LMFDB - Embedded newform 9522.2.a.x.1.1

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,-2,0,0,0,0,8,0,0,2,0,0,0,0,0,0,0,0,-10,-8,0,0, -6] 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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3174)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.73205 q^{7} -1.00000 q^{8} +1.73205 q^{11} +4.00000 q^{13} +1.73205 q^{14} +1.00000 q^{16} -3.46410 q^{17} -1.73205 q^{22} -5.00000 q^{25} -4.00000 q^{26} -1.73205 q^{28} -3.00000 q^{29} +7.00000 q^{31} -1.00000 q^{32} +3.46410 q^{34} -12.0000 q^{41} +10.3923 q^{43} +1.73205 q^{44} -4.00000 q^{49} +5.00000 q^{50} +4.00000 q^{52} +8.66025 q^{53} +1.73205 q^{56} +3.00000 q^{58} -9.00000 q^{59} +6.92820 q^{61} -7.00000 q^{62} +1.00000 q^{64} -3.46410 q^{68} +6.00000 q^{71} -11.0000 q^{73} -3.00000 q^{77} -5.19615 q^{79} +12.0000 q^{82} -1.73205 q^{83} -10.3923 q^{86} -1.73205 q^{88} -10.3923 q^{89} -6.92820 q^{91} -15.5885 q^{97} +4.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 8 q^{13} + 2 q^{16} - 10 q^{25} - 8 q^{26} - 6 q^{29} + 14 q^{31} - 2 q^{32} - 24 q^{41} - 8 q^{49} + 10 q^{50} + 8 q^{52} + 6 q^{58} - 18 q^{59} - 14 q^{62} + 2 q^{64}+ \cdots + 8 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 8 * q^13 + 2 * q^16 - 10 * q^25 - 8 * q^26 - 6 * q^29 + 14 * q^31 - 2 * q^32 - 24 * q^41 - 8 * q^49 + 10 * q^50 + 8 * q^52 + 6 * q^58 - 18 * q^59 - 14 * q^62 + 2 * q^64 + 12 * q^71 - 22 * q^73 - 6 * q^77 + 24 * q^82 + 8 * 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
$7$	−1.73205	−0.654654	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	−1.73205	−0.654654	−0.327327	+	0.944911i	$0.606148\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	1.73205	0.522233	0.261116	−	0.965307i	$-0.415909\pi$
$11$	1.73205	0.522233	0.261116	+	0.965307i	$0.415909\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	1.73205	0.462910
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	−1.73205	−0.369274
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	−1.73205	−0.327327
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	3.46410	0.594089
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	10.3923	1.58481	0.792406	−	0.609994i	$-0.208828\pi$
$43$	10.3923	1.58481	0.792406	+	0.609994i	$0.208828\pi$
$44$	1.73205	0.261116
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	5.00000	0.707107
$51$	0	0
$52$	4.00000	0.554700
$53$	8.66025	1.18958	0.594789	−	0.803882i	$-0.297236\pi$
$53$	8.66025	1.18958	0.594789	+	0.803882i	$0.297236\pi$
$54$	0	0
$55$	0	0
$56$	1.73205	0.231455
$57$	0	0
$58$	3.00000	0.393919
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	6.92820	0.887066	0.443533	−	0.896258i	$-0.353725\pi$
$61$	6.92820	0.887066	0.443533	+	0.896258i	$0.353725\pi$
$62$	−7.00000	−0.889001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−3.46410	−0.420084
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−5.19615	−0.584613	−0.292306	−	0.956325i	$-0.594423\pi$
$79$	−5.19615	−0.584613	−0.292306	+	0.956325i	$0.594423\pi$
$80$	0	0
$81$	0	0
$82$	12.0000	1.32518
$83$	−1.73205	−0.190117	−0.0950586	−	0.995472i	$-0.530304\pi$
$83$	−1.73205	−0.190117	−0.0950586	+	0.995472i	$0.530304\pi$
$84$	0	0
$85$	0	0
$86$	−10.3923	−1.12063
$87$	0	0
$88$	−1.73205	−0.184637
$89$	−10.3923	−1.10158	−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923	−1.10158	−0.550791	+	0.834643i	$0.685674\pi$
$90$	0	0
$91$	−6.92820	−0.726273
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.5885	−1.58277	−0.791384	−	0.611319i	$-0.790639\pi$
$97$	−15.5885	−1.58277	−0.791384	+	0.611319i	$0.790639\pi$
$98$	4.00000	0.404061
$99$	0	0
$100$	−5.00000	−0.500000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	19.0526	1.87730	0.938652	−	0.344865i	$-0.112075\pi$
$103$	19.0526	1.87730	0.938652	+	0.344865i	$0.112075\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−8.66025	−0.841158
$107$	−17.3205	−1.67444	−0.837218	−	0.546869i	$-0.815820\pi$
$107$	−17.3205	−1.67444	−0.837218	+	0.546869i	$0.815820\pi$
$108$	0	0
$109$	6.92820	0.663602	0.331801	−	0.943349i	$-0.392344\pi$
$109$	6.92820	0.663602	0.331801	+	0.943349i	$0.392344\pi$
$110$	0	0
$111$	0	0
$112$	−1.73205	−0.163663
$113$	20.7846	1.95525	0.977626	−	0.210352i	$-0.0674609\pi$
$113$	20.7846	1.95525	0.977626	+	0.210352i	$0.0674609\pi$
$114$	0	0
$115$	0	0
$116$	−3.00000	−0.278543
$117$	0	0
$118$	9.00000	0.828517
$119$	6.00000	0.550019
$120$	0	0
$121$	−8.00000	−0.727273
$122$	−6.92820	−0.627250
$123$	0	0
$124$	7.00000	0.628619
$125$	0	0
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	3.46410	0.297044
$137$	−6.92820	−0.591916	−0.295958	−	0.955201i	$-0.595639\pi$
$137$	−6.92820	−0.591916	−0.295958	+	0.955201i	$0.595639\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	0	0
$142$	−6.00000	−0.503509
$143$	6.92820	0.579365
$144$	0	0
$145$	0	0
$146$	11.0000	0.910366
$147$	0	0
$148$	0	0
$149$	13.8564	1.13516	0.567581	−	0.823318i	$-0.307880\pi$
$149$	13.8564	1.13516	0.567581	+	0.823318i	$0.307880\pi$
$150$	0	0
$151$	17.0000	1.38344	0.691720	−	0.722166i	$-0.256853\pi$
$151$	17.0000	1.38344	0.691720	+	0.722166i	$0.256853\pi$
$152$	0	0
$153$	0	0
$154$	3.00000	0.241747
$155$	0	0
$156$	0	0
$157$	−13.8564	−1.10586	−0.552931	−	0.833227i	$-0.686491\pi$
$157$	−13.8564	−1.10586	−0.552931	+	0.833227i	$0.686491\pi$
$158$	5.19615	0.413384
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−14.0000	−1.09656	−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000	−1.09656	−0.548282	+	0.836293i	$0.684718\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	1.73205	0.134433
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	10.3923	0.792406
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	8.66025	0.654654
$176$	1.73205	0.130558
$177$	0	0
$178$	10.3923	0.778936
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−13.8564	−1.02994	−0.514969	−	0.857209i	$-0.672197\pi$
$181$	−13.8564	−1.02994	−0.514969	+	0.857209i	$0.672197\pi$
$182$	6.92820	0.513553
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.00000	−0.438763
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.2487	1.75458	0.877288	−	0.479965i	$-0.159351\pi$
$191$	24.2487	1.75458	0.877288	+	0.479965i	$0.159351\pi$
$192$	0	0
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	15.5885	1.11919
$195$	0	0
$196$	−4.00000	−0.285714
$197$	15.0000	1.06871	0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000	1.06871	0.534353	+	0.845262i	$0.320555\pi$
$198$	0	0
$199$	8.66025	0.613909	0.306955	−	0.951724i	$-0.400690\pi$
$199$	8.66025	0.613909	0.306955	+	0.951724i	$0.400690\pi$
$200$	5.00000	0.353553
$201$	0	0
$202$	3.00000	0.211079
$203$	5.19615	0.364698
$204$	0	0
$205$	0	0
$206$	−19.0526	−1.32745
$207$	0	0
$208$	4.00000	0.277350
$209$	0	0
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	8.66025	0.594789
$213$	0	0
$214$	17.3205	1.18401
$215$	0	0
$216$	0	0
$217$	−12.1244	−0.823055
$218$	−6.92820	−0.469237
$219$	0	0
$220$	0	0
$221$	−13.8564	−0.932083
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	1.73205	0.115728
$225$	0	0
$226$	−20.7846	−1.38257
$227$	−15.5885	−1.03464	−0.517321	−	0.855791i	$-0.673071\pi$
$227$	−15.5885	−1.03464	−0.517321	+	0.855791i	$0.673071\pi$
$228$	0	0
$229$	−20.7846	−1.37349	−0.686743	−	0.726900i	$-0.740960\pi$
$229$	−20.7846	−1.37349	−0.686743	+	0.726900i	$0.740960\pi$
$230$	0	0
$231$	0	0
$232$	3.00000	0.196960
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	0	0
$236$	−9.00000	−0.585850
$237$	0	0
$238$	−6.00000	−0.388922
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	20.7846	1.33885	0.669427	−	0.742878i	$-0.266540\pi$
$241$	20.7846	1.33885	0.669427	+	0.742878i	$0.266540\pi$
$242$	8.00000	0.514259
$243$	0	0
$244$	6.92820	0.443533
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−7.00000	−0.444500
$249$	0	0
$250$	0	0
$251$	3.46410	0.218652	0.109326	−	0.994006i	$-0.465131\pi$
$251$	3.46410	0.218652	0.109326	+	0.994006i	$0.465131\pi$
$252$	0	0
$253$	0	0
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	15.0000	0.926703
$263$	−13.8564	−0.854423	−0.427211	−	0.904152i	$-0.640504\pi$
$263$	−13.8564	−0.854423	−0.427211	+	0.904152i	$0.640504\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	−11.0000	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$272$	−3.46410	−0.210042
$273$	0	0
$274$	6.92820	0.418548
$275$	−8.66025	−0.522233
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−8.00000	−0.479808
$279$	0	0
$280$	0	0
$281$	6.92820	0.413302	0.206651	−	0.978415i	$-0.433744\pi$
$281$	6.92820	0.413302	0.206651	+	0.978415i	$0.433744\pi$
$282$	0	0
$283$	−20.7846	−1.23552	−0.617758	−	0.786368i	$-0.711959\pi$
$283$	−20.7846	−1.23552	−0.617758	+	0.786368i	$0.711959\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	−6.92820	−0.409673
$287$	20.7846	1.22688
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	−11.0000	−0.643726
$293$	−19.0526	−1.11306	−0.556531	−	0.830827i	$-0.687868\pi$
$293$	−19.0526	−1.11306	−0.556531	+	0.830827i	$0.687868\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−13.8564	−0.802680
$299$	0	0
$300$	0	0
$301$	−18.0000	−1.03750
$302$	−17.0000	−0.978240
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	−3.00000	−0.170941
$309$	0	0
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	20.7846	1.17482	0.587408	−	0.809291i	$-0.300148\pi$
$313$	20.7846	1.17482	0.587408	+	0.809291i	$0.300148\pi$
$314$	13.8564	0.781962
$315$	0	0
$316$	−5.19615	−0.292306
$317$	9.00000	0.505490	0.252745	−	0.967533i	$-0.418667\pi$
$317$	9.00000	0.505490	0.252745	+	0.967533i	$0.418667\pi$
$318$	0	0
$319$	−5.19615	−0.290929
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−20.0000	−1.10940
$326$	14.0000	0.775388
$327$	0	0
$328$	12.0000	0.662589
$329$	0	0
$330$	0	0
$331$	−26.0000	−1.42909	−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000	−1.42909	−0.714545	+	0.699590i	$0.753366\pi$
$332$	−1.73205	−0.0950586
$333$	0	0
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	34.6410	1.88702	0.943508	−	0.331349i	$-0.107504\pi$
$337$	34.6410	1.88702	0.943508	+	0.331349i	$0.107504\pi$
$338$	−3.00000	−0.163178
$339$	0	0
$340$	0	0
$341$	12.1244	0.656571
$342$	0	0
$343$	19.0526	1.02874
$344$	−10.3923	−0.560316
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−27.0000	−1.44944	−0.724718	−	0.689046i	$-0.758030\pi$
$347$	−27.0000	−1.44944	−0.724718	+	0.689046i	$0.758030\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	−8.66025	−0.462910
$351$	0	0
$352$	−1.73205	−0.0923186
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	−10.3923	−0.550791
$357$	0	0
$358$	0	0
$359$	27.7128	1.46263	0.731313	−	0.682042i	$-0.238908\pi$
$359$	27.7128	1.46263	0.731313	+	0.682042i	$0.238908\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	13.8564	0.728277
$363$	0	0
$364$	−6.92820	−0.363137
$365$	0	0
$366$	0	0
$367$	−19.0526	−0.994535	−0.497268	−	0.867597i	$-0.665663\pi$
$367$	−19.0526	−0.994535	−0.497268	+	0.867597i	$0.665663\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−15.0000	−0.778761
$372$	0	0
$373$	−24.2487	−1.25555	−0.627775	−	0.778395i	$-0.716034\pi$
$373$	−24.2487	−1.25555	−0.627775	+	0.778395i	$0.716034\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−10.3923	−0.533817	−0.266908	−	0.963722i	$-0.586002\pi$
$379$	−10.3923	−0.533817	−0.266908	+	0.963722i	$0.586002\pi$
$380$	0	0
$381$	0	0
$382$	−24.2487	−1.24067
$383$	−38.1051	−1.94708	−0.973540	−	0.228515i	$-0.926613\pi$
$383$	−38.1051	−1.94708	−0.973540	+	0.228515i	$0.926613\pi$
$384$	0	0
$385$	0	0
$386$	7.00000	0.356291
$387$	0	0
$388$	−15.5885	−0.791384
$389$	−29.4449	−1.49291	−0.746457	−	0.665434i	$-0.768247\pi$
$389$	−29.4449	−1.49291	−0.746457	+	0.665434i	$0.768247\pi$
$390$	0	0
$391$	0	0
$392$	4.00000	0.202031
$393$	0	0
$394$	−15.0000	−0.755689
$395$	0	0
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	−8.66025	−0.434099
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−6.92820	−0.345978	−0.172989	−	0.984924i	$-0.555343\pi$
$401$	−6.92820	−0.345978	−0.172989	+	0.984924i	$0.555343\pi$
$402$	0	0
$403$	28.0000	1.39478
$404$	−3.00000	−0.149256
$405$	0	0
$406$	−5.19615	−0.257881
$407$	0	0
$408$	0	0
$409$	−13.0000	−0.642809	−0.321404	−	0.946942i	$-0.604155\pi$
$409$	−13.0000	−0.642809	−0.321404	+	0.946942i	$0.604155\pi$
$410$	0	0
$411$	0	0
$412$	19.0526	0.938652
$413$	15.5885	0.767058
$414$	0	0
$415$	0	0
$416$	−4.00000	−0.196116
$417$	0	0
$418$	0	0
$419$	−15.5885	−0.761546	−0.380773	−	0.924669i	$-0.624342\pi$
$419$	−15.5885	−0.761546	−0.380773	+	0.924669i	$0.624342\pi$
$420$	0	0
$421$	17.3205	0.844150	0.422075	−	0.906561i	$-0.361302\pi$
$421$	17.3205	0.844150	0.422075	+	0.906561i	$0.361302\pi$
$422$	14.0000	0.681509
$423$	0	0
$424$	−8.66025	−0.420579
$425$	17.3205	0.840168
$426$	0	0
$427$	−12.0000	−0.580721
$428$	−17.3205	−0.837218
$429$	0	0
$430$	0	0
$431$	20.7846	1.00116	0.500580	−	0.865690i	$-0.333120\pi$
$431$	20.7846	1.00116	0.500580	+	0.865690i	$0.333120\pi$
$432$	0	0
$433$	−6.92820	−0.332948	−0.166474	−	0.986046i	$-0.553238\pi$
$433$	−6.92820	−0.332948	−0.166474	+	0.986046i	$0.553238\pi$
$434$	12.1244	0.581988
$435$	0	0
$436$	6.92820	0.331801
$437$	0	0
$438$	0	0
$439$	−19.0000	−0.906821	−0.453410	−	0.891302i	$-0.649793\pi$
$439$	−19.0000	−0.906821	−0.453410	+	0.891302i	$0.649793\pi$
$440$	0	0
$441$	0	0
$442$	13.8564	0.659082
$443$	−9.00000	−0.427603	−0.213801	−	0.976877i	$-0.568585\pi$
$443$	−9.00000	−0.427603	−0.213801	+	0.976877i	$0.568585\pi$
$444$	0	0
$445$	0	0
$446$	16.0000	0.757622
$447$	0	0
$448$	−1.73205	−0.0818317
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	0	0
$451$	−20.7846	−0.978709
$452$	20.7846	0.977626
$453$	0	0
$454$	15.5885	0.731603
$455$	0	0
$456$	0	0
$457$	−15.5885	−0.729197	−0.364599	−	0.931165i	$-0.618794\pi$
$457$	−15.5885	−0.729197	−0.364599	+	0.931165i	$0.618794\pi$
$458$	20.7846	0.971201
$459$	0	0
$460$	0	0
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	0	0
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−18.0000	−0.833834
$467$	8.66025	0.400749	0.200374	−	0.979719i	$-0.435784\pi$
$467$	8.66025	0.400749	0.200374	+	0.979719i	$0.435784\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	9.00000	0.414259
$473$	18.0000	0.827641
$474$	0	0
$475$	0	0
$476$	6.00000	0.275010
$477$	0	0
$478$	0	0
$479$	−38.1051	−1.74107	−0.870534	−	0.492109i	$-0.836226\pi$
$479$	−38.1051	−1.74107	−0.870534	+	0.492109i	$0.836226\pi$
$480$	0	0
$481$	0	0
$482$	−20.7846	−0.946713
$483$	0	0
$484$	−8.00000	−0.363636
$485$	0	0
$486$	0	0
$487$	−7.00000	−0.317200	−0.158600	−	0.987343i	$-0.550698\pi$
$487$	−7.00000	−0.317200	−0.158600	+	0.987343i	$0.550698\pi$
$488$	−6.92820	−0.313625
$489$	0	0
$490$	0	0
$491$	−15.0000	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$492$	0	0
$493$	10.3923	0.468046
$494$	0	0
$495$	0	0
$496$	7.00000	0.314309
$497$	−10.3923	−0.466159
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	0	0
$501$	0	0
$502$	−3.46410	−0.154610
$503$	−17.3205	−0.772283	−0.386142	−	0.922440i	$-0.626192\pi$
$503$	−17.3205	−0.772283	−0.386142	+	0.922440i	$0.626192\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	4.00000	0.177471
$509$	9.00000	0.398918	0.199459	−	0.979906i	$-0.436082\pi$
$509$	9.00000	0.398918	0.199459	+	0.979906i	$0.436082\pi$
$510$	0	0
$511$	19.0526	0.842836
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.2487	1.06236	0.531178	−	0.847260i	$-0.321750\pi$
$521$	24.2487	1.06236	0.531178	+	0.847260i	$0.321750\pi$
$522$	0	0
$523$	−3.46410	−0.151475	−0.0757373	−	0.997128i	$-0.524131\pi$
$523$	−3.46410	−0.151475	−0.0757373	+	0.997128i	$0.524131\pi$
$524$	−15.0000	−0.655278
$525$	0	0
$526$	13.8564	0.604168
$527$	−24.2487	−1.05629
$528$	0	0
$529$	0	0
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−48.0000	−2.07911
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	9.00000	0.388018
$539$	−6.92820	−0.298419
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	11.0000	0.472490
$543$	0	0
$544$	3.46410	0.148522
$545$	0	0
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−6.92820	−0.295958
$549$	0	0
$550$	8.66025	0.369274
$551$	0	0
$552$	0	0
$553$	9.00000	0.382719
$554$	−10.0000	−0.424859
$555$	0	0
$556$	8.00000	0.339276
$557$	−32.9090	−1.39440	−0.697199	−	0.716878i	$-0.745571\pi$
$557$	−32.9090	−1.39440	−0.697199	+	0.716878i	$0.745571\pi$
$558$	0	0
$559$	41.5692	1.75819
$560$	0	0
$561$	0	0
$562$	−6.92820	−0.292249
$563$	25.9808	1.09496	0.547479	−	0.836819i	$-0.315587\pi$
$563$	25.9808	1.09496	0.547479	+	0.836819i	$0.315587\pi$
$564$	0	0
$565$	0	0
$566$	20.7846	0.873642
$567$	0	0
$568$	−6.00000	−0.251754
$569$	24.2487	1.01656	0.508279	−	0.861192i	$-0.330282\pi$
$569$	24.2487	1.01656	0.508279	+	0.861192i	$0.330282\pi$
$570$	0	0
$571$	20.7846	0.869809	0.434904	−	0.900477i	$-0.356782\pi$
$571$	20.7846	0.869809	0.434904	+	0.900477i	$0.356782\pi$
$572$	6.92820	0.289683
$573$	0	0
$574$	−20.7846	−0.867533
$575$	0	0
$576$	0	0
$577$	17.0000	0.707719	0.353860	−	0.935299i	$-0.384869\pi$
$577$	17.0000	0.707719	0.353860	+	0.935299i	$0.384869\pi$
$578$	5.00000	0.207973
$579$	0	0
$580$	0	0
$581$	3.00000	0.124461
$582$	0	0
$583$	15.0000	0.621237
$584$	11.0000	0.455183
$585$	0	0
$586$	19.0526	0.787054
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	0	0
$595$	0	0
$596$	13.8564	0.567581
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	18.0000	0.733625
$603$	0	0
$604$	17.0000	0.691720
$605$	0	0
$606$	0	0
$607$	−43.0000	−1.74532	−0.872658	−	0.488332i	$-0.837606\pi$
$607$	−43.0000	−1.74532	−0.872658	+	0.488332i	$0.837606\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−27.7128	−1.11931	−0.559655	−	0.828726i	$-0.689066\pi$
$613$	−27.7128	−1.11931	−0.559655	+	0.828726i	$0.689066\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	3.00000	0.120873
$617$	−6.92820	−0.278919	−0.139459	−	0.990228i	$-0.544536\pi$
$617$	−6.92820	−0.278919	−0.139459	+	0.990228i	$0.544536\pi$
$618$	0	0
$619$	20.7846	0.835404	0.417702	−	0.908584i	$-0.362836\pi$
$619$	20.7846	0.835404	0.417702	+	0.908584i	$0.362836\pi$
$620$	0	0
$621$	0	0
$622$	−12.0000	−0.481156
$623$	18.0000	0.721155
$624$	0	0
$625$	25.0000	1.00000
$626$	−20.7846	−0.830720
$627$	0	0
$628$	−13.8564	−0.552931
$629$	0	0
$630$	0	0
$631$	−38.1051	−1.51694	−0.758470	−	0.651707i	$-0.774053\pi$
$631$	−38.1051	−1.51694	−0.758470	+	0.651707i	$0.774053\pi$
$632$	5.19615	0.206692
$633$	0	0
$634$	−9.00000	−0.357436
$635$	0	0
$636$	0	0
$637$	−16.0000	−0.633943
$638$	5.19615	0.205718
$639$	0	0
$640$	0	0
$641$	13.8564	0.547295	0.273648	−	0.961830i	$-0.411770\pi$
$641$	13.8564	0.547295	0.273648	+	0.961830i	$0.411770\pi$
$642$	0	0
$643$	−6.92820	−0.273222	−0.136611	−	0.990625i	$-0.543621\pi$
$643$	−6.92820	−0.273222	−0.136611	+	0.990625i	$0.543621\pi$
$644$	0	0
$645$	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
$649$	−15.5885	−0.611900
$650$	20.0000	0.784465
$651$	0	0
$652$	−14.0000	−0.548282
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	0	0
$655$	0	0
$656$	−12.0000	−0.468521
$657$	0	0
$658$	0	0
$659$	45.0333	1.75425	0.877125	−	0.480263i	$-0.159459\pi$
$659$	45.0333	1.75425	0.877125	+	0.480263i	$0.159459\pi$
$660$	0	0
$661$	17.3205	0.673690	0.336845	−	0.941560i	$-0.390640\pi$
$661$	17.3205	0.673690	0.336845	+	0.941560i	$0.390640\pi$
$662$	26.0000	1.01052
$663$	0	0
$664$	1.73205	0.0672166
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−12.0000	−0.464294
$669$	0	0
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	−34.6410	−1.33432
$675$	0	0
$676$	3.00000	0.115385
$677$	−46.7654	−1.79734	−0.898670	−	0.438626i	$-0.855465\pi$
$677$	−46.7654	−1.79734	−0.898670	+	0.438626i	$0.855465\pi$
$678$	0	0
$679$	27.0000	1.03616
$680$	0	0
$681$	0	0
$682$	−12.1244	−0.464266
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	0	0
$686$	−19.0526	−0.727430
$687$	0	0
$688$	10.3923	0.396203
$689$	34.6410	1.31972
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	27.0000	1.02491
$695$	0	0
$696$	0	0
$697$	41.5692	1.57455
$698$	−14.0000	−0.529908
$699$	0	0
$700$	8.66025	0.327327
$701$	5.19615	0.196256	0.0981280	−	0.995174i	$-0.468715\pi$
$701$	5.19615	0.196256	0.0981280	+	0.995174i	$0.468715\pi$
$702$	0	0
$703$	0	0
$704$	1.73205	0.0652791
$705$	0	0
$706$	0	0
$707$	5.19615	0.195421
$708$	0	0
$709$	24.2487	0.910679	0.455340	−	0.890318i	$-0.349518\pi$
$709$	24.2487	0.910679	0.455340	+	0.890318i	$0.349518\pi$
$710$	0	0
$711$	0	0
$712$	10.3923	0.389468
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−27.7128	−1.03423
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	−33.0000	−1.22898
$722$	19.0000	0.707107
$723$	0	0
$724$	−13.8564	−0.514969
$725$	15.0000	0.557086
$726$	0	0
$727$	−51.9615	−1.92715	−0.963573	−	0.267445i	$-0.913821\pi$
$727$	−51.9615	−1.92715	−0.963573	+	0.267445i	$0.913821\pi$
$728$	6.92820	0.256776
$729$	0	0
$730$	0	0
$731$	−36.0000	−1.33151
$732$	0	0
$733$	−10.3923	−0.383849	−0.191924	−	0.981410i	$-0.561473\pi$
$733$	−10.3923	−0.383849	−0.191924	+	0.981410i	$0.561473\pi$
$734$	19.0526	0.703243
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	32.0000	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$740$	0	0
$741$	0	0
$742$	15.0000	0.550667
$743$	34.6410	1.27086	0.635428	−	0.772160i	$-0.280824\pi$
$743$	34.6410	1.27086	0.635428	+	0.772160i	$0.280824\pi$
$744$	0	0
$745$	0	0
$746$	24.2487	0.887808
$747$	0	0
$748$	−6.00000	−0.219382
$749$	30.0000	1.09618
$750$	0	0
$751$	−32.9090	−1.20087	−0.600433	−	0.799675i	$-0.705005\pi$
$751$	−32.9090	−1.20087	−0.600433	+	0.799675i	$0.705005\pi$
$752$	0	0
$753$	0	0
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	−17.3205	−0.629525	−0.314762	−	0.949171i	$-0.601925\pi$
$757$	−17.3205	−0.629525	−0.314762	+	0.949171i	$0.601925\pi$
$758$	10.3923	0.377466
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	−12.0000	−0.434429
$764$	24.2487	0.877288
$765$	0	0
$766$	38.1051	1.37679
$767$	−36.0000	−1.29988
$768$	0	0
$769$	22.5167	0.811972	0.405986	−	0.913879i	$-0.366928\pi$
$769$	22.5167	0.811972	0.405986	+	0.913879i	$0.366928\pi$
$770$	0	0
$771$	0	0
$772$	−7.00000	−0.251936
$773$	15.5885	0.560678	0.280339	−	0.959901i	$-0.409553\pi$
$773$	15.5885	0.560678	0.280339	+	0.959901i	$0.409553\pi$
$774$	0	0
$775$	−35.0000	−1.25724
$776$	15.5885	0.559593
$777$	0	0
$778$	29.4449	1.05565
$779$	0	0
$780$	0	0
$781$	10.3923	0.371866
$782$	0	0
$783$	0	0
$784$	−4.00000	−0.142857
$785$	0	0
$786$	0	0
$787$	−6.92820	−0.246964	−0.123482	−	0.992347i	$-0.539406\pi$
$787$	−6.92820	−0.246964	−0.123482	+	0.992347i	$0.539406\pi$
$788$	15.0000	0.534353
$789$	0	0
$790$	0	0
$791$	−36.0000	−1.28001
$792$	0	0
$793$	27.7128	0.984111
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	8.66025	0.306955
$797$	27.7128	0.981638	0.490819	−	0.871262i	$-0.336698\pi$
$797$	27.7128	0.981638	0.490819	+	0.871262i	$0.336698\pi$
$798$	0	0
$799$	0	0
$800$	5.00000	0.176777
$801$	0	0
$802$	6.92820	0.244643
$803$	−19.0526	−0.672350
$804$	0	0
$805$	0	0
$806$	−28.0000	−0.986258
$807$	0	0
$808$	3.00000	0.105540
$809$	−48.0000	−1.68759	−0.843795	−	0.536666i	$-0.819684\pi$
$809$	−48.0000	−1.68759	−0.843795	+	0.536666i	$0.819684\pi$
$810$	0	0
$811$	10.0000	0.351147	0.175574	−	0.984466i	$-0.443822\pi$
$811$	10.0000	0.351147	0.175574	+	0.984466i	$0.443822\pi$
$812$	5.19615	0.182349
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	13.0000	0.454534
$819$	0	0
$820$	0	0
$821$	33.0000	1.15171	0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000	1.15171	0.575854	+	0.817553i	$0.304670\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	−19.0526	−0.663727
$825$	0	0
$826$	−15.5885	−0.542392
$827$	−10.3923	−0.361376	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923	−0.361376	−0.180688	+	0.983540i	$0.557832\pi$
$828$	0	0
$829$	−52.0000	−1.80603	−0.903017	−	0.429604i	$-0.858653\pi$
$829$	−52.0000	−1.80603	−0.903017	+	0.429604i	$0.858653\pi$
$830$	0	0
$831$	0	0
$832$	4.00000	0.138675
$833$	13.8564	0.480096
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	15.5885	0.538494
$839$	−27.7128	−0.956753	−0.478376	−	0.878155i	$-0.658774\pi$
$839$	−27.7128	−0.956753	−0.478376	+	0.878155i	$0.658774\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−17.3205	−0.596904
$843$	0	0
$844$	−14.0000	−0.481900
$845$	0	0
$846$	0	0
$847$	13.8564	0.476112
$848$	8.66025	0.297394
$849$	0	0
$850$	−17.3205	−0.594089
$851$	0	0
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	12.0000	0.410632
$855$	0	0
$856$	17.3205	0.592003
$857$	−24.0000	−0.819824	−0.409912	−	0.912125i	$-0.634441\pi$
$857$	−24.0000	−0.819824	−0.409912	+	0.912125i	$0.634441\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	0	0
$862$	−20.7846	−0.707927
$863$	−30.0000	−1.02121	−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000	−1.02121	−0.510606	+	0.859815i	$0.670579\pi$
$864$	0	0
$865$	0	0
$866$	6.92820	0.235430
$867$	0	0
$868$	−12.1244	−0.411527
$869$	−9.00000	−0.305304
$870$	0	0
$871$	0	0
$872$	−6.92820	−0.234619
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	19.0000	0.641219
$879$	0	0
$880$	0	0
$881$	−48.4974	−1.63392	−0.816960	−	0.576695i	$-0.804342\pi$
$881$	−48.4974	−1.63392	−0.816960	+	0.576695i	$0.804342\pi$
$882$	0	0
$883$	−2.00000	−0.0673054	−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000	−0.0673054	−0.0336527	+	0.999434i	$0.510714\pi$
$884$	−13.8564	−0.466041
$885$	0	0
$886$	9.00000	0.302361
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	0	0
$889$	−6.92820	−0.232364
$890$	0	0
$891$	0	0
$892$	−16.0000	−0.535720
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.73205	0.0578638
$897$	0	0
$898$	−12.0000	−0.400445
$899$	−21.0000	−0.700389
$900$	0	0
$901$	−30.0000	−0.999445
$902$	20.7846	0.692052
$903$	0	0
$904$	−20.7846	−0.691286
$905$	0	0
$906$	0	0
$907$	−3.46410	−0.115024	−0.0575118	−	0.998345i	$-0.518317\pi$
$907$	−3.46410	−0.115024	−0.0575118	+	0.998345i	$0.518317\pi$
$908$	−15.5885	−0.517321
$909$	0	0
$910$	0	0
$911$	−17.3205	−0.573854	−0.286927	−	0.957952i	$-0.592634\pi$
$911$	−17.3205	−0.573854	−0.286927	+	0.957952i	$0.592634\pi$
$912$	0	0
$913$	−3.00000	−0.0992855
$914$	15.5885	0.515620
$915$	0	0
$916$	−20.7846	−0.686743
$917$	25.9808	0.857960
$918$	0	0
$919$	24.2487	0.799891	0.399946	−	0.916539i	$-0.369029\pi$
$919$	24.2487	0.799891	0.399946	+	0.916539i	$0.369029\pi$
$920$	0	0
$921$	0	0
$922$	−33.0000	−1.08680
$923$	24.0000	0.789970
$924$	0	0
$925$	0	0
$926$	31.0000	1.01872
$927$	0	0
$928$	3.00000	0.0984798
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	0	0
$932$	18.0000	0.589610
$933$	0	0
$934$	−8.66025	−0.283372
$935$	0	0
$936$	0	0
$937$	36.3731	1.18826	0.594128	−	0.804370i	$-0.297497\pi$
$937$	36.3731	1.18826	0.594128	+	0.804370i	$0.297497\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.73205	0.0564632	0.0282316	−	0.999601i	$-0.491012\pi$
$941$	1.73205	0.0564632	0.0282316	+	0.999601i	$0.491012\pi$
$942$	0	0
$943$	0	0
$944$	−9.00000	−0.292925
$945$	0	0
$946$	−18.0000	−0.585230
$947$	−39.0000	−1.26733	−0.633665	−	0.773608i	$-0.718450\pi$
$947$	−39.0000	−1.26733	−0.633665	+	0.773608i	$0.718450\pi$
$948$	0	0
$949$	−44.0000	−1.42830
$950$	0	0
$951$	0	0
$952$	−6.00000	−0.194461
$953$	−6.92820	−0.224427	−0.112213	−	0.993684i	$-0.535794\pi$
$953$	−6.92820	−0.224427	−0.112213	+	0.993684i	$0.535794\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	38.1051	1.23112
$959$	12.0000	0.387500
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	0	0
$964$	20.7846	0.669427
$965$	0	0
$966$	0	0
$967$	−5.00000	−0.160789	−0.0803946	−	0.996763i	$-0.525618\pi$
$967$	−5.00000	−0.160789	−0.0803946	+	0.996763i	$0.525618\pi$
$968$	8.00000	0.257130
$969$	0	0
$970$	0	0
$971$	−45.0333	−1.44519	−0.722594	−	0.691273i	$-0.757050\pi$
$971$	−45.0333	−1.44519	−0.722594	+	0.691273i	$0.757050\pi$
$972$	0	0
$973$	−13.8564	−0.444216
$974$	7.00000	0.224294
$975$	0	0
$976$	6.92820	0.221766
$977$	−24.2487	−0.775785	−0.387893	−	0.921705i	$-0.626797\pi$
$977$	−24.2487	−0.775785	−0.387893	+	0.921705i	$0.626797\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	0	0
$982$	15.0000	0.478669
$983$	13.8564	0.441951	0.220975	−	0.975279i	$-0.429076\pi$
$983$	13.8564	0.441951	0.220975	+	0.975279i	$0.429076\pi$
$984$	0	0
$985$	0	0
$986$	−10.3923	−0.330958
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	37.0000	1.17534	0.587672	−	0.809099i	$-0.300045\pi$
$991$	37.0000	1.17534	0.587672	+	0.809099i	$0.300045\pi$
$992$	−7.00000	−0.222250
$993$	0	0
$994$	10.3923	0.329624
$995$	0	0
$996$	0	0
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	32.0000	1.01294
$999$	0	0

Twists

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

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

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.73205 q^{7} -1.00000 q^{8} +1.73205 q^{11} +4.00000 q^{13} +1.73205 q^{14} +1.00000 q^{16} -3.46410 q^{17} -1.73205 q^{22} -5.00000 q^{25} -4.00000 q^{26} -1.73205 q^{28} -3.00000 q^{29} +7.00000 q^{31} -1.00000 q^{32} +3.46410 q^{34} -12.0000 q^{41} +10.3923 q^{43} +1.73205 q^{44} -4.00000 q^{49} +5.00000 q^{50} +4.00000 q^{52} +8.66025 q^{53} +1.73205 q^{56} +3.00000 q^{58} -9.00000 q^{59} +6.92820 q^{61} -7.00000 q^{62} +1.00000 q^{64} -3.46410 q^{68} +6.00000 q^{71} -11.0000 q^{73} -3.00000 q^{77} -5.19615 q^{79} +12.0000 q^{82} -1.73205 q^{83} -10.3923 q^{86} -1.73205 q^{88} -10.3923 q^{89} -6.92820 q^{91} -15.5885 q^{97} +4.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 8 q^{13} + 2 q^{16} - 10 q^{25} - 8 q^{26} - 6 q^{29} + 14 q^{31} - 2 q^{32} - 24 q^{41} - 8 q^{49} + 10 q^{50} + 8 q^{52} + 6 q^{58} - 18 q^{59} - 14 q^{62} + 2 q^{64}+ \cdots + 8 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 8 * q^13 + 2 * q^16 - 10 * q^25 - 8 * q^26 - 6 * q^29 + 14 * q^31 - 2 * q^32 - 24 * q^41 - 8 * q^49 + 10 * q^50 + 8 * q^52 + 6 * q^58 - 18 * q^59 - 14 * q^62 + 2 * q^64 + 12 * q^71 - 22 * q^73 - 6 * q^77 + 24 * q^82 + 8 * q^98

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