LMFDB - Embedded newform 3249.2.a.ba.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3249,2,Mod(1,3249)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3249.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3249, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,2,-2,0,-2,0,0,0,-8,0,0,0,0,-14,8,0,0,-6,0,0,24,0,-14,20, 0,-6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,-24,6,0,0,-38,0,-22,0,0,0,0,0,-6,0,0, -20,0,0,-20,-20,0,8,0,0,0,-6,0,0,0,0,-24,-40,0,0,-6,0,0,12,0,0,4,0,6,0, 0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.9433956167$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.90211$ of defining polynomial
Character	$\chi$	$=$	3249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.90211 q^{2} +1.61803 q^{4} -1.61803 q^{5} -1.61803 q^{7} +0.726543 q^{8} +3.07768 q^{10} +0.236068 q^{11} -4.97980 q^{13} +3.07768 q^{14} -4.61803 q^{16} -0.236068 q^{17} -2.61803 q^{20} -0.449028 q^{22} +8.23607 q^{23} -2.38197 q^{25} +9.47214 q^{26} -2.61803 q^{28} +8.50651 q^{29} +7.33094 q^{31} +7.33094 q^{32} +0.449028 q^{34} +2.61803 q^{35} +5.25731 q^{37} -1.17557 q^{40} -3.52671 q^{41} -8.23607 q^{43} +0.381966 q^{44} -15.6659 q^{46} -8.38197 q^{47} -4.38197 q^{49} +4.53077 q^{50} -8.05748 q^{52} +12.1392 q^{53} -0.381966 q^{55} -1.17557 q^{56} -16.1803 q^{58} -13.2088 q^{59} -2.76393 q^{61} -13.9443 q^{62} -4.70820 q^{64} +8.05748 q^{65} +8.95554 q^{67} -0.381966 q^{68} -4.97980 q^{70} +8.78402 q^{71} +0.708204 q^{73} -10.0000 q^{74} -0.381966 q^{77} -0.171513 q^{79} +7.47214 q^{80} +6.70820 q^{82} +9.94427 q^{83} +0.381966 q^{85} +15.6659 q^{86} +0.171513 q^{88} +0.898056 q^{89} +8.05748 q^{91} +13.3262 q^{92} +15.9434 q^{94} +12.3107 q^{97} +8.33499 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{5} - 2 q^{7} - 8 q^{11} - 14 q^{16} + 8 q^{17} - 6 q^{20} + 24 q^{23} - 14 q^{25} + 20 q^{26} - 6 q^{28} + 6 q^{35} - 24 q^{43} + 6 q^{44} - 38 q^{47} - 22 q^{49} - 6 q^{55} - 20 q^{58}+ \cdots + 22 q^{92}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^5 - 2 * q^7 - 8 * q^11 - 14 * q^16 + 8 * q^17 - 6 * q^20 + 24 * q^23 - 14 * q^25 + 20 * q^26 - 6 * q^28 + 6 * q^35 - 24 * q^43 + 6 * q^44 - 38 * q^47 - 22 * q^49 - 6 * q^55 - 20 * q^58 - 20 * q^61 - 20 * q^62 + 8 * q^64 - 6 * q^68 - 24 * q^73 - 40 * q^74 - 6 * q^77 + 12 * q^80 + 4 * q^83 + 6 * q^85 + 22 * q^92

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 $

$2$	−1.90211	−1.34500	−0.672499	−	0.740098i	$-0.734779\pi$
$2$	−1.90211	−1.34500	−0.672499	+	0.740098i	$0.734779\pi$
$3$	0	0
$3$	0	0
$4$	1.61803	0.809017
$5$	−1.61803	−0.723607	−0.361803	−	0.932254i	$-0.617839\pi$
$5$	−1.61803	−0.723607	−0.361803	+	0.932254i	$0.617839\pi$
$6$	0	0
$7$	−1.61803	−0.611559	−0.305780	−	0.952102i	$-0.598917\pi$
$7$	−1.61803	−0.611559	−0.305780	+	0.952102i	$0.598917\pi$
$8$	0.726543	0.256872
$9$	0	0
$10$	3.07768	0.973249
$11$	0.236068	0.0711772	0.0355886	−	0.999367i	$-0.488669\pi$
$11$	0.236068	0.0711772	0.0355886	+	0.999367i	$0.488669\pi$
$12$	0	0
$13$	−4.97980	−1.38115	−0.690574	−	0.723262i	$-0.742642\pi$
$13$	−4.97980	−1.38115	−0.690574	+	0.723262i	$0.742642\pi$
$14$	3.07768	0.822546
$15$	0	0
$16$	−4.61803	−1.15451
$17$	−0.236068	−0.0572549	−0.0286274	−	0.999590i	$-0.509114\pi$
$17$	−0.236068	−0.0572549	−0.0286274	+	0.999590i	$0.509114\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−2.61803	−0.585410
$21$	0	0
$22$	−0.449028	−0.0957331
$23$	8.23607	1.71734	0.858669	−	0.512530i	$-0.171292\pi$
$23$	8.23607	1.71734	0.858669	+	0.512530i	$0.171292\pi$
$24$	0	0
$25$	−2.38197	−0.476393
$26$	9.47214	1.85764
$27$	0	0
$28$	−2.61803	−0.494762
$29$	8.50651	1.57962	0.789809	−	0.613352i	$-0.210179\pi$
$29$	8.50651	1.57962	0.789809	+	0.613352i	$0.210179\pi$
$30$	0	0
$31$	7.33094	1.31668	0.658338	−	0.752723i	$-0.271260\pi$
$31$	7.33094	1.31668	0.658338	+	0.752723i	$0.271260\pi$
$32$	7.33094	1.29594
$33$	0	0
$34$	0.449028	0.0770077
$35$	2.61803	0.442529
$36$	0	0
$37$	5.25731	0.864297	0.432148	−	0.901803i	$-0.357756\pi$
$37$	5.25731	0.864297	0.432148	+	0.901803i	$0.357756\pi$
$38$	0	0
$39$	0	0
$40$	−1.17557	−0.185874
$41$	−3.52671	−0.550780	−0.275390	−	0.961333i	$-0.588807\pi$
$41$	−3.52671	−0.550780	−0.275390	+	0.961333i	$0.588807\pi$
$42$	0	0
$43$	−8.23607	−1.25599	−0.627994	−	0.778218i	$-0.716124\pi$
$43$	−8.23607	−1.25599	−0.627994	+	0.778218i	$0.716124\pi$
$44$	0.381966	0.0575835
$45$	0	0
$46$	−15.6659	−2.30982
$47$	−8.38197	−1.22264	−0.611318	−	0.791385i	$-0.709360\pi$
$47$	−8.38197	−1.22264	−0.611318	+	0.791385i	$0.709360\pi$
$48$	0	0
$49$	−4.38197	−0.625995
$50$	4.53077	0.640747
$51$	0	0
$52$	−8.05748	−1.11737
$53$	12.1392	1.66745	0.833725	−	0.552180i	$-0.186204\pi$
$53$	12.1392	1.66745	0.833725	+	0.552180i	$0.186204\pi$
$54$	0	0
$55$	−0.381966	−0.0515043
$56$	−1.17557	−0.157092
$57$	0	0
$58$	−16.1803	−2.12458
$59$	−13.2088	−1.71964	−0.859819	−	0.510599i	$-0.829424\pi$
$59$	−13.2088	−1.71964	−0.859819	+	0.510599i	$0.829424\pi$
$60$	0	0
$61$	−2.76393	−0.353885	−0.176943	−	0.984221i	$-0.556621\pi$
$61$	−2.76393	−0.353885	−0.176943	+	0.984221i	$0.556621\pi$
$62$	−13.9443	−1.77092
$63$	0	0
$64$	−4.70820	−0.588525
$65$	8.05748	0.999407
$66$	0	0
$67$	8.95554	1.09409	0.547046	−	0.837102i	$-0.315752\pi$
$67$	8.95554	1.09409	0.547046	+	0.837102i	$0.315752\pi$
$68$	−0.381966	−0.0463202
$69$	0	0
$70$	−4.97980	−0.595200
$71$	8.78402	1.04247	0.521236	−	0.853413i	$-0.325471\pi$
$71$	8.78402	1.04247	0.521236	+	0.853413i	$0.325471\pi$
$72$	0	0
$73$	0.708204	0.0828890	0.0414445	−	0.999141i	$-0.486804\pi$
$73$	0.708204	0.0828890	0.0414445	+	0.999141i	$0.486804\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	0	0
$77$	−0.381966	−0.0435291
$78$	0	0
$79$	−0.171513	−0.0192968	−0.00964838	−	0.999953i	$-0.503071\pi$
$79$	−0.171513	−0.0192968	−0.00964838	+	0.999953i	$0.503071\pi$
$80$	7.47214	0.835410
$81$	0	0
$82$	6.70820	0.740797
$83$	9.94427	1.09153	0.545763	−	0.837940i	$-0.316240\pi$
$83$	9.94427	1.09153	0.545763	+	0.837940i	$0.316240\pi$
$84$	0	0
$85$	0.381966	0.0414300
$86$	15.6659	1.68930
$87$	0	0
$88$	0.171513	0.0182834
$89$	0.898056	0.0951937	0.0475969	−	0.998867i	$-0.484844\pi$
$89$	0.898056	0.0951937	0.0475969	+	0.998867i	$0.484844\pi$
$90$	0	0
$91$	8.05748	0.844653
$92$	13.3262	1.38936
$93$	0	0
$94$	15.9434	1.64444
$95$	0	0
$96$	0	0
$97$	12.3107	1.24997	0.624983	−	0.780638i	$-0.285106\pi$
$97$	12.3107	1.24997	0.624983	+	0.780638i	$0.285106\pi$
$98$	8.33499	0.841962
$99$	0	0
$100$	−3.85410	−0.385410
$101$	−13.6525	−1.35847	−0.679236	−	0.733920i	$-0.737689\pi$
$101$	−13.6525	−1.35847	−0.679236	+	0.733920i	$0.737689\pi$
$102$	0	0
$103$	5.15131	0.507574	0.253787	−	0.967260i	$-0.418324\pi$
$103$	5.15131	0.507574	0.253787	+	0.967260i	$0.418324\pi$
$104$	−3.61803	−0.354777
$105$	0	0
$106$	−23.0902	−2.24272
$107$	1.62460	0.157056	0.0785279	−	0.996912i	$-0.474978\pi$
$107$	1.62460	0.157056	0.0785279	+	0.996912i	$0.474978\pi$
$108$	0	0
$109$	−15.3884	−1.47394	−0.736972	−	0.675924i	$-0.763745\pi$
$109$	−15.3884	−1.47394	−0.736972	+	0.675924i	$0.763745\pi$
$110$	0.726543	0.0692731
$111$	0	0
$112$	7.47214	0.706050
$113$	−12.7598	−1.20034	−0.600169	−	0.799873i	$-0.704900\pi$
$113$	−12.7598	−1.20034	−0.600169	+	0.799873i	$0.704900\pi$
$114$	0	0
$115$	−13.3262	−1.24268
$116$	13.7638	1.27794
$117$	0	0
$118$	25.1246	2.31291
$119$	0.381966	0.0350148
$120$	0	0
$121$	−10.9443	−0.994934
$122$	5.25731	0.475975
$123$	0	0
$124$	11.8617	1.06521
$125$	11.9443	1.06833
$126$	0	0
$127$	−7.60845	−0.675141	−0.337570	−	0.941300i	$-0.609605\pi$
$127$	−7.60845	−0.675141	−0.337570	+	0.941300i	$0.609605\pi$
$128$	−5.70634	−0.504374
$129$	0	0
$130$	−15.3262	−1.34420
$131$	7.76393	0.678338	0.339169	−	0.940725i	$-0.389854\pi$
$131$	7.76393	0.678338	0.339169	+	0.940725i	$0.389854\pi$
$132$	0	0
$133$	0	0
$134$	−17.0344	−1.47155
$135$	0	0
$136$	−0.171513	−0.0147072
$137$	2.14590	0.183336	0.0916682	−	0.995790i	$-0.470780\pi$
$137$	2.14590	0.183336	0.0916682	+	0.995790i	$0.470780\pi$
$138$	0	0
$139$	0.180340	0.0152962	0.00764811	−	0.999971i	$-0.497566\pi$
$139$	0.180340	0.0152962	0.00764811	+	0.999971i	$0.497566\pi$
$140$	4.23607	0.358013
$141$	0	0
$142$	−16.7082	−1.40212
$143$	−1.17557	−0.0983061
$144$	0	0
$145$	−13.7638	−1.14302
$146$	−1.34708	−0.111485
$147$	0	0
$148$	8.50651	0.699231
$149$	1.38197	0.113215	0.0566075	−	0.998397i	$-0.481972\pi$
$149$	1.38197	0.113215	0.0566075	+	0.998397i	$0.481972\pi$
$150$	0	0
$151$	−11.7557	−0.956666	−0.478333	−	0.878179i	$-0.658759\pi$
$151$	−11.7557	−0.956666	−0.478333	+	0.878179i	$0.658759\pi$
$152$	0	0
$153$	0	0
$154$	0.726543	0.0585465
$155$	−11.8617	−0.952755
$156$	0	0
$157$	20.4164	1.62941	0.814703	−	0.579878i	$-0.196900\pi$
$157$	20.4164	1.62941	0.814703	+	0.579878i	$0.196900\pi$
$158$	0.326238	0.0259541
$159$	0	0
$160$	−11.8617	−0.937750
$161$	−13.3262	−1.05025
$162$	0	0
$163$	−0.673762	−0.0527731	−0.0263866	−	0.999652i	$-0.508400\pi$
$163$	−0.673762	−0.0527731	−0.0263866	+	0.999652i	$0.508400\pi$
$164$	−5.70634	−0.445590
$165$	0	0
$166$	−18.9151	−1.46810
$167$	10.8576	0.840190	0.420095	−	0.907480i	$-0.361997\pi$
$167$	10.8576	0.840190	0.420095	+	0.907480i	$0.361997\pi$
$168$	0	0
$169$	11.7984	0.907567
$170$	−0.726543	−0.0557233
$171$	0	0
$172$	−13.3262	−1.01612
$173$	−16.6700	−1.26740	−0.633698	−	0.773581i	$-0.718464\pi$
$173$	−16.6700	−1.26740	−0.633698	+	0.773581i	$0.718464\pi$
$174$	0	0
$175$	3.85410	0.291343
$176$	−1.09017	−0.0821747
$177$	0	0
$178$	−1.70820	−0.128035
$179$	1.90211	0.142171	0.0710853	−	0.997470i	$-0.477354\pi$
$179$	1.90211	0.142171	0.0710853	+	0.997470i	$0.477354\pi$
$180$	0	0
$181$	0.620541	0.0461245	0.0230622	−	0.999734i	$-0.492658\pi$
$181$	0.620541	0.0461245	0.0230622	+	0.999734i	$0.492658\pi$
$182$	−15.3262	−1.13606
$183$	0	0
$184$	5.98385	0.441136
$185$	−8.50651	−0.625411
$186$	0	0
$187$	−0.0557281	−0.00407524
$188$	−13.5623	−0.989133
$189$	0	0
$190$	0	0
$191$	18.4164	1.33256	0.666282	−	0.745700i	$-0.267885\pi$
$191$	18.4164	1.33256	0.666282	+	0.745700i	$0.267885\pi$
$192$	0	0
$193$	−8.50651	−0.612312	−0.306156	−	0.951981i	$-0.599043\pi$
$193$	−8.50651	−0.612312	−0.306156	+	0.951981i	$0.599043\pi$
$194$	−23.4164	−1.68120
$195$	0	0
$196$	−7.09017	−0.506441
$197$	−26.7426	−1.90533	−0.952667	−	0.304015i	$-0.901673\pi$
$197$	−26.7426	−1.90533	−0.952667	+	0.304015i	$0.901673\pi$
$198$	0	0
$199$	−21.1803	−1.50143	−0.750717	−	0.660624i	$-0.770292\pi$
$199$	−21.1803	−1.50143	−0.750717	+	0.660624i	$0.770292\pi$
$200$	−1.73060	−0.122372
$201$	0	0
$202$	25.9686	1.82714
$203$	−13.7638	−0.966031
$204$	0	0
$205$	5.70634	0.398548
$206$	−9.79837	−0.682685
$207$	0	0
$208$	22.9969	1.59455
$209$	0	0
$210$	0	0
$211$	7.77997	0.535595	0.267797	−	0.963475i	$-0.413704\pi$
$211$	7.77997	0.535595	0.267797	+	0.963475i	$0.413704\pi$
$212$	19.6417	1.34900
$213$	0	0
$214$	−3.09017	−0.211240
$215$	13.3262	0.908842
$216$	0	0
$217$	−11.8617	−0.805225
$218$	29.2705	1.98245
$219$	0	0
$220$	−0.618034	−0.0416678
$221$	1.17557	0.0790774
$222$	0	0
$223$	−14.9394	−1.00042	−0.500208	−	0.865905i	$-0.666743\pi$
$223$	−14.9394	−1.00042	−0.500208	+	0.865905i	$0.666743\pi$
$224$	−11.8617	−0.792544
$225$	0	0
$226$	24.2705	1.61445
$227$	8.67802	0.575981	0.287990	−	0.957633i	$-0.407013\pi$
$227$	8.67802	0.575981	0.287990	+	0.957633i	$0.407013\pi$
$228$	0	0
$229$	−21.6525	−1.43084	−0.715418	−	0.698697i	$-0.753764\pi$
$229$	−21.6525	−1.43084	−0.715418	+	0.698697i	$0.753764\pi$
$230$	25.3480	1.67140
$231$	0	0
$232$	6.18034	0.405759
$233$	11.8885	0.778844	0.389422	−	0.921059i	$-0.372675\pi$
$233$	11.8885	0.778844	0.389422	+	0.921059i	$0.372675\pi$
$234$	0	0
$235$	13.5623	0.884707
$236$	−21.3723	−1.39122
$237$	0	0
$238$	−0.726543	−0.0470948
$239$	6.85410	0.443355	0.221677	−	0.975120i	$-0.428847\pi$
$239$	6.85410	0.443355	0.221677	+	0.975120i	$0.428847\pi$
$240$	0	0
$241$	−4.08174	−0.262928	−0.131464	−	0.991321i	$-0.541968\pi$
$241$	−4.08174	−0.262928	−0.131464	+	0.991321i	$0.541968\pi$
$242$	20.8172	1.33818
$243$	0	0
$244$	−4.47214	−0.286299
$245$	7.09017	0.452974
$246$	0	0
$247$	0	0
$248$	5.32624	0.338216
$249$	0	0
$250$	−22.7194	−1.43690
$251$	−1.58359	−0.0999554	−0.0499777	−	0.998750i	$-0.515915\pi$
$251$	−1.58359	−0.0999554	−0.0499777	+	0.998750i	$0.515915\pi$
$252$	0	0
$253$	1.94427	0.122235
$254$	14.4721	0.908063
$255$	0	0
$256$	20.2705	1.26691
$257$	−20.0907	−1.25322	−0.626612	−	0.779332i	$-0.715559\pi$
$257$	−20.0907	−1.25322	−0.626612	+	0.779332i	$0.715559\pi$
$258$	0	0
$259$	−8.50651	−0.528569
$260$	13.0373	0.808538
$261$	0	0
$262$	−14.7679	−0.912362
$263$	−19.5279	−1.20414	−0.602070	−	0.798443i	$-0.705657\pi$
$263$	−19.5279	−1.20414	−0.602070	+	0.798443i	$0.705657\pi$
$264$	0	0
$265$	−19.6417	−1.20658
$266$	0	0
$267$	0	0
$268$	14.4904	0.885140
$269$	−21.2008	−1.29263	−0.646317	−	0.763069i	$-0.723691\pi$
$269$	−21.2008	−1.29263	−0.646317	+	0.763069i	$0.723691\pi$
$270$	0	0
$271$	−14.1459	−0.859302	−0.429651	−	0.902995i	$-0.641363\pi$
$271$	−14.1459	−0.859302	−0.429651	+	0.902995i	$0.641363\pi$
$272$	1.09017	0.0661013
$273$	0	0
$274$	−4.08174	−0.246587
$275$	−0.562306	−0.0339083
$276$	0	0
$277$	11.7984	0.708896	0.354448	−	0.935076i	$-0.384669\pi$
$277$	11.7984	0.708896	0.354448	+	0.935076i	$0.384669\pi$
$278$	−0.343027	−0.0205734
$279$	0	0
$280$	1.90211	0.113673
$281$	−25.7315	−1.53501	−0.767507	−	0.641040i	$-0.778503\pi$
$281$	−25.7315	−1.53501	−0.767507	+	0.641040i	$0.778503\pi$
$282$	0	0
$283$	−32.2148	−1.91497	−0.957485	−	0.288483i	$-0.906849\pi$
$283$	−32.2148	−1.91497	−0.957485	+	0.288483i	$0.906849\pi$
$284$	14.2128	0.843377
$285$	0	0
$286$	2.23607	0.132221
$287$	5.70634	0.336835
$288$	0	0
$289$	−16.9443	−0.996722
$290$	26.1803	1.53736
$291$	0	0
$292$	1.14590	0.0670586
$293$	−5.53483	−0.323348	−0.161674	−	0.986844i	$-0.551689\pi$
$293$	−5.53483	−0.323348	−0.161674	+	0.986844i	$0.551689\pi$
$294$	0	0
$295$	21.3723	1.24434
$296$	3.81966	0.222013
$297$	0	0
$298$	−2.62866	−0.152274
$299$	−41.0139	−2.37190
$300$	0	0
$301$	13.3262	0.768112
$302$	22.3607	1.28671
$303$	0	0
$304$	0	0
$305$	4.47214	0.256074
$306$	0	0
$307$	26.8011	1.52962	0.764810	−	0.644256i	$-0.222833\pi$
$307$	26.8011	1.52962	0.764810	+	0.644256i	$0.222833\pi$
$308$	−0.618034	−0.0352158
$309$	0	0
$310$	22.5623	1.28145
$311$	−25.9443	−1.47116	−0.735582	−	0.677435i	$-0.763091\pi$
$311$	−25.9443	−1.47116	−0.735582	+	0.677435i	$0.763091\pi$
$312$	0	0
$313$	−0.180340	−0.0101934	−0.00509671	−	0.999987i	$-0.501622\pi$
$313$	−0.180340	−0.0101934	−0.00509671	+	0.999987i	$0.501622\pi$
$314$	−38.8343	−2.19155
$315$	0	0
$316$	−0.277515	−0.0156114
$317$	−15.6659	−0.879886	−0.439943	−	0.898026i	$-0.645001\pi$
$317$	−15.6659	−0.879886	−0.439943	+	0.898026i	$0.645001\pi$
$318$	0	0
$319$	2.00811	0.112433
$320$	7.61803	0.425861
$321$	0	0
$322$	25.3480	1.41259
$323$	0	0
$324$	0	0
$325$	11.8617	0.657969
$326$	1.28157	0.0709797
$327$	0	0
$328$	−2.56231	−0.141480
$329$	13.5623	0.747714
$330$	0	0
$331$	−3.97574	−0.218526	−0.109263	−	0.994013i	$-0.534849\pi$
$331$	−3.97574	−0.218526	−0.109263	+	0.994013i	$0.534849\pi$
$332$	16.0902	0.883063
$333$	0	0
$334$	−20.6525	−1.13005
$335$	−14.4904	−0.791693
$336$	0	0
$337$	4.08174	0.222347	0.111173	−	0.993801i	$-0.464539\pi$
$337$	4.08174	0.222347	0.111173	+	0.993801i	$0.464539\pi$
$338$	−22.4418	−1.22068
$339$	0	0
$340$	0.618034	0.0335176
$341$	1.73060	0.0937172
$342$	0	0
$343$	18.4164	0.994393
$344$	−5.98385	−0.322628
$345$	0	0
$346$	31.7082	1.70464
$347$	11.4164	0.612865	0.306432	−	0.951892i	$-0.400865\pi$
$347$	11.4164	0.612865	0.306432	+	0.951892i	$0.400865\pi$
$348$	0	0
$349$	−29.7984	−1.59507	−0.797535	−	0.603272i	$-0.793863\pi$
$349$	−29.7984	−1.59507	−0.797535	+	0.603272i	$0.793863\pi$
$350$	−7.33094	−0.391855
$351$	0	0
$352$	1.73060	0.0922413
$353$	−23.3607	−1.24336	−0.621682	−	0.783270i	$-0.713550\pi$
$353$	−23.3607	−1.24336	−0.621682	+	0.783270i	$0.713550\pi$
$354$	0	0
$355$	−14.2128	−0.754340
$356$	1.45309	0.0770134
$357$	0	0
$358$	−3.61803	−0.191219
$359$	31.5066	1.66285	0.831427	−	0.555634i	$-0.187525\pi$
$359$	31.5066	1.66285	0.831427	+	0.555634i	$0.187525\pi$
$360$	0	0
$361$	0	0
$362$	−1.18034	−0.0620373
$363$	0	0
$364$	13.0373	0.683339
$365$	−1.14590	−0.0599790
$366$	0	0
$367$	−30.8885	−1.61237	−0.806184	−	0.591664i	$-0.798471\pi$
$367$	−30.8885	−1.61237	−0.806184	+	0.591664i	$0.798471\pi$
$368$	−38.0344	−1.98268
$369$	0	0
$370$	16.1803	0.841176
$371$	−19.6417	−1.01974
$372$	0	0
$373$	11.2412	0.582045	0.291023	−	0.956716i	$-0.406004\pi$
$373$	11.2412	0.582045	0.291023	+	0.956716i	$0.406004\pi$
$374$	0.106001	0.00548119
$375$	0	0
$376$	−6.08985	−0.314060
$377$	−42.3607	−2.18169
$378$	0	0
$379$	−1.45309	−0.0746400	−0.0373200	−	0.999303i	$-0.511882\pi$
$379$	−1.45309	−0.0746400	−0.0373200	+	0.999303i	$0.511882\pi$
$380$	0	0
$381$	0	0
$382$	−35.0301	−1.79230
$383$	16.9070	0.863908	0.431954	−	0.901896i	$-0.357824\pi$
$383$	16.9070	0.863908	0.431954	+	0.901896i	$0.357824\pi$
$384$	0	0
$385$	0.618034	0.0314979
$386$	16.1803	0.823558
$387$	0	0
$388$	19.9192	1.01124
$389$	12.8197	0.649983	0.324991	−	0.945717i	$-0.394639\pi$
$389$	12.8197	0.649983	0.324991	+	0.945717i	$0.394639\pi$
$390$	0	0
$391$	−1.94427	−0.0983261
$392$	−3.18368	−0.160800
$393$	0	0
$394$	50.8675	2.56267
$395$	0.277515	0.0139633
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	40.2874	2.01942
$399$	0	0
$400$	11.0000	0.550000
$401$	−4.53077	−0.226256	−0.113128	−	0.993580i	$-0.536087\pi$
$401$	−4.53077	−0.226256	−0.113128	+	0.993580i	$0.536087\pi$
$402$	0	0
$403$	−36.5066	−1.81852
$404$	−22.0902	−1.09903
$405$	0	0
$406$	26.1803	1.29931
$407$	1.24108	0.0615182
$408$	0	0
$409$	−17.2250	−0.851722	−0.425861	−	0.904789i	$-0.640029\pi$
$409$	−17.2250	−0.851722	−0.425861	+	0.904789i	$0.640029\pi$
$410$	−10.8541	−0.536046
$411$	0	0
$412$	8.33499	0.410636
$413$	21.3723	1.05166
$414$	0	0
$415$	−16.0902	−0.789835
$416$	−36.5066	−1.78988
$417$	0	0
$418$	0	0
$419$	20.7082	1.01166	0.505831	−	0.862633i	$-0.331186\pi$
$419$	20.7082	1.01166	0.505831	+	0.862633i	$0.331186\pi$
$420$	0	0
$421$	10.7516	0.524003	0.262002	−	0.965067i	$-0.415617\pi$
$421$	10.7516	0.524003	0.262002	+	0.965067i	$0.415617\pi$
$422$	−14.7984	−0.720374
$423$	0	0
$424$	8.81966	0.428321
$425$	0.562306	0.0272758
$426$	0	0
$427$	4.47214	0.216422
$428$	2.62866	0.127061
$429$	0	0
$430$	−25.3480	−1.22239
$431$	−12.2047	−0.587881	−0.293941	−	0.955824i	$-0.594967\pi$
$431$	−12.2047	−0.587881	−0.293941	+	0.955824i	$0.594967\pi$
$432$	0	0
$433$	28.0827	1.34957	0.674783	−	0.738016i	$-0.264237\pi$
$433$	28.0827	1.34957	0.674783	+	0.738016i	$0.264237\pi$
$434$	22.5623	1.08303
$435$	0	0
$436$	−24.8990	−1.19245
$437$	0	0
$438$	0	0
$439$	−21.2008	−1.01186	−0.505928	−	0.862575i	$-0.668850\pi$
$439$	−21.2008	−1.01186	−0.505928	+	0.862575i	$0.668850\pi$
$440$	−0.277515	−0.0132300
$441$	0	0
$442$	−2.23607	−0.106359
$443$	−27.1803	−1.29138	−0.645688	−	0.763601i	$-0.723430\pi$
$443$	−27.1803	−1.29138	−0.645688	+	0.763601i	$0.723430\pi$
$444$	0	0
$445$	−1.45309	−0.0688828
$446$	28.4164	1.34556
$447$	0	0
$448$	7.61803	0.359918
$449$	20.7112	0.977424	0.488712	−	0.872445i	$-0.337467\pi$
$449$	20.7112	0.977424	0.488712	+	0.872445i	$0.337467\pi$
$450$	0	0
$451$	−0.832544	−0.0392029
$452$	−20.6457	−0.971093
$453$	0	0
$454$	−16.5066	−0.774692
$455$	−13.0373	−0.611197
$456$	0	0
$457$	15.5623	0.727974	0.363987	−	0.931404i	$-0.381415\pi$
$457$	15.5623	0.727974	0.363987	+	0.931404i	$0.381415\pi$
$458$	41.1855	1.92447
$459$	0	0
$460$	−21.5623	−1.00535
$461$	33.0902	1.54116	0.770581	−	0.637342i	$-0.219966\pi$
$461$	33.0902	1.54116	0.770581	+	0.637342i	$0.219966\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	−39.2833	−1.82368
$465$	0	0
$466$	−22.6134	−1.04754
$467$	22.3262	1.03314	0.516568	−	0.856246i	$-0.327210\pi$
$467$	22.3262	1.03314	0.516568	+	0.856246i	$0.327210\pi$
$468$	0	0
$469$	−14.4904	−0.669103
$470$	−25.7970	−1.18993
$471$	0	0
$472$	−9.59675	−0.441726
$473$	−1.94427	−0.0893977
$474$	0	0
$475$	0	0
$476$	0.618034	0.0283275
$477$	0	0
$478$	−13.0373	−0.596311
$479$	−28.2148	−1.28917	−0.644583	−	0.764534i	$-0.722969\pi$
$479$	−28.2148	−1.28917	−0.644583	+	0.764534i	$0.722969\pi$
$480$	0	0
$481$	−26.1803	−1.19372
$482$	7.76393	0.353637
$483$	0	0
$484$	−17.7082	−0.804918
$485$	−19.9192	−0.904484
$486$	0	0
$487$	24.1069	1.09239	0.546195	−	0.837658i	$-0.316076\pi$
$487$	24.1069	1.09239	0.546195	+	0.837658i	$0.316076\pi$
$488$	−2.00811	−0.0909031
$489$	0	0
$490$	−13.4863	−0.609249
$491$	−11.7082	−0.528384	−0.264192	−	0.964470i	$-0.585105\pi$
$491$	−11.7082	−0.528384	−0.264192	+	0.964470i	$0.585105\pi$
$492$	0	0
$493$	−2.00811	−0.0904409
$494$	0	0
$495$	0	0
$496$	−33.8545	−1.52011
$497$	−14.2128	−0.637533
$498$	0	0
$499$	−18.2361	−0.816359	−0.408179	−	0.912902i	$-0.633836\pi$
$499$	−18.2361	−0.816359	−0.408179	+	0.912902i	$0.633836\pi$
$500$	19.3262	0.864296
$501$	0	0
$502$	3.01217	0.134440
$503$	−33.0344	−1.47293	−0.736466	−	0.676474i	$-0.763507\pi$
$503$	−33.0344	−1.47293	−0.736466	+	0.676474i	$0.763507\pi$
$504$	0	0
$505$	22.0902	0.983000
$506$	−3.69822	−0.164406
$507$	0	0
$508$	−12.3107	−0.546201
$509$	−4.08174	−0.180920	−0.0904600	−	0.995900i	$-0.528834\pi$
$509$	−4.08174	−0.180920	−0.0904600	+	0.995900i	$0.528834\pi$
$510$	0	0
$511$	−1.14590	−0.0506915
$512$	−27.1441	−1.19961
$513$	0	0
$514$	38.2148	1.68558
$515$	−8.33499	−0.367284
$516$	0	0
$517$	−1.97871	−0.0870237
$518$	16.1803	0.710923
$519$	0	0
$520$	5.85410	0.256719
$521$	22.6538	0.992483	0.496241	−	0.868185i	$-0.334713\pi$
$521$	22.6538	0.992483	0.496241	+	0.868185i	$0.334713\pi$
$522$	0	0
$523$	−16.4580	−0.719658	−0.359829	−	0.933018i	$-0.617165\pi$
$523$	−16.4580	−0.719658	−0.359829	+	0.933018i	$0.617165\pi$
$524$	12.5623	0.548787
$525$	0	0
$526$	37.1442	1.61956
$527$	−1.73060	−0.0753861
$528$	0	0
$529$	44.8328	1.94925
$530$	37.3607	1.62284
$531$	0	0
$532$	0	0
$533$	17.5623	0.760708
$534$	0	0
$535$	−2.62866	−0.113647
$536$	6.50658	0.281041
$537$	0	0
$538$	40.3262	1.73859
$539$	−1.03444	−0.0445566
$540$	0	0
$541$	−13.4164	−0.576816	−0.288408	−	0.957508i	$-0.593126\pi$
$541$	−13.4164	−0.576816	−0.288408	+	0.957508i	$0.593126\pi$
$542$	26.9071	1.15576
$543$	0	0
$544$	−1.73060	−0.0741988
$545$	24.8990	1.06656
$546$	0	0
$547$	−33.8545	−1.44751	−0.723757	−	0.690055i	$-0.757587\pi$
$547$	−33.8545	−1.44751	−0.723757	+	0.690055i	$0.757587\pi$
$548$	3.47214	0.148322
$549$	0	0
$550$	1.06957	0.0456066
$551$	0	0
$552$	0	0
$553$	0.277515	0.0118011
$554$	−22.4418	−0.953462
$555$	0	0
$556$	0.291796	0.0123749
$557$	16.9443	0.717952	0.358976	−	0.933347i	$-0.383126\pi$
$557$	16.9443	0.717952	0.358976	+	0.933347i	$0.383126\pi$
$558$	0	0
$559$	41.0139	1.73470
$560$	−12.0902	−0.510903
$561$	0	0
$562$	48.9443	2.06459
$563$	−12.2047	−0.514368	−0.257184	−	0.966362i	$-0.582795\pi$
$563$	−12.2047	−0.514368	−0.257184	+	0.966362i	$0.582795\pi$
$564$	0	0
$565$	20.6457	0.868572
$566$	61.2762	2.57563
$567$	0	0
$568$	6.38197	0.267781
$569$	−8.26948	−0.346675	−0.173337	−	0.984863i	$-0.555455\pi$
$569$	−8.26948	−0.346675	−0.173337	+	0.984863i	$0.555455\pi$
$570$	0	0
$571$	3.41641	0.142972	0.0714861	−	0.997442i	$-0.477226\pi$
$571$	3.41641	0.142972	0.0714861	+	0.997442i	$0.477226\pi$
$572$	−1.90211	−0.0795313
$573$	0	0
$574$	−10.8541	−0.453041
$575$	−19.6180	−0.818129
$576$	0	0
$577$	22.1246	0.921060	0.460530	−	0.887644i	$-0.347659\pi$
$577$	22.1246	0.921060	0.460530	+	0.887644i	$0.347659\pi$
$578$	32.2299	1.34059
$579$	0	0
$580$	−22.2703	−0.924725
$581$	−16.0902	−0.667533
$582$	0	0
$583$	2.86568	0.118684
$584$	0.514540	0.0212918
$585$	0	0
$586$	10.5279	0.434902
$587$	−33.5066	−1.38296	−0.691482	−	0.722393i	$-0.743042\pi$
$587$	−33.5066	−1.38296	−0.691482	+	0.722393i	$0.743042\pi$
$588$	0	0
$589$	0	0
$590$	−40.6525	−1.67364
$591$	0	0
$592$	−24.2784	−0.997838
$593$	19.4164	0.797336	0.398668	−	0.917095i	$-0.369473\pi$
$593$	19.4164	0.797336	0.398668	+	0.917095i	$0.369473\pi$
$594$	0	0
$595$	−0.618034	−0.0253369
$596$	2.23607	0.0915929
$597$	0	0
$598$	78.0132	3.19020
$599$	6.22088	0.254178	0.127089	−	0.991891i	$-0.459437\pi$
$599$	6.22088	0.254178	0.127089	+	0.991891i	$0.459437\pi$
$600$	0	0
$601$	−15.4539	−0.630379	−0.315189	−	0.949029i	$-0.602068\pi$
$601$	−15.4539	−0.630379	−0.315189	+	0.949029i	$0.602068\pi$
$602$	−25.3480	−1.03311
$603$	0	0
$604$	−19.0211	−0.773959
$605$	17.7082	0.719941
$606$	0	0
$607$	−47.5528	−1.93011	−0.965055	−	0.262048i	$-0.915602\pi$
$607$	−47.5528	−1.93011	−0.965055	+	0.262048i	$0.915602\pi$
$608$	0	0
$609$	0	0
$610$	−8.50651	−0.344418
$611$	41.7405	1.68864
$612$	0	0
$613$	14.5279	0.586775	0.293387	−	0.955994i	$-0.405217\pi$
$613$	14.5279	0.586775	0.293387	+	0.955994i	$0.405217\pi$
$614$	−50.9787	−2.05733
$615$	0	0
$616$	−0.277515	−0.0111814
$617$	−2.87539	−0.115759	−0.0578794	−	0.998324i	$-0.518434\pi$
$617$	−2.87539	−0.115759	−0.0578794	+	0.998324i	$0.518434\pi$
$618$	0	0
$619$	−9.09017	−0.365365	−0.182682	−	0.983172i	$-0.558478\pi$
$619$	−9.09017	−0.365365	−0.182682	+	0.983172i	$0.558478\pi$
$620$	−19.1926	−0.770795
$621$	0	0
$622$	49.3489	1.97871
$623$	−1.45309	−0.0582166
$624$	0	0
$625$	−7.41641	−0.296656
$626$	0.343027	0.0137101
$627$	0	0
$628$	33.0344	1.31822
$629$	−1.24108	−0.0494852
$630$	0	0
$631$	−36.8328	−1.46629	−0.733146	−	0.680072i	$-0.761949\pi$
$631$	−36.8328	−1.46629	−0.733146	+	0.680072i	$0.761949\pi$
$632$	−0.124612	−0.00495679
$633$	0	0
$634$	29.7984	1.18344
$635$	12.3107	0.488537
$636$	0	0
$637$	21.8213	0.864591
$638$	−3.81966	−0.151222
$639$	0	0
$640$	9.23305	0.364968
$641$	−20.4337	−0.807084	−0.403542	−	0.914961i	$-0.632221\pi$
$641$	−20.4337	−0.807084	−0.403542	+	0.914961i	$0.632221\pi$
$642$	0	0
$643$	13.4377	0.529931	0.264965	−	0.964258i	$-0.414639\pi$
$643$	13.4377	0.529931	0.264965	+	0.964258i	$0.414639\pi$
$644$	−21.5623	−0.849674
$645$	0	0
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	−3.11817	−0.122399
$650$	−22.5623	−0.884966
$651$	0	0
$652$	−1.09017	−0.0426944
$653$	0.0557281	0.00218081	0.00109040	−	0.999999i	$-0.499653\pi$
$653$	0.0557281	0.00218081	0.00109040	+	0.999999i	$0.499653\pi$
$654$	0	0
$655$	−12.5623	−0.490850
$656$	16.2865	0.635880
$657$	0	0
$658$	−25.7970	−1.00567
$659$	−11.7962	−0.459514	−0.229757	−	0.973248i	$-0.573793\pi$
$659$	−11.7962	−0.459514	−0.229757	+	0.973248i	$0.573793\pi$
$660$	0	0
$661$	47.5528	1.84959	0.924795	−	0.380465i	$-0.124236\pi$
$661$	47.5528	1.84959	0.924795	+	0.380465i	$0.124236\pi$
$662$	7.56231	0.293917
$663$	0	0
$664$	7.22494	0.280382
$665$	0	0
$666$	0	0
$667$	70.0602	2.71274
$668$	17.5680	0.679728
$669$	0	0
$670$	27.5623	1.06482
$671$	−0.652476	−0.0251886
$672$	0	0
$673$	33.8950	1.30656	0.653278	−	0.757118i	$-0.273393\pi$
$673$	33.8950	1.30656	0.653278	+	0.757118i	$0.273393\pi$
$674$	−7.76393	−0.299055
$675$	0	0
$676$	19.0902	0.734237
$677$	−4.70228	−0.180723	−0.0903617	−	0.995909i	$-0.528802\pi$
$677$	−4.70228	−0.180723	−0.0903617	+	0.995909i	$0.528802\pi$
$678$	0	0
$679$	−19.9192	−0.764428
$680$	0.277515	0.0106422
$681$	0	0
$682$	−3.29180	−0.126049
$683$	−13.8293	−0.529165	−0.264582	−	0.964363i	$-0.585234\pi$
$683$	−13.8293	−0.529165	−0.264582	+	0.964363i	$0.585234\pi$
$684$	0	0
$685$	−3.47214	−0.132663
$686$	−35.0301	−1.33746
$687$	0	0
$688$	38.0344	1.45005
$689$	−60.4508	−2.30299
$690$	0	0
$691$	20.7771	0.790398	0.395199	−	0.918596i	$-0.370676\pi$
$691$	20.7771	0.790398	0.395199	+	0.918596i	$0.370676\pi$
$692$	−26.9726	−1.02534
$693$	0	0
$694$	−21.7153	−0.824301
$695$	−0.291796	−0.0110685
$696$	0	0
$697$	0.832544	0.0315348
$698$	56.6799	2.14536
$699$	0	0
$700$	6.23607	0.235701
$701$	−3.18034	−0.120120	−0.0600599	−	0.998195i	$-0.519129\pi$
$701$	−3.18034	−0.120120	−0.0600599	+	0.998195i	$0.519129\pi$
$702$	0	0
$703$	0	0
$704$	−1.11146	−0.0418896
$705$	0	0
$706$	44.4347	1.67232
$707$	22.0902	0.830786
$708$	0	0
$709$	−22.6869	−0.852025	−0.426013	−	0.904717i	$-0.640082\pi$
$709$	−22.6869	−0.852025	−0.426013	+	0.904717i	$0.640082\pi$
$710$	27.0344	1.01458
$711$	0	0
$712$	0.652476	0.0244526
$713$	60.3781	2.26118
$714$	0	0
$715$	1.90211	0.0711350
$716$	3.07768	0.115018
$717$	0	0
$718$	−59.9291	−2.23653
$719$	29.6738	1.10664	0.553322	−	0.832967i	$-0.313360\pi$
$719$	29.6738	1.10664	0.553322	+	0.832967i	$0.313360\pi$
$720$	0	0
$721$	−8.33499	−0.310411
$722$	0	0
$723$	0	0
$724$	1.00406	0.0373155
$725$	−20.2622	−0.752520
$726$	0	0
$727$	29.1591	1.08145	0.540725	−	0.841200i	$-0.318150\pi$
$727$	29.1591	1.08145	0.540725	+	0.841200i	$0.318150\pi$
$728$	5.85410	0.216967
$729$	0	0
$730$	2.17963	0.0806716
$731$	1.94427	0.0719115
$732$	0	0
$733$	12.8197	0.473505	0.236752	−	0.971570i	$-0.423917\pi$
$733$	12.8197	0.473505	0.236752	+	0.971570i	$0.423917\pi$
$734$	58.7535	2.16863
$735$	0	0
$736$	60.3781	2.22557
$737$	2.11412	0.0778744
$738$	0	0
$739$	−4.79837	−0.176511	−0.0882555	−	0.996098i	$-0.528129\pi$
$739$	−4.79837	−0.176511	−0.0882555	+	0.996098i	$0.528129\pi$
$740$	−13.7638	−0.505968
$741$	0	0
$742$	37.3607	1.37155
$743$	6.88191	0.252473	0.126236	−	0.992000i	$-0.459710\pi$
$743$	6.88191	0.252473	0.126236	+	0.992000i	$0.459710\pi$
$744$	0	0
$745$	−2.23607	−0.0819232
$746$	−21.3820	−0.782849
$747$	0	0
$748$	−0.0901699	−0.00329694
$749$	−2.62866	−0.0960490
$750$	0	0
$751$	−44.6467	−1.62918	−0.814590	−	0.580037i	$-0.803038\pi$
$751$	−44.6467	−1.62918	−0.814590	+	0.580037i	$0.803038\pi$
$752$	38.7082	1.41154
$753$	0	0
$754$	80.5748	2.93436
$755$	19.0211	0.692250
$756$	0	0
$757$	28.1246	1.02221	0.511103	−	0.859519i	$-0.329237\pi$
$757$	28.1246	1.02221	0.511103	+	0.859519i	$0.329237\pi$
$758$	2.76393	0.100391
$759$	0	0
$760$	0	0
$761$	−19.3050	−0.699804	−0.349902	−	0.936786i	$-0.613785\pi$
$761$	−19.3050	−0.699804	−0.349902	+	0.936786i	$0.613785\pi$
$762$	0	0
$763$	24.8990	0.901404
$764$	29.7984	1.07807
$765$	0	0
$766$	−32.1591	−1.16195
$767$	65.7771	2.37507
$768$	0	0
$769$	31.0344	1.11913	0.559565	−	0.828786i	$-0.310968\pi$
$769$	31.0344	1.11913	0.559565	+	0.828786i	$0.310968\pi$
$770$	−1.17557	−0.0423646
$771$	0	0
$772$	−13.7638	−0.495371
$773$	22.3763	0.804821	0.402410	−	0.915459i	$-0.368173\pi$
$773$	22.3763	0.804821	0.402410	+	0.915459i	$0.368173\pi$
$774$	0	0
$775$	−17.4620	−0.627255
$776$	8.94427	0.321081
$777$	0	0
$778$	−24.3844	−0.874225
$779$	0	0
$780$	0	0
$781$	2.07363	0.0742002
$782$	3.69822	0.132248
$783$	0	0
$784$	20.2361	0.722717
$785$	−33.0344	−1.17905
$786$	0	0
$787$	13.6578	0.486849	0.243424	−	0.969920i	$-0.421729\pi$
$787$	13.6578	0.486849	0.243424	+	0.969920i	$0.421729\pi$
$788$	−43.2705	−1.54145
$789$	0	0
$790$	−0.527864	−0.0187806
$791$	20.6457	0.734078
$792$	0	0
$793$	13.7638	0.488768
$794$	13.3148	0.472524
$795$	0	0
$796$	−34.2705	−1.21469
$797$	−11.4782	−0.406578	−0.203289	−	0.979119i	$-0.565163\pi$
$797$	−11.4782	−0.406578	−0.203289	+	0.979119i	$0.565163\pi$
$798$	0	0
$799$	1.97871	0.0700019
$800$	−17.4620	−0.617376
$801$	0	0
$802$	8.61803	0.304313
$803$	0.167184	0.00589980
$804$	0	0
$805$	21.5623	0.759971
$806$	69.4396	2.44591
$807$	0	0
$808$	−9.91910	−0.348953
$809$	37.5410	1.31987	0.659936	−	0.751322i	$-0.270583\pi$
$809$	37.5410	1.31987	0.659936	+	0.751322i	$0.270583\pi$
$810$	0	0
$811$	15.0454	0.528315	0.264158	−	0.964480i	$-0.414906\pi$
$811$	15.0454	0.528315	0.264158	+	0.964480i	$0.414906\pi$
$812$	−22.2703	−0.781535
$813$	0	0
$814$	−2.36068	−0.0827418
$815$	1.09017	0.0381870
$816$	0	0
$817$	0	0
$818$	32.7639	1.14556
$819$	0	0
$820$	9.23305	0.322432
$821$	29.6738	1.03562	0.517811	−	0.855495i	$-0.326747\pi$
$821$	29.6738	1.03562	0.517811	+	0.855495i	$0.326747\pi$
$822$	0	0
$823$	40.3050	1.40494	0.702471	−	0.711712i	$-0.252080\pi$
$823$	40.3050	1.40494	0.702471	+	0.711712i	$0.252080\pi$
$824$	3.74265	0.130381
$825$	0	0
$826$	−40.6525	−1.41448
$827$	30.8828	1.07390	0.536951	−	0.843614i	$-0.319576\pi$
$827$	30.8828	1.07390	0.536951	+	0.843614i	$0.319576\pi$
$828$	0	0
$829$	−49.8635	−1.73183	−0.865915	−	0.500191i	$-0.833263\pi$
$829$	−49.8635	−1.73183	−0.865915	+	0.500191i	$0.833263\pi$
$830$	30.6053	1.06233
$831$	0	0
$832$	23.4459	0.812840
$833$	1.03444	0.0358413
$834$	0	0
$835$	−17.5680	−0.607967
$836$	0	0
$837$	0	0
$838$	−39.3893	−1.36068
$839$	−51.3821	−1.77391	−0.886953	−	0.461859i	$-0.847182\pi$
$839$	−51.3821	−1.77391	−0.886953	+	0.461859i	$0.847182\pi$
$840$	0	0
$841$	43.3607	1.49520
$842$	−20.4508	−0.704783
$843$	0	0
$844$	12.5882	0.433305
$845$	−19.0902	−0.656722
$846$	0	0
$847$	17.7082	0.608461
$848$	−56.0593	−1.92509
$849$	0	0
$850$	−1.06957	−0.0366859
$851$	43.2996	1.48429
$852$	0	0
$853$	33.5967	1.15033	0.575165	−	0.818037i	$-0.304938\pi$
$853$	33.5967	1.15033	0.575165	+	0.818037i	$0.304938\pi$
$854$	−8.50651	−0.291087
$855$	0	0
$856$	1.18034	0.0403432
$857$	−15.3884	−0.525658	−0.262829	−	0.964842i	$-0.584656\pi$
$857$	−15.3884	−0.525658	−0.262829	+	0.964842i	$0.584656\pi$
$858$	0	0
$859$	−27.1591	−0.926655	−0.463327	−	0.886187i	$-0.653345\pi$
$859$	−27.1591	−0.926655	−0.463327	+	0.886187i	$0.653345\pi$
$860$	21.5623	0.735269
$861$	0	0
$862$	23.2148	0.790699
$863$	10.8981	0.370977	0.185488	−	0.982646i	$-0.440613\pi$
$863$	10.8981	0.370977	0.185488	+	0.982646i	$0.440613\pi$
$864$	0	0
$865$	26.9726	0.917096
$866$	−53.4164	−1.81516
$867$	0	0
$868$	−19.1926	−0.651441
$869$	−0.0404888	−0.00137349
$870$	0	0
$871$	−44.5967	−1.51110
$872$	−11.1803	−0.378614
$873$	0	0
$874$	0	0
$875$	−19.3262	−0.653346
$876$	0	0
$877$	−29.3642	−0.991560	−0.495780	−	0.868448i	$-0.665118\pi$
$877$	−29.3642	−0.991560	−0.495780	+	0.868448i	$0.665118\pi$
$878$	40.3262	1.36094
$879$	0	0
$880$	1.76393	0.0594621
$881$	4.36068	0.146915	0.0734575	−	0.997298i	$-0.476597\pi$
$881$	4.36068	0.146915	0.0734575	+	0.997298i	$0.476597\pi$
$882$	0	0
$883$	4.52786	0.152375	0.0761874	−	0.997094i	$-0.475725\pi$
$883$	4.52786	0.152375	0.0761874	+	0.997094i	$0.475725\pi$
$884$	1.90211	0.0639750
$885$	0	0
$886$	51.7001	1.73690
$887$	18.8091	0.631549	0.315774	−	0.948834i	$-0.397736\pi$
$887$	18.8091	0.631549	0.315774	+	0.948834i	$0.397736\pi$
$888$	0	0
$889$	12.3107	0.412889
$890$	2.76393	0.0926472
$891$	0	0
$892$	−24.1724	−0.809353
$893$	0	0
$894$	0	0
$895$	−3.07768	−0.102876
$896$	9.23305	0.308455
$897$	0	0
$898$	−39.3951	−1.31463
$899$	62.3607	2.07985
$900$	0	0
$901$	−2.86568	−0.0954697
$902$	1.58359	0.0527279
$903$	0	0
$904$	−9.27051	−0.308333
$905$	−1.00406	−0.0333760
$906$	0	0
$907$	−24.1069	−0.800457	−0.400229	−	0.916415i	$-0.631069\pi$
$907$	−24.1069	−0.800457	−0.400229	+	0.916415i	$0.631069\pi$
$908$	14.0413	0.465978
$909$	0	0
$910$	24.7984	0.822058
$911$	39.0463	1.29366	0.646831	−	0.762633i	$-0.276094\pi$
$911$	39.0463	1.29366	0.646831	+	0.762633i	$0.276094\pi$
$912$	0	0
$913$	2.34752	0.0776917
$914$	−29.6013	−0.979123
$915$	0	0
$916$	−35.0344	−1.15757
$917$	−12.5623	−0.414844
$918$	0	0
$919$	−33.9443	−1.11972	−0.559859	−	0.828588i	$-0.689145\pi$
$919$	−33.9443	−1.11972	−0.559859	+	0.828588i	$0.689145\pi$
$920$	−9.68208	−0.319209
$921$	0	0
$922$	−62.9412	−2.07286
$923$	−43.7426	−1.43981
$924$	0	0
$925$	−12.5227	−0.411745
$926$	45.6507	1.50018
$927$	0	0
$928$	62.3607	2.04709
$929$	−14.8197	−0.486217	−0.243109	−	0.969999i	$-0.578167\pi$
$929$	−14.8197	−0.486217	−0.243109	+	0.969999i	$0.578167\pi$
$930$	0	0
$931$	0	0
$932$	19.2361	0.630098
$933$	0	0
$934$	−42.4670	−1.38956
$935$	0.0901699	0.00294887
$936$	0	0
$937$	−18.1803	−0.593926	−0.296963	−	0.954889i	$-0.595974\pi$
$937$	−18.1803	−0.593926	−0.296963	+	0.954889i	$0.595974\pi$
$938$	27.5623	0.899941
$939$	0	0
$940$	21.9443	0.715743
$941$	16.3925	0.534379	0.267190	−	0.963644i	$-0.413905\pi$
$941$	16.3925	0.534379	0.267190	+	0.963644i	$0.413905\pi$
$942$	0	0
$943$	−29.0462	−0.945876
$944$	60.9986	1.98534
$945$	0	0
$946$	3.69822	0.120240
$947$	19.8885	0.646291	0.323145	−	0.946349i	$-0.395260\pi$
$947$	19.8885	0.646291	0.323145	+	0.946349i	$0.395260\pi$
$948$	0	0
$949$	−3.52671	−0.114482
$950$	0	0
$951$	0	0
$952$	0.277515	0.00899430
$953$	56.1248	1.81806	0.909031	−	0.416728i	$-0.136823\pi$
$953$	56.1248	1.81806	0.909031	+	0.416728i	$0.136823\pi$
$954$	0	0
$955$	−29.7984	−0.964253
$956$	11.0902	0.358682
$957$	0	0
$958$	53.6677	1.73392
$959$	−3.47214	−0.112121
$960$	0	0
$961$	22.7426	0.733634
$962$	49.7980	1.60555
$963$	0	0
$964$	−6.60440	−0.212713
$965$	13.7638	0.443073
$966$	0	0
$967$	−6.27051	−0.201646	−0.100823	−	0.994904i	$-0.532148\pi$
$967$	−6.27051	−0.201646	−0.100823	+	0.994904i	$0.532148\pi$
$968$	−7.95148	−0.255570
$969$	0	0
$970$	37.8885	1.21653
$971$	4.53077	0.145399	0.0726996	−	0.997354i	$-0.476839\pi$
$971$	4.53077	0.145399	0.0726996	+	0.997354i	$0.476839\pi$
$972$	0	0
$973$	−0.291796	−0.00935455
$974$	−45.8541	−1.46926
$975$	0	0
$976$	12.7639	0.408564
$977$	−16.4580	−0.526538	−0.263269	−	0.964723i	$-0.584801\pi$
$977$	−16.4580	−0.526538	−0.263269	+	0.964723i	$0.584801\pi$
$978$	0	0
$979$	0.212002	0.00677562
$980$	11.4721	0.366464
$981$	0	0
$982$	22.2703	0.710675
$983$	25.7315	0.820708	0.410354	−	0.911926i	$-0.365405\pi$
$983$	25.7315	0.820708	0.410354	+	0.911926i	$0.365405\pi$
$984$	0	0
$985$	43.2705	1.37871
$986$	3.81966	0.121643
$987$	0	0
$988$	0	0
$989$	−67.8328	−2.15696
$990$	0	0
$991$	−15.3884	−0.488829	−0.244415	−	0.969671i	$-0.578596\pi$
$991$	−15.3884	−0.488829	−0.244415	+	0.969671i	$0.578596\pi$
$992$	53.7426	1.70633
$993$	0	0
$994$	27.0344	0.857480
$995$	34.2705	1.08645
$996$	0	0
$997$	−12.1246	−0.383990	−0.191995	−	0.981396i	$-0.561496\pi$
$997$	−12.1246	−0.383990	−0.191995	+	0.981396i	$0.561496\pi$
$998$	34.6871	1.09800
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.ba.1.1	✓	4
3.2	odd	2		3249.2.a.bb.1.4	yes	4
19.18	odd	2	inner	3249.2.a.ba.1.4	yes	4
57.56	even	2		3249.2.a.bb.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3249.2.a.ba.1.1	✓	4	1.1	even	1	trivial
3249.2.a.ba.1.4	yes	4	19.18	odd	2	inner
3249.2.a.bb.1.1	yes	4	57.56	even	2
3249.2.a.bb.1.4	yes	4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q-1.90211 q^{2} +1.61803 q^{4} -1.61803 q^{5} -1.61803 q^{7} +0.726543 q^{8} +3.07768 q^{10} +0.236068 q^{11} -4.97980 q^{13} +3.07768 q^{14} -4.61803 q^{16} -0.236068 q^{17} -2.61803 q^{20} -0.449028 q^{22} +8.23607 q^{23} -2.38197 q^{25} +9.47214 q^{26} -2.61803 q^{28} +8.50651 q^{29} +7.33094 q^{31} +7.33094 q^{32} +0.449028 q^{34} +2.61803 q^{35} +5.25731 q^{37} -1.17557 q^{40} -3.52671 q^{41} -8.23607 q^{43} +0.381966 q^{44} -15.6659 q^{46} -8.38197 q^{47} -4.38197 q^{49} +4.53077 q^{50} -8.05748 q^{52} +12.1392 q^{53} -0.381966 q^{55} -1.17557 q^{56} -16.1803 q^{58} -13.2088 q^{59} -2.76393 q^{61} -13.9443 q^{62} -4.70820 q^{64} +8.05748 q^{65} +8.95554 q^{67} -0.381966 q^{68} -4.97980 q^{70} +8.78402 q^{71} +0.708204 q^{73} -10.0000 q^{74} -0.381966 q^{77} -0.171513 q^{79} +7.47214 q^{80} +6.70820 q^{82} +9.94427 q^{83} +0.381966 q^{85} +15.6659 q^{86} +0.171513 q^{88} +0.898056 q^{89} +8.05748 q^{91} +13.3262 q^{92} +15.9434 q^{94} +12.3107 q^{97} +8.33499 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{5} - 2 q^{7} - 8 q^{11} - 14 q^{16} + 8 q^{17} - 6 q^{20} + 24 q^{23} - 14 q^{25} + 20 q^{26} - 6 q^{28} + 6 q^{35} - 24 q^{43} + 6 q^{44} - 38 q^{47} - 22 q^{49} - 6 q^{55} - 20 q^{58}+ \cdots + 22 q^{92}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^5 - 2 * q^7 - 8 * q^11 - 14 * q^16 + 8 * q^17 - 6 * q^20 + 24 * q^23 - 14 * q^25 + 20 * q^26 - 6 * q^28 + 6 * q^35 - 24 * q^43 + 6 * q^44 - 38 * q^47 - 22 * q^49 - 6 * q^55 - 20 * q^58 - 20 * q^61 - 20 * q^62 + 8 * q^64 - 6 * q^68 - 24 * q^73 - 40 * q^74 - 6 * q^77 + 12 * q^80 + 4 * q^83 + 6 * q^85 + 22 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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