LMFDB - Embedded newform 9025.2.a.bz.1.1

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.41289040.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 8x^{4} + 21x^{3} + 18x^{2} - 25x - 5 $ x^6 - 3x^5 - 8x^4 + 21x^3 + 18x^2 - 25*x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.85674$ 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)

$f(q)$	$=$	\(q-1.48119 q^{2} -0.181608 q^{3} +0.193937 q^{4} +0.268996 q^{6} -1.30422 q^{7} +2.67513 q^{8} -2.96702 q^{9} +4.98247 q^{11} -0.0352204 q^{12} -0.406135 q^{13} +1.93180 q^{14} -4.35026 q^{16} -2.75019 q^{17} +4.39473 q^{18} +0.236856 q^{21} -7.38001 q^{22} +6.95719 q^{23} -0.485824 q^{24} +0.601564 q^{26} +1.08366 q^{27} -0.252935 q^{28} -4.01457 q^{29} -2.57321 q^{31} +1.09332 q^{32} -0.904855 q^{33} +4.07357 q^{34} -0.575413 q^{36} +3.71348 q^{37} +0.0737572 q^{39} -1.21593 q^{41} -0.350829 q^{42} +3.12261 q^{43} +0.966284 q^{44} -10.3050 q^{46} -6.50809 q^{47} +0.790041 q^{48} -5.29902 q^{49} +0.499456 q^{51} -0.0787643 q^{52} +6.32187 q^{53} -1.60511 q^{54} -3.48895 q^{56} +5.94636 q^{58} +11.2327 q^{59} +0.934143 q^{61} +3.81143 q^{62} +3.86964 q^{63} +7.08110 q^{64} +1.34027 q^{66} +5.29529 q^{67} -0.533363 q^{68} -1.26348 q^{69} -1.63532 q^{71} -7.93716 q^{72} +7.68792 q^{73} -5.50038 q^{74} -6.49822 q^{77} -0.109249 q^{78} -14.5540 q^{79} +8.70426 q^{81} +1.80103 q^{82} -15.2643 q^{83} +0.0459350 q^{84} -4.62519 q^{86} +0.729076 q^{87} +13.3288 q^{88} +14.2145 q^{89} +0.529687 q^{91} +1.34925 q^{92} +0.467315 q^{93} +9.63975 q^{94} -0.198556 q^{96} +18.2822 q^{97} +7.84888 q^{98} -14.7831 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{6} + 2 q^{7} + 6 q^{8} + 7 q^{9} - q^{11} + 7 q^{12} + 5 q^{13} - 6 q^{14} - 6 q^{16} - 3 q^{17} + 7 q^{18} + 3 q^{21} + 9 q^{22} - 6 q^{23} + 11 q^{24} + 19 q^{26}+ \cdots - 20 q^{99}+O(q^{100}) $ 6 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - q^6 + 2 * q^7 + 6 * q^8 + 7 * q^9 - q^11 + 7 * q^12 + 5 * q^13 - 6 * q^14 - 6 * q^16 - 3 * q^17 + 7 * q^18 + 3 * q^21 + 9 * q^22 - 6 * q^23 + 11 * q^24 + 19 * q^26 + 18 * q^27 - 4 * q^28 + 3 * q^29 - 3 * q^31 - 6 * q^32 - 18 * q^33 - q^34 + 13 * q^36 - 6 * q^37 + 8 * q^39 + 11 * q^41 - 11 * q^42 + 13 * q^43 + 21 * q^44 - 12 * q^46 - 6 * q^47 - 19 * q^48 + 4 * q^49 - 17 * q^51 - q^52 + 18 * q^53 + 18 * q^54 + 4 * q^56 + 5 * q^58 + 4 * q^59 + 25 * q^61 - 21 * q^62 + 43 * q^63 - 22 * q^64 + 34 * q^66 + 6 * q^67 - q^68 + 13 * q^69 + 18 * q^71 + 13 * q^72 + q^73 - 6 * q^74 - 11 * q^77 + 72 * q^78 + 3 * q^79 + 2 * q^81 + 31 * q^82 - 23 * q^83 + 37 * q^84 + 9 * q^86 + 11 * q^87 + 11 * q^88 + 12 * q^89 - 11 * q^91 + 28 * q^92 - 13 * q^93 + 8 * q^94 - 13 * q^96 + 3 * q^97 - 22 * q^98 - 20 * 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 $

$2$	−1.48119	−1.04736	−0.523681	−	0.851914i	$-0.675442\pi$
$2$	−1.48119	−1.04736	−0.523681	+	0.851914i	$0.675442\pi$
$3$	−0.181608	−0.104851	−0.0524256	−	0.998625i	$-0.516695\pi$
$3$	−0.181608	−0.104851	−0.0524256	+	0.998625i	$0.516695\pi$
$4$	0.193937	0.0969683
$5$	0	0
$5$	0	0
$6$	0.268996	0.109817
$7$	−1.30422	−0.492948	−0.246474	−	0.969149i	$-0.579272\pi$
$7$	−1.30422	−0.492948	−0.246474	+	0.969149i	$0.579272\pi$
$8$	2.67513	0.945802
$9$	−2.96702	−0.989006
$10$	0	0
$11$	4.98247	1.50227	0.751136	−	0.660147i	$-0.229506\pi$
$11$	4.98247	1.50227	0.751136	+	0.660147i	$0.229506\pi$
$12$	−0.0352204	−0.0101672
$13$	−0.406135	−0.112641	−0.0563207	−	0.998413i	$-0.517937\pi$
$13$	−0.406135	−0.112641	−0.0563207	+	0.998413i	$0.517937\pi$
$14$	1.93180	0.516295
$15$	0	0
$16$	−4.35026	−1.08757
$17$	−2.75019	−0.667019	−0.333510	−	0.942747i	$-0.608233\pi$
$17$	−2.75019	−0.667019	−0.333510	+	0.942747i	$0.608233\pi$
$18$	4.39473	1.03585
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0.236856	0.0516862
$22$	−7.38001	−1.57342
$23$	6.95719	1.45067	0.725337	−	0.688394i	$-0.241684\pi$
$23$	6.95719	1.45067	0.725337	+	0.688394i	$0.241684\pi$
$24$	−0.485824	−0.0991685
$25$	0	0
$26$	0.601564	0.117976
$27$	1.08366	0.208550
$28$	−0.252935	−0.0478003
$29$	−4.01457	−0.745487	−0.372743	−	0.927935i	$-0.621583\pi$
$29$	−4.01457	−0.745487	−0.372743	+	0.927935i	$0.621583\pi$
$30$	0	0
$31$	−2.57321	−0.462163	−0.231081	−	0.972934i	$-0.574226\pi$
$31$	−2.57321	−0.462163	−0.231081	+	0.972934i	$0.574226\pi$
$32$	1.09332	0.193274
$33$	−0.904855	−0.157515
$34$	4.07357	0.698611
$35$	0	0
$36$	−0.575413	−0.0959022
$37$	3.71348	0.610492	0.305246	−	0.952274i	$-0.401261\pi$
$37$	3.71348	0.610492	0.305246	+	0.952274i	$0.401261\pi$
$38$	0	0
$39$	0.0737572	0.0118106
$40$	0	0
$41$	−1.21593	−0.189896	−0.0949482	−	0.995482i	$-0.530269\pi$
$41$	−1.21593	−0.189896	−0.0949482	+	0.995482i	$0.530269\pi$
$42$	−0.350829	−0.0541342
$43$	3.12261	0.476193	0.238097	−	0.971241i	$-0.423476\pi$
$43$	3.12261	0.476193	0.238097	+	0.971241i	$0.423476\pi$
$44$	0.966284	0.145673
$45$	0	0
$46$	−10.3050	−1.51938
$47$	−6.50809	−0.949303	−0.474651	−	0.880174i	$-0.657426\pi$
$47$	−6.50809	−0.949303	−0.474651	+	0.880174i	$0.657426\pi$
$48$	0.790041	0.114033
$49$	−5.29902	−0.757003
$50$	0	0
$51$	0.499456	0.0699378
$52$	−0.0787643	−0.0109226
$53$	6.32187	0.868376	0.434188	−	0.900822i	$-0.357035\pi$
$53$	6.32187	0.868376	0.434188	+	0.900822i	$0.357035\pi$
$54$	−1.60511	−0.218427
$55$	0	0
$56$	−3.48895	−0.466231
$57$	0	0
$58$	5.94636	0.780795
$59$	11.2327	1.46238	0.731188	−	0.682176i	$-0.238966\pi$
$59$	11.2327	1.46238	0.731188	+	0.682176i	$0.238966\pi$
$60$	0	0
$61$	0.934143	0.119605	0.0598024	−	0.998210i	$-0.480953\pi$
$61$	0.934143	0.119605	0.0598024	+	0.998210i	$0.480953\pi$
$62$	3.81143	0.484052
$63$	3.86964	0.487528
$64$	7.08110	0.885138
$65$	0	0
$66$	1.34027	0.164975
$67$	5.29529	0.646922	0.323461	−	0.946241i	$-0.395154\pi$
$67$	5.29529	0.646922	0.323461	+	0.946241i	$0.395154\pi$
$68$	−0.533363	−0.0646797
$69$	−1.26348	−0.152105
$70$	0	0
$71$	−1.63532	−0.194077	−0.0970383	−	0.995281i	$-0.530937\pi$
$71$	−1.63532	−0.194077	−0.0970383	+	0.995281i	$0.530937\pi$
$72$	−7.93716	−0.935404
$73$	7.68792	0.899803	0.449902	−	0.893078i	$-0.351459\pi$
$73$	7.68792	0.899803	0.449902	+	0.893078i	$0.351459\pi$
$74$	−5.50038	−0.639406
$75$	0	0
$76$	0	0
$77$	−6.49822	−0.740541
$78$	−0.109249	−0.0123700
$79$	−14.5540	−1.63745	−0.818725	−	0.574186i	$-0.805319\pi$
$79$	−14.5540	−1.63745	−0.818725	+	0.574186i	$0.805319\pi$
$80$	0	0
$81$	8.70426	0.967140
$82$	1.80103	0.198890
$83$	−15.2643	−1.67547	−0.837735	−	0.546077i	$-0.816121\pi$
$83$	−15.2643	−1.67547	−0.837735	+	0.546077i	$0.816121\pi$
$84$	0.0459350	0.00501192
$85$	0	0
$86$	−4.62519	−0.498747
$87$	0.729076	0.0781652
$88$	13.3288	1.42085
$89$	14.2145	1.50674	0.753369	−	0.657598i	$-0.228427\pi$
$89$	14.2145	1.50674	0.753369	+	0.657598i	$0.228427\pi$
$90$	0	0
$91$	0.529687	0.0555263
$92$	1.34925	0.140669
$93$	0.467315	0.0484583
$94$	9.63975	0.994264
$95$	0	0
$96$	−0.198556	−0.0202650
$97$	18.2822	1.85628	0.928139	−	0.372233i	$-0.121408\pi$
$97$	18.2822	1.85628	0.928139	+	0.372233i	$0.121408\pi$
$98$	7.84888	0.792856
$99$	−14.7831	−1.48576
$100$	0	0
$101$	−15.9953	−1.59159	−0.795794	−	0.605567i	$-0.792946\pi$
$101$	−15.9953	−1.59159	−0.795794	+	0.605567i	$0.792946\pi$
$102$	−0.739791	−0.0732502
$103$	5.30941	0.523152	0.261576	−	0.965183i	$-0.415758\pi$
$103$	5.30941	0.523152	0.261576	+	0.965183i	$0.415758\pi$
$104$	−1.08646	−0.106536
$105$	0	0
$106$	−9.36392	−0.909504
$107$	−13.0024	−1.25699	−0.628494	−	0.777814i	$-0.716329\pi$
$107$	−13.0024	−1.25699	−0.628494	+	0.777814i	$0.716329\pi$
$108$	0.210161	0.0202227
$109$	−8.19394	−0.784837	−0.392418	−	0.919787i	$-0.628361\pi$
$109$	−8.19394	−0.784837	−0.392418	+	0.919787i	$0.628361\pi$
$110$	0	0
$111$	−0.674396	−0.0640108
$112$	5.67368	0.536113
$113$	−6.54779	−0.615964	−0.307982	−	0.951392i	$-0.599654\pi$
$113$	−6.54779	−0.615964	−0.307982	+	0.951392i	$0.599654\pi$
$114$	0	0
$115$	0	0
$116$	−0.778572	−0.0722886
$117$	1.20501	0.111403
$118$	−16.6378	−1.53164
$119$	3.58684	0.328805
$120$	0	0
$121$	13.8250	1.25682
$122$	−1.38365	−0.125270
$123$	0.220822	0.0199109
$124$	−0.499040	−0.0448151
$125$	0	0
$126$	−5.73168	−0.510619
$127$	15.1086	1.34067	0.670336	−	0.742057i	$-0.266150\pi$
$127$	15.1086	1.34067	0.670336	+	0.742057i	$0.266150\pi$
$128$	−12.6751	−1.12033
$129$	−0.567090	−0.0499295
$130$	0	0
$131$	−2.49358	−0.217865	−0.108933	−	0.994049i	$-0.534743\pi$
$131$	−2.49358	−0.217865	−0.108933	+	0.994049i	$0.534743\pi$
$132$	−0.175485	−0.0152740
$133$	0	0
$134$	−7.84335	−0.677562
$135$	0	0
$136$	−7.35712	−0.630868
$137$	−17.6081	−1.50436	−0.752181	−	0.658956i	$-0.770998\pi$
$137$	−17.6081	−1.50436	−0.752181	+	0.658956i	$0.770998\pi$
$138$	1.87146	0.159309
$139$	3.15472	0.267580	0.133790	−	0.991010i	$-0.457285\pi$
$139$	3.15472	0.267580	0.133790	+	0.991010i	$0.457285\pi$
$140$	0	0
$141$	1.18192	0.0995356
$142$	2.42223	0.203269
$143$	−2.02355	−0.169218
$144$	12.9073	1.07561
$145$	0	0
$146$	−11.3873	−0.942420
$147$	0.962343	0.0793727
$148$	0.720179	0.0591983
$149$	−6.37388	−0.522168	−0.261084	−	0.965316i	$-0.584080\pi$
$149$	−6.37388	−0.522168	−0.261084	+	0.965316i	$0.584080\pi$
$150$	0	0
$151$	4.96340	0.403915	0.201958	−	0.979394i	$-0.435270\pi$
$151$	4.96340	0.403915	0.201958	+	0.979394i	$0.435270\pi$
$152$	0	0
$153$	8.15987	0.659686
$154$	9.62513	0.775615
$155$	0	0
$156$	0.0143042	0.00114525
$157$	−2.09499	−0.167199	−0.0835993	−	0.996499i	$-0.526642\pi$
$157$	−2.09499	−0.167199	−0.0835993	+	0.996499i	$0.526642\pi$
$158$	21.5573	1.71500
$159$	−1.14810	−0.0910503
$160$	0	0
$161$	−9.07368	−0.715106
$162$	−12.8927	−1.01295
$163$	9.55821	0.748656	0.374328	−	0.927296i	$-0.377873\pi$
$163$	9.55821	0.748656	0.374328	+	0.927296i	$0.377873\pi$
$164$	−0.235813	−0.0184139
$165$	0	0
$166$	22.6093	1.75482
$167$	16.7395	1.29534	0.647670	−	0.761921i	$-0.275744\pi$
$167$	16.7395	1.29534	0.647670	+	0.761921i	$0.275744\pi$
$168$	0.633620	0.0488849
$169$	−12.8351	−0.987312
$170$	0	0
$171$	0	0
$172$	0.605588	0.0461757
$173$	13.9754	1.06253	0.531264	−	0.847206i	$-0.321717\pi$
$173$	13.9754	1.06253	0.531264	+	0.847206i	$0.321717\pi$
$174$	−1.07990	−0.0818673
$175$	0	0
$176$	−21.6751	−1.63382
$177$	−2.03995	−0.153332
$178$	−21.0545	−1.57810
$179$	−9.86133	−0.737071	−0.368535	−	0.929614i	$-0.620141\pi$
$179$	−9.86133	−0.737071	−0.368535	+	0.929614i	$0.620141\pi$
$180$	0	0
$181$	14.7726	1.09804	0.549019	−	0.835810i	$-0.315002\pi$
$181$	14.7726	1.09804	0.549019	+	0.835810i	$0.315002\pi$
$182$	−0.784570	−0.0581562
$183$	−0.169648	−0.0125407
$184$	18.6114	1.37205
$185$	0	0
$186$	−0.692185	−0.0507534
$187$	−13.7028	−1.00204
$188$	−1.26216	−0.0920522
$189$	−1.41332	−0.102804
$190$	0	0
$191$	−20.6797	−1.49633	−0.748166	−	0.663512i	$-0.769065\pi$
$191$	−20.6797	−1.49633	−0.748166	+	0.663512i	$0.769065\pi$
$192$	−1.28598	−0.0928078
$193$	20.2783	1.45966	0.729831	−	0.683628i	$-0.239599\pi$
$193$	20.2783	1.45966	0.729831	+	0.683628i	$0.239599\pi$
$194$	−27.0795	−1.94420
$195$	0	0
$196$	−1.02767	−0.0734053
$197$	15.9172	1.13406	0.567028	−	0.823698i	$-0.308093\pi$
$197$	15.9172	1.13406	0.567028	+	0.823698i	$0.308093\pi$
$198$	21.8966	1.55613
$199$	−13.2357	−0.938254	−0.469127	−	0.883131i	$-0.655431\pi$
$199$	−13.2357	−0.938254	−0.469127	+	0.883131i	$0.655431\pi$
$200$	0	0
$201$	−0.961665	−0.0678306
$202$	23.6921	1.66697
$203$	5.23587	0.367486
$204$	0.0968627	0.00678175
$205$	0	0
$206$	−7.86427	−0.547930
$207$	−20.6421	−1.43473
$208$	1.76679	0.122505
$209$	0	0
$210$	0	0
$211$	−9.77834	−0.673168	−0.336584	−	0.941653i	$-0.609272\pi$
$211$	−9.77834	−0.673168	−0.336584	+	0.941653i	$0.609272\pi$
$212$	1.22604	0.0842049
$213$	0.296987	0.0203492
$214$	19.2591	1.31652
$215$	0	0
$216$	2.89892	0.197247
$217$	3.35603	0.227822
$218$	12.1368	0.822009
$219$	−1.39619	−0.0943455
$220$	0	0
$221$	1.11695	0.0751340
$222$	0.998912	0.0670426
$223$	−25.8620	−1.73185	−0.865923	−	0.500178i	$-0.833268\pi$
$223$	−25.8620	−1.73185	−0.865923	+	0.500178i	$0.833268\pi$
$224$	−1.42593	−0.0952738
$225$	0	0
$226$	9.69854	0.645137
$227$	2.28573	0.151709	0.0758545	−	0.997119i	$-0.475832\pi$
$227$	2.28573	0.151709	0.0758545	+	0.997119i	$0.475832\pi$
$228$	0	0
$229$	−21.8805	−1.44590	−0.722952	−	0.690898i	$-0.757215\pi$
$229$	−21.8805	−1.44590	−0.722952	+	0.690898i	$0.757215\pi$
$230$	0	0
$231$	1.18013	0.0776467
$232$	−10.7395	−0.705082
$233$	−2.05384	−0.134552	−0.0672758	−	0.997734i	$-0.521431\pi$
$233$	−2.05384	−0.134552	−0.0672758	+	0.997734i	$0.521431\pi$
$234$	−1.78485	−0.116679
$235$	0	0
$236$	2.17844	0.141804
$237$	2.64311	0.171689
$238$	−5.31281	−0.344379
$239$	7.68110	0.496849	0.248425	−	0.968651i	$-0.420087\pi$
$239$	7.68110	0.496849	0.248425	+	0.968651i	$0.420087\pi$
$240$	0	0
$241$	14.5366	0.936387	0.468194	−	0.883626i	$-0.344905\pi$
$241$	14.5366	0.936387	0.468194	+	0.883626i	$0.344905\pi$
$242$	−20.4776	−1.31635
$243$	−4.83173	−0.309956
$244$	0.181165	0.0115979
$245$	0	0
$246$	−0.327081	−0.0208539
$247$	0	0
$248$	−6.88368	−0.437114
$249$	2.77211	0.175675
$250$	0	0
$251$	7.18731	0.453659	0.226829	−	0.973935i	$-0.427164\pi$
$251$	7.18731	0.453659	0.226829	+	0.973935i	$0.427164\pi$
$252$	0.750464	0.0472748
$253$	34.6640	2.17931
$254$	−22.3788	−1.40417
$255$	0	0
$256$	4.61213	0.288258
$257$	−8.70588	−0.543058	−0.271529	−	0.962430i	$-0.587529\pi$
$257$	−8.70588	−0.543058	−0.271529	+	0.962430i	$0.587529\pi$
$258$	0.839970	0.0522943
$259$	−4.84318	−0.300940
$260$	0	0
$261$	11.9113	0.737291
$262$	3.69348	0.228184
$263$	30.5232	1.88214	0.941072	−	0.338208i	$-0.109821\pi$
$263$	30.5232	1.88214	0.941072	+	0.338208i	$0.109821\pi$
$264$	−2.42061	−0.148978
$265$	0	0
$266$	0	0
$267$	−2.58147	−0.157983
$268$	1.02695	0.0627309
$269$	−18.4076	−1.12233	−0.561165	−	0.827704i	$-0.689647\pi$
$269$	−18.4076	−1.12233	−0.561165	+	0.827704i	$0.689647\pi$
$270$	0	0
$271$	−11.4739	−0.696989	−0.348494	−	0.937311i	$-0.613307\pi$
$271$	−11.4739	−0.696989	−0.348494	+	0.937311i	$0.613307\pi$
$272$	11.9640	0.725427
$273$	−0.0961953	−0.00582201
$274$	26.0810	1.57561
$275$	0	0
$276$	−0.245035	−0.0147494
$277$	1.79689	0.107965	0.0539823	−	0.998542i	$-0.482809\pi$
$277$	1.79689	0.107965	0.0539823	+	0.998542i	$0.482809\pi$
$278$	−4.67276	−0.280253
$279$	7.63477	0.457082
$280$	0	0
$281$	−14.1910	−0.846564	−0.423282	−	0.905998i	$-0.639122\pi$
$281$	−14.1910	−0.846564	−0.423282	+	0.905998i	$0.639122\pi$
$282$	−1.75065	−0.104250
$283$	−17.9087	−1.06456	−0.532281	−	0.846568i	$-0.678665\pi$
$283$	−17.9087	−1.06456	−0.532281	+	0.846568i	$0.678665\pi$
$284$	−0.317148	−0.0188193
$285$	0	0
$286$	2.99728	0.177233
$287$	1.58584	0.0936090
$288$	−3.24390	−0.191149
$289$	−9.43645	−0.555085
$290$	0	0
$291$	−3.32019	−0.194633
$292$	1.49097	0.0872524
$293$	33.3088	1.94592	0.972961	−	0.230970i	$-0.0741900\pi$
$293$	33.3088	1.94592	0.972961	+	0.230970i	$0.0741900\pi$
$294$	−1.42542	−0.0831320
$295$	0	0
$296$	9.93404	0.577404
$297$	5.39929	0.313299
$298$	9.44095	0.546900
$299$	−2.82556	−0.163406
$300$	0	0
$301$	−4.07256	−0.234738
$302$	−7.35175	−0.423046
$303$	2.90486	0.166880
$304$	0	0
$305$	0	0
$306$	−12.0863	−0.690931
$307$	−15.5571	−0.887890	−0.443945	−	0.896054i	$-0.646421\pi$
$307$	−15.5571	−0.887890	−0.443945	+	0.896054i	$0.646421\pi$
$308$	−1.26024	−0.0718090
$309$	−0.964230	−0.0548532
$310$	0	0
$311$	29.1959	1.65555	0.827773	−	0.561063i	$-0.189608\pi$
$311$	29.1959	1.65555	0.827773	+	0.561063i	$0.189608\pi$
$312$	0.197310	0.0111705
$313$	−31.8880	−1.80241	−0.901207	−	0.433388i	$-0.857318\pi$
$313$	−31.8880	−1.80241	−0.901207	+	0.433388i	$0.857318\pi$
$314$	3.10309	0.175118
$315$	0	0
$316$	−2.82255	−0.158781
$317$	13.5701	0.762173	0.381086	−	0.924539i	$-0.375550\pi$
$317$	13.5701	0.762173	0.381086	+	0.924539i	$0.375550\pi$
$318$	1.70056	0.0953627
$319$	−20.0025	−1.11992
$320$	0	0
$321$	2.36133	0.131797
$322$	13.4399	0.748976
$323$	0	0
$324$	1.68807	0.0937819
$325$	0	0
$326$	−14.1576	−0.784115
$327$	1.48808	0.0822911
$328$	−3.25277	−0.179604
$329$	8.48796	0.467956
$330$	0	0
$331$	−29.4274	−1.61747	−0.808737	−	0.588171i	$-0.799848\pi$
$331$	−29.4274	−1.61747	−0.808737	+	0.588171i	$0.799848\pi$
$332$	−2.96030	−0.162467
$333$	−11.0180	−0.603780
$334$	−24.7944	−1.35669
$335$	0	0
$336$	−1.03038	−0.0562121
$337$	15.5118	0.844979	0.422489	−	0.906368i	$-0.361156\pi$
$337$	15.5118	0.844979	0.422489	+	0.906368i	$0.361156\pi$
$338$	19.0112	1.03407
$339$	1.18913	0.0645846
$340$	0	0
$341$	−12.8210	−0.694294
$342$	0	0
$343$	16.0406	0.866110
$344$	8.35339	0.450384
$345$	0	0
$346$	−20.7003	−1.11285
$347$	22.5877	1.21257	0.606285	−	0.795247i	$-0.292659\pi$
$347$	22.5877	1.21257	0.606285	+	0.795247i	$0.292659\pi$
$348$	0.141395	0.00757954
$349$	25.4885	1.36437	0.682184	−	0.731181i	$-0.261030\pi$
$349$	25.4885	1.36437	0.682184	+	0.731181i	$0.261030\pi$
$350$	0	0
$351$	−0.440110	−0.0234914
$352$	5.44744	0.290350
$353$	−10.2959	−0.547994	−0.273997	−	0.961730i	$-0.588346\pi$
$353$	−10.2959	−0.547994	−0.273997	+	0.961730i	$0.588346\pi$
$354$	3.02156	0.160594
$355$	0	0
$356$	2.75672	0.146106
$357$	−0.651399	−0.0344757
$358$	14.6066	0.771980
$359$	1.98251	0.104633	0.0523165	−	0.998631i	$-0.483340\pi$
$359$	1.98251	0.104633	0.0523165	+	0.998631i	$0.483340\pi$
$360$	0	0
$361$	0	0
$362$	−21.8811	−1.15004
$363$	−2.51073	−0.131779
$364$	0.102726	0.00538429
$365$	0	0
$366$	0.251281	0.0131347
$367$	5.27424	0.275313	0.137657	−	0.990480i	$-0.456043\pi$
$367$	5.27424	0.275313	0.137657	+	0.990480i	$0.456043\pi$
$368$	−30.2656	−1.57770
$369$	3.60769	0.187809
$370$	0	0
$371$	−8.24509	−0.428064
$372$	0.0906295	0.00469892
$373$	31.7192	1.64236	0.821179	−	0.570670i	$-0.193317\pi$
$373$	31.7192	1.64236	0.821179	+	0.570670i	$0.193317\pi$
$374$	20.2964	1.04950
$375$	0	0
$376$	−17.4100	−0.897852
$377$	1.63045	0.0839727
$378$	2.09341	0.107673
$379$	35.5119	1.82412	0.912062	−	0.410052i	$-0.134490\pi$
$379$	35.5119	1.82412	0.912062	+	0.410052i	$0.134490\pi$
$380$	0	0
$381$	−2.74384	−0.140571
$382$	30.6307	1.56720
$383$	−13.9302	−0.711799	−0.355899	−	0.934524i	$-0.615825\pi$
$383$	−13.9302	−0.711799	−0.355899	+	0.934524i	$0.615825\pi$
$384$	2.30190	0.117468
$385$	0	0
$386$	−30.0361	−1.52880
$387$	−9.26484	−0.470958
$388$	3.54559	0.180000
$389$	−0.942869	−0.0478053	−0.0239027	−	0.999714i	$-0.507609\pi$
$389$	−0.942869	−0.0478053	−0.0239027	+	0.999714i	$0.507609\pi$
$390$	0	0
$391$	−19.1336	−0.967628
$392$	−14.1756	−0.715974
$393$	0.452853	0.0228434
$394$	−23.5765	−1.18777
$395$	0	0
$396$	−2.86698	−0.144071
$397$	28.5656	1.43367	0.716834	−	0.697244i	$-0.245591\pi$
$397$	28.5656	1.43367	0.716834	+	0.697244i	$0.245591\pi$
$398$	19.6047	0.982693
$399$	0	0
$400$	0	0
$401$	−7.16731	−0.357918	−0.178959	−	0.983857i	$-0.557273\pi$
$401$	−7.16731	−0.357918	−0.178959	+	0.983857i	$0.557273\pi$
$402$	1.42441	0.0710432
$403$	1.04507	0.0520587
$404$	−3.10207	−0.154334
$405$	0	0
$406$	−7.75534	−0.384891
$407$	18.5023	0.917125
$408$	1.33611	0.0661473
$409$	−10.5325	−0.520798	−0.260399	−	0.965501i	$-0.583854\pi$
$409$	−10.5325	−0.520798	−0.260399	+	0.965501i	$0.583854\pi$
$410$	0	0
$411$	3.19777	0.157734
$412$	1.02969	0.0507292
$413$	−14.6499	−0.720875
$414$	30.5750	1.50268
$415$	0	0
$416$	−0.444036	−0.0217706
$417$	−0.572922	−0.0280561
$418$	0	0
$419$	2.52693	0.123448	0.0617242	−	0.998093i	$-0.480340\pi$
$419$	2.52693	0.123448	0.0617242	+	0.998093i	$0.480340\pi$
$420$	0	0
$421$	14.8669	0.724569	0.362285	−	0.932068i	$-0.381997\pi$
$421$	14.8669	0.724569	0.362285	+	0.932068i	$0.381997\pi$
$422$	14.4836	0.705051
$423$	19.3096	0.938866
$424$	16.9118	0.821311
$425$	0	0
$426$	−0.439895	−0.0213130
$427$	−1.21833	−0.0589589
$428$	−2.52164	−0.121888
$429$	0.367493	0.0177427
$430$	0	0
$431$	−4.18068	−0.201376	−0.100688	−	0.994918i	$-0.532104\pi$
$431$	−4.18068	−0.201376	−0.100688	+	0.994918i	$0.532104\pi$
$432$	−4.71419	−0.226812
$433$	25.3053	1.21610	0.608048	−	0.793900i	$-0.291953\pi$
$433$	25.3053	1.21610	0.608048	+	0.793900i	$0.291953\pi$
$434$	−4.97093	−0.238612
$435$	0	0
$436$	−1.58910	−0.0761043
$437$	0	0
$438$	2.06802	0.0988139
$439$	28.9121	1.37990	0.689950	−	0.723857i	$-0.257633\pi$
$439$	28.9121	1.37990	0.689950	+	0.723857i	$0.257633\pi$
$440$	0	0
$441$	15.7223	0.748680
$442$	−1.65442	−0.0786926
$443$	4.58864	0.218013	0.109007	−	0.994041i	$-0.465233\pi$
$443$	4.58864	0.218013	0.109007	+	0.994041i	$0.465233\pi$
$444$	−0.130790	−0.00620702
$445$	0	0
$446$	38.3066	1.81387
$447$	1.15755	0.0547500
$448$	−9.23529	−0.436327
$449$	29.2171	1.37884	0.689420	−	0.724362i	$-0.257865\pi$
$449$	29.2171	1.37884	0.689420	+	0.724362i	$0.257865\pi$
$450$	0	0
$451$	−6.05834	−0.285276
$452$	−1.26986	−0.0597290
$453$	−0.901391	−0.0423510
$454$	−3.38560	−0.158894
$455$	0	0
$456$	0	0
$457$	23.7113	1.10917	0.554584	−	0.832128i	$-0.312878\pi$
$457$	23.7113	1.10917	0.554584	+	0.832128i	$0.312878\pi$
$458$	32.4093	1.51439
$459$	−2.98026	−0.139107
$460$	0	0
$461$	35.0218	1.63113	0.815565	−	0.578666i	$-0.196426\pi$
$461$	35.0218	1.63113	0.815565	+	0.578666i	$0.196426\pi$
$462$	−1.74800	−0.0813242
$463$	−20.4936	−0.952417	−0.476209	−	0.879332i	$-0.657989\pi$
$463$	−20.4936	−0.952417	−0.476209	+	0.879332i	$0.657989\pi$
$464$	17.4644	0.810765
$465$	0	0
$466$	3.04214	0.140924
$467$	−9.24346	−0.427736	−0.213868	−	0.976863i	$-0.568606\pi$
$467$	−9.24346	−0.427736	−0.213868	+	0.976863i	$0.568606\pi$
$468$	0.233695	0.0108026
$469$	−6.90620	−0.318899
$470$	0	0
$471$	0.380467	0.0175310
$472$	30.0490	1.38312
$473$	15.5583	0.715372
$474$	−3.91496	−0.179820
$475$	0	0
$476$	0.695620	0.0318837
$477$	−18.7571	−0.858829
$478$	−11.3772	−0.520381
$479$	3.76753	0.172143	0.0860715	−	0.996289i	$-0.472569\pi$
$479$	3.76753	0.172143	0.0860715	+	0.996289i	$0.472569\pi$
$480$	0	0
$481$	−1.50817	−0.0687667
$482$	−21.5316	−0.980737
$483$	1.64785	0.0749798
$484$	2.68118	0.121872
$485$	0	0
$486$	7.15673	0.324636
$487$	−11.7981	−0.534621	−0.267311	−	0.963610i	$-0.586135\pi$
$487$	−11.7981	−0.534621	−0.267311	+	0.963610i	$0.586135\pi$
$488$	2.49896	0.113122
$489$	−1.73584	−0.0784976
$490$	0	0
$491$	7.83563	0.353617	0.176809	−	0.984245i	$-0.443423\pi$
$491$	7.83563	0.353617	0.176809	+	0.984245i	$0.443423\pi$
$492$	0.0428255	0.00193072
$493$	11.0408	0.497254
$494$	0	0
$495$	0	0
$496$	11.1941	0.502632
$497$	2.13281	0.0956696
$498$	−4.10603	−0.183996
$499$	1.05959	0.0474338	0.0237169	−	0.999719i	$-0.492450\pi$
$499$	1.05959	0.0474338	0.0237169	+	0.999719i	$0.492450\pi$
$500$	0	0
$501$	−3.04002	−0.135818
$502$	−10.6458	−0.475145
$503$	−5.37711	−0.239753	−0.119877	−	0.992789i	$-0.538250\pi$
$503$	−5.37711	−0.239753	−0.119877	+	0.992789i	$0.538250\pi$
$504$	10.3518	0.461105
$505$	0	0
$506$	−51.3441	−2.28253
$507$	2.33094	0.103521
$508$	2.93011	0.130003
$509$	33.8480	1.50028	0.750142	−	0.661276i	$-0.229985\pi$
$509$	33.8480	1.50028	0.750142	+	0.661276i	$0.229985\pi$
$510$	0	0
$511$	−10.0267	−0.443556
$512$	18.5188	0.818423
$513$	0	0
$514$	12.8951	0.568778
$515$	0	0
$516$	−0.109979	−0.00484158
$517$	−32.4264	−1.42611
$518$	7.17369	0.315194
$519$	−2.53804	−0.111407
$520$	0	0
$521$	−29.5742	−1.29567	−0.647834	−	0.761782i	$-0.724325\pi$
$521$	−29.5742	−1.29567	−0.647834	+	0.761782i	$0.724325\pi$
$522$	−17.6429	−0.772211
$523$	24.0446	1.05140	0.525698	−	0.850671i	$-0.323804\pi$
$523$	24.0446	1.05140	0.525698	+	0.850671i	$0.323804\pi$
$524$	−0.483596	−0.0211260
$525$	0	0
$526$	−45.2108	−1.97129
$527$	7.07683	0.308271
$528$	3.93636	0.171308
$529$	25.4025	1.10446
$530$	0	0
$531$	−33.3277	−1.44630
$532$	0	0
$533$	0.493831	0.0213902
$534$	3.82366	0.165466
$535$	0	0
$536$	14.1656	0.611860
$537$	1.79089	0.0772828
$538$	27.2652	1.17549
$539$	−26.4022	−1.13722
$540$	0	0
$541$	−26.1917	−1.12607	−0.563035	−	0.826433i	$-0.690366\pi$
$541$	−26.1917	−1.12607	−0.563035	+	0.826433i	$0.690366\pi$
$542$	16.9951	0.730000
$543$	−2.68282	−0.115131
$544$	−3.00684	−0.128917
$545$	0	0
$546$	0.142484	0.00609775
$547$	−14.9284	−0.638293	−0.319146	−	0.947705i	$-0.603396\pi$
$547$	−14.9284	−0.638293	−0.319146	+	0.947705i	$0.603396\pi$
$548$	−3.41486	−0.145875
$549$	−2.77162	−0.118290
$550$	0	0
$551$	0	0
$552$	−3.37997	−0.143861
$553$	18.9815	0.807177
$554$	−2.66154	−0.113078
$555$	0	0
$556$	0.611816	0.0259468
$557$	−5.27514	−0.223515	−0.111758	−	0.993736i	$-0.535648\pi$
$557$	−5.27514	−0.223515	−0.111758	+	0.993736i	$0.535648\pi$
$558$	−11.3086	−0.478730
$559$	−1.26820	−0.0536391
$560$	0	0
$561$	2.48853	0.105066
$562$	21.0196	0.886660
$563$	17.8950	0.754186	0.377093	−	0.926175i	$-0.376924\pi$
$563$	17.8950	0.754186	0.377093	+	0.926175i	$0.376924\pi$
$564$	0.229217	0.00965179
$565$	0	0
$566$	26.5263	1.11498
$567$	−11.3522	−0.476749
$568$	−4.37469	−0.183558
$569$	6.81848	0.285846	0.142923	−	0.989734i	$-0.454350\pi$
$569$	6.81848	0.285846	0.142923	+	0.989734i	$0.454350\pi$
$570$	0	0
$571$	5.16915	0.216322	0.108161	−	0.994133i	$-0.465504\pi$
$571$	5.16915	0.216322	0.108161	+	0.994133i	$0.465504\pi$
$572$	−0.392441	−0.0164088
$573$	3.75560	0.156892
$574$	−2.34893	−0.0980425
$575$	0	0
$576$	−21.0098	−0.875407
$577$	−28.5621	−1.18905	−0.594527	−	0.804076i	$-0.702661\pi$
$577$	−28.5621	−1.18905	−0.594527	+	0.804076i	$0.702661\pi$
$578$	13.9772	0.581376
$579$	−3.68269	−0.153047
$580$	0	0
$581$	19.9079	0.825919
$582$	4.91785	0.203851
$583$	31.4986	1.30454
$584$	20.5662	0.851035
$585$	0	0
$586$	−49.3368	−2.03809
$587$	−40.3254	−1.66441	−0.832204	−	0.554469i	$-0.812921\pi$
$587$	−40.3254	−1.66441	−0.832204	+	0.554469i	$0.812921\pi$
$588$	0.186633	0.00769663
$589$	0	0
$590$	0	0
$591$	−2.89069	−0.118907
$592$	−16.1546	−0.663950
$593$	−27.6905	−1.13711	−0.568555	−	0.822645i	$-0.692498\pi$
$593$	−27.6905	−1.13711	−0.568555	+	0.822645i	$0.692498\pi$
$594$	−7.99740	−0.328137
$595$	0	0
$596$	−1.23613	−0.0506338
$597$	2.40371	0.0983771
$598$	4.18520	0.171145
$599$	−2.62932	−0.107431	−0.0537156	−	0.998556i	$-0.517106\pi$
$599$	−2.62932	−0.107431	−0.0537156	+	0.998556i	$0.517106\pi$
$600$	0	0
$601$	27.7009	1.12994	0.564972	−	0.825110i	$-0.308887\pi$
$601$	27.7009	1.12994	0.564972	+	0.825110i	$0.308887\pi$
$602$	6.03225	0.245856
$603$	−15.7112	−0.639810
$604$	0.962584	0.0391670
$605$	0	0
$606$	−4.30267	−0.174784
$607$	−32.0668	−1.30155	−0.650775	−	0.759271i	$-0.725556\pi$
$607$	−32.0668	−1.30155	−0.650775	+	0.759271i	$0.725556\pi$
$608$	0	0
$609$	−0.950874	−0.0385313
$610$	0	0
$611$	2.64316	0.106931
$612$	1.58250	0.0639686
$613$	12.4603	0.503266	0.251633	−	0.967823i	$-0.419032\pi$
$613$	12.4603	0.503266	0.251633	+	0.967823i	$0.419032\pi$
$614$	23.0431	0.929943
$615$	0	0
$616$	−17.3836	−0.700405
$617$	−26.4739	−1.06580	−0.532900	−	0.846178i	$-0.678898\pi$
$617$	−26.4739	−1.06580	−0.532900	+	0.846178i	$0.678898\pi$
$618$	1.42821	0.0574511
$619$	33.3766	1.34152	0.670760	−	0.741674i	$-0.265968\pi$
$619$	33.3766	1.34152	0.670760	+	0.741674i	$0.265968\pi$
$620$	0	0
$621$	7.53921	0.302538
$622$	−43.2448	−1.73396
$623$	−18.5388	−0.742743
$624$	−0.320863	−0.0128448
$625$	0	0
$626$	47.2323	1.88778
$627$	0	0
$628$	−0.406296	−0.0162130
$629$	−10.2128	−0.407210
$630$	0	0
$631$	40.1201	1.59716	0.798578	−	0.601891i	$-0.205586\pi$
$631$	40.1201	1.59716	0.798578	+	0.601891i	$0.205586\pi$
$632$	−38.9338	−1.54870
$633$	1.77582	0.0705826
$634$	−20.1000	−0.798271
$635$	0	0
$636$	−0.222659	−0.00882899
$637$	2.15211	0.0852699
$638$	29.6276	1.17297
$639$	4.85202	0.191943
$640$	0	0
$641$	0.406546	0.0160576	0.00802880	−	0.999968i	$-0.497444\pi$
$641$	0.406546	0.0160576	0.00802880	+	0.999968i	$0.497444\pi$
$642$	−3.49760	−0.138039
$643$	−25.1696	−0.992593	−0.496296	−	0.868153i	$-0.665307\pi$
$643$	−25.1696	−0.992593	−0.496296	+	0.868153i	$0.665307\pi$
$644$	−1.75972	−0.0693426
$645$	0	0
$646$	0	0
$647$	18.3604	0.721821	0.360911	−	0.932600i	$-0.382466\pi$
$647$	18.3604	0.721821	0.360911	+	0.932600i	$0.382466\pi$
$648$	23.2850	0.914722
$649$	55.9668	2.19689
$650$	0	0
$651$	−0.609480	−0.0238874
$652$	1.85369	0.0725959
$653$	33.0301	1.29257	0.646284	−	0.763097i	$-0.276322\pi$
$653$	33.0301	1.29257	0.646284	+	0.763097i	$0.276322\pi$
$654$	−2.20414	−0.0861886
$655$	0	0
$656$	5.28961	0.206525
$657$	−22.8102	−0.889911
$658$	−12.5723	−0.490120
$659$	−26.4620	−1.03081	−0.515406	−	0.856946i	$-0.672359\pi$
$659$	−26.4620	−1.03081	−0.515406	+	0.856946i	$0.672359\pi$
$660$	0	0
$661$	−15.5935	−0.606515	−0.303258	−	0.952909i	$-0.598074\pi$
$661$	−15.5935	−0.606515	−0.303258	+	0.952909i	$0.598074\pi$
$662$	43.5876	1.69408
$663$	−0.202846	−0.00787790
$664$	−40.8339	−1.58466
$665$	0	0
$666$	16.3197	0.632377
$667$	−27.9301	−1.08146
$668$	3.24640	0.125607
$669$	4.69673	0.181586
$670$	0	0
$671$	4.65434	0.179679
$672$	0.258959	0.00998958
$673$	−6.13645	−0.236543	−0.118271	−	0.992981i	$-0.537735\pi$
$673$	−6.13645	−0.236543	−0.118271	+	0.992981i	$0.537735\pi$
$674$	−22.9759	−0.884999
$675$	0	0
$676$	−2.48919	−0.0957379
$677$	22.2754	0.856112	0.428056	−	0.903752i	$-0.359199\pi$
$677$	22.2754	0.856112	0.428056	+	0.903752i	$0.359199\pi$
$678$	−1.76133	−0.0676435
$679$	−23.8440	−0.915048
$680$	0	0
$681$	−0.415105	−0.0159069
$682$	18.9903	0.727178
$683$	16.7397	0.640527	0.320264	−	0.947328i	$-0.396228\pi$
$683$	16.7397	0.640527	0.320264	+	0.947328i	$0.396228\pi$
$684$	0	0
$685$	0	0
$686$	−23.7592	−0.907131
$687$	3.97367	0.151605
$688$	−13.5842	−0.517891
$689$	−2.56753	−0.0978151
$690$	0	0
$691$	36.7557	1.39825	0.699126	−	0.714999i	$-0.253573\pi$
$691$	36.7557	1.39825	0.699126	+	0.714999i	$0.253573\pi$
$692$	2.71034	0.103032
$693$	19.2804	0.732400
$694$	−33.4567	−1.27000
$695$	0	0
$696$	1.95037	0.0739288
$697$	3.34404	0.126665
$698$	−37.7534	−1.42899
$699$	0.372993	0.0141079
$700$	0	0
$701$	35.4106	1.33744	0.668721	−	0.743514i	$-0.266842\pi$
$701$	35.4106	1.33744	0.668721	+	0.743514i	$0.266842\pi$
$702$	0.651889	0.0246040
$703$	0	0
$704$	35.2814	1.32972
$705$	0	0
$706$	15.2502	0.573949
$707$	20.8613	0.784570
$708$	−0.395621	−0.0148683
$709$	33.8369	1.27077	0.635386	−	0.772195i	$-0.280841\pi$
$709$	33.8369	1.27077	0.635386	+	0.772195i	$0.280841\pi$
$710$	0	0
$711$	43.1819	1.61945
$712$	38.0257	1.42508
$713$	−17.9023	−0.670448
$714$	0.964848	0.0361085
$715$	0	0
$716$	−1.91247	−0.0714725
$717$	−1.39495	−0.0520953
$718$	−2.93649	−0.109589
$719$	26.9940	1.00671	0.503353	−	0.864081i	$-0.332100\pi$
$719$	26.9940	1.00671	0.503353	+	0.864081i	$0.332100\pi$
$720$	0	0
$721$	−6.92463	−0.257887
$722$	0	0
$723$	−2.63997	−0.0981814
$724$	2.86494	0.106475
$725$	0	0
$726$	3.71888	0.138021
$727$	21.6033	0.801221	0.400610	−	0.916249i	$-0.368798\pi$
$727$	21.6033	0.801221	0.400610	+	0.916249i	$0.368798\pi$
$728$	1.41698	0.0525169
$729$	−25.2353	−0.934640
$730$	0	0
$731$	−8.58777	−0.317630
$732$	−0.0329009	−0.00121605
$733$	17.2551	0.637332	0.318666	−	0.947867i	$-0.396765\pi$
$733$	17.2551	0.637332	0.318666	+	0.947867i	$0.396765\pi$
$734$	−7.81217	−0.288353
$735$	0	0
$736$	7.60644	0.280377
$737$	26.3836	0.971853
$738$	−5.34369	−0.196704
$739$	4.79106	0.176242	0.0881210	−	0.996110i	$-0.471914\pi$
$739$	4.79106	0.176242	0.0881210	+	0.996110i	$0.471914\pi$
$740$	0	0
$741$	0	0
$742$	12.2126	0.448338
$743$	−14.4875	−0.531493	−0.265747	−	0.964043i	$-0.585618\pi$
$743$	−14.4875	−0.531493	−0.265747	+	0.964043i	$0.585618\pi$
$744$	1.25013	0.0458320
$745$	0	0
$746$	−46.9823	−1.72014
$747$	45.2893	1.65705
$748$	−2.65746	−0.0971665
$749$	16.9579	0.619630
$750$	0	0
$751$	6.45274	0.235464	0.117732	−	0.993045i	$-0.462438\pi$
$751$	6.45274	0.235464	0.117732	+	0.993045i	$0.462438\pi$
$752$	28.3119	1.03243
$753$	−1.30527	−0.0475667
$754$	−2.41502	−0.0879499
$755$	0	0
$756$	−0.274095	−0.00996874
$757$	−36.9590	−1.34330	−0.671649	−	0.740870i	$-0.734414\pi$
$757$	−36.9590	−1.34330	−0.671649	+	0.740870i	$0.734414\pi$
$758$	−52.6000	−1.91052
$759$	−6.29525	−0.228503
$760$	0	0
$761$	48.3893	1.75411	0.877055	−	0.480389i	$-0.159505\pi$
$761$	48.3893	1.75411	0.877055	+	0.480389i	$0.159505\pi$
$762$	4.06416	0.147229
$763$	10.6867	0.386883
$764$	−4.01055	−0.145097
$765$	0	0
$766$	20.6333	0.745511
$767$	−4.56200	−0.164724
$768$	−0.837598	−0.0302242
$769$	41.4353	1.49420	0.747098	−	0.664713i	$-0.231446\pi$
$769$	41.4353	1.49420	0.747098	+	0.664713i	$0.231446\pi$
$770$	0	0
$771$	1.58105	0.0569403
$772$	3.93270	0.141541
$773$	38.8309	1.39665	0.698325	−	0.715781i	$-0.253929\pi$
$773$	38.8309	1.39665	0.698325	+	0.715781i	$0.253929\pi$
$774$	13.7230	0.493264
$775$	0	0
$776$	48.9074	1.75567
$777$	0.879559	0.0315540
$778$	1.39657	0.0500695
$779$	0	0
$780$	0	0
$781$	−8.14793	−0.291556
$782$	28.3406	1.01346
$783$	−4.35041	−0.155471
$784$	23.0521	0.823290
$785$	0	0
$786$	−0.670764	−0.0239254
$787$	51.4952	1.83561	0.917803	−	0.397037i	$-0.129962\pi$
$787$	51.4952	1.83561	0.917803	+	0.397037i	$0.129962\pi$
$788$	3.08694	0.109968
$789$	−5.54325	−0.197345
$790$	0	0
$791$	8.53973	0.303638
$792$	−39.5467	−1.40523
$793$	−0.379388	−0.0134725
$794$	−42.3112	−1.50157
$795$	0	0
$796$	−2.56689	−0.0909809
$797$	−7.50433	−0.265817	−0.132908	−	0.991128i	$-0.542432\pi$
$797$	−7.50433	−0.265817	−0.132908	+	0.991128i	$0.542432\pi$
$798$	0	0
$799$	17.8985	0.633203
$800$	0	0
$801$	−42.1748	−1.49017
$802$	10.6162	0.374870
$803$	38.3049	1.35175
$804$	−0.186502	−0.00657742
$805$	0	0
$806$	−1.54795	−0.0545243
$807$	3.34296	0.117678
$808$	−42.7894	−1.50533
$809$	−17.9109	−0.629714	−0.314857	−	0.949139i	$-0.601957\pi$
$809$	−17.9109	−0.629714	−0.314857	+	0.949139i	$0.601957\pi$
$810$	0	0
$811$	5.15032	0.180852	0.0904260	−	0.995903i	$-0.471177\pi$
$811$	5.15032	0.180852	0.0904260	+	0.995903i	$0.471177\pi$
$812$	1.01543	0.0356345
$813$	2.08375	0.0730802
$814$	−27.4055	−0.960562
$815$	0	0
$816$	−2.17276	−0.0760619
$817$	0	0
$818$	15.6007	0.545464
$819$	−1.57159	−0.0549159
$820$	0	0
$821$	−1.38105	−0.0481990	−0.0240995	−	0.999710i	$-0.507672\pi$
$821$	−1.38105	−0.0481990	−0.0240995	+	0.999710i	$0.507672\pi$
$822$	−4.73652	−0.165205
$823$	−14.4160	−0.502509	−0.251254	−	0.967921i	$-0.580843\pi$
$823$	−14.4160	−0.502509	−0.251254	+	0.967921i	$0.580843\pi$
$824$	14.2034	0.494798
$825$	0	0
$826$	21.6994	0.755017
$827$	−9.47321	−0.329416	−0.164708	−	0.986342i	$-0.552668\pi$
$827$	−9.47321	−0.329416	−0.164708	+	0.986342i	$0.552668\pi$
$828$	−4.00326	−0.139123
$829$	−12.3950	−0.430497	−0.215248	−	0.976559i	$-0.569056\pi$
$829$	−12.3950	−0.430497	−0.215248	+	0.976559i	$0.569056\pi$
$830$	0	0
$831$	−0.326329	−0.0113202
$832$	−2.87588	−0.0997032
$833$	14.5733	0.504935
$834$	0.848608	0.0293849
$835$	0	0
$836$	0	0
$837$	−2.78848	−0.0963839
$838$	−3.74287	−0.129295
$839$	12.5298	0.432578	0.216289	−	0.976329i	$-0.430605\pi$
$839$	12.5298	0.432578	0.216289	+	0.976329i	$0.430605\pi$
$840$	0	0
$841$	−12.8832	−0.444250
$842$	−22.0208	−0.758887
$843$	2.57720	0.0887633
$844$	−1.89638	−0.0652760
$845$	0	0
$846$	−28.6013	−0.983333
$847$	−18.0308	−0.619547
$848$	−27.5018	−0.944416
$849$	3.25236	0.111621
$850$	0	0
$851$	25.8354	0.885625
$852$	0.0575965	0.00197323
$853$	−3.47300	−0.118913	−0.0594567	−	0.998231i	$-0.518937\pi$
$853$	−3.47300	−0.118913	−0.0594567	+	0.998231i	$0.518937\pi$
$854$	1.80458	0.0617513
$855$	0	0
$856$	−34.7831	−1.18886
$857$	16.6617	0.569153	0.284577	−	0.958653i	$-0.408147\pi$
$857$	16.6617	0.569153	0.284577	+	0.958653i	$0.408147\pi$
$858$	−0.544329	−0.0185831
$859$	40.8085	1.39237	0.696183	−	0.717864i	$-0.254880\pi$
$859$	40.8085	1.39237	0.696183	+	0.717864i	$0.254880\pi$
$860$	0	0
$861$	−0.288000	−0.00981502
$862$	6.19239	0.210914
$863$	−4.07050	−0.138561	−0.0692806	−	0.997597i	$-0.522070\pi$
$863$	−4.07050	−0.138561	−0.0692806	+	0.997597i	$0.522070\pi$
$864$	1.18478	0.0403072
$865$	0	0
$866$	−37.4821	−1.27369
$867$	1.71373	0.0582014
$868$	0.650856	0.0220915
$869$	−72.5148	−2.45990
$870$	0	0
$871$	−2.15060	−0.0728703
$872$	−21.9199	−0.742300
$873$	−54.2437	−1.83587
$874$	0	0
$875$	0	0
$876$	−0.270771	−0.00914852
$877$	26.5144	0.895327	0.447663	−	0.894202i	$-0.352256\pi$
$877$	26.5144	0.895327	0.447663	+	0.894202i	$0.352256\pi$
$878$	−42.8244	−1.44526
$879$	−6.04914	−0.204032
$880$	0	0
$881$	−22.3730	−0.753766	−0.376883	−	0.926261i	$-0.623004\pi$
$881$	−22.3730	−0.753766	−0.376883	+	0.926261i	$0.623004\pi$
$882$	−23.2878	−0.784140
$883$	−19.7138	−0.663422	−0.331711	−	0.943381i	$-0.607626\pi$
$883$	−19.7138	−0.663422	−0.331711	+	0.943381i	$0.607626\pi$
$884$	0.216617	0.00728562
$885$	0	0
$886$	−6.79667	−0.228339
$887$	−11.4216	−0.383500	−0.191750	−	0.981444i	$-0.561416\pi$
$887$	−11.4216	−0.383500	−0.191750	+	0.981444i	$0.561416\pi$
$888$	−1.80410	−0.0605416
$889$	−19.7049	−0.660881
$890$	0	0
$891$	43.3687	1.45291
$892$	−5.01558	−0.167934
$893$	0	0
$894$	−1.71455	−0.0573431
$895$	0	0
$896$	16.5311	0.552266
$897$	0.513143	0.0171333
$898$	−43.2762	−1.44415
$899$	10.3303	0.344536
$900$	0	0
$901$	−17.3864	−0.579223
$902$	8.97358	0.298787
$903$	0.739608	0.0246126
$904$	−17.5162	−0.582580
$905$	0	0
$906$	1.33514	0.0443569
$907$	10.7383	0.356559	0.178280	−	0.983980i	$-0.442947\pi$
$907$	10.7383	0.356559	0.178280	+	0.983980i	$0.442947\pi$
$908$	0.443286	0.0147110
$909$	47.4582	1.57409
$910$	0	0
$911$	−27.5952	−0.914269	−0.457134	−	0.889398i	$-0.651124\pi$
$911$	−27.5952	−0.914269	−0.457134	+	0.889398i	$0.651124\pi$
$912$	0	0
$913$	−76.0538	−2.51701
$914$	−35.1210	−1.16170
$915$	0	0
$916$	−4.24343	−0.140207
$917$	3.25217	0.107396
$918$	4.41435	0.145695
$919$	−49.2639	−1.62507	−0.812533	−	0.582915i	$-0.801912\pi$
$919$	−49.2639	−1.62507	−0.812533	+	0.582915i	$0.801912\pi$
$920$	0	0
$921$	2.82529	0.0930964
$922$	−51.8741	−1.70838
$923$	0.664160	0.0218611
$924$	0.228870	0.00752927
$925$	0	0
$926$	30.3550	0.997526
$927$	−15.7531	−0.517401
$928$	−4.38921	−0.144083
$929$	43.0748	1.41324	0.706619	−	0.707594i	$-0.250220\pi$
$929$	43.0748	1.41324	0.706619	+	0.707594i	$0.250220\pi$
$930$	0	0
$931$	0	0
$932$	−0.398315	−0.0130472
$933$	−5.30219	−0.173586
$934$	13.6914	0.447995
$935$	0	0
$936$	3.22356	0.105365
$937$	43.4444	1.41927	0.709633	−	0.704571i	$-0.248861\pi$
$937$	43.4444	1.41927	0.709633	+	0.704571i	$0.248861\pi$
$938$	10.2294	0.334003
$939$	5.79110	0.188985
$940$	0	0
$941$	29.2941	0.954961	0.477480	−	0.878642i	$-0.341550\pi$
$941$	29.2941	0.954961	0.477480	+	0.878642i	$0.341550\pi$
$942$	−0.563545	−0.0183613
$943$	−8.45946	−0.275478
$944$	−48.8653	−1.59043
$945$	0	0
$946$	−23.0449	−0.749254
$947$	−2.00589	−0.0651826	−0.0325913	−	0.999469i	$-0.510376\pi$
$947$	−2.00589	−0.0651826	−0.0325913	+	0.999469i	$0.510376\pi$
$948$	0.512596	0.0166484
$949$	−3.12233	−0.101355
$950$	0	0
$951$	−2.46443	−0.0799148
$952$	9.59528	0.310985
$953$	−19.2389	−0.623210	−0.311605	−	0.950212i	$-0.600867\pi$
$953$	−19.2389	−0.623210	−0.311605	+	0.950212i	$0.600867\pi$
$954$	27.7829	0.899505
$955$	0	0
$956$	1.48965	0.0481786
$957$	3.63260	0.117425
$958$	−5.58045	−0.180296
$959$	22.9648	0.741572
$960$	0	0
$961$	−24.3786	−0.786406
$962$	2.23390	0.0720237
$963$	38.5783	1.24317
$964$	2.81919	0.0907999
$965$	0	0
$966$	−2.44079	−0.0785310
$967$	28.4143	0.913742	0.456871	−	0.889533i	$-0.348970\pi$
$967$	28.4143	0.913742	0.456871	+	0.889533i	$0.348970\pi$
$968$	36.9838	1.18870
$969$	0	0
$970$	0	0
$971$	17.9762	0.576885	0.288442	−	0.957497i	$-0.406863\pi$
$971$	17.9762	0.576885	0.288442	+	0.957497i	$0.406863\pi$
$972$	−0.937049	−0.0300559
$973$	−4.11444	−0.131903
$974$	17.4752	0.559942
$975$	0	0
$976$	−4.06377	−0.130078
$977$	13.2288	0.423226	0.211613	−	0.977354i	$-0.432128\pi$
$977$	13.2288	0.423226	0.211613	+	0.977354i	$0.432128\pi$
$978$	2.57112	0.0822154
$979$	70.8235	2.26353
$980$	0	0
$981$	24.3116	0.776208
$982$	−11.6061	−0.370365
$983$	48.5234	1.54766	0.773828	−	0.633396i	$-0.218339\pi$
$983$	48.5234	1.54766	0.773828	+	0.633396i	$0.218339\pi$
$984$	0.590728	0.0188317
$985$	0	0
$986$	−16.3536	−0.520805
$987$	−1.54148	−0.0490658
$988$	0	0
$989$	21.7246	0.690802
$990$	0	0
$991$	40.2803	1.27955	0.639773	−	0.768564i	$-0.279028\pi$
$991$	40.2803	1.27955	0.639773	+	0.768564i	$0.279028\pi$
$992$	−2.81335	−0.0893239
$993$	5.34423	0.169594
$994$	−3.15911	−0.100201
$995$	0	0
$996$	0.537613	0.0170349
$997$	27.5590	0.872803	0.436401	−	0.899752i	$-0.356253\pi$
$997$	27.5590	0.872803	0.436401	+	0.899752i	$0.356253\pi$
$998$	−1.56946	−0.0496804
$999$	4.02413	0.127318

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bz.1.1		6
5.4	even	2		9025.2.a.br.1.6		6
19.7	even	3		475.2.e.f.201.6	yes	12
19.11	even	3		475.2.e.f.26.6	✓	12
19.18	odd	2		9025.2.a.bs.1.6		6
95.7	odd	12		475.2.j.d.49.9		24
95.49	even	6		475.2.e.h.26.1	yes	12
95.64	even	6		475.2.e.h.201.1	yes	12
95.68	odd	12		475.2.j.d.349.9		24
95.83	odd	12		475.2.j.d.49.4		24
95.87	odd	12		475.2.j.d.349.4		24
95.94	odd	2		9025.2.a.by.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.e.f.26.6	✓	12	19.11	even	3
475.2.e.f.201.6	yes	12	19.7	even	3
475.2.e.h.26.1	yes	12	95.49	even	6
475.2.e.h.201.1	yes	12	95.64	even	6
475.2.j.d.49.4		24	95.83	odd	12
475.2.j.d.49.9		24	95.7	odd	12
475.2.j.d.349.4		24	95.87	odd	12
475.2.j.d.349.9		24	95.68	odd	12
9025.2.a.br.1.6		6	5.4	even	2
9025.2.a.bs.1.6		6	19.18	odd	2
9025.2.a.by.1.1		6	95.94	odd	2
9025.2.a.bz.1.1		6	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.48119 q^{2} -0.181608 q^{3} +0.193937 q^{4} +0.268996 q^{6} -1.30422 q^{7} +2.67513 q^{8} -2.96702 q^{9} +4.98247 q^{11} -0.0352204 q^{12} -0.406135 q^{13} +1.93180 q^{14} -4.35026 q^{16} -2.75019 q^{17} +4.39473 q^{18} +0.236856 q^{21} -7.38001 q^{22} +6.95719 q^{23} -0.485824 q^{24} +0.601564 q^{26} +1.08366 q^{27} -0.252935 q^{28} -4.01457 q^{29} -2.57321 q^{31} +1.09332 q^{32} -0.904855 q^{33} +4.07357 q^{34} -0.575413 q^{36} +3.71348 q^{37} +0.0737572 q^{39} -1.21593 q^{41} -0.350829 q^{42} +3.12261 q^{43} +0.966284 q^{44} -10.3050 q^{46} -6.50809 q^{47} +0.790041 q^{48} -5.29902 q^{49} +0.499456 q^{51} -0.0787643 q^{52} +6.32187 q^{53} -1.60511 q^{54} -3.48895 q^{56} +5.94636 q^{58} +11.2327 q^{59} +0.934143 q^{61} +3.81143 q^{62} +3.86964 q^{63} +7.08110 q^{64} +1.34027 q^{66} +5.29529 q^{67} -0.533363 q^{68} -1.26348 q^{69} -1.63532 q^{71} -7.93716 q^{72} +7.68792 q^{73} -5.50038 q^{74} -6.49822 q^{77} -0.109249 q^{78} -14.5540 q^{79} +8.70426 q^{81} +1.80103 q^{82} -15.2643 q^{83} +0.0459350 q^{84} -4.62519 q^{86} +0.729076 q^{87} +13.3288 q^{88} +14.2145 q^{89} +0.529687 q^{91} +1.34925 q^{92} +0.467315 q^{93} +9.63975 q^{94} -0.198556 q^{96} +18.2822 q^{97} +7.84888 q^{98} -14.7831 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - q^{6} + 2 q^{7} + 6 q^{8} + 7 q^{9} - q^{11} + 7 q^{12} + 5 q^{13} - 6 q^{14} - 6 q^{16} - 3 q^{17} + 7 q^{18} + 3 q^{21} + 9 q^{22} - 6 q^{23} + 11 q^{24} + 19 q^{26}+ \cdots - 20 q^{99}+O(q^{100}) \) 6 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - q^6 + 2 * q^7 + 6 * q^8 + 7 * q^9 - q^11 + 7 * q^12 + 5 * q^13 - 6 * q^14 - 6 * q^16 - 3 * q^17 + 7 * q^18 + 3 * q^21 + 9 * q^22 - 6 * q^23 + 11 * q^24 + 19 * q^26 + 18 * q^27 - 4 * q^28 + 3 * q^29 - 3 * q^31 - 6 * q^32 - 18 * q^33 - q^34 + 13 * q^36 - 6 * q^37 + 8 * q^39 + 11 * q^41 - 11 * q^42 + 13 * q^43 + 21 * q^44 - 12 * q^46 - 6 * q^47 - 19 * q^48 + 4 * q^49 - 17 * q^51 - q^52 + 18 * q^53 + 18 * q^54 + 4 * q^56 + 5 * q^58 + 4 * q^59 + 25 * q^61 - 21 * q^62 + 43 * q^63 - 22 * q^64 + 34 * q^66 + 6 * q^67 - q^68 + 13 * q^69 + 18 * q^71 + 13 * q^72 + q^73 - 6 * q^74 - 11 * q^77 + 72 * q^78 + 3 * q^79 + 2 * q^81 + 31 * q^82 - 23 * q^83 + 37 * q^84 + 9 * q^86 + 11 * q^87 + 11 * q^88 + 12 * q^89 - 11 * q^91 + 28 * q^92 - 13 * q^93 + 8 * q^94 - 13 * q^96 + 3 * q^97 - 22 * q^98 - 20 * q^99

Introduction

L-functions

Modular forms

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Database

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