LMFDB - Embedded newform 9522.2.a.ca.1.4

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 = [5,5,0,5,11,0,1,5,0,11,11,0,10,1,0,5,11,0,1,11,0,11,0,0,30,10, 0,1,-3] 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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +4.16140 q^{5} +2.94408 q^{7} +1.00000 q^{8} +4.16140 q^{10} -3.28400 q^{11} -0.140551 q^{13} +2.94408 q^{14} +1.00000 q^{16} +6.25297 q^{17} -2.37482 q^{19} +4.16140 q^{20} -3.28400 q^{22} +12.3172 q^{25} -0.140551 q^{26} +2.94408 q^{28} +9.93789 q^{29} +2.28666 q^{31} +1.00000 q^{32} +6.25297 q^{34} +12.2515 q^{35} -6.20786 q^{37} -2.37482 q^{38} +4.16140 q^{40} -6.45186 q^{41} +1.28173 q^{43} -3.28400 q^{44} -2.84018 q^{47} +1.66759 q^{49} +12.3172 q^{50} -0.140551 q^{52} +4.35073 q^{53} -13.6660 q^{55} +2.94408 q^{56} +9.93789 q^{58} -0.0732426 q^{59} -1.92202 q^{61} +2.28666 q^{62} +1.00000 q^{64} -0.584891 q^{65} -4.33350 q^{67} +6.25297 q^{68} +12.2515 q^{70} -9.78735 q^{71} +5.48540 q^{73} -6.20786 q^{74} -2.37482 q^{76} -9.66835 q^{77} -9.62324 q^{79} +4.16140 q^{80} -6.45186 q^{82} -11.0085 q^{83} +26.0211 q^{85} +1.28173 q^{86} -3.28400 q^{88} +13.1038 q^{89} -0.413795 q^{91} -2.84018 q^{94} -9.88257 q^{95} +12.2102 q^{97} +1.66759 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} + 11 q^{5} + q^{7} + 5 q^{8} + 11 q^{10} + 11 q^{11} + 10 q^{13} + q^{14} + 5 q^{16} + 11 q^{17} + q^{19} + 11 q^{20} + 11 q^{22} + 30 q^{25} + 10 q^{26} + q^{28} - 3 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 + 11 * q^5 + q^7 + 5 * q^8 + 11 * q^10 + 11 * q^11 + 10 * q^13 + q^14 + 5 * q^16 + 11 * q^17 + q^19 + 11 * q^20 + 11 * q^22 + 30 * q^25 + 10 * q^26 + q^28 - 3 * q^29 + 2 * q^31 + 5 * q^32 + 11 * q^34 + 11 * q^35 - 16 * q^37 + q^38 + 11 * q^40 + 8 * q^41 + 12 * q^43 + 11 * q^44 - 4 * q^49 + 30 * q^50 + 10 * q^52 + 27 * q^53 + q^56 - 3 * q^58 + 6 * q^59 - 5 * q^61 + 2 * q^62 + 5 * q^64 + 22 * q^65 + 7 * q^67 + 11 * q^68 + 11 * q^70 + 11 * q^71 - 12 * q^73 - 16 * q^74 + q^76 - 11 * q^77 - 16 * q^79 + 11 * q^80 + 8 * q^82 + 22 * q^83 - 11 * q^85 + 12 * q^86 + 11 * q^88 + 38 * q^89 - 9 * q^91 - 11 * q^95 - 5 * q^97 - 4 * 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
$5$	4.16140	1.86103	0.930517	−	0.366249i	$-0.119358\pi$
$5$	4.16140	1.86103	0.930517	+	0.366249i	$0.119358\pi$
$6$	0	0
$7$	2.94408	1.11276	0.556378	−	0.830929i	$-0.312191\pi$
$7$	2.94408	1.11276	0.556378	+	0.830929i	$0.312191\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	4.16140	1.31595
$11$	−3.28400	−0.990163	−0.495082	−	0.868846i	$-0.664862\pi$
$11$	−3.28400	−0.990163	−0.495082	+	0.868846i	$0.664862\pi$
$12$	0	0
$13$	−0.140551	−0.0389820	−0.0194910	−	0.999810i	$-0.506205\pi$
$13$	−0.140551	−0.0389820	−0.0194910	+	0.999810i	$0.506205\pi$
$14$	2.94408	0.786838
$15$	0	0
$16$	1.00000	0.250000
$17$	6.25297	1.51657	0.758285	−	0.651924i	$-0.226038\pi$
$17$	6.25297	1.51657	0.758285	+	0.651924i	$0.226038\pi$
$18$	0	0
$19$	−2.37482	−0.544821	−0.272410	−	0.962181i	$-0.587821\pi$
$19$	−2.37482	−0.544821	−0.272410	+	0.962181i	$0.587821\pi$
$20$	4.16140	0.930517
$21$	0	0
$22$	−3.28400	−0.700151
$23$	0	0
$23$	0	0
$24$	0	0
$25$	12.3172	2.46345
$26$	−0.140551	−0.0275644
$27$	0	0
$28$	2.94408	0.556378
$29$	9.93789	1.84542	0.922710	−	0.385495i	$-0.125969\pi$
$29$	9.93789	1.84542	0.922710	+	0.385495i	$0.125969\pi$
$30$	0	0
$31$	2.28666	0.410697	0.205348	−	0.978689i	$-0.434167\pi$
$31$	2.28666	0.410697	0.205348	+	0.978689i	$0.434167\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.25297	1.07238
$35$	12.2515	2.07088
$36$	0	0
$37$	−6.20786	−1.02057	−0.510283	−	0.860007i	$-0.670459\pi$
$37$	−6.20786	−1.02057	−0.510283	+	0.860007i	$0.670459\pi$
$38$	−2.37482	−0.385246
$39$	0	0
$40$	4.16140	0.657975
$41$	−6.45186	−1.00761	−0.503805	−	0.863817i	$-0.668067\pi$
$41$	−6.45186	−1.00761	−0.503805	+	0.863817i	$0.668067\pi$
$42$	0	0
$43$	1.28173	0.195462	0.0977312	−	0.995213i	$-0.468841\pi$
$43$	1.28173	0.195462	0.0977312	+	0.995213i	$0.468841\pi$
$44$	−3.28400	−0.495082
$45$	0	0
$46$	0	0
$47$	−2.84018	−0.414283	−0.207142	−	0.978311i	$-0.566416\pi$
$47$	−2.84018	−0.414283	−0.207142	+	0.978311i	$0.566416\pi$
$48$	0	0
$49$	1.66759	0.238228
$50$	12.3172	1.74192
$51$	0	0
$52$	−0.140551	−0.0194910
$53$	4.35073	0.597619	0.298809	−	0.954313i	$-0.403411\pi$
$53$	4.35073	0.597619	0.298809	+	0.954313i	$0.403411\pi$
$54$	0	0
$55$	−13.6660	−1.84273
$56$	2.94408	0.393419
$57$	0	0
$58$	9.93789	1.30491
$59$	−0.0732426	−0.00953537	−0.00476769	−	0.999989i	$-0.501518\pi$
$59$	−0.0732426	−0.00953537	−0.00476769	+	0.999989i	$0.501518\pi$
$60$	0	0
$61$	−1.92202	−0.246090	−0.123045	−	0.992401i	$-0.539266\pi$
$61$	−1.92202	−0.246090	−0.123045	+	0.992401i	$0.539266\pi$
$62$	2.28666	0.290406
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.584891	−0.0725468
$66$	0	0
$67$	−4.33350	−0.529421	−0.264711	−	0.964328i	$-0.585276\pi$
$67$	−4.33350	−0.529421	−0.264711	+	0.964328i	$0.585276\pi$
$68$	6.25297	0.758285
$69$	0	0
$70$	12.2515	1.46433
$71$	−9.78735	−1.16155	−0.580773	−	0.814066i	$-0.697249\pi$
$71$	−9.78735	−1.16155	−0.580773	+	0.814066i	$0.697249\pi$
$72$	0	0
$73$	5.48540	0.642018	0.321009	−	0.947076i	$-0.395978\pi$
$73$	5.48540	0.642018	0.321009	+	0.947076i	$0.395978\pi$
$74$	−6.20786	−0.721649
$75$	0	0
$76$	−2.37482	−0.272410
$77$	−9.66835	−1.10181
$78$	0	0
$79$	−9.62324	−1.08270	−0.541349	−	0.840798i	$-0.682086\pi$
$79$	−9.62324	−1.08270	−0.541349	+	0.840798i	$0.682086\pi$
$80$	4.16140	0.465258
$81$	0	0
$82$	−6.45186	−0.712488
$83$	−11.0085	−1.20834	−0.604168	−	0.796857i	$-0.706495\pi$
$83$	−11.0085	−1.20834	−0.604168	+	0.796857i	$0.706495\pi$
$84$	0	0
$85$	26.0211	2.82239
$86$	1.28173	0.138213
$87$	0	0
$88$	−3.28400	−0.350076
$89$	13.1038	1.38900	0.694499	−	0.719493i	$-0.255626\pi$
$89$	13.1038	1.38900	0.694499	+	0.719493i	$0.255626\pi$
$90$	0	0
$91$	−0.413795	−0.0433775
$92$	0	0
$93$	0	0
$94$	−2.84018	−0.292943
$95$	−9.88257	−1.01393
$96$	0	0
$97$	12.2102	1.23975	0.619877	−	0.784699i	$-0.287182\pi$
$97$	12.2102	1.23975	0.619877	+	0.784699i	$0.287182\pi$
$98$	1.66759	0.168452
$99$	0	0
$100$	12.3172	1.23172
$101$	1.40824	0.140125	0.0700623	−	0.997543i	$-0.477680\pi$
$101$	1.40824	0.140125	0.0700623	+	0.997543i	$0.477680\pi$
$102$	0	0
$103$	−8.03671	−0.791881	−0.395940	−	0.918276i	$-0.629581\pi$
$103$	−8.03671	−0.791881	−0.395940	+	0.918276i	$0.629581\pi$
$104$	−0.140551	−0.0137822
$105$	0	0
$106$	4.35073	0.422580
$107$	19.3300	1.86870	0.934352	−	0.356351i	$-0.115979\pi$
$107$	19.3300	1.86870	0.934352	+	0.356351i	$0.115979\pi$
$108$	0	0
$109$	−6.47928	−0.620602	−0.310301	−	0.950638i	$-0.600430\pi$
$109$	−6.47928	−0.620602	−0.310301	+	0.950638i	$0.600430\pi$
$110$	−13.6660	−1.30301
$111$	0	0
$112$	2.94408	0.278189
$113$	−10.4243	−0.980635	−0.490318	−	0.871544i	$-0.663119\pi$
$113$	−10.4243	−0.980635	−0.490318	+	0.871544i	$0.663119\pi$
$114$	0	0
$115$	0	0
$116$	9.93789	0.922710
$117$	0	0
$118$	−0.0732426	−0.00674253
$119$	18.4092	1.68757
$120$	0	0
$121$	−0.215343	−0.0195767
$122$	−1.92202	−0.174012
$123$	0	0
$124$	2.28666	0.205348
$125$	30.4499	2.72353
$126$	0	0
$127$	−6.16595	−0.547140	−0.273570	−	0.961852i	$-0.588205\pi$
$127$	−6.16595	−0.547140	−0.273570	+	0.961852i	$0.588205\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−0.584891	−0.0512983
$131$	6.44554	0.563149	0.281575	−	0.959539i	$-0.409143\pi$
$131$	6.44554	0.563149	0.281575	+	0.959539i	$0.409143\pi$
$132$	0	0
$133$	−6.99165	−0.606253
$134$	−4.33350	−0.374357
$135$	0	0
$136$	6.25297	0.536188
$137$	14.3939	1.22975	0.614877	−	0.788623i	$-0.289206\pi$
$137$	14.3939	1.22975	0.614877	+	0.788623i	$0.289206\pi$
$138$	0	0
$139$	−10.9441	−0.928268	−0.464134	−	0.885765i	$-0.653634\pi$
$139$	−10.9441	−0.928268	−0.464134	+	0.885765i	$0.653634\pi$
$140$	12.2515	1.03544
$141$	0	0
$142$	−9.78735	−0.821336
$143$	0.461571	0.0385985
$144$	0	0
$145$	41.3555	3.43439
$146$	5.48540	0.453975
$147$	0	0
$148$	−6.20786	−0.510283
$149$	4.04605	0.331465	0.165732	−	0.986171i	$-0.447001\pi$
$149$	4.04605	0.331465	0.165732	+	0.986171i	$0.447001\pi$
$150$	0	0
$151$	−17.5087	−1.42483	−0.712417	−	0.701756i	$-0.752400\pi$
$151$	−17.5087	−1.42483	−0.712417	+	0.701756i	$0.752400\pi$
$152$	−2.37482	−0.192623
$153$	0	0
$154$	−9.66835	−0.779098
$155$	9.51571	0.764321
$156$	0	0
$157$	2.84274	0.226875	0.113438	−	0.993545i	$-0.463814\pi$
$157$	2.84274	0.226875	0.113438	+	0.993545i	$0.463814\pi$
$158$	−9.62324	−0.765584
$159$	0	0
$160$	4.16140	0.328987
$161$	0	0
$162$	0	0
$163$	−2.00952	−0.157397	−0.0786987	−	0.996898i	$-0.525077\pi$
$163$	−2.00952	−0.157397	−0.0786987	+	0.996898i	$0.525077\pi$
$164$	−6.45186	−0.503805
$165$	0	0
$166$	−11.0085	−0.854423
$167$	9.22759	0.714052	0.357026	−	0.934094i	$-0.383791\pi$
$167$	9.22759	0.714052	0.357026	+	0.934094i	$0.383791\pi$
$168$	0	0
$169$	−12.9802	−0.998480
$170$	26.0211	1.99573
$171$	0	0
$172$	1.28173	0.0977312
$173$	−9.39646	−0.714399	−0.357200	−	0.934028i	$-0.616268\pi$
$173$	−9.39646	−0.714399	−0.357200	+	0.934028i	$0.616268\pi$
$174$	0	0
$175$	36.2629	2.74122
$176$	−3.28400	−0.247541
$177$	0	0
$178$	13.1038	0.982170
$179$	20.4263	1.52674	0.763368	−	0.645964i	$-0.223544\pi$
$179$	20.4263	1.52674	0.763368	+	0.645964i	$0.223544\pi$
$180$	0	0
$181$	−14.9334	−1.10999	−0.554994	−	0.831854i	$-0.687279\pi$
$181$	−14.9334	−1.10999	−0.554994	+	0.831854i	$0.687279\pi$
$182$	−0.413795	−0.0306725
$183$	0	0
$184$	0	0
$185$	−25.8334	−1.89931
$186$	0	0
$187$	−20.5348	−1.50165
$188$	−2.84018	−0.207142
$189$	0	0
$190$	−9.88257	−0.716957
$191$	23.6544	1.71157	0.855785	−	0.517332i	$-0.173075\pi$
$191$	23.6544	1.71157	0.855785	+	0.517332i	$0.173075\pi$
$192$	0	0
$193$	−1.95483	−0.140712	−0.0703559	−	0.997522i	$-0.522413\pi$
$193$	−1.95483	−0.140712	−0.0703559	+	0.997522i	$0.522413\pi$
$194$	12.2102	0.876638
$195$	0	0
$196$	1.66759	0.119114
$197$	−1.87950	−0.133909	−0.0669546	−	0.997756i	$-0.521328\pi$
$197$	−1.87950	−0.133909	−0.0669546	+	0.997756i	$0.521328\pi$
$198$	0	0
$199$	16.5039	1.16993	0.584964	−	0.811059i	$-0.301109\pi$
$199$	16.5039	1.16993	0.584964	+	0.811059i	$0.301109\pi$
$200$	12.3172	0.870960
$201$	0	0
$202$	1.40824	0.0990831
$203$	29.2579	2.05350
$204$	0	0
$205$	−26.8487	−1.87520
$206$	−8.03671	−0.559944
$207$	0	0
$208$	−0.140551	−0.00974549
$209$	7.79890	0.539461
$210$	0	0
$211$	5.97819	0.411555	0.205778	−	0.978599i	$-0.434028\pi$
$211$	5.97819	0.411555	0.205778	+	0.978599i	$0.434028\pi$
$212$	4.35073	0.298809
$213$	0	0
$214$	19.3300	1.32137
$215$	5.33380	0.363762
$216$	0	0
$217$	6.73211	0.457006
$218$	−6.47928	−0.438832
$219$	0	0
$220$	−13.6660	−0.921364
$221$	−0.878865	−0.0591189
$222$	0	0
$223$	−20.1193	−1.34729	−0.673643	−	0.739057i	$-0.735271\pi$
$223$	−20.1193	−1.34729	−0.673643	+	0.739057i	$0.735271\pi$
$224$	2.94408	0.196709
$225$	0	0
$226$	−10.4243	−0.693414
$227$	13.0539	0.866418	0.433209	−	0.901294i	$-0.357381\pi$
$227$	13.0539	0.866418	0.433209	+	0.901294i	$0.357381\pi$
$228$	0	0
$229$	−17.7990	−1.17619	−0.588095	−	0.808792i	$-0.700122\pi$
$229$	−17.7990	−1.17619	−0.588095	+	0.808792i	$0.700122\pi$
$230$	0	0
$231$	0	0
$232$	9.93789	0.652454
$233$	−9.79827	−0.641906	−0.320953	−	0.947095i	$-0.604003\pi$
$233$	−9.79827	−0.641906	−0.320953	+	0.947095i	$0.604003\pi$
$234$	0	0
$235$	−11.8191	−0.770995
$236$	−0.0732426	−0.00476769
$237$	0	0
$238$	18.4092	1.19329
$239$	−2.99564	−0.193772	−0.0968859	−	0.995295i	$-0.530888\pi$
$239$	−2.99564	−0.193772	−0.0968859	+	0.995295i	$0.530888\pi$
$240$	0	0
$241$	20.4314	1.31610	0.658051	−	0.752973i	$-0.271381\pi$
$241$	20.4314	1.31610	0.658051	+	0.752973i	$0.271381\pi$
$242$	−0.215343	−0.0138428
$243$	0	0
$244$	−1.92202	−0.123045
$245$	6.93952	0.443350
$246$	0	0
$247$	0.333784	0.0212382
$248$	2.28666	0.145203
$249$	0	0
$250$	30.4499	1.92582
$251$	−0.994410	−0.0627666	−0.0313833	−	0.999507i	$-0.509991\pi$
$251$	−0.994410	−0.0627666	−0.0313833	+	0.999507i	$0.509991\pi$
$252$	0	0
$253$	0	0
$254$	−6.16595	−0.386886
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.02074	0.437942	0.218971	−	0.975731i	$-0.429730\pi$
$257$	7.02074	0.437942	0.218971	+	0.975731i	$0.429730\pi$
$258$	0	0
$259$	−18.2764	−1.13564
$260$	−0.584891	−0.0362734
$261$	0	0
$262$	6.44554	0.398207
$263$	−17.3724	−1.07123	−0.535613	−	0.844464i	$-0.679919\pi$
$263$	−17.3724	−1.07123	−0.535613	+	0.844464i	$0.679919\pi$
$264$	0	0
$265$	18.1051	1.11219
$266$	−6.99165	−0.428686
$267$	0	0
$268$	−4.33350	−0.264711
$269$	−30.4092	−1.85408	−0.927041	−	0.374960i	$-0.877656\pi$
$269$	−30.4092	−1.85408	−0.927041	+	0.374960i	$0.877656\pi$
$270$	0	0
$271$	−6.05563	−0.367854	−0.183927	−	0.982940i	$-0.558881\pi$
$271$	−6.05563	−0.367854	−0.183927	+	0.982940i	$0.558881\pi$
$272$	6.25297	0.379142
$273$	0	0
$274$	14.3939	0.869567
$275$	−40.4498	−2.43922
$276$	0	0
$277$	7.58051	0.455469	0.227734	−	0.973723i	$-0.426868\pi$
$277$	7.58051	0.455469	0.227734	+	0.973723i	$0.426868\pi$
$278$	−10.9441	−0.656385
$279$	0	0
$280$	12.2515	0.732166
$281$	−15.5093	−0.925205	−0.462603	−	0.886566i	$-0.653084\pi$
$281$	−15.5093	−0.925205	−0.462603	+	0.886566i	$0.653084\pi$
$282$	0	0
$283$	9.42266	0.560119	0.280059	−	0.959983i	$-0.409646\pi$
$283$	9.42266	0.560119	0.280059	+	0.959983i	$0.409646\pi$
$284$	−9.78735	−0.580773
$285$	0	0
$286$	0.461571	0.0272933
$287$	−18.9948	−1.12123
$288$	0	0
$289$	22.0997	1.29998
$290$	41.3555	2.42848
$291$	0	0
$292$	5.48540	0.321009
$293$	−6.05714	−0.353862	−0.176931	−	0.984223i	$-0.556617\pi$
$293$	−6.05714	−0.353862	−0.176931	+	0.984223i	$0.556617\pi$
$294$	0	0
$295$	−0.304792	−0.0177457
$296$	−6.20786	−0.360825
$297$	0	0
$298$	4.04605	0.234381
$299$	0	0
$300$	0	0
$301$	3.77352	0.217502
$302$	−17.5087	−1.00751
$303$	0	0
$304$	−2.37482	−0.136205
$305$	−7.99830	−0.457981
$306$	0	0
$307$	−21.2457	−1.21255	−0.606277	−	0.795253i	$-0.707338\pi$
$307$	−21.2457	−1.21255	−0.606277	+	0.795253i	$0.707338\pi$
$308$	−9.66835	−0.550905
$309$	0	0
$310$	9.51571	0.540456
$311$	−8.87563	−0.503291	−0.251645	−	0.967820i	$-0.580972\pi$
$311$	−8.87563	−0.503291	−0.251645	+	0.967820i	$0.580972\pi$
$312$	0	0
$313$	−10.8762	−0.614758	−0.307379	−	0.951587i	$-0.599452\pi$
$313$	−10.8762	−0.614758	−0.307379	+	0.951587i	$0.599452\pi$
$314$	2.84274	0.160425
$315$	0	0
$316$	−9.62324	−0.541349
$317$	32.8164	1.84315	0.921576	−	0.388197i	$-0.126902\pi$
$317$	32.8164	1.84315	0.921576	+	0.388197i	$0.126902\pi$
$318$	0	0
$319$	−32.6360	−1.82727
$320$	4.16140	0.232629
$321$	0	0
$322$	0	0
$323$	−14.8497	−0.826258
$324$	0	0
$325$	−1.73121	−0.0960300
$326$	−2.00952	−0.111297
$327$	0	0
$328$	−6.45186	−0.356244
$329$	−8.36172	−0.460997
$330$	0	0
$331$	−17.6130	−0.968100	−0.484050	−	0.875040i	$-0.660835\pi$
$331$	−17.6130	−0.968100	−0.484050	+	0.875040i	$0.660835\pi$
$332$	−11.0085	−0.604168
$333$	0	0
$334$	9.22759	0.504911
$335$	−18.0334	−0.985271
$336$	0	0
$337$	21.5047	1.17144	0.585718	−	0.810515i	$-0.300812\pi$
$337$	21.5047	1.17144	0.585718	+	0.810515i	$0.300812\pi$
$338$	−12.9802	−0.706032
$339$	0	0
$340$	26.0211	1.41119
$341$	−7.50940	−0.406657
$342$	0	0
$343$	−15.6990	−0.847667
$344$	1.28173	0.0691064
$345$	0	0
$346$	−9.39646	−0.505157
$347$	−21.4058	−1.14912	−0.574562	−	0.818461i	$-0.694828\pi$
$347$	−21.4058	−1.14912	−0.574562	+	0.818461i	$0.694828\pi$
$348$	0	0
$349$	29.5309	1.58075	0.790376	−	0.612622i	$-0.209885\pi$
$349$	29.5309	1.58075	0.790376	+	0.612622i	$0.209885\pi$
$350$	36.2629	1.93833
$351$	0	0
$352$	−3.28400	−0.175038
$353$	8.89518	0.473442	0.236721	−	0.971578i	$-0.423927\pi$
$353$	8.89518	0.473442	0.236721	+	0.971578i	$0.423927\pi$
$354$	0	0
$355$	−40.7291	−2.16168
$356$	13.1038	0.694499
$357$	0	0
$358$	20.4263	1.07957
$359$	22.5098	1.18802	0.594011	−	0.804457i	$-0.297544\pi$
$359$	22.5098	1.18802	0.594011	+	0.804457i	$0.297544\pi$
$360$	0	0
$361$	−13.3602	−0.703170
$362$	−14.9334	−0.784880
$363$	0	0
$364$	−0.413795	−0.0216887
$365$	22.8270	1.19482
$366$	0	0
$367$	7.06807	0.368950	0.184475	−	0.982837i	$-0.440942\pi$
$367$	7.06807	0.368950	0.184475	+	0.982837i	$0.440942\pi$
$368$	0	0
$369$	0	0
$370$	−25.8334	−1.34301
$371$	12.8089	0.665004
$372$	0	0
$373$	−17.8360	−0.923511	−0.461756	−	0.887007i	$-0.652780\pi$
$373$	−17.8360	−0.923511	−0.461756	+	0.887007i	$0.652780\pi$
$374$	−20.5348	−1.06183
$375$	0	0
$376$	−2.84018	−0.146471
$377$	−1.39679	−0.0719381
$378$	0	0
$379$	21.0931	1.08348	0.541740	−	0.840546i	$-0.317766\pi$
$379$	21.0931	1.08348	0.541740	+	0.840546i	$0.317766\pi$
$380$	−9.88257	−0.506965
$381$	0	0
$382$	23.6544	1.21026
$383$	−1.12641	−0.0575570	−0.0287785	−	0.999586i	$-0.509162\pi$
$383$	−1.12641	−0.0575570	−0.0287785	+	0.999586i	$0.509162\pi$
$384$	0	0
$385$	−40.2339	−2.05051
$386$	−1.95483	−0.0994982
$387$	0	0
$388$	12.2102	0.619877
$389$	3.11938	0.158159	0.0790795	−	0.996868i	$-0.474802\pi$
$389$	3.11938	0.158159	0.0790795	+	0.996868i	$0.474802\pi$
$390$	0	0
$391$	0	0
$392$	1.66759	0.0842262
$393$	0	0
$394$	−1.87950	−0.0946880
$395$	−40.0461	−2.01494
$396$	0	0
$397$	−20.6576	−1.03678	−0.518388	−	0.855146i	$-0.673468\pi$
$397$	−20.6576	−1.03678	−0.518388	+	0.855146i	$0.673468\pi$
$398$	16.5039	0.827264
$399$	0	0
$400$	12.3172	0.615862
$401$	−10.3034	−0.514529	−0.257264	−	0.966341i	$-0.582821\pi$
$401$	−10.3034	−0.514529	−0.257264	+	0.966341i	$0.582821\pi$
$402$	0	0
$403$	−0.321394	−0.0160098
$404$	1.40824	0.0700623
$405$	0	0
$406$	29.2579	1.45205
$407$	20.3866	1.01053
$408$	0	0
$409$	3.60277	0.178145	0.0890727	−	0.996025i	$-0.471610\pi$
$409$	3.60277	0.178145	0.0890727	+	0.996025i	$0.471610\pi$
$410$	−26.8487	−1.32596
$411$	0	0
$412$	−8.03671	−0.395940
$413$	−0.215632	−0.0106106
$414$	0	0
$415$	−45.8106	−2.24876
$416$	−0.140551	−0.00689110
$417$	0	0
$418$	7.79890	0.381457
$419$	13.4970	0.659370	0.329685	−	0.944091i	$-0.393057\pi$
$419$	13.4970	0.659370	0.329685	+	0.944091i	$0.393057\pi$
$420$	0	0
$421$	−16.6474	−0.811342	−0.405671	−	0.914019i	$-0.632962\pi$
$421$	−16.6474	−0.811342	−0.405671	+	0.914019i	$0.632962\pi$
$422$	5.97819	0.291014
$423$	0	0
$424$	4.35073	0.211290
$425$	77.0194	3.73599
$426$	0	0
$427$	−5.65858	−0.273838
$428$	19.3300	0.934352
$429$	0	0
$430$	5.33380	0.257219
$431$	−21.0164	−1.01232	−0.506162	−	0.862438i	$-0.668936\pi$
$431$	−21.0164	−1.01232	−0.506162	+	0.862438i	$0.668936\pi$
$432$	0	0
$433$	2.07347	0.0996445	0.0498222	−	0.998758i	$-0.484135\pi$
$433$	2.07347	0.0996445	0.0498222	+	0.998758i	$0.484135\pi$
$434$	6.73211	0.323152
$435$	0	0
$436$	−6.47928	−0.310301
$437$	0	0
$438$	0	0
$439$	2.26950	0.108317	0.0541587	−	0.998532i	$-0.482752\pi$
$439$	2.26950	0.108317	0.0541587	+	0.998532i	$0.482752\pi$
$440$	−13.6660	−0.651503
$441$	0	0
$442$	−0.878865	−0.0418033
$443$	−26.8178	−1.27415	−0.637075	−	0.770802i	$-0.719856\pi$
$443$	−26.8178	−1.27415	−0.637075	+	0.770802i	$0.719856\pi$
$444$	0	0
$445$	54.5301	2.58497
$446$	−20.1193	−0.952675
$447$	0	0
$448$	2.94408	0.139095
$449$	−4.56941	−0.215644	−0.107822	−	0.994170i	$-0.534388\pi$
$449$	−4.56941	−0.215644	−0.107822	+	0.994170i	$0.534388\pi$
$450$	0	0
$451$	21.1879	0.997699
$452$	−10.4243	−0.490318
$453$	0	0
$454$	13.0539	0.612650
$455$	−1.72196	−0.0807269
$456$	0	0
$457$	26.8144	1.25433	0.627163	−	0.778888i	$-0.284216\pi$
$457$	26.8144	1.25433	0.627163	+	0.778888i	$0.284216\pi$
$458$	−17.7990	−0.831691
$459$	0	0
$460$	0	0
$461$	1.33658	0.0622507	0.0311253	−	0.999515i	$-0.490091\pi$
$461$	1.33658	0.0622507	0.0311253	+	0.999515i	$0.490091\pi$
$462$	0	0
$463$	−36.2702	−1.68562	−0.842809	−	0.538213i	$-0.819100\pi$
$463$	−36.2702	−1.68562	−0.842809	+	0.538213i	$0.819100\pi$
$464$	9.93789	0.461355
$465$	0	0
$466$	−9.79827	−0.453896
$467$	−13.3242	−0.616570	−0.308285	−	0.951294i	$-0.599755\pi$
$467$	−13.3242	−0.616570	−0.308285	+	0.951294i	$0.599755\pi$
$468$	0	0
$469$	−12.7582	−0.589117
$470$	−11.8191	−0.545176
$471$	0	0
$472$	−0.0732426	−0.00337126
$473$	−4.20921	−0.193540
$474$	0	0
$475$	−29.2512	−1.34214
$476$	18.4092	0.843786
$477$	0	0
$478$	−2.99564	−0.137017
$479$	26.5551	1.21333	0.606666	−	0.794957i	$-0.292506\pi$
$479$	26.5551	1.21333	0.606666	+	0.794957i	$0.292506\pi$
$480$	0	0
$481$	0.872524	0.0397837
$482$	20.4314	0.930625
$483$	0	0
$484$	−0.215343	−0.00978834
$485$	50.8113	2.30722
$486$	0	0
$487$	−13.0668	−0.592114	−0.296057	−	0.955170i	$-0.595672\pi$
$487$	−13.0668	−0.592114	−0.296057	+	0.955170i	$0.595672\pi$
$488$	−1.92202	−0.0870059
$489$	0	0
$490$	6.93952	0.313496
$491$	33.6063	1.51663	0.758315	−	0.651888i	$-0.226023\pi$
$491$	33.6063	1.51663	0.758315	+	0.651888i	$0.226023\pi$
$492$	0	0
$493$	62.1414	2.79871
$494$	0.333784	0.0150177
$495$	0	0
$496$	2.28666	0.102674
$497$	−28.8147	−1.29252
$498$	0	0
$499$	24.2163	1.08407	0.542035	−	0.840356i	$-0.317654\pi$
$499$	24.2163	1.08407	0.542035	+	0.840356i	$0.317654\pi$
$500$	30.4499	1.36176
$501$	0	0
$502$	−0.994410	−0.0443827
$503$	−40.8112	−1.81968	−0.909841	−	0.414956i	$-0.863797\pi$
$503$	−40.8112	−1.81968	−0.909841	+	0.414956i	$0.863797\pi$
$504$	0	0
$505$	5.86023	0.260777
$506$	0	0
$507$	0	0
$508$	−6.16595	−0.273570
$509$	6.50479	0.288320	0.144160	−	0.989554i	$-0.453952\pi$
$509$	6.50479	0.288320	0.144160	+	0.989554i	$0.453952\pi$
$510$	0	0
$511$	16.1495	0.714410
$512$	1.00000	0.0441942
$513$	0	0
$514$	7.02074	0.309672
$515$	−33.4440	−1.47372
$516$	0	0
$517$	9.32716	0.410208
$518$	−18.2764	−0.803020
$519$	0	0
$520$	−0.584891	−0.0256492
$521$	40.1764	1.76016	0.880080	−	0.474825i	$-0.157489\pi$
$521$	40.1764	1.76016	0.880080	+	0.474825i	$0.157489\pi$
$522$	0	0
$523$	12.3036	0.538000	0.269000	−	0.963140i	$-0.413307\pi$
$523$	12.3036	0.538000	0.269000	+	0.963140i	$0.413307\pi$
$524$	6.44554	0.281575
$525$	0	0
$526$	−17.3724	−0.757471
$527$	14.2984	0.622850
$528$	0	0
$529$	0	0
$530$	18.1051	0.786436
$531$	0	0
$532$	−6.99165	−0.303126
$533$	0.906818	0.0392786
$534$	0	0
$535$	80.4399	3.47772
$536$	−4.33350	−0.187179
$537$	0	0
$538$	−30.4092	−1.31103
$539$	−5.47638	−0.235884
$540$	0	0
$541$	2.90280	0.124801	0.0624006	−	0.998051i	$-0.480124\pi$
$541$	2.90280	0.124801	0.0624006	+	0.998051i	$0.480124\pi$
$542$	−6.05563	−0.260112
$543$	0	0
$544$	6.25297	0.268094
$545$	−26.9628	−1.15496
$546$	0	0
$547$	26.7027	1.14172	0.570862	−	0.821046i	$-0.306609\pi$
$547$	26.7027	1.14172	0.570862	+	0.821046i	$0.306609\pi$
$548$	14.3939	0.614877
$549$	0	0
$550$	−40.4498	−1.72479
$551$	−23.6007	−1.00542
$552$	0	0
$553$	−28.3316	−1.20478
$554$	7.58051	0.322065
$555$	0	0
$556$	−10.9441	−0.464134
$557$	−24.9858	−1.05868	−0.529340	−	0.848410i	$-0.677560\pi$
$557$	−24.9858	−1.05868	−0.529340	+	0.848410i	$0.677560\pi$
$558$	0	0
$559$	−0.180149	−0.00761951
$560$	12.2515	0.517720
$561$	0	0
$562$	−15.5093	−0.654219
$563$	31.1889	1.31445	0.657227	−	0.753693i	$-0.271729\pi$
$563$	31.1889	1.31445	0.657227	+	0.753693i	$0.271729\pi$
$564$	0	0
$565$	−43.3797	−1.82500
$566$	9.42266	0.396064
$567$	0	0
$568$	−9.78735	−0.410668
$569$	6.06836	0.254399	0.127199	−	0.991877i	$-0.459401\pi$
$569$	6.06836	0.254399	0.127199	+	0.991877i	$0.459401\pi$
$570$	0	0
$571$	35.4907	1.48524	0.742620	−	0.669712i	$-0.233583\pi$
$571$	35.4907	1.48524	0.742620	+	0.669712i	$0.233583\pi$
$572$	0.461571	0.0192993
$573$	0	0
$574$	−18.9948	−0.792826
$575$	0	0
$576$	0	0
$577$	5.97865	0.248894	0.124447	−	0.992226i	$-0.460284\pi$
$577$	5.97865	0.248894	0.124447	+	0.992226i	$0.460284\pi$
$578$	22.0997	0.919226
$579$	0	0
$580$	41.3555	1.71719
$581$	−32.4098	−1.34458
$582$	0	0
$583$	−14.2878	−0.591740
$584$	5.48540	0.226988
$585$	0	0
$586$	−6.05714	−0.250218
$587$	7.73412	0.319221	0.159611	−	0.987180i	$-0.448976\pi$
$587$	7.73412	0.319221	0.159611	+	0.987180i	$0.448976\pi$
$588$	0	0
$589$	−5.43041	−0.223756
$590$	−0.304792	−0.0125481
$591$	0	0
$592$	−6.20786	−0.255141
$593$	−19.3172	−0.793263	−0.396632	−	0.917978i	$-0.629821\pi$
$593$	−19.3172	−0.793263	−0.396632	+	0.917978i	$0.629821\pi$
$594$	0	0
$595$	76.6082	3.14063
$596$	4.04605	0.165732
$597$	0	0
$598$	0	0
$599$	−18.1025	−0.739648	−0.369824	−	0.929102i	$-0.620582\pi$
$599$	−18.1025	−0.739648	−0.369824	+	0.929102i	$0.620582\pi$
$600$	0	0
$601$	−31.5227	−1.28584	−0.642918	−	0.765935i	$-0.722277\pi$
$601$	−31.5227	−1.28584	−0.642918	+	0.765935i	$0.722277\pi$
$602$	3.77352	0.153797
$603$	0	0
$604$	−17.5087	−0.712417
$605$	−0.896130	−0.0364329
$606$	0	0
$607$	9.53426	0.386984	0.193492	−	0.981102i	$-0.438019\pi$
$607$	9.53426	0.386984	0.193492	+	0.981102i	$0.438019\pi$
$608$	−2.37482	−0.0963116
$609$	0	0
$610$	−7.99830	−0.323842
$611$	0.399192	0.0161496
$612$	0	0
$613$	14.2612	0.576003	0.288001	−	0.957630i	$-0.407009\pi$
$613$	14.2612	0.576003	0.288001	+	0.957630i	$0.407009\pi$
$614$	−21.2457	−0.857405
$615$	0	0
$616$	−9.66835	−0.389549
$617$	−3.10089	−0.124837	−0.0624186	−	0.998050i	$-0.519881\pi$
$617$	−3.10089	−0.124837	−0.0624186	+	0.998050i	$0.519881\pi$
$618$	0	0
$619$	−45.2548	−1.81894	−0.909472	−	0.415765i	$-0.863514\pi$
$619$	−45.2548	−1.81894	−0.909472	+	0.415765i	$0.863514\pi$
$620$	9.51571	0.382160
$621$	0	0
$622$	−8.87563	−0.355880
$623$	38.5786	1.54562
$624$	0	0
$625$	65.1282	2.60513
$626$	−10.8762	−0.434699
$627$	0	0
$628$	2.84274	0.113438
$629$	−38.8176	−1.54776
$630$	0	0
$631$	−22.9711	−0.914465	−0.457233	−	0.889347i	$-0.651159\pi$
$631$	−22.9711	−0.914465	−0.457233	+	0.889347i	$0.651159\pi$
$632$	−9.62324	−0.382792
$633$	0	0
$634$	32.8164	1.30331
$635$	−25.6590	−1.01825
$636$	0	0
$637$	−0.234383	−0.00928658
$638$	−32.6360	−1.29207
$639$	0	0
$640$	4.16140	0.164494
$641$	−19.6746	−0.777101	−0.388550	−	0.921428i	$-0.627024\pi$
$641$	−19.6746	−0.777101	−0.388550	+	0.921428i	$0.627024\pi$
$642$	0	0
$643$	23.1768	0.914005	0.457003	−	0.889465i	$-0.348923\pi$
$643$	23.1768	0.914005	0.457003	+	0.889465i	$0.348923\pi$
$644$	0	0
$645$	0	0
$646$	−14.8497	−0.584253
$647$	−37.5092	−1.47464	−0.737319	−	0.675545i	$-0.763909\pi$
$647$	−37.5092	−1.47464	−0.737319	+	0.675545i	$0.763909\pi$
$648$	0	0
$649$	0.240529	0.00944158
$650$	−1.73121	−0.0679035
$651$	0	0
$652$	−2.00952	−0.0786987
$653$	13.3580	0.522739	0.261369	−	0.965239i	$-0.415826\pi$
$653$	13.3580	0.522739	0.261369	+	0.965239i	$0.415826\pi$
$654$	0	0
$655$	26.8225	1.04804
$656$	−6.45186	−0.251903
$657$	0	0
$658$	−8.36172	−0.325974
$659$	−14.0656	−0.547919	−0.273959	−	0.961741i	$-0.588333\pi$
$659$	−14.0656	−0.547919	−0.273959	+	0.961741i	$0.588333\pi$
$660$	0	0
$661$	7.69284	0.299217	0.149608	−	0.988745i	$-0.452199\pi$
$661$	7.69284	0.299217	0.149608	+	0.988745i	$0.452199\pi$
$662$	−17.6130	−0.684550
$663$	0	0
$664$	−11.0085	−0.427211
$665$	−29.0950	−1.12826
$666$	0	0
$667$	0	0
$668$	9.22759	0.357026
$669$	0	0
$670$	−18.0334	−0.696692
$671$	6.31192	0.243669
$672$	0	0
$673$	−49.2611	−1.89888	−0.949438	−	0.313954i	$-0.898346\pi$
$673$	−49.2611	−1.89888	−0.949438	+	0.313954i	$0.898346\pi$
$674$	21.5047	0.828331
$675$	0	0
$676$	−12.9802	−0.499240
$677$	−29.0927	−1.11812	−0.559062	−	0.829126i	$-0.688839\pi$
$677$	−29.0927	−1.11812	−0.559062	+	0.829126i	$0.688839\pi$
$678$	0	0
$679$	35.9477	1.37954
$680$	26.0211	0.997864
$681$	0	0
$682$	−7.50940	−0.287550
$683$	8.45035	0.323344	0.161672	−	0.986845i	$-0.448311\pi$
$683$	8.45035	0.323344	0.161672	+	0.986845i	$0.448311\pi$
$684$	0	0
$685$	59.8988	2.28861
$686$	−15.6990	−0.599391
$687$	0	0
$688$	1.28173	0.0488656
$689$	−0.611502	−0.0232964
$690$	0	0
$691$	16.9073	0.643183	0.321592	−	0.946878i	$-0.395782\pi$
$691$	16.9073	0.643183	0.321592	+	0.946878i	$0.395782\pi$
$692$	−9.39646	−0.357200
$693$	0	0
$694$	−21.4058	−0.812554
$695$	−45.5428	−1.72754
$696$	0	0
$697$	−40.3433	−1.52811
$698$	29.5309	1.11776
$699$	0	0
$700$	36.2629	1.37061
$701$	−27.3451	−1.03281	−0.516405	−	0.856345i	$-0.672730\pi$
$701$	−27.3451	−1.03281	−0.516405	+	0.856345i	$0.672730\pi$
$702$	0	0
$703$	14.7425	0.556025
$704$	−3.28400	−0.123770
$705$	0	0
$706$	8.89518	0.334774
$707$	4.14595	0.155925
$708$	0	0
$709$	0.711433	0.0267184	0.0133592	−	0.999911i	$-0.495748\pi$
$709$	0.711433	0.0267184	0.0133592	+	0.999911i	$0.495748\pi$
$710$	−40.7291	−1.52854
$711$	0	0
$712$	13.1038	0.491085
$713$	0	0
$714$	0	0
$715$	1.92078	0.0718331
$716$	20.4263	0.763368
$717$	0	0
$718$	22.5098	0.840058
$719$	19.7815	0.737727	0.368863	−	0.929484i	$-0.379747\pi$
$719$	19.7815	0.737727	0.368863	+	0.929484i	$0.379747\pi$
$720$	0	0
$721$	−23.6607	−0.881171
$722$	−13.3602	−0.497217
$723$	0	0
$724$	−14.9334	−0.554994
$725$	122.407	4.54609
$726$	0	0
$727$	34.5177	1.28019	0.640096	−	0.768295i	$-0.278895\pi$
$727$	34.5177	1.28019	0.640096	+	0.768295i	$0.278895\pi$
$728$	−0.413795	−0.0153362
$729$	0	0
$730$	22.8270	0.844863
$731$	8.01464	0.296432
$732$	0	0
$733$	9.13217	0.337304	0.168652	−	0.985676i	$-0.446059\pi$
$733$	9.13217	0.337304	0.168652	+	0.985676i	$0.446059\pi$
$734$	7.06807	0.260887
$735$	0	0
$736$	0	0
$737$	14.2312	0.524213
$738$	0	0
$739$	6.29732	0.231651	0.115825	−	0.993270i	$-0.463049\pi$
$739$	6.29732	0.231651	0.115825	+	0.993270i	$0.463049\pi$
$740$	−25.8334	−0.949654
$741$	0	0
$742$	12.8089	0.470229
$743$	23.5496	0.863953	0.431976	−	0.901885i	$-0.357816\pi$
$743$	23.5496	0.863953	0.431976	+	0.901885i	$0.357816\pi$
$744$	0	0
$745$	16.8372	0.616868
$746$	−17.8360	−0.653021
$747$	0	0
$748$	−20.5348	−0.750826
$749$	56.9091	2.07941
$750$	0	0
$751$	−32.1484	−1.17311	−0.586556	−	0.809909i	$-0.699517\pi$
$751$	−32.1484	−1.17311	−0.586556	+	0.809909i	$0.699517\pi$
$752$	−2.84018	−0.103571
$753$	0	0
$754$	−1.39679	−0.0508679
$755$	−72.8605	−2.65167
$756$	0	0
$757$	4.54012	0.165013	0.0825067	−	0.996591i	$-0.473707\pi$
$757$	4.54012	0.165013	0.0825067	+	0.996591i	$0.473707\pi$
$758$	21.0931	0.766137
$759$	0	0
$760$	−9.88257	−0.358478
$761$	−39.3062	−1.42485	−0.712425	−	0.701748i	$-0.752403\pi$
$761$	−39.3062	−1.42485	−0.712425	+	0.701748i	$0.752403\pi$
$762$	0	0
$763$	−19.0755	−0.690579
$764$	23.6544	0.855785
$765$	0	0
$766$	−1.12641	−0.0406989
$767$	0.0102944	0.000371708 0
$768$	0	0
$769$	3.99101	0.143920	0.0719598	−	0.997408i	$-0.477075\pi$
$769$	3.99101	0.143920	0.0719598	+	0.997408i	$0.477075\pi$
$770$	−40.2339	−1.44993
$771$	0	0
$772$	−1.95483	−0.0703559
$773$	28.1865	1.01380	0.506899	−	0.862006i	$-0.330792\pi$
$773$	28.1865	1.01380	0.506899	+	0.862006i	$0.330792\pi$
$774$	0	0
$775$	28.1654	1.01173
$776$	12.2102	0.438319
$777$	0	0
$778$	3.11938	0.111835
$779$	15.3220	0.548967
$780$	0	0
$781$	32.1417	1.15012
$782$	0	0
$783$	0	0
$784$	1.66759	0.0595569
$785$	11.8298	0.422222
$786$	0	0
$787$	20.2518	0.721900	0.360950	−	0.932585i	$-0.382453\pi$
$787$	20.2518	0.721900	0.360950	+	0.932585i	$0.382453\pi$
$788$	−1.87950	−0.0669546
$789$	0	0
$790$	−40.0461	−1.42478
$791$	−30.6899	−1.09121
$792$	0	0
$793$	0.270143	0.00959306
$794$	−20.6576	−0.733111
$795$	0	0
$796$	16.5039	0.584964
$797$	−0.725198	−0.0256878	−0.0128439	−	0.999918i	$-0.504088\pi$
$797$	−0.725198	−0.0256878	−0.0128439	+	0.999918i	$0.504088\pi$
$798$	0	0
$799$	−17.7596	−0.628289
$800$	12.3172	0.435480
$801$	0	0
$802$	−10.3034	−0.363827
$803$	−18.0141	−0.635703
$804$	0	0
$805$	0	0
$806$	−0.321394	−0.0113206
$807$	0	0
$808$	1.40824	0.0495415
$809$	45.5541	1.60160	0.800799	−	0.598934i	$-0.204409\pi$
$809$	45.5541	1.60160	0.800799	+	0.598934i	$0.204409\pi$
$810$	0	0
$811$	−13.1651	−0.462288	−0.231144	−	0.972920i	$-0.574247\pi$
$811$	−13.1651	−0.462288	−0.231144	+	0.972920i	$0.574247\pi$
$812$	29.2579	1.02675
$813$	0	0
$814$	20.3866	0.714550
$815$	−8.36240	−0.292922
$816$	0	0
$817$	−3.04388	−0.106492
$818$	3.60277	0.125968
$819$	0	0
$820$	−26.8487	−0.937599
$821$	24.0276	0.838569	0.419285	−	0.907855i	$-0.362281\pi$
$821$	24.0276	0.838569	0.419285	+	0.907855i	$0.362281\pi$
$822$	0	0
$823$	4.93230	0.171929	0.0859645	−	0.996298i	$-0.472603\pi$
$823$	4.93230	0.171929	0.0859645	+	0.996298i	$0.472603\pi$
$824$	−8.03671	−0.279972
$825$	0	0
$826$	−0.215632	−0.00750279
$827$	7.46011	0.259413	0.129707	−	0.991552i	$-0.458596\pi$
$827$	7.46011	0.259413	0.129707	+	0.991552i	$0.458596\pi$
$828$	0	0
$829$	41.5609	1.44347	0.721735	−	0.692170i	$-0.243345\pi$
$829$	41.5609	1.44347	0.721735	+	0.692170i	$0.243345\pi$
$830$	−45.8106	−1.59011
$831$	0	0
$832$	−0.140551	−0.00487275
$833$	10.4274	0.361289
$834$	0	0
$835$	38.3997	1.32888
$836$	7.79890	0.269731
$837$	0	0
$838$	13.4970	0.466245
$839$	−41.1489	−1.42062	−0.710309	−	0.703890i	$-0.751445\pi$
$839$	−41.1489	−1.42062	−0.710309	+	0.703890i	$0.751445\pi$
$840$	0	0
$841$	69.7616	2.40557
$842$	−16.6474	−0.573706
$843$	0	0
$844$	5.97819	0.205778
$845$	−54.0160	−1.85821
$846$	0	0
$847$	−0.633988	−0.0217841
$848$	4.35073	0.149405
$849$	0	0
$850$	77.0194	2.64174
$851$	0	0
$852$	0	0
$853$	−16.0683	−0.550168	−0.275084	−	0.961420i	$-0.588706\pi$
$853$	−16.0683	−0.550168	−0.275084	+	0.961420i	$0.588706\pi$
$854$	−5.65858	−0.193633
$855$	0	0
$856$	19.3300	0.660687
$857$	26.0246	0.888983	0.444491	−	0.895783i	$-0.353384\pi$
$857$	26.0246	0.888983	0.444491	+	0.895783i	$0.353384\pi$
$858$	0	0
$859$	−11.7835	−0.402049	−0.201025	−	0.979586i	$-0.564427\pi$
$859$	−11.7835	−0.402049	−0.201025	+	0.979586i	$0.564427\pi$
$860$	5.33380	0.181881
$861$	0	0
$862$	−21.0164	−0.715821
$863$	17.6743	0.601639	0.300820	−	0.953681i	$-0.402740\pi$
$863$	17.6743	0.601639	0.300820	+	0.953681i	$0.402740\pi$
$864$	0	0
$865$	−39.1024	−1.32952
$866$	2.07347	0.0704593
$867$	0	0
$868$	6.73211	0.228503
$869$	31.6027	1.07205
$870$	0	0
$871$	0.609080	0.0206379
$872$	−6.47928	−0.219416
$873$	0	0
$874$	0	0
$875$	89.6470	3.03062
$876$	0	0
$877$	−32.2732	−1.08979	−0.544895	−	0.838505i	$-0.683430\pi$
$877$	−32.2732	−1.08979	−0.544895	+	0.838505i	$0.683430\pi$
$878$	2.26950	0.0765920
$879$	0	0
$880$	−13.6660	−0.460682
$881$	−10.3585	−0.348987	−0.174494	−	0.984658i	$-0.555829\pi$
$881$	−10.3585	−0.348987	−0.174494	+	0.984658i	$0.555829\pi$
$882$	0	0
$883$	56.8041	1.91161	0.955805	−	0.294001i	$-0.0949868\pi$
$883$	56.8041	1.91161	0.955805	+	0.294001i	$0.0949868\pi$
$884$	−0.878865	−0.0295594
$885$	0	0
$886$	−26.8178	−0.900961
$887$	−41.9582	−1.40882	−0.704409	−	0.709795i	$-0.748788\pi$
$887$	−41.9582	−1.40882	−0.704409	+	0.709795i	$0.748788\pi$
$888$	0	0
$889$	−18.1530	−0.608834
$890$	54.5301	1.82785
$891$	0	0
$892$	−20.1193	−0.673643
$893$	6.74492	0.225710
$894$	0	0
$895$	85.0021	2.84131
$896$	2.94408	0.0983547
$897$	0	0
$898$	−4.56941	−0.152483
$899$	22.7246	0.757908
$900$	0	0
$901$	27.2050	0.906330
$902$	21.1879	0.705480
$903$	0	0
$904$	−10.4243	−0.346707
$905$	−62.1437	−2.06572
$906$	0	0
$907$	20.9112	0.694344	0.347172	−	0.937801i	$-0.387142\pi$
$907$	20.9112	0.694344	0.347172	+	0.937801i	$0.387142\pi$
$908$	13.0539	0.433209
$909$	0	0
$910$	−1.72196	−0.0570825
$911$	−2.38186	−0.0789144	−0.0394572	−	0.999221i	$-0.512563\pi$
$911$	−2.38186	−0.0789144	−0.0394572	+	0.999221i	$0.512563\pi$
$912$	0	0
$913$	36.1518	1.19645
$914$	26.8144	0.886943
$915$	0	0
$916$	−17.7990	−0.588095
$917$	18.9762	0.626648
$918$	0	0
$919$	−33.7332	−1.11275	−0.556377	−	0.830930i	$-0.687809\pi$
$919$	−33.7332	−1.11275	−0.556377	+	0.830930i	$0.687809\pi$
$920$	0	0
$921$	0	0
$922$	1.33658	0.0440179
$923$	1.37563	0.0452793
$924$	0	0
$925$	−76.4637	−2.51411
$926$	−36.2702	−1.19191
$927$	0	0
$928$	9.93789	0.326227
$929$	−5.53540	−0.181611	−0.0908053	−	0.995869i	$-0.528944\pi$
$929$	−5.53540	−0.181611	−0.0908053	+	0.995869i	$0.528944\pi$
$930$	0	0
$931$	−3.96023	−0.129791
$932$	−9.79827	−0.320953
$933$	0	0
$934$	−13.3242	−0.435981
$935$	−85.4534	−2.79462
$936$	0	0
$937$	23.4571	0.766309	0.383154	−	0.923684i	$-0.374838\pi$
$937$	23.4571	0.766309	0.383154	+	0.923684i	$0.374838\pi$
$938$	−12.7582	−0.416569
$939$	0	0
$940$	−11.8191	−0.385498
$941$	13.9706	0.455428	0.227714	−	0.973728i	$-0.426875\pi$
$941$	13.9706	0.455428	0.227714	+	0.973728i	$0.426875\pi$
$942$	0	0
$943$	0	0
$944$	−0.0732426	−0.00238384
$945$	0	0
$946$	−4.20921	−0.136853
$947$	−49.1714	−1.59786	−0.798928	−	0.601427i	$-0.794599\pi$
$947$	−49.1714	−1.59786	−0.798928	+	0.601427i	$0.794599\pi$
$948$	0	0
$949$	−0.770982	−0.0250271
$950$	−29.2512	−0.949034
$951$	0	0
$952$	18.4092	0.596647
$953$	−8.63836	−0.279824	−0.139912	−	0.990164i	$-0.544682\pi$
$953$	−8.63836	−0.279824	−0.139912	+	0.990164i	$0.544682\pi$
$954$	0	0
$955$	98.4352	3.18529
$956$	−2.99564	−0.0968859
$957$	0	0
$958$	26.5551	0.857956
$959$	42.3768	1.36842
$960$	0	0
$961$	−25.7712	−0.831328
$962$	0.872524	0.0281313
$963$	0	0
$964$	20.4314	0.658051
$965$	−8.13483	−0.261869
$966$	0	0
$967$	−36.8290	−1.18434	−0.592170	−	0.805813i	$-0.701728\pi$
$967$	−36.8290	−1.18434	−0.592170	+	0.805813i	$0.701728\pi$
$968$	−0.215343	−0.00692140
$969$	0	0
$970$	50.8113	1.63145
$971$	−20.9794	−0.673260	−0.336630	−	0.941637i	$-0.609287\pi$
$971$	−20.9794	−0.673260	−0.336630	+	0.941637i	$0.609287\pi$
$972$	0	0
$973$	−32.2203	−1.03294
$974$	−13.0668	−0.418688
$975$	0	0
$976$	−1.92202	−0.0615224
$977$	6.69406	0.214162	0.107081	−	0.994250i	$-0.465850\pi$
$977$	6.69406	0.214162	0.107081	+	0.994250i	$0.465850\pi$
$978$	0	0
$979$	−43.0328	−1.37534
$980$	6.93952	0.221675
$981$	0	0
$982$	33.6063	1.07242
$983$	4.95104	0.157914	0.0789568	−	0.996878i	$-0.474841\pi$
$983$	4.95104	0.157914	0.0789568	+	0.996878i	$0.474841\pi$
$984$	0	0
$985$	−7.82136	−0.249209
$986$	62.1414	1.97898
$987$	0	0
$988$	0.333784	0.0106191
$989$	0	0
$990$	0	0
$991$	−20.7885	−0.660369	−0.330184	−	0.943916i	$-0.607111\pi$
$991$	−20.7885	−0.660369	−0.330184	+	0.943916i	$0.607111\pi$
$992$	2.28666	0.0726016
$993$	0	0
$994$	−28.8147	−0.913948
$995$	68.6791	2.17727
$996$	0	0
$997$	−35.2435	−1.11617	−0.558086	−	0.829783i	$-0.688464\pi$
$997$	−35.2435	−1.11617	−0.558086	+	0.829783i	$0.688464\pi$
$998$	24.2163	0.766553
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.ca.1.4		5
3.2	odd	2		3174.2.a.y.1.2		5
23.15	odd	22		414.2.i.b.271.1		10
23.20	odd	22		414.2.i.b.55.1		10
23.22	odd	2		9522.2.a.bv.1.2		5
69.20	even	22		138.2.e.c.55.1	✓	10
69.38	even	22		138.2.e.c.133.1	yes	10
69.68	even	2		3174.2.a.z.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.c.55.1	✓	10	69.20	even	22
138.2.e.c.133.1	yes	10	69.38	even	22
414.2.i.b.55.1		10	23.20	odd	22
414.2.i.b.271.1		10	23.15	odd	22
3174.2.a.y.1.2		5	3.2	odd	2
3174.2.a.z.1.4		5	69.68	even	2
9522.2.a.bv.1.2		5	23.22	odd	2
9522.2.a.ca.1.4		5	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.16140 q^{5} +2.94408 q^{7} +1.00000 q^{8} +4.16140 q^{10} -3.28400 q^{11} -0.140551 q^{13} +2.94408 q^{14} +1.00000 q^{16} +6.25297 q^{17} -2.37482 q^{19} +4.16140 q^{20} -3.28400 q^{22} +12.3172 q^{25} -0.140551 q^{26} +2.94408 q^{28} +9.93789 q^{29} +2.28666 q^{31} +1.00000 q^{32} +6.25297 q^{34} +12.2515 q^{35} -6.20786 q^{37} -2.37482 q^{38} +4.16140 q^{40} -6.45186 q^{41} +1.28173 q^{43} -3.28400 q^{44} -2.84018 q^{47} +1.66759 q^{49} +12.3172 q^{50} -0.140551 q^{52} +4.35073 q^{53} -13.6660 q^{55} +2.94408 q^{56} +9.93789 q^{58} -0.0732426 q^{59} -1.92202 q^{61} +2.28666 q^{62} +1.00000 q^{64} -0.584891 q^{65} -4.33350 q^{67} +6.25297 q^{68} +12.2515 q^{70} -9.78735 q^{71} +5.48540 q^{73} -6.20786 q^{74} -2.37482 q^{76} -9.66835 q^{77} -9.62324 q^{79} +4.16140 q^{80} -6.45186 q^{82} -11.0085 q^{83} +26.0211 q^{85} +1.28173 q^{86} -3.28400 q^{88} +13.1038 q^{89} -0.413795 q^{91} -2.84018 q^{94} -9.88257 q^{95} +12.2102 q^{97} +1.66759 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} + 11 q^{5} + q^{7} + 5 q^{8} + 11 q^{10} + 11 q^{11} + 10 q^{13} + q^{14} + 5 q^{16} + 11 q^{17} + q^{19} + 11 q^{20} + 11 q^{22} + 30 q^{25} + 10 q^{26} + q^{28} - 3 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 + 11 * q^5 + q^7 + 5 * q^8 + 11 * q^10 + 11 * q^11 + 10 * q^13 + q^14 + 5 * q^16 + 11 * q^17 + q^19 + 11 * q^20 + 11 * q^22 + 30 * q^25 + 10 * q^26 + q^28 - 3 * q^29 + 2 * q^31 + 5 * q^32 + 11 * q^34 + 11 * q^35 - 16 * q^37 + q^38 + 11 * q^40 + 8 * q^41 + 12 * q^43 + 11 * q^44 - 4 * q^49 + 30 * q^50 + 10 * q^52 + 27 * q^53 + q^56 - 3 * q^58 + 6 * q^59 - 5 * q^61 + 2 * q^62 + 5 * q^64 + 22 * q^65 + 7 * q^67 + 11 * q^68 + 11 * q^70 + 11 * q^71 - 12 * q^73 - 16 * q^74 + q^76 - 11 * q^77 - 16 * q^79 + 11 * q^80 + 8 * q^82 + 22 * q^83 - 11 * q^85 + 12 * q^86 + 11 * q^88 + 38 * q^89 - 9 * q^91 - 11 * q^95 - 5 * q^97 - 4 * q^98

Introduction

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