LMFDB - Embedded newform 9522.2.a.ca.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)

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			1.3
Root			$1.30972$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.89389 q^{5} +2.77066 q^{7} +1.00000 q^{8} +2.89389 q^{10} +5.45027 q^{11} +0.365644 q^{13} +2.77066 q^{14} +1.00000 q^{16} -6.29760 q^{17} -6.13568 q^{19} +2.89389 q^{20} +5.45027 q^{22} +3.37462 q^{25} +0.365644 q^{26} +2.77066 q^{28} -3.41027 q^{29} +7.68188 q^{31} +1.00000 q^{32} -6.29760 q^{34} +8.01801 q^{35} -4.82862 q^{37} -6.13568 q^{38} +2.89389 q^{40} +4.23685 q^{41} +7.03648 q^{43} +5.45027 q^{44} +6.68409 q^{47} +0.676573 q^{49} +3.37462 q^{50} +0.365644 q^{52} +7.13462 q^{53} +15.7725 q^{55} +2.77066 q^{56} -3.41027 q^{58} -0.440095 q^{59} -8.13399 q^{61} +7.68188 q^{62} +1.00000 q^{64} +1.05813 q^{65} -4.44927 q^{67} -6.29760 q^{68} +8.01801 q^{70} +1.41293 q^{71} -3.94118 q^{73} -4.82862 q^{74} -6.13568 q^{76} +15.1009 q^{77} +3.97465 q^{79} +2.89389 q^{80} +4.23685 q^{82} +16.1023 q^{83} -18.2246 q^{85} +7.03648 q^{86} +5.45027 q^{88} +13.7607 q^{89} +1.01308 q^{91} +6.68409 q^{94} -17.7560 q^{95} +4.33160 q^{97} +0.676573 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.89389	1.29419	0.647094	−	0.762410i	$-0.275984\pi$
$5$	2.89389	1.29419	0.647094	+	0.762410i	$0.275984\pi$
$6$	0	0
$7$	2.77066	1.04721	0.523606	−	0.851960i	$-0.324586\pi$
$7$	2.77066	1.04721	0.523606	+	0.851960i	$0.324586\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.89389	0.915130
$11$	5.45027	1.64332	0.821660	−	0.569979i	$-0.193049\pi$
$11$	5.45027	1.64332	0.821660	+	0.569979i	$0.193049\pi$
$12$	0	0
$13$	0.365644	0.101411	0.0507057	−	0.998714i	$-0.483853\pi$
$13$	0.365644	0.101411	0.0507057	+	0.998714i	$0.483853\pi$
$14$	2.77066	0.740491
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.29760	−1.52739	−0.763696	−	0.645576i	$-0.776617\pi$
$17$	−6.29760	−1.52739	−0.763696	+	0.645576i	$0.776617\pi$
$18$	0	0
$19$	−6.13568	−1.40762	−0.703810	−	0.710388i	$-0.748520\pi$
$19$	−6.13568	−1.40762	−0.703810	+	0.710388i	$0.748520\pi$
$20$	2.89389	0.647094
$21$	0	0
$22$	5.45027	1.16200
$23$	0	0
$23$	0	0
$24$	0	0
$25$	3.37462	0.674925
$26$	0.365644	0.0717086
$27$	0	0
$28$	2.77066	0.523606
$29$	−3.41027	−0.633271	−0.316635	−	0.948547i	$-0.602553\pi$
$29$	−3.41027	−0.633271	−0.316635	+	0.948547i	$0.602553\pi$
$30$	0	0
$31$	7.68188	1.37971	0.689853	−	0.723950i	$-0.257675\pi$
$31$	7.68188	1.37971	0.689853	+	0.723950i	$0.257675\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.29760	−1.08003
$35$	8.01801	1.35529
$36$	0	0
$37$	−4.82862	−0.793820	−0.396910	−	0.917858i	$-0.629917\pi$
$37$	−4.82862	−0.793820	−0.396910	+	0.917858i	$0.629917\pi$
$38$	−6.13568	−0.995338
$39$	0	0
$40$	2.89389	0.457565
$41$	4.23685	0.661685	0.330843	−	0.943686i	$-0.392667\pi$
$41$	4.23685	0.661685	0.330843	+	0.943686i	$0.392667\pi$
$42$	0	0
$43$	7.03648	1.07305	0.536526	−	0.843884i	$-0.319736\pi$
$43$	7.03648	1.07305	0.536526	+	0.843884i	$0.319736\pi$
$44$	5.45027	0.821660
$45$	0	0
$46$	0	0
$47$	6.68409	0.974975	0.487487	−	0.873130i	$-0.337914\pi$
$47$	6.68409	0.974975	0.487487	+	0.873130i	$0.337914\pi$
$48$	0	0
$49$	0.676573	0.0966533
$50$	3.37462	0.477244
$51$	0	0
$52$	0.365644	0.0507057
$53$	7.13462	0.980015	0.490008	−	0.871718i	$-0.336994\pi$
$53$	7.13462	0.980015	0.490008	+	0.871718i	$0.336994\pi$
$54$	0	0
$55$	15.7725	2.12677
$56$	2.77066	0.370245
$57$	0	0
$58$	−3.41027	−0.447790
$59$	−0.440095	−0.0572954	−0.0286477	−	0.999590i	$-0.509120\pi$
$59$	−0.440095	−0.0572954	−0.0286477	+	0.999590i	$0.509120\pi$
$60$	0	0
$61$	−8.13399	−1.04145	−0.520725	−	0.853724i	$-0.674338\pi$
$61$	−8.13399	−1.04145	−0.520725	+	0.853724i	$0.674338\pi$
$62$	7.68188	0.975599
$63$	0	0
$64$	1.00000	0.125000
$65$	1.05813	0.131245
$66$	0	0
$67$	−4.44927	−0.543565	−0.271782	−	0.962359i	$-0.587613\pi$
$67$	−4.44927	−0.543565	−0.271782	+	0.962359i	$0.587613\pi$
$68$	−6.29760	−0.763696
$69$	0	0
$70$	8.01801	0.958335
$71$	1.41293	0.167684	0.0838420	−	0.996479i	$-0.473281\pi$
$71$	1.41293	0.167684	0.0838420	+	0.996479i	$0.473281\pi$
$72$	0	0
$73$	−3.94118	−0.461280	−0.230640	−	0.973039i	$-0.574082\pi$
$73$	−3.94118	−0.461280	−0.230640	+	0.973039i	$0.574082\pi$
$74$	−4.82862	−0.561315
$75$	0	0
$76$	−6.13568	−0.703810
$77$	15.1009	1.72090
$78$	0	0
$79$	3.97465	0.447183	0.223592	−	0.974683i	$-0.428222\pi$
$79$	3.97465	0.447183	0.223592	+	0.974683i	$0.428222\pi$
$80$	2.89389	0.323547
$81$	0	0
$82$	4.23685	0.467882
$83$	16.1023	1.76746	0.883730	−	0.467997i	$-0.155024\pi$
$83$	16.1023	1.76746	0.883730	+	0.467997i	$0.155024\pi$
$84$	0	0
$85$	−18.2246	−1.97673
$86$	7.03648	0.758763
$87$	0	0
$88$	5.45027	0.581001
$89$	13.7607	1.45863	0.729315	−	0.684179i	$-0.239839\pi$
$89$	13.7607	1.45863	0.729315	+	0.684179i	$0.239839\pi$
$90$	0	0
$91$	1.01308	0.106199
$92$	0	0
$93$	0	0
$94$	6.68409	0.689411
$95$	−17.7560	−1.82173
$96$	0	0
$97$	4.33160	0.439807	0.219904	−	0.975522i	$-0.429426\pi$
$97$	4.33160	0.439807	0.219904	+	0.975522i	$0.429426\pi$
$98$	0.676573	0.0683442
$99$	0	0
$100$	3.37462	0.337462
$101$	−6.42624	−0.639435	−0.319717	−	0.947513i	$-0.603588\pi$
$101$	−6.42624	−0.639435	−0.319717	+	0.947513i	$0.603588\pi$
$102$	0	0
$103$	12.2847	1.21045	0.605225	−	0.796055i	$-0.293083\pi$
$103$	12.2847	1.21045	0.605225	+	0.796055i	$0.293083\pi$
$104$	0.365644	0.0358543
$105$	0	0
$106$	7.13462	0.692975
$107$	9.93617	0.960565	0.480283	−	0.877114i	$-0.340534\pi$
$107$	9.93617	0.960565	0.480283	+	0.877114i	$0.340534\pi$
$108$	0	0
$109$	10.4065	0.996761	0.498381	−	0.866958i	$-0.333928\pi$
$109$	10.4065	0.996761	0.498381	+	0.866958i	$0.333928\pi$
$110$	15.7725	1.50385
$111$	0	0
$112$	2.77066	0.261803
$113$	10.7510	1.01137	0.505685	−	0.862718i	$-0.331240\pi$
$113$	10.7510	1.01137	0.505685	+	0.862718i	$0.331240\pi$
$114$	0	0
$115$	0	0
$116$	−3.41027	−0.316635
$117$	0	0
$118$	−0.440095	−0.0405140
$119$	−17.4485	−1.59950
$120$	0	0
$121$	18.7055	1.70050
$122$	−8.13399	−0.736417
$123$	0	0
$124$	7.68188	0.689853
$125$	−4.70367	−0.420709
$126$	0	0
$127$	−9.70663	−0.861324	−0.430662	−	0.902513i	$-0.641720\pi$
$127$	−9.70663	−0.861324	−0.430662	+	0.902513i	$0.641720\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.05813	0.0928045
$131$	13.1274	1.14695	0.573474	−	0.819224i	$-0.305595\pi$
$131$	13.1274	1.14695	0.573474	+	0.819224i	$0.305595\pi$
$132$	0	0
$133$	−16.9999	−1.47408
$134$	−4.44927	−0.384358
$135$	0	0
$136$	−6.29760	−0.540015
$137$	−3.83057	−0.327268	−0.163634	−	0.986521i	$-0.552322\pi$
$137$	−3.83057	−0.327268	−0.163634	+	0.986521i	$0.552322\pi$
$138$	0	0
$139$	17.3000	1.46736	0.733682	−	0.679494i	$-0.237800\pi$
$139$	17.3000	1.46736	0.733682	+	0.679494i	$0.237800\pi$
$140$	8.01801	0.677645
$141$	0	0
$142$	1.41293	0.118570
$143$	1.99286	0.166651
$144$	0	0
$145$	−9.86895	−0.819572
$146$	−3.94118	−0.326174
$147$	0	0
$148$	−4.82862	−0.396910
$149$	14.5551	1.19240	0.596200	−	0.802836i	$-0.296676\pi$
$149$	14.5551	1.19240	0.596200	+	0.802836i	$0.296676\pi$
$150$	0	0
$151$	−9.51541	−0.774353	−0.387177	−	0.922006i	$-0.626550\pi$
$151$	−9.51541	−0.774353	−0.387177	+	0.922006i	$0.626550\pi$
$152$	−6.13568	−0.497669
$153$	0	0
$154$	15.1009	1.21686
$155$	22.2305	1.78560
$156$	0	0
$157$	−23.8636	−1.90452	−0.952261	−	0.305286i	$-0.901248\pi$
$157$	−23.8636	−1.90452	−0.952261	+	0.305286i	$0.901248\pi$
$158$	3.97465	0.316206
$159$	0	0
$160$	2.89389	0.228782
$161$	0	0
$162$	0	0
$163$	8.16971	0.639901	0.319950	−	0.947434i	$-0.396334\pi$
$163$	8.16971	0.639901	0.319950	+	0.947434i	$0.396334\pi$
$164$	4.23685	0.330843
$165$	0	0
$166$	16.1023	1.24978
$167$	10.2364	0.792119	0.396059	−	0.918225i	$-0.370377\pi$
$167$	10.2364	0.792119	0.396059	+	0.918225i	$0.370377\pi$
$168$	0	0
$169$	−12.8663	−0.989716
$170$	−18.2246	−1.39776
$171$	0	0
$172$	7.03648	0.536526
$173$	3.39250	0.257927	0.128963	−	0.991649i	$-0.458835\pi$
$173$	3.39250	0.257927	0.128963	+	0.991649i	$0.458835\pi$
$174$	0	0
$175$	9.34994	0.706789
$176$	5.45027	0.410830
$177$	0	0
$178$	13.7607	1.03141
$179$	−6.82941	−0.510454	−0.255227	−	0.966881i	$-0.582150\pi$
$179$	−6.82941	−0.510454	−0.255227	+	0.966881i	$0.582150\pi$
$180$	0	0
$181$	−0.940518	−0.0699082	−0.0349541	−	0.999389i	$-0.511128\pi$
$181$	−0.940518	−0.0699082	−0.0349541	+	0.999389i	$0.511128\pi$
$182$	1.01308	0.0750942
$183$	0	0
$184$	0	0
$185$	−13.9735	−1.02735
$186$	0	0
$187$	−34.3236	−2.50999
$188$	6.68409	0.487487
$189$	0	0
$190$	−17.7560	−1.28816
$191$	−2.04470	−0.147949	−0.0739745	−	0.997260i	$-0.523568\pi$
$191$	−2.04470	−0.147949	−0.0739745	+	0.997260i	$0.523568\pi$
$192$	0	0
$193$	−23.5905	−1.69808	−0.849042	−	0.528325i	$-0.822820\pi$
$193$	−23.5905	−1.69808	−0.849042	+	0.528325i	$0.822820\pi$
$194$	4.33160	0.310991
$195$	0	0
$196$	0.676573	0.0483267
$197$	3.37234	0.240269	0.120134	−	0.992758i	$-0.461667\pi$
$197$	3.37234	0.240269	0.120134	+	0.992758i	$0.461667\pi$
$198$	0	0
$199$	−21.6752	−1.53651	−0.768257	−	0.640141i	$-0.778876\pi$
$199$	−21.6752	−1.53651	−0.768257	+	0.640141i	$0.778876\pi$
$200$	3.37462	0.238622
$201$	0	0
$202$	−6.42624	−0.452149
$203$	−9.44870	−0.663169
$204$	0	0
$205$	12.2610	0.856346
$206$	12.2847	0.855917
$207$	0	0
$208$	0.365644	0.0253528
$209$	−33.4411	−2.31317
$210$	0	0
$211$	−22.9306	−1.57861	−0.789303	−	0.614003i	$-0.789558\pi$
$211$	−22.9306	−1.57861	−0.789303	+	0.614003i	$0.789558\pi$
$212$	7.13462	0.490008
$213$	0	0
$214$	9.93617	0.679222
$215$	20.3628	1.38873
$216$	0	0
$217$	21.2839	1.44484
$218$	10.4065	0.704817
$219$	0	0
$220$	15.7725	1.06338
$221$	−2.30268	−0.154895
$222$	0	0
$223$	−19.8265	−1.32768	−0.663841	−	0.747873i	$-0.731075\pi$
$223$	−19.8265	−1.32768	−0.663841	+	0.747873i	$0.731075\pi$
$224$	2.77066	0.185123
$225$	0	0
$226$	10.7510	0.715147
$227$	−19.9316	−1.32290	−0.661452	−	0.749988i	$-0.730059\pi$
$227$	−19.9316	−1.32290	−0.661452	+	0.749988i	$0.730059\pi$
$228$	0	0
$229$	16.2991	1.07708	0.538539	−	0.842601i	$-0.318977\pi$
$229$	16.2991	1.07708	0.538539	+	0.842601i	$0.318977\pi$
$230$	0	0
$231$	0	0
$232$	−3.41027	−0.223895
$233$	−18.6367	−1.22093	−0.610465	−	0.792043i	$-0.709018\pi$
$233$	−18.6367	−1.22093	−0.610465	+	0.792043i	$0.709018\pi$
$234$	0	0
$235$	19.3430	1.26180
$236$	−0.440095	−0.0286477
$237$	0	0
$238$	−17.4485	−1.13102
$239$	−4.40675	−0.285049	−0.142524	−	0.989791i	$-0.545522\pi$
$239$	−4.40675	−0.285049	−0.142524	+	0.989791i	$0.545522\pi$
$240$	0	0
$241$	−13.1178	−0.844990	−0.422495	−	0.906365i	$-0.638846\pi$
$241$	−13.1178	−0.844990	−0.422495	+	0.906365i	$0.638846\pi$
$242$	18.7055	1.20243
$243$	0	0
$244$	−8.13399	−0.520725
$245$	1.95793	0.125088
$246$	0	0
$247$	−2.24347	−0.142749
$248$	7.68188	0.487800
$249$	0	0
$250$	−4.70367	−0.297486
$251$	−9.21467	−0.581625	−0.290813	−	0.956780i	$-0.593926\pi$
$251$	−9.21467	−0.581625	−0.290813	+	0.956780i	$0.593926\pi$
$252$	0	0
$253$	0	0
$254$	−9.70663	−0.609048
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.43454	−0.151862	−0.0759311	−	0.997113i	$-0.524193\pi$
$257$	−2.43454	−0.151862	−0.0759311	+	0.997113i	$0.524193\pi$
$258$	0	0
$259$	−13.3785	−0.831298
$260$	1.05813	0.0656227
$261$	0	0
$262$	13.1274	0.811015
$263$	−3.60644	−0.222383	−0.111191	−	0.993799i	$-0.535467\pi$
$263$	−3.60644	−0.222383	−0.111191	+	0.993799i	$0.535467\pi$
$264$	0	0
$265$	20.6468	1.26832
$266$	−16.9999	−1.04233
$267$	0	0
$268$	−4.44927	−0.271782
$269$	−22.6221	−1.37929	−0.689647	−	0.724146i	$-0.742234\pi$
$269$	−22.6221	−1.37929	−0.689647	+	0.724146i	$0.742234\pi$
$270$	0	0
$271$	13.6955	0.831942	0.415971	−	0.909378i	$-0.363442\pi$
$271$	13.6955	0.831942	0.415971	+	0.909378i	$0.363442\pi$
$272$	−6.29760	−0.381848
$273$	0	0
$274$	−3.83057	−0.231413
$275$	18.3926	1.10912
$276$	0	0
$277$	−10.2577	−0.616324	−0.308162	−	0.951334i	$-0.599714\pi$
$277$	−10.2577	−0.616324	−0.308162	+	0.951334i	$0.599714\pi$
$278$	17.3000	1.03758
$279$	0	0
$280$	8.01801	0.479167
$281$	−3.11451	−0.185796	−0.0928981	−	0.995676i	$-0.529613\pi$
$281$	−3.11451	−0.185796	−0.0928981	+	0.995676i	$0.529613\pi$
$282$	0	0
$283$	9.50350	0.564925	0.282462	−	0.959278i	$-0.408849\pi$
$283$	9.50350	0.564925	0.282462	+	0.959278i	$0.408849\pi$
$284$	1.41293	0.0838420
$285$	0	0
$286$	1.99286	0.117840
$287$	11.7389	0.692925
$288$	0	0
$289$	22.6597	1.33293
$290$	−9.86895	−0.579525
$291$	0	0
$292$	−3.94118	−0.230640
$293$	25.1900	1.47162	0.735809	−	0.677190i	$-0.236802\pi$
$293$	25.1900	1.47162	0.735809	+	0.677190i	$0.236802\pi$
$294$	0	0
$295$	−1.27359	−0.0741511
$296$	−4.82862	−0.280658
$297$	0	0
$298$	14.5551	0.843155
$299$	0	0
$300$	0	0
$301$	19.4957	1.12371
$302$	−9.51541	−0.547550
$303$	0	0
$304$	−6.13568	−0.351905
$305$	−23.5389	−1.34783
$306$	0	0
$307$	1.33026	0.0759218	0.0379609	−	0.999279i	$-0.487914\pi$
$307$	1.33026	0.0759218	0.0379609	+	0.999279i	$0.487914\pi$
$308$	15.1009	0.860452
$309$	0	0
$310$	22.2305	1.26261
$311$	8.70784	0.493776	0.246888	−	0.969044i	$-0.420592\pi$
$311$	8.70784	0.493776	0.246888	+	0.969044i	$0.420592\pi$
$312$	0	0
$313$	−12.7984	−0.723406	−0.361703	−	0.932293i	$-0.617805\pi$
$313$	−12.7984	−0.723406	−0.361703	+	0.932293i	$0.617805\pi$
$314$	−23.8636	−1.34670
$315$	0	0
$316$	3.97465	0.223592
$317$	−13.1795	−0.740237	−0.370118	−	0.928985i	$-0.620683\pi$
$317$	−13.1795	−0.740237	−0.370118	+	0.928985i	$0.620683\pi$
$318$	0	0
$319$	−18.5869	−1.04067
$320$	2.89389	0.161774
$321$	0	0
$322$	0	0
$323$	38.6400	2.14999
$324$	0	0
$325$	1.23391	0.0684450
$326$	8.16971	0.452478
$327$	0	0
$328$	4.23685	0.233941
$329$	18.5194	1.02101
$330$	0	0
$331$	−31.6717	−1.74083	−0.870417	−	0.492315i	$-0.836151\pi$
$331$	−31.6717	−1.74083	−0.870417	+	0.492315i	$0.836151\pi$
$332$	16.1023	0.883730
$333$	0	0
$334$	10.2364	0.560112
$335$	−12.8757	−0.703475
$336$	0	0
$337$	−9.14100	−0.497942	−0.248971	−	0.968511i	$-0.580092\pi$
$337$	−9.14100	−0.497942	−0.248971	+	0.968511i	$0.580092\pi$
$338$	−12.8663	−0.699835
$339$	0	0
$340$	−18.2246	−0.988367
$341$	41.8683	2.26730
$342$	0	0
$343$	−17.5201	−0.945996
$344$	7.03648	0.379381
$345$	0	0
$346$	3.39250	0.182382
$347$	−17.8143	−0.956324	−0.478162	−	0.878272i	$-0.658697\pi$
$347$	−17.8143	−0.956324	−0.478162	+	0.878272i	$0.658697\pi$
$348$	0	0
$349$	−20.8462	−1.11587	−0.557935	−	0.829885i	$-0.688406\pi$
$349$	−20.8462	−1.11587	−0.557935	+	0.829885i	$0.688406\pi$
$350$	9.34994	0.499776
$351$	0	0
$352$	5.45027	0.290501
$353$	−22.1361	−1.17818	−0.589092	−	0.808066i	$-0.700515\pi$
$353$	−22.1361	−1.17818	−0.589092	+	0.808066i	$0.700515\pi$
$354$	0	0
$355$	4.08887	0.217015
$356$	13.7607	0.729315
$357$	0	0
$358$	−6.82941	−0.360945
$359$	−27.8721	−1.47104	−0.735518	−	0.677505i	$-0.763061\pi$
$359$	−27.8721	−1.47104	−0.735518	+	0.677505i	$0.763061\pi$
$360$	0	0
$361$	18.6465	0.981397
$362$	−0.940518	−0.0494325
$363$	0	0
$364$	1.01308	0.0530996
$365$	−11.4054	−0.596984
$366$	0	0
$367$	−17.5832	−0.917835	−0.458917	−	0.888479i	$-0.651763\pi$
$367$	−17.5832	−0.917835	−0.458917	+	0.888479i	$0.651763\pi$
$368$	0	0
$369$	0	0
$370$	−13.9735	−0.726448
$371$	19.7676	1.02628
$372$	0	0
$373$	−26.4065	−1.36727	−0.683637	−	0.729822i	$-0.739603\pi$
$373$	−26.4065	−1.36727	−0.683637	+	0.729822i	$0.739603\pi$
$374$	−34.3236	−1.77483
$375$	0	0
$376$	6.68409	0.344706
$377$	−1.24694	−0.0642208
$378$	0	0
$379$	−2.53379	−0.130152	−0.0650760	−	0.997880i	$-0.520729\pi$
$379$	−2.53379	−0.130152	−0.0650760	+	0.997880i	$0.520729\pi$
$380$	−17.7560	−0.910864
$381$	0	0
$382$	−2.04470	−0.104616
$383$	3.55050	0.181422	0.0907109	−	0.995877i	$-0.471086\pi$
$383$	3.55050	0.181422	0.0907109	+	0.995877i	$0.471086\pi$
$384$	0	0
$385$	43.7003	2.22717
$386$	−23.5905	−1.20073
$387$	0	0
$388$	4.33160	0.219904
$389$	27.7005	1.40447	0.702237	−	0.711944i	$-0.252185\pi$
$389$	27.7005	1.40447	0.702237	+	0.711944i	$0.252185\pi$
$390$	0	0
$391$	0	0
$392$	0.676573	0.0341721
$393$	0	0
$394$	3.37234	0.169896
$395$	11.5022	0.578740
$396$	0	0
$397$	−21.3083	−1.06943	−0.534716	−	0.845032i	$-0.679581\pi$
$397$	−21.3083	−1.06943	−0.534716	+	0.845032i	$0.679581\pi$
$398$	−21.6752	−1.08648
$399$	0	0
$400$	3.37462	0.168731
$401$	10.9560	0.547117	0.273559	−	0.961855i	$-0.411799\pi$
$401$	10.9560	0.547117	0.273559	+	0.961855i	$0.411799\pi$
$402$	0	0
$403$	2.80883	0.139918
$404$	−6.42624	−0.319717
$405$	0	0
$406$	−9.44870	−0.468931
$407$	−26.3173	−1.30450
$408$	0	0
$409$	29.0013	1.43402	0.717010	−	0.697062i	$-0.245510\pi$
$409$	29.0013	1.43402	0.717010	+	0.697062i	$0.245510\pi$
$410$	12.2610	0.605528
$411$	0	0
$412$	12.2847	0.605225
$413$	−1.21935	−0.0600005
$414$	0	0
$415$	46.5984	2.28743
$416$	0.365644	0.0179272
$417$	0	0
$418$	−33.4411	−1.63566
$419$	−8.36307	−0.408563	−0.204281	−	0.978912i	$-0.565486\pi$
$419$	−8.36307	−0.408563	−0.204281	+	0.978912i	$0.565486\pi$
$420$	0	0
$421$	10.6014	0.516682	0.258341	−	0.966054i	$-0.416824\pi$
$421$	10.6014	0.516682	0.258341	+	0.966054i	$0.416824\pi$
$422$	−22.9306	−1.11624
$423$	0	0
$424$	7.13462	0.346488
$425$	−21.2520	−1.03087
$426$	0	0
$427$	−22.5365	−1.09062
$428$	9.93617	0.480283
$429$	0	0
$430$	20.3628	0.981983
$431$	0.470563	0.0226662	0.0113331	−	0.999936i	$-0.496392\pi$
$431$	0.470563	0.0226662	0.0113331	+	0.999936i	$0.496392\pi$
$432$	0	0
$433$	29.0030	1.39380	0.696898	−	0.717170i	$-0.254563\pi$
$433$	29.0030	1.39380	0.696898	+	0.717170i	$0.254563\pi$
$434$	21.2839	1.02166
$435$	0	0
$436$	10.4065	0.498381
$437$	0	0
$438$	0	0
$439$	−15.0844	−0.719937	−0.359969	−	0.932964i	$-0.617213\pi$
$439$	−15.0844	−0.719937	−0.359969	+	0.932964i	$0.617213\pi$
$440$	15.7725	0.751925
$441$	0	0
$442$	−2.30268	−0.109527
$443$	26.5727	1.26250	0.631252	−	0.775577i	$-0.282541\pi$
$443$	26.5727	1.26250	0.631252	+	0.775577i	$0.282541\pi$
$444$	0	0
$445$	39.8220	1.88774
$446$	−19.8265	−0.938814
$447$	0	0
$448$	2.77066	0.130902
$449$	−3.87601	−0.182920	−0.0914601	−	0.995809i	$-0.529153\pi$
$449$	−3.87601	−0.182920	−0.0914601	+	0.995809i	$0.529153\pi$
$450$	0	0
$451$	23.0920	1.08736
$452$	10.7510	0.505685
$453$	0	0
$454$	−19.9316	−0.935434
$455$	2.93173	0.137442
$456$	0	0
$457$	−3.22201	−0.150719	−0.0753597	−	0.997156i	$-0.524010\pi$
$457$	−3.22201	−0.150719	−0.0753597	+	0.997156i	$0.524010\pi$
$458$	16.2991	0.761609
$459$	0	0
$460$	0	0
$461$	25.7463	1.19912	0.599562	−	0.800329i	$-0.295342\pi$
$461$	25.7463	1.19912	0.599562	+	0.800329i	$0.295342\pi$
$462$	0	0
$463$	18.5671	0.862889	0.431444	−	0.902140i	$-0.358004\pi$
$463$	18.5671	0.862889	0.431444	+	0.902140i	$0.358004\pi$
$464$	−3.41027	−0.158318
$465$	0	0
$466$	−18.6367	−0.863328
$467$	6.22189	0.287915	0.143957	−	0.989584i	$-0.454017\pi$
$467$	6.22189	0.287915	0.143957	+	0.989584i	$0.454017\pi$
$468$	0	0
$469$	−12.3274	−0.569228
$470$	19.3430	0.892228
$471$	0	0
$472$	−0.440095	−0.0202570
$473$	38.3507	1.76337
$474$	0	0
$475$	−20.7056	−0.950038
$476$	−17.4485	−0.799752
$477$	0	0
$478$	−4.40675	−0.201560
$479$	−10.3248	−0.471753	−0.235877	−	0.971783i	$-0.575796\pi$
$479$	−10.3248	−0.471753	−0.235877	+	0.971783i	$0.575796\pi$
$480$	0	0
$481$	−1.76555	−0.0805023
$482$	−13.1178	−0.597498
$483$	0	0
$484$	18.7055	0.850249
$485$	12.5352	0.569193
$486$	0	0
$487$	25.1187	1.13824	0.569119	−	0.822255i	$-0.307284\pi$
$487$	25.1187	1.13824	0.569119	+	0.822255i	$0.307284\pi$
$488$	−8.13399	−0.368208
$489$	0	0
$490$	1.95793	0.0884503
$491$	−1.53268	−0.0691689	−0.0345845	−	0.999402i	$-0.511011\pi$
$491$	−1.53268	−0.0691689	−0.0345845	+	0.999402i	$0.511011\pi$
$492$	0	0
$493$	21.4765	0.967253
$494$	−2.24347	−0.100939
$495$	0	0
$496$	7.68188	0.344926
$497$	3.91475	0.175601
$498$	0	0
$499$	−23.4321	−1.04896	−0.524482	−	0.851421i	$-0.675741\pi$
$499$	−23.4321	−1.04896	−0.524482	+	0.851421i	$0.675741\pi$
$500$	−4.70367	−0.210354
$501$	0	0
$502$	−9.21467	−0.411271
$503$	9.41759	0.419909	0.209955	−	0.977711i	$-0.432668\pi$
$503$	9.41759	0.419909	0.209955	+	0.977711i	$0.432668\pi$
$504$	0	0
$505$	−18.5969	−0.827549
$506$	0	0
$507$	0	0
$508$	−9.70663	−0.430662
$509$	−9.99388	−0.442971	−0.221485	−	0.975164i	$-0.571091\pi$
$509$	−9.99388	−0.442971	−0.221485	+	0.975164i	$0.571091\pi$
$510$	0	0
$511$	−10.9197	−0.483058
$512$	1.00000	0.0441942
$513$	0	0
$514$	−2.43454	−0.107383
$515$	35.5507	1.56655
$516$	0	0
$517$	36.4301	1.60219
$518$	−13.3785	−0.587816
$519$	0	0
$520$	1.05813	0.0464023
$521$	−38.5925	−1.69077	−0.845384	−	0.534159i	$-0.820628\pi$
$521$	−38.5925	−1.69077	−0.845384	+	0.534159i	$0.820628\pi$
$522$	0	0
$523$	9.61719	0.420530	0.210265	−	0.977644i	$-0.432567\pi$
$523$	9.61719	0.420530	0.210265	+	0.977644i	$0.432567\pi$
$524$	13.1274	0.573474
$525$	0	0
$526$	−3.60644	−0.157248
$527$	−48.3774	−2.10735
$528$	0	0
$529$	0	0
$530$	20.6468	0.896841
$531$	0	0
$532$	−16.9999	−0.737039
$533$	1.54918	0.0671024
$534$	0	0
$535$	28.7542	1.24315
$536$	−4.44927	−0.192179
$537$	0	0
$538$	−22.6221	−0.975308
$539$	3.68751	0.158832
$540$	0	0
$541$	14.4164	0.619811	0.309906	−	0.950767i	$-0.399703\pi$
$541$	14.4164	0.619811	0.309906	+	0.950767i	$0.399703\pi$
$542$	13.6955	0.588272
$543$	0	0
$544$	−6.29760	−0.270007
$545$	30.1153	1.29000
$546$	0	0
$547$	−14.1450	−0.604797	−0.302398	−	0.953182i	$-0.597787\pi$
$547$	−14.1450	−0.604797	−0.302398	+	0.953182i	$0.597787\pi$
$548$	−3.83057	−0.163634
$549$	0	0
$550$	18.3926	0.784264
$551$	20.9243	0.891405
$552$	0	0
$553$	11.0124	0.468296
$554$	−10.2577	−0.435807
$555$	0	0
$556$	17.3000	0.733682
$557$	10.4965	0.444751	0.222375	−	0.974961i	$-0.428619\pi$
$557$	10.4965	0.444751	0.222375	+	0.974961i	$0.428619\pi$
$558$	0	0
$559$	2.57284	0.108820
$560$	8.01801	0.338823
$561$	0	0
$562$	−3.11451	−0.131378
$563$	−23.7052	−0.999057	−0.499528	−	0.866298i	$-0.666493\pi$
$563$	−23.7052	−0.999057	−0.499528	+	0.866298i	$0.666493\pi$
$564$	0	0
$565$	31.1123	1.30890
$566$	9.50350	0.399462
$567$	0	0
$568$	1.41293	0.0592852
$569$	9.77315	0.409712	0.204856	−	0.978792i	$-0.434327\pi$
$569$	9.77315	0.409712	0.204856	+	0.978792i	$0.434327\pi$
$570$	0	0
$571$	24.7004	1.03368	0.516840	−	0.856082i	$-0.327108\pi$
$571$	24.7004	1.03368	0.516840	+	0.856082i	$0.327108\pi$
$572$	1.99286	0.0833256
$573$	0	0
$574$	11.7389	0.489972
$575$	0	0
$576$	0	0
$577$	−0.949796	−0.0395405	−0.0197703	−	0.999805i	$-0.506293\pi$
$577$	−0.949796	−0.0395405	−0.0197703	+	0.999805i	$0.506293\pi$
$578$	22.6597	0.942521
$579$	0	0
$580$	−9.86895	−0.409786
$581$	44.6141	1.85091
$582$	0	0
$583$	38.8856	1.61048
$584$	−3.94118	−0.163087
$585$	0	0
$586$	25.1900	1.04059
$587$	−4.13796	−0.170792	−0.0853959	−	0.996347i	$-0.527215\pi$
$587$	−4.13796	−0.170792	−0.0853959	+	0.996347i	$0.527215\pi$
$588$	0	0
$589$	−47.1335	−1.94210
$590$	−1.27359	−0.0524327
$591$	0	0
$592$	−4.82862	−0.198455
$593$	36.8544	1.51343	0.756714	−	0.653746i	$-0.226803\pi$
$593$	36.8544	1.51343	0.756714	+	0.653746i	$0.226803\pi$
$594$	0	0
$595$	−50.4942	−2.07006
$596$	14.5551	0.596200
$597$	0	0
$598$	0	0
$599$	−1.06163	−0.0433771	−0.0216885	−	0.999765i	$-0.506904\pi$
$599$	−1.06163	−0.0433771	−0.0216885	+	0.999765i	$0.506904\pi$
$600$	0	0
$601$	−7.83752	−0.319699	−0.159850	−	0.987141i	$-0.551101\pi$
$601$	−7.83752	−0.319699	−0.159850	+	0.987141i	$0.551101\pi$
$602$	19.4957	0.794586
$603$	0	0
$604$	−9.51541	−0.387177
$605$	54.1317	2.20077
$606$	0	0
$607$	−1.63562	−0.0663877	−0.0331938	−	0.999449i	$-0.510568\pi$
$607$	−1.63562	−0.0663877	−0.0331938	+	0.999449i	$0.510568\pi$
$608$	−6.13568	−0.248835
$609$	0	0
$610$	−23.5389	−0.953062
$611$	2.44400	0.0988735
$612$	0	0
$613$	27.7008	1.11882	0.559412	−	0.828890i	$-0.311027\pi$
$613$	27.7008	1.11882	0.559412	+	0.828890i	$0.311027\pi$
$614$	1.33026	0.0536848
$615$	0	0
$616$	15.1009	0.608431
$617$	−11.4797	−0.462154	−0.231077	−	0.972935i	$-0.574225\pi$
$617$	−11.4797	−0.462154	−0.231077	+	0.972935i	$0.574225\pi$
$618$	0	0
$619$	41.9154	1.68472	0.842361	−	0.538914i	$-0.181165\pi$
$619$	41.9154	1.68472	0.842361	+	0.538914i	$0.181165\pi$
$620$	22.2305	0.892800
$621$	0	0
$622$	8.70784	0.349152
$623$	38.1262	1.52749
$624$	0	0
$625$	−30.4850	−1.21940
$626$	−12.7984	−0.511526
$627$	0	0
$628$	−23.8636	−0.952261
$629$	30.4087	1.21247
$630$	0	0
$631$	24.3716	0.970218	0.485109	−	0.874454i	$-0.338780\pi$
$631$	24.3716	0.970218	0.485109	+	0.874454i	$0.338780\pi$
$632$	3.97465	0.158103
$633$	0	0
$634$	−13.1795	−0.523426
$635$	−28.0899	−1.11472
$636$	0	0
$637$	0.247385	0.00980174
$638$	−18.5869	−0.735862
$639$	0	0
$640$	2.89389	0.114391
$641$	−37.8396	−1.49457	−0.747287	−	0.664501i	$-0.768644\pi$
$641$	−37.8396	−1.49457	−0.747287	+	0.664501i	$0.768644\pi$
$642$	0	0
$643$	5.62757	0.221930	0.110965	−	0.993824i	$-0.464606\pi$
$643$	5.62757	0.221930	0.110965	+	0.993824i	$0.464606\pi$
$644$	0	0
$645$	0	0
$646$	38.6400	1.52027
$647$	24.6512	0.969138	0.484569	−	0.874753i	$-0.338976\pi$
$647$	24.6512	0.969138	0.484569	+	0.874753i	$0.338976\pi$
$648$	0	0
$649$	−2.39864	−0.0941547
$650$	1.23391	0.0483979
$651$	0	0
$652$	8.16971	0.319950
$653$	−41.4052	−1.62031	−0.810155	−	0.586215i	$-0.800617\pi$
$653$	−41.4052	−1.62031	−0.810155	+	0.586215i	$0.800617\pi$
$654$	0	0
$655$	37.9894	1.48437
$656$	4.23685	0.165421
$657$	0	0
$658$	18.5194	0.721960
$659$	−15.2904	−0.595629	−0.297814	−	0.954624i	$-0.596258\pi$
$659$	−15.2904	−0.595629	−0.297814	+	0.954624i	$0.596258\pi$
$660$	0	0
$661$	16.3453	0.635759	0.317879	−	0.948131i	$-0.397029\pi$
$661$	16.3453	0.635759	0.317879	+	0.948131i	$0.397029\pi$
$662$	−31.6717	−1.23096
$663$	0	0
$664$	16.1023	0.624891
$665$	−49.1959	−1.90773
$666$	0	0
$667$	0	0
$668$	10.2364	0.396059
$669$	0	0
$670$	−12.8757	−0.497432
$671$	−44.3325	−1.71144
$672$	0	0
$673$	−40.3634	−1.55590	−0.777948	−	0.628329i	$-0.783739\pi$
$673$	−40.3634	−1.55590	−0.777948	+	0.628329i	$0.783739\pi$
$674$	−9.14100	−0.352098
$675$	0	0
$676$	−12.8663	−0.494858
$677$	4.69864	0.180583	0.0902916	−	0.995915i	$-0.471220\pi$
$677$	4.69864	0.180583	0.0902916	+	0.995915i	$0.471220\pi$
$678$	0	0
$679$	12.0014	0.460571
$680$	−18.2246	−0.698881
$681$	0	0
$682$	41.8683	1.60322
$683$	−28.7965	−1.10187	−0.550934	−	0.834549i	$-0.685729\pi$
$683$	−28.7965	−1.10187	−0.550934	+	0.834549i	$0.685729\pi$
$684$	0	0
$685$	−11.0853	−0.423547
$686$	−17.5201	−0.668920
$687$	0	0
$688$	7.03648	0.268263
$689$	2.60873	0.0993846
$690$	0	0
$691$	−27.0579	−1.02933	−0.514666	−	0.857391i	$-0.672084\pi$
$691$	−27.0579	−1.02933	−0.514666	+	0.857391i	$0.672084\pi$
$692$	3.39250	0.128963
$693$	0	0
$694$	−17.8143	−0.676223
$695$	50.0642	1.89904
$696$	0	0
$697$	−26.6820	−1.01065
$698$	−20.8462	−0.789039
$699$	0	0
$700$	9.34994	0.353395
$701$	−32.6337	−1.23256	−0.616278	−	0.787528i	$-0.711360\pi$
$701$	−32.6337	−1.23256	−0.616278	+	0.787528i	$0.711360\pi$
$702$	0	0
$703$	29.6268	1.11740
$704$	5.45027	0.205415
$705$	0	0
$706$	−22.1361	−0.833102
$707$	−17.8049	−0.669624
$708$	0	0
$709$	−42.5915	−1.59956	−0.799778	−	0.600296i	$-0.795050\pi$
$709$	−42.5915	−1.59956	−0.799778	+	0.600296i	$0.795050\pi$
$710$	4.08887	0.153453
$711$	0	0
$712$	13.7607	0.515703
$713$	0	0
$714$	0	0
$715$	5.76712	0.215678
$716$	−6.82941	−0.255227
$717$	0	0
$718$	−27.8721	−1.04018
$719$	−18.1927	−0.678473	−0.339237	−	0.940701i	$-0.610169\pi$
$719$	−18.1927	−0.678473	−0.339237	+	0.940701i	$0.610169\pi$
$720$	0	0
$721$	34.0368	1.26760
$722$	18.6465	0.693952
$723$	0	0
$724$	−0.940518	−0.0349541
$725$	−11.5084	−0.427410
$726$	0	0
$727$	−1.80888	−0.0670876	−0.0335438	−	0.999437i	$-0.510679\pi$
$727$	−1.80888	−0.0670876	−0.0335438	+	0.999437i	$0.510679\pi$
$728$	1.01308	0.0375471
$729$	0	0
$730$	−11.4054	−0.422131
$731$	−44.3129	−1.63897
$732$	0	0
$733$	17.0822	0.630944	0.315472	−	0.948935i	$-0.397837\pi$
$733$	17.0822	0.630944	0.315472	+	0.948935i	$0.397837\pi$
$734$	−17.5832	−0.649007
$735$	0	0
$736$	0	0
$737$	−24.2497	−0.893250
$738$	0	0
$739$	−40.8754	−1.50363	−0.751813	−	0.659377i	$-0.770820\pi$
$739$	−40.8754	−1.50363	−0.751813	+	0.659377i	$0.770820\pi$
$740$	−13.9735	−0.513676
$741$	0	0
$742$	19.7676	0.725692
$743$	−36.8404	−1.35154	−0.675772	−	0.737111i	$-0.736190\pi$
$743$	−36.8404	−1.35154	−0.675772	+	0.737111i	$0.736190\pi$
$744$	0	0
$745$	42.1209	1.54319
$746$	−26.4065	−0.966809
$747$	0	0
$748$	−34.3236	−1.25500
$749$	27.5298	1.00592
$750$	0	0
$751$	−1.92778	−0.0703458	−0.0351729	−	0.999381i	$-0.511198\pi$
$751$	−1.92778	−0.0703458	−0.0351729	+	0.999381i	$0.511198\pi$
$752$	6.68409	0.243744
$753$	0	0
$754$	−1.24694	−0.0454110
$755$	−27.5366	−1.00216
$756$	0	0
$757$	23.5286	0.855161	0.427581	−	0.903977i	$-0.359366\pi$
$757$	23.5286	0.855161	0.427581	+	0.903977i	$0.359366\pi$
$758$	−2.53379	−0.0920313
$759$	0	0
$760$	−17.7560	−0.644078
$761$	−19.2372	−0.697349	−0.348674	−	0.937244i	$-0.613368\pi$
$761$	−19.2372	−0.697349	−0.348674	+	0.937244i	$0.613368\pi$
$762$	0	0
$763$	28.8329	1.04382
$764$	−2.04470	−0.0739745
$765$	0	0
$766$	3.55050	0.128285
$767$	−0.160918	−0.00581040
$768$	0	0
$769$	20.6749	0.745556	0.372778	−	0.927921i	$-0.378405\pi$
$769$	20.6749	0.745556	0.372778	+	0.927921i	$0.378405\pi$
$770$	43.7003	1.57485
$771$	0	0
$772$	−23.5905	−0.849042
$773$	−11.2960	−0.406289	−0.203144	−	0.979149i	$-0.565116\pi$
$773$	−11.2960	−0.406289	−0.203144	+	0.979149i	$0.565116\pi$
$774$	0	0
$775$	25.9234	0.931198
$776$	4.33160	0.155495
$777$	0	0
$778$	27.7005	0.993113
$779$	−25.9960	−0.931402
$780$	0	0
$781$	7.70086	0.275558
$782$	0	0
$783$	0	0
$784$	0.676573	0.0241633
$785$	−69.0587	−2.46481
$786$	0	0
$787$	29.3047	1.04460	0.522300	−	0.852762i	$-0.325074\pi$
$787$	29.3047	1.04460	0.522300	+	0.852762i	$0.325074\pi$
$788$	3.37234	0.120134
$789$	0	0
$790$	11.5022	0.409231
$791$	29.7874	1.05912
$792$	0	0
$793$	−2.97414	−0.105615
$794$	−21.3083	−0.756202
$795$	0	0
$796$	−21.6752	−0.768257
$797$	33.2393	1.17740	0.588698	−	0.808353i	$-0.299641\pi$
$797$	33.2393	1.17740	0.588698	+	0.808353i	$0.299641\pi$
$798$	0	0
$799$	−42.0937	−1.48917
$800$	3.37462	0.119311
$801$	0	0
$802$	10.9560	0.386870
$803$	−21.4805	−0.758031
$804$	0	0
$805$	0	0
$806$	2.80883	0.0989368
$807$	0	0
$808$	−6.42624	−0.226074
$809$	−37.1758	−1.30703	−0.653516	−	0.756913i	$-0.726707\pi$
$809$	−37.1758	−1.30703	−0.653516	+	0.756913i	$0.726707\pi$
$810$	0	0
$811$	−1.82998	−0.0642591	−0.0321296	−	0.999484i	$-0.510229\pi$
$811$	−1.82998	−0.0642591	−0.0321296	+	0.999484i	$0.510229\pi$
$812$	−9.44870	−0.331584
$813$	0	0
$814$	−26.3173	−0.922420
$815$	23.6423	0.828153
$816$	0	0
$817$	−43.1736	−1.51045
$818$	29.0013	1.01401
$819$	0	0
$820$	12.2610	0.428173
$821$	30.4875	1.06402	0.532010	−	0.846738i	$-0.321437\pi$
$821$	30.4875	1.06402	0.532010	+	0.846738i	$0.321437\pi$
$822$	0	0
$823$	50.2124	1.75029	0.875146	−	0.483858i	$-0.160765\pi$
$823$	50.2124	1.75029	0.875146	+	0.483858i	$0.160765\pi$
$824$	12.2847	0.427958
$825$	0	0
$826$	−1.21935	−0.0424267
$827$	−47.7273	−1.65964	−0.829821	−	0.558030i	$-0.811557\pi$
$827$	−47.7273	−1.65964	−0.829821	+	0.558030i	$0.811557\pi$
$828$	0	0
$829$	−53.8283	−1.86953	−0.934767	−	0.355262i	$-0.884392\pi$
$829$	−53.8283	−1.86953	−0.934767	+	0.355262i	$0.884392\pi$
$830$	46.5984	1.61745
$831$	0	0
$832$	0.365644	0.0126764
$833$	−4.26079	−0.147628
$834$	0	0
$835$	29.6231	1.02515
$836$	−33.4411	−1.15659
$837$	0	0
$838$	−8.36307	−0.288897
$839$	−7.11207	−0.245536	−0.122768	−	0.992435i	$-0.539177\pi$
$839$	−7.11207	−0.245536	−0.122768	+	0.992435i	$0.539177\pi$
$840$	0	0
$841$	−17.3701	−0.598968
$842$	10.6014	0.365349
$843$	0	0
$844$	−22.9306	−0.789303
$845$	−37.2337	−1.28088
$846$	0	0
$847$	51.8266	1.78078
$848$	7.13462	0.245004
$849$	0	0
$850$	−21.2520	−0.728938
$851$	0	0
$852$	0	0
$853$	−12.9685	−0.444032	−0.222016	−	0.975043i	$-0.571264\pi$
$853$	−12.9685	−0.444032	−0.222016	+	0.975043i	$0.571264\pi$
$854$	−22.5365	−0.771185
$855$	0	0
$856$	9.93617	0.339611
$857$	−24.8914	−0.850275	−0.425137	−	0.905129i	$-0.639774\pi$
$857$	−24.8914	−0.850275	−0.425137	+	0.905129i	$0.639774\pi$
$858$	0	0
$859$	−20.4110	−0.696413	−0.348207	−	0.937418i	$-0.613209\pi$
$859$	−20.4110	−0.696413	−0.348207	+	0.937418i	$0.613209\pi$
$860$	20.3628	0.694367
$861$	0	0
$862$	0.470563	0.0160274
$863$	−35.8151	−1.21916	−0.609579	−	0.792725i	$-0.708662\pi$
$863$	−35.8151	−1.21916	−0.609579	+	0.792725i	$0.708662\pi$
$864$	0	0
$865$	9.81753	0.333806
$866$	29.0030	0.985563
$867$	0	0
$868$	21.2839	0.722422
$869$	21.6629	0.734865
$870$	0	0
$871$	−1.62685	−0.0551236
$872$	10.4065	0.352408
$873$	0	0
$874$	0	0
$875$	−13.0323	−0.440571
$876$	0	0
$877$	4.42754	0.149507	0.0747537	−	0.997202i	$-0.476183\pi$
$877$	4.42754	0.149507	0.0747537	+	0.997202i	$0.476183\pi$
$878$	−15.0844	−0.509072
$879$	0	0
$880$	15.7725	0.531691
$881$	−29.3054	−0.987325	−0.493663	−	0.869654i	$-0.664342\pi$
$881$	−29.3054	−0.987325	−0.493663	+	0.869654i	$0.664342\pi$
$882$	0	0
$883$	11.6067	0.390597	0.195299	−	0.980744i	$-0.437432\pi$
$883$	11.6067	0.390597	0.195299	+	0.980744i	$0.437432\pi$
$884$	−2.30268	−0.0774474
$885$	0	0
$886$	26.5727	0.892726
$887$	10.1680	0.341410	0.170705	−	0.985322i	$-0.445396\pi$
$887$	10.1680	0.341410	0.170705	+	0.985322i	$0.445396\pi$
$888$	0	0
$889$	−26.8938	−0.901989
$890$	39.8220	1.33483
$891$	0	0
$892$	−19.8265	−0.663841
$893$	−41.0114	−1.37239
$894$	0	0
$895$	−19.7636	−0.660624
$896$	2.77066	0.0925614
$897$	0	0
$898$	−3.87601	−0.129344
$899$	−26.1973	−0.873728
$900$	0	0
$901$	−44.9310	−1.49687
$902$	23.0920	0.768880
$903$	0	0
$904$	10.7510	0.357573
$905$	−2.72176	−0.0904744
$906$	0	0
$907$	30.4063	1.00962	0.504812	−	0.863229i	$-0.331562\pi$
$907$	30.4063	1.00962	0.504812	+	0.863229i	$0.331562\pi$
$908$	−19.9316	−0.661452
$909$	0	0
$910$	2.93173	0.0971860
$911$	13.3731	0.443069	0.221535	−	0.975153i	$-0.428893\pi$
$911$	13.3731	0.443069	0.221535	+	0.975153i	$0.428893\pi$
$912$	0	0
$913$	87.7621	2.90450
$914$	−3.22201	−0.106575
$915$	0	0
$916$	16.2991	0.538539
$917$	36.3717	1.20110
$918$	0	0
$919$	28.5942	0.943237	0.471618	−	0.881803i	$-0.343670\pi$
$919$	28.5942	0.943237	0.471618	+	0.881803i	$0.343670\pi$
$920$	0	0
$921$	0	0
$922$	25.7463	0.847908
$923$	0.516629	0.0170051
$924$	0	0
$925$	−16.2948	−0.535769
$926$	18.5671	0.610154
$927$	0	0
$928$	−3.41027	−0.111948
$929$	53.3717	1.75107	0.875535	−	0.483154i	$-0.160509\pi$
$929$	53.3717	1.75107	0.875535	+	0.483154i	$0.160509\pi$
$930$	0	0
$931$	−4.15124	−0.136051
$932$	−18.6367	−0.610465
$933$	0	0
$934$	6.22189	0.203586
$935$	−99.3290	−3.24840
$936$	0	0
$937$	41.9499	1.37044	0.685222	−	0.728334i	$-0.259705\pi$
$937$	41.9499	1.37044	0.685222	+	0.728334i	$0.259705\pi$
$938$	−12.3274	−0.402505
$939$	0	0
$940$	19.3430	0.630901
$941$	20.7950	0.677897	0.338949	−	0.940805i	$-0.389929\pi$
$941$	20.7950	0.677897	0.338949	+	0.940805i	$0.389929\pi$
$942$	0	0
$943$	0	0
$944$	−0.440095	−0.0143239
$945$	0	0
$946$	38.3507	1.24689
$947$	−39.2571	−1.27568	−0.637841	−	0.770168i	$-0.720172\pi$
$947$	−39.2571	−1.27568	−0.637841	+	0.770168i	$0.720172\pi$
$948$	0	0
$949$	−1.44107	−0.0467790
$950$	−20.7056	−0.671778
$951$	0	0
$952$	−17.4485	−0.565510
$953$	20.6219	0.668008	0.334004	−	0.942572i	$-0.391600\pi$
$953$	20.6219	0.668008	0.334004	+	0.942572i	$0.391600\pi$
$954$	0	0
$955$	−5.91714	−0.191474
$956$	−4.40675	−0.142524
$957$	0	0
$958$	−10.3248	−0.333580
$959$	−10.6132	−0.342719
$960$	0	0
$961$	28.0112	0.903588
$962$	−1.76555	−0.0569237
$963$	0	0
$964$	−13.1178	−0.422495
$965$	−68.2685	−2.19764
$966$	0	0
$967$	−12.2331	−0.393391	−0.196696	−	0.980465i	$-0.563021\pi$
$967$	−12.2331	−0.393391	−0.196696	+	0.980465i	$0.563021\pi$
$968$	18.7055	0.601217
$969$	0	0
$970$	12.5352	0.402481
$971$	−14.5172	−0.465878	−0.232939	−	0.972491i	$-0.574834\pi$
$971$	−14.5172	−0.465878	−0.232939	+	0.972491i	$0.574834\pi$
$972$	0	0
$973$	47.9323	1.53664
$974$	25.1187	0.804855
$975$	0	0
$976$	−8.13399	−0.260363
$977$	4.71117	0.150724	0.0753618	−	0.997156i	$-0.475989\pi$
$977$	4.71117	0.150724	0.0753618	+	0.997156i	$0.475989\pi$
$978$	0	0
$979$	74.9995	2.39699
$980$	1.95793	0.0625438
$981$	0	0
$982$	−1.53268	−0.0489098
$983$	29.0494	0.926533	0.463267	−	0.886219i	$-0.346677\pi$
$983$	29.0494	0.926533	0.463267	+	0.886219i	$0.346677\pi$
$984$	0	0
$985$	9.75918	0.310953
$986$	21.4765	0.683951
$987$	0	0
$988$	−2.24347	−0.0713743
$989$	0	0
$990$	0	0
$991$	−18.2547	−0.579881	−0.289941	−	0.957045i	$-0.593636\pi$
$991$	−18.2547	−0.579881	−0.289941	+	0.957045i	$0.593636\pi$
$992$	7.68188	0.243900
$993$	0	0
$994$	3.91475	0.124168
$995$	−62.7257	−1.98854
$996$	0	0
$997$	19.3682	0.613396	0.306698	−	0.951807i	$-0.400776\pi$
$997$	19.3682	0.613396	0.306698	+	0.951807i	$0.400776\pi$
$998$	−23.4321	−0.741730
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.ca.1.3		5
3.2	odd	2		3174.2.a.y.1.3		5
23.11	odd	22		414.2.i.b.397.1		10
23.21	odd	22		414.2.i.b.73.1		10
23.22	odd	2		9522.2.a.bv.1.3		5
69.11	even	22		138.2.e.c.121.1	yes	10
69.44	even	22		138.2.e.c.73.1	✓	10
69.68	even	2		3174.2.a.z.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.c.73.1	✓	10	69.44	even	22
138.2.e.c.121.1	yes	10	69.11	even	22
414.2.i.b.73.1		10	23.21	odd	22
414.2.i.b.397.1		10	23.11	odd	22
3174.2.a.y.1.3		5	3.2	odd	2
3174.2.a.z.1.3		5	69.68	even	2
9522.2.a.bv.1.3		5	23.22	odd	2
9522.2.a.ca.1.3		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.89389 q^{5} +2.77066 q^{7} +1.00000 q^{8} +2.89389 q^{10} +5.45027 q^{11} +0.365644 q^{13} +2.77066 q^{14} +1.00000 q^{16} -6.29760 q^{17} -6.13568 q^{19} +2.89389 q^{20} +5.45027 q^{22} +3.37462 q^{25} +0.365644 q^{26} +2.77066 q^{28} -3.41027 q^{29} +7.68188 q^{31} +1.00000 q^{32} -6.29760 q^{34} +8.01801 q^{35} -4.82862 q^{37} -6.13568 q^{38} +2.89389 q^{40} +4.23685 q^{41} +7.03648 q^{43} +5.45027 q^{44} +6.68409 q^{47} +0.676573 q^{49} +3.37462 q^{50} +0.365644 q^{52} +7.13462 q^{53} +15.7725 q^{55} +2.77066 q^{56} -3.41027 q^{58} -0.440095 q^{59} -8.13399 q^{61} +7.68188 q^{62} +1.00000 q^{64} +1.05813 q^{65} -4.44927 q^{67} -6.29760 q^{68} +8.01801 q^{70} +1.41293 q^{71} -3.94118 q^{73} -4.82862 q^{74} -6.13568 q^{76} +15.1009 q^{77} +3.97465 q^{79} +2.89389 q^{80} +4.23685 q^{82} +16.1023 q^{83} -18.2246 q^{85} +7.03648 q^{86} +5.45027 q^{88} +13.7607 q^{89} +1.01308 q^{91} +6.68409 q^{94} -17.7560 q^{95} +4.33160 q^{97} +0.676573 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} + 11 q^{5} + q^{7} + 5 q^{8} + 11 q^{10} + 11 q^{11} + 10 q^{13} + q^{14} + 5 q^{16} + 11 q^{17} + q^{19} + 11 q^{20} + 11 q^{22} + 30 q^{25} + 10 q^{26} + q^{28} - 3 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) \) 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

L-functions

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