286	−0.616430	−	0.787410i	$-0.711422\pi$
$439$	−25.8313	−1.23286	−0.616430	+	0.787410i	$0.711422\pi$
$440$	0	0
$441$	7.98958	0.380456
$442$	37.7224	1.79427
$443$	39.8319	1.89247	0.946235	−	0.323480i	$-0.104853\pi$
$443$	39.8319	1.89247	0.946235	+	0.323480i	$0.104853\pi$
$444$	−5.09787	−0.241934
$445$	0	0
$446$	−26.3385	−1.24716
$447$	16.0970	0.761363
$448$	−50.2715	−2.37511
$449$	−34.7543	−1.64016	−0.820078	−	0.572252i	$-0.806070\pi$
$449$	−34.7543	−1.64016	−0.820078	+	0.572252i	$0.806070\pi$
$450$	0	0
$451$	−0.0106246	−0.000500292 0
$452$	10.2195	0.480687
$453$	−40.1530	−1.88655
$454$	−9.03907	−0.424225
$455$	0	0
$456$	−7.87240	−0.368659
$457$	17.2797	0.808310	0.404155	−	0.914691i	$-0.367566\pi$
$457$	17.2797	0.808310	0.404155	+	0.914691i	$0.367566\pi$
$458$	5.10728	0.238648
$459$	17.3651	0.810534
$460$	0	0
$461$	−6.98823	−0.325474	−0.162737	−	0.986669i	$-0.552032\pi$
$461$	−6.98823	−0.325474	−0.162737	+	0.986669i	$0.552032\pi$
$462$	15.5682	0.724299
$463$	−30.0184	−1.39507	−0.697537	−	0.716549i	$-0.745721\pi$
$463$	−30.0184	−1.39507	−0.697537	+	0.716549i	$0.745721\pi$
$464$	−0.0821414	−0.00381332
$465$	0	0
$466$	45.2006	2.09388
$467$	18.5779	0.859685	0.429842	−	0.902904i	$-0.358569\pi$
$467$	18.5779	0.859685	0.429842	+	0.902904i	$0.358569\pi$
$468$	−8.73590	−0.403817
$469$	29.1372	1.34543
$470$	0	0
$471$	9.71025	0.447425
$472$	−5.09462	−0.234499
$473$	−11.4620	−0.527023
$474$	−31.3843	−1.44153
$475$	0	0
$476$	−44.0169	−2.01751
$477$	6.12339	0.280371
$478$	−24.0867	−1.10170
$479$	−4.63021	−0.211560	−0.105780	−	0.994390i	$-0.533734\pi$
$479$	−4.63021	−0.211560	−0.105780	+	0.994390i	$0.533734\pi$
$480$	0	0
$481$	4.38953	0.200145
$482$	6.66339	0.303509
$483$	27.8778	1.26849
$484$	−26.8449	−1.22022
$485$	0	0
$486$	−16.5467	−0.750572
$487$	−20.5716	−0.932187	−0.466093	−	0.884735i	$-0.654339\pi$
$487$	−20.5716	−0.932187	−0.466093	+	0.884735i	$0.654339\pi$
$488$	18.3038	0.828575
$489$	−11.5029	−0.520179
$490$	0	0
$491$	−5.71743	−0.258024	−0.129012	−	0.991643i	$-0.541181\pi$
$491$	−5.71743	−0.258024	−0.129012	+	0.991643i	$0.541181\pi$
$492$	−0.0608012	−0.00274113
$493$	−0.140046	−0.00630736
$494$	28.2910	1.27287
$495$	0	0
$496$	−13.4228	−0.602700
$497$	−44.9049	−2.01426
$498$	39.3046	1.76128
$499$	−16.5695	−0.741751	−0.370876	−	0.928683i	$-0.620942\pi$
$499$	−16.5695	−0.741751	−0.370876	+	0.928683i	$0.620942\pi$
$500$	0	0
$501$	−0.994920	−0.0444498
$502$	−4.53332	−0.202332
$503$	37.3268	1.66432	0.832159	−	0.554536i	$-0.187104\pi$
$503$	37.3268	1.66432	0.832159	+	0.554536i	$0.187104\pi$
$504$	4.29958	0.191519
$505$	0	0
$506$	6.57956	0.292497
$507$	12.1486	0.539538
$508$	−49.3992	−2.19174
$509$	6.65148	0.294822	0.147411	−	0.989075i	$-0.452906\pi$
$509$	6.65148	0.294822	0.147411	+	0.989075i	$0.452906\pi$
$510$	0	0
$511$	46.6090	2.06186
$512$	24.5131	1.08334
$513$	13.0235	0.575001
$514$	−12.7300	−0.561496
$515$	0	0
$516$	−65.5935	−2.88759
$517$	6.37914	0.280554
$518$	−9.01674	−0.396173
$519$	−0.344688	−0.0151301
$520$	0	0
$521$	7.20458	0.315638	0.157819	−	0.987468i	$-0.449554\pi$
$521$	7.20458	0.315638	0.157819	+	0.987468i	$0.449554\pi$
$522$	0.0570941	0.00249894
$523$	22.9924	1.00539	0.502695	−	0.864464i	$-0.332342\pi$
$523$	22.9924	1.00539	0.502695	+	0.864464i	$0.332342\pi$
$524$	0.986553	0.0430978
$525$	0	0
$526$	−11.9998	−0.523218
$527$	−22.8850	−0.996887
$528$	−4.04448	−0.176013
$529$	−11.2181	−0.487742
$530$	0	0
$531$	−2.84275	−0.123365
$532$	−33.0118	−1.43124
$533$	0.0523529	0.00226766
$534$	−46.7443	−2.02283
$535$	0	0
$536$	9.42915	0.407277
$537$	46.7218	2.01619
$538$	−19.9605	−0.860559
$539$	9.40613	0.405151
$540$	0	0
$541$	−22.8649	−0.983037	−0.491519	−	0.870867i	$-0.663558\pi$
$541$	−22.8649	−0.983037	−0.491519	+	0.870867i	$0.663558\pi$
$542$	56.8217	2.44070
$543$	−34.3136	−1.47254
$544$	30.9619	1.32748
$545$	0	0
$546$	−76.7128	−3.28301
$547$	−1.23359	−0.0527444	−0.0263722	−	0.999652i	$-0.508396\pi$
$547$	−1.23359	−0.0527444	−0.0263722	+	0.999652i	$0.508396\pi$
$548$	−36.2675	−1.54927
$549$	10.2134	0.435896
$550$	0	0
$551$	−0.105032	−0.00447450
$552$	9.02162	0.383985
$553$	−31.5328	−1.34091
$554$	−0.256747	−0.0109081
$555$	0	0
$556$	−27.9388	−1.18487
$557$	32.4356	1.37434	0.687170	−	0.726497i	$-0.258853\pi$
$557$	32.4356	1.37434	0.687170	+	0.726497i	$0.258853\pi$
$558$	9.32977	0.394961
$559$	56.4793	2.38882
$560$	0	0
$561$	−6.89559	−0.291132
$562$	−20.4079	−0.860854
$563$	−15.9666	−0.672914	−0.336457	−	0.941699i	$-0.609229\pi$
$563$	−15.9666	−0.672914	−0.336457	+	0.941699i	$0.609229\pi$
$564$	36.5058	1.53717
$565$	0	0
$566$	34.0124	1.42965
$567$	−44.8260	−1.88251
$568$	−14.5318	−0.609740
$569$	44.6043	1.86991	0.934954	−	0.354769i	$-0.115440\pi$
$569$	44.6043	1.86991	0.934954	+	0.354769i	$0.115440\pi$
$570$	0	0
$571$	7.68699	0.321690	0.160845	−	0.986980i	$-0.448578\pi$
$571$	7.68699	0.321690	0.160845	+	0.986980i	$0.448578\pi$
$572$	−10.2848	−0.430028
$573$	−5.92722	−0.247613
$574$	−0.107541	−0.00448866
$575$	0	0
$576$	−9.07767	−0.378236
$577$	35.3204	1.47041	0.735204	−	0.677846i	$-0.237086\pi$
$577$	35.3204	1.47041	0.735204	+	0.677846i	$0.237086\pi$
$578$	−2.25913	−0.0939676
$579$	21.7948	0.905763
$580$	0	0
$581$	39.4905	1.63834
$582$	−14.1829	−0.587901
$583$	7.20906	0.298569
$584$	15.0832	0.624149
$585$	0	0
$586$	46.8456	1.93517
$587$	20.2513	0.835860	0.417930	−	0.908479i	$-0.362756\pi$
$587$	20.2513	0.835860	0.417930	+	0.908479i	$0.362756\pi$
$588$	53.8284	2.21984
$589$	−17.1633	−0.707201
$590$	0	0
$591$	47.6889	1.96166
$592$	2.34246	0.0962747
$593$	−5.54331	−0.227636	−0.113818	−	0.993502i	$-0.536308\pi$
$593$	−5.54331	−0.227636	−0.113818	+	0.993502i	$0.536308\pi$
$594$	−8.33459	−0.341972
$595$	0	0
$596$	21.8440	0.894766
$597$	21.1563	0.865870
$598$	−32.4210	−1.32579
$599$	−10.5067	−0.429292	−0.214646	−	0.976692i	$-0.568860\pi$
$599$	−10.5067	−0.429292	−0.214646	+	0.976692i	$0.568860\pi$
$600$	0	0
$601$	−20.4154	−0.832763	−0.416381	−	0.909190i	$-0.636702\pi$
$601$	−20.4154	−0.832763	−0.416381	+	0.909190i	$0.636702\pi$
$602$	−116.017	−4.72849
$603$	5.26138	0.214260
$604$	−54.4885	−2.21711
$605$	0	0
$606$	30.8617	1.25367
$607$	7.54513	0.306248	0.153124	−	0.988207i	$-0.451067\pi$
$607$	7.54513	0.306248	0.153124	+	0.988207i	$0.451067\pi$
$608$	23.2208	0.941729
$609$	0.284800	0.0115407
$610$	0	0
$611$	−31.4334	−1.27166
$612$	−7.94826	−0.321289
$613$	17.8383	0.720481	0.360241	−	0.932859i	$-0.382695\pi$
$613$	17.8383	0.720481	0.360241	+	0.932859i	$0.382695\pi$
$614$	30.8538	1.24516
$615$	0	0
$616$	5.06190	0.203950
$617$	−19.4501	−0.783030	−0.391515	−	0.920172i	$-0.628049\pi$
$617$	−19.4501	−0.783030	−0.391515	+	0.920172i	$0.628049\pi$
$618$	12.0228	0.483627
$619$	−30.5238	−1.22686	−0.613428	−	0.789750i	$-0.710210\pi$
$619$	−30.5238	−1.22686	−0.613428	+	0.789750i	$0.710210\pi$
$620$	0	0
$621$	−14.9247	−0.598906
$622$	−32.1573	−1.28939
$623$	−46.9654	−1.88163
$624$	19.9293	0.797810
$625$	0	0
$626$	−27.5769	−1.10220
$627$	−5.17155	−0.206532
$628$	13.1770	0.525821
$629$	3.99376	0.159242
$630$	0	0
$631$	−21.4049	−0.852116	−0.426058	−	0.904696i	$-0.640098\pi$
$631$	−21.4049	−0.852116	−0.426058	+	0.904696i	$0.640098\pi$
$632$	−10.2044	−0.405909
$633$	11.4813	0.456341
$634$	−30.2806	−1.20259
$635$	0	0
$636$	41.2552	1.63588
$637$	−46.3489	−1.83641
$638$	0.0672168	0.00266114
$639$	−8.10860	−0.320771
$640$	0	0
$641$	48.4290	1.91283	0.956415	−	0.292011i	$-0.0943242\pi$
$641$	48.4290	1.91283	0.956415	+	0.292011i	$0.0943242\pi$
$642$	81.3485	3.21057
$643$	35.8989	1.41571	0.707856	−	0.706356i	$-0.249662\pi$
$643$	35.8989	1.41571	0.707856	+	0.706356i	$0.249662\pi$
$644$	37.8309	1.49074
$645$	0	0
$646$	25.7403	1.01274
$647$	15.5320	0.610628	0.305314	−	0.952252i	$-0.401239\pi$
$647$	15.5320	0.610628	0.305314	+	0.952252i	$0.401239\pi$
$648$	−14.5062	−0.569859
$649$	−3.34677	−0.131372
$650$	0	0
$651$	46.5393	1.82402
$652$	−15.6097	−0.611322
$653$	−32.5029	−1.27194	−0.635968	−	0.771716i	$-0.719399\pi$
$653$	−32.5029	−1.27194	−0.635968	+	0.771716i	$0.719399\pi$
$654$	13.9383	0.545032
$655$	0	0
$656$	0.0279381	0.00109080
$657$	8.41632	0.328352
$658$	64.5688	2.51715
$659$	−22.8242	−0.889104	−0.444552	−	0.895753i	$-0.646637\pi$
$659$	−22.8242	−0.889104	−0.444552	+	0.895753i	$0.646637\pi$
$660$	0	0
$661$	−29.2556	−1.13791	−0.568955	−	0.822369i	$-0.692652\pi$
$661$	−29.2556	−1.13791	−0.568955	+	0.822369i	$0.692652\pi$
$662$	31.7181	1.23276
$663$	33.9782	1.31960
$664$	12.7796	0.495945
$665$	0	0
$666$	−1.62818	−0.0630906
$667$	0.120364	0.00466053
$668$	−1.35013	−0.0522381
$669$	−23.7243	−0.917233
$670$	0	0
$671$	12.0242	0.464189
$672$	−62.9647	−2.42892
$673$	−27.0638	−1.04323	−0.521616	−	0.853180i	$-0.674671\pi$
$673$	−27.0638	−1.04323	−0.521616	+	0.853180i	$0.674671\pi$
$674$	−56.5392	−2.17781
$675$	0	0
$676$	16.4859	0.634074
$677$	40.4369	1.55412	0.777059	−	0.629428i	$-0.216711\pi$
$677$	40.4369	1.55412	0.777059	+	0.629428i	$0.216711\pi$
$678$	16.2048	0.622342
$679$	−14.2500	−0.546865
$680$	0	0
$681$	−8.14189	−0.311998
$682$	10.9839	0.420596
$683$	−44.8108	−1.71464	−0.857319	−	0.514786i	$-0.827871\pi$
$683$	−44.8108	−1.71464	−0.857319	+	0.514786i	$0.827871\pi$
$684$	−5.96103	−0.227926
$685$	0	0
$686$	32.0904	1.22522
$687$	4.60035	0.175514
$688$	30.1401	1.14908
$689$	−35.5228	−1.35331
$690$	0	0
$691$	−24.2835	−0.923786	−0.461893	−	0.886936i	$-0.652830\pi$
$691$	−24.2835	−0.923786	−0.461893	+	0.886936i	$0.652830\pi$
$692$	−0.467749	−0.0177812
$693$	2.82449	0.107294
$694$	28.2028	1.07056
$695$	0	0
$696$	0.0921649	0.00349350
$697$	0.0476327	0.00180422
$698$	70.3146	2.66145
$699$	40.7141	1.53995
$700$	0	0
$701$	11.6405	0.439657	0.219828	−	0.975539i	$-0.429450\pi$
$701$	11.6405	0.439657	0.219828	+	0.975539i	$0.429450\pi$
$702$	41.0689	1.55005
$703$	2.99524	0.112968
$704$	−10.6871	−0.402786
$705$	0	0
$706$	22.3883	0.842594
$707$	31.0077	1.16616
$708$	−19.1525	−0.719797
$709$	−16.3873	−0.615438	−0.307719	−	0.951477i	$-0.599566\pi$
$709$	−16.3873	−0.615438	−0.307719	+	0.951477i	$0.599566\pi$
$710$	0	0
$711$	−5.69396	−0.213540
$712$	−15.1986	−0.569591
$713$	19.6688	0.736602
$714$	−69.7963	−2.61206
$715$	0	0
$716$	63.4025	2.36946
$717$	−21.6960	−0.810251
$718$	−3.69601	−0.137934
$719$	−6.95213	−0.259271	−0.129635	−	0.991562i	$-0.541381\pi$
$719$	−6.95213	−0.259271	−0.129635	+	0.991562i	$0.541381\pi$
$720$	0	0
$721$	12.0796	0.449869
$722$	−21.5793	−0.803099
$723$	6.00201	0.223217
$724$	−46.5643	−1.73055
$725$	0	0
$726$	−42.5672	−1.57982
$727$	−38.0524	−1.41128	−0.705642	−	0.708569i	$-0.749341\pi$
$727$	−38.0524	−1.41128	−0.705642	+	0.708569i	$0.749341\pi$
$728$	−24.9426	−0.924435
$729$	17.1880	0.636594
$730$	0	0
$731$	51.3870	1.90062
$732$	68.8107	2.54332
$733$	19.2460	0.710868	0.355434	−	0.934701i	$-0.384333\pi$
$733$	19.2460	0.710868	0.355434	+	0.934701i	$0.384333\pi$
$734$	18.1010	0.668119
$735$	0	0
$736$	−26.6106	−0.980880
$737$	6.19422	0.228167
$738$	−0.0194189	−0.000714820 0
$739$	21.4641	0.789571	0.394785	−	0.918773i	$-0.370819\pi$
$739$	21.4641	0.789571	0.394785	+	0.918773i	$0.370819\pi$
$740$	0	0
$741$	25.4830	0.936140
$742$	72.9691	2.67878
$743$	−22.4355	−0.823077	−0.411539	−	0.911392i	$-0.635008\pi$
$743$	−22.4355	−0.823077	−0.411539	+	0.911392i	$0.635008\pi$
$744$	15.0607	0.552152
$745$	0	0
$746$	28.7223	1.05160
$747$	7.13091	0.260906
$748$	−9.35747	−0.342143
$749$	81.7332	2.98647
$750$	0	0
$751$	−2.27609	−0.0830557	−0.0415278	−	0.999137i	$-0.513223\pi$
$751$	−2.27609	−0.0830557	−0.0415278	+	0.999137i	$0.513223\pi$
$752$	−16.7744	−0.611698
$753$	−4.08337	−0.148806
$754$	−0.331213	−0.0120621
$755$	0	0
$756$	−47.9218	−1.74290
$757$	−32.2028	−1.17043	−0.585216	−	0.810877i	$-0.698990\pi$
$757$	−32.2028	−1.17043	−0.585216	+	0.810877i	$0.698990\pi$
$758$	−38.7432	−1.40722
$759$	5.92650	0.215118
$760$	0	0
$761$	−6.49880	−0.235581	−0.117791	−	0.993038i	$-0.537581\pi$
$761$	−6.49880	−0.235581	−0.117791	+	0.993038i	$0.537581\pi$
$762$	−78.3309	−2.83763
$763$	14.0042	0.506988
$764$	−8.04337	−0.290999
$765$	0	0
$766$	−19.6111	−0.708577
$767$	16.4913	0.595466
$768$	3.50701	0.126548
$769$	1.94137	0.0700077	0.0350039	−	0.999387i	$-0.488856\pi$
$769$	1.94137	0.0700077	0.0350039	+	0.999387i	$0.488856\pi$
$770$	0	0
$771$	−11.4665	−0.412955
$772$	29.5761	1.06447
$773$	−6.96315	−0.250447	−0.125224	−	0.992129i	$-0.539965\pi$
$773$	−6.96315	−0.250447	−0.125224	+	0.992129i	$0.539965\pi$
$774$	−20.9495	−0.753014
$775$	0	0
$776$	−4.61148	−0.165542
$777$	−8.12177	−0.291367
$778$	27.1453	0.973206
$779$	0.0357235	0.00127993
$780$	0	0
$781$	−9.54625	−0.341592
$782$	−29.4978	−1.05484
$783$	−0.152470	−0.00544884
$784$	−24.7340	−0.883358
$785$	0	0
$786$	1.56435	0.0557984
$787$	−41.6122	−1.48332	−0.741658	−	0.670779i	$-0.765960\pi$
$787$	−41.6122	−1.48332	−0.741658	+	0.670779i	$0.765960\pi$
$788$	64.7149	2.30537
$789$	−10.8088	−0.384803
$790$	0	0
$791$	16.2815	0.578902
$792$	0.914041	0.0324790
$793$	−59.2495	−2.10401
$794$	−52.6547	−1.86865
$795$	0	0
$796$	28.7096	1.01758
$797$	−48.6470	−1.72316	−0.861582	−	0.507618i	$-0.830526\pi$
$797$	−48.6470	−1.72316	−0.861582	+	0.507618i	$0.830526\pi$
$798$	−52.3458	−1.85302
$799$	−28.5993	−1.01177
$800$	0	0
$801$	−8.48067	−0.299650
$802$	−14.4390	−0.509860
$803$	9.90852	0.349664
$804$	35.4476	1.25014
$805$	0	0
$806$	−54.1236	−1.90642
$807$	−17.9793	−0.632902
$808$	10.0345	0.353011
$809$	31.1289	1.09443	0.547216	−	0.836991i	$-0.315687\pi$
$809$	31.1289	1.09443	0.547216	+	0.836991i	$0.315687\pi$
$810$	0	0
$811$	12.1233	0.425708	0.212854	−	0.977084i	$-0.431724\pi$
$811$	12.1233	0.425708	0.212854	+	0.977084i	$0.431724\pi$
$812$	0.386480	0.0135628
$813$	51.1818	1.79502
$814$	−1.91685	−0.0671856
$815$	0	0
$816$	18.1324	0.634762
$817$	38.5392	1.34832
$818$	−1.97442	−0.0690341
$819$	−13.9178	−0.486326
$820$	0	0
$821$	−42.3718	−1.47879	−0.739393	−	0.673274i	$-0.764887\pi$
$821$	−42.3718	−1.47879	−0.739393	+	0.673274i	$0.764887\pi$
$822$	−57.5082	−2.00583
$823$	−8.91190	−0.310649	−0.155325	−	0.987863i	$-0.549642\pi$
$823$	−8.91190	−0.310649	−0.155325	+	0.987863i	$0.549642\pi$
$824$	3.90912	0.136181
$825$	0	0
$826$	−33.8756	−1.17868
$827$	20.0414	0.696909	0.348454	−	0.937326i	$-0.386707\pi$
$827$	20.0414	0.696909	0.348454	+	0.937326i	$0.386707\pi$
$828$	6.83122	0.237401
$829$	−4.61854	−0.160408	−0.0802042	−	0.996778i	$-0.525557\pi$
$829$	−4.61854	−0.160408	−0.0802042	+	0.996778i	$0.525557\pi$
$830$	0	0
$831$	−0.231264	−0.00802245
$832$	52.6611	1.82570
$833$	−42.1700	−1.46111
$834$	−44.3017	−1.53404
$835$	0	0
$836$	−7.01791	−0.242720
$837$	−24.9152	−0.861197
$838$	19.8392	0.685333
$839$	10.6805	0.368732	0.184366	−	0.982858i	$-0.440977\pi$
$839$	10.6805	0.368732	0.184366	+	0.982858i	$0.440977\pi$
$840$	0	0
$841$	−28.9988	−0.999958
$842$	6.79866	0.234297
$843$	−18.3823	−0.633119
$844$	15.5804	0.536299
$845$	0	0
$846$	11.6594	0.400857
$847$	−42.7685	−1.46954
$848$	−18.9567	−0.650975
$849$	30.6365	1.05144
$850$	0	0
$851$	−3.43248	−0.117664
$852$	−54.6302	−1.87160
$853$	−10.2330	−0.350371	−0.175186	−	0.984535i	$-0.556053\pi$
$853$	−10.2330	−0.350371	−0.175186	+	0.984535i	$0.556053\pi$
$854$	121.707	4.16474
$855$	0	0
$856$	26.4499	0.904038
$857$	36.7790	1.25635	0.628174	−	0.778073i	$-0.283803\pi$
$857$	36.7790	1.25635	0.628174	+	0.778073i	$0.283803\pi$
$858$	−16.3082	−0.556754
$859$	45.2427	1.54366	0.771831	−	0.635828i	$-0.219341\pi$
$859$	45.2427	1.54366	0.771831	+	0.635828i	$0.219341\pi$
$860$	0	0
$861$	−0.0968666	−0.00330120
$862$	−21.7105	−0.739464
$863$	8.43897	0.287266	0.143633	−	0.989631i	$-0.454122\pi$
$863$	8.43897	0.287266	0.143633	+	0.989631i	$0.454122\pi$
$864$	33.7087	1.14679
$865$	0	0
$866$	39.0747	1.32781
$867$	−2.03490	−0.0691089
$868$	63.1549	2.14362
$869$	−6.70349	−0.227400
$870$	0	0
$871$	−30.5222	−1.03420
$872$	4.53195	0.153471
$873$	−2.57316	−0.0870884
$874$	−22.1228	−0.748314
$875$	0	0
$876$	56.7034	1.91583
$877$	−4.35345	−0.147006	−0.0735029	−	0.997295i	$-0.523418\pi$
$877$	−4.35345	−0.147006	−0.0735029	+	0.997295i	$0.523418\pi$
$878$	−55.5835	−1.87585
$879$	42.1959	1.42323
$880$	0	0
$881$	−18.2086	−0.613464	−0.306732	−	0.951796i	$-0.599236\pi$
$881$	−18.2086	−0.613464	−0.306732	+	0.951796i	$0.599236\pi$
$882$	17.1919	0.578881
$883$	−0.677160	−0.0227883	−0.0113941	−	0.999935i	$-0.503627\pi$
$883$	−0.677160	−0.0227883	−0.0113941	+	0.999935i	$0.503627\pi$
$884$	46.1092	1.55082
$885$	0	0
$886$	85.7098	2.87948
$887$	9.33431	0.313415	0.156708	−	0.987645i	$-0.449912\pi$
$887$	9.33431	0.313415	0.156708	+	0.987645i	$0.449912\pi$
$888$	−2.62831	−0.0882002
$889$	−78.7013	−2.63956
$890$	0	0
$891$	−9.52947	−0.319249
$892$	−32.1943	−1.07795
$893$	−21.4489	−0.717759
$894$	34.6374	1.15845
$895$	0	0
$896$	−43.2017	−1.44327
$897$	−29.2030	−0.975059
$898$	−74.7839	−2.49557
$899$	0.200937	0.00670161
$900$	0	0
$901$	−32.3200	−1.07674
$902$	−0.0228619	−0.000761217 0
$903$	−104.501	−3.47759
$904$	5.26888	0.175240
$905$	0	0
$906$	−86.4008	−2.87048
$907$	−57.2782	−1.90189	−0.950946	−	0.309356i	$-0.899887\pi$
$907$	−57.2782	−1.90189	−0.950946	+	0.309356i	$0.899887\pi$
$908$	−11.0487	−0.366665
$909$	5.59914	0.185712
$910$	0	0
$911$	19.0860	0.632348	0.316174	−	0.948701i	$-0.397602\pi$
$911$	19.0860	0.632348	0.316174	+	0.948701i	$0.397602\pi$
$912$	13.5989	0.450306
$913$	8.39521	0.277841
$914$	37.1822	1.22988
$915$	0	0
$916$	6.24278	0.206267
$917$	1.57174	0.0519036
$918$	37.3661	1.23326
$919$	−4.13733	−0.136478	−0.0682389	−	0.997669i	$-0.521738\pi$
$919$	−4.13733	−0.136478	−0.0682389	+	0.997669i	$0.521738\pi$
$920$	0	0
$921$	27.7914	0.915758
$922$	−15.0372	−0.495224
$923$	47.0394	1.54832
$924$	19.0295	0.626025
$925$	0	0
$926$	−64.5933	−2.12267
$927$	2.18125	0.0716418
$928$	−0.271854	−0.00892405
$929$	−40.3149	−1.32269	−0.661344	−	0.750082i	$-0.730014\pi$
$929$	−40.3149	−1.32269	−0.661344	+	0.750082i	$0.730014\pi$
$930$	0	0
$931$	−31.6267	−1.03652
$932$	55.2500	1.80977
$933$	−28.9655	−0.948288
$934$	39.9758	1.30805
$935$	0	0
$936$	−4.50396	−0.147216
$937$	−4.45592	−0.145569	−0.0727843	−	0.997348i	$-0.523188\pi$
$937$	−4.45592	−0.145569	−0.0727843	+	0.997348i	$0.523188\pi$
$938$	62.6970	2.04713
$939$	−24.8398	−0.810615
$940$	0	0
$941$	−20.6455	−0.673023	−0.336511	−	0.941679i	$-0.609247\pi$
$941$	−20.6455	−0.673023	−0.336511	+	0.941679i	$0.609247\pi$
$942$	20.8944	0.680777
$943$	−0.0409385	−0.00133314
$944$	8.80056	0.286434
$945$	0	0
$946$	−24.6638	−0.801889
$947$	37.8088	1.22862	0.614311	−	0.789064i	$-0.289434\pi$
$947$	37.8088	1.22862	0.614311	+	0.789064i	$0.289434\pi$
$948$	−38.3620	−1.24594
$949$	−48.8245	−1.58491
$950$	0	0
$951$	−27.2750	−0.884454
$952$	−22.6938	−0.735509
$953$	−57.7180	−1.86967	−0.934835	−	0.355082i	$-0.884453\pi$
$953$	−57.7180	−1.86967	−0.934835	+	0.355082i	$0.884453\pi$
$954$	13.1762	0.426596
$955$	0	0
$956$	−29.4419	−0.952219
$957$	0.0605451	0.00195715
$958$	−9.96323	−0.321897
$959$	−57.7802	−1.86582
$960$	0	0
$961$	1.83513	0.0591977
$962$	9.44533	0.304530
$963$	14.7588	0.475595
$964$	8.14486	0.262328
$965$	0	0
$966$	59.9872	1.93006
$967$	−31.1156	−1.00061	−0.500305	−	0.865850i	$-0.666779\pi$
$967$	−31.1156	−1.00061	−0.500305	+	0.865850i	$0.666779\pi$
$968$	−13.8404	−0.444848
$969$	23.1854	0.744822
$970$	0	0
$971$	−3.19664	−0.102585	−0.0512925	−	0.998684i	$-0.516334\pi$
$971$	−3.19664	−0.102585	−0.0512925	+	0.998684i	$0.516334\pi$
$972$	−20.2255	−0.648733
$973$	−44.5113	−1.42697
$974$	−44.2657	−1.41836
$975$	0	0
$976$	−31.6184	−1.01208
$977$	−52.6545	−1.68457	−0.842283	−	0.539036i	$-0.818789\pi$
$977$	−52.6545	−1.68457	−0.842283	+	0.539036i	$0.818789\pi$
$978$	−24.7518	−0.791475
$979$	−9.98429	−0.319099
$980$	0	0
$981$	2.52878	0.0807379
$982$	−12.3027	−0.392595
$983$	19.0510	0.607632	0.303816	−	0.952731i	$-0.401739\pi$
$983$	19.0510	0.607632	0.303816	+	0.952731i	$0.401739\pi$
$984$	−0.0313472	−0.000999313 0
$985$	0	0
$986$	−0.301350	−0.00959694
$987$	58.1599	1.85125
$988$	34.5809	1.10017
$989$	−44.1652	−1.40437
$990$	0	0
$991$	46.0922	1.46417	0.732083	−	0.681215i	$-0.238548\pi$
$991$	46.0922	1.46417	0.732083	+	0.681215i	$0.238548\pi$
$992$	−44.4238	−1.41046
$993$	28.5699	0.906639
$994$	−96.6259	−3.06479
$995$	0	0
$996$	48.0432	1.52231
$997$	25.1633	0.796930	0.398465	−	0.917184i	$-0.369543\pi$
$997$	25.1633	0.796930	0.398465	+	0.917184i	$0.369543\pi$
$998$	−35.6540	−1.12861
$999$	4.34806	0.137567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	925.2.a.m.1.7		9
3.2	odd	2		8325.2.a.cq.1.3		9
5.2	odd	4		185.2.b.a.149.16	yes	18
5.3	odd	4		185.2.b.a.149.3	✓	18
5.4	even	2		925.2.a.l.1.3		9
15.2	even	4		1665.2.c.e.334.3		18
15.8	even	4		1665.2.c.e.334.16		18
15.14	odd	2		8325.2.a.cr.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.b.a.149.3	✓	18	5.3	odd	4
185.2.b.a.149.16	yes	18	5.2	odd	4
925.2.a.l.1.3		9	5.4	even	2
925.2.a.m.1.7		9	1.1	even	1	trivial
1665.2.c.e.334.3		18	15.2	even	4
1665.2.c.e.334.16		18	15.8	even	4
8325.2.a.cq.1.3		9	3.2	odd	2
8325.2.a.cr.1.7		9	15.14	odd	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:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.a (trivial)

\(f(q)\)	\(=\)	\(q+2.15179 q^{2} +1.93821 q^{3} +2.63020 q^{4} +4.17062 q^{6} +4.19034 q^{7} +1.35605 q^{8} +0.756662 q^{9} +0.890817 q^{11} +5.09787 q^{12} -4.38953 q^{13} +9.01674 q^{14} -2.34246 q^{16} -3.99376 q^{17} +1.62818 q^{18} -2.99524 q^{19} +8.12177 q^{21} +1.91685 q^{22} +3.43248 q^{23} +2.62831 q^{24} -9.44533 q^{26} -4.34806 q^{27} +11.0214 q^{28} +0.0350663 q^{29} +5.73019 q^{31} -7.75258 q^{32} +1.72659 q^{33} -8.59373 q^{34} +1.99017 q^{36} -1.00000 q^{37} -6.44512 q^{38} -8.50783 q^{39} -0.0119268 q^{41} +17.4763 q^{42} -12.8668 q^{43} +2.34302 q^{44} +7.38598 q^{46} +7.16099 q^{47} -4.54019 q^{48} +10.5590 q^{49} -7.74075 q^{51} -11.5453 q^{52} +8.09263 q^{53} -9.35611 q^{54} +5.68230 q^{56} -5.80540 q^{57} +0.0754552 q^{58} -3.75697 q^{59} +13.4979 q^{61} +12.3302 q^{62} +3.17067 q^{63} -11.9970 q^{64} +3.71526 q^{66} +6.95341 q^{67} -10.5044 q^{68} +6.65288 q^{69} -10.7163 q^{71} +1.02607 q^{72} +11.1230 q^{73} -2.15179 q^{74} -7.87806 q^{76} +3.73283 q^{77} -18.3071 q^{78} -7.52510 q^{79} -10.6974 q^{81} -0.0256639 q^{82} +9.42416 q^{83} +21.3618 q^{84} -27.6867 q^{86} +0.0679658 q^{87} +1.20799 q^{88} -11.2080 q^{89} -18.3936 q^{91} +9.02810 q^{92} +11.1063 q^{93} +15.4089 q^{94} -15.0261 q^{96} -3.40068 q^{97} +22.7207 q^{98} +0.674048 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 8 q^{3} + 11 q^{4} + 2 q^{6} + 8 q^{7} + 15 q^{8} + 13 q^{9} + 16 q^{12} + 6 q^{13} - 4 q^{14} + 11 q^{16} + 18 q^{17} - 3 q^{18} - 4 q^{19} + 4 q^{21} + 6 q^{22} + 16 q^{23} + 6 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) \) 9 * q + 5 * q^2 + 8 * q^3 + 11 * q^4 + 2 * q^6 + 8 * q^7 + 15 * q^8 + 13 * q^9 + 16 * q^12 + 6 * q^13 - 4 * q^14 + 11 * q^16 + 18 * q^17 - 3 * q^18 - 4 * q^19 + 4 * q^21 + 6 * q^22 + 16 * q^23 + 6 * q^24 - 6 * q^26 + 32 * q^27 - 20 * q^28 - 2 * q^29 - 6 * q^31 + 35 * q^32 + 6 * q^33 + 6 * q^34 + 11 * q^36 - 9 * q^37 + 4 * q^38 - 16 * q^39 + 2 * q^41 + 4 * q^42 + 14 * q^43 + 4 * q^44 + 16 * q^46 + 34 * q^47 + 6 * q^48 - 7 * q^49 + 8 * q^51 + 16 * q^52 - 4 * q^53 - 18 * q^54 - 24 * q^56 - 4 * q^57 - 22 * q^58 + 10 * q^59 + 8 * q^61 + 26 * q^62 + 18 * q^63 + 15 * q^64 - 34 * q^66 + 16 * q^67 + 36 * q^68 - 10 * q^69 + 8 * q^71 + 9 * q^72 + 8 * q^73 - 5 * q^74 - 48 * q^76 + 14 * q^77 - 76 * q^78 - 26 * q^79 + 13 * q^81 - 4 * q^82 + 70 * q^83 - 24 * q^84 - 16 * q^86 - 12 * q^87 - 36 * q^88 - 8 * q^89 - 18 * q^91 + 16 * q^92 + 2 * q^93 - 26 * q^94 + 52 * q^96 - 2 * q^97 + 21 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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