LMFDB - Embedded newform 9522.2.a.bp.1.3

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)

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")

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);

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 = [5,-5,0,5,-8,0,7,-5,0,8,-5,0,-7,-7,0,5,-13,0,12,-8,0,5,0,0,1, 7,0,7,4] 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:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 46)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.68251$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.37279 q^{5} -0.397877 q^{7} -1.00000 q^{8} +2.37279 q^{10} +3.21076 q^{11} +0.155465 q^{13} +0.397877 q^{14} +1.00000 q^{16} -3.28242 q^{17} +2.84574 q^{19} -2.37279 q^{20} -3.21076 q^{22} +0.630111 q^{25} -0.155465 q^{26} -0.397877 q^{28} -0.378719 q^{29} +2.74692 q^{31} -1.00000 q^{32} +3.28242 q^{34} +0.944078 q^{35} +6.42128 q^{37} -2.84574 q^{38} +2.37279 q^{40} +4.00379 q^{41} +7.43011 q^{43} +3.21076 q^{44} -2.97017 q^{47} -6.84169 q^{49} -0.630111 q^{50} +0.155465 q^{52} -13.0325 q^{53} -7.61844 q^{55} +0.397877 q^{56} +0.378719 q^{58} -8.64612 q^{59} +11.1436 q^{61} -2.74692 q^{62} +1.00000 q^{64} -0.368885 q^{65} +8.40761 q^{67} -3.28242 q^{68} -0.944078 q^{70} -0.356673 q^{71} +5.63594 q^{73} -6.42128 q^{74} +2.84574 q^{76} -1.27749 q^{77} -7.53843 q^{79} -2.37279 q^{80} -4.00379 q^{82} -13.3899 q^{83} +7.78847 q^{85} -7.43011 q^{86} -3.21076 q^{88} -2.16071 q^{89} -0.0618559 q^{91} +2.97017 q^{94} -6.75234 q^{95} -2.63000 q^{97} +6.84169 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} - 8 q^{5} + 7 q^{7} - 5 q^{8} + 8 q^{10} - 5 q^{11} - 7 q^{13} - 7 q^{14} + 5 q^{16} - 13 q^{17} + 12 q^{19} - 8 q^{20} + 5 q^{22} + q^{25} + 7 q^{26} + 7 q^{28} + 4 q^{29} + 6 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 - 8 * q^5 + 7 * q^7 - 5 * q^8 + 8 * q^10 - 5 * q^11 - 7 * q^13 - 7 * q^14 + 5 * q^16 - 13 * q^17 + 12 * q^19 - 8 * q^20 + 5 * q^22 + q^25 + 7 * q^26 + 7 * q^28 + 4 * q^29 + 6 * q^31 - 5 * q^32 + 13 * q^34 - 9 * q^35 + 14 * q^37 - 12 * q^38 + 8 * q^40 - q^41 + 11 * q^43 - 5 * q^44 - 9 * q^47 - 12 * q^49 - q^50 - 7 * q^52 - 6 * q^53 - 14 * q^55 - 7 * q^56 - 4 * q^58 - 3 * q^59 + 13 * q^61 - 6 * q^62 + 5 * q^64 + 9 * q^65 + 10 * q^67 - 13 * q^68 + 9 * q^70 - 8 * q^71 - 4 * q^73 - 14 * q^74 + 12 * q^76 - 18 * q^77 - 18 * q^79 - 8 * q^80 + q^82 - 2 * q^83 + 23 * q^85 - 11 * q^86 + 5 * q^88 - 10 * q^89 - 12 * q^91 + 9 * q^94 - 28 * q^95 + 17 * q^97 + 12 * 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
$5$	−2.37279	−1.06114	−0.530571	−	0.847641i	$-0.678022\pi$
$5$	−2.37279	−1.06114	−0.530571	+	0.847641i	$0.678022\pi$
$6$	0	0
$7$	−0.397877	−0.150384	−0.0751918	−	0.997169i	$-0.523957\pi$
$7$	−0.397877	−0.150384	−0.0751918	+	0.997169i	$0.523957\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.37279	0.750341
$11$	3.21076	0.968080	0.484040	−	0.875046i	$-0.339169\pi$
$11$	3.21076	0.968080	0.484040	+	0.875046i	$0.339169\pi$
$12$	0	0
$13$	0.155465	0.0431182	0.0215591	−	0.999768i	$-0.493137\pi$
$13$	0.155465	0.0431182	0.0215591	+	0.999768i	$0.493137\pi$
$14$	0.397877	0.106337
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.28242	−0.796103	−0.398052	−	0.917363i	$-0.630314\pi$
$17$	−3.28242	−0.796103	−0.398052	+	0.917363i	$0.630314\pi$
$18$	0	0
$19$	2.84574	0.652858	0.326429	−	0.945222i	$-0.394155\pi$
$19$	2.84574	0.652858	0.326429	+	0.945222i	$0.394155\pi$
$20$	−2.37279	−0.530571
$21$	0	0
$22$	−3.21076	−0.684536
$23$	0	0
$23$	0	0
$24$	0	0
$25$	0.630111	0.126022
$26$	−0.155465	−0.0304892
$27$	0	0
$28$	−0.397877	−0.0751918
$29$	−0.378719	−0.0703264	−0.0351632	−	0.999382i	$-0.511195\pi$
$29$	−0.378719	−0.0703264	−0.0351632	+	0.999382i	$0.511195\pi$
$30$	0	0
$31$	2.74692	0.493361	0.246681	−	0.969097i	$-0.420660\pi$
$31$	2.74692	0.493361	0.246681	+	0.969097i	$0.420660\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.28242	0.562930
$35$	0.944078	0.159578
$36$	0	0
$37$	6.42128	1.05565	0.527826	−	0.849353i	$-0.323007\pi$
$37$	6.42128	1.05565	0.527826	+	0.849353i	$0.323007\pi$
$38$	−2.84574	−0.461641
$39$	0	0
$40$	2.37279	0.375170
$41$	4.00379	0.625288	0.312644	−	0.949870i	$-0.398785\pi$
$41$	4.00379	0.625288	0.312644	+	0.949870i	$0.398785\pi$
$42$	0	0
$43$	7.43011	1.13308	0.566541	−	0.824034i	$-0.308281\pi$
$43$	7.43011	1.13308	0.566541	+	0.824034i	$0.308281\pi$
$44$	3.21076	0.484040
$45$	0	0
$46$	0	0
$47$	−2.97017	−0.433244	−0.216622	−	0.976256i	$-0.569504\pi$
$47$	−2.97017	−0.433244	−0.216622	+	0.976256i	$0.569504\pi$
$48$	0	0
$49$	−6.84169	−0.977385
$50$	−0.630111	−0.0891112
$51$	0	0
$52$	0.155465	0.0215591
$53$	−13.0325	−1.79015	−0.895076	−	0.445914i	$-0.852879\pi$
$53$	−13.0325	−1.79015	−0.895076	+	0.445914i	$0.852879\pi$
$54$	0	0
$55$	−7.61844	−1.02727
$56$	0.397877	0.0531686
$57$	0	0
$58$	0.378719	0.0497283
$59$	−8.64612	−1.12563	−0.562814	−	0.826583i	$-0.690281\pi$
$59$	−8.64612	−1.12563	−0.562814	+	0.826583i	$0.690281\pi$
$60$	0	0
$61$	11.1436	1.42680	0.713399	−	0.700758i	$-0.247155\pi$
$61$	11.1436	1.42680	0.713399	+	0.700758i	$0.247155\pi$
$62$	−2.74692	−0.348859
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.368885	−0.0457545
$66$	0	0
$67$	8.40761	1.02715	0.513576	−	0.858044i	$-0.328320\pi$
$67$	8.40761	1.02715	0.513576	+	0.858044i	$0.328320\pi$
$68$	−3.28242	−0.398052
$69$	0	0
$70$	−0.944078	−0.112839
$71$	−0.356673	−0.0423294	−0.0211647	−	0.999776i	$-0.506737\pi$
$71$	−0.356673	−0.0423294	−0.0211647	+	0.999776i	$0.506737\pi$
$72$	0	0
$73$	5.63594	0.659637	0.329818	−	0.944044i	$-0.393012\pi$
$73$	5.63594	0.659637	0.329818	+	0.944044i	$0.393012\pi$
$74$	−6.42128	−0.746459
$75$	0	0
$76$	2.84574	0.326429
$77$	−1.27749	−0.145583
$78$	0	0
$79$	−7.53843	−0.848140	−0.424070	−	0.905630i	$-0.639399\pi$
$79$	−7.53843	−0.848140	−0.424070	+	0.905630i	$0.639399\pi$
$80$	−2.37279	−0.265285
$81$	0	0
$82$	−4.00379	−0.442145
$83$	−13.3899	−1.46973	−0.734864	−	0.678215i	$-0.762754\pi$
$83$	−13.3899	−1.46973	−0.734864	+	0.678215i	$0.762754\pi$
$84$	0	0
$85$	7.78847	0.844779
$86$	−7.43011	−0.801209
$87$	0	0
$88$	−3.21076	−0.342268
$89$	−2.16071	−0.229035	−0.114518	−	0.993421i	$-0.536532\pi$
$89$	−2.16071	−0.229035	−0.114518	+	0.993421i	$0.536532\pi$
$90$	0	0
$91$	−0.0618559	−0.00648426
$92$	0	0
$93$	0	0
$94$	2.97017	0.306350
$95$	−6.75234	−0.692775
$96$	0	0
$97$	−2.63000	−0.267037	−0.133518	−	0.991046i	$-0.542627\pi$
$97$	−2.63000	−0.267037	−0.133518	+	0.991046i	$0.542627\pi$
$98$	6.84169	0.691115
$99$	0	0
$100$	0.630111	0.0630111
$101$	8.73694	0.869358	0.434679	−	0.900586i	$-0.356862\pi$
$101$	8.73694	0.869358	0.434679	+	0.900586i	$0.356862\pi$
$102$	0	0
$103$	4.28331	0.422048	0.211024	−	0.977481i	$-0.432320\pi$
$103$	4.28331	0.422048	0.211024	+	0.977481i	$0.432320\pi$
$104$	−0.155465	−0.0152446
$105$	0	0
$106$	13.0325	1.26583
$107$	18.0123	1.74132	0.870659	−	0.491887i	$-0.163693\pi$
$107$	18.0123	1.74132	0.870659	+	0.491887i	$0.163693\pi$
$108$	0	0
$109$	−9.53659	−0.913440	−0.456720	−	0.889611i	$-0.650976\pi$
$109$	−9.53659	−0.913440	−0.456720	+	0.889611i	$0.650976\pi$
$110$	7.61844	0.726390
$111$	0	0
$112$	−0.397877	−0.0375959
$113$	6.28331	0.591085	0.295542	−	0.955330i	$-0.404500\pi$
$113$	6.28331	0.591085	0.295542	+	0.955330i	$0.404500\pi$
$114$	0	0
$115$	0	0
$116$	−0.378719	−0.0351632
$117$	0	0
$118$	8.64612	0.795940
$119$	1.30600	0.119721
$120$	0	0
$121$	−0.691036	−0.0628215
$122$	−11.1436	−1.00890
$123$	0	0
$124$	2.74692	0.246681
$125$	10.3688	0.927414
$126$	0	0
$127$	−8.86900	−0.786996	−0.393498	−	0.919325i	$-0.628735\pi$
$127$	−8.86900	−0.786996	−0.393498	+	0.919325i	$0.628735\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0.368885	0.0323533
$131$	−5.20178	−0.454481	−0.227241	−	0.973839i	$-0.572970\pi$
$131$	−5.20178	−0.454481	−0.227241	+	0.973839i	$0.572970\pi$
$132$	0	0
$133$	−1.13226	−0.0981791
$134$	−8.40761	−0.726306
$135$	0	0
$136$	3.28242	0.281465
$137$	0.501086	0.0428107	0.0214053	−	0.999771i	$-0.493186\pi$
$137$	0.501086	0.0428107	0.0214053	+	0.999771i	$0.493186\pi$
$138$	0	0
$139$	−15.7509	−1.33598	−0.667989	−	0.744171i	$-0.732845\pi$
$139$	−15.7509	−1.33598	−0.667989	+	0.744171i	$0.732845\pi$
$140$	0.944078	0.0797891
$141$	0	0
$142$	0.356673	0.0299314
$143$	0.499160	0.0417418
$144$	0	0
$145$	0.898620	0.0746263
$146$	−5.63594	−0.466434
$147$	0	0
$148$	6.42128	0.527826
$149$	3.86997	0.317040	0.158520	−	0.987356i	$-0.449328\pi$
$149$	3.86997	0.317040	0.158520	+	0.987356i	$0.449328\pi$
$150$	0	0
$151$	−15.8518	−1.29000	−0.644999	−	0.764183i	$-0.723142\pi$
$151$	−15.8518	−1.29000	−0.644999	+	0.764183i	$0.723142\pi$
$152$	−2.84574	−0.230820
$153$	0	0
$154$	1.27749	0.102943
$155$	−6.51785	−0.523526
$156$	0	0
$157$	20.8710	1.66568	0.832842	−	0.553511i	$-0.186712\pi$
$157$	20.8710	1.66568	0.832842	+	0.553511i	$0.186712\pi$
$158$	7.53843	0.599725
$159$	0	0
$160$	2.37279	0.187585
$161$	0	0
$162$	0	0
$163$	23.7839	1.86290	0.931450	−	0.363869i	$-0.118545\pi$
$163$	23.7839	1.86290	0.931450	+	0.363869i	$0.118545\pi$
$164$	4.00379	0.312644
$165$	0	0
$166$	13.3899	1.03925
$167$	−5.58734	−0.432361	−0.216181	−	0.976353i	$-0.569360\pi$
$167$	−5.58734	−0.432361	−0.216181	+	0.976353i	$0.569360\pi$
$168$	0	0
$169$	−12.9758	−0.998141
$170$	−7.78847	−0.597349
$171$	0	0
$172$	7.43011	0.566541
$173$	22.7623	1.73059	0.865294	−	0.501264i	$-0.167132\pi$
$173$	22.7623	1.73059	0.865294	+	0.501264i	$0.167132\pi$
$174$	0	0
$175$	−0.250707	−0.0189517
$176$	3.21076	0.242020
$177$	0	0
$178$	2.16071	0.161952
$179$	−2.49235	−0.186287	−0.0931435	−	0.995653i	$-0.529692\pi$
$179$	−2.49235	−0.186287	−0.0931435	+	0.995653i	$0.529692\pi$
$180$	0	0
$181$	10.1004	0.750754	0.375377	−	0.926872i	$-0.377513\pi$
$181$	10.1004	0.750754	0.375377	+	0.926872i	$0.377513\pi$
$182$	0.0618559	0.00458507
$183$	0	0
$184$	0	0
$185$	−15.2363	−1.12020
$186$	0	0
$187$	−10.5390	−0.770691
$188$	−2.97017	−0.216622
$189$	0	0
$190$	6.75234	0.489866
$191$	−2.57056	−0.185999	−0.0929995	−	0.995666i	$-0.529645\pi$
$191$	−2.57056	−0.185999	−0.0929995	+	0.995666i	$0.529645\pi$
$192$	0	0
$193$	−22.5870	−1.62585	−0.812923	−	0.582371i	$-0.802125\pi$
$193$	−22.5870	−1.62585	−0.812923	+	0.582371i	$0.802125\pi$
$194$	2.63000	0.188823
$195$	0	0
$196$	−6.84169	−0.488692
$197$	−0.512278	−0.0364983	−0.0182491	−	0.999833i	$-0.505809\pi$
$197$	−0.512278	−0.0364983	−0.0182491	+	0.999833i	$0.505809\pi$
$198$	0	0
$199$	−1.31542	−0.0932477	−0.0466238	−	0.998913i	$-0.514846\pi$
$199$	−1.31542	−0.0932477	−0.0466238	+	0.998913i	$0.514846\pi$
$200$	−0.630111	−0.0445556
$201$	0	0
$202$	−8.73694	−0.614729
$203$	0.150684	0.0105759
$204$	0	0
$205$	−9.50015	−0.663519
$206$	−4.28331	−0.298433
$207$	0	0
$208$	0.155465	0.0107795
$209$	9.13699	0.632019
$210$	0	0
$211$	−7.01387	−0.482855	−0.241427	−	0.970419i	$-0.577616\pi$
$211$	−7.01387	−0.482855	−0.241427	+	0.970419i	$0.577616\pi$
$212$	−13.0325	−0.895076
$213$	0	0
$214$	−18.0123	−1.23130
$215$	−17.6301	−1.20236
$216$	0	0
$217$	−1.09294	−0.0741934
$218$	9.53659	0.645899
$219$	0	0
$220$	−7.61844	−0.513635
$221$	−0.510301	−0.0343265
$222$	0	0
$223$	12.8998	0.863837	0.431919	−	0.901913i	$-0.357837\pi$
$223$	12.8998	0.863837	0.431919	+	0.901913i	$0.357837\pi$
$224$	0.397877	0.0265843
$225$	0	0
$226$	−6.28331	−0.417960
$227$	0.770428	0.0511351	0.0255676	−	0.999673i	$-0.491861\pi$
$227$	0.770428	0.0511351	0.0255676	+	0.999673i	$0.491861\pi$
$228$	0	0
$229$	−19.2701	−1.27341	−0.636703	−	0.771109i	$-0.719702\pi$
$229$	−19.2701	−1.27341	−0.636703	+	0.771109i	$0.719702\pi$
$230$	0	0
$231$	0	0
$232$	0.378719	0.0248641
$233$	−8.46744	−0.554721	−0.277360	−	0.960766i	$-0.589460\pi$
$233$	−8.46744	−0.554721	−0.277360	+	0.960766i	$0.589460\pi$
$234$	0	0
$235$	7.04758	0.459734
$236$	−8.64612	−0.562814
$237$	0	0
$238$	−1.30600	−0.0846554
$239$	−7.94660	−0.514023	−0.257011	−	0.966408i	$-0.582738\pi$
$239$	−7.94660	−0.514023	−0.257011	+	0.966408i	$0.582738\pi$
$240$	0	0
$241$	13.4446	0.866043	0.433022	−	0.901384i	$-0.357447\pi$
$241$	13.4446	0.866043	0.433022	+	0.901384i	$0.357447\pi$
$242$	0.691036	0.0444215
$243$	0	0
$244$	11.1436	0.713399
$245$	16.2339	1.03714
$246$	0	0
$247$	0.442413	0.0281501
$248$	−2.74692	−0.174430
$249$	0	0
$250$	−10.3688	−0.655781
$251$	−15.0551	−0.950268	−0.475134	−	0.879913i	$-0.657600\pi$
$251$	−15.0551	−0.950268	−0.475134	+	0.879913i	$0.657600\pi$
$252$	0	0
$253$	0	0
$254$	8.86900	0.556490
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.70307	0.293369	0.146685	−	0.989183i	$-0.453140\pi$
$257$	4.70307	0.293369	0.146685	+	0.989183i	$0.453140\pi$
$258$	0	0
$259$	−2.55488	−0.158753
$260$	−0.368885	−0.0228773
$261$	0	0
$262$	5.20178	0.321367
$263$	20.6640	1.27420	0.637098	−	0.770782i	$-0.280134\pi$
$263$	20.6640	1.27420	0.637098	+	0.770782i	$0.280134\pi$
$264$	0	0
$265$	30.9233	1.89961
$266$	1.13226	0.0694231
$267$	0	0
$268$	8.40761	0.513576
$269$	−16.7791	−1.02304	−0.511519	−	0.859272i	$-0.670917\pi$
$269$	−16.7791	−1.02304	−0.511519	+	0.859272i	$0.670917\pi$
$270$	0	0
$271$	10.2655	0.623586	0.311793	−	0.950150i	$-0.399070\pi$
$271$	10.2655	0.623586	0.311793	+	0.950150i	$0.399070\pi$
$272$	−3.28242	−0.199026
$273$	0	0
$274$	−0.501086	−0.0302717
$275$	2.02314	0.122000
$276$	0	0
$277$	25.2836	1.51914	0.759572	−	0.650423i	$-0.225408\pi$
$277$	25.2836	1.51914	0.759572	+	0.650423i	$0.225408\pi$
$278$	15.7509	0.944679
$279$	0	0
$280$	−0.944078	−0.0564194
$281$	13.4458	0.802110	0.401055	−	0.916054i	$-0.368644\pi$
$281$	13.4458	0.802110	0.401055	+	0.916054i	$0.368644\pi$
$282$	0	0
$283$	−4.38852	−0.260870	−0.130435	−	0.991457i	$-0.541637\pi$
$283$	−4.38852	−0.260870	−0.130435	+	0.991457i	$0.541637\pi$
$284$	−0.356673	−0.0211647
$285$	0	0
$286$	−0.499160	−0.0295159
$287$	−1.59302	−0.0940330
$288$	0	0
$289$	−6.22573	−0.366220
$290$	−0.898620	−0.0527688
$291$	0	0
$292$	5.63594	0.329818
$293$	5.50794	0.321777	0.160889	−	0.986973i	$-0.448564\pi$
$293$	5.50794	0.321777	0.160889	+	0.986973i	$0.448564\pi$
$294$	0	0
$295$	20.5154	1.19445
$296$	−6.42128	−0.373229
$297$	0	0
$298$	−3.86997	−0.224181
$299$	0	0
$300$	0	0
$301$	−2.95627	−0.170397
$302$	15.8518	0.912167
$303$	0	0
$304$	2.84574	0.163215
$305$	−26.4415	−1.51403
$306$	0	0
$307$	−25.5133	−1.45612	−0.728060	−	0.685513i	$-0.759578\pi$
$307$	−25.5133	−1.45612	−0.728060	+	0.685513i	$0.759578\pi$
$308$	−1.27749	−0.0727916
$309$	0	0
$310$	6.51785	0.370189
$311$	−10.0559	−0.570219	−0.285110	−	0.958495i	$-0.592030\pi$
$311$	−10.0559	−0.570219	−0.285110	+	0.958495i	$0.592030\pi$
$312$	0	0
$313$	−3.51695	−0.198790	−0.0993949	−	0.995048i	$-0.531691\pi$
$313$	−3.51695	−0.198790	−0.0993949	+	0.995048i	$0.531691\pi$
$314$	−20.8710	−1.17782
$315$	0	0
$316$	−7.53843	−0.424070
$317$	34.9502	1.96300	0.981499	−	0.191466i	$-0.0613241\pi$
$317$	34.9502	1.96300	0.981499	+	0.191466i	$0.0613241\pi$
$318$	0	0
$319$	−1.21598	−0.0680816
$320$	−2.37279	−0.132643
$321$	0	0
$322$	0	0
$323$	−9.34092	−0.519743
$324$	0	0
$325$	0.0979602	0.00543385
$326$	−23.7839	−1.31727
$327$	0	0
$328$	−4.00379	−0.221073
$329$	1.18176	0.0651528
$330$	0	0
$331$	2.35635	0.129517	0.0647584	−	0.997901i	$-0.479372\pi$
$331$	2.35635	0.129517	0.0647584	+	0.997901i	$0.479372\pi$
$332$	−13.3899	−0.734864
$333$	0	0
$334$	5.58734	0.305726
$335$	−19.9494	−1.08995
$336$	0	0
$337$	17.8166	0.970530	0.485265	−	0.874367i	$-0.338723\pi$
$337$	17.8166	0.970530	0.485265	+	0.874367i	$0.338723\pi$
$338$	12.9758	0.705792
$339$	0	0
$340$	7.78847	0.422389
$341$	8.81969	0.477613
$342$	0	0
$343$	5.50730	0.297366
$344$	−7.43011	−0.400605
$345$	0	0
$346$	−22.7623	−1.22371
$347$	23.1166	1.24097	0.620483	−	0.784220i	$-0.286937\pi$
$347$	23.1166	1.24097	0.620483	+	0.784220i	$0.286937\pi$
$348$	0	0
$349$	9.92941	0.531509	0.265755	−	0.964041i	$-0.414379\pi$
$349$	9.92941	0.531509	0.265755	+	0.964041i	$0.414379\pi$
$350$	0.250707	0.0134009
$351$	0	0
$352$	−3.21076	−0.171134
$353$	9.79362	0.521262	0.260631	−	0.965439i	$-0.416069\pi$
$353$	9.79362	0.521262	0.260631	+	0.965439i	$0.416069\pi$
$354$	0	0
$355$	0.846310	0.0449175
$356$	−2.16071	−0.114518
$357$	0	0
$358$	2.49235	0.131725
$359$	26.9003	1.41974	0.709872	−	0.704331i	$-0.248753\pi$
$359$	26.9003	1.41974	0.709872	+	0.704331i	$0.248753\pi$
$360$	0	0
$361$	−10.9017	−0.573776
$362$	−10.1004	−0.530864
$363$	0	0
$364$	−0.0618559	−0.00324213
$365$	−13.3729	−0.699968
$366$	0	0
$367$	20.9617	1.09419	0.547097	−	0.837069i	$-0.315733\pi$
$367$	20.9617	1.09419	0.547097	+	0.837069i	$0.315733\pi$
$368$	0	0
$369$	0	0
$370$	15.2363	0.792099
$371$	5.18534	0.269209
$372$	0	0
$373$	27.4107	1.41927	0.709637	−	0.704568i	$-0.248859\pi$
$373$	27.4107	1.41927	0.709637	+	0.704568i	$0.248859\pi$
$374$	10.5390	0.544961
$375$	0	0
$376$	2.97017	0.153175
$377$	−0.0588775	−0.00303235
$378$	0	0
$379$	−4.06405	−0.208756	−0.104378	−	0.994538i	$-0.533285\pi$
$379$	−4.06405	−0.208756	−0.104378	+	0.994538i	$0.533285\pi$
$380$	−6.75234	−0.346388
$381$	0	0
$382$	2.57056	0.131521
$383$	17.4544	0.891879	0.445940	−	0.895063i	$-0.352870\pi$
$383$	17.4544	0.891879	0.445940	+	0.895063i	$0.352870\pi$
$384$	0	0
$385$	3.03120	0.154484
$386$	22.5870	1.14965
$387$	0	0
$388$	−2.63000	−0.133518
$389$	−11.0073	−0.558094	−0.279047	−	0.960277i	$-0.590018\pi$
$389$	−11.0073	−0.558094	−0.279047	+	0.960277i	$0.590018\pi$
$390$	0	0
$391$	0	0
$392$	6.84169	0.345558
$393$	0	0
$394$	0.512278	0.0258082
$395$	17.8871	0.899996
$396$	0	0
$397$	22.4799	1.12824	0.564118	−	0.825694i	$-0.309216\pi$
$397$	22.4799	1.12824	0.564118	+	0.825694i	$0.309216\pi$
$398$	1.31542	0.0659361
$399$	0	0
$400$	0.630111	0.0315056
$401$	0.647541	0.0323367	0.0161683	−	0.999869i	$-0.494853\pi$
$401$	0.647541	0.0323367	0.0161683	+	0.999869i	$0.494853\pi$
$402$	0	0
$403$	0.427049	0.0212728
$404$	8.73694	0.434679
$405$	0	0
$406$	−0.150684	−0.00747831
$407$	20.6172	1.02196
$408$	0	0
$409$	23.9213	1.18283	0.591416	−	0.806366i	$-0.298569\pi$
$409$	23.9213	1.18283	0.591416	+	0.806366i	$0.298569\pi$
$410$	9.50015	0.469179
$411$	0	0
$412$	4.28331	0.211024
$413$	3.44009	0.169276
$414$	0	0
$415$	31.7712	1.55959
$416$	−0.155465	−0.00762229
$417$	0	0
$418$	−9.13699	−0.446905
$419$	32.7060	1.59779	0.798897	−	0.601468i	$-0.205417\pi$
$419$	32.7060	1.59779	0.798897	+	0.601468i	$0.205417\pi$
$420$	0	0
$421$	−27.9916	−1.36423	−0.682114	−	0.731245i	$-0.738939\pi$
$421$	−27.9916	−1.36423	−0.682114	+	0.731245i	$0.738939\pi$
$422$	7.01387	0.341430
$423$	0	0
$424$	13.0325	0.632914
$425$	−2.06829	−0.100327
$426$	0	0
$427$	−4.43380	−0.214567
$428$	18.0123	0.870659
$429$	0	0
$430$	17.6301	0.850197
$431$	−6.00349	−0.289178	−0.144589	−	0.989492i	$-0.546186\pi$
$431$	−6.00349	−0.289178	−0.144589	+	0.989492i	$0.546186\pi$
$432$	0	0
$433$	10.6707	0.512800	0.256400	−	0.966571i	$-0.417464\pi$
$433$	10.6707	0.512800	0.256400	+	0.966571i	$0.417464\pi$
$434$	1.09294	0.0524627
$435$	0	0
$436$	−9.53659	−0.456720
$437$	0	0
$438$	0	0
$439$	32.8435	1.56754	0.783768	−	0.621054i	$-0.213295\pi$
$439$	32.8435	1.56754	0.783768	+	0.621054i	$0.213295\pi$
$440$	7.61844	0.363195
$441$	0	0
$442$	0.510301	0.0242725
$443$	0.412897	0.0196173	0.00980867	−	0.999952i	$-0.496878\pi$
$443$	0.412897	0.0196173	0.00980867	+	0.999952i	$0.496878\pi$
$444$	0	0
$445$	5.12691	0.243039
$446$	−12.8998	−0.610825
$447$	0	0
$448$	−0.397877	−0.0187979
$449$	9.42926	0.444994	0.222497	−	0.974933i	$-0.428579\pi$
$449$	9.42926	0.444994	0.222497	+	0.974933i	$0.428579\pi$
$450$	0	0
$451$	12.8552	0.605328
$452$	6.28331	0.295542
$453$	0	0
$454$	−0.770428	−0.0361580
$455$	0.146771	0.00688073
$456$	0	0
$457$	−31.0334	−1.45168	−0.725840	−	0.687864i	$-0.758549\pi$
$457$	−31.0334	−1.45168	−0.725840	+	0.687864i	$0.758549\pi$
$458$	19.2701	0.900434
$459$	0	0
$460$	0	0
$461$	33.0117	1.53751	0.768753	−	0.639546i	$-0.220878\pi$
$461$	33.0117	1.53751	0.768753	+	0.639546i	$0.220878\pi$
$462$	0	0
$463$	−17.6701	−0.821201	−0.410600	−	0.911815i	$-0.634681\pi$
$463$	−17.6701	−0.821201	−0.410600	+	0.911815i	$0.634681\pi$
$464$	−0.378719	−0.0175816
$465$	0	0
$466$	8.46744	0.392247
$467$	11.1053	0.513892	0.256946	−	0.966426i	$-0.417284\pi$
$467$	11.1053	0.513892	0.256946	+	0.966426i	$0.417284\pi$
$468$	0	0
$469$	−3.34520	−0.154467
$470$	−7.04758	−0.325081
$471$	0	0
$472$	8.64612	0.397970
$473$	23.8563	1.09691
$474$	0	0
$475$	1.79314	0.0822747
$476$	1.30600	0.0598604
$477$	0	0
$478$	7.94660	0.363469
$479$	10.3037	0.470789	0.235394	−	0.971900i	$-0.424362\pi$
$479$	10.3037	0.470789	0.235394	+	0.971900i	$0.424362\pi$
$480$	0	0
$481$	0.998283	0.0455178
$482$	−13.4446	−0.612385
$483$	0	0
$484$	−0.691036	−0.0314107
$485$	6.24044	0.283364
$486$	0	0
$487$	−11.2887	−0.511538	−0.255769	−	0.966738i	$-0.582329\pi$
$487$	−11.2887	−0.511538	−0.255769	+	0.966738i	$0.582329\pi$
$488$	−11.1436	−0.504449
$489$	0	0
$490$	−16.2339	−0.733372
$491$	9.15794	0.413292	0.206646	−	0.978416i	$-0.433745\pi$
$491$	9.15794	0.413292	0.206646	+	0.978416i	$0.433745\pi$
$492$	0	0
$493$	1.24311	0.0559871
$494$	−0.442413	−0.0199051
$495$	0	0
$496$	2.74692	0.123340
$497$	0.141912	0.00636564
$498$	0	0
$499$	−31.2250	−1.39782	−0.698912	−	0.715208i	$-0.746332\pi$
$499$	−31.2250	−1.39782	−0.698912	+	0.715208i	$0.746332\pi$
$500$	10.3688	0.463707
$501$	0	0
$502$	15.0551	0.671941
$503$	−17.0084	−0.758367	−0.379183	−	0.925322i	$-0.623795\pi$
$503$	−17.0084	−0.758367	−0.379183	+	0.925322i	$0.623795\pi$
$504$	0	0
$505$	−20.7309	−0.922512
$506$	0	0
$507$	0	0
$508$	−8.86900	−0.393498
$509$	−18.3615	−0.813861	−0.406931	−	0.913459i	$-0.633401\pi$
$509$	−18.3615	−0.813861	−0.406931	+	0.913459i	$0.633401\pi$
$510$	0	0
$511$	−2.24241	−0.0991985
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−4.70307	−0.207443
$515$	−10.1634	−0.447852
$516$	0	0
$517$	−9.53651	−0.419415
$518$	2.55488	0.112255
$519$	0	0
$520$	0.368885	0.0161767
$521$	26.2822	1.15144	0.575721	−	0.817646i	$-0.304721\pi$
$521$	26.2822	1.15144	0.575721	+	0.817646i	$0.304721\pi$
$522$	0	0
$523$	−34.2705	−1.49855	−0.749273	−	0.662262i	$-0.769597\pi$
$523$	−34.2705	−1.49855	−0.749273	+	0.662262i	$0.769597\pi$
$524$	−5.20178	−0.227241
$525$	0	0
$526$	−20.6640	−0.900993
$527$	−9.01654	−0.392766
$528$	0	0
$529$	0	0
$530$	−30.9233	−1.34322
$531$	0	0
$532$	−1.13226	−0.0490896
$533$	0.622449	0.0269613
$534$	0	0
$535$	−42.7394	−1.84779
$536$	−8.40761	−0.363153
$537$	0	0
$538$	16.7791	0.723397
$539$	−21.9670	−0.946186
$540$	0	0
$541$	8.01966	0.344792	0.172396	−	0.985028i	$-0.444849\pi$
$541$	8.01966	0.344792	0.172396	+	0.985028i	$0.444849\pi$
$542$	−10.2655	−0.440942
$543$	0	0
$544$	3.28242	0.140732
$545$	22.6283	0.969289
$546$	0	0
$547$	14.1044	0.603059	0.301530	−	0.953457i	$-0.402503\pi$
$547$	14.1044	0.603059	0.301530	+	0.953457i	$0.402503\pi$
$548$	0.501086	0.0214053
$549$	0	0
$550$	−2.02314	−0.0862668
$551$	−1.07774	−0.0459132
$552$	0	0
$553$	2.99937	0.127546
$554$	−25.2836	−1.07420
$555$	0	0
$556$	−15.7509	−0.667989
$557$	6.54420	0.277286	0.138643	−	0.990342i	$-0.455726\pi$
$557$	6.54420	0.277286	0.138643	+	0.990342i	$0.455726\pi$
$558$	0	0
$559$	1.15512	0.0488564
$560$	0.944078	0.0398946
$561$	0	0
$562$	−13.4458	−0.567177
$563$	18.1561	0.765190	0.382595	−	0.923916i	$-0.375030\pi$
$563$	18.1561	0.765190	0.382595	+	0.923916i	$0.375030\pi$
$564$	0	0
$565$	−14.9090	−0.627225
$566$	4.38852	0.184463
$567$	0	0
$568$	0.356673	0.0149657
$569$	3.87946	0.162636	0.0813178	−	0.996688i	$-0.474087\pi$
$569$	3.87946	0.162636	0.0813178	+	0.996688i	$0.474087\pi$
$570$	0	0
$571$	33.0657	1.38376	0.691878	−	0.722014i	$-0.256783\pi$
$571$	33.0657	1.38376	0.691878	+	0.722014i	$0.256783\pi$
$572$	0.499160	0.0208709
$573$	0	0
$574$	1.59302	0.0664913
$575$	0	0
$576$	0	0
$577$	19.8949	0.828237	0.414118	−	0.910223i	$-0.364090\pi$
$577$	19.8949	0.828237	0.414118	+	0.910223i	$0.364090\pi$
$578$	6.22573	0.258956
$579$	0	0
$580$	0.898620	0.0373131
$581$	5.32752	0.221023
$582$	0	0
$583$	−41.8442	−1.73301
$584$	−5.63594	−0.233217
$585$	0	0
$586$	−5.50794	−0.227531
$587$	2.76600	0.114165	0.0570826	−	0.998369i	$-0.481820\pi$
$587$	2.76600	0.114165	0.0570826	+	0.998369i	$0.481820\pi$
$588$	0	0
$589$	7.81703	0.322095
$590$	−20.5154	−0.844605
$591$	0	0
$592$	6.42128	0.263913
$593$	−42.6402	−1.75102	−0.875511	−	0.483197i	$-0.839475\pi$
$593$	−42.6402	−1.75102	−0.875511	+	0.483197i	$0.839475\pi$
$594$	0	0
$595$	−3.09886	−0.127041
$596$	3.86997	0.158520
$597$	0	0
$598$	0	0
$599$	20.1889	0.824895	0.412448	−	0.910981i	$-0.364674\pi$
$599$	20.1889	0.824895	0.412448	+	0.910981i	$0.364674\pi$
$600$	0	0
$601$	−14.9028	−0.607899	−0.303950	−	0.952688i	$-0.598305\pi$
$601$	−14.9028	−0.607899	−0.303950	+	0.952688i	$0.598305\pi$
$602$	2.95627	0.120489
$603$	0	0
$604$	−15.8518	−0.644999
$605$	1.63968	0.0666625
$606$	0	0
$607$	30.7143	1.24666	0.623328	−	0.781961i	$-0.285780\pi$
$607$	30.7143	1.24666	0.623328	+	0.781961i	$0.285780\pi$
$608$	−2.84574	−0.115410
$609$	0	0
$610$	26.4415	1.07058
$611$	−0.461757	−0.0186807
$612$	0	0
$613$	18.9197	0.764157	0.382079	−	0.924130i	$-0.375208\pi$
$613$	18.9197	0.764157	0.382079	+	0.924130i	$0.375208\pi$
$614$	25.5133	1.02963
$615$	0	0
$616$	1.27749	0.0514715
$617$	21.3603	0.859934	0.429967	−	0.902845i	$-0.358525\pi$
$617$	21.3603	0.859934	0.429967	+	0.902845i	$0.358525\pi$
$618$	0	0
$619$	8.05295	0.323675	0.161838	−	0.986817i	$-0.448258\pi$
$619$	8.05295	0.323675	0.161838	+	0.986817i	$0.448258\pi$
$620$	−6.51785	−0.261763
$621$	0	0
$622$	10.0559	0.403206
$623$	0.859699	0.0344431
$624$	0	0
$625$	−27.7535	−1.11014
$626$	3.51695	0.140566
$627$	0	0
$628$	20.8710	0.832842
$629$	−21.0773	−0.840408
$630$	0	0
$631$	27.0590	1.07720	0.538600	−	0.842561i	$-0.318953\pi$
$631$	27.0590	1.07720	0.538600	+	0.842561i	$0.318953\pi$
$632$	7.53843	0.299863
$633$	0	0
$634$	−34.9502	−1.38805
$635$	21.0442	0.835115
$636$	0	0
$637$	−1.06364	−0.0421431
$638$	1.21598	0.0481409
$639$	0	0
$640$	2.37279	0.0937926
$641$	39.7357	1.56946	0.784732	−	0.619835i	$-0.212800\pi$
$641$	39.7357	1.56946	0.784732	+	0.619835i	$0.212800\pi$
$642$	0	0
$643$	−1.75237	−0.0691066	−0.0345533	−	0.999403i	$-0.511001\pi$
$643$	−1.75237	−0.0691066	−0.0345533	+	0.999403i	$0.511001\pi$
$644$	0	0
$645$	0	0
$646$	9.34092	0.367514
$647$	−8.90302	−0.350014	−0.175007	−	0.984567i	$-0.555995\pi$
$647$	−8.90302	−0.350014	−0.175007	+	0.984567i	$0.555995\pi$
$648$	0	0
$649$	−27.7606	−1.08970
$650$	−0.0979602	−0.00384231
$651$	0	0
$652$	23.7839	0.931450
$653$	22.9984	0.899995	0.449998	−	0.893030i	$-0.351425\pi$
$653$	22.9984	0.899995	0.449998	+	0.893030i	$0.351425\pi$
$654$	0	0
$655$	12.3427	0.482269
$656$	4.00379	0.156322
$657$	0	0
$658$	−1.18176	−0.0460700
$659$	−33.1406	−1.29097	−0.645486	−	0.763772i	$-0.723345\pi$
$659$	−33.1406	−1.29097	−0.645486	+	0.763772i	$0.723345\pi$
$660$	0	0
$661$	−21.4839	−0.835627	−0.417813	−	0.908533i	$-0.637203\pi$
$661$	−21.4839	−0.835627	−0.417813	+	0.908533i	$0.637203\pi$
$662$	−2.35635	−0.0915823
$663$	0	0
$664$	13.3899	0.519627
$665$	2.68660	0.104182
$666$	0	0
$667$	0	0
$668$	−5.58734	−0.216181
$669$	0	0
$670$	19.9494	0.770714
$671$	35.7795	1.38125
$672$	0	0
$673$	25.5429	0.984605	0.492302	−	0.870424i	$-0.336155\pi$
$673$	25.5429	0.984605	0.492302	+	0.870424i	$0.336155\pi$
$674$	−17.8166	−0.686269
$675$	0	0
$676$	−12.9758	−0.499070
$677$	48.1558	1.85078	0.925390	−	0.379017i	$-0.123738\pi$
$677$	48.1558	1.85078	0.925390	+	0.379017i	$0.123738\pi$
$678$	0	0
$679$	1.04642	0.0401579
$680$	−7.78847	−0.298674
$681$	0	0
$682$	−8.81969	−0.337723
$683$	4.70614	0.180075	0.0900377	−	0.995938i	$-0.471301\pi$
$683$	4.70614	0.180075	0.0900377	+	0.995938i	$0.471301\pi$
$684$	0	0
$685$	−1.18897	−0.0454282
$686$	−5.50730	−0.210270
$687$	0	0
$688$	7.43011	0.283270
$689$	−2.02610	−0.0771881
$690$	0	0
$691$	−7.93027	−0.301682	−0.150841	−	0.988558i	$-0.548198\pi$
$691$	−7.93027	−0.301682	−0.150841	+	0.988558i	$0.548198\pi$
$692$	22.7623	0.865294
$693$	0	0
$694$	−23.1166	−0.877495
$695$	37.3736	1.41766
$696$	0	0
$697$	−13.1421	−0.497793
$698$	−9.92941	−0.375834
$699$	0	0
$700$	−0.250707	−0.00947584
$701$	−32.6814	−1.23436	−0.617181	−	0.786822i	$-0.711725\pi$
$701$	−32.6814	−1.23436	−0.617181	+	0.786822i	$0.711725\pi$
$702$	0	0
$703$	18.2733	0.689191
$704$	3.21076	0.121010
$705$	0	0
$706$	−9.79362	−0.368588
$707$	−3.47623	−0.130737
$708$	0	0
$709$	−49.4619	−1.85758	−0.928790	−	0.370608i	$-0.879149\pi$
$709$	−49.4619	−1.85758	−0.928790	+	0.370608i	$0.879149\pi$
$710$	−0.846310	−0.0317614
$711$	0	0
$712$	2.16071	0.0809762
$713$	0	0
$714$	0	0
$715$	−1.18440	−0.0442940
$716$	−2.49235	−0.0931435
$717$	0	0
$718$	−26.9003	−1.00391
$719$	28.1208	1.04873	0.524364	−	0.851494i	$-0.324303\pi$
$719$	28.1208	1.04873	0.524364	+	0.851494i	$0.324303\pi$
$720$	0	0
$721$	−1.70423	−0.0634690
$722$	10.9017	0.405721
$723$	0	0
$724$	10.1004	0.375377
$725$	−0.238635	−0.00886270
$726$	0	0
$727$	15.6528	0.580529	0.290264	−	0.956947i	$-0.406257\pi$
$727$	15.6528	0.580529	0.290264	+	0.956947i	$0.406257\pi$
$728$	0.0618559	0.00229253
$729$	0	0
$730$	13.3729	0.494952
$731$	−24.3887	−0.902050
$732$	0	0
$733$	−10.1204	−0.373807	−0.186904	−	0.982378i	$-0.559845\pi$
$733$	−10.1204	−0.373807	−0.186904	+	0.982378i	$0.559845\pi$
$734$	−20.9617	−0.773712
$735$	0	0
$736$	0	0
$737$	26.9948	0.994366
$738$	0	0
$739$	12.7821	0.470197	0.235098	−	0.971972i	$-0.424459\pi$
$739$	12.7821	0.470197	0.235098	+	0.971972i	$0.424459\pi$
$740$	−15.2363	−0.560098
$741$	0	0
$742$	−5.18534	−0.190360
$743$	36.1598	1.32657	0.663287	−	0.748365i	$-0.269161\pi$
$743$	36.1598	1.32657	0.663287	+	0.748365i	$0.269161\pi$
$744$	0	0
$745$	−9.18261	−0.336425
$746$	−27.4107	−1.00358
$747$	0	0
$748$	−10.5390	−0.385346
$749$	−7.16670	−0.261866
$750$	0	0
$751$	17.6518	0.644124	0.322062	−	0.946719i	$-0.395624\pi$
$751$	17.6518	0.644124	0.322062	+	0.946719i	$0.395624\pi$
$752$	−2.97017	−0.108311
$753$	0	0
$754$	0.0588775	0.00214419
$755$	37.6128	1.36887
$756$	0	0
$757$	49.4769	1.79827	0.899134	−	0.437673i	$-0.144197\pi$
$757$	49.4769	1.79827	0.899134	+	0.437673i	$0.144197\pi$
$758$	4.06405	0.147613
$759$	0	0
$760$	6.75234	0.244933
$761$	23.2497	0.842802	0.421401	−	0.906874i	$-0.361538\pi$
$761$	23.2497	0.842802	0.421401	+	0.906874i	$0.361538\pi$
$762$	0	0
$763$	3.79439	0.137366
$764$	−2.57056	−0.0929995
$765$	0	0
$766$	−17.4544	−0.630654
$767$	−1.34417	−0.0485351
$768$	0	0
$769$	42.2272	1.52275	0.761377	−	0.648310i	$-0.224524\pi$
$769$	42.2272	1.52275	0.761377	+	0.648310i	$0.224524\pi$
$770$	−3.03120	−0.109237
$771$	0	0
$772$	−22.5870	−0.812923
$773$	−13.1605	−0.473350	−0.236675	−	0.971589i	$-0.576058\pi$
$773$	−13.1605	−0.473350	−0.236675	+	0.971589i	$0.576058\pi$
$774$	0	0
$775$	1.73087	0.0621745
$776$	2.63000	0.0944117
$777$	0	0
$778$	11.0073	0.394632
$779$	11.3938	0.408224
$780$	0	0
$781$	−1.14519	−0.0409782
$782$	0	0
$783$	0	0
$784$	−6.84169	−0.244346
$785$	−49.5223	−1.76753
$786$	0	0
$787$	37.6413	1.34177	0.670884	−	0.741562i	$-0.265915\pi$
$787$	37.6413	1.34177	0.670884	+	0.741562i	$0.265915\pi$
$788$	−0.512278	−0.0182491
$789$	0	0
$790$	−17.8871	−0.636394
$791$	−2.49999	−0.0888894
$792$	0	0
$793$	1.73244	0.0615209
$794$	−22.4799	−0.797783
$795$	0	0
$796$	−1.31542	−0.0466238
$797$	−6.37725	−0.225894	−0.112947	−	0.993601i	$-0.536029\pi$
$797$	−6.37725	−0.225894	−0.112947	+	0.993601i	$0.536029\pi$
$798$	0	0
$799$	9.74935	0.344907
$800$	−0.630111	−0.0222778
$801$	0	0
$802$	−0.647541	−0.0228655
$803$	18.0956	0.638581
$804$	0	0
$805$	0	0
$806$	−0.427049	−0.0150422
$807$	0	0
$808$	−8.73694	−0.307364
$809$	−26.2928	−0.924407	−0.462203	−	0.886774i	$-0.652941\pi$
$809$	−26.2928	−0.924407	−0.462203	+	0.886774i	$0.652941\pi$
$810$	0	0
$811$	−12.6663	−0.444773	−0.222387	−	0.974959i	$-0.571385\pi$
$811$	−12.6663	−0.444773	−0.222387	+	0.974959i	$0.571385\pi$
$812$	0.150684	0.00528797
$813$	0	0
$814$	−20.6172	−0.722632
$815$	−56.4341	−1.97680
$816$	0	0
$817$	21.1442	0.739742
$818$	−23.9213	−0.836389
$819$	0	0
$820$	−9.50015	−0.331759
$821$	−49.5832	−1.73047	−0.865234	−	0.501369i	$-0.832830\pi$
$821$	−49.5832	−1.73047	−0.865234	+	0.501369i	$0.832830\pi$
$822$	0	0
$823$	−29.9411	−1.04368	−0.521840	−	0.853043i	$-0.674754\pi$
$823$	−29.9411	−1.04368	−0.521840	+	0.853043i	$0.674754\pi$
$824$	−4.28331	−0.149216
$825$	0	0
$826$	−3.44009	−0.119696
$827$	−19.8427	−0.689998	−0.344999	−	0.938603i	$-0.612121\pi$
$827$	−19.8427	−0.689998	−0.344999	+	0.938603i	$0.612121\pi$
$828$	0	0
$829$	38.9517	1.35285	0.676425	−	0.736512i	$-0.263528\pi$
$829$	38.9517	1.35285	0.676425	+	0.736512i	$0.263528\pi$
$830$	−31.7712	−1.10280
$831$	0	0
$832$	0.155465	0.00538977
$833$	22.4573	0.778099
$834$	0	0
$835$	13.2576	0.458797
$836$	9.13699	0.316009
$837$	0	0
$838$	−32.7060	−1.12981
$839$	46.7337	1.61343	0.806713	−	0.590944i	$-0.201244\pi$
$839$	46.7337	1.61343	0.806713	+	0.590944i	$0.201244\pi$
$840$	0	0
$841$	−28.8566	−0.995054
$842$	27.9916	0.964655
$843$	0	0
$844$	−7.01387	−0.241427
$845$	30.7889	1.05917
$846$	0	0
$847$	0.274948	0.00944732
$848$	−13.0325	−0.447538
$849$	0	0
$850$	2.06829	0.0709417
$851$	0	0
$852$	0	0
$853$	17.7689	0.608395	0.304197	−	0.952609i	$-0.401612\pi$
$853$	17.7689	0.608395	0.304197	+	0.952609i	$0.401612\pi$
$854$	4.43380	0.151722
$855$	0	0
$856$	−18.0123	−0.615649
$857$	23.5901	0.805822	0.402911	−	0.915239i	$-0.367998\pi$
$857$	23.5901	0.805822	0.402911	+	0.915239i	$0.367998\pi$
$858$	0	0
$859$	23.1877	0.791154	0.395577	−	0.918433i	$-0.370545\pi$
$859$	23.1877	0.791154	0.395577	+	0.918433i	$0.370545\pi$
$860$	−17.6301	−0.601180
$861$	0	0
$862$	6.00349	0.204480
$863$	−35.2374	−1.19950	−0.599748	−	0.800189i	$-0.704732\pi$
$863$	−35.2374	−1.19950	−0.599748	+	0.800189i	$0.704732\pi$
$864$	0	0
$865$	−54.0101	−1.83640
$866$	−10.6707	−0.362604
$867$	0	0
$868$	−1.09294	−0.0370967
$869$	−24.2041	−0.821067
$870$	0	0
$871$	1.30709	0.0442890
$872$	9.53659	0.322950
$873$	0	0
$874$	0	0
$875$	−4.12551	−0.139468
$876$	0	0
$877$	33.2073	1.12133	0.560665	−	0.828043i	$-0.310545\pi$
$877$	33.2073	1.12133	0.560665	+	0.828043i	$0.310545\pi$
$878$	−32.8435	−1.10841
$879$	0	0
$880$	−7.61844	−0.256818
$881$	−12.4825	−0.420547	−0.210274	−	0.977643i	$-0.567436\pi$
$881$	−12.4825	−0.420547	−0.210274	+	0.977643i	$0.567436\pi$
$882$	0	0
$883$	−20.6454	−0.694774	−0.347387	−	0.937722i	$-0.612931\pi$
$883$	−20.6454	−0.694774	−0.347387	+	0.937722i	$0.612931\pi$
$884$	−0.510301	−0.0171633
$885$	0	0
$886$	−0.412897	−0.0138716
$887$	−22.9594	−0.770900	−0.385450	−	0.922729i	$-0.625954\pi$
$887$	−22.9594	−0.770900	−0.385450	+	0.922729i	$0.625954\pi$
$888$	0	0
$889$	3.52877	0.118351
$890$	−5.12691	−0.171854
$891$	0	0
$892$	12.8998	0.431919
$893$	−8.45235	−0.282847
$894$	0	0
$895$	5.91381	0.197677
$896$	0.397877	0.0132922
$897$	0	0
$898$	−9.42926	−0.314659
$899$	−1.04031	−0.0346963
$900$	0	0
$901$	42.7781	1.42515
$902$	−12.8552	−0.428032
$903$	0	0
$904$	−6.28331	−0.208980
$905$	−23.9660	−0.796657
$906$	0	0
$907$	25.2028	0.836847	0.418423	−	0.908252i	$-0.362583\pi$
$907$	25.2028	0.836847	0.418423	+	0.908252i	$0.362583\pi$
$908$	0.770428	0.0255676
$909$	0	0
$910$	−0.146771	−0.00486541
$911$	−29.1090	−0.964423	−0.482212	−	0.876055i	$-0.660166\pi$
$911$	−29.1090	−0.964423	−0.482212	+	0.876055i	$0.660166\pi$
$912$	0	0
$913$	−42.9916	−1.42281
$914$	31.0334	1.02649
$915$	0	0
$916$	−19.2701	−0.636703
$917$	2.06967	0.0683465
$918$	0	0
$919$	22.9832	0.758147	0.379074	−	0.925367i	$-0.376243\pi$
$919$	22.9832	0.758147	0.379074	+	0.925367i	$0.376243\pi$
$920$	0	0
$921$	0	0
$922$	−33.0117	−1.08718
$923$	−0.0554502	−0.00182516
$924$	0	0
$925$	4.04612	0.133036
$926$	17.6701	0.580676
$927$	0	0
$928$	0.378719	0.0124321
$929$	17.5736	0.576571	0.288285	−	0.957545i	$-0.406915\pi$
$929$	17.5736	0.576571	0.288285	+	0.957545i	$0.406915\pi$
$930$	0	0
$931$	−19.4697	−0.638094
$932$	−8.46744	−0.277360
$933$	0	0
$934$	−11.1053	−0.363377
$935$	25.0069	0.817813
$936$	0	0
$937$	18.9264	0.618298	0.309149	−	0.951014i	$-0.399956\pi$
$937$	18.9264	0.618298	0.309149	+	0.951014i	$0.399956\pi$
$938$	3.34520	0.109225
$939$	0	0
$940$	7.04758	0.229867
$941$	40.7284	1.32771	0.663853	−	0.747863i	$-0.268920\pi$
$941$	40.7284	1.32771	0.663853	+	0.747863i	$0.268920\pi$
$942$	0	0
$943$	0	0
$944$	−8.64612	−0.281407
$945$	0	0
$946$	−23.8563	−0.775635
$947$	15.8069	0.513656	0.256828	−	0.966457i	$-0.417323\pi$
$947$	15.8069	0.513656	0.256828	+	0.966457i	$0.417323\pi$
$948$	0	0
$949$	0.876190	0.0284423
$950$	−1.79314	−0.0581770
$951$	0	0
$952$	−1.30600	−0.0423277
$953$	16.2387	0.526024	0.263012	−	0.964792i	$-0.415284\pi$
$953$	16.2387	0.526024	0.263012	+	0.964792i	$0.415284\pi$
$954$	0	0
$955$	6.09938	0.197371
$956$	−7.94660	−0.257011
$957$	0	0
$958$	−10.3037	−0.332898
$959$	−0.199371	−0.00643802
$960$	0	0
$961$	−23.4544	−0.756595
$962$	−0.998283	−0.0321859
$963$	0	0
$964$	13.4446	0.433022
$965$	53.5941	1.72525
$966$	0	0
$967$	−22.9012	−0.736453	−0.368226	−	0.929736i	$-0.620035\pi$
$967$	−22.9012	−0.736453	−0.368226	+	0.929736i	$0.620035\pi$
$968$	0.691036	0.0222107
$969$	0	0
$970$	−6.24044	−0.200368
$971$	−26.9582	−0.865129	−0.432564	−	0.901603i	$-0.642391\pi$
$971$	−26.9582	−0.865129	−0.432564	+	0.901603i	$0.642391\pi$
$972$	0	0
$973$	6.26695	0.200909
$974$	11.2887	0.361712
$975$	0	0
$976$	11.1436	0.356699
$977$	−27.7321	−0.887229	−0.443615	−	0.896218i	$-0.646304\pi$
$977$	−27.7321	−0.887229	−0.443615	+	0.896218i	$0.646304\pi$
$978$	0	0
$979$	−6.93753	−0.221724
$980$	16.2339	0.518572
$981$	0	0
$982$	−9.15794	−0.292242
$983$	−44.7433	−1.42709	−0.713545	−	0.700609i	$-0.752912\pi$
$983$	−44.7433	−1.42709	−0.713545	+	0.700609i	$0.752912\pi$
$984$	0	0
$985$	1.21553	0.0387299
$986$	−1.24311	−0.0395888
$987$	0	0
$988$	0.442413	0.0140750
$989$	0	0
$990$	0	0
$991$	2.16949	0.0689161	0.0344581	−	0.999406i	$-0.489029\pi$
$991$	2.16949	0.0689161	0.0344581	+	0.999406i	$0.489029\pi$
$992$	−2.74692	−0.0872148
$993$	0	0
$994$	−0.141912	−0.00450119
$995$	3.12121	0.0989490
$996$	0	0
$997$	10.9055	0.345381	0.172691	−	0.984976i	$-0.444754\pi$
$997$	10.9055	0.345381	0.172691	+	0.984976i	$0.444754\pi$
$998$	31.2250	0.988410
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.bp.1.3		5
3.2	odd	2		1058.2.a.m.1.1		5
12.11	even	2		8464.2.a.bx.1.5		5
23.4	even	11		414.2.i.f.361.1		10
23.6	even	11		414.2.i.f.289.1		10
23.22	odd	2		9522.2.a.bu.1.3		5
69.29	odd	22		46.2.c.a.13.1	✓	10
69.50	odd	22		46.2.c.a.39.1	yes	10
69.68	even	2		1058.2.a.l.1.1		5
276.119	even	22		368.2.m.b.177.1		10
276.167	even	22		368.2.m.b.289.1		10
276.275	odd	2		8464.2.a.bw.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.2.c.a.13.1	✓	10	69.29	odd	22
46.2.c.a.39.1	yes	10	69.50	odd	22
368.2.m.b.177.1		10	276.119	even	22
368.2.m.b.289.1		10	276.167	even	22
414.2.i.f.289.1		10	23.6	even	11
414.2.i.f.361.1		10	23.4	even	11
1058.2.a.l.1.1		5	69.68	even	2
1058.2.a.m.1.1		5	3.2	odd	2
8464.2.a.bw.1.5		5	276.275	odd	2
8464.2.a.bx.1.5		5	12.11	even	2
9522.2.a.bp.1.3		5	1.1	even	1	trivial
9522.2.a.bu.1.3		5	23.22	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:	\( N \)	\(=\)	\( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9522.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.37279 q^{5} -0.397877 q^{7} -1.00000 q^{8} +2.37279 q^{10} +3.21076 q^{11} +0.155465 q^{13} +0.397877 q^{14} +1.00000 q^{16} -3.28242 q^{17} +2.84574 q^{19} -2.37279 q^{20} -3.21076 q^{22} +0.630111 q^{25} -0.155465 q^{26} -0.397877 q^{28} -0.378719 q^{29} +2.74692 q^{31} -1.00000 q^{32} +3.28242 q^{34} +0.944078 q^{35} +6.42128 q^{37} -2.84574 q^{38} +2.37279 q^{40} +4.00379 q^{41} +7.43011 q^{43} +3.21076 q^{44} -2.97017 q^{47} -6.84169 q^{49} -0.630111 q^{50} +0.155465 q^{52} -13.0325 q^{53} -7.61844 q^{55} +0.397877 q^{56} +0.378719 q^{58} -8.64612 q^{59} +11.1436 q^{61} -2.74692 q^{62} +1.00000 q^{64} -0.368885 q^{65} +8.40761 q^{67} -3.28242 q^{68} -0.944078 q^{70} -0.356673 q^{71} +5.63594 q^{73} -6.42128 q^{74} +2.84574 q^{76} -1.27749 q^{77} -7.53843 q^{79} -2.37279 q^{80} -4.00379 q^{82} -13.3899 q^{83} +7.78847 q^{85} -7.43011 q^{86} -3.21076 q^{88} -2.16071 q^{89} -0.0618559 q^{91} +2.97017 q^{94} -6.75234 q^{95} -2.63000 q^{97} +6.84169 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} - 8 q^{5} + 7 q^{7} - 5 q^{8} + 8 q^{10} - 5 q^{11} - 7 q^{13} - 7 q^{14} + 5 q^{16} - 13 q^{17} + 12 q^{19} - 8 q^{20} + 5 q^{22} + q^{25} + 7 q^{26} + 7 q^{28} + 4 q^{29} + 6 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 - 8 * q^5 + 7 * q^7 - 5 * q^8 + 8 * q^10 - 5 * q^11 - 7 * q^13 - 7 * q^14 + 5 * q^16 - 13 * q^17 + 12 * q^19 - 8 * q^20 + 5 * q^22 + q^25 + 7 * q^26 + 7 * q^28 + 4 * q^29 + 6 * q^31 - 5 * q^32 + 13 * q^34 - 9 * q^35 + 14 * q^37 - 12 * q^38 + 8 * q^40 - q^41 + 11 * q^43 - 5 * q^44 - 9 * q^47 - 12 * q^49 - q^50 - 7 * q^52 - 6 * q^53 - 14 * q^55 - 7 * q^56 - 4 * q^58 - 3 * q^59 + 13 * q^61 - 6 * q^62 + 5 * q^64 + 9 * q^65 + 10 * q^67 - 13 * q^68 + 9 * q^70 - 8 * q^71 - 4 * q^73 - 14 * q^74 + 12 * q^76 - 18 * q^77 - 18 * q^79 - 8 * q^80 + q^82 - 2 * q^83 + 23 * q^85 - 11 * q^86 + 5 * q^88 - 10 * q^89 - 12 * q^91 + 9 * q^94 - 28 * q^95 + 17 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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