LMFDB - Embedded newform 9522.2.a.bf.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9522,2,Mod(1,9522)] 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("9522.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9522, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9522 = 2 \cdot 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9522.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,0,0,0,2,0,0,0,0,-4,0,0,2,0,0,0,0,0,0,0,0,-10,-4,0,0, 12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3174)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	9522.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +4.89898 q^{7} +1.00000 q^{8} +4.89898 q^{11} -2.00000 q^{13} +4.89898 q^{14} +1.00000 q^{16} +4.89898 q^{17} +4.89898 q^{22} -5.00000 q^{25} -2.00000 q^{26} +4.89898 q^{28} +6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +4.89898 q^{34} +4.89898 q^{37} +6.00000 q^{41} -9.79796 q^{43} +4.89898 q^{44} +17.0000 q^{49} -5.00000 q^{50} -2.00000 q^{52} +9.79796 q^{53} +4.89898 q^{56} +6.00000 q^{58} +12.0000 q^{59} -4.89898 q^{61} -8.00000 q^{62} +1.00000 q^{64} -9.79796 q^{67} +4.89898 q^{68} -2.00000 q^{73} +4.89898 q^{74} +24.0000 q^{77} +14.6969 q^{79} +6.00000 q^{82} -4.89898 q^{83} -9.79796 q^{86} +4.89898 q^{88} -14.6969 q^{89} -9.79796 q^{91} +17.0000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 4 q^{13} + 2 q^{16} - 10 q^{25} - 4 q^{26} + 12 q^{29} - 16 q^{31} + 2 q^{32} + 12 q^{41} + 34 q^{49} - 10 q^{50} - 4 q^{52} + 12 q^{58} + 24 q^{59} - 16 q^{62} + 2 q^{64}+ \cdots + 34 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 4 * q^13 + 2 * q^16 - 10 * q^25 - 4 * q^26 + 12 * q^29 - 16 * q^31 + 2 * q^32 + 12 * q^41 + 34 * q^49 - 10 * q^50 - 4 * q^52 + 12 * q^58 + 24 * q^59 - 16 * q^62 + 2 * q^64 - 4 * q^73 + 48 * q^77 + 12 * q^82 + 34 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	4.89898	1.85164	0.925820	−	0.377964i	$-0.123376\pi$
$7$	4.89898	1.85164	0.925820	+	0.377964i	$0.123376\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	4.89898	1.47710	0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898	1.47710	0.738549	+	0.674200i	$0.235511\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	4.89898	1.30931
$15$	0	0
$16$	1.00000	0.250000
$17$	4.89898	1.18818	0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898	1.18818	0.594089	+	0.804400i	$0.297513\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	4.89898	1.04447
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	4.89898	0.925820
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.89898	0.840168
$35$	0	0
$36$	0	0
$37$	4.89898	0.805387	0.402694	−	0.915335i	$-0.368074\pi$
$37$	4.89898	0.805387	0.402694	+	0.915335i	$0.368074\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−9.79796	−1.49417	−0.747087	−	0.664726i	$-0.768548\pi$
$43$	−9.79796	−1.49417	−0.747087	+	0.664726i	$0.768548\pi$
$44$	4.89898	0.738549
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	17.0000	2.42857
$50$	−5.00000	−0.707107
$51$	0	0
$52$	−2.00000	−0.277350
$53$	9.79796	1.34585	0.672927	−	0.739709i	$-0.265037\pi$
$53$	9.79796	1.34585	0.672927	+	0.739709i	$0.265037\pi$
$54$	0	0
$55$	0	0
$56$	4.89898	0.654654
$57$	0	0
$58$	6.00000	0.787839
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−4.89898	−0.627250	−0.313625	−	0.949547i	$-0.601543\pi$
$61$	−4.89898	−0.627250	−0.313625	+	0.949547i	$0.601543\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−9.79796	−1.19701	−0.598506	−	0.801119i	$-0.704239\pi$
$67$	−9.79796	−1.19701	−0.598506	+	0.801119i	$0.704239\pi$
$68$	4.89898	0.594089
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	4.89898	0.569495
$75$	0	0
$76$	0	0
$77$	24.0000	2.73505
$78$	0	0
$79$	14.6969	1.65353	0.826767	−	0.562544i	$-0.190177\pi$
$79$	14.6969	1.65353	0.826767	+	0.562544i	$0.190177\pi$
$80$	0	0
$81$	0	0
$82$	6.00000	0.662589
$83$	−4.89898	−0.537733	−0.268866	−	0.963177i	$-0.586649\pi$
$83$	−4.89898	−0.537733	−0.268866	+	0.963177i	$0.586649\pi$
$84$	0	0
$85$	0	0
$86$	−9.79796	−1.05654
$87$	0	0
$88$	4.89898	0.522233
$89$	−14.6969	−1.55787	−0.778936	−	0.627103i	$-0.784240\pi$
$89$	−14.6969	−1.55787	−0.778936	+	0.627103i	$0.784240\pi$
$90$	0	0
$91$	−9.79796	−1.02711
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	17.0000	1.71726
$99$	0	0
$100$	−5.00000	−0.500000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	4.89898	0.482711	0.241355	−	0.970437i	$-0.422408\pi$
$103$	4.89898	0.482711	0.241355	+	0.970437i	$0.422408\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	9.79796	0.951662
$107$	−4.89898	−0.473602	−0.236801	−	0.971558i	$-0.576099\pi$
$107$	−4.89898	−0.473602	−0.236801	+	0.971558i	$0.576099\pi$
$108$	0	0
$109$	4.89898	0.469237	0.234619	−	0.972088i	$-0.424616\pi$
$109$	4.89898	0.469237	0.234619	+	0.972088i	$0.424616\pi$
$110$	0	0
$111$	0	0
$112$	4.89898	0.462910
$113$	−14.6969	−1.38257	−0.691286	−	0.722581i	$-0.742955\pi$
$113$	−14.6969	−1.38257	−0.691286	+	0.722581i	$0.742955\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	12.0000	1.10469
$119$	24.0000	2.20008
$120$	0	0
$121$	13.0000	1.18182
$122$	−4.89898	−0.443533
$123$	0	0
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	−9.79796	−0.846415
$135$	0	0
$136$	4.89898	0.420084
$137$	−4.89898	−0.418548	−0.209274	−	0.977857i	$-0.567110\pi$
$137$	−4.89898	−0.418548	−0.209274	+	0.977857i	$0.567110\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.79796	−0.819346
$144$	0	0
$145$	0	0
$146$	−2.00000	−0.165521
$147$	0	0
$148$	4.89898	0.402694
$149$	−19.5959	−1.60536	−0.802680	−	0.596410i	$-0.796593\pi$
$149$	−19.5959	−1.60536	−0.802680	+	0.596410i	$0.796593\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	0	0
$154$	24.0000	1.93398
$155$	0	0
$156$	0	0
$157$	4.89898	0.390981	0.195491	−	0.980706i	$-0.437370\pi$
$157$	4.89898	0.390981	0.195491	+	0.980706i	$0.437370\pi$
$158$	14.6969	1.16923
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−4.89898	−0.380235
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−9.79796	−0.747087
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−24.4949	−1.85164
$176$	4.89898	0.369274
$177$	0	0
$178$	−14.6969	−1.10158
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	4.89898	0.364138	0.182069	−	0.983286i	$-0.441721\pi$
$181$	4.89898	0.364138	0.182069	+	0.983286i	$0.441721\pi$
$182$	−9.79796	−0.726273
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	24.0000	1.75505
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.5959	−1.41791	−0.708955	−	0.705253i	$-0.750833\pi$
$191$	−19.5959	−1.41791	−0.708955	+	0.705253i	$0.750833\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	0	0
$196$	17.0000	1.21429
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	14.6969	1.04184	0.520919	−	0.853606i	$-0.325589\pi$
$199$	14.6969	1.04184	0.520919	+	0.853606i	$0.325589\pi$
$200$	−5.00000	−0.353553
$201$	0	0
$202$	−6.00000	−0.422159
$203$	29.3939	2.06305
$204$	0	0
$205$	0	0
$206$	4.89898	0.341328
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	9.79796	0.672927
$213$	0	0
$214$	−4.89898	−0.334887
$215$	0	0
$216$	0	0
$217$	−39.1918	−2.66052
$218$	4.89898	0.331801
$219$	0	0
$220$	0	0
$221$	−9.79796	−0.659082
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	4.89898	0.327327
$225$	0	0
$226$	−14.6969	−0.977626
$227$	14.6969	0.975470	0.487735	−	0.872992i	$-0.337823\pi$
$227$	14.6969	0.975470	0.487735	+	0.872992i	$0.337823\pi$
$228$	0	0
$229$	24.4949	1.61867	0.809334	−	0.587348i	$-0.199828\pi$
$229$	24.4949	1.61867	0.809334	+	0.587348i	$0.199828\pi$
$230$	0	0
$231$	0	0
$232$	6.00000	0.393919
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	12.0000	0.781133
$237$	0	0
$238$	24.0000	1.55569
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−19.5959	−1.26228	−0.631142	−	0.775667i	$-0.717413\pi$
$241$	−19.5959	−1.26228	−0.631142	+	0.775667i	$0.717413\pi$
$242$	13.0000	0.835672
$243$	0	0
$244$	−4.89898	−0.313625
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−8.00000	−0.508001
$249$	0	0
$250$	0	0
$251$	24.4949	1.54610	0.773052	−	0.634343i	$-0.218729\pi$
$251$	24.4949	1.54610	0.773052	+	0.634343i	$0.218729\pi$
$252$	0	0
$253$	0	0
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	0	0
$259$	24.0000	1.49129
$260$	0	0
$261$	0	0
$262$	−12.0000	−0.741362
$263$	−9.79796	−0.604168	−0.302084	−	0.953281i	$-0.597682\pi$
$263$	−9.79796	−0.604168	−0.302084	+	0.953281i	$0.597682\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−9.79796	−0.598506
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	4.89898	0.297044
$273$	0	0
$274$	−4.89898	−0.295958
$275$	−24.4949	−1.47710
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	0	0
$281$	4.89898	0.292249	0.146124	−	0.989266i	$-0.453320\pi$
$281$	4.89898	0.292249	0.146124	+	0.989266i	$0.453320\pi$
$282$	0	0
$283$	−19.5959	−1.16486	−0.582428	−	0.812882i	$-0.697897\pi$
$283$	−19.5959	−1.16486	−0.582428	+	0.812882i	$0.697897\pi$
$284$	0	0
$285$	0	0
$286$	−9.79796	−0.579365
$287$	29.3939	1.73507
$288$	0	0
$289$	7.00000	0.411765
$290$	0	0
$291$	0	0
$292$	−2.00000	−0.117041
$293$	−9.79796	−0.572403	−0.286201	−	0.958169i	$-0.592393\pi$
$293$	−9.79796	−0.572403	−0.286201	+	0.958169i	$0.592393\pi$
$294$	0	0
$295$	0	0
$296$	4.89898	0.284747
$297$	0	0
$298$	−19.5959	−1.13516
$299$	0	0
$300$	0	0
$301$	−48.0000	−2.76667
$302$	−16.0000	−0.920697
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	24.0000	1.36753
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	9.79796	0.553813	0.276907	−	0.960897i	$-0.410691\pi$
$313$	9.79796	0.553813	0.276907	+	0.960897i	$0.410691\pi$
$314$	4.89898	0.276465
$315$	0	0
$316$	14.6969	0.826767
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	29.3939	1.64574
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	10.0000	0.554700
$326$	−20.0000	−1.10770
$327$	0	0
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−4.89898	−0.268866
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−9.79796	−0.533729	−0.266864	−	0.963734i	$-0.585988\pi$
$337$	−9.79796	−0.533729	−0.266864	+	0.963734i	$0.585988\pi$
$338$	−9.00000	−0.489535
$339$	0	0
$340$	0	0
$341$	−39.1918	−2.12236
$342$	0	0
$343$	48.9898	2.64520
$344$	−9.79796	−0.528271
$345$	0	0
$346$	6.00000	0.322562
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	−24.4949	−1.30931
$351$	0	0
$352$	4.89898	0.261116
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	0	0
$356$	−14.6969	−0.778936
$357$	0	0
$358$	12.0000	0.634220
$359$	19.5959	1.03423	0.517116	−	0.855915i	$-0.327005\pi$
$359$	19.5959	1.03423	0.517116	+	0.855915i	$0.327005\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	4.89898	0.257485
$363$	0	0
$364$	−9.79796	−0.513553
$365$	0	0
$366$	0	0
$367$	−24.4949	−1.27862	−0.639312	−	0.768948i	$-0.720781\pi$
$367$	−24.4949	−1.27862	−0.639312	+	0.768948i	$0.720781\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	48.0000	2.49204
$372$	0	0
$373$	−14.6969	−0.760979	−0.380489	−	0.924785i	$-0.624244\pi$
$373$	−14.6969	−0.760979	−0.380489	+	0.924785i	$0.624244\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	19.5959	1.00657	0.503287	−	0.864119i	$-0.332124\pi$
$379$	19.5959	1.00657	0.503287	+	0.864119i	$0.332124\pi$
$380$	0	0
$381$	0	0
$382$	−19.5959	−1.00261
$383$	9.79796	0.500652	0.250326	−	0.968162i	$-0.419462\pi$
$383$	9.79796	0.500652	0.250326	+	0.968162i	$0.419462\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	0	0
$389$	19.5959	0.993552	0.496776	−	0.867879i	$-0.334517\pi$
$389$	19.5959	0.993552	0.496776	+	0.867879i	$0.334517\pi$
$390$	0	0
$391$	0	0
$392$	17.0000	0.858630
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	14.6969	0.736691
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−34.2929	−1.71250	−0.856252	−	0.516559i	$-0.827213\pi$
$401$	−34.2929	−1.71250	−0.856252	+	0.516559i	$0.827213\pi$
$402$	0	0
$403$	16.0000	0.797017
$404$	−6.00000	−0.298511
$405$	0	0
$406$	29.3939	1.45879
$407$	24.0000	1.18964
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	4.89898	0.241355
$413$	58.7878	2.89276
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	0	0
$419$	14.6969	0.717992	0.358996	−	0.933339i	$-0.383119\pi$
$419$	14.6969	0.717992	0.358996	+	0.933339i	$0.383119\pi$
$420$	0	0
$421$	−4.89898	−0.238762	−0.119381	−	0.992849i	$-0.538091\pi$
$421$	−4.89898	−0.238762	−0.119381	+	0.992849i	$0.538091\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	9.79796	0.475831
$425$	−24.4949	−1.18818
$426$	0	0
$427$	−24.0000	−1.16144
$428$	−4.89898	−0.236801
$429$	0	0
$430$	0	0
$431$	29.3939	1.41585	0.707927	−	0.706286i	$-0.249631\pi$
$431$	29.3939	1.41585	0.707927	+	0.706286i	$0.249631\pi$
$432$	0	0
$433$	−9.79796	−0.470860	−0.235430	−	0.971891i	$-0.575650\pi$
$433$	−9.79796	−0.470860	−0.235430	+	0.971891i	$0.575650\pi$
$434$	−39.1918	−1.88127
$435$	0	0
$436$	4.89898	0.234619
$437$	0	0
$438$	0	0
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	0	0
$441$	0	0
$442$	−9.79796	−0.466041
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	0	0
$446$	8.00000	0.378811
$447$	0	0
$448$	4.89898	0.231455
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	29.3939	1.38410
$452$	−14.6969	−0.691286
$453$	0	0
$454$	14.6969	0.689761
$455$	0	0
$456$	0	0
$457$	−19.5959	−0.916658	−0.458329	−	0.888783i	$-0.651552\pi$
$457$	−19.5959	−0.916658	−0.458329	+	0.888783i	$0.651552\pi$
$458$	24.4949	1.14457
$459$	0	0
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	24.4949	1.13349	0.566744	−	0.823894i	$-0.308203\pi$
$467$	24.4949	1.13349	0.566744	+	0.823894i	$0.308203\pi$
$468$	0	0
$469$	−48.0000	−2.21643
$470$	0	0
$471$	0	0
$472$	12.0000	0.552345
$473$	−48.0000	−2.20704
$474$	0	0
$475$	0	0
$476$	24.0000	1.10004
$477$	0	0
$478$	−24.0000	−1.09773
$479$	−19.5959	−0.895360	−0.447680	−	0.894194i	$-0.647750\pi$
$479$	−19.5959	−0.895360	−0.447680	+	0.894194i	$0.647750\pi$
$480$	0	0
$481$	−9.79796	−0.446748
$482$	−19.5959	−0.892570
$483$	0	0
$484$	13.0000	0.590909
$485$	0	0
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−4.89898	−0.221766
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	29.3939	1.32383
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	24.4949	1.09326
$503$	9.79796	0.436869	0.218435	−	0.975852i	$-0.429905\pi$
$503$	9.79796	0.436869	0.218435	+	0.975852i	$0.429905\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	16.0000	0.709885
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−9.79796	−0.433436
$512$	1.00000	0.0441942
$513$	0	0
$514$	30.0000	1.32324
$515$	0	0
$516$	0	0
$517$	0	0
$518$	24.0000	1.05450
$519$	0	0
$520$	0	0
$521$	−4.89898	−0.214628	−0.107314	−	0.994225i	$-0.534225\pi$
$521$	−4.89898	−0.214628	−0.107314	+	0.994225i	$0.534225\pi$
$522$	0	0
$523$	−9.79796	−0.428435	−0.214217	−	0.976786i	$-0.568720\pi$
$523$	−9.79796	−0.428435	−0.214217	+	0.976786i	$0.568720\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−9.79796	−0.427211
$527$	−39.1918	−1.70722
$528$	0	0
$529$	0	0
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	0	0
$536$	−9.79796	−0.423207
$537$	0	0
$538$	−18.0000	−0.776035
$539$	83.2827	3.58724
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	16.0000	0.687259
$543$	0	0
$544$	4.89898	0.210042
$545$	0	0
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−4.89898	−0.209274
$549$	0	0
$550$	−24.4949	−1.04447
$551$	0	0
$552$	0	0
$553$	72.0000	3.06175
$554$	10.0000	0.424859
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−19.5959	−0.830306	−0.415153	−	0.909752i	$-0.636272\pi$
$557$	−19.5959	−0.830306	−0.415153	+	0.909752i	$0.636272\pi$
$558$	0	0
$559$	19.5959	0.828819
$560$	0	0
$561$	0	0
$562$	4.89898	0.206651
$563$	14.6969	0.619402	0.309701	−	0.950834i	$-0.399771\pi$
$563$	14.6969	0.619402	0.309701	+	0.950834i	$0.399771\pi$
$564$	0	0
$565$	0	0
$566$	−19.5959	−0.823678
$567$	0	0
$568$	0	0
$569$	−34.2929	−1.43763	−0.718816	−	0.695201i	$-0.755315\pi$
$569$	−34.2929	−1.43763	−0.718816	+	0.695201i	$0.755315\pi$
$570$	0	0
$571$	9.79796	0.410032	0.205016	−	0.978759i	$-0.434275\pi$
$571$	9.79796	0.410032	0.205016	+	0.978759i	$0.434275\pi$
$572$	−9.79796	−0.409673
$573$	0	0
$574$	29.3939	1.22688
$575$	0	0
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	7.00000	0.291162
$579$	0	0
$580$	0	0
$581$	−24.0000	−0.995688
$582$	0	0
$583$	48.0000	1.98796
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−9.79796	−0.404750
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	4.89898	0.201347
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	−19.5959	−0.802680
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−48.0000	−1.95633
$603$	0	0
$604$	−16.0000	−0.651031
$605$	0	0
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−24.4949	−0.989340	−0.494670	−	0.869081i	$-0.664711\pi$
$613$	−24.4949	−0.989340	−0.494670	+	0.869081i	$0.664711\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	24.0000	0.966988
$617$	24.4949	0.986127	0.493064	−	0.869993i	$-0.335877\pi$
$617$	24.4949	0.986127	0.493064	+	0.869993i	$0.335877\pi$
$618$	0	0
$619$	−39.1918	−1.57525	−0.787626	−	0.616153i	$-0.788690\pi$
$619$	−39.1918	−1.57525	−0.787626	+	0.616153i	$0.788690\pi$
$620$	0	0
$621$	0	0
$622$	24.0000	0.962312
$623$	−72.0000	−2.88462
$624$	0	0
$625$	25.0000	1.00000
$626$	9.79796	0.391605
$627$	0	0
$628$	4.89898	0.195491
$629$	24.0000	0.956943
$630$	0	0
$631$	4.89898	0.195025	0.0975126	−	0.995234i	$-0.468911\pi$
$631$	4.89898	0.195025	0.0975126	+	0.995234i	$0.468911\pi$
$632$	14.6969	0.584613
$633$	0	0
$634$	6.00000	0.238290
$635$	0	0
$636$	0	0
$637$	−34.0000	−1.34713
$638$	29.3939	1.16371
$639$	0	0
$640$	0	0
$641$	24.4949	0.967490	0.483745	−	0.875209i	$-0.339276\pi$
$641$	24.4949	0.967490	0.483745	+	0.875209i	$0.339276\pi$
$642$	0	0
$643$	29.3939	1.15918	0.579591	−	0.814908i	$-0.303212\pi$
$643$	29.3939	1.15918	0.579591	+	0.814908i	$0.303212\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	0	0
$649$	58.7878	2.30762
$650$	10.0000	0.392232
$651$	0	0
$652$	−20.0000	−0.783260
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	0	0
$656$	6.00000	0.234261
$657$	0	0
$658$	0	0
$659$	24.4949	0.954186	0.477093	−	0.878853i	$-0.341691\pi$
$659$	24.4949	0.954186	0.477093	+	0.878853i	$0.341691\pi$
$660$	0	0
$661$	−24.4949	−0.952741	−0.476371	−	0.879245i	$-0.658048\pi$
$661$	−24.4949	−0.952741	−0.476371	+	0.879245i	$0.658048\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−4.89898	−0.190117
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−9.79796	−0.377403
$675$	0	0
$676$	−9.00000	−0.346154
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	−39.1918	−1.50073
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	0	0
$686$	48.9898	1.87044
$687$	0	0
$688$	−9.79796	−0.373544
$689$	−19.5959	−0.746545
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	29.3939	1.11337
$698$	2.00000	0.0757011
$699$	0	0
$700$	−24.4949	−0.925820
$701$	29.3939	1.11019	0.555096	−	0.831786i	$-0.312682\pi$
$701$	29.3939	1.11019	0.555096	+	0.831786i	$0.312682\pi$
$702$	0	0
$703$	0	0
$704$	4.89898	0.184637
$705$	0	0
$706$	18.0000	0.677439
$707$	−29.3939	−1.10547
$708$	0	0
$709$	−44.0908	−1.65587	−0.827933	−	0.560828i	$-0.810483\pi$
$709$	−44.0908	−1.65587	−0.827933	+	0.560828i	$0.810483\pi$
$710$	0	0
$711$	0	0
$712$	−14.6969	−0.550791
$713$	0	0
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	19.5959	0.731313
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	24.0000	0.893807
$722$	−19.0000	−0.707107
$723$	0	0
$724$	4.89898	0.182069
$725$	−30.0000	−1.11417
$726$	0	0
$727$	−14.6969	−0.545079	−0.272540	−	0.962145i	$-0.587864\pi$
$727$	−14.6969	−0.545079	−0.272540	+	0.962145i	$0.587864\pi$
$728$	−9.79796	−0.363137
$729$	0	0
$730$	0	0
$731$	−48.0000	−1.77534
$732$	0	0
$733$	−24.4949	−0.904740	−0.452370	−	0.891830i	$-0.649421\pi$
$733$	−24.4949	−0.904740	−0.452370	+	0.891830i	$0.649421\pi$
$734$	−24.4949	−0.904123
$735$	0	0
$736$	0	0
$737$	−48.0000	−1.76810
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	48.0000	1.76214
$743$	−48.9898	−1.79726	−0.898631	−	0.438706i	$-0.855437\pi$
$743$	−48.9898	−1.79726	−0.898631	+	0.438706i	$0.855437\pi$
$744$	0	0
$745$	0	0
$746$	−14.6969	−0.538093
$747$	0	0
$748$	24.0000	0.877527
$749$	−24.0000	−0.876941
$750$	0	0
$751$	−4.89898	−0.178766	−0.0893832	−	0.995997i	$-0.528490\pi$
$751$	−4.89898	−0.178766	−0.0893832	+	0.995997i	$0.528490\pi$
$752$	0	0
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	−34.2929	−1.24640	−0.623198	−	0.782064i	$-0.714167\pi$
$757$	−34.2929	−1.24640	−0.623198	+	0.782064i	$0.714167\pi$
$758$	19.5959	0.711756
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	24.0000	0.868858
$764$	−19.5959	−0.708955
$765$	0	0
$766$	9.79796	0.354015
$767$	−24.0000	−0.866590
$768$	0	0
$769$	29.3939	1.05997	0.529985	−	0.848007i	$-0.322197\pi$
$769$	29.3939	1.05997	0.529985	+	0.848007i	$0.322197\pi$
$770$	0	0
$771$	0	0
$772$	−10.0000	−0.359908
$773$	29.3939	1.05722	0.528612	−	0.848863i	$-0.322713\pi$
$773$	29.3939	1.05722	0.528612	+	0.848863i	$0.322713\pi$
$774$	0	0
$775$	40.0000	1.43684
$776$	0	0
$777$	0	0
$778$	19.5959	0.702548
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	17.0000	0.607143
$785$	0	0
$786$	0	0
$787$	9.79796	0.349260	0.174630	−	0.984634i	$-0.444127\pi$
$787$	9.79796	0.349260	0.174630	+	0.984634i	$0.444127\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	0	0
$791$	−72.0000	−2.56003
$792$	0	0
$793$	9.79796	0.347936
$794$	14.0000	0.496841
$795$	0	0
$796$	14.6969	0.520919
$797$	−39.1918	−1.38825	−0.694123	−	0.719856i	$-0.744208\pi$
$797$	−39.1918	−1.38825	−0.694123	+	0.719856i	$0.744208\pi$
$798$	0	0
$799$	0	0
$800$	−5.00000	−0.176777
$801$	0	0
$802$	−34.2929	−1.21092
$803$	−9.79796	−0.345762
$804$	0	0
$805$	0	0
$806$	16.0000	0.563576
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	29.3939	1.03152
$813$	0	0
$814$	24.0000	0.841200
$815$	0	0
$816$	0	0
$817$	0	0
$818$	14.0000	0.489499
$819$	0	0
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	4.89898	0.170664
$825$	0	0
$826$	58.7878	2.04549
$827$	−14.6969	−0.511063	−0.255531	−	0.966801i	$-0.582250\pi$
$827$	−14.6969	−0.511063	−0.255531	+	0.966801i	$0.582250\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	83.2827	2.88557
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	14.6969	0.507697
$839$	−48.9898	−1.69132	−0.845658	−	0.533726i	$-0.820792\pi$
$839$	−48.9898	−1.69132	−0.845658	+	0.533726i	$0.820792\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−4.89898	−0.168830
$843$	0	0
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	63.6867	2.18830
$848$	9.79796	0.336463
$849$	0	0
$850$	−24.4949	−0.840168
$851$	0	0
$852$	0	0
$853$	38.0000	1.30110	0.650548	−	0.759465i	$-0.274539\pi$
$853$	38.0000	1.30110	0.650548	+	0.759465i	$0.274539\pi$
$854$	−24.0000	−0.821263
$855$	0	0
$856$	−4.89898	−0.167444
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	29.3939	1.00116
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	−9.79796	−0.332948
$867$	0	0
$868$	−39.1918	−1.33026
$869$	72.0000	2.44243
$870$	0	0
$871$	19.5959	0.663982
$872$	4.89898	0.165900
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	−40.0000	−1.34993
$879$	0	0
$880$	0	0
$881$	24.4949	0.825254	0.412627	−	0.910900i	$-0.364611\pi$
$881$	24.4949	0.825254	0.412627	+	0.910900i	$0.364611\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	−9.79796	−0.329541
$885$	0	0
$886$	−12.0000	−0.403148
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	78.3837	2.62890
$890$	0	0
$891$	0	0
$892$	8.00000	0.267860
$893$	0	0
$894$	0	0
$895$	0	0
$896$	4.89898	0.163663
$897$	0	0
$898$	30.0000	1.00111
$899$	−48.0000	−1.60089
$900$	0	0
$901$	48.0000	1.59911
$902$	29.3939	0.978709
$903$	0	0
$904$	−14.6969	−0.488813
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	14.6969	0.487735
$909$	0	0
$910$	0	0
$911$	9.79796	0.324621	0.162310	−	0.986740i	$-0.448105\pi$
$911$	9.79796	0.324621	0.162310	+	0.986740i	$0.448105\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	−19.5959	−0.648175
$915$	0	0
$916$	24.4949	0.809334
$917$	−58.7878	−1.94134
$918$	0	0
$919$	14.6969	0.484807	0.242404	−	0.970175i	$-0.422064\pi$
$919$	14.6969	0.484807	0.242404	+	0.970175i	$0.422064\pi$
$920$	0	0
$921$	0	0
$922$	−18.0000	−0.592798
$923$	0	0
$924$	0	0
$925$	−24.4949	−0.805387
$926$	−16.0000	−0.525793
$927$	0	0
$928$	6.00000	0.196960
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	6.00000	0.196537
$933$	0	0
$934$	24.4949	0.801498
$935$	0	0
$936$	0	0
$937$	19.5959	0.640171	0.320085	−	0.947389i	$-0.396288\pi$
$937$	19.5959	0.640171	0.320085	+	0.947389i	$0.396288\pi$
$938$	−48.0000	−1.56726
$939$	0	0
$940$	0	0
$941$	19.5959	0.638809	0.319404	−	0.947619i	$-0.396517\pi$
$941$	19.5959	0.638809	0.319404	+	0.947619i	$0.396517\pi$
$942$	0	0
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	−48.0000	−1.56061
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	0	0
$952$	24.0000	0.777844
$953$	−34.2929	−1.11085	−0.555427	−	0.831565i	$-0.687445\pi$
$953$	−34.2929	−1.11085	−0.555427	+	0.831565i	$0.687445\pi$
$954$	0	0
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−19.5959	−0.633115
$959$	−24.0000	−0.775000
$960$	0	0
$961$	33.0000	1.06452
$962$	−9.79796	−0.315899
$963$	0	0
$964$	−19.5959	−0.631142
$965$	0	0
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	13.0000	0.417836
$969$	0	0
$970$	0	0
$971$	−24.4949	−0.786079	−0.393039	−	0.919522i	$-0.628576\pi$
$971$	−24.4949	−0.786079	−0.393039	+	0.919522i	$0.628576\pi$
$972$	0	0
$973$	−19.5959	−0.628216
$974$	8.00000	0.256337
$975$	0	0
$976$	−4.89898	−0.156813
$977$	−24.4949	−0.783661	−0.391831	−	0.920037i	$-0.628158\pi$
$977$	−24.4949	−0.783661	−0.391831	+	0.920037i	$0.628158\pi$
$978$	0	0
$979$	−72.0000	−2.30113
$980$	0	0
$981$	0	0
$982$	−12.0000	−0.382935
$983$	9.79796	0.312506	0.156253	−	0.987717i	$-0.450058\pi$
$983$	9.79796	0.312506	0.156253	+	0.987717i	$0.450058\pi$
$984$	0	0
$985$	0	0
$986$	29.3939	0.936092
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	−8.00000	−0.254000
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	−20.0000	−0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.bf.1.2		2
3.2	odd	2		3174.2.a.k.1.2	yes	2
23.22	odd	2	inner	9522.2.a.bf.1.1		2
69.68	even	2		3174.2.a.k.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3174.2.a.k.1.1	✓	2	69.68	even	2
3174.2.a.k.1.2	yes	2	3.2	odd	2
9522.2.a.bf.1.1		2	23.22	odd	2	inner
9522.2.a.bf.1.2		2	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.89898 q^{7} +1.00000 q^{8} +4.89898 q^{11} -2.00000 q^{13} +4.89898 q^{14} +1.00000 q^{16} +4.89898 q^{17} +4.89898 q^{22} -5.00000 q^{25} -2.00000 q^{26} +4.89898 q^{28} +6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +4.89898 q^{34} +4.89898 q^{37} +6.00000 q^{41} -9.79796 q^{43} +4.89898 q^{44} +17.0000 q^{49} -5.00000 q^{50} -2.00000 q^{52} +9.79796 q^{53} +4.89898 q^{56} +6.00000 q^{58} +12.0000 q^{59} -4.89898 q^{61} -8.00000 q^{62} +1.00000 q^{64} -9.79796 q^{67} +4.89898 q^{68} -2.00000 q^{73} +4.89898 q^{74} +24.0000 q^{77} +14.6969 q^{79} +6.00000 q^{82} -4.89898 q^{83} -9.79796 q^{86} +4.89898 q^{88} -14.6969 q^{89} -9.79796 q^{91} +17.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 4 q^{13} + 2 q^{16} - 10 q^{25} - 4 q^{26} + 12 q^{29} - 16 q^{31} + 2 q^{32} + 12 q^{41} + 34 q^{49} - 10 q^{50} - 4 q^{52} + 12 q^{58} + 24 q^{59} - 16 q^{62} + 2 q^{64}+ \cdots + 34 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 4 * q^13 + 2 * q^16 - 10 * q^25 - 4 * q^26 + 12 * q^29 - 16 * q^31 + 2 * q^32 + 12 * q^41 + 34 * q^49 - 10 * q^50 - 4 * q^52 + 12 * q^58 + 24 * q^59 - 16 * q^62 + 2 * q^64 - 4 * q^73 + 48 * q^77 + 12 * q^82 + 34 * q^98

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