LMFDB - Embedded newform 9025.2.a.cd.1.2

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 12x^{7} - 4x^{6} + 48x^{5} + 27x^{4} - 72x^{3} - 51x^{2} + 27x + 19 $ x^9 - 12x^7 - 4x^6 + 48x^5 + 27x^4 - 72x^3 - 51x^2 + 27*x + 19
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.57047$ 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)

$f(q)$	$=$	$q-1.57047 q^{2} +2.28502 q^{3} +0.466387 q^{4} -3.58856 q^{6} +4.01337 q^{7} +2.40850 q^{8} +2.22131 q^{9} +O(q^{10})$	\(q-1.57047 q^{2} +2.28502 q^{3} +0.466387 q^{4} -3.58856 q^{6} +4.01337 q^{7} +2.40850 q^{8} +2.22131 q^{9} +2.76620 q^{11} +1.06570 q^{12} +2.78732 q^{13} -6.30289 q^{14} -4.71526 q^{16} +3.60673 q^{17} -3.48851 q^{18} +9.17062 q^{21} -4.34425 q^{22} -1.36746 q^{23} +5.50347 q^{24} -4.37742 q^{26} -1.77932 q^{27} +1.87178 q^{28} +9.50940 q^{29} -1.55371 q^{31} +2.58819 q^{32} +6.32083 q^{33} -5.66428 q^{34} +1.03599 q^{36} +8.51183 q^{37} +6.36909 q^{39} +6.60899 q^{41} -14.4022 q^{42} +6.25476 q^{43} +1.29012 q^{44} +2.14756 q^{46} -9.10988 q^{47} -10.7745 q^{48} +9.10711 q^{49} +8.24145 q^{51} +1.29997 q^{52} -3.94456 q^{53} +2.79437 q^{54} +9.66619 q^{56} -14.9343 q^{58} -3.61183 q^{59} -7.48908 q^{61} +2.44005 q^{62} +8.91494 q^{63} +5.36583 q^{64} -9.92669 q^{66} -14.5720 q^{67} +1.68213 q^{68} -3.12468 q^{69} +3.31888 q^{71} +5.35003 q^{72} +10.0077 q^{73} -13.3676 q^{74} +11.1018 q^{77} -10.0025 q^{78} +1.94735 q^{79} -10.7297 q^{81} -10.3792 q^{82} -1.61868 q^{83} +4.27705 q^{84} -9.82294 q^{86} +21.7292 q^{87} +6.66240 q^{88} +12.2510 q^{89} +11.1866 q^{91} -0.637766 q^{92} -3.55025 q^{93} +14.3068 q^{94} +5.91406 q^{96} -11.5912 q^{97} -14.3025 q^{98} +6.14460 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{3} + 6 q^{4} - 12 q^{6} + 12 q^{8} + 6 q^{9}+O(q^{10}) $ 9 * q - 3 * q^3 + 6 * q^4 - 12 * q^6 + 12 * q^8 + 6 * q^9	$ 9 q - 3 q^{3} + 6 q^{4} - 12 q^{6} + 12 q^{8} + 6 q^{9} - 6 q^{12} + 3 q^{13} + 12 q^{14} - 12 q^{16} + 9 q^{17} - 6 q^{18} + 12 q^{21} - 12 q^{22} + 15 q^{24} + 21 q^{26} - 6 q^{27} + 15 q^{28} + 15 q^{29} + 30 q^{31} + 9 q^{32} - 9 q^{33} - 6 q^{36} - 30 q^{37} + 6 q^{39} + 18 q^{41} - 36 q^{42} + 6 q^{43} - 24 q^{44} + 21 q^{46} - 21 q^{47} - 15 q^{48} + 3 q^{49} + 18 q^{51} + 3 q^{52} + 9 q^{53} - 9 q^{54} + 36 q^{56} - 18 q^{58} + 27 q^{59} + 12 q^{61} + 6 q^{62} + 15 q^{63} + 24 q^{64} + 3 q^{66} - 36 q^{67} - 3 q^{68} + 27 q^{69} - 6 q^{71} - 12 q^{72} + 9 q^{73} - 9 q^{74} - 12 q^{77} - 54 q^{78} + 45 q^{79} - 15 q^{81} + 48 q^{82} - 12 q^{84} - 9 q^{86} - 45 q^{87} - 39 q^{88} - 9 q^{89} + 51 q^{91} + 54 q^{92} - 9 q^{93} + 33 q^{94} - 9 q^{96} - 45 q^{97} - 33 q^{98} + 12 q^{99}+O(q^{100}) $ 9 * q - 3 * q^3 + 6 * q^4 - 12 * q^6 + 12 * q^8 + 6 * q^9 - 6 * q^12 + 3 * q^13 + 12 * q^14 - 12 * q^16 + 9 * q^17 - 6 * q^18 + 12 * q^21 - 12 * q^22 + 15 * q^24 + 21 * q^26 - 6 * q^27 + 15 * q^28 + 15 * q^29 + 30 * q^31 + 9 * q^32 - 9 * q^33 - 6 * q^36 - 30 * q^37 + 6 * q^39 + 18 * q^41 - 36 * q^42 + 6 * q^43 - 24 * q^44 + 21 * q^46 - 21 * q^47 - 15 * q^48 + 3 * q^49 + 18 * q^51 + 3 * q^52 + 9 * q^53 - 9 * q^54 + 36 * q^56 - 18 * q^58 + 27 * q^59 + 12 * q^61 + 6 * q^62 + 15 * q^63 + 24 * q^64 + 3 * q^66 - 36 * q^67 - 3 * q^68 + 27 * q^69 - 6 * q^71 - 12 * q^72 + 9 * q^73 - 9 * q^74 - 12 * q^77 - 54 * q^78 + 45 * q^79 - 15 * q^81 + 48 * q^82 - 12 * q^84 - 9 * q^86 - 45 * q^87 - 39 * q^88 - 9 * q^89 + 51 * q^91 + 54 * q^92 - 9 * q^93 + 33 * q^94 - 9 * q^96 - 45 * q^97 - 33 * q^98 + 12 * 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 $

$2$	−1.57047	−1.11049	−0.555246	−	0.831686i	$-0.687376\pi$
$2$	−1.57047	−1.11049	−0.555246	+	0.831686i	$0.687376\pi$
$3$	2.28502	1.31926	0.659628	−	0.751592i	$-0.270714\pi$
$3$	2.28502	1.31926	0.659628	+	0.751592i	$0.270714\pi$
$4$	0.466387	0.233193
$5$	0	0
$5$	0	0
$6$	−3.58856	−1.46502
$7$	4.01337	1.51691	0.758455	−	0.651725i	$-0.225955\pi$
$7$	4.01337	1.51691	0.758455	+	0.651725i	$0.225955\pi$
$8$	2.40850	0.851533
$9$	2.22131	0.740437
$10$	0	0
$11$	2.76620	0.834041	0.417021	−	0.908897i	$-0.363074\pi$
$11$	2.76620	0.834041	0.417021	+	0.908897i	$0.363074\pi$
$12$	1.06570	0.307642
$13$	2.78732	0.773065	0.386532	−	0.922276i	$-0.373673\pi$
$13$	2.78732	0.773065	0.386532	+	0.922276i	$0.373673\pi$
$14$	−6.30289	−1.68452
$15$	0	0
$16$	−4.71526	−1.17881
$17$	3.60673	0.874761	0.437381	−	0.899277i	$-0.355906\pi$
$17$	3.60673	0.874761	0.437381	+	0.899277i	$0.355906\pi$
$18$	−3.48851	−0.822250
$19$	0	0
$19$	0	0
$20$	0	0
$21$	9.17062	2.00119
$22$	−4.34425	−0.926197
$23$	−1.36746	−0.285136	−0.142568	−	0.989785i	$-0.545536\pi$
$23$	−1.36746	−0.285136	−0.142568	+	0.989785i	$0.545536\pi$
$24$	5.50347	1.12339
$25$	0	0
$26$	−4.37742	−0.858482
$27$	−1.77932	−0.342430
$28$	1.87178	0.353733
$29$	9.50940	1.76585	0.882926	−	0.469512i	$-0.155570\pi$
$29$	9.50940	1.76585	0.882926	+	0.469512i	$0.155570\pi$
$30$	0	0
$31$	−1.55371	−0.279054	−0.139527	−	0.990218i	$-0.544558\pi$
$31$	−1.55371	−0.279054	−0.139527	+	0.990218i	$0.544558\pi$
$32$	2.58819	0.457531
$33$	6.32083	1.10031
$34$	−5.66428	−0.971415
$35$	0	0
$36$	1.03599	0.172665
$37$	8.51183	1.39934	0.699668	−	0.714468i	$-0.253331\pi$
$37$	8.51183	1.39934	0.699668	+	0.714468i	$0.253331\pi$
$38$	0	0
$39$	6.36909	1.01987
$40$	0	0
$41$	6.60899	1.03215	0.516075	−	0.856543i	$-0.327393\pi$
$41$	6.60899	1.03215	0.516075	+	0.856543i	$0.327393\pi$
$42$	−14.4022	−2.22231
$43$	6.25476	0.953842	0.476921	−	0.878946i	$-0.341753\pi$
$43$	6.25476	0.953842	0.476921	+	0.878946i	$0.341753\pi$
$44$	1.29012	0.194493
$45$	0	0
$46$	2.14756	0.316641
$47$	−9.10988	−1.32881	−0.664406	−	0.747371i	$-0.731316\pi$
$47$	−9.10988	−1.32881	−0.664406	+	0.747371i	$0.731316\pi$
$48$	−10.7745	−1.55516
$49$	9.10711	1.30102
$50$	0	0
$51$	8.24145	1.15403
$52$	1.29997	0.180274
$53$	−3.94456	−0.541827	−0.270913	−	0.962604i	$-0.587326\pi$
$53$	−3.94456	−0.541827	−0.270913	+	0.962604i	$0.587326\pi$
$54$	2.79437	0.380265
$55$	0	0
$56$	9.66619	1.29170
$57$	0	0
$58$	−14.9343	−1.96096
$59$	−3.61183	−0.470220	−0.235110	−	0.971969i	$-0.575545\pi$
$59$	−3.61183	−0.470220	−0.235110	+	0.971969i	$0.575545\pi$
$60$	0	0
$61$	−7.48908	−0.958879	−0.479439	−	0.877575i	$-0.659160\pi$
$61$	−7.48908	−0.958879	−0.479439	+	0.877575i	$0.659160\pi$
$62$	2.44005	0.309887
$63$	8.91494	1.12318
$64$	5.36583	0.670729
$65$	0	0
$66$	−9.92669	−1.22189
$67$	−14.5720	−1.78025	−0.890125	−	0.455717i	$-0.849383\pi$
$67$	−14.5720	−1.78025	−0.890125	+	0.455717i	$0.849383\pi$
$68$	1.68213	0.203988
$69$	−3.12468	−0.376167
$70$	0	0
$71$	3.31888	0.393879	0.196940	−	0.980416i	$-0.436900\pi$
$71$	3.31888	0.393879	0.196940	+	0.980416i	$0.436900\pi$
$72$	5.35003	0.630507
$73$	10.0077	1.17132	0.585659	−	0.810558i	$-0.300836\pi$
$73$	10.0077	1.17132	0.585659	+	0.810558i	$0.300836\pi$
$74$	−13.3676	−1.55395
$75$	0	0
$76$	0	0
$77$	11.1018	1.26517
$78$	−10.0025	−1.13256
$79$	1.94735	0.219094	0.109547	−	0.993982i	$-0.465060\pi$
$79$	1.94735	0.219094	0.109547	+	0.993982i	$0.465060\pi$
$80$	0	0
$81$	−10.7297	−1.19219
$82$	−10.3792	−1.14619
$83$	−1.61868	−0.177673	−0.0888365	−	0.996046i	$-0.528315\pi$
$83$	−1.61868	−0.177673	−0.0888365	+	0.996046i	$0.528315\pi$
$84$	4.27705	0.466665
$85$	0	0
$86$	−9.82294	−1.05923
$87$	21.7292	2.32961
$88$	6.66240	0.710214
$89$	12.2510	1.29860	0.649300	−	0.760532i	$-0.275062\pi$
$89$	12.2510	1.29860	0.649300	+	0.760532i	$0.275062\pi$
$90$	0	0
$91$	11.1866	1.17267
$92$	−0.637766	−0.0664917
$93$	−3.55025	−0.368143
$94$	14.3068	1.47564
$95$	0	0
$96$	5.91406	0.603601
$97$	−11.5912	−1.17690	−0.588452	−	0.808532i	$-0.700262\pi$
$97$	−11.5912	−1.17690	−0.588452	+	0.808532i	$0.700262\pi$
$98$	−14.3025	−1.44477
$99$	6.14460	0.617555
$100$	0	0
$101$	−6.98240	−0.694774	−0.347387	−	0.937722i	$-0.612931\pi$
$101$	−6.98240	−0.694774	−0.347387	+	0.937722i	$0.612931\pi$
$102$	−12.9430	−1.28155
$103$	10.1338	0.998510	0.499255	−	0.866455i	$-0.333607\pi$
$103$	10.1338	0.998510	0.499255	+	0.866455i	$0.333607\pi$
$104$	6.71327	0.658290
$105$	0	0
$106$	6.19482	0.601694
$107$	−11.7835	−1.13916	−0.569578	−	0.821937i	$-0.692893\pi$
$107$	−11.7835	−1.13916	−0.569578	+	0.821937i	$0.692893\pi$
$108$	−0.829849	−0.0798523
$109$	1.60407	0.153642	0.0768212	−	0.997045i	$-0.475523\pi$
$109$	1.60407	0.153642	0.0768212	+	0.997045i	$0.475523\pi$
$110$	0	0
$111$	19.4497	1.84608
$112$	−18.9241	−1.78815
$113$	12.9449	1.21775	0.608876	−	0.793266i	$-0.291621\pi$
$113$	12.9449	1.21775	0.608876	+	0.793266i	$0.291621\pi$
$114$	0	0
$115$	0	0
$116$	4.43506	0.411785
$117$	6.19152	0.572406
$118$	5.67228	0.522175
$119$	14.4751	1.32693
$120$	0	0
$121$	−3.34812	−0.304375
$122$	11.7614	1.06483
$123$	15.1017	1.36167
$124$	−0.724627	−0.0650735
$125$	0	0
$126$	−14.0007	−1.24728
$127$	−8.29802	−0.736331	−0.368165	−	0.929760i	$-0.620014\pi$
$127$	−8.29802	−0.736331	−0.368165	+	0.929760i	$0.620014\pi$
$128$	−13.6033	−1.20237
$129$	14.2923	1.25836
$130$	0	0
$131$	−6.51185	−0.568943	−0.284471	−	0.958685i	$-0.591818\pi$
$131$	−6.51185	−0.568943	−0.284471	+	0.958685i	$0.591818\pi$
$132$	2.94795	0.256586
$133$	0	0
$134$	22.8849	1.97695
$135$	0	0
$136$	8.68681	0.744888
$137$	−14.8686	−1.27031	−0.635156	−	0.772384i	$-0.719064\pi$
$137$	−14.8686	−1.27031	−0.635156	+	0.772384i	$0.719064\pi$
$138$	4.90722	0.417731
$139$	9.95823	0.844646	0.422323	−	0.906445i	$-0.361215\pi$
$139$	9.95823	0.844646	0.422323	+	0.906445i	$0.361215\pi$
$140$	0	0
$141$	−20.8163	−1.75304
$142$	−5.21222	−0.437400
$143$	7.71030	0.644768
$144$	−10.4741	−0.872838
$145$	0	0
$146$	−15.7169	−1.30074
$147$	20.8099	1.71637
$148$	3.96980	0.326316
$149$	−1.43162	−0.117283	−0.0586416	−	0.998279i	$-0.518677\pi$
$149$	−1.43162	−0.117283	−0.0586416	+	0.998279i	$0.518677\pi$
$150$	0	0
$151$	8.18383	0.665990	0.332995	−	0.942929i	$-0.391941\pi$
$151$	8.18383	0.665990	0.332995	+	0.942929i	$0.391941\pi$
$152$	0	0
$153$	8.01168	0.647706
$154$	−17.4351	−1.40496
$155$	0	0
$156$	2.97046	0.237827
$157$	7.83512	0.625311	0.312655	−	0.949867i	$-0.398782\pi$
$157$	7.83512	0.625311	0.312655	+	0.949867i	$0.398782\pi$
$158$	−3.05827	−0.243303
$159$	−9.01339	−0.714808
$160$	0	0
$161$	−5.48813	−0.432525
$162$	16.8507	1.32392
$163$	19.2994	1.51164	0.755821	−	0.654778i	$-0.227238\pi$
$163$	19.2994	1.51164	0.755821	+	0.654778i	$0.227238\pi$
$164$	3.08234	0.240690
$165$	0	0
$166$	2.54209	0.197304
$167$	9.38968	0.726595	0.363297	−	0.931673i	$-0.381651\pi$
$167$	9.38968	0.726595	0.363297	+	0.931673i	$0.381651\pi$
$168$	22.0874	1.70408
$169$	−5.23082	−0.402371
$170$	0	0
$171$	0	0
$172$	2.91714	0.222430
$173$	8.31084	0.631861	0.315931	−	0.948782i	$-0.397683\pi$
$173$	8.31084	0.631861	0.315931	+	0.948782i	$0.397683\pi$
$174$	−34.1251	−2.58702
$175$	0	0
$176$	−13.0434	−0.983180
$177$	−8.25309	−0.620340
$178$	−19.2398	−1.44209
$179$	18.5319	1.38514	0.692568	−	0.721352i	$-0.256479\pi$
$179$	18.5319	1.38514	0.692568	+	0.721352i	$0.256479\pi$
$180$	0	0
$181$	−11.3292	−0.842094	−0.421047	−	0.907039i	$-0.638337\pi$
$181$	−11.3292	−0.842094	−0.421047	+	0.907039i	$0.638337\pi$
$182$	−17.5682	−1.30224
$183$	−17.1127	−1.26501
$184$	−3.29353	−0.242802
$185$	0	0
$186$	5.57557	0.408820
$187$	9.97695	0.729587
$188$	−4.24873	−0.309870
$189$	−7.14105	−0.519435
$190$	0	0
$191$	11.8879	0.860175	0.430088	−	0.902787i	$-0.358483\pi$
$191$	11.8879	0.860175	0.430088	+	0.902787i	$0.358483\pi$
$192$	12.2610	0.884864
$193$	−0.957834	−0.0689464	−0.0344732	−	0.999406i	$-0.510975\pi$
$193$	−0.957834	−0.0689464	−0.0344732	+	0.999406i	$0.510975\pi$
$194$	18.2036	1.30694
$195$	0	0
$196$	4.24743	0.303388
$197$	−14.2776	−1.01724	−0.508619	−	0.860991i	$-0.669844\pi$
$197$	−14.2776	−1.01724	−0.508619	+	0.860991i	$0.669844\pi$
$198$	−9.64993	−0.685791
$199$	−7.76571	−0.550497	−0.275248	−	0.961373i	$-0.588760\pi$
$199$	−7.76571	−0.550497	−0.275248	+	0.961373i	$0.588760\pi$
$200$	0	0
$201$	−33.2972	−2.34861
$202$	10.9657	0.771542
$203$	38.1647	2.67864
$204$	3.84370	0.269113
$205$	0	0
$206$	−15.9148	−1.10884
$207$	−3.03756	−0.211125
$208$	−13.1430	−0.911300
$209$	0	0
$210$	0	0
$211$	−7.62274	−0.524771	−0.262386	−	0.964963i	$-0.584509\pi$
$211$	−7.62274	−0.524771	−0.262386	+	0.964963i	$0.584509\pi$
$212$	−1.83969	−0.126350
$213$	7.58371	0.519627
$214$	18.5057	1.26502
$215$	0	0
$216$	−4.28548	−0.291590
$217$	−6.23559	−0.423299
$218$	−2.51916	−0.170619
$219$	22.8679	1.54527
$220$	0	0
$221$	10.0531	0.676247
$222$	−30.5452	−2.05006
$223$	−8.00331	−0.535942	−0.267971	−	0.963427i	$-0.586353\pi$
$223$	−8.00331	−0.535942	−0.267971	+	0.963427i	$0.586353\pi$
$224$	10.3873	0.694034
$225$	0	0
$226$	−20.3296	−1.35230
$227$	−17.7409	−1.17750	−0.588752	−	0.808314i	$-0.700381\pi$
$227$	−17.7409	−1.17750	−0.588752	+	0.808314i	$0.700381\pi$
$228$	0	0
$229$	−24.2231	−1.60071	−0.800354	−	0.599528i	$-0.795355\pi$
$229$	−24.2231	−1.60071	−0.800354	+	0.599528i	$0.795355\pi$
$230$	0	0
$231$	25.3678	1.66908
$232$	22.9034	1.50368
$233$	−13.6491	−0.894183	−0.447092	−	0.894488i	$-0.647540\pi$
$233$	−13.6491	−0.894183	−0.447092	+	0.894488i	$0.647540\pi$
$234$	−9.72361	−0.635653
$235$	0	0
$236$	−1.68451	−0.109652
$237$	4.44974	0.289042
$238$	−22.7328	−1.47355
$239$	3.73764	0.241768	0.120884	−	0.992667i	$-0.461427\pi$
$239$	3.73764	0.241768	0.120884	+	0.992667i	$0.461427\pi$
$240$	0	0
$241$	5.28146	0.340209	0.170104	−	0.985426i	$-0.445590\pi$
$241$	5.28146	0.340209	0.170104	+	0.985426i	$0.445590\pi$
$242$	5.25814	0.338006
$243$	−19.1796	−1.23037
$244$	−3.49281	−0.223604
$245$	0	0
$246$	−23.7168	−1.51212
$247$	0	0
$248$	−3.74210	−0.237623
$249$	−3.69871	−0.234396
$250$	0	0
$251$	−12.5445	−0.791801	−0.395900	−	0.918293i	$-0.629567\pi$
$251$	−12.5445	−0.791801	−0.395900	+	0.918293i	$0.629567\pi$
$252$	4.15781	0.261917
$253$	−3.78268	−0.237815
$254$	13.0318	0.817689
$255$	0	0
$256$	10.6319	0.664494
$257$	8.59982	0.536442	0.268221	−	0.963357i	$-0.413564\pi$
$257$	8.59982	0.536442	0.268221	+	0.963357i	$0.413564\pi$
$258$	−22.4456	−1.39740
$259$	34.1611	2.12267
$260$	0	0
$261$	21.1234	1.30750
$262$	10.2267	0.631806
$263$	−26.6903	−1.64580	−0.822898	−	0.568189i	$-0.807644\pi$
$263$	−26.6903	−1.64580	−0.822898	+	0.568189i	$0.807644\pi$
$264$	15.2237	0.936954
$265$	0	0
$266$	0	0
$267$	27.9937	1.71319
$268$	−6.79617	−0.415142
$269$	3.30320	0.201400	0.100700	−	0.994917i	$-0.467892\pi$
$269$	3.30320	0.201400	0.100700	+	0.994917i	$0.467892\pi$
$270$	0	0
$271$	−18.3875	−1.11696	−0.558482	−	0.829517i	$-0.688616\pi$
$271$	−18.3875	−1.11696	−0.558482	+	0.829517i	$0.688616\pi$
$272$	−17.0067	−1.03118
$273$	25.5615	1.54705
$274$	23.3508	1.41067
$275$	0	0
$276$	−1.45731	−0.0877196
$277$	−26.7787	−1.60898	−0.804488	−	0.593969i	$-0.797560\pi$
$277$	−26.7787	−1.60898	−0.804488	+	0.593969i	$0.797560\pi$
$278$	−15.6391	−0.937973
$279$	−3.45126	−0.206622
$280$	0	0
$281$	−21.5462	−1.28534	−0.642669	−	0.766144i	$-0.722173\pi$
$281$	−21.5462	−1.28534	−0.642669	+	0.766144i	$0.722173\pi$
$282$	32.6914	1.94674
$283$	−18.3471	−1.09062	−0.545311	−	0.838234i	$-0.683588\pi$
$283$	−18.3471	−1.09062	−0.545311	+	0.838234i	$0.683588\pi$
$284$	1.54788	0.0918499
$285$	0	0
$286$	−12.1088	−0.716010
$287$	26.5243	1.56568
$288$	5.74917	0.338773
$289$	−3.99148	−0.234793
$290$	0	0
$291$	−26.4860	−1.55264
$292$	4.66748	0.273143
$293$	−9.28872	−0.542653	−0.271326	−	0.962487i	$-0.587462\pi$
$293$	−9.28872	−0.542653	−0.271326	+	0.962487i	$0.587462\pi$
$294$	−32.6814	−1.90602
$295$	0	0
$296$	20.5007	1.19158
$297$	−4.92195	−0.285600
$298$	2.24833	0.130242
$299$	−3.81156	−0.220428
$300$	0	0
$301$	25.1027	1.44689
$302$	−12.8525	−0.739577
$303$	−15.9549	−0.916585
$304$	0	0
$305$	0	0
$306$	−12.5821	−0.719272
$307$	−11.0151	−0.628666	−0.314333	−	0.949313i	$-0.601781\pi$
$307$	−11.0151	−0.628666	−0.314333	+	0.949313i	$0.601781\pi$
$308$	5.17772	0.295028
$309$	23.1559	1.31729
$310$	0	0
$311$	25.3821	1.43929	0.719644	−	0.694343i	$-0.244305\pi$
$311$	25.3821	1.43929	0.719644	+	0.694343i	$0.244305\pi$
$312$	15.3399	0.868453
$313$	−21.0559	−1.19015	−0.595074	−	0.803671i	$-0.702877\pi$
$313$	−21.0559	−1.19015	−0.595074	+	0.803671i	$0.702877\pi$
$314$	−12.3048	−0.694403
$315$	0	0
$316$	0.908220	0.0510914
$317$	4.00986	0.225216	0.112608	−	0.993639i	$-0.464080\pi$
$317$	4.00986	0.225216	0.112608	+	0.993639i	$0.464080\pi$
$318$	14.1553	0.793789
$319$	26.3049	1.47279
$320$	0	0
$321$	−26.9256	−1.50284
$322$	8.61896	0.480316
$323$	0	0
$324$	−5.00419	−0.278011
$325$	0	0
$326$	−30.3091	−1.67867
$327$	3.66534	0.202694
$328$	15.9177	0.878910
$329$	−36.5613	−2.01569
$330$	0	0
$331$	32.1360	1.76635	0.883176	−	0.469042i	$-0.155401\pi$
$331$	32.1360	1.76635	0.883176	+	0.469042i	$0.155401\pi$
$332$	−0.754929	−0.0414321
$333$	18.9074	1.03612
$334$	−14.7462	−0.806878
$335$	0	0
$336$	−43.2418	−2.35903
$337$	23.4476	1.27727	0.638636	−	0.769509i	$-0.279499\pi$
$337$	23.4476	1.27727	0.638636	+	0.769509i	$0.279499\pi$
$338$	8.21487	0.446830
$339$	29.5793	1.60653
$340$	0	0
$341$	−4.29786	−0.232742
$342$	0	0
$343$	8.45661	0.456614
$344$	15.0646	0.812228
$345$	0	0
$346$	−13.0519	−0.701677
$347$	−6.49353	−0.348591	−0.174295	−	0.984693i	$-0.555765\pi$
$347$	−6.49353	−0.348591	−0.174295	+	0.984693i	$0.555765\pi$
$348$	10.1342	0.543250
$349$	−27.2008	−1.45603	−0.728013	−	0.685563i	$-0.759556\pi$
$349$	−27.2008	−1.45603	−0.728013	+	0.685563i	$0.759556\pi$
$350$	0	0
$351$	−4.95953	−0.264720
$352$	7.15945	0.381600
$353$	6.56125	0.349220	0.174610	−	0.984638i	$-0.444134\pi$
$353$	6.56125	0.349220	0.174610	+	0.984638i	$0.444134\pi$
$354$	12.9613	0.688883
$355$	0	0
$356$	5.71369	0.302825
$357$	33.0760	1.75057
$358$	−29.1038	−1.53818
$359$	−2.73194	−0.144186	−0.0720932	−	0.997398i	$-0.522968\pi$
$359$	−2.73194	−0.144186	−0.0720932	+	0.997398i	$0.522968\pi$
$360$	0	0
$361$	0	0
$362$	17.7922	0.935139
$363$	−7.65053	−0.401549
$364$	5.21726	0.273459
$365$	0	0
$366$	26.8750	1.40478
$367$	−6.57903	−0.343423	−0.171711	−	0.985147i	$-0.554930\pi$
$367$	−6.57903	−0.343423	−0.171711	+	0.985147i	$0.554930\pi$
$368$	6.44794	0.336122
$369$	14.6806	0.764243
$370$	0	0
$371$	−15.8310	−0.821902
$372$	−1.65579	−0.0858486
$373$	−8.55993	−0.443216	−0.221608	−	0.975136i	$-0.571131\pi$
$373$	−8.55993	−0.443216	−0.221608	+	0.975136i	$0.571131\pi$
$374$	−15.6685	−0.810201
$375$	0	0
$376$	−21.9411	−1.13153
$377$	26.5058	1.36512
$378$	11.2148	0.576828
$379$	−14.2477	−0.731856	−0.365928	−	0.930643i	$-0.619248\pi$
$379$	−14.2477	−0.731856	−0.365928	+	0.930643i	$0.619248\pi$
$380$	0	0
$381$	−18.9611	−0.971409
$382$	−18.6696	−0.955218
$383$	21.3664	1.09177	0.545885	−	0.837860i	$-0.316193\pi$
$383$	21.3664	1.09177	0.545885	+	0.837860i	$0.316193\pi$
$384$	−31.0837	−1.58624
$385$	0	0
$386$	1.50425	0.0765644
$387$	13.8938	0.706261
$388$	−5.40596	−0.274446
$389$	15.2926	0.775367	0.387683	−	0.921793i	$-0.373275\pi$
$389$	15.2926	0.775367	0.387683	+	0.921793i	$0.373275\pi$
$390$	0	0
$391$	−4.93207	−0.249426
$392$	21.9345	1.10786
$393$	−14.8797	−0.750581
$394$	22.4226	1.12964
$395$	0	0
$396$	2.86576	0.144010
$397$	−32.6020	−1.63625	−0.818124	−	0.575042i	$-0.804986\pi$
$397$	−32.6020	−1.63625	−0.818124	+	0.575042i	$0.804986\pi$
$398$	12.1958	0.611322
$399$	0	0
$400$	0	0
$401$	17.3119	0.864517	0.432258	−	0.901750i	$-0.357717\pi$
$401$	17.3119	0.864517	0.432258	+	0.901750i	$0.357717\pi$
$402$	52.2924	2.60811
$403$	−4.33068	−0.215727
$404$	−3.25650	−0.162017
$405$	0	0
$406$	−59.9367	−2.97461
$407$	23.5455	1.16710
$408$	19.8495	0.982698
$409$	10.8272	0.535370	0.267685	−	0.963506i	$-0.413741\pi$
$409$	10.8272	0.535370	0.267685	+	0.963506i	$0.413741\pi$
$410$	0	0
$411$	−33.9751	−1.67587
$412$	4.72625	0.232846
$413$	−14.4956	−0.713281
$414$	4.77041	0.234453
$415$	0	0
$416$	7.21412	0.353701
$417$	22.7547	1.11430
$418$	0	0
$419$	33.9763	1.65985	0.829925	−	0.557876i	$-0.188383\pi$
$419$	33.9763	1.65985	0.829925	+	0.557876i	$0.188383\pi$
$420$	0	0
$421$	−8.30825	−0.404919	−0.202460	−	0.979291i	$-0.564894\pi$
$421$	−8.30825	−0.404919	−0.202460	+	0.979291i	$0.564894\pi$
$422$	11.9713	0.582754
$423$	−20.2359	−0.983903
$424$	−9.50046	−0.461383
$425$	0	0
$426$	−11.9100	−0.577042
$427$	−30.0564	−1.45453
$428$	−5.49568	−0.265644
$429$	17.6182	0.850614
$430$	0	0
$431$	12.1298	0.584271	0.292135	−	0.956377i	$-0.405634\pi$
$431$	12.1298	0.584271	0.292135	+	0.956377i	$0.405634\pi$
$432$	8.38993	0.403661
$433$	−5.43541	−0.261209	−0.130605	−	0.991435i	$-0.541692\pi$
$433$	−5.43541	−0.261209	−0.130605	+	0.991435i	$0.541692\pi$
$434$	9.79283	0.470071
$435$	0	0
$436$	0.748119	0.0358284
$437$	0	0
$438$	−35.9134	−1.71601
$439$	25.3807	1.21136	0.605678	−	0.795710i	$-0.292902\pi$
$439$	25.3807	1.21136	0.605678	+	0.795710i	$0.292902\pi$
$440$	0	0
$441$	20.2297	0.963321
$442$	−15.7882	−0.750967
$443$	4.92125	0.233816	0.116908	−	0.993143i	$-0.462702\pi$
$443$	4.92125	0.233816	0.116908	+	0.993143i	$0.462702\pi$
$444$	9.07108	0.430494
$445$	0	0
$446$	12.5690	0.595159
$447$	−3.27129	−0.154727
$448$	21.5351	1.01744
$449$	−4.06649	−0.191910	−0.0959548	−	0.995386i	$-0.530590\pi$
$449$	−4.06649	−0.191910	−0.0959548	+	0.995386i	$0.530590\pi$
$450$	0	0
$451$	18.2818	0.860856
$452$	6.03732	0.283971
$453$	18.7002	0.878612
$454$	27.8616	1.30761
$455$	0	0
$456$	0	0
$457$	13.5517	0.633922	0.316961	−	0.948439i	$-0.397338\pi$
$457$	13.5517	0.633922	0.316961	+	0.948439i	$0.397338\pi$
$458$	38.0418	1.77757
$459$	−6.41752	−0.299544
$460$	0	0
$461$	17.8812	0.832812	0.416406	−	0.909179i	$-0.363290\pi$
$461$	17.8812	0.832812	0.416406	+	0.909179i	$0.363290\pi$
$462$	−39.8394	−1.85350
$463$	−1.30695	−0.0607393	−0.0303696	−	0.999539i	$-0.509668\pi$
$463$	−1.30695	−0.0607393	−0.0303696	+	0.999539i	$0.509668\pi$
$464$	−44.8393	−2.08161
$465$	0	0
$466$	21.4356	0.992983
$467$	−27.0266	−1.25064	−0.625321	−	0.780368i	$-0.715032\pi$
$467$	−27.0266	−1.25064	−0.625321	+	0.780368i	$0.715032\pi$
$468$	2.88764	0.133481
$469$	−58.4826	−2.70048
$470$	0	0
$471$	17.9034	0.824945
$472$	−8.69908	−0.400408
$473$	17.3019	0.795544
$474$	−6.98820	−0.320979
$475$	0	0
$476$	6.75101	0.309432
$477$	−8.76210	−0.401189
$478$	−5.86987	−0.268481
$479$	0.289474	0.0132264	0.00661321	−	0.999978i	$-0.497895\pi$
$479$	0.289474	0.0132264	0.00661321	+	0.999978i	$0.497895\pi$
$480$	0	0
$481$	23.7252	1.08178
$482$	−8.29439	−0.377799
$483$	−12.5405	−0.570611
$484$	−1.56152	−0.0709782
$485$	0	0
$486$	30.1211	1.36632
$487$	15.3711	0.696530	0.348265	−	0.937396i	$-0.386771\pi$
$487$	15.3711	0.696530	0.348265	+	0.937396i	$0.386771\pi$
$488$	−18.0374	−0.816517
$489$	44.0994	1.99424
$490$	0	0
$491$	14.5101	0.654834	0.327417	−	0.944880i	$-0.393822\pi$
$491$	14.5101	0.654834	0.327417	+	0.944880i	$0.393822\pi$
$492$	7.04321	0.317532
$493$	34.2979	1.54470
$494$	0	0
$495$	0	0
$496$	7.32612	0.328952
$497$	13.3199	0.597479
$498$	5.80872	0.260295
$499$	−14.7379	−0.659761	−0.329880	−	0.944023i	$-0.607008\pi$
$499$	−14.7379	−0.659761	−0.329880	+	0.944023i	$0.607008\pi$
$500$	0	0
$501$	21.4556	0.958565
$502$	19.7008	0.879289
$503$	−4.35253	−0.194070	−0.0970348	−	0.995281i	$-0.530936\pi$
$503$	−4.35253	−0.194070	−0.0970348	+	0.995281i	$0.530936\pi$
$504$	21.4716	0.956422
$505$	0	0
$506$	5.94060	0.264092
$507$	−11.9525	−0.530830
$508$	−3.87009	−0.171707
$509$	−8.59364	−0.380906	−0.190453	−	0.981696i	$-0.560996\pi$
$509$	−8.59364	−0.380906	−0.190453	+	0.981696i	$0.560996\pi$
$510$	0	0
$511$	40.1647	1.77678
$512$	10.5094	0.464455
$513$	0	0
$514$	−13.5058	−0.595715
$515$	0	0
$516$	6.66571	0.293442
$517$	−25.1998	−1.10828
$518$	−53.6491	−2.35721
$519$	18.9904	0.833587
$520$	0	0
$521$	−2.06951	−0.0906668	−0.0453334	−	0.998972i	$-0.514435\pi$
$521$	−2.06951	−0.0906668	−0.0453334	+	0.998972i	$0.514435\pi$
$522$	−33.1737	−1.45197
$523$	14.0120	0.612702	0.306351	−	0.951919i	$-0.400892\pi$
$523$	14.0120	0.612702	0.306351	+	0.951919i	$0.400892\pi$
$524$	−3.03704	−0.132674
$525$	0	0
$526$	41.9164	1.82764
$527$	−5.60380	−0.244105
$528$	−29.8043	−1.29707
$529$	−21.1300	−0.918698
$530$	0	0
$531$	−8.02299	−0.348168
$532$	0	0
$533$	18.4214	0.797919
$534$	−43.9634	−1.90248
$535$	0	0
$536$	−35.0966	−1.51594
$537$	42.3457	1.82735
$538$	−5.18759	−0.223653
$539$	25.1921	1.08510
$540$	0	0
$541$	−16.6193	−0.714520	−0.357260	−	0.934005i	$-0.616289\pi$
$541$	−16.6193	−0.714520	−0.357260	+	0.934005i	$0.616289\pi$
$542$	28.8771	1.24038
$543$	−25.8875	−1.11094
$544$	9.33490	0.400230
$545$	0	0
$546$	−40.1436	−1.71799
$547$	−6.03456	−0.258019	−0.129010	−	0.991643i	$-0.541180\pi$
$547$	−6.03456	−0.258019	−0.129010	+	0.991643i	$0.541180\pi$
$548$	−6.93453	−0.296228
$549$	−16.6356	−0.709990
$550$	0	0
$551$	0	0
$552$	−7.52578	−0.320319
$553$	7.81545	0.332347
$554$	42.0552	1.78676
$555$	0	0
$556$	4.64439	0.196966
$557$	28.1274	1.19180	0.595898	−	0.803060i	$-0.296796\pi$
$557$	28.1274	1.19180	0.595898	+	0.803060i	$0.296796\pi$
$558$	5.42012	0.229452
$559$	17.4341	0.737382
$560$	0	0
$561$	22.7975	0.962512
$562$	33.8377	1.42736
$563$	−1.21155	−0.0510609	−0.0255305	−	0.999674i	$-0.508127\pi$
$563$	−1.21155	−0.0510609	−0.0255305	+	0.999674i	$0.508127\pi$
$564$	−9.70842	−0.408798
$565$	0	0
$566$	28.8136	1.21113
$567$	−43.0623	−1.80844
$568$	7.99353	0.335401
$569$	17.5814	0.737049	0.368524	−	0.929618i	$-0.379863\pi$
$569$	17.5814	0.737049	0.368524	+	0.929618i	$0.379863\pi$
$570$	0	0
$571$	42.6982	1.78686	0.893432	−	0.449198i	$-0.148290\pi$
$571$	42.6982	1.78686	0.893432	+	0.449198i	$0.148290\pi$
$572$	3.59598	0.150356
$573$	27.1640	1.13479
$574$	−41.6557	−1.73867
$575$	0	0
$576$	11.9192	0.496633
$577$	−5.49356	−0.228700	−0.114350	−	0.993441i	$-0.536479\pi$
$577$	−5.49356	−0.228700	−0.114350	+	0.993441i	$0.536479\pi$
$578$	6.26852	0.260736
$579$	−2.18867	−0.0909579
$580$	0	0
$581$	−6.49635	−0.269514
$582$	41.5956	1.72419
$583$	−10.9114	−0.451906
$584$	24.1036	0.997415
$585$	0	0
$586$	14.5877	0.602611
$587$	12.4236	0.512778	0.256389	−	0.966574i	$-0.417467\pi$
$587$	12.4236	0.512778	0.256389	+	0.966574i	$0.417467\pi$
$588$	9.70547	0.400247
$589$	0	0
$590$	0	0
$591$	−32.6247	−1.34200
$592$	−40.1355	−1.64956
$593$	1.11776	0.0459009	0.0229505	−	0.999737i	$-0.492694\pi$
$593$	1.11776	0.0459009	0.0229505	+	0.999737i	$0.492694\pi$
$594$	7.72979	0.317157
$595$	0	0
$596$	−0.667690	−0.0273497
$597$	−17.7448	−0.726246
$598$	5.98596	0.244784
$599$	27.5096	1.12401	0.562007	−	0.827133i	$-0.310030\pi$
$599$	27.5096	1.12401	0.562007	+	0.827133i	$0.310030\pi$
$600$	0	0
$601$	−14.0537	−0.573263	−0.286631	−	0.958041i	$-0.592536\pi$
$601$	−14.0537	−0.573263	−0.286631	+	0.958041i	$0.592536\pi$
$602$	−39.4230	−1.60676
$603$	−32.3689	−1.31816
$604$	3.81683	0.155305
$605$	0	0
$606$	25.0568	1.01786
$607$	−16.1942	−0.657304	−0.328652	−	0.944451i	$-0.606594\pi$
$607$	−16.1942	−0.657304	−0.328652	+	0.944451i	$0.606594\pi$
$608$	0	0
$609$	87.2071	3.53381
$610$	0	0
$611$	−25.3922	−1.02726
$612$	3.73654	0.151041
$613$	−27.6779	−1.11790	−0.558949	−	0.829202i	$-0.688795\pi$
$613$	−27.6779	−1.11790	−0.558949	+	0.829202i	$0.688795\pi$
$614$	17.2990	0.698129
$615$	0	0
$616$	26.7386	1.07733
$617$	−6.09559	−0.245399	−0.122700	−	0.992444i	$-0.539155\pi$
$617$	−6.09559	−0.245399	−0.122700	+	0.992444i	$0.539155\pi$
$618$	−36.3656	−1.46284
$619$	−11.7187	−0.471014	−0.235507	−	0.971873i	$-0.575675\pi$
$619$	−11.7187	−0.471014	−0.235507	+	0.971873i	$0.575675\pi$
$620$	0	0
$621$	2.43315	0.0976389
$622$	−39.8619	−1.59832
$623$	49.1677	1.96986
$624$	−30.0319	−1.20224
$625$	0	0
$626$	33.0677	1.32165
$627$	0	0
$628$	3.65419	0.145818
$629$	30.6999	1.22409
$630$	0	0
$631$	20.9211	0.832856	0.416428	−	0.909169i	$-0.363282\pi$
$631$	20.9211	0.832856	0.416428	+	0.909169i	$0.363282\pi$
$632$	4.69020	0.186566
$633$	−17.4181	−0.692308
$634$	−6.29738	−0.250101
$635$	0	0
$636$	−4.20372	−0.166689
$637$	25.3845	1.00577
$638$	−41.3112	−1.63553
$639$	7.37228	0.291643
$640$	0	0
$641$	9.48683	0.374707	0.187354	−	0.982293i	$-0.440009\pi$
$641$	9.48683	0.374707	0.187354	+	0.982293i	$0.440009\pi$
$642$	42.2859	1.66889
$643$	−12.5393	−0.494500	−0.247250	−	0.968952i	$-0.579527\pi$
$643$	−12.5393	−0.494500	−0.247250	+	0.968952i	$0.579527\pi$
$644$	−2.55959	−0.100862
$645$	0	0
$646$	0	0
$647$	15.6504	0.615281	0.307640	−	0.951503i	$-0.400461\pi$
$647$	15.6504	0.615281	0.307640	+	0.951503i	$0.400461\pi$
$648$	−25.8425	−1.01519
$649$	−9.99104	−0.392183
$650$	0	0
$651$	−14.2484	−0.558440
$652$	9.00096	0.352505
$653$	−19.0906	−0.747071	−0.373536	−	0.927616i	$-0.621855\pi$
$653$	−19.0906	−0.747071	−0.373536	+	0.927616i	$0.621855\pi$
$654$	−5.75632	−0.225090
$655$	0	0
$656$	−31.1631	−1.21671
$657$	22.2303	0.867287
$658$	57.4185	2.23841
$659$	10.4682	0.407783	0.203891	−	0.978994i	$-0.434641\pi$
$659$	10.4682	0.407783	0.203891	+	0.978994i	$0.434641\pi$
$660$	0	0
$661$	29.1420	1.13349	0.566745	−	0.823893i	$-0.308202\pi$
$661$	29.1420	1.13349	0.566745	+	0.823893i	$0.308202\pi$
$662$	−50.4687	−1.96152
$663$	22.9716	0.892143
$664$	−3.89858	−0.151294
$665$	0	0
$666$	−29.6936	−1.15060
$667$	−13.0038	−0.503507
$668$	4.37922	0.169437
$669$	−18.2877	−0.707044
$670$	0	0
$671$	−20.7163	−0.799745
$672$	23.7353	0.915608
$673$	−14.5550	−0.561052	−0.280526	−	0.959846i	$-0.590509\pi$
$673$	−14.5550	−0.561052	−0.280526	+	0.959846i	$0.590509\pi$
$674$	−36.8239	−1.41840
$675$	0	0
$676$	−2.43959	−0.0938302
$677$	−10.9800	−0.421995	−0.210998	−	0.977487i	$-0.567671\pi$
$677$	−10.9800	−0.421995	−0.210998	+	0.977487i	$0.567671\pi$
$678$	−46.4535	−1.78404
$679$	−46.5196	−1.78526
$680$	0	0
$681$	−40.5383	−1.55343
$682$	6.74968	0.258459
$683$	−43.3449	−1.65854	−0.829272	−	0.558844i	$-0.811245\pi$
$683$	−43.3449	−1.65854	−0.829272	+	0.558844i	$0.811245\pi$
$684$	0	0
$685$	0	0
$686$	−13.2809	−0.507066
$687$	−55.3503	−2.11174
$688$	−29.4928	−1.12440
$689$	−10.9948	−0.418867
$690$	0	0
$691$	30.4057	1.15669	0.578344	−	0.815793i	$-0.303699\pi$
$691$	30.4057	1.15669	0.578344	+	0.815793i	$0.303699\pi$
$692$	3.87606	0.147346
$693$	24.6605	0.936776
$694$	10.1979	0.387107
$695$	0	0
$696$	52.3347	1.98374
$697$	23.8368	0.902885
$698$	42.7181	1.61691
$699$	−31.1885	−1.17966
$700$	0	0
$701$	21.3688	0.807088	0.403544	−	0.914960i	$-0.367778\pi$
$701$	21.3688	0.807088	0.403544	+	0.914960i	$0.367778\pi$
$702$	7.78881	0.293970
$703$	0	0
$704$	14.8430	0.559416
$705$	0	0
$706$	−10.3043	−0.387806
$707$	−28.0229	−1.05391
$708$	−3.84913	−0.144659
$709$	−20.0113	−0.751541	−0.375770	−	0.926713i	$-0.622622\pi$
$709$	−20.0113	−0.751541	−0.375770	+	0.926713i	$0.622622\pi$
$710$	0	0
$711$	4.32568	0.162226
$712$	29.5065	1.10580
$713$	2.12463	0.0795682
$714$	−51.9449	−1.94399
$715$	0	0
$716$	8.64301	0.323005
$717$	8.54058	0.318954
$718$	4.29045	0.160118
$719$	−4.10144	−0.152958	−0.0764790	−	0.997071i	$-0.524368\pi$
$719$	−4.10144	−0.152958	−0.0764790	+	0.997071i	$0.524368\pi$
$720$	0	0
$721$	40.6705	1.51465
$722$	0	0
$723$	12.0682	0.448822
$724$	−5.28379	−0.196371
$725$	0	0
$726$	12.0149	0.445917
$727$	35.7495	1.32588	0.662938	−	0.748674i	$-0.269309\pi$
$727$	35.7495	1.32588	0.662938	+	0.748674i	$0.269309\pi$
$728$	26.9428	0.998567
$729$	−11.6367	−0.430990
$730$	0	0
$731$	22.5593	0.834384
$732$	−7.98113	−0.294991
$733$	31.2944	1.15589	0.577943	−	0.816077i	$-0.303855\pi$
$733$	31.2944	1.15589	0.577943	+	0.816077i	$0.303855\pi$
$734$	10.3322	0.381368
$735$	0	0
$736$	−3.53925	−0.130458
$737$	−40.3090	−1.48480
$738$	−23.0555	−0.848686
$739$	32.5127	1.19600	0.597999	−	0.801497i	$-0.295963\pi$
$739$	32.5127	1.19600	0.597999	+	0.801497i	$0.295963\pi$
$740$	0	0
$741$	0	0
$742$	24.8621	0.912716
$743$	38.8471	1.42516	0.712580	−	0.701590i	$-0.247526\pi$
$743$	38.8471	1.42516	0.712580	+	0.701590i	$0.247526\pi$
$744$	−8.55076	−0.313486
$745$	0	0
$746$	13.4431	0.492188
$747$	−3.59559	−0.131556
$748$	4.65312	0.170135
$749$	−47.2916	−1.72800
$750$	0	0
$751$	1.80781	0.0659681	0.0329840	−	0.999456i	$-0.489499\pi$
$751$	1.80781	0.0659681	0.0329840	+	0.999456i	$0.489499\pi$
$752$	42.9554	1.56642
$753$	−28.6644	−1.04459
$754$	−41.6266	−1.51595
$755$	0	0
$756$	−3.33049	−0.121129
$757$	−22.5632	−0.820074	−0.410037	−	0.912069i	$-0.634484\pi$
$757$	−22.5632	−0.820074	−0.410037	+	0.912069i	$0.634484\pi$
$758$	22.3757	0.812721
$759$	−8.64349	−0.313739
$760$	0	0
$761$	2.85080	0.103341	0.0516707	−	0.998664i	$-0.483545\pi$
$761$	2.85080	0.103341	0.0516707	+	0.998664i	$0.483545\pi$
$762$	29.7780	1.07874
$763$	6.43774	0.233062
$764$	5.54434	0.200587
$765$	0	0
$766$	−33.5553	−1.21240
$767$	−10.0673	−0.363510
$768$	24.2941	0.876639
$769$	6.11973	0.220683	0.110342	−	0.993894i	$-0.464806\pi$
$769$	6.11973	0.220683	0.110342	+	0.993894i	$0.464806\pi$
$770$	0	0
$771$	19.6508	0.707705
$772$	−0.446721	−0.0160778
$773$	50.1577	1.80405	0.902023	−	0.431688i	$-0.142082\pi$
$773$	50.1577	1.80405	0.902023	+	0.431688i	$0.142082\pi$
$774$	−21.8198	−0.784297
$775$	0	0
$776$	−27.9173	−1.00217
$777$	78.0588	2.80034
$778$	−24.0167	−0.861039
$779$	0	0
$780$	0	0
$781$	9.18070	0.328511
$782$	7.74569	0.276985
$783$	−16.9202	−0.604680
$784$	−42.9424	−1.53366
$785$	0	0
$786$	23.3682	0.833515
$787$	−33.0203	−1.17705	−0.588524	−	0.808480i	$-0.700291\pi$
$787$	−33.0203	−1.17705	−0.588524	+	0.808480i	$0.700291\pi$
$788$	−6.65889	−0.237213
$789$	−60.9879	−2.17123
$790$	0	0
$791$	51.9525	1.84722
$792$	14.7993	0.525869
$793$	−20.8745	−0.741275
$794$	51.2006	1.81704
$795$	0	0
$796$	−3.62182	−0.128372
$797$	47.2358	1.67318	0.836590	−	0.547830i	$-0.184546\pi$
$797$	47.2358	1.67318	0.836590	+	0.547830i	$0.184546\pi$
$798$	0	0
$799$	−32.8569	−1.16239
$800$	0	0
$801$	27.2132	0.961533
$802$	−27.1879	−0.960039
$803$	27.6834	0.976927
$804$	−15.5294	−0.547679
$805$	0	0
$806$	6.80122	0.239563
$807$	7.54788	0.265698
$808$	−16.8171	−0.591623
$809$	4.38236	0.154076	0.0770378	−	0.997028i	$-0.475454\pi$
$809$	4.38236	0.154076	0.0770378	+	0.997028i	$0.475454\pi$
$810$	0	0
$811$	33.9613	1.19254	0.596271	−	0.802783i	$-0.296649\pi$
$811$	33.9613	1.19254	0.596271	+	0.802783i	$0.296649\pi$
$812$	17.7995	0.624640
$813$	−42.0159	−1.47356
$814$	−36.9775	−1.29606
$815$	0	0
$816$	−38.8606	−1.36039
$817$	0	0
$818$	−17.0038	−0.594525
$819$	24.8488	0.868288
$820$	0	0
$821$	40.1364	1.40077	0.700385	−	0.713766i	$-0.253012\pi$
$821$	40.1364	1.40077	0.700385	+	0.713766i	$0.253012\pi$
$822$	53.3570	1.86104
$823$	−6.31582	−0.220156	−0.110078	−	0.993923i	$-0.535110\pi$
$823$	−6.31582	−0.220156	−0.110078	+	0.993923i	$0.535110\pi$
$824$	24.4072	0.850264
$825$	0	0
$826$	22.7649	0.792093
$827$	21.3614	0.742808	0.371404	−	0.928471i	$-0.378877\pi$
$827$	21.3614	0.742808	0.371404	+	0.928471i	$0.378877\pi$
$828$	−1.41668	−0.0492330
$829$	−49.0602	−1.70393	−0.851966	−	0.523597i	$-0.824590\pi$
$829$	−49.0602	−1.70393	−0.851966	+	0.523597i	$0.824590\pi$
$830$	0	0
$831$	−61.1898	−2.12265
$832$	14.9563	0.518517
$833$	32.8469	1.13808
$834$	−35.7357	−1.23743
$835$	0	0
$836$	0	0
$837$	2.76453	0.0955562
$838$	−53.3588	−1.84325
$839$	7.38997	0.255130	0.127565	−	0.991830i	$-0.459284\pi$
$839$	7.38997	0.255130	0.127565	+	0.991830i	$0.459284\pi$
$840$	0	0
$841$	61.4287	2.11823
$842$	13.0479	0.449660
$843$	−49.2334	−1.69569
$844$	−3.55514	−0.122373
$845$	0	0
$846$	31.7799	1.09262
$847$	−13.4372	−0.461709
$848$	18.5996	0.638713
$849$	−41.9235	−1.43881
$850$	0	0
$851$	−11.6396	−0.399001
$852$	3.53694	0.121174
$853$	−2.75310	−0.0942642	−0.0471321	−	0.998889i	$-0.515008\pi$
$853$	−2.75310	−0.0942642	−0.0471321	+	0.998889i	$0.515008\pi$
$854$	47.2028	1.61525
$855$	0	0
$856$	−28.3806	−0.970029
$857$	−44.1261	−1.50732	−0.753659	−	0.657266i	$-0.771713\pi$
$857$	−44.1261	−1.50732	−0.753659	+	0.657266i	$0.771713\pi$
$858$	−27.6689	−0.944601
$859$	−11.7531	−0.401010	−0.200505	−	0.979693i	$-0.564258\pi$
$859$	−11.7531	−0.401010	−0.200505	+	0.979693i	$0.564258\pi$
$860$	0	0
$861$	60.6085	2.06553
$862$	−19.0495	−0.648828
$863$	−34.2831	−1.16701	−0.583504	−	0.812110i	$-0.698319\pi$
$863$	−34.2831	−1.16701	−0.583504	+	0.812110i	$0.698319\pi$
$864$	−4.60520	−0.156672
$865$	0	0
$866$	8.53617	0.290071
$867$	−9.12061	−0.309752
$868$	−2.90820	−0.0987106
$869$	5.38678	0.182734
$870$	0	0
$871$	−40.6168	−1.37625
$872$	3.86341	0.130832
$873$	−25.7476	−0.871423
$874$	0	0
$875$	0	0
$876$	10.6653	0.360346
$877$	−24.2041	−0.817313	−0.408656	−	0.912688i	$-0.634003\pi$
$877$	−24.2041	−0.817313	−0.408656	+	0.912688i	$0.634003\pi$
$878$	−39.8597	−1.34520
$879$	−21.2249	−0.715898
$880$	0	0
$881$	−39.1733	−1.31978	−0.659892	−	0.751361i	$-0.729398\pi$
$881$	−39.1733	−1.31978	−0.659892	+	0.751361i	$0.729398\pi$
$882$	−31.7703	−1.06976
$883$	−32.3695	−1.08932	−0.544660	−	0.838657i	$-0.683341\pi$
$883$	−32.3695	−1.08932	−0.544660	+	0.838657i	$0.683341\pi$
$884$	4.68865	0.157696
$885$	0	0
$886$	−7.72870	−0.259651
$887$	0.380819	0.0127866	0.00639332	−	0.999980i	$-0.497965\pi$
$887$	0.380819	0.0127866	0.00639332	+	0.999980i	$0.497965\pi$
$888$	46.8446	1.57200
$889$	−33.3030	−1.11695
$890$	0	0
$891$	−29.6805	−0.994336
$892$	−3.73264	−0.124978
$893$	0	0
$894$	5.13747	0.171823
$895$	0	0
$896$	−54.5949	−1.82389
$897$	−8.70949	−0.290801
$898$	6.38632	0.213114
$899$	−14.7748	−0.492767
$900$	0	0
$901$	−14.2270	−0.473969
$902$	−28.7111	−0.955974
$903$	57.3600	1.90882
$904$	31.1777	1.03696
$905$	0	0
$906$	−29.3682	−0.975692
$907$	−4.92616	−0.163571	−0.0817853	−	0.996650i	$-0.526062\pi$
$907$	−4.92616	−0.163571	−0.0817853	+	0.996650i	$0.526062\pi$
$908$	−8.27411	−0.274586
$909$	−15.5101	−0.514437
$910$	0	0
$911$	−20.6775	−0.685075	−0.342537	−	0.939504i	$-0.611286\pi$
$911$	−20.6775	−0.685075	−0.342537	+	0.939504i	$0.611286\pi$
$912$	0	0
$913$	−4.47759	−0.148187
$914$	−21.2826	−0.703965
$915$	0	0
$916$	−11.2973	−0.373274
$917$	−26.1344	−0.863035
$918$	10.0785	0.332641
$919$	−60.5868	−1.99857	−0.999287	−	0.0377591i	$-0.987978\pi$
$919$	−60.5868	−1.99857	−0.999287	+	0.0377591i	$0.987978\pi$
$920$	0	0
$921$	−25.1698	−0.829372
$922$	−28.0820	−0.924831
$923$	9.25080	0.304494
$924$	11.8312	0.389218
$925$	0	0
$926$	2.05254	0.0674505
$927$	22.5103	0.739334
$928$	24.6121	0.807932
$929$	−0.290620	−0.00953494	−0.00476747	−	0.999989i	$-0.501518\pi$
$929$	−0.290620	−0.00953494	−0.00476747	+	0.999989i	$0.501518\pi$
$930$	0	0
$931$	0	0
$932$	−6.36576	−0.208517
$933$	57.9986	1.89879
$934$	42.4446	1.38883
$935$	0	0
$936$	14.9123	0.487423
$937$	−5.30585	−0.173335	−0.0866673	−	0.996237i	$-0.527622\pi$
$937$	−5.30585	−0.173335	−0.0866673	+	0.996237i	$0.527622\pi$
$938$	91.8454	2.99886
$939$	−48.1131	−1.57011
$940$	0	0
$941$	28.4255	0.926646	0.463323	−	0.886189i	$-0.346657\pi$
$941$	28.4255	0.926646	0.463323	+	0.886189i	$0.346657\pi$
$942$	−28.1168	−0.916095
$943$	−9.03754	−0.294303
$944$	17.0307	0.554302
$945$	0	0
$946$	−27.1722	−0.883446
$947$	49.9790	1.62410	0.812049	−	0.583589i	$-0.198352\pi$
$947$	49.9790	1.62410	0.812049	+	0.583589i	$0.198352\pi$
$948$	2.07530	0.0674026
$949$	27.8948	0.905504
$950$	0	0
$951$	9.16261	0.297118
$952$	34.8634	1.12993
$953$	−12.1003	−0.391966	−0.195983	−	0.980607i	$-0.562790\pi$
$953$	−12.1003	−0.391966	−0.195983	+	0.980607i	$0.562790\pi$
$954$	13.7606	0.445517
$955$	0	0
$956$	1.74319	0.0563787
$957$	60.1073	1.94299
$958$	−0.454611	−0.0146878
$959$	−59.6732	−1.92695
$960$	0	0
$961$	−28.5860	−0.922129
$962$	−37.2599	−1.20131
$963$	−26.1749	−0.843474
$964$	2.46320	0.0793344
$965$	0	0
$966$	19.6945	0.633660
$967$	−13.1688	−0.423480	−0.211740	−	0.977326i	$-0.567913\pi$
$967$	−13.1688	−0.423480	−0.211740	+	0.977326i	$0.567913\pi$
$968$	−8.06395	−0.259185
$969$	0	0
$970$	0	0
$971$	17.5135	0.562035	0.281018	−	0.959703i	$-0.409328\pi$
$971$	17.5135	0.562035	0.281018	+	0.959703i	$0.409328\pi$
$972$	−8.94513	−0.286915
$973$	39.9660	1.28125
$974$	−24.1399	−0.773491
$975$	0	0
$976$	35.3129	1.13034
$977$	−10.5361	−0.337081	−0.168541	−	0.985695i	$-0.553905\pi$
$977$	−10.5361	−0.337081	−0.168541	+	0.985695i	$0.553905\pi$
$978$	−69.2569	−2.21459
$979$	33.8887	1.08309
$980$	0	0
$981$	3.56315	0.113763
$982$	−22.7878	−0.727188
$983$	51.9781	1.65784	0.828922	−	0.559364i	$-0.188954\pi$
$983$	51.9781	1.65784	0.828922	+	0.559364i	$0.188954\pi$
$984$	36.3723	1.15951
$985$	0	0
$986$	−53.8639	−1.71538
$987$	−83.5433	−2.65921
$988$	0	0
$989$	−8.55315	−0.271974
$990$	0	0
$991$	−6.77628	−0.215256	−0.107628	−	0.994191i	$-0.534326\pi$
$991$	−6.77628	−0.215256	−0.107628	+	0.994191i	$0.534326\pi$
$992$	−4.02128	−0.127676
$993$	73.4313	2.33027
$994$	−20.9185	−0.663496
$995$	0	0
$996$	−1.72503	−0.0546596
$997$	40.2693	1.27534	0.637671	−	0.770309i	$-0.279898\pi$
$997$	40.2693	1.27534	0.637671	+	0.770309i	$0.279898\pi$
$998$	23.1455	0.732659
$999$	−15.1452	−0.479174

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.cd.1.2		9
5.4	even	2		1805.2.a.u.1.8		9
19.14	odd	18		475.2.l.b.101.3		18
19.15	odd	18		475.2.l.b.301.3		18
19.18	odd	2		9025.2.a.ce.1.8		9
95.14	odd	18		95.2.k.b.6.1	✓	18
95.33	even	36		475.2.u.c.424.2		36
95.34	odd	18		95.2.k.b.16.1	yes	18
95.52	even	36		475.2.u.c.424.5		36
95.53	even	36		475.2.u.c.149.5		36
95.72	even	36		475.2.u.c.149.2		36
95.94	odd	2		1805.2.a.t.1.2		9
285.14	even	18		855.2.bs.b.766.3		18
285.224	even	18		855.2.bs.b.586.3		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.k.b.6.1	✓	18	95.14	odd	18
95.2.k.b.16.1	yes	18	95.34	odd	18
475.2.l.b.101.3		18	19.14	odd	18
475.2.l.b.301.3		18	19.15	odd	18
475.2.u.c.149.2		36	95.72	even	36
475.2.u.c.149.5		36	95.53	even	36
475.2.u.c.424.2		36	95.33	even	36
475.2.u.c.424.5		36	95.52	even	36
855.2.bs.b.586.3		18	285.224	even	18
855.2.bs.b.766.3		18	285.14	even	18
1805.2.a.t.1.2		9	95.94	odd	2
1805.2.a.u.1.8		9	5.4	even	2
9025.2.a.cd.1.2		9	1.1	even	1	trivial
9025.2.a.ce.1.8		9	19.18	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)

\(f(q)\)	\(=\)	\(q-1.57047 q^{2} +2.28502 q^{3} +0.466387 q^{4} -3.58856 q^{6} +4.01337 q^{7} +2.40850 q^{8} +2.22131 q^{9} +O(q^{10})\)	\(q-1.57047 q^{2} +2.28502 q^{3} +0.466387 q^{4} -3.58856 q^{6} +4.01337 q^{7} +2.40850 q^{8} +2.22131 q^{9} +2.76620 q^{11} +1.06570 q^{12} +2.78732 q^{13} -6.30289 q^{14} -4.71526 q^{16} +3.60673 q^{17} -3.48851 q^{18} +9.17062 q^{21} -4.34425 q^{22} -1.36746 q^{23} +5.50347 q^{24} -4.37742 q^{26} -1.77932 q^{27} +1.87178 q^{28} +9.50940 q^{29} -1.55371 q^{31} +2.58819 q^{32} +6.32083 q^{33} -5.66428 q^{34} +1.03599 q^{36} +8.51183 q^{37} +6.36909 q^{39} +6.60899 q^{41} -14.4022 q^{42} +6.25476 q^{43} +1.29012 q^{44} +2.14756 q^{46} -9.10988 q^{47} -10.7745 q^{48} +9.10711 q^{49} +8.24145 q^{51} +1.29997 q^{52} -3.94456 q^{53} +2.79437 q^{54} +9.66619 q^{56} -14.9343 q^{58} -3.61183 q^{59} -7.48908 q^{61} +2.44005 q^{62} +8.91494 q^{63} +5.36583 q^{64} -9.92669 q^{66} -14.5720 q^{67} +1.68213 q^{68} -3.12468 q^{69} +3.31888 q^{71} +5.35003 q^{72} +10.0077 q^{73} -13.3676 q^{74} +11.1018 q^{77} -10.0025 q^{78} +1.94735 q^{79} -10.7297 q^{81} -10.3792 q^{82} -1.61868 q^{83} +4.27705 q^{84} -9.82294 q^{86} +21.7292 q^{87} +6.66240 q^{88} +12.2510 q^{89} +11.1866 q^{91} -0.637766 q^{92} -3.55025 q^{93} +14.3068 q^{94} +5.91406 q^{96} -11.5912 q^{97} -14.3025 q^{98} +6.14460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{3} + 6 q^{4} - 12 q^{6} + 12 q^{8} + 6 q^{9}+O(q^{10}) \) 9 * q - 3 * q^3 + 6 * q^4 - 12 * q^6 + 12 * q^8 + 6 * q^9	\( 9 q - 3 q^{3} + 6 q^{4} - 12 q^{6} + 12 q^{8} + 6 q^{9} - 6 q^{12} + 3 q^{13} + 12 q^{14} - 12 q^{16} + 9 q^{17} - 6 q^{18} + 12 q^{21} - 12 q^{22} + 15 q^{24} + 21 q^{26} - 6 q^{27} + 15 q^{28} + 15 q^{29} + 30 q^{31} + 9 q^{32} - 9 q^{33} - 6 q^{36} - 30 q^{37} + 6 q^{39} + 18 q^{41} - 36 q^{42} + 6 q^{43} - 24 q^{44} + 21 q^{46} - 21 q^{47} - 15 q^{48} + 3 q^{49} + 18 q^{51} + 3 q^{52} + 9 q^{53} - 9 q^{54} + 36 q^{56} - 18 q^{58} + 27 q^{59} + 12 q^{61} + 6 q^{62} + 15 q^{63} + 24 q^{64} + 3 q^{66} - 36 q^{67} - 3 q^{68} + 27 q^{69} - 6 q^{71} - 12 q^{72} + 9 q^{73} - 9 q^{74} - 12 q^{77} - 54 q^{78} + 45 q^{79} - 15 q^{81} + 48 q^{82} - 12 q^{84} - 9 q^{86} - 45 q^{87} - 39 q^{88} - 9 q^{89} + 51 q^{91} + 54 q^{92} - 9 q^{93} + 33 q^{94} - 9 q^{96} - 45 q^{97} - 33 q^{98} + 12 q^{99}+O(q^{100}) \) 9 * q - 3 * q^3 + 6 * q^4 - 12 * q^6 + 12 * q^8 + 6 * q^9 - 6 * q^12 + 3 * q^13 + 12 * q^14 - 12 * q^16 + 9 * q^17 - 6 * q^18 + 12 * q^21 - 12 * q^22 + 15 * q^24 + 21 * q^26 - 6 * q^27 + 15 * q^28 + 15 * q^29 + 30 * q^31 + 9 * q^32 - 9 * q^33 - 6 * q^36 - 30 * q^37 + 6 * q^39 + 18 * q^41 - 36 * q^42 + 6 * q^43 - 24 * q^44 + 21 * q^46 - 21 * q^47 - 15 * q^48 + 3 * q^49 + 18 * q^51 + 3 * q^52 + 9 * q^53 - 9 * q^54 + 36 * q^56 - 18 * q^58 + 27 * q^59 + 12 * q^61 + 6 * q^62 + 15 * q^63 + 24 * q^64 + 3 * q^66 - 36 * q^67 - 3 * q^68 + 27 * q^69 - 6 * q^71 - 12 * q^72 + 9 * q^73 - 9 * q^74 - 12 * q^77 - 54 * q^78 + 45 * q^79 - 15 * q^81 + 48 * q^82 - 12 * q^84 - 9 * q^86 - 45 * q^87 - 39 * q^88 - 9 * q^89 + 51 * q^91 + 54 * q^92 - 9 * q^93 + 33 * q^94 - 9 * q^96 - 45 * q^97 - 33 * q^98 + 12 * q^99

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