LMFDB - Embedded newform 9025.2.a.cn.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 32 x^{18} + 426 x^{16} - 3061 x^{14} + 12909 x^{12} - 32678 x^{10} + 49159 x^{8} - 42549 x^{6} + \cdots + 405 $ x^20 - 32x^18 + 426x^16 - 3061x^14 + 12909x^12 - 32678x^10 + 49159x^8 - 42549x^6 + 20056x^4 - 4635*x^2 + 405
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$1.75426$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.75426 q^{2} +0.822481 q^{3} +1.07744 q^{4} +1.44285 q^{6} +3.28692 q^{7} -1.61842 q^{8} -2.32353 q^{9} +O(q^{10})$	\(q+1.75426 q^{2} +0.822481 q^{3} +1.07744 q^{4} +1.44285 q^{6} +3.28692 q^{7} -1.61842 q^{8} -2.32353 q^{9} +1.18053 q^{11} +0.886172 q^{12} -3.78272 q^{13} +5.76611 q^{14} -4.99400 q^{16} -2.75438 q^{17} -4.07607 q^{18} +2.70342 q^{21} +2.07096 q^{22} +3.31465 q^{23} -1.33112 q^{24} -6.63589 q^{26} -4.37850 q^{27} +3.54145 q^{28} -9.75933 q^{29} +1.66191 q^{31} -5.52396 q^{32} +0.970961 q^{33} -4.83191 q^{34} -2.50345 q^{36} +8.47172 q^{37} -3.11122 q^{39} +5.81804 q^{41} +4.74252 q^{42} -4.63105 q^{43} +1.27195 q^{44} +5.81477 q^{46} -8.63909 q^{47} -4.10747 q^{48} +3.80382 q^{49} -2.26543 q^{51} -4.07565 q^{52} -0.0828551 q^{53} -7.68103 q^{54} -5.31960 q^{56} -17.1204 q^{58} -4.80786 q^{59} +1.01041 q^{61} +2.91542 q^{62} -7.63723 q^{63} +0.297527 q^{64} +1.70332 q^{66} +8.59954 q^{67} -2.96767 q^{68} +2.72623 q^{69} +5.01790 q^{71} +3.76043 q^{72} -14.7801 q^{73} +14.8616 q^{74} +3.88030 q^{77} -5.45789 q^{78} -13.9448 q^{79} +3.36935 q^{81} +10.2064 q^{82} -11.8189 q^{83} +2.91277 q^{84} -8.12409 q^{86} -8.02686 q^{87} -1.91059 q^{88} -15.1049 q^{89} -12.4335 q^{91} +3.57133 q^{92} +1.36689 q^{93} -15.1552 q^{94} -4.54335 q^{96} -10.9874 q^{97} +6.67290 q^{98} -2.74299 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 24 q^{4} - 20 q^{6} - 8 q^{7} + 16 q^{9}+O(q^{10}) $ 20 * q + 24 * q^4 - 20 * q^6 - 8 * q^7 + 16 * q^9	$ 20 q + 24 q^{4} - 20 q^{6} - 8 q^{7} + 16 q^{9} + 10 q^{11} + 40 q^{16} - 38 q^{17} - 56 q^{23} - 54 q^{24} - 22 q^{26} + 16 q^{28} - 32 q^{36} + 80 q^{39} - 44 q^{43} - 16 q^{44} - 98 q^{47} - 4 q^{49} - 54 q^{54} - 50 q^{58} + 40 q^{61} - 90 q^{62} - 6 q^{63} + 46 q^{64} + 8 q^{66} - 4 q^{68} - 34 q^{73} + 20 q^{74} - 80 q^{77} + 60 q^{81} + 10 q^{82} - 84 q^{83} - 90 q^{87} - 78 q^{92} - 20 q^{93} - 120 q^{96} - 76 q^{99}+O(q^{100}) $ 20 * q + 24 * q^4 - 20 * q^6 - 8 * q^7 + 16 * q^9 + 10 * q^11 + 40 * q^16 - 38 * q^17 - 56 * q^23 - 54 * q^24 - 22 * q^26 + 16 * q^28 - 32 * q^36 + 80 * q^39 - 44 * q^43 - 16 * q^44 - 98 * q^47 - 4 * q^49 - 54 * q^54 - 50 * q^58 + 40 * q^61 - 90 * q^62 - 6 * q^63 + 46 * q^64 + 8 * q^66 - 4 * q^68 - 34 * q^73 + 20 * q^74 - 80 * q^77 + 60 * q^81 + 10 * q^82 - 84 * q^83 - 90 * q^87 - 78 * q^92 - 20 * q^93 - 120 * q^96 - 76 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	1.75426	1.24045	0.620226	−	0.784424i	$-0.287041\pi$
$2$	1.75426	1.24045	0.620226	+	0.784424i	$0.287041\pi$
$3$	0.822481	0.474859	0.237430	−	0.971405i	$-0.423695\pi$
$3$	0.822481	0.474859	0.237430	+	0.971405i	$0.423695\pi$
$4$	1.07744	0.538719
$5$	0	0
$5$	0	0
$6$	1.44285	0.589040
$7$	3.28692	1.24234	0.621169	−	0.783677i	$-0.286658\pi$
$7$	3.28692	1.24234	0.621169	+	0.783677i	$0.286658\pi$
$8$	−1.61842	−0.572197
$9$	−2.32353	−0.774509
$10$	0	0
$11$	1.18053	0.355942	0.177971	−	0.984036i	$-0.443047\pi$
$11$	1.18053	0.355942	0.177971	+	0.984036i	$0.443047\pi$
$12$	0.886172	0.255816
$13$	−3.78272	−1.04914	−0.524569	−	0.851368i	$-0.675774\pi$
$13$	−3.78272	−1.04914	−0.524569	+	0.851368i	$0.675774\pi$
$14$	5.76611	1.54106
$15$	0	0
$16$	−4.99400	−1.24850
$17$	−2.75438	−0.668036	−0.334018	−	0.942567i	$-0.608405\pi$
$17$	−2.75438	−0.668036	−0.334018	+	0.942567i	$0.608405\pi$
$18$	−4.07607	−0.960740
$19$	0	0
$19$	0	0
$20$	0	0
$21$	2.70342	0.589936
$22$	2.07096	0.441529
$23$	3.31465	0.691152	0.345576	−	0.938391i	$-0.387684\pi$
$23$	3.31465	0.691152	0.345576	+	0.938391i	$0.387684\pi$
$24$	−1.33112	−0.271713
$25$	0	0
$26$	−6.63589	−1.30141
$27$	−4.37850	−0.842642
$28$	3.54145	0.669271
$29$	−9.75933	−1.81226	−0.906131	−	0.422996i	$-0.860978\pi$
$29$	−9.75933	−1.81226	−0.906131	+	0.422996i	$0.860978\pi$
$30$	0	0
$31$	1.66191	0.298488	0.149244	−	0.988800i	$-0.452316\pi$
$31$	1.66191	0.298488	0.149244	+	0.988800i	$0.452316\pi$
$32$	−5.52396	−0.976508
$33$	0.970961	0.169023
$34$	−4.83191	−0.828665
$35$	0	0
$36$	−2.50345	−0.417242
$37$	8.47172	1.39274	0.696371	−	0.717682i	$-0.254797\pi$
$37$	8.47172	1.39274	0.696371	+	0.717682i	$0.254797\pi$
$38$	0	0
$39$	−3.11122	−0.498193
$40$	0	0
$41$	5.81804	0.908624	0.454312	−	0.890843i	$-0.349885\pi$
$41$	5.81804	0.908624	0.454312	+	0.890843i	$0.349885\pi$
$42$	4.74252	0.731786
$43$	−4.63105	−0.706229	−0.353115	−	0.935580i	$-0.614877\pi$
$43$	−4.63105	−0.706229	−0.353115	+	0.935580i	$0.614877\pi$
$44$	1.27195	0.191753
$45$	0	0
$46$	5.81477	0.857340
$47$	−8.63909	−1.26014	−0.630071	−	0.776538i	$-0.716974\pi$
$47$	−8.63909	−1.26014	−0.630071	+	0.776538i	$0.716974\pi$
$48$	−4.10747	−0.592862
$49$	3.80382	0.543403
$50$	0	0
$51$	−2.26543	−0.317223
$52$	−4.07565	−0.565191
$53$	−0.0828551	−0.0113810	−0.00569051	−	0.999984i	$-0.501811\pi$
$53$	−0.0828551	−0.0113810	−0.00569051	+	0.999984i	$0.501811\pi$
$54$	−7.68103	−1.04526
$55$	0	0
$56$	−5.31960	−0.710861
$57$	0	0
$58$	−17.1204	−2.24802
$59$	−4.80786	−0.625930	−0.312965	−	0.949765i	$-0.601322\pi$
$59$	−4.80786	−0.625930	−0.312965	+	0.949765i	$0.601322\pi$
$60$	0	0
$61$	1.01041	0.129370	0.0646850	−	0.997906i	$-0.479396\pi$
$61$	1.01041	0.129370	0.0646850	+	0.997906i	$0.479396\pi$
$62$	2.91542	0.370259
$63$	−7.63723	−0.962201
$64$	0.297527	0.0371909
$65$	0	0
$66$	1.70332	0.209664
$67$	8.59954	1.05060	0.525301	−	0.850917i	$-0.323953\pi$
$67$	8.59954	1.05060	0.525301	+	0.850917i	$0.323953\pi$
$68$	−2.96767	−0.359883
$69$	2.72623	0.328200
$70$	0	0
$71$	5.01790	0.595515	0.297758	−	0.954642i	$-0.403761\pi$
$71$	5.01790	0.595515	0.297758	+	0.954642i	$0.403761\pi$
$72$	3.76043	0.443171
$73$	−14.7801	−1.72988	−0.864940	−	0.501876i	$-0.832644\pi$
$73$	−14.7801	−1.72988	−0.864940	+	0.501876i	$0.832644\pi$
$74$	14.8616	1.72763
$75$	0	0
$76$	0	0
$77$	3.88030	0.442201
$78$	−5.45789	−0.617984
$79$	−13.9448	−1.56891	−0.784455	−	0.620185i	$-0.787057\pi$
$79$	−13.9448	−1.56891	−0.784455	+	0.620185i	$0.787057\pi$
$80$	0	0
$81$	3.36935	0.374372
$82$	10.2064	1.12710
$83$	−11.8189	−1.29729	−0.648646	−	0.761090i	$-0.724665\pi$
$83$	−11.8189	−1.29729	−0.648646	+	0.761090i	$0.724665\pi$
$84$	2.91277	0.317810
$85$	0	0
$86$	−8.12409	−0.876043
$87$	−8.02686	−0.860570
$88$	−1.91059	−0.203669
$89$	−15.1049	−1.60111	−0.800556	−	0.599258i	$-0.795462\pi$
$89$	−15.1049	−1.60111	−0.800556	+	0.599258i	$0.795462\pi$
$90$	0	0
$91$	−12.4335	−1.30338
$92$	3.57133	0.372337
$93$	1.36689	0.141740
$94$	−15.1552	−1.56314
$95$	0	0
$96$	−4.54335	−0.463704
$97$	−10.9874	−1.11560	−0.557802	−	0.829974i	$-0.688355\pi$
$97$	−10.9874	−1.11560	−0.557802	+	0.829974i	$0.688355\pi$
$98$	6.67290	0.674064
$99$	−2.74299	−0.275680
$100$	0	0
$101$	−12.2361	−1.21753	−0.608767	−	0.793349i	$-0.708336\pi$
$101$	−12.2361	−1.21753	−0.608767	+	0.793349i	$0.708336\pi$
$102$	−3.97415	−0.393500
$103$	−7.75916	−0.764533	−0.382266	−	0.924052i	$-0.624856\pi$
$103$	−7.75916	−0.764533	−0.382266	+	0.924052i	$0.624856\pi$
$104$	6.12202	0.600314
$105$	0	0
$106$	−0.145350	−0.0141176
$107$	−6.28028	−0.607138	−0.303569	−	0.952809i	$-0.598178\pi$
$107$	−6.28028	−0.607138	−0.303569	+	0.952809i	$0.598178\pi$
$108$	−4.71756	−0.453947
$109$	20.2061	1.93539	0.967695	−	0.252125i	$-0.0811295\pi$
$109$	20.2061	1.93539	0.967695	+	0.252125i	$0.0811295\pi$
$110$	0	0
$111$	6.96783	0.661357
$112$	−16.4149	−1.55106
$113$	9.73195	0.915505	0.457752	−	0.889080i	$-0.348655\pi$
$113$	9.73195	0.915505	0.457752	+	0.889080i	$0.348655\pi$
$114$	0	0
$115$	0	0
$116$	−10.5151	−0.976300
$117$	8.78925	0.812567
$118$	−8.43424	−0.776435
$119$	−9.05342	−0.829926
$120$	0	0
$121$	−9.60635	−0.873305
$122$	1.77253	0.160477
$123$	4.78522	0.431469
$124$	1.79060	0.160801
$125$	0	0
$126$	−13.3977	−1.19356
$127$	11.1672	0.990927	0.495464	−	0.868629i	$-0.334998\pi$
$127$	11.1672	0.990927	0.495464	+	0.868629i	$0.334998\pi$
$128$	11.5699	1.02264
$129$	−3.80895	−0.335360
$130$	0	0
$131$	−12.2556	−1.07077	−0.535387	−	0.844607i	$-0.679834\pi$
$131$	−12.2556	−1.07077	−0.535387	+	0.844607i	$0.679834\pi$
$132$	1.04615	0.0910557
$133$	0	0
$134$	15.0859	1.30322
$135$	0	0
$136$	4.45774	0.382248
$137$	−18.5506	−1.58488	−0.792440	−	0.609949i	$-0.791190\pi$
$137$	−18.5506	−1.58488	−0.792440	+	0.609949i	$0.791190\pi$
$138$	4.78253	0.407116
$139$	−11.1960	−0.949635	−0.474817	−	0.880084i	$-0.657486\pi$
$139$	−11.1960	−0.949635	−0.474817	+	0.880084i	$0.657486\pi$
$140$	0	0
$141$	−7.10549	−0.598390
$142$	8.80271	0.738707
$143$	−4.46561	−0.373433
$144$	11.6037	0.966975
$145$	0	0
$146$	−25.9282	−2.14583
$147$	3.12857	0.258040
$148$	9.12775	0.750297
$149$	−18.2403	−1.49430	−0.747150	−	0.664656i	$-0.768578\pi$
$149$	−18.2403	−1.49430	−0.747150	+	0.664656i	$0.768578\pi$
$150$	0	0
$151$	7.19473	0.585499	0.292749	−	0.956189i	$-0.405430\pi$
$151$	7.19473	0.585499	0.292749	+	0.956189i	$0.405430\pi$
$152$	0	0
$153$	6.39988	0.517399
$154$	6.80706	0.548528
$155$	0	0
$156$	−3.35214	−0.268386
$157$	7.73363	0.617211	0.308605	−	0.951190i	$-0.400138\pi$
$157$	7.73363	0.617211	0.308605	+	0.951190i	$0.400138\pi$
$158$	−24.4628	−1.94616
$159$	−0.0681467	−0.00540439
$160$	0	0
$161$	10.8950	0.858644
$162$	5.91072	0.464390
$163$	22.4050	1.75490	0.877448	−	0.479671i	$-0.159244\pi$
$163$	22.4050	1.75490	0.877448	+	0.479671i	$0.159244\pi$
$164$	6.26857	0.489493
$165$	0	0
$166$	−20.7335	−1.60923
$167$	24.5234	1.89768	0.948839	−	0.315759i	$-0.102259\pi$
$167$	24.5234	1.89768	0.948839	+	0.315759i	$0.102259\pi$
$168$	−4.37527	−0.337559
$169$	1.30899	0.100692
$170$	0	0
$171$	0	0
$172$	−4.98967	−0.380459
$173$	14.0400	1.06744	0.533721	−	0.845661i	$-0.320793\pi$
$173$	14.0400	1.06744	0.533721	+	0.845661i	$0.320793\pi$
$174$	−14.0812	−1.06750
$175$	0	0
$176$	−5.89556	−0.444394
$177$	−3.95437	−0.297229
$178$	−26.4979	−1.98610
$179$	6.63809	0.496154	0.248077	−	0.968740i	$-0.420201\pi$
$179$	6.63809	0.496154	0.248077	+	0.968740i	$0.420201\pi$
$180$	0	0
$181$	−14.7329	−1.09509	−0.547546	−	0.836776i	$-0.684438\pi$
$181$	−14.7329	−1.09509	−0.547546	+	0.836776i	$0.684438\pi$
$182$	−21.8116	−1.61678
$183$	0.831044	0.0614325
$184$	−5.36448	−0.395475
$185$	0	0
$186$	2.39788	0.175821
$187$	−3.25162	−0.237782
$188$	−9.30809	−0.678862
$189$	−14.3918	−1.04685
$190$	0	0
$191$	0.363561	0.0263064	0.0131532	−	0.999913i	$-0.495813\pi$
$191$	0.363561	0.0263064	0.0131532	+	0.999913i	$0.495813\pi$
$192$	0.244710	0.0176604
$193$	4.06780	0.292807	0.146403	−	0.989225i	$-0.453230\pi$
$193$	4.06780	0.292807	0.146403	+	0.989225i	$0.453230\pi$
$194$	−19.2748	−1.38385
$195$	0	0
$196$	4.09838	0.292741
$197$	2.78703	0.198568	0.0992840	−	0.995059i	$-0.468345\pi$
$197$	2.78703	0.198568	0.0992840	+	0.995059i	$0.468345\pi$
$198$	−4.81192	−0.341968
$199$	15.7467	1.11625	0.558126	−	0.829756i	$-0.311521\pi$
$199$	15.7467	1.11625	0.558126	+	0.829756i	$0.311521\pi$
$200$	0	0
$201$	7.07296	0.498888
$202$	−21.4653	−1.51029
$203$	−32.0781	−2.25144
$204$	−2.44085	−0.170894
$205$	0	0
$206$	−13.6116	−0.948365
$207$	−7.70167	−0.535303
$208$	18.8909	1.30985
$209$	0	0
$210$	0	0
$211$	1.88685	0.129896	0.0649481	−	0.997889i	$-0.479312\pi$
$211$	1.88685	0.129896	0.0649481	+	0.997889i	$0.479312\pi$
$212$	−0.0892713	−0.00613117
$213$	4.12712	0.282786
$214$	−11.0173	−0.753125
$215$	0	0
$216$	7.08623	0.482157
$217$	5.46255	0.370822
$218$	35.4467	2.40076
$219$	−12.1563	−0.821450
$220$	0	0
$221$	10.4191	0.700862
$222$	12.2234	0.820381
$223$	−21.5427	−1.44261	−0.721305	−	0.692618i	$-0.756457\pi$
$223$	−21.5427	−1.44261	−0.721305	+	0.692618i	$0.756457\pi$
$224$	−18.1568	−1.21315
$225$	0	0
$226$	17.0724	1.13564
$227$	3.39937	0.225624	0.112812	−	0.993616i	$-0.464014\pi$
$227$	3.39937	0.225624	0.112812	+	0.993616i	$0.464014\pi$
$228$	0	0
$229$	21.5113	1.42151	0.710753	−	0.703442i	$-0.248355\pi$
$229$	21.5113	1.42151	0.710753	+	0.703442i	$0.248355\pi$
$230$	0	0
$231$	3.19147	0.209983
$232$	15.7947	1.03697
$233$	1.52013	0.0995871	0.0497935	−	0.998760i	$-0.484144\pi$
$233$	1.52013	0.0995871	0.0497935	+	0.998760i	$0.484144\pi$
$234$	15.4187	1.00795
$235$	0	0
$236$	−5.18017	−0.337200
$237$	−11.4693	−0.745012
$238$	−15.8821	−1.02948
$239$	24.8318	1.60624	0.803118	−	0.595820i	$-0.203173\pi$
$239$	24.8318	1.60624	0.803118	+	0.595820i	$0.203173\pi$
$240$	0	0
$241$	−10.3790	−0.668572	−0.334286	−	0.942472i	$-0.608495\pi$
$241$	−10.3790	−0.668572	−0.334286	+	0.942472i	$0.608495\pi$
$242$	−16.8521	−1.08329
$243$	15.9067	1.02042
$244$	1.08866	0.0696940
$245$	0	0
$246$	8.39454	0.535216
$247$	0	0
$248$	−2.68966	−0.170794
$249$	−9.72082	−0.616032
$250$	0	0
$251$	−22.3378	−1.40995	−0.704974	−	0.709233i	$-0.749041\pi$
$251$	−22.3378	−1.40995	−0.704974	+	0.709233i	$0.749041\pi$
$252$	−8.22865	−0.518356
$253$	3.91303	0.246010
$254$	19.5902	1.22920
$255$	0	0
$256$	19.7015	1.23135
$257$	−16.4731	−1.02756	−0.513780	−	0.857922i	$-0.671755\pi$
$257$	−16.4731	−1.02756	−0.513780	+	0.857922i	$0.671755\pi$
$258$	−6.68190	−0.415997
$259$	27.8458	1.73026
$260$	0	0
$261$	22.6761	1.40361
$262$	−21.4995	−1.32824
$263$	−6.76045	−0.416867	−0.208434	−	0.978037i	$-0.566837\pi$
$263$	−6.76045	−0.416867	−0.208434	+	0.978037i	$0.566837\pi$
$264$	−1.57142	−0.0967142
$265$	0	0
$266$	0	0
$267$	−12.4235	−0.760303
$268$	9.26548	0.565979
$269$	−13.1958	−0.804562	−0.402281	−	0.915516i	$-0.631783\pi$
$269$	−13.1958	−0.804562	−0.402281	+	0.915516i	$0.631783\pi$
$270$	0	0
$271$	31.0774	1.88782	0.943908	−	0.330209i	$-0.107119\pi$
$271$	31.0774	1.88782	0.943908	+	0.330209i	$0.107119\pi$
$272$	13.7554	0.834043
$273$	−10.2263	−0.618924
$274$	−32.5425	−1.96597
$275$	0	0
$276$	2.93735	0.176808
$277$	−10.8765	−0.653505	−0.326753	−	0.945110i	$-0.605954\pi$
$277$	−10.8765	−0.653505	−0.326753	+	0.945110i	$0.605954\pi$
$278$	−19.6408	−1.17798
$279$	−3.86149	−0.231181
$280$	0	0
$281$	10.3306	0.616271	0.308135	−	0.951343i	$-0.400295\pi$
$281$	10.3306	0.616271	0.308135	+	0.951343i	$0.400295\pi$
$282$	−12.4649	−0.742273
$283$	15.4014	0.915517	0.457758	−	0.889077i	$-0.348653\pi$
$283$	15.4014	0.915517	0.457758	+	0.889077i	$0.348653\pi$
$284$	5.40647	0.320815
$285$	0	0
$286$	−7.83385	−0.463225
$287$	19.1234	1.12882
$288$	12.8351	0.756314
$289$	−9.41338	−0.553728
$290$	0	0
$291$	−9.03695	−0.529755
$292$	−15.9246	−0.931919
$293$	−4.61862	−0.269823	−0.134911	−	0.990858i	$-0.543075\pi$
$293$	−4.61862	−0.269823	−0.134911	+	0.990858i	$0.543075\pi$
$294$	5.48833	0.320086
$295$	0	0
$296$	−13.7108	−0.796922
$297$	−5.16894	−0.299932
$298$	−31.9982	−1.85361
$299$	−12.5384	−0.725114
$300$	0	0
$301$	−15.2219	−0.877375
$302$	12.6215	0.726283
$303$	−10.0639	−0.578157
$304$	0	0
$305$	0	0
$306$	11.2271	0.641808
$307$	23.4626	1.33908	0.669541	−	0.742775i	$-0.266491\pi$
$307$	23.4626	1.33908	0.669541	+	0.742775i	$0.266491\pi$
$308$	4.18078	0.238222
$309$	−6.38176	−0.363045
$310$	0	0
$311$	−11.7514	−0.666363	−0.333181	−	0.942863i	$-0.608122\pi$
$311$	−11.7514	−0.666363	−0.333181	+	0.942863i	$0.608122\pi$
$312$	5.03524	0.285065
$313$	10.6986	0.604722	0.302361	−	0.953193i	$-0.402225\pi$
$313$	10.6986	0.604722	0.302361	+	0.953193i	$0.402225\pi$
$314$	13.5668	0.765620
$315$	0	0
$316$	−15.0246	−0.845202
$317$	−0.144196	−0.00809886	−0.00404943	−	0.999992i	$-0.501289\pi$
$317$	−0.144196	−0.00809886	−0.00404943	+	0.999992i	$0.501289\pi$
$318$	−0.119547	−0.00670388
$319$	−11.5212	−0.645061
$320$	0	0
$321$	−5.16541	−0.288305
$322$	19.1126	1.06511
$323$	0	0
$324$	3.63026	0.201681
$325$	0	0
$326$	39.3043	2.17686
$327$	16.6191	0.919038
$328$	−9.41600	−0.519912
$329$	−28.3960	−1.56552
$330$	0	0
$331$	−20.5397	−1.12896	−0.564481	−	0.825446i	$-0.690924\pi$
$331$	−20.5397	−1.12896	−0.564481	+	0.825446i	$0.690924\pi$
$332$	−12.7341	−0.698876
$333$	−19.6843	−1.07869
$334$	43.0205	2.35398
$335$	0	0
$336$	−13.5009	−0.736535
$337$	−14.8076	−0.806620	−0.403310	−	0.915063i	$-0.632140\pi$
$337$	−14.8076	−0.806620	−0.403310	+	0.915063i	$0.632140\pi$
$338$	2.29632	0.124903
$339$	8.00434	0.434736
$340$	0	0
$341$	1.96193	0.106244
$342$	0	0
$343$	−10.5056	−0.567248
$344$	7.49498	0.404102
$345$	0	0
$346$	24.6298	1.32411
$347$	5.25982	0.282362	0.141181	−	0.989984i	$-0.454910\pi$
$347$	5.25982	0.282362	0.141181	+	0.989984i	$0.454910\pi$
$348$	−8.64845	−0.463605
$349$	8.86326	0.474439	0.237220	−	0.971456i	$-0.423764\pi$
$349$	8.86326	0.474439	0.237220	+	0.971456i	$0.423764\pi$
$350$	0	0
$351$	16.5626	0.884048
$352$	−6.52119	−0.347581
$353$	−18.5459	−0.987099	−0.493550	−	0.869718i	$-0.664301\pi$
$353$	−18.5459	−0.987099	−0.493550	+	0.869718i	$0.664301\pi$
$354$	−6.93700	−0.368698
$355$	0	0
$356$	−16.2745	−0.862549
$357$	−7.44626	−0.394098
$358$	11.6450	0.615455
$359$	21.6459	1.14243	0.571214	−	0.820801i	$-0.306473\pi$
$359$	21.6459	1.14243	0.571214	+	0.820801i	$0.306473\pi$
$360$	0	0
$361$	0	0
$362$	−25.8455	−1.35841
$363$	−7.90104	−0.414697
$364$	−13.3963	−0.702158
$365$	0	0
$366$	1.45787	0.0762041
$367$	−18.0042	−0.939813	−0.469907	−	0.882716i	$-0.655712\pi$
$367$	−18.0042	−0.939813	−0.469907	+	0.882716i	$0.655712\pi$
$368$	−16.5534	−0.862904
$369$	−13.5184	−0.703737
$370$	0	0
$371$	−0.272338	−0.0141391
$372$	1.47274	0.0763578
$373$	−15.5878	−0.807104	−0.403552	−	0.914957i	$-0.632225\pi$
$373$	−15.5878	−0.807104	−0.403552	+	0.914957i	$0.632225\pi$
$374$	−5.70420	−0.294957
$375$	0	0
$376$	13.9817	0.721049
$377$	36.9169	1.90131
$378$	−25.2469	−1.29856
$379$	−19.1906	−0.985753	−0.492876	−	0.870099i	$-0.664055\pi$
$379$	−19.1906	−0.985753	−0.492876	+	0.870099i	$0.664055\pi$
$380$	0	0
$381$	9.18479	0.470551
$382$	0.637782	0.0326317
$383$	−1.19446	−0.0610343	−0.0305171	−	0.999534i	$-0.509715\pi$
$383$	−1.19446	−0.0610343	−0.0305171	+	0.999534i	$0.509715\pi$
$384$	9.51599	0.485611
$385$	0	0
$386$	7.13599	0.363212
$387$	10.7604	0.546981
$388$	−11.8383	−0.600997
$389$	−18.9387	−0.960232	−0.480116	−	0.877205i	$-0.659405\pi$
$389$	−18.9387	−0.960232	−0.480116	+	0.877205i	$0.659405\pi$
$390$	0	0
$391$	−9.12981	−0.461714
$392$	−6.15616	−0.310933
$393$	−10.0800	−0.508467
$394$	4.88919	0.246314
$395$	0	0
$396$	−2.95540	−0.148514
$397$	−7.08835	−0.355754	−0.177877	−	0.984053i	$-0.556923\pi$
$397$	−7.08835	−0.355754	−0.177877	+	0.984053i	$0.556923\pi$
$398$	27.6238	1.38466
$399$	0	0
$400$	0	0
$401$	17.6177	0.879786	0.439893	−	0.898050i	$-0.355016\pi$
$401$	17.6177	0.879786	0.439893	+	0.898050i	$0.355016\pi$
$402$	12.4078	0.618846
$403$	−6.28654	−0.313155
$404$	−13.1836	−0.655909
$405$	0	0
$406$	−56.2734	−2.79280
$407$	10.0011	0.495736
$408$	3.66640	0.181514
$409$	−6.28149	−0.310600	−0.155300	−	0.987867i	$-0.549634\pi$
$409$	−6.28149	−0.310600	−0.155300	+	0.987867i	$0.549634\pi$
$410$	0	0
$411$	−15.2575	−0.752596
$412$	−8.36001	−0.411868
$413$	−15.8030	−0.777616
$414$	−13.5108	−0.664017
$415$	0	0
$416$	20.8956	1.02449
$417$	−9.20851	−0.450943
$418$	0	0
$419$	26.7472	1.30669	0.653343	−	0.757062i	$-0.273366\pi$
$419$	26.7472	1.30669	0.653343	+	0.757062i	$0.273366\pi$
$420$	0	0
$421$	−35.1919	−1.71515	−0.857573	−	0.514362i	$-0.828029\pi$
$421$	−35.1919	−1.71515	−0.857573	+	0.514362i	$0.828029\pi$
$422$	3.31003	0.161130
$423$	20.0732	0.975990
$424$	0.134094	0.00651219
$425$	0	0
$426$	7.24006	0.350782
$427$	3.32114	0.160721
$428$	−6.76661	−0.327077
$429$	−3.67288	−0.177328
$430$	0	0
$431$	16.4602	0.792860	0.396430	−	0.918065i	$-0.370249\pi$
$431$	16.4602	0.792860	0.396430	+	0.918065i	$0.370249\pi$
$432$	21.8662	1.05204
$433$	13.3712	0.642579	0.321289	−	0.946981i	$-0.395884\pi$
$433$	13.3712	0.642579	0.321289	+	0.946981i	$0.395884\pi$
$434$	9.58275	0.459987
$435$	0	0
$436$	21.7708	1.04263
$437$	0	0
$438$	−21.3254	−1.01897
$439$	18.1159	0.864623	0.432312	−	0.901724i	$-0.357698\pi$
$439$	18.1159	0.864623	0.432312	+	0.901724i	$0.357698\pi$
$440$	0	0
$441$	−8.83827	−0.420870
$442$	18.2778	0.869385
$443$	0.868020	0.0412409	0.0206204	−	0.999787i	$-0.493436\pi$
$443$	0.868020	0.0412409	0.0206204	+	0.999787i	$0.493436\pi$
$444$	7.50740	0.356285
$445$	0	0
$446$	−37.7916	−1.78949
$447$	−15.0023	−0.709582
$448$	0.977946	0.0462036
$449$	16.9428	0.799578	0.399789	−	0.916607i	$-0.369083\pi$
$449$	16.9428	0.799578	0.399789	+	0.916607i	$0.369083\pi$
$450$	0	0
$451$	6.86835	0.323418
$452$	10.4856	0.493200
$453$	5.91753	0.278030
$454$	5.96339	0.279876
$455$	0	0
$456$	0	0
$457$	22.8692	1.06977	0.534887	−	0.844924i	$-0.320354\pi$
$457$	22.8692	1.06977	0.534887	+	0.844924i	$0.320354\pi$
$458$	37.7364	1.76331
$459$	12.0600	0.562915
$460$	0	0
$461$	−30.5780	−1.42416	−0.712079	−	0.702100i	$-0.752246\pi$
$461$	−30.5780	−1.42416	−0.712079	+	0.702100i	$0.752246\pi$
$462$	5.59867	0.260474
$463$	−22.5471	−1.04785	−0.523927	−	0.851763i	$-0.675533\pi$
$463$	−22.5471	−1.04785	−0.523927	+	0.851763i	$0.675533\pi$
$464$	48.7381	2.26261
$465$	0	0
$466$	2.66671	0.123533
$467$	−8.45394	−0.391202	−0.195601	−	0.980684i	$-0.562666\pi$
$467$	−8.45394	−0.391202	−0.195601	+	0.980684i	$0.562666\pi$
$468$	9.46987	0.437745
$469$	28.2660	1.30520
$470$	0	0
$471$	6.36076	0.293088
$472$	7.78112	0.358155
$473$	−5.46709	−0.251377
$474$	−20.1202	−0.924151
$475$	0	0
$476$	−9.75450	−0.447097
$477$	0.192516	0.00881470
$478$	43.5615	1.99246
$479$	9.94732	0.454505	0.227252	−	0.973836i	$-0.427026\pi$
$479$	9.94732	0.454505	0.227252	+	0.973836i	$0.427026\pi$
$480$	0	0
$481$	−32.0462	−1.46118
$482$	−18.2075	−0.829331
$483$	8.96091	0.407735
$484$	−10.3503	−0.470466
$485$	0	0
$486$	27.9046	1.26578
$487$	30.2632	1.37136	0.685679	−	0.727904i	$-0.259506\pi$
$487$	30.2632	1.37136	0.685679	+	0.727904i	$0.259506\pi$
$488$	−1.63527	−0.0740251
$489$	18.4277	0.833329
$490$	0	0
$491$	17.7209	0.799732	0.399866	−	0.916574i	$-0.369057\pi$
$491$	17.7209	0.799732	0.399866	+	0.916574i	$0.369057\pi$
$492$	5.15578	0.232440
$493$	26.8809	1.21066
$494$	0	0
$495$	0	0
$496$	−8.29958	−0.372662
$497$	16.4934	0.739831
$498$	−17.0529	−0.764157
$499$	38.8961	1.74123	0.870614	−	0.491967i	$-0.163722\pi$
$499$	38.8961	1.74123	0.870614	+	0.491967i	$0.163722\pi$
$500$	0	0
$501$	20.1700	0.901131
$502$	−39.1863	−1.74897
$503$	18.7202	0.834692	0.417346	−	0.908748i	$-0.362960\pi$
$503$	18.7202	0.834692	0.417346	+	0.908748i	$0.362960\pi$
$504$	12.3602	0.550568
$505$	0	0
$506$	6.86449	0.305164
$507$	1.07662	0.0478144
$508$	12.0319	0.533831
$509$	9.43885	0.418370	0.209185	−	0.977876i	$-0.432919\pi$
$509$	9.43885	0.418370	0.209185	+	0.977876i	$0.432919\pi$
$510$	0	0
$511$	−48.5809	−2.14909
$512$	11.4219	0.504783
$513$	0	0
$514$	−28.8981	−1.27464
$515$	0	0
$516$	−4.10391	−0.180665
$517$	−10.1987	−0.448538
$518$	48.8489	2.14630
$519$	11.5476	0.506884
$520$	0	0
$521$	−3.91656	−0.171587	−0.0857937	−	0.996313i	$-0.527343\pi$
$521$	−3.91656	−0.171587	−0.0857937	+	0.996313i	$0.527343\pi$
$522$	39.7798	1.74111
$523$	11.9016	0.520419	0.260210	−	0.965552i	$-0.416208\pi$
$523$	11.9016	0.520419	0.260210	+	0.965552i	$0.416208\pi$
$524$	−13.2046	−0.576847
$525$	0	0
$526$	−11.8596	−0.517103
$527$	−4.57753	−0.199400
$528$	−4.84898	−0.211025
$529$	−12.0131	−0.522309
$530$	0	0
$531$	11.1712	0.484788
$532$	0	0
$533$	−22.0080	−0.953273
$534$	−21.7940	−0.943118
$535$	0	0
$536$	−13.9176	−0.601151
$537$	5.45970	0.235604
$538$	−23.1489	−0.998020
$539$	4.49051	0.193420
$540$	0	0
$541$	21.7564	0.935381	0.467691	−	0.883892i	$-0.345086\pi$
$541$	21.7564	0.935381	0.467691	+	0.883892i	$0.345086\pi$
$542$	54.5179	2.34174
$543$	−12.1176	−0.520014
$544$	15.2151	0.652342
$545$	0	0
$546$	−17.9396	−0.767745
$547$	22.2776	0.952521	0.476261	−	0.879304i	$-0.341992\pi$
$547$	22.2776	0.952521	0.476261	+	0.879304i	$0.341992\pi$
$548$	−19.9871	−0.853805
$549$	−2.34772	−0.100198
$550$	0	0
$551$	0	0
$552$	−4.41218	−0.187795
$553$	−45.8353	−1.94912
$554$	−19.0802	−0.810641
$555$	0	0
$556$	−12.0630	−0.511586
$557$	−43.9900	−1.86392	−0.931958	−	0.362566i	$-0.881901\pi$
$557$	−43.9900	−1.86392	−0.931958	+	0.362566i	$0.881901\pi$
$558$	−6.77406	−0.286769
$559$	17.5180	0.740932
$560$	0	0
$561$	−2.67440	−0.112913
$562$	18.1226	0.764454
$563$	4.06538	0.171335	0.0856677	−	0.996324i	$-0.472698\pi$
$563$	4.06538	0.171335	0.0856677	+	0.996324i	$0.472698\pi$
$564$	−7.65572	−0.322364
$565$	0	0
$566$	27.0181	1.13565
$567$	11.0748	0.465096
$568$	−8.12105	−0.340752
$569$	−12.6422	−0.529988	−0.264994	−	0.964250i	$-0.585370\pi$
$569$	−12.6422	−0.529988	−0.264994	+	0.964250i	$0.585370\pi$
$570$	0	0
$571$	−28.4993	−1.19266	−0.596329	−	0.802740i	$-0.703375\pi$
$571$	−28.4993	−1.19266	−0.596329	+	0.802740i	$0.703375\pi$
$572$	−4.81142	−0.201175
$573$	0.299022	0.0124918
$574$	33.5475	1.40024
$575$	0	0
$576$	−0.691312	−0.0288047
$577$	−35.2404	−1.46708	−0.733538	−	0.679649i	$-0.762132\pi$
$577$	−35.2404	−1.46708	−0.733538	+	0.679649i	$0.762132\pi$
$578$	−16.5135	−0.686873
$579$	3.34569	0.139042
$580$	0	0
$581$	−38.8477	−1.61168
$582$	−15.8532	−0.657135
$583$	−0.0978128	−0.00405099
$584$	23.9204	0.989831
$585$	0	0
$586$	−8.10227	−0.334702
$587$	3.38822	0.139847	0.0699235	−	0.997552i	$-0.477724\pi$
$587$	3.38822	0.139847	0.0699235	+	0.997552i	$0.477724\pi$
$588$	3.37084	0.139011
$589$	0	0
$590$	0	0
$591$	2.29228	0.0942919
$592$	−42.3078	−1.73884
$593$	−10.8479	−0.445471	−0.222735	−	0.974879i	$-0.571499\pi$
$593$	−10.8479	−0.445471	−0.222735	+	0.974879i	$0.571499\pi$
$594$	−9.06767	−0.372051
$595$	0	0
$596$	−19.6527	−0.805007
$597$	12.9513	0.530063
$598$	−21.9956	−0.899469
$599$	13.3422	0.545148	0.272574	−	0.962135i	$-0.412125\pi$
$599$	13.3422	0.545148	0.272574	+	0.962135i	$0.412125\pi$
$600$	0	0
$601$	−22.4985	−0.917734	−0.458867	−	0.888505i	$-0.651744\pi$
$601$	−22.4985	−0.917734	−0.458867	+	0.888505i	$0.651744\pi$
$602$	−26.7032	−1.08834
$603$	−19.9813	−0.813700
$604$	7.75188	0.315419
$605$	0	0
$606$	−17.6548	−0.717176
$607$	−10.0129	−0.406411	−0.203206	−	0.979136i	$-0.565136\pi$
$607$	−10.0129	−0.406411	−0.203206	+	0.979136i	$0.565136\pi$
$608$	0	0
$609$	−26.3836	−1.06912
$610$	0	0
$611$	32.6793	1.32206
$612$	6.89547	0.278733
$613$	−17.0655	−0.689268	−0.344634	−	0.938737i	$-0.611997\pi$
$613$	−17.0655	−0.689268	−0.344634	+	0.938737i	$0.611997\pi$
$614$	41.1596	1.66106
$615$	0	0
$616$	−6.27993	−0.253026
$617$	9.23533	0.371800	0.185900	−	0.982569i	$-0.440480\pi$
$617$	9.23533	0.371800	0.185900	+	0.982569i	$0.440480\pi$
$618$	−11.1953	−0.450340
$619$	−43.9538	−1.76665	−0.883325	−	0.468761i	$-0.844701\pi$
$619$	−43.9538	−1.76665	−0.883325	+	0.468761i	$0.844701\pi$
$620$	0	0
$621$	−14.5132	−0.582394
$622$	−20.6151	−0.826591
$623$	−49.6484	−1.98912
$624$	15.5374	0.621995
$625$	0	0
$626$	18.7682	0.750128
$627$	0	0
$628$	8.33250	0.332503
$629$	−23.3343	−0.930401
$630$	0	0
$631$	49.7923	1.98220	0.991101	−	0.133114i	$-0.0424975\pi$
$631$	49.7923	1.98220	0.991101	+	0.133114i	$0.0424975\pi$
$632$	22.5685	0.897725
$633$	1.55190	0.0616824
$634$	−0.252958	−0.0100462
$635$	0	0
$636$	−0.0734239	−0.00291145
$637$	−14.3888	−0.570105
$638$	−20.2111	−0.800167
$639$	−11.6592	−0.461232
$640$	0	0
$641$	29.5597	1.16754	0.583769	−	0.811920i	$-0.301577\pi$
$641$	29.5597	1.16754	0.583769	+	0.811920i	$0.301577\pi$
$642$	−9.06149	−0.357628
$643$	3.65094	0.143979	0.0719895	−	0.997405i	$-0.477065\pi$
$643$	3.65094	0.143979	0.0719895	+	0.997405i	$0.477065\pi$
$644$	11.7387	0.462568
$645$	0	0
$646$	0	0
$647$	−11.2901	−0.443858	−0.221929	−	0.975063i	$-0.571235\pi$
$647$	−11.2901	−0.443858	−0.221929	+	0.975063i	$0.571235\pi$
$648$	−5.45301	−0.214214
$649$	−5.67581	−0.222795
$650$	0	0
$651$	4.49284	0.176088
$652$	24.1400	0.945396
$653$	−28.5362	−1.11671	−0.558353	−	0.829603i	$-0.688567\pi$
$653$	−28.5362	−1.11671	−0.558353	+	0.829603i	$0.688567\pi$
$654$	29.1542	1.14002
$655$	0	0
$656$	−29.0553	−1.13442
$657$	34.3419	1.33981
$658$	−49.8140	−1.94195
$659$	42.7200	1.66413	0.832067	−	0.554675i	$-0.187157\pi$
$659$	42.7200	1.66413	0.832067	+	0.554675i	$0.187157\pi$
$660$	0	0
$661$	15.0393	0.584961	0.292480	−	0.956271i	$-0.405519\pi$
$661$	15.0393	0.584961	0.292480	+	0.956271i	$0.405519\pi$
$662$	−36.0320	−1.40042
$663$	8.56948	0.332811
$664$	19.1279	0.742307
$665$	0	0
$666$	−34.5314	−1.33806
$667$	−32.3488	−1.25255
$668$	26.4225	1.02232
$669$	−17.7185	−0.685037
$670$	0	0
$671$	1.19282	0.0460483
$672$	−14.9336	−0.576077
$673$	16.0734	0.619584	0.309792	−	0.950804i	$-0.399741\pi$
$673$	16.0734	0.619584	0.309792	+	0.950804i	$0.399741\pi$
$674$	−25.9764	−1.00057
$675$	0	0
$676$	1.41036	0.0542445
$677$	41.6435	1.60049	0.800244	−	0.599674i	$-0.204703\pi$
$677$	41.6435	1.60049	0.800244	+	0.599674i	$0.204703\pi$
$678$	14.0417	0.539269
$679$	−36.1148	−1.38596
$680$	0	0
$681$	2.79591	0.107140
$682$	3.44174	0.131791
$683$	−13.5982	−0.520322	−0.260161	−	0.965565i	$-0.583776\pi$
$683$	−13.5982	−0.520322	−0.260161	+	0.965565i	$0.583776\pi$
$684$	0	0
$685$	0	0
$686$	−18.4295	−0.703643
$687$	17.6926	0.675015
$688$	23.1275	0.881728
$689$	0.313418	0.0119403
$690$	0	0
$691$	40.7185	1.54900	0.774502	−	0.632571i	$-0.218000\pi$
$691$	40.7185	1.54900	0.774502	+	0.632571i	$0.218000\pi$
$692$	15.1272	0.575051
$693$	−9.01597	−0.342488
$694$	9.22711	0.350256
$695$	0	0
$696$	12.9908	0.492415
$697$	−16.0251	−0.606993
$698$	15.5485	0.588519
$699$	1.25028	0.0472899
$700$	0	0
$701$	−3.66455	−0.138408	−0.0692040	−	0.997603i	$-0.522046\pi$
$701$	−3.66455	−0.138408	−0.0692040	+	0.997603i	$0.522046\pi$
$702$	29.0552	1.09662
$703$	0	0
$704$	0.351239	0.0132378
$705$	0	0
$706$	−32.5344	−1.22445
$707$	−40.2189	−1.51259
$708$	−4.26059	−0.160123
$709$	10.2722	0.385779	0.192890	−	0.981220i	$-0.438214\pi$
$709$	10.2722	0.385779	0.192890	+	0.981220i	$0.438214\pi$
$710$	0	0
$711$	32.4011	1.21513
$712$	24.4459	0.916151
$713$	5.50864	0.206300
$714$	−13.0627	−0.488859
$715$	0	0
$716$	7.15213	0.267288
$717$	20.4237	0.762737
$718$	37.9726	1.41713
$719$	31.4259	1.17199	0.585995	−	0.810315i	$-0.300704\pi$
$719$	31.4259	1.17199	0.585995	+	0.810315i	$0.300704\pi$
$720$	0	0
$721$	−25.5037	−0.949807
$722$	0	0
$723$	−8.53655	−0.317478
$724$	−15.8738	−0.589947
$725$	0	0
$726$	−13.8605	−0.514411
$727$	−15.0853	−0.559484	−0.279742	−	0.960075i	$-0.590249\pi$
$727$	−15.0853	−0.559484	−0.279742	+	0.960075i	$0.590249\pi$
$728$	20.1226	0.745792
$729$	2.97492	0.110182
$730$	0	0
$731$	12.7557	0.471786
$732$	0.895398	0.0330949
$733$	−23.6607	−0.873930	−0.436965	−	0.899479i	$-0.643947\pi$
$733$	−23.6607	−0.873930	−0.436965	+	0.899479i	$0.643947\pi$
$734$	−31.5841	−1.16579
$735$	0	0
$736$	−18.3100	−0.674915
$737$	10.1520	0.373954
$738$	−23.7147	−0.872952
$739$	15.4199	0.567231	0.283615	−	0.958938i	$-0.408466\pi$
$739$	15.4199	0.567231	0.283615	+	0.958938i	$0.408466\pi$
$740$	0	0
$741$	0	0
$742$	−0.477752	−0.0175388
$743$	−25.9872	−0.953378	−0.476689	−	0.879072i	$-0.658163\pi$
$743$	−25.9872	−0.953378	−0.476689	+	0.879072i	$0.658163\pi$
$744$	−2.21219	−0.0811029
$745$	0	0
$746$	−27.3450	−1.00117
$747$	27.4615	1.00476
$748$	−3.50342	−0.128098
$749$	−20.6428	−0.754270
$750$	0	0
$751$	6.25003	0.228067	0.114033	−	0.993477i	$-0.463623\pi$
$751$	6.25003	0.228067	0.114033	+	0.993477i	$0.463623\pi$
$752$	43.1437	1.57329
$753$	−18.3724	−0.669527
$754$	64.7619	2.35849
$755$	0	0
$756$	−15.5062	−0.563956
$757$	−1.86179	−0.0676678	−0.0338339	−	0.999427i	$-0.510772\pi$
$757$	−1.86179	−0.0676678	−0.0338339	+	0.999427i	$0.510772\pi$
$758$	−33.6653	−1.22278
$759$	3.21840	0.116820
$760$	0	0
$761$	−21.8059	−0.790463	−0.395231	−	0.918582i	$-0.629336\pi$
$761$	−21.8059	−0.790463	−0.395231	+	0.918582i	$0.629336\pi$
$762$	16.1125	0.583696
$763$	66.4156	2.40441
$764$	0.391714	0.0141717
$765$	0	0
$766$	−2.09540	−0.0757100
$767$	18.1868	0.656687
$768$	16.2041	0.584716
$769$	−30.2405	−1.09050	−0.545250	−	0.838273i	$-0.683565\pi$
$769$	−30.2405	−1.09050	−0.545250	+	0.838273i	$0.683565\pi$
$770$	0	0
$771$	−13.5488	−0.487947
$772$	4.38280	0.157740
$773$	23.8034	0.856149	0.428074	−	0.903744i	$-0.359192\pi$
$773$	23.8034	0.856149	0.428074	+	0.903744i	$0.359192\pi$
$774$	18.8765	0.678503
$775$	0	0
$776$	17.7822	0.638345
$777$	22.9027	0.821628
$778$	−33.2235	−1.19112
$779$	0	0
$780$	0	0
$781$	5.92377	0.211969
$782$	−16.0161	−0.572734
$783$	42.7312	1.52709
$784$	−18.9963	−0.678439
$785$	0	0
$786$	−17.6829	−0.630729
$787$	−0.635412	−0.0226500	−0.0113250	−	0.999936i	$-0.503605\pi$
$787$	−0.635412	−0.0226500	−0.0113250	+	0.999936i	$0.503605\pi$
$788$	3.00286	0.106972
$789$	−5.56034	−0.197953
$790$	0	0
$791$	31.9881	1.13737
$792$	4.43929	0.157743
$793$	−3.82211	−0.135727
$794$	−12.4348	−0.441295
$795$	0	0
$796$	16.9661	0.601346
$797$	−29.6702	−1.05097	−0.525486	−	0.850802i	$-0.676117\pi$
$797$	−29.6702	−1.05097	−0.525486	+	0.850802i	$0.676117\pi$
$798$	0	0
$799$	23.7954	0.841819
$800$	0	0
$801$	35.0965	1.24007
$802$	30.9061	1.09133
$803$	−17.4483	−0.615738
$804$	7.62067	0.268760
$805$	0	0
$806$	−11.0282	−0.388453
$807$	−10.8533	−0.382054
$808$	19.8030	0.696669
$809$	−23.2880	−0.818764	−0.409382	−	0.912363i	$-0.634256\pi$
$809$	−23.2880	−0.818764	−0.409382	+	0.912363i	$0.634256\pi$
$810$	0	0
$811$	52.6829	1.84995	0.924974	−	0.380031i	$-0.124087\pi$
$811$	52.6829	1.84995	0.924974	+	0.380031i	$0.124087\pi$
$812$	−34.5622	−1.21289
$813$	25.5605	0.896447
$814$	17.5446	0.614936
$815$	0	0
$816$	11.3135	0.396053
$817$	0	0
$818$	−11.0194	−0.385284
$819$	28.8895	1.00948
$820$	0	0
$821$	25.2205	0.880202	0.440101	−	0.897948i	$-0.354943\pi$
$821$	25.2205	0.880202	0.440101	+	0.897948i	$0.354943\pi$
$822$	−26.7656	−0.933558
$823$	−4.78467	−0.166783	−0.0833916	−	0.996517i	$-0.526575\pi$
$823$	−4.78467	−0.166783	−0.0833916	+	0.996517i	$0.526575\pi$
$824$	12.5575	0.437463
$825$	0	0
$826$	−27.7227	−0.964595
$827$	−11.2608	−0.391578	−0.195789	−	0.980646i	$-0.562727\pi$
$827$	−11.2608	−0.391578	−0.195789	+	0.980646i	$0.562727\pi$
$828$	−8.29807	−0.288378
$829$	11.7132	0.406816	0.203408	−	0.979094i	$-0.434798\pi$
$829$	11.7132	0.406816	0.203408	+	0.979094i	$0.434798\pi$
$830$	0	0
$831$	−8.94570	−0.310323
$832$	−1.12546	−0.0390184
$833$	−10.4772	−0.363012
$834$	−16.1542	−0.559373
$835$	0	0
$836$	0	0
$837$	−7.27666	−0.251518
$838$	46.9216	1.62088
$839$	51.9732	1.79431	0.897157	−	0.441712i	$-0.145629\pi$
$839$	51.9732	1.79431	0.897157	+	0.441712i	$0.145629\pi$
$840$	0	0
$841$	66.2446	2.28430
$842$	−61.7358	−2.12755
$843$	8.49670	0.292642
$844$	2.03297	0.0699775
$845$	0	0
$846$	35.2136	1.21067
$847$	−31.5753	−1.08494
$848$	0.413779	0.0142092
$849$	12.6673	0.434742
$850$	0	0
$851$	28.0808	0.962597
$852$	4.44672	0.152342
$853$	−2.23758	−0.0766132	−0.0383066	−	0.999266i	$-0.512196\pi$
$853$	−2.23758	−0.0766132	−0.0383066	+	0.999266i	$0.512196\pi$
$854$	5.82615	0.199367
$855$	0	0
$856$	10.1641	0.347402
$857$	49.8445	1.70266	0.851328	−	0.524634i	$-0.175798\pi$
$857$	49.8445	1.70266	0.851328	+	0.524634i	$0.175798\pi$
$858$	−6.44319	−0.219967
$859$	13.4945	0.460426	0.230213	−	0.973140i	$-0.426058\pi$
$859$	13.4945	0.460426	0.230213	+	0.973140i	$0.426058\pi$
$860$	0	0
$861$	15.7286	0.536030
$862$	28.8755	0.983504
$863$	−15.7231	−0.535221	−0.267610	−	0.963527i	$-0.586234\pi$
$863$	−15.7231	−0.535221	−0.267610	+	0.963527i	$0.586234\pi$
$864$	24.1866	0.822846
$865$	0	0
$866$	23.4566	0.797087
$867$	−7.74233	−0.262943
$868$	5.88556	0.199769
$869$	−16.4622	−0.558442
$870$	0	0
$871$	−32.5297	−1.10223
$872$	−32.7018	−1.10742
$873$	25.5296	0.864045
$874$	0	0
$875$	0	0
$876$	−13.0977	−0.442530
$877$	−5.92134	−0.199949	−0.0999747	−	0.994990i	$-0.531876\pi$
$877$	−5.92134	−0.199949	−0.0999747	+	0.994990i	$0.531876\pi$
$878$	31.7800	1.07252
$879$	−3.79872	−0.128128
$880$	0	0
$881$	−11.9843	−0.403761	−0.201880	−	0.979410i	$-0.564705\pi$
$881$	−11.9843	−0.403761	−0.201880	+	0.979410i	$0.564705\pi$
$882$	−15.5046	−0.522069
$883$	−42.2882	−1.42311	−0.711556	−	0.702630i	$-0.752009\pi$
$883$	−42.2882	−1.42311	−0.711556	+	0.702630i	$0.752009\pi$
$884$	11.2259	0.377568
$885$	0	0
$886$	1.52274	0.0511573
$887$	−54.7820	−1.83940	−0.919700	−	0.392622i	$-0.871568\pi$
$887$	−54.7820	−1.83940	−0.919700	+	0.392622i	$0.871568\pi$
$888$	−11.2768	−0.378426
$889$	36.7056	1.23107
$890$	0	0
$891$	3.97761	0.133255
$892$	−23.2110	−0.777161
$893$	0	0
$894$	−26.3179	−0.880202
$895$	0	0
$896$	38.0292	1.27047
$897$	−10.3126	−0.344327
$898$	29.7220	0.991838
$899$	−16.2191	−0.540938
$900$	0	0
$901$	0.228215	0.00760293
$902$	12.0489	0.401184
$903$	−12.5197	−0.416630
$904$	−15.7504	−0.523849
$905$	0	0
$906$	10.3809	0.344882
$907$	10.4933	0.348424	0.174212	−	0.984708i	$-0.444262\pi$
$907$	10.4933	0.348424	0.174212	+	0.984708i	$0.444262\pi$
$908$	3.66261	0.121548
$909$	28.4308	0.942990
$910$	0	0
$911$	−18.3295	−0.607282	−0.303641	−	0.952786i	$-0.598202\pi$
$911$	−18.3295	−0.607282	−0.303641	+	0.952786i	$0.598202\pi$
$912$	0	0
$913$	−13.9525	−0.461762
$914$	40.1185	1.32700
$915$	0	0
$916$	23.1771	0.765792
$917$	−40.2831	−1.33026
$918$	21.1565	0.698268
$919$	9.34747	0.308345	0.154172	−	0.988044i	$-0.450729\pi$
$919$	9.34747	0.308345	0.154172	+	0.988044i	$0.450729\pi$
$920$	0	0
$921$	19.2975	0.635875
$922$	−53.6418	−1.76660
$923$	−18.9813	−0.624778
$924$	3.43861	0.113122
$925$	0	0
$926$	−39.5536	−1.29981
$927$	18.0286	0.592137
$928$	53.9102	1.76969
$929$	−49.4310	−1.62178	−0.810889	−	0.585199i	$-0.801016\pi$
$929$	−49.4310	−1.62178	−0.810889	+	0.585199i	$0.801016\pi$
$930$	0	0
$931$	0	0
$932$	1.63785	0.0536495
$933$	−9.66533	−0.316429
$934$	−14.8304	−0.485267
$935$	0	0
$936$	−14.2247	−0.464948
$937$	−47.5613	−1.55376	−0.776879	−	0.629649i	$-0.783199\pi$
$937$	−47.5613	−1.55376	−0.776879	+	0.629649i	$0.783199\pi$
$938$	49.5860	1.61904
$939$	8.79941	0.287158
$940$	0	0
$941$	29.3057	0.955338	0.477669	−	0.878540i	$-0.341482\pi$
$941$	29.3057	0.955338	0.477669	+	0.878540i	$0.341482\pi$
$942$	11.1584	0.363562
$943$	19.2847	0.627998
$944$	24.0105	0.781474
$945$	0	0
$946$	−9.59071	−0.311821
$947$	−51.1324	−1.66158	−0.830790	−	0.556586i	$-0.812111\pi$
$947$	−51.1324	−1.66158	−0.830790	+	0.556586i	$0.812111\pi$
$948$	−12.3575	−0.401352
$949$	55.9090	1.81488
$950$	0	0
$951$	−0.118598	−0.00384582
$952$	14.6522	0.474881
$953$	−46.8750	−1.51843	−0.759214	−	0.650841i	$-0.774417\pi$
$953$	−46.8750	−1.51843	−0.759214	+	0.650841i	$0.774417\pi$
$954$	0.337724	0.0109342
$955$	0	0
$956$	26.7547	0.865310
$957$	−9.47593	−0.306313
$958$	17.4502	0.563791
$959$	−60.9741	−1.96896
$960$	0	0
$961$	−28.2381	−0.910905
$962$	−56.2174	−1.81252
$963$	14.5924	0.470233
$964$	−11.1828	−0.360172
$965$	0	0
$966$	15.7198	0.505776
$967$	−16.2364	−0.522127	−0.261064	−	0.965322i	$-0.584073\pi$
$967$	−16.2364	−0.522127	−0.261064	+	0.965322i	$0.584073\pi$
$968$	15.5471	0.499702
$969$	0	0
$970$	0	0
$971$	42.0596	1.34976	0.674879	−	0.737928i	$-0.264196\pi$
$971$	42.0596	1.34976	0.674879	+	0.737928i	$0.264196\pi$
$972$	17.1385	0.549717
$973$	−36.8004	−1.17977
$974$	53.0896	1.70110
$975$	0	0
$976$	−5.04600	−0.161519
$977$	−21.1766	−0.677498	−0.338749	−	0.940877i	$-0.610004\pi$
$977$	−21.1766	−0.677498	−0.338749	+	0.940877i	$0.610004\pi$
$978$	32.3270	1.03370
$979$	−17.8317	−0.569904
$980$	0	0
$981$	−46.9493	−1.49898
$982$	31.0871	0.992028
$983$	−57.8160	−1.84404	−0.922022	−	0.387138i	$-0.873464\pi$
$983$	−57.8160	−1.84404	−0.922022	+	0.387138i	$0.873464\pi$
$984$	−7.74448	−0.246885
$985$	0	0
$986$	47.1562	1.50176
$987$	−23.3551	−0.743402
$988$	0	0
$989$	−15.3503	−0.488112
$990$	0	0
$991$	−2.78874	−0.0885872	−0.0442936	−	0.999019i	$-0.514104\pi$
$991$	−2.78874	−0.0885872	−0.0442936	+	0.999019i	$0.514104\pi$
$992$	−9.18032	−0.291475
$993$	−16.8935	−0.536098
$994$	28.9338	0.917724
$995$	0	0
$996$	−10.4736	−0.331868
$997$	11.9770	0.379316	0.189658	−	0.981850i	$-0.439262\pi$
$997$	11.9770	0.379316	0.189658	+	0.981850i	$0.439262\pi$
$998$	68.2339	2.15991
$999$	−37.0934	−1.17358

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.cn.1.16	yes	20
5.4	even	2		9025.2.a.co.1.5	yes	20
19.18	odd	2	inner	9025.2.a.cn.1.5	✓	20
95.94	odd	2		9025.2.a.co.1.16	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9025.2.a.cn.1.5	✓	20	19.18	odd	2	inner
9025.2.a.cn.1.16	yes	20	1.1	even	1	trivial
9025.2.a.co.1.5	yes	20	5.4	even	2
9025.2.a.co.1.16	yes	20	95.94	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.75426 q^{2} +0.822481 q^{3} +1.07744 q^{4} +1.44285 q^{6} +3.28692 q^{7} -1.61842 q^{8} -2.32353 q^{9} +O(q^{10})\)	\(q+1.75426 q^{2} +0.822481 q^{3} +1.07744 q^{4} +1.44285 q^{6} +3.28692 q^{7} -1.61842 q^{8} -2.32353 q^{9} +1.18053 q^{11} +0.886172 q^{12} -3.78272 q^{13} +5.76611 q^{14} -4.99400 q^{16} -2.75438 q^{17} -4.07607 q^{18} +2.70342 q^{21} +2.07096 q^{22} +3.31465 q^{23} -1.33112 q^{24} -6.63589 q^{26} -4.37850 q^{27} +3.54145 q^{28} -9.75933 q^{29} +1.66191 q^{31} -5.52396 q^{32} +0.970961 q^{33} -4.83191 q^{34} -2.50345 q^{36} +8.47172 q^{37} -3.11122 q^{39} +5.81804 q^{41} +4.74252 q^{42} -4.63105 q^{43} +1.27195 q^{44} +5.81477 q^{46} -8.63909 q^{47} -4.10747 q^{48} +3.80382 q^{49} -2.26543 q^{51} -4.07565 q^{52} -0.0828551 q^{53} -7.68103 q^{54} -5.31960 q^{56} -17.1204 q^{58} -4.80786 q^{59} +1.01041 q^{61} +2.91542 q^{62} -7.63723 q^{63} +0.297527 q^{64} +1.70332 q^{66} +8.59954 q^{67} -2.96767 q^{68} +2.72623 q^{69} +5.01790 q^{71} +3.76043 q^{72} -14.7801 q^{73} +14.8616 q^{74} +3.88030 q^{77} -5.45789 q^{78} -13.9448 q^{79} +3.36935 q^{81} +10.2064 q^{82} -11.8189 q^{83} +2.91277 q^{84} -8.12409 q^{86} -8.02686 q^{87} -1.91059 q^{88} -15.1049 q^{89} -12.4335 q^{91} +3.57133 q^{92} +1.36689 q^{93} -15.1552 q^{94} -4.54335 q^{96} -10.9874 q^{97} +6.67290 q^{98} -2.74299 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 24 q^{4} - 20 q^{6} - 8 q^{7} + 16 q^{9}+O(q^{10}) \) 20 * q + 24 * q^4 - 20 * q^6 - 8 * q^7 + 16 * q^9	\( 20 q + 24 q^{4} - 20 q^{6} - 8 q^{7} + 16 q^{9} + 10 q^{11} + 40 q^{16} - 38 q^{17} - 56 q^{23} - 54 q^{24} - 22 q^{26} + 16 q^{28} - 32 q^{36} + 80 q^{39} - 44 q^{43} - 16 q^{44} - 98 q^{47} - 4 q^{49} - 54 q^{54} - 50 q^{58} + 40 q^{61} - 90 q^{62} - 6 q^{63} + 46 q^{64} + 8 q^{66} - 4 q^{68} - 34 q^{73} + 20 q^{74} - 80 q^{77} + 60 q^{81} + 10 q^{82} - 84 q^{83} - 90 q^{87} - 78 q^{92} - 20 q^{93} - 120 q^{96} - 76 q^{99}+O(q^{100}) \) 20 * q + 24 * q^4 - 20 * q^6 - 8 * q^7 + 16 * q^9 + 10 * q^11 + 40 * q^16 - 38 * q^17 - 56 * q^23 - 54 * q^24 - 22 * q^26 + 16 * q^28 - 32 * q^36 + 80 * q^39 - 44 * q^43 - 16 * q^44 - 98 * q^47 - 4 * q^49 - 54 * q^54 - 50 * q^58 + 40 * q^61 - 90 * q^62 - 6 * q^63 + 46 * q^64 + 8 * q^66 - 4 * q^68 - 34 * q^73 + 20 * q^74 - 80 * q^77 + 60 * q^81 + 10 * q^82 - 84 * q^83 - 90 * q^87 - 78 * q^92 - 20 * q^93 - 120 * q^96 - 76 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more