LMFDB - Embedded newform 7225.2.a.bo.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,-3,3,21,0,9,6] 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:	$0$
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} - 3 x^{11} - 18 x^{10} + 55 x^{9} + 114 x^{8} - 354 x^{7} - 309 x^{6} + 936 x^{5} + 396 x^{4} + \cdots + 9 $ x^12 - 3x^11 - 18x^10 + 55x^9 + 114x^8 - 354x^7 - 309x^6 + 936x^5 + 396x^4 - 899x^3 - 297x^2 + 153*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.9
Root			$-1.02921$ 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)

$f(q)$	$=$	\(q+1.02921 q^{2} +1.23410 q^{3} -0.940729 q^{4} +1.27015 q^{6} +0.979236 q^{7} -3.02662 q^{8} -1.47699 q^{9} -6.35575 q^{11} -1.16096 q^{12} +2.29653 q^{13} +1.00784 q^{14} -1.23357 q^{16} -1.52013 q^{18} -1.91355 q^{19} +1.20848 q^{21} -6.54140 q^{22} +8.96509 q^{23} -3.73517 q^{24} +2.36361 q^{26} -5.52507 q^{27} -0.921196 q^{28} -2.21557 q^{29} +5.74037 q^{31} +4.78365 q^{32} -7.84366 q^{33} +1.38945 q^{36} +5.49260 q^{37} -1.96944 q^{38} +2.83416 q^{39} -9.14368 q^{41} +1.24378 q^{42} +4.57764 q^{43} +5.97904 q^{44} +9.22695 q^{46} +1.95008 q^{47} -1.52235 q^{48} -6.04110 q^{49} -2.16041 q^{52} +4.00762 q^{53} -5.68645 q^{54} -2.96378 q^{56} -2.36152 q^{57} -2.28028 q^{58} +12.6197 q^{59} -5.87984 q^{61} +5.90804 q^{62} -1.44632 q^{63} +7.39051 q^{64} -8.07276 q^{66} -6.72054 q^{67} +11.0638 q^{69} +5.08034 q^{71} +4.47029 q^{72} +3.68032 q^{73} +5.65303 q^{74} +1.80013 q^{76} -6.22378 q^{77} +2.91694 q^{78} +1.08356 q^{79} -2.38754 q^{81} -9.41075 q^{82} -3.74449 q^{83} -1.13685 q^{84} +4.71135 q^{86} -2.73424 q^{87} +19.2365 q^{88} +9.03669 q^{89} +2.24885 q^{91} -8.43372 q^{92} +7.08421 q^{93} +2.00704 q^{94} +5.90351 q^{96} +2.17633 q^{97} -6.21755 q^{98} +9.38737 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 3 q^{3} + 21 q^{4} + 9 q^{6} + 6 q^{7} - 12 q^{8} + 21 q^{9} + 6 q^{11} + 6 q^{12} - 9 q^{13} + 18 q^{14} + 39 q^{16} + 9 q^{18} + 27 q^{19} + 6 q^{21} + 15 q^{22} + 18 q^{23} + 36 q^{24}+ \cdots + 6 q^{99}+O(q^{100}) $ 12 * q - 3 * q^2 + 3 * q^3 + 21 * q^4 + 9 * q^6 + 6 * q^7 - 12 * q^8 + 21 * q^9 + 6 * q^11 + 6 * q^12 - 9 * q^13 + 18 * q^14 + 39 * q^16 + 9 * q^18 + 27 * q^19 + 6 * q^21 + 15 * q^22 + 18 * q^23 + 36 * q^24 + 6 * q^26 + 12 * q^27 + 39 * q^28 + 6 * q^31 - 12 * q^32 + 48 * q^36 + 21 * q^37 + 3 * q^38 + 9 * q^39 + 15 * q^41 - 30 * q^42 - 45 * q^43 + 42 * q^44 + 15 * q^46 - 3 * q^47 - 63 * q^48 + 12 * q^49 - 51 * q^52 + 12 * q^53 + 27 * q^54 - 3 * q^56 + 6 * q^57 - 51 * q^58 + 36 * q^59 - 3 * q^61 + 36 * q^62 + 72 * q^63 + 42 * q^64 - 39 * q^66 - 78 * q^67 - 66 * q^69 + 30 * q^71 + 12 * q^72 - 33 * q^73 - 9 * q^74 + 27 * q^76 + 39 * q^77 - 24 * q^78 + 3 * q^79 + 12 * q^81 - 51 * q^82 - 27 * q^83 + 27 * q^84 + 18 * q^86 - 24 * q^87 + 36 * q^88 + 6 * q^89 + 24 * q^91 - 3 * q^92 + 12 * q^93 + 21 * q^94 + 39 * q^96 - 6 * q^97 + 42 * q^98 + 6 * 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 $

$2$	1.02921	0.727761	0.363880	−	0.931446i	$-0.381452\pi$
$2$	1.02921	0.727761	0.363880	+	0.931446i	$0.381452\pi$
$3$	1.23410	0.712510	0.356255	−	0.934389i	$-0.384053\pi$
$3$	1.23410	0.712510	0.356255	+	0.934389i	$0.384053\pi$
$4$	−0.940729	−0.470364
$5$	0	0
$5$	0	0
$6$	1.27015	0.518537
$7$	0.979236	0.370116	0.185058	−	0.982728i	$-0.440753\pi$
$7$	0.979236	0.370116	0.185058	+	0.982728i	$0.440753\pi$
$8$	−3.02662	−1.07007
$9$	−1.47699	−0.492330
$10$	0	0
$11$	−6.35575	−1.91633	−0.958166	−	0.286214i	$-0.907603\pi$
$11$	−6.35575	−1.91633	−0.958166	+	0.286214i	$0.907603\pi$
$12$	−1.16096	−0.335139
$13$	2.29653	0.636943	0.318472	−	0.947932i	$-0.396830\pi$
$13$	2.29653	0.636943	0.318472	+	0.947932i	$0.396830\pi$
$14$	1.00784	0.269356
$15$	0	0
$16$	−1.23357	−0.308393
$17$	0	0
$17$	0	0
$18$	−1.52013	−0.358298
$19$	−1.91355	−0.438999	−0.219499	−	0.975613i	$-0.570442\pi$
$19$	−1.91355	−0.438999	−0.219499	+	0.975613i	$0.570442\pi$
$20$	0	0
$21$	1.20848	0.263712
$22$	−6.54140	−1.39463
$23$	8.96509	1.86935	0.934675	−	0.355503i	$-0.115690\pi$
$23$	8.96509	1.86935	0.934675	+	0.355503i	$0.115690\pi$
$24$	−3.73517	−0.762438
$25$	0	0
$26$	2.36361	0.463542
$27$	−5.52507	−1.06330
$28$	−0.921196	−0.174090
$29$	−2.21557	−0.411421	−0.205710	−	0.978613i	$-0.565950\pi$
$29$	−2.21557	−0.411421	−0.205710	+	0.978613i	$0.565950\pi$
$30$	0	0
$31$	5.74037	1.03100	0.515501	−	0.856889i	$-0.327606\pi$
$31$	5.74037	1.03100	0.515501	+	0.856889i	$0.327606\pi$
$32$	4.78365	0.845637
$33$	−7.84366	−1.36541
$34$	0	0
$35$	0	0
$36$	1.38945	0.231574
$37$	5.49260	0.902977	0.451489	−	0.892277i	$-0.350893\pi$
$37$	5.49260	0.902977	0.451489	+	0.892277i	$0.350893\pi$
$38$	−1.96944	−0.319486
$39$	2.83416	0.453828
$40$	0	0
$41$	−9.14368	−1.42800	−0.714001	−	0.700144i	$-0.753119\pi$
$41$	−9.14368	−1.42800	−0.714001	+	0.700144i	$0.753119\pi$
$42$	1.24378	0.191919
$43$	4.57764	0.698083	0.349042	−	0.937107i	$-0.386507\pi$
$43$	4.57764	0.698083	0.349042	+	0.937107i	$0.386507\pi$
$44$	5.97904	0.901374
$45$	0	0
$46$	9.22695	1.36044
$47$	1.95008	0.284449	0.142224	−	0.989834i	$-0.454575\pi$
$47$	1.95008	0.284449	0.142224	+	0.989834i	$0.454575\pi$
$48$	−1.52235	−0.219733
$49$	−6.04110	−0.863014
$50$	0	0
$51$	0	0
$52$	−2.16041	−0.299595
$53$	4.00762	0.550488	0.275244	−	0.961374i	$-0.411241\pi$
$53$	4.00762	0.550488	0.275244	+	0.961374i	$0.411241\pi$
$54$	−5.68645	−0.773828
$55$	0	0
$56$	−2.96378	−0.396052
$57$	−2.36152	−0.312791
$58$	−2.28028	−0.299416
$59$	12.6197	1.64294	0.821471	−	0.570250i	$-0.193154\pi$
$59$	12.6197	1.64294	0.821471	+	0.570250i	$0.193154\pi$
$60$	0	0
$61$	−5.87984	−0.752837	−0.376418	−	0.926450i	$-0.622844\pi$
$61$	−5.87984	−0.752837	−0.376418	+	0.926450i	$0.622844\pi$
$62$	5.90804	0.750322
$63$	−1.44632	−0.182219
$64$	7.39051	0.923814
$65$	0	0
$66$	−8.07276	−0.993688
$67$	−6.72054	−0.821044	−0.410522	−	0.911851i	$-0.634654\pi$
$67$	−6.72054	−0.821044	−0.410522	+	0.911851i	$0.634654\pi$
$68$	0	0
$69$	11.0638	1.33193
$70$	0	0
$71$	5.08034	0.602926	0.301463	−	0.953478i	$-0.402525\pi$
$71$	5.08034	0.602926	0.301463	+	0.953478i	$0.402525\pi$
$72$	4.47029	0.526829
$73$	3.68032	0.430749	0.215374	−	0.976532i	$-0.430903\pi$
$73$	3.68032	0.430749	0.215374	+	0.976532i	$0.430903\pi$
$74$	5.65303	0.657151
$75$	0	0
$76$	1.80013	0.206489
$77$	−6.22378	−0.709266
$78$	2.91694	0.330278
$79$	1.08356	0.121910	0.0609548	−	0.998141i	$-0.480585\pi$
$79$	1.08356	0.121910	0.0609548	+	0.998141i	$0.480585\pi$
$80$	0	0
$81$	−2.38754	−0.265282
$82$	−9.41075	−1.03924
$83$	−3.74449	−0.411011	−0.205506	−	0.978656i	$-0.565884\pi$
$83$	−3.74449	−0.411011	−0.205506	+	0.978656i	$0.565884\pi$
$84$	−1.13685	−0.124041
$85$	0	0
$86$	4.71135	0.508038
$87$	−2.73424	−0.293141
$88$	19.2365	2.05062
$89$	9.03669	0.957888	0.478944	−	0.877846i	$-0.341020\pi$
$89$	9.03669	0.957888	0.478944	+	0.877846i	$0.341020\pi$
$90$	0	0
$91$	2.24885	0.235743
$92$	−8.43372	−0.879276
$93$	7.08421	0.734599
$94$	2.00704	0.207011
$95$	0	0
$96$	5.90351	0.602525
$97$	2.17633	0.220973	0.110487	−	0.993878i	$-0.464759\pi$
$97$	2.17633	0.220973	0.110487	+	0.993878i	$0.464759\pi$
$98$	−6.21755	−0.628067
$99$	9.38737	0.943467
$100$	0	0
$101$	15.2485	1.51729	0.758644	−	0.651506i	$-0.225862\pi$
$101$	15.2485	1.51729	0.758644	+	0.651506i	$0.225862\pi$
$102$	0	0
$103$	7.04871	0.694530	0.347265	−	0.937767i	$-0.387110\pi$
$103$	7.04871	0.694530	0.347265	+	0.937767i	$0.387110\pi$
$104$	−6.95074	−0.681576
$105$	0	0
$106$	4.12467	0.400624
$107$	−10.1983	−0.985911	−0.492956	−	0.870054i	$-0.664084\pi$
$107$	−10.1983	−0.985911	−0.492956	+	0.870054i	$0.664084\pi$
$108$	5.19759	0.500138
$109$	6.04775	0.579269	0.289635	−	0.957137i	$-0.406466\pi$
$109$	6.04775	0.579269	0.289635	+	0.957137i	$0.406466\pi$
$110$	0	0
$111$	6.77843	0.643380
$112$	−1.20796	−0.114141
$113$	15.5915	1.46673	0.733363	−	0.679838i	$-0.237950\pi$
$113$	15.5915	1.46673	0.733363	+	0.679838i	$0.237950\pi$
$114$	−2.43050	−0.227637
$115$	0	0
$116$	2.08425	0.193518
$117$	−3.39195	−0.313586
$118$	12.9883	1.19567
$119$	0	0
$120$	0	0
$121$	29.3956	2.67233
$122$	−6.05159	−0.547885
$123$	−11.2842	−1.01747
$124$	−5.40013	−0.484946
$125$	0	0
$126$	−1.48857	−0.132612
$127$	−18.7365	−1.66259	−0.831296	−	0.555830i	$-0.812401\pi$
$127$	−18.7365	−1.66259	−0.831296	+	0.555830i	$0.812401\pi$
$128$	−1.96091	−0.173322
$129$	5.64928	0.497391
$130$	0	0
$131$	1.78563	0.156011	0.0780055	−	0.996953i	$-0.475145\pi$
$131$	1.78563	0.156011	0.0780055	+	0.996953i	$0.475145\pi$
$132$	7.37875	0.642238
$133$	−1.87382	−0.162481
$134$	−6.91684	−0.597524
$135$	0	0
$136$	0	0
$137$	18.7053	1.59810	0.799051	−	0.601263i	$-0.205336\pi$
$137$	18.7053	1.59810	0.799051	+	0.601263i	$0.205336\pi$
$138$	11.3870	0.969327
$139$	20.7640	1.76118	0.880590	−	0.473880i	$-0.157147\pi$
$139$	20.7640	1.76118	0.880590	+	0.473880i	$0.157147\pi$
$140$	0	0
$141$	2.40660	0.202673
$142$	5.22873	0.438786
$143$	−14.5962	−1.22059
$144$	1.82197	0.151831
$145$	0	0
$146$	3.78782	0.313482
$147$	−7.45534	−0.614906
$148$	−5.16705	−0.424729
$149$	−10.5671	−0.865692	−0.432846	−	0.901468i	$-0.642491\pi$
$149$	−10.5671	−0.865692	−0.432846	+	0.901468i	$0.642491\pi$
$150$	0	0
$151$	0.726074	0.0590871	0.0295435	−	0.999563i	$-0.490595\pi$
$151$	0.726074	0.0590871	0.0295435	+	0.999563i	$0.490595\pi$
$152$	5.79160	0.469761
$153$	0	0
$154$	−6.40557	−0.516176
$155$	0	0
$156$	−2.66617	−0.213465
$157$	−15.5125	−1.23803	−0.619016	−	0.785378i	$-0.712469\pi$
$157$	−15.5125	−1.23803	−0.619016	+	0.785378i	$0.712469\pi$
$158$	1.11521	0.0887211
$159$	4.94581	0.392228
$160$	0	0
$161$	8.77894	0.691877
$162$	−2.45728	−0.193062
$163$	20.3277	1.59219	0.796094	−	0.605173i	$-0.206896\pi$
$163$	20.3277	1.59219	0.796094	+	0.605173i	$0.206896\pi$
$164$	8.60172	0.671682
$165$	0	0
$166$	−3.85386	−0.299118
$167$	14.0573	1.08779	0.543894	−	0.839154i	$-0.316949\pi$
$167$	14.0573	1.08779	0.543894	+	0.839154i	$0.316949\pi$
$168$	−3.65761	−0.282191
$169$	−7.72594	−0.594303
$170$	0	0
$171$	2.82629	0.216132
$172$	−4.30632	−0.328354
$173$	23.1932	1.76335	0.881674	−	0.471858i	$-0.156417\pi$
$173$	23.1932	1.76335	0.881674	+	0.471858i	$0.156417\pi$
$174$	−2.81410	−0.213337
$175$	0	0
$176$	7.84027	0.590983
$177$	15.5740	1.17061
$178$	9.30064	0.697113
$179$	−13.1808	−0.985176	−0.492588	−	0.870263i	$-0.663949\pi$
$179$	−13.1808	−0.985176	−0.492588	+	0.870263i	$0.663949\pi$
$180$	0	0
$181$	19.6975	1.46410	0.732052	−	0.681249i	$-0.238563\pi$
$181$	19.6975	1.46410	0.732052	+	0.681249i	$0.238563\pi$
$182$	2.31453	0.171565
$183$	−7.25633	−0.536404
$184$	−27.1340	−2.00034
$185$	0	0
$186$	7.29114	0.534612
$187$	0	0
$188$	−1.83450	−0.133795
$189$	−5.41035	−0.393545
$190$	0	0
$191$	−20.7793	−1.50354	−0.751768	−	0.659427i	$-0.770799\pi$
$191$	−20.7793	−1.50354	−0.751768	+	0.659427i	$0.770799\pi$
$192$	9.12066	0.658227
$193$	1.42071	0.102265	0.0511326	−	0.998692i	$-0.483717\pi$
$193$	1.42071	0.102265	0.0511326	+	0.998692i	$0.483717\pi$
$194$	2.23990	0.160816
$195$	0	0
$196$	5.68303	0.405931
$197$	8.19934	0.584179	0.292089	−	0.956391i	$-0.405649\pi$
$197$	8.19934	0.584179	0.292089	+	0.956391i	$0.405649\pi$
$198$	9.66157	0.686618
$199$	−11.9339	−0.845972	−0.422986	−	0.906136i	$-0.639018\pi$
$199$	−11.9339	−0.845972	−0.422986	+	0.906136i	$0.639018\pi$
$200$	0	0
$201$	−8.29384	−0.585002
$202$	15.6939	1.10422
$203$	−2.16956	−0.152274
$204$	0	0
$205$	0	0
$206$	7.25460	0.505452
$207$	−13.2413	−0.920336
$208$	−2.83293	−0.196429
$209$	12.1621	0.841267
$210$	0	0
$211$	−1.88689	−0.129899	−0.0649494	−	0.997889i	$-0.520689\pi$
$211$	−1.88689	−0.129899	−0.0649494	+	0.997889i	$0.520689\pi$
$212$	−3.77008	−0.258930
$213$	6.26967	0.429591
$214$	−10.4962	−0.717508
$215$	0	0
$216$	16.7223	1.13781
$217$	5.62118	0.381591
$218$	6.22439	0.421569
$219$	4.54190	0.306913
$220$	0	0
$221$	0	0
$222$	6.97642	0.468227
$223$	7.42720	0.497362	0.248681	−	0.968585i	$-0.420003\pi$
$223$	7.42720	0.497362	0.248681	+	0.968585i	$0.420003\pi$
$224$	4.68432	0.312984
$225$	0	0
$226$	16.0469	1.06742
$227$	−13.3575	−0.886568	−0.443284	−	0.896381i	$-0.646187\pi$
$227$	−13.3575	−0.886568	−0.443284	+	0.896381i	$0.646187\pi$
$228$	2.22155	0.147126
$229$	−4.04244	−0.267132	−0.133566	−	0.991040i	$-0.542643\pi$
$229$	−4.04244	−0.267132	−0.133566	+	0.991040i	$0.542643\pi$
$230$	0	0
$231$	−7.68079	−0.505359
$232$	6.70569	0.440250
$233$	16.8618	1.10465	0.552327	−	0.833628i	$-0.313740\pi$
$233$	16.8618	1.10465	0.552327	+	0.833628i	$0.313740\pi$
$234$	−3.49103	−0.228216
$235$	0	0
$236$	−11.8717	−0.772782
$237$	1.33722	0.0868619
$238$	0	0
$239$	11.6307	0.752325	0.376162	−	0.926554i	$-0.377243\pi$
$239$	11.6307	0.752325	0.376162	+	0.926554i	$0.377243\pi$
$240$	0	0
$241$	17.1784	1.10656	0.553280	−	0.832996i	$-0.313376\pi$
$241$	17.1784	1.10656	0.553280	+	0.832996i	$0.313376\pi$
$242$	30.2542	1.94481
$243$	13.6287	0.874284
$244$	5.53134	0.354108
$245$	0	0
$246$	−11.6138	−0.740472
$247$	−4.39453	−0.279617
$248$	−17.3740	−1.10325
$249$	−4.62109	−0.292850
$250$	0	0
$251$	7.44998	0.470239	0.235119	−	0.971967i	$-0.424452\pi$
$251$	7.44998	0.470239	0.235119	+	0.971967i	$0.424452\pi$
$252$	1.36060	0.0857095
$253$	−56.9799	−3.58230
$254$	−19.2837	−1.20997
$255$	0	0
$256$	−16.7992	−1.04995
$257$	−3.73906	−0.233236	−0.116618	−	0.993177i	$-0.537205\pi$
$257$	−3.73906	−0.233236	−0.116618	+	0.993177i	$0.537205\pi$
$258$	5.81429	0.361982
$259$	5.37855	0.334207
$260$	0	0
$261$	3.27237	0.202555
$262$	1.83778	0.113539
$263$	15.1304	0.932979	0.466489	−	0.884527i	$-0.345519\pi$
$263$	15.1304	0.932979	0.466489	+	0.884527i	$0.345519\pi$
$264$	23.7398	1.46108
$265$	0	0
$266$	−1.92855	−0.118247
$267$	11.1522	0.682504
$268$	6.32220	0.386190
$269$	16.0552	0.978902	0.489451	−	0.872031i	$-0.337197\pi$
$269$	16.0552	0.978902	0.489451	+	0.872031i	$0.337197\pi$
$270$	0	0
$271$	−13.1259	−0.797343	−0.398671	−	0.917094i	$-0.630529\pi$
$271$	−13.1259	−0.797343	−0.398671	+	0.917094i	$0.630529\pi$
$272$	0	0
$273$	2.77531	0.167969
$274$	19.2517	1.16304
$275$	0	0
$276$	−10.4081	−0.626493
$277$	−5.65874	−0.340001	−0.170000	−	0.985444i	$-0.554377\pi$
$277$	−5.65874	−0.340001	−0.170000	+	0.985444i	$0.554377\pi$
$278$	21.3705	1.28172
$279$	−8.47847	−0.507592
$280$	0	0
$281$	−20.3992	−1.21691	−0.608456	−	0.793588i	$-0.708211\pi$
$281$	−20.3992	−1.21691	−0.608456	+	0.793588i	$0.708211\pi$
$282$	2.47690	0.147497
$283$	−9.85417	−0.585770	−0.292885	−	0.956148i	$-0.594615\pi$
$283$	−9.85417	−0.585770	−0.292885	+	0.956148i	$0.594615\pi$
$284$	−4.77923	−0.283595
$285$	0	0
$286$	−15.0225	−0.888301
$287$	−8.95382	−0.528527
$288$	−7.06539	−0.416332
$289$	0	0
$290$	0	0
$291$	2.68582	0.157446
$292$	−3.46218	−0.202609
$293$	21.7306	1.26952	0.634758	−	0.772711i	$-0.281100\pi$
$293$	21.7306	1.26952	0.634758	+	0.772711i	$0.281100\pi$
$294$	−7.67310	−0.447504
$295$	0	0
$296$	−16.6240	−0.966252
$297$	35.1160	2.03763
$298$	−10.8758	−0.630016
$299$	20.5886	1.19067
$300$	0	0
$301$	4.48259	0.258372
$302$	0.747282	0.0430012
$303$	18.8183	1.08108
$304$	2.36050	0.135384
$305$	0	0
$306$	0	0
$307$	−24.1970	−1.38100	−0.690499	−	0.723333i	$-0.742609\pi$
$307$	−24.1970	−1.38100	−0.690499	+	0.723333i	$0.742609\pi$
$308$	5.85489	0.333613
$309$	8.69884	0.494860
$310$	0	0
$311$	16.7204	0.948126	0.474063	−	0.880491i	$-0.342787\pi$
$311$	16.7204	0.948126	0.474063	+	0.880491i	$0.342787\pi$
$312$	−8.57793	−0.485630
$313$	−31.1252	−1.75930	−0.879649	−	0.475623i	$-0.842223\pi$
$313$	−31.1252	−1.75930	−0.879649	+	0.475623i	$0.842223\pi$
$314$	−15.9656	−0.900991
$315$	0	0
$316$	−1.01933	−0.0573420
$317$	−5.43596	−0.305314	−0.152657	−	0.988279i	$-0.548783\pi$
$317$	−5.43596	−0.305314	−0.152657	+	0.988279i	$0.548783\pi$
$318$	5.09028	0.285448
$319$	14.0816	0.788418
$320$	0	0
$321$	−12.5858	−0.702472
$322$	9.03536	0.503521
$323$	0	0
$324$	2.24603	0.124779
$325$	0	0
$326$	20.9214	1.15873
$327$	7.46354	0.412735
$328$	27.6745	1.52807
$329$	1.90959	0.105279
$330$	0	0
$331$	−5.45030	−0.299575	−0.149788	−	0.988718i	$-0.547859\pi$
$331$	−5.45030	−0.299575	−0.149788	+	0.988718i	$0.547859\pi$
$332$	3.52255	0.193325
$333$	−8.11250	−0.444562
$334$	14.4679	0.791649
$335$	0	0
$336$	−1.49074	−0.0813268
$337$	11.9722	0.652170	0.326085	−	0.945340i	$-0.394270\pi$
$337$	11.9722	0.652170	0.326085	+	0.945340i	$0.394270\pi$
$338$	−7.95161	−0.432511
$339$	19.2415	1.04506
$340$	0	0
$341$	−36.4844	−1.97574
$342$	2.90885	0.157292
$343$	−12.7703	−0.689532
$344$	−13.8548	−0.747001
$345$	0	0
$346$	23.8707	1.28330
$347$	9.78974	0.525541	0.262770	−	0.964858i	$-0.415364\pi$
$347$	9.78974	0.525541	0.262770	+	0.964858i	$0.415364\pi$
$348$	2.57218	0.137883
$349$	−27.6429	−1.47969	−0.739846	−	0.672776i	$-0.765102\pi$
$349$	−27.6429	−1.47969	−0.739846	+	0.672776i	$0.765102\pi$
$350$	0	0
$351$	−12.6885	−0.677262
$352$	−30.4037	−1.62052
$353$	−5.25206	−0.279539	−0.139769	−	0.990184i	$-0.544636\pi$
$353$	−5.25206	−0.279539	−0.139769	+	0.990184i	$0.544636\pi$
$354$	16.0289	0.851926
$355$	0	0
$356$	−8.50108	−0.450556
$357$	0	0
$358$	−13.5658	−0.716972
$359$	−15.2594	−0.805360	−0.402680	−	0.915341i	$-0.631921\pi$
$359$	−15.2594	−0.805360	−0.402680	+	0.915341i	$0.631921\pi$
$360$	0	0
$361$	−15.3383	−0.807280
$362$	20.2728	1.06552
$363$	36.2772	1.90406
$364$	−2.11556	−0.110885
$365$	0	0
$366$	−7.46828	−0.390373
$367$	20.0338	1.04575	0.522877	−	0.852408i	$-0.324859\pi$
$367$	20.0338	1.04575	0.522877	+	0.852408i	$0.324859\pi$
$368$	−11.0591	−0.576494
$369$	13.5051	0.703048
$370$	0	0
$371$	3.92440	0.203745
$372$	−6.66432	−0.345529
$373$	6.72325	0.348117	0.174058	−	0.984735i	$-0.444312\pi$
$373$	6.72325	0.348117	0.174058	+	0.984735i	$0.444312\pi$
$374$	0	0
$375$	0	0
$376$	−5.90217	−0.304381
$377$	−5.08812	−0.262052
$378$	−5.56838	−0.286406
$379$	−4.93364	−0.253424	−0.126712	−	0.991940i	$-0.540442\pi$
$379$	−4.93364	−0.253424	−0.126712	+	0.991940i	$0.540442\pi$
$380$	0	0
$381$	−23.1227	−1.18461
$382$	−21.3862	−1.09421
$383$	23.0814	1.17940	0.589702	−	0.807621i	$-0.299245\pi$
$383$	23.0814	1.17940	0.589702	+	0.807621i	$0.299245\pi$
$384$	−2.41997	−0.123493
$385$	0	0
$386$	1.46221	0.0744246
$387$	−6.76112	−0.343687
$388$	−2.04734	−0.103938
$389$	0.937529	0.0475346	0.0237673	−	0.999718i	$-0.492434\pi$
$389$	0.937529	0.0475346	0.0237673	+	0.999718i	$0.492434\pi$
$390$	0	0
$391$	0	0
$392$	18.2841	0.923488
$393$	2.20365	0.111159
$394$	8.43884	0.425142
$395$	0	0
$396$	−8.83098	−0.443773
$397$	−26.1623	−1.31305	−0.656525	−	0.754304i	$-0.727974\pi$
$397$	−26.1623	−1.31305	−0.656525	+	0.754304i	$0.727974\pi$
$398$	−12.2825	−0.615665
$399$	−2.31249	−0.115769
$400$	0	0
$401$	−10.5093	−0.524811	−0.262405	−	0.964958i	$-0.584516\pi$
$401$	−10.5093	−0.524811	−0.262405	+	0.964958i	$0.584516\pi$
$402$	−8.53609	−0.425742
$403$	13.1829	0.656689
$404$	−14.3448	−0.713678
$405$	0	0
$406$	−2.23293	−0.110819
$407$	−34.9096	−1.73040
$408$	0	0
$409$	−26.3801	−1.30441	−0.652206	−	0.758042i	$-0.726156\pi$
$409$	−26.3801	−1.30441	−0.652206	+	0.758042i	$0.726156\pi$
$410$	0	0
$411$	23.0843	1.13866
$412$	−6.63093	−0.326682
$413$	12.3576	0.608080
$414$	−13.6281	−0.669785
$415$	0	0
$416$	10.9858	0.538623
$417$	25.6249	1.25486
$418$	12.5173	0.612241
$419$	−3.97802	−0.194339	−0.0971695	−	0.995268i	$-0.530979\pi$
$419$	−3.97802	−0.194339	−0.0971695	+	0.995268i	$0.530979\pi$
$420$	0	0
$421$	−11.0109	−0.536638	−0.268319	−	0.963330i	$-0.586468\pi$
$421$	−11.0109	−0.536638	−0.268319	+	0.963330i	$0.586468\pi$
$422$	−1.94200	−0.0945352
$423$	−2.88025	−0.140043
$424$	−12.1295	−0.589063
$425$	0	0
$426$	6.45280	0.312639
$427$	−5.75775	−0.278637
$428$	9.59388	0.463738
$429$	−18.0132	−0.869686
$430$	0	0
$431$	35.6219	1.71585	0.857923	−	0.513779i	$-0.171755\pi$
$431$	35.6219	1.71585	0.857923	+	0.513779i	$0.171755\pi$
$432$	6.81556	0.327914
$433$	8.53691	0.410258	0.205129	−	0.978735i	$-0.434239\pi$
$433$	8.53691	0.410258	0.205129	+	0.978735i	$0.434239\pi$
$434$	5.78537	0.277707
$435$	0	0
$436$	−5.68929	−0.272468
$437$	−17.1552	−0.820642
$438$	4.67456	0.223359
$439$	−15.6854	−0.748625	−0.374313	−	0.927303i	$-0.622121\pi$
$439$	−15.6854	−0.748625	−0.374313	+	0.927303i	$0.622121\pi$
$440$	0	0
$441$	8.92263	0.424887
$442$	0	0
$443$	1.36560	0.0648815	0.0324408	−	0.999474i	$-0.489672\pi$
$443$	1.36560	0.0648815	0.0324408	+	0.999474i	$0.489672\pi$
$444$	−6.37667	−0.302623
$445$	0	0
$446$	7.64414	0.361961
$447$	−13.0409	−0.616814
$448$	7.23706	0.341919
$449$	−7.14027	−0.336970	−0.168485	−	0.985704i	$-0.553888\pi$
$449$	−7.14027	−0.336970	−0.168485	+	0.985704i	$0.553888\pi$
$450$	0	0
$451$	58.1150	2.73653
$452$	−14.6674	−0.689895
$453$	0.896051	0.0421001
$454$	−13.7477	−0.645209
$455$	0	0
$456$	7.14743	0.334709
$457$	29.4258	1.37648	0.688240	−	0.725483i	$-0.258383\pi$
$457$	29.4258	1.37648	0.688240	+	0.725483i	$0.258383\pi$
$458$	−4.16052	−0.194408
$459$	0	0
$460$	0	0
$461$	−13.1230	−0.611197	−0.305599	−	0.952160i	$-0.598857\pi$
$461$	−13.1230	−0.611197	−0.305599	+	0.952160i	$0.598857\pi$
$462$	−7.90514	−0.367780
$463$	−24.8827	−1.15640	−0.578198	−	0.815896i	$-0.696244\pi$
$463$	−24.8827	−1.15640	−0.578198	+	0.815896i	$0.696244\pi$
$464$	2.73306	0.126879
$465$	0	0
$466$	17.3543	0.803923
$467$	33.2280	1.53761	0.768805	−	0.639483i	$-0.220852\pi$
$467$	33.2280	1.53761	0.768805	+	0.639483i	$0.220852\pi$
$468$	3.19091	0.147500
$469$	−6.58099	−0.303882
$470$	0	0
$471$	−19.1440	−0.882111
$472$	−38.1950	−1.75807
$473$	−29.0943	−1.33776
$474$	1.37628	0.0632146
$475$	0	0
$476$	0	0
$477$	−5.91920	−0.271022
$478$	11.9704	0.547512
$479$	1.31256	0.0599722	0.0299861	−	0.999550i	$-0.490454\pi$
$479$	1.31256	0.0599722	0.0299861	+	0.999550i	$0.490454\pi$
$480$	0	0
$481$	12.6139	0.575145
$482$	17.6802	0.805310
$483$	10.8341	0.492970
$484$	−27.6533	−1.25697
$485$	0	0
$486$	14.0268	0.636269
$487$	−23.1429	−1.04870	−0.524352	−	0.851502i	$-0.675692\pi$
$487$	−23.1429	−1.04870	−0.524352	+	0.851502i	$0.675692\pi$
$488$	17.7961	0.805591
$489$	25.0865	1.13445
$490$	0	0
$491$	−7.02201	−0.316899	−0.158449	−	0.987367i	$-0.550649\pi$
$491$	−7.02201	−0.316899	−0.158449	+	0.987367i	$0.550649\pi$
$492$	10.6154	0.478580
$493$	0	0
$494$	−4.52289	−0.203494
$495$	0	0
$496$	−7.08116	−0.317953
$497$	4.97485	0.223153
$498$	−4.75606	−0.213124
$499$	1.20807	0.0540806	0.0270403	−	0.999634i	$-0.491392\pi$
$499$	1.20807	0.0540806	0.0270403	+	0.999634i	$0.491392\pi$
$500$	0	0
$501$	17.3482	0.775059
$502$	7.66759	0.342221
$503$	−18.5304	−0.826229	−0.413114	−	0.910679i	$-0.635559\pi$
$503$	−18.5304	−0.826229	−0.413114	+	0.910679i	$0.635559\pi$
$504$	4.37747	0.194988
$505$	0	0
$506$	−58.6442	−2.60705
$507$	−9.53461	−0.423447
$508$	17.6259	0.782024
$509$	1.45296	0.0644015	0.0322008	−	0.999481i	$-0.489748\pi$
$509$	1.45296	0.0644015	0.0322008	+	0.999481i	$0.489748\pi$
$510$	0	0
$511$	3.60390	0.159427
$512$	−13.3681	−0.590791
$513$	10.5725	0.466787
$514$	−3.84827	−0.169740
$515$	0	0
$516$	−5.31444	−0.233955
$517$	−12.3942	−0.545098
$518$	5.53565	0.243223
$519$	28.6229	1.25640
$520$	0	0
$521$	−12.9375	−0.566802	−0.283401	−	0.959001i	$-0.591463\pi$
$521$	−12.9375	−0.566802	−0.283401	+	0.959001i	$0.591463\pi$
$522$	3.36795	0.147411
$523$	−29.1271	−1.27364	−0.636819	−	0.771013i	$-0.719750\pi$
$523$	−29.1271	−1.27364	−0.636819	+	0.771013i	$0.719750\pi$
$524$	−1.67979	−0.0733820
$525$	0	0
$526$	15.5723	0.678985
$527$	0	0
$528$	9.67571	0.421081
$529$	57.3728	2.49447
$530$	0	0
$531$	−18.6391	−0.808869
$532$	1.76275	0.0764251
$533$	−20.9987	−0.909557
$534$	11.4780	0.496700
$535$	0	0
$536$	20.3405	0.878578
$537$	−16.2664	−0.701948
$538$	16.5241	0.712406
$539$	38.3957	1.65382
$540$	0	0
$541$	12.9238	0.555639	0.277820	−	0.960633i	$-0.410388\pi$
$541$	12.9238	0.555639	0.277820	+	0.960633i	$0.410388\pi$
$542$	−13.5093	−0.580275
$543$	24.3087	1.04319
$544$	0	0
$545$	0	0
$546$	2.85637	0.122241
$547$	3.74868	0.160282	0.0801410	−	0.996784i	$-0.474463\pi$
$547$	3.74868	0.160282	0.0801410	+	0.996784i	$0.474463\pi$
$548$	−17.5966	−0.751690
$549$	8.68446	0.370644
$550$	0	0
$551$	4.23960	0.180613
$552$	−33.4861	−1.42526
$553$	1.06106	0.0451208
$554$	−5.82403	−0.247439
$555$	0	0
$556$	−19.5333	−0.828396
$557$	39.1300	1.65799	0.828996	−	0.559255i	$-0.188913\pi$
$557$	39.1300	1.65799	0.828996	+	0.559255i	$0.188913\pi$
$558$	−8.72611	−0.369406
$559$	10.5127	0.444640
$560$	0	0
$561$	0	0
$562$	−20.9950	−0.885621
$563$	39.8934	1.68131	0.840653	−	0.541574i	$-0.182171\pi$
$563$	39.8934	1.68131	0.840653	+	0.541574i	$0.182171\pi$
$564$	−2.26396	−0.0953300
$565$	0	0
$566$	−10.1420	−0.426300
$567$	−2.33796	−0.0981853
$568$	−15.3763	−0.645175
$569$	29.7213	1.24598	0.622991	−	0.782229i	$-0.285917\pi$
$569$	29.7213	1.24598	0.622991	+	0.782229i	$0.285917\pi$
$570$	0	0
$571$	16.8380	0.704650	0.352325	−	0.935878i	$-0.385391\pi$
$571$	16.8380	0.704650	0.352325	+	0.935878i	$0.385391\pi$
$572$	13.7311	0.574124
$573$	−25.6438	−1.07128
$574$	−9.21535	−0.384641
$575$	0	0
$576$	−10.9157	−0.454821
$577$	0.136830	0.00569631	0.00284816	−	0.999996i	$-0.499093\pi$
$577$	0.136830	0.00569631	0.00284816	+	0.999996i	$0.499093\pi$
$578$	0	0
$579$	1.75331	0.0728650
$580$	0	0
$581$	−3.66674	−0.152122
$582$	2.76427	0.114583
$583$	−25.4714	−1.05492
$584$	−11.1389	−0.460933
$585$	0	0
$586$	22.3654	0.923904
$587$	−23.8523	−0.984492	−0.492246	−	0.870456i	$-0.663824\pi$
$587$	−23.8523	−0.984492	−0.492246	+	0.870456i	$0.663824\pi$
$588$	7.01345	0.289230
$589$	−10.9845	−0.452608
$590$	0	0
$591$	10.1188	0.416233
$592$	−6.77551	−0.278472
$593$	1.47081	0.0603990	0.0301995	−	0.999544i	$-0.490386\pi$
$593$	1.47081	0.0603990	0.0301995	+	0.999544i	$0.490386\pi$
$594$	36.1417	1.48291
$595$	0	0
$596$	9.94079	0.407191
$597$	−14.7277	−0.602763
$598$	21.1900	0.866523
$599$	25.1103	1.02598	0.512990	−	0.858394i	$-0.328538\pi$
$599$	25.1103	1.02598	0.512990	+	0.858394i	$0.328538\pi$
$600$	0	0
$601$	−17.1247	−0.698529	−0.349265	−	0.937024i	$-0.613569\pi$
$601$	−17.1247	−0.698529	−0.349265	+	0.937024i	$0.613569\pi$
$602$	4.61352	0.188033
$603$	9.92616	0.404224
$604$	−0.683039	−0.0277925
$605$	0	0
$606$	19.3679	0.786769
$607$	16.7901	0.681487	0.340744	−	0.940156i	$-0.389321\pi$
$607$	16.7901	0.681487	0.340744	+	0.940156i	$0.389321\pi$
$608$	−9.15375	−0.371234
$609$	−2.67747	−0.108496
$610$	0	0
$611$	4.47843	0.181178
$612$	0	0
$613$	−22.7069	−0.917125	−0.458562	−	0.888662i	$-0.651635\pi$
$613$	−22.7069	−0.917125	−0.458562	+	0.888662i	$0.651635\pi$
$614$	−24.9038	−1.00504
$615$	0	0
$616$	18.8371	0.758966
$617$	6.33001	0.254837	0.127418	−	0.991849i	$-0.459331\pi$
$617$	6.33001	0.254837	0.127418	+	0.991849i	$0.459331\pi$
$618$	8.95292	0.360139
$619$	17.9393	0.721043	0.360522	−	0.932751i	$-0.382599\pi$
$619$	17.9393	0.721043	0.360522	+	0.932751i	$0.382599\pi$
$620$	0	0
$621$	−49.5327	−1.98768
$622$	17.2088	0.690009
$623$	8.84906	0.354530
$624$	−3.49613	−0.139957
$625$	0	0
$626$	−32.0343	−1.28035
$627$	15.0092	0.599411
$628$	14.5931	0.582327
$629$	0	0
$630$	0	0
$631$	17.4118	0.693152	0.346576	−	0.938022i	$-0.387344\pi$
$631$	17.4118	0.693152	0.346576	+	0.938022i	$0.387344\pi$
$632$	−3.27952	−0.130452
$633$	−2.32862	−0.0925542
$634$	−5.59474	−0.222195
$635$	0	0
$636$	−4.65267	−0.184490
$637$	−13.8736	−0.549691
$638$	14.4929	0.573780
$639$	−7.50361	−0.296838
$640$	0	0
$641$	−39.1880	−1.54783	−0.773917	−	0.633287i	$-0.781705\pi$
$641$	−39.1880	−1.54783	−0.773917	+	0.633287i	$0.781705\pi$
$642$	−12.9534	−0.511231
$643$	18.5219	0.730432	0.365216	−	0.930923i	$-0.380995\pi$
$643$	18.5219	0.730432	0.365216	+	0.930923i	$0.380995\pi$
$644$	−8.25860	−0.325435
$645$	0	0
$646$	0	0
$647$	−35.7438	−1.40523	−0.702617	−	0.711569i	$-0.747985\pi$
$647$	−35.7438	−1.40523	−0.702617	+	0.711569i	$0.747985\pi$
$648$	7.22618	0.283871
$649$	−80.2075	−3.14842
$650$	0	0
$651$	6.93712	0.271887
$652$	−19.1229	−0.748909
$653$	12.8668	0.503516	0.251758	−	0.967790i	$-0.418991\pi$
$653$	12.8668	0.503516	0.251758	+	0.967790i	$0.418991\pi$
$654$	7.68155	0.300372
$655$	0	0
$656$	11.2794	0.440386
$657$	−5.43579	−0.212070
$658$	1.96537	0.0766181
$659$	10.2310	0.398542	0.199271	−	0.979944i	$-0.436143\pi$
$659$	10.2310	0.398542	0.199271	+	0.979944i	$0.436143\pi$
$660$	0	0
$661$	−8.64646	−0.336309	−0.168154	−	0.985761i	$-0.553781\pi$
$661$	−8.64646	−0.336309	−0.168154	+	0.985761i	$0.553781\pi$
$662$	−5.60949	−0.218019
$663$	0	0
$664$	11.3332	0.439812
$665$	0	0
$666$	−8.34946	−0.323535
$667$	−19.8628	−0.769089
$668$	−13.2241	−0.511657
$669$	9.16594	0.354376
$670$	0	0
$671$	37.3708	1.44268
$672$	5.78094	0.223004
$673$	11.8516	0.456846	0.228423	−	0.973562i	$-0.426643\pi$
$673$	11.8516	0.456846	0.228423	+	0.973562i	$0.426643\pi$
$674$	12.3219	0.474624
$675$	0	0
$676$	7.26802	0.279539
$677$	−11.8491	−0.455400	−0.227700	−	0.973731i	$-0.573121\pi$
$677$	−11.8491	−0.455400	−0.227700	+	0.973731i	$0.573121\pi$
$678$	19.8035	0.760551
$679$	2.13115	0.0817858
$680$	0	0
$681$	−16.4845	−0.631689
$682$	−37.5501	−1.43787
$683$	−9.39813	−0.359610	−0.179805	−	0.983702i	$-0.557547\pi$
$683$	−9.39813	−0.359610	−0.179805	+	0.983702i	$0.557547\pi$
$684$	−2.65878	−0.101661
$685$	0	0
$686$	−13.1433	−0.501814
$687$	−4.98879	−0.190334
$688$	−5.64684	−0.215284
$689$	9.20362	0.350630
$690$	0	0
$691$	20.8031	0.791385	0.395693	−	0.918383i	$-0.370505\pi$
$691$	20.8031	0.791385	0.395693	+	0.918383i	$0.370505\pi$
$692$	−21.8185	−0.829417
$693$	9.19246	0.349193
$694$	10.0757	0.382468
$695$	0	0
$696$	8.27552	0.313683
$697$	0	0
$698$	−28.4504	−1.07686
$699$	20.8092	0.787076
$700$	0	0
$701$	20.1807	0.762214	0.381107	−	0.924531i	$-0.375543\pi$
$701$	20.1807	0.762214	0.381107	+	0.924531i	$0.375543\pi$
$702$	−13.0591	−0.492884
$703$	−10.5104	−0.396406
$704$	−46.9723	−1.77033
$705$	0	0
$706$	−5.40547	−0.203437
$707$	14.9319	0.561573
$708$	−14.6509	−0.550615
$709$	40.7711	1.53119	0.765595	−	0.643323i	$-0.222445\pi$
$709$	40.7711	1.53119	0.765595	+	0.643323i	$0.222445\pi$
$710$	0	0
$711$	−1.60040	−0.0600197
$712$	−27.3507	−1.02501
$713$	51.4630	1.92730
$714$	0	0
$715$	0	0
$716$	12.3995	0.463392
$717$	14.3534	0.536039
$718$	−15.7051	−0.586109
$719$	10.3493	0.385964	0.192982	−	0.981202i	$-0.438184\pi$
$719$	10.3493	0.385964	0.192982	+	0.981202i	$0.438184\pi$
$720$	0	0
$721$	6.90235	0.257057
$722$	−15.7863	−0.587507
$723$	21.1999	0.788435
$724$	−18.5300	−0.688662
$725$	0	0
$726$	37.3368	1.38570
$727$	−3.42797	−0.127136	−0.0635682	−	0.997977i	$-0.520248\pi$
$727$	−3.42797	−0.127136	−0.0635682	+	0.997977i	$0.520248\pi$
$728$	−6.80641	−0.252262
$729$	23.9819	0.888218
$730$	0	0
$731$	0	0
$732$	6.82624	0.252305
$733$	−10.9959	−0.406144	−0.203072	−	0.979164i	$-0.565092\pi$
$733$	−10.9959	−0.406144	−0.203072	+	0.979164i	$0.565092\pi$
$734$	20.6189	0.761058
$735$	0	0
$736$	42.8858	1.58079
$737$	42.7141	1.57339
$738$	13.8996	0.511651
$739$	33.9094	1.24738	0.623690	−	0.781672i	$-0.285633\pi$
$739$	33.9094	1.24738	0.623690	+	0.781672i	$0.285633\pi$
$740$	0	0
$741$	−5.42330	−0.199230
$742$	4.03903	0.148277
$743$	−7.10950	−0.260822	−0.130411	−	0.991460i	$-0.541630\pi$
$743$	−7.10950	−0.260822	−0.130411	+	0.991460i	$0.541630\pi$
$744$	−21.4413	−0.786075
$745$	0	0
$746$	6.91963	0.253346
$747$	5.53057	0.202353
$748$	0	0
$749$	−9.98659	−0.364902
$750$	0	0
$751$	1.52759	0.0557427	0.0278714	−	0.999612i	$-0.491127\pi$
$751$	1.52759	0.0557427	0.0278714	+	0.999612i	$0.491127\pi$
$752$	−2.40557	−0.0877220
$753$	9.19405	0.335050
$754$	−5.23674	−0.190711
$755$	0	0
$756$	5.08967	0.185109
$757$	−41.2064	−1.49767	−0.748835	−	0.662756i	$-0.769387\pi$
$757$	−41.2064	−1.49767	−0.748835	+	0.662756i	$0.769387\pi$
$758$	−5.07774	−0.184432
$759$	−70.3191	−2.55242
$760$	0	0
$761$	29.5046	1.06954	0.534771	−	0.844997i	$-0.320398\pi$
$761$	29.5046	1.06954	0.534771	+	0.844997i	$0.320398\pi$
$762$	−23.7981	−0.862115
$763$	5.92217	0.214397
$764$	19.5477	0.707210
$765$	0	0
$766$	23.7556	0.858323
$767$	28.9815	1.04646
$768$	−20.7320	−0.748100
$769$	36.3744	1.31170	0.655848	−	0.754893i	$-0.272311\pi$
$769$	36.3744	1.31170	0.655848	+	0.754893i	$0.272311\pi$
$770$	0	0
$771$	−4.61439	−0.166183
$772$	−1.33651	−0.0481019
$773$	−14.6994	−0.528700	−0.264350	−	0.964427i	$-0.585157\pi$
$773$	−14.6994	−0.528700	−0.264350	+	0.964427i	$0.585157\pi$
$774$	−6.95861	−0.250122
$775$	0	0
$776$	−6.58695	−0.236458
$777$	6.63769	0.238126
$778$	0.964913	0.0345938
$779$	17.4969	0.626891
$780$	0	0
$781$	−32.2894	−1.15541
$782$	0	0
$783$	12.2412	0.437463
$784$	7.45212	0.266147
$785$	0	0
$786$	2.26801	0.0808974
$787$	21.2080	0.755983	0.377991	−	0.925809i	$-0.376615\pi$
$787$	21.2080	0.755983	0.377991	+	0.925809i	$0.376615\pi$
$788$	−7.71336	−0.274777
$789$	18.6724	0.664756
$790$	0	0
$791$	15.2678	0.542859
$792$	−28.4121	−1.00958
$793$	−13.5032	−0.479514
$794$	−26.9265	−0.955586
$795$	0	0
$796$	11.2266	0.397915
$797$	28.5869	1.01260	0.506300	−	0.862358i	$-0.331013\pi$
$797$	28.5869	1.01260	0.506300	+	0.862358i	$0.331013\pi$
$798$	−2.38003	−0.0842522
$799$	0	0
$800$	0	0
$801$	−13.3471	−0.471596
$802$	−10.8163	−0.381937
$803$	−23.3912	−0.825458
$804$	7.80225	0.275164
$805$	0	0
$806$	13.5680	0.477913
$807$	19.8138	0.697477
$808$	−46.1516	−1.62361
$809$	5.51809	0.194006	0.0970028	−	0.995284i	$-0.469074\pi$
$809$	5.51809	0.194006	0.0970028	+	0.995284i	$0.469074\pi$
$810$	0	0
$811$	30.7170	1.07862	0.539309	−	0.842108i	$-0.318685\pi$
$811$	30.7170	1.07862	0.539309	+	0.842108i	$0.318685\pi$
$812$	2.04097	0.0716241
$813$	−16.1987	−0.568115
$814$	−35.9293	−1.25932
$815$	0	0
$816$	0	0
$817$	−8.75955	−0.306458
$818$	−27.1506	−0.949299
$819$	−3.32152	−0.116063
$820$	0	0
$821$	−3.81816	−0.133255	−0.0666274	−	0.997778i	$-0.521224\pi$
$821$	−3.81816	−0.133255	−0.0666274	+	0.997778i	$0.521224\pi$
$822$	23.7585	0.828674
$823$	−3.04341	−0.106087	−0.0530434	−	0.998592i	$-0.516892\pi$
$823$	−3.04341	−0.106087	−0.0530434	+	0.998592i	$0.516892\pi$
$824$	−21.3338	−0.743198
$825$	0	0
$826$	12.7186	0.442537
$827$	47.0851	1.63731	0.818654	−	0.574287i	$-0.194721\pi$
$827$	47.0851	1.63731	0.818654	+	0.574287i	$0.194721\pi$
$828$	12.4565	0.432894
$829$	−43.3282	−1.50485	−0.752425	−	0.658678i	$-0.771116\pi$
$829$	−43.3282	−1.50485	−0.752425	+	0.658678i	$0.771116\pi$
$830$	0	0
$831$	−6.98347	−0.242254
$832$	16.9725	0.588417
$833$	0	0
$834$	26.3734	0.913236
$835$	0	0
$836$	−11.4412	−0.395702
$837$	−31.7159	−1.09626
$838$	−4.09421	−0.141432
$839$	−18.8359	−0.650289	−0.325145	−	0.945664i	$-0.605413\pi$
$839$	−18.8359	−0.650289	−0.325145	+	0.945664i	$0.605413\pi$
$840$	0	0
$841$	−24.0913	−0.830733
$842$	−11.3325	−0.390544
$843$	−25.1747	−0.867062
$844$	1.77505	0.0610998
$845$	0	0
$846$	−2.96438	−0.101917
$847$	28.7852	0.989072
$848$	−4.94368	−0.169767
$849$	−12.1611	−0.417367
$850$	0	0
$851$	49.2416	1.68798
$852$	−5.89806	−0.202064
$853$	−5.51855	−0.188951	−0.0944757	−	0.995527i	$-0.530117\pi$
$853$	−5.51855	−0.188951	−0.0944757	+	0.995527i	$0.530117\pi$
$854$	−5.92593	−0.202781
$855$	0	0
$856$	30.8666	1.05500
$857$	56.0071	1.91317	0.956583	−	0.291460i	$-0.0941410\pi$
$857$	56.0071	1.91317	0.956583	+	0.291460i	$0.0941410\pi$
$858$	−18.5394	−0.632923
$859$	−4.95424	−0.169036	−0.0845182	−	0.996422i	$-0.526935\pi$
$859$	−4.95424	−0.169036	−0.0845182	+	0.996422i	$0.526935\pi$
$860$	0	0
$861$	−11.0499	−0.376581
$862$	36.6624	1.24872
$863$	−16.6702	−0.567461	−0.283730	−	0.958904i	$-0.591572\pi$
$863$	−16.6702	−0.567461	−0.283730	+	0.958904i	$0.591572\pi$
$864$	−26.4300	−0.899166
$865$	0	0
$866$	8.78626	0.298569
$867$	0	0
$868$	−5.28801	−0.179487
$869$	−6.88682	−0.233619
$870$	0	0
$871$	−15.4339	−0.522959
$872$	−18.3043	−0.619860
$873$	−3.21442	−0.108792
$874$	−17.6562	−0.597231
$875$	0	0
$876$	−4.27269	−0.144361
$877$	−0.897652	−0.0303116	−0.0151558	−	0.999885i	$-0.504824\pi$
$877$	−0.897652	−0.0303116	−0.0151558	+	0.999885i	$0.504824\pi$
$878$	−16.1436	−0.544820
$879$	26.8178	0.904543
$880$	0	0
$881$	41.9218	1.41238	0.706191	−	0.708022i	$-0.250412\pi$
$881$	41.9218	1.41238	0.706191	+	0.708022i	$0.250412\pi$
$882$	9.18325	0.309216
$883$	−33.8488	−1.13910	−0.569552	−	0.821956i	$-0.692883\pi$
$883$	−33.8488	−1.13910	−0.569552	+	0.821956i	$0.692883\pi$
$884$	0	0
$885$	0	0
$886$	1.40549	0.0472182
$887$	24.3798	0.818594	0.409297	−	0.912401i	$-0.365774\pi$
$887$	24.3798	0.818594	0.409297	+	0.912401i	$0.365774\pi$
$888$	−20.5158	−0.688464
$889$	−18.3474	−0.615353
$890$	0	0
$891$	15.1746	0.508368
$892$	−6.98699	−0.233942
$893$	−3.73158	−0.124873
$894$	−13.4218	−0.448893
$895$	0	0
$896$	−1.92019	−0.0641492
$897$	25.4085	0.848364
$898$	−7.34883	−0.245234
$899$	−12.7182	−0.424175
$900$	0	0
$901$	0	0
$902$	59.8124	1.99154
$903$	5.53198	0.184093
$904$	−47.1896	−1.56950
$905$	0	0
$906$	0.922223	0.0306388
$907$	−26.5406	−0.881267	−0.440633	−	0.897687i	$-0.645246\pi$
$907$	−26.5406	−0.881267	−0.440633	+	0.897687i	$0.645246\pi$
$908$	12.5658	0.417010
$909$	−22.5219	−0.747005
$910$	0	0
$911$	−3.08111	−0.102082	−0.0510408	−	0.998697i	$-0.516254\pi$
$911$	−3.08111	−0.102082	−0.0510408	+	0.998697i	$0.516254\pi$
$912$	2.91310	0.0964624
$913$	23.7990	0.787634
$914$	30.2853	1.00175
$915$	0	0
$916$	3.80284	0.125650
$917$	1.74855	0.0577422
$918$	0	0
$919$	−24.4292	−0.805844	−0.402922	−	0.915234i	$-0.632005\pi$
$919$	−24.4292	−0.805844	−0.402922	+	0.915234i	$0.632005\pi$
$920$	0	0
$921$	−29.8617	−0.983975
$922$	−13.5063	−0.444805
$923$	11.6672	0.384029
$924$	7.22554	0.237703
$925$	0	0
$926$	−25.6095	−0.841580
$927$	−10.4109	−0.341938
$928$	−10.5985	−0.347913
$929$	22.7545	0.746551	0.373275	−	0.927721i	$-0.378235\pi$
$929$	22.7545	0.746551	0.373275	+	0.927721i	$0.378235\pi$
$930$	0	0
$931$	11.5599	0.378862
$932$	−15.8624	−0.519590
$933$	20.6347	0.675549
$934$	34.1986	1.11901
$935$	0	0
$936$	10.2662	0.335560
$937$	44.9890	1.46973	0.734863	−	0.678216i	$-0.237246\pi$
$937$	44.9890	1.46973	0.734863	+	0.678216i	$0.237246\pi$
$938$	−6.77322	−0.221153
$939$	−38.4117	−1.25352
$940$	0	0
$941$	12.9456	0.422015	0.211008	−	0.977484i	$-0.432326\pi$
$941$	12.9456	0.422015	0.211008	+	0.977484i	$0.432326\pi$
$942$	−19.7032	−0.641965
$943$	−81.9739	−2.66944
$944$	−15.5673	−0.506671
$945$	0	0
$946$	−29.9442	−0.973569
$947$	−26.7281	−0.868545	−0.434273	−	0.900781i	$-0.642995\pi$
$947$	−26.7281	−0.868545	−0.434273	+	0.900781i	$0.642995\pi$
$948$	−1.25796	−0.0408567
$949$	8.45197	0.274363
$950$	0	0
$951$	−6.70853	−0.217539
$952$	0	0
$953$	−49.4696	−1.60248	−0.801239	−	0.598344i	$-0.795825\pi$
$953$	−49.4696	−1.60248	−0.801239	+	0.598344i	$0.795825\pi$
$954$	−6.09210	−0.197239
$955$	0	0
$956$	−10.9413	−0.353867
$957$	17.3782	0.561756
$958$	1.35089	0.0436454
$959$	18.3169	0.591484
$960$	0	0
$961$	1.95188	0.0629637
$962$	12.9824	0.418568
$963$	15.0628	0.485393
$964$	−16.1602	−0.520486
$965$	0	0
$966$	11.1506	0.358764
$967$	−14.3279	−0.460754	−0.230377	−	0.973101i	$-0.573996\pi$
$967$	−14.3279	−0.460754	−0.230377	+	0.973101i	$0.573996\pi$
$968$	−88.9694	−2.85959
$969$	0	0
$970$	0	0
$971$	−33.6600	−1.08020	−0.540101	−	0.841600i	$-0.681614\pi$
$971$	−33.6600	−1.08020	−0.540101	+	0.841600i	$0.681614\pi$
$972$	−12.8209	−0.411232
$973$	20.3329	0.651841
$974$	−23.8189	−0.763205
$975$	0	0
$976$	7.25320	0.232169
$977$	28.6485	0.916545	0.458273	−	0.888812i	$-0.348468\pi$
$977$	28.6485	0.916545	0.458273	+	0.888812i	$0.348468\pi$
$978$	25.8192	0.825608
$979$	−57.4350	−1.83563
$980$	0	0
$981$	−8.93245	−0.285191
$982$	−7.22711	−0.230626
$983$	29.9884	0.956482	0.478241	−	0.878229i	$-0.341274\pi$
$983$	29.9884	0.956482	0.478241	+	0.878229i	$0.341274\pi$
$984$	34.1532	1.08876
$985$	0	0
$986$	0	0
$987$	2.35663	0.0750125
$988$	4.13406	0.131522
$989$	41.0389	1.30496
$990$	0	0
$991$	−26.1900	−0.831951	−0.415976	−	0.909376i	$-0.636560\pi$
$991$	−26.1900	−0.831951	−0.415976	+	0.909376i	$0.636560\pi$
$992$	27.4599	0.871853
$993$	−6.72623	−0.213450
$994$	5.12016	0.162402
$995$	0	0
$996$	4.34719	0.137746
$997$	−36.2778	−1.14893	−0.574464	−	0.818530i	$-0.694789\pi$
$997$	−36.2778	−1.14893	−0.574464	+	0.818530i	$0.694789\pi$
$998$	1.24335	0.0393577
$999$	−30.3470	−0.960136

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bo.1.9		12
5.4	even	2		1445.2.a.r.1.4	✓	12
17.16	even	2		7225.2.a.bn.1.9		12
85.4	even	4		1445.2.d.i.866.17		24
85.64	even	4		1445.2.d.i.866.18		24
85.84	even	2		1445.2.a.s.1.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1445.2.a.r.1.4	✓	12	5.4	even	2
1445.2.a.s.1.4	yes	12	85.84	even	2
1445.2.d.i.866.17		24	85.4	even	4
1445.2.d.i.866.18		24	85.64	even	4
7225.2.a.bn.1.9		12	17.16	even	2
7225.2.a.bo.1.9		12	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.02921 q^{2} +1.23410 q^{3} -0.940729 q^{4} +1.27015 q^{6} +0.979236 q^{7} -3.02662 q^{8} -1.47699 q^{9} -6.35575 q^{11} -1.16096 q^{12} +2.29653 q^{13} +1.00784 q^{14} -1.23357 q^{16} -1.52013 q^{18} -1.91355 q^{19} +1.20848 q^{21} -6.54140 q^{22} +8.96509 q^{23} -3.73517 q^{24} +2.36361 q^{26} -5.52507 q^{27} -0.921196 q^{28} -2.21557 q^{29} +5.74037 q^{31} +4.78365 q^{32} -7.84366 q^{33} +1.38945 q^{36} +5.49260 q^{37} -1.96944 q^{38} +2.83416 q^{39} -9.14368 q^{41} +1.24378 q^{42} +4.57764 q^{43} +5.97904 q^{44} +9.22695 q^{46} +1.95008 q^{47} -1.52235 q^{48} -6.04110 q^{49} -2.16041 q^{52} +4.00762 q^{53} -5.68645 q^{54} -2.96378 q^{56} -2.36152 q^{57} -2.28028 q^{58} +12.6197 q^{59} -5.87984 q^{61} +5.90804 q^{62} -1.44632 q^{63} +7.39051 q^{64} -8.07276 q^{66} -6.72054 q^{67} +11.0638 q^{69} +5.08034 q^{71} +4.47029 q^{72} +3.68032 q^{73} +5.65303 q^{74} +1.80013 q^{76} -6.22378 q^{77} +2.91694 q^{78} +1.08356 q^{79} -2.38754 q^{81} -9.41075 q^{82} -3.74449 q^{83} -1.13685 q^{84} +4.71135 q^{86} -2.73424 q^{87} +19.2365 q^{88} +9.03669 q^{89} +2.24885 q^{91} -8.43372 q^{92} +7.08421 q^{93} +2.00704 q^{94} +5.90351 q^{96} +2.17633 q^{97} -6.21755 q^{98} +9.38737 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 3 q^{3} + 21 q^{4} + 9 q^{6} + 6 q^{7} - 12 q^{8} + 21 q^{9} + 6 q^{11} + 6 q^{12} - 9 q^{13} + 18 q^{14} + 39 q^{16} + 9 q^{18} + 27 q^{19} + 6 q^{21} + 15 q^{22} + 18 q^{23} + 36 q^{24}+ \cdots + 6 q^{99}+O(q^{100}) \) 12 * q - 3 * q^2 + 3 * q^3 + 21 * q^4 + 9 * q^6 + 6 * q^7 - 12 * q^8 + 21 * q^9 + 6 * q^11 + 6 * q^12 - 9 * q^13 + 18 * q^14 + 39 * q^16 + 9 * q^18 + 27 * q^19 + 6 * q^21 + 15 * q^22 + 18 * q^23 + 36 * q^24 + 6 * q^26 + 12 * q^27 + 39 * q^28 + 6 * q^31 - 12 * q^32 + 48 * q^36 + 21 * q^37 + 3 * q^38 + 9 * q^39 + 15 * q^41 - 30 * q^42 - 45 * q^43 + 42 * q^44 + 15 * q^46 - 3 * q^47 - 63 * q^48 + 12 * q^49 - 51 * q^52 + 12 * q^53 + 27 * q^54 - 3 * q^56 + 6 * q^57 - 51 * q^58 + 36 * q^59 - 3 * q^61 + 36 * q^62 + 72 * q^63 + 42 * q^64 - 39 * q^66 - 78 * q^67 - 66 * q^69 + 30 * q^71 + 12 * q^72 - 33 * q^73 - 9 * q^74 + 27 * q^76 + 39 * q^77 - 24 * q^78 + 3 * q^79 + 12 * q^81 - 51 * q^82 - 27 * q^83 + 27 * q^84 + 18 * q^86 - 24 * q^87 + 36 * q^88 + 6 * q^89 + 24 * q^91 - 3 * q^92 + 12 * q^93 + 21 * q^94 + 39 * q^96 - 6 * q^97 + 42 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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