LMFDB - Embedded newform 9522.2.a.ch.1.9

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 = [10,-10,0,10,10,0,2,-10,0,-10,12,0,0,-2,0,10,24,0,-8,10,0,-12, 0,0,8,0,0,2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	yes
Analytic conductor:	$76.0335528047$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.10.52900342088704.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 18x^{8} + 123x^{6} - 390x^{4} + 548x^{2} - 241 $ x^10 - 18x^8 + 123x^6 - 390x^4 + 548x^2 - 241
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 414)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.878233$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.83556 q^{5} +5.08193 q^{7} -1.00000 q^{8} -3.83556 q^{10} +4.51702 q^{11} -3.32949 q^{13} -5.08193 q^{14} +1.00000 q^{16} -0.154095 q^{17} +5.25031 q^{19} +3.83556 q^{20} -4.51702 q^{22} +9.71149 q^{25} +3.32949 q^{26} +5.08193 q^{28} +5.23379 q^{29} +1.26824 q^{31} -1.00000 q^{32} +0.154095 q^{34} +19.4920 q^{35} -4.45795 q^{37} -5.25031 q^{38} -3.83556 q^{40} -2.21152 q^{41} -2.63867 q^{43} +4.51702 q^{44} -1.10906 q^{47} +18.8261 q^{49} -9.71149 q^{50} -3.32949 q^{52} +4.28457 q^{53} +17.3253 q^{55} -5.08193 q^{56} -5.23379 q^{58} -3.22790 q^{59} -13.1399 q^{61} -1.26824 q^{62} +1.00000 q^{64} -12.7704 q^{65} -10.4839 q^{67} -0.154095 q^{68} -19.4920 q^{70} -14.2586 q^{71} +6.20430 q^{73} +4.45795 q^{74} +5.25031 q^{76} +22.9552 q^{77} -2.91689 q^{79} +3.83556 q^{80} +2.21152 q^{82} +2.33395 q^{83} -0.591040 q^{85} +2.63867 q^{86} -4.51702 q^{88} +5.50801 q^{89} -16.9202 q^{91} +1.10906 q^{94} +20.1378 q^{95} -13.1549 q^{97} -18.8261 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{2} + 10 q^{4} + 10 q^{5} + 2 q^{7} - 10 q^{8} - 10 q^{10} + 12 q^{11} - 2 q^{14} + 10 q^{16} + 24 q^{17} - 8 q^{19} + 10 q^{20} - 12 q^{22} + 8 q^{25} + 2 q^{28} - 4 q^{29} + 18 q^{31} - 10 q^{32}+ \cdots - 36 q^{98}+O(q^{100}) $ 10 * q - 10 * q^2 + 10 * q^4 + 10 * q^5 + 2 * q^7 - 10 * q^8 - 10 * q^10 + 12 * q^11 - 2 * q^14 + 10 * q^16 + 24 * q^17 - 8 * q^19 + 10 * q^20 - 12 * q^22 + 8 * q^25 + 2 * q^28 - 4 * q^29 + 18 * q^31 - 10 * q^32 - 24 * q^34 + 24 * q^35 - 12 * q^37 + 8 * q^38 - 10 * q^40 - 28 * q^41 - 8 * q^43 + 12 * q^44 + 16 * q^47 + 36 * q^49 - 8 * q^50 + 34 * q^53 + 30 * q^55 - 2 * q^56 + 4 * q^58 + 22 * q^59 - 30 * q^61 - 18 * q^62 + 10 * q^64 + 36 * q^65 - 18 * q^67 + 24 * q^68 - 24 * q^70 + 28 * q^71 - 20 * q^73 + 12 * q^74 - 8 * q^76 + 20 * q^77 - 2 * q^79 + 10 * q^80 + 28 * q^82 + 44 * q^83 + 16 * q^85 + 8 * q^86 - 12 * q^88 + 44 * q^89 - 22 * q^91 - 16 * q^94 + 10 * q^95 - 18 * q^97 - 36 * 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
$5$	3.83556	1.71531	0.857656	−	0.514223i	$-0.171920\pi$
$5$	3.83556	1.71531	0.857656	+	0.514223i	$0.171920\pi$
$6$	0	0
$7$	5.08193	1.92079	0.960395	−	0.278641i	$-0.0898839\pi$
$7$	5.08193	1.92079	0.960395	+	0.278641i	$0.0898839\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.83556	−1.21291
$11$	4.51702	1.36193	0.680966	−	0.732315i	$-0.261560\pi$
$11$	4.51702	1.36193	0.680966	+	0.732315i	$0.261560\pi$
$12$	0	0
$13$	−3.32949	−0.923433	−0.461717	−	0.887028i	$-0.652766\pi$
$13$	−3.32949	−0.923433	−0.461717	+	0.887028i	$0.652766\pi$
$14$	−5.08193	−1.35820
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.154095	−0.0373735	−0.0186868	−	0.999825i	$-0.505949\pi$
$17$	−0.154095	−0.0373735	−0.0186868	+	0.999825i	$0.505949\pi$
$18$	0	0
$19$	5.25031	1.20450	0.602252	−	0.798306i	$-0.294270\pi$
$19$	5.25031	1.20450	0.602252	+	0.798306i	$0.294270\pi$
$20$	3.83556	0.857656
$21$	0	0
$22$	−4.51702	−0.963031
$23$	0	0
$23$	0	0
$24$	0	0
$25$	9.71149	1.94230
$26$	3.32949	0.652966
$27$	0	0
$28$	5.08193	0.960395
$29$	5.23379	0.971891	0.485946	−	0.873989i	$-0.338475\pi$
$29$	5.23379	0.971891	0.485946	+	0.873989i	$0.338475\pi$
$30$	0	0
$31$	1.26824	0.227782	0.113891	−	0.993493i	$-0.463669\pi$
$31$	1.26824	0.227782	0.113891	+	0.993493i	$0.463669\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.154095	0.0264271
$35$	19.4920	3.29476
$36$	0	0
$37$	−4.45795	−0.732883	−0.366442	−	0.930441i	$-0.619424\pi$
$37$	−4.45795	−0.732883	−0.366442	+	0.930441i	$0.619424\pi$
$38$	−5.25031	−0.851712
$39$	0	0
$40$	−3.83556	−0.606455
$41$	−2.21152	−0.345382	−0.172691	−	0.984976i	$-0.555246\pi$
$41$	−2.21152	−0.345382	−0.172691	+	0.984976i	$0.555246\pi$
$42$	0	0
$43$	−2.63867	−0.402393	−0.201196	−	0.979551i	$-0.564483\pi$
$43$	−2.63867	−0.402393	−0.201196	+	0.979551i	$0.564483\pi$
$44$	4.51702	0.680966
$45$	0	0
$46$	0	0
$47$	−1.10906	−0.161773	−0.0808865	−	0.996723i	$-0.525775\pi$
$47$	−1.10906	−0.161773	−0.0808865	+	0.996723i	$0.525775\pi$
$48$	0	0
$49$	18.8261	2.68944
$50$	−9.71149	−1.37341
$51$	0	0
$52$	−3.32949	−0.461717
$53$	4.28457	0.588530	0.294265	−	0.955724i	$-0.404925\pi$
$53$	4.28457	0.588530	0.294265	+	0.955724i	$0.404925\pi$
$54$	0	0
$55$	17.3253	2.33614
$56$	−5.08193	−0.679102
$57$	0	0
$58$	−5.23379	−0.687231
$59$	−3.22790	−0.420236	−0.210118	−	0.977676i	$-0.567385\pi$
$59$	−3.22790	−0.420236	−0.210118	+	0.977676i	$0.567385\pi$
$60$	0	0
$61$	−13.1399	−1.68239	−0.841197	−	0.540729i	$-0.818149\pi$
$61$	−13.1399	−1.68239	−0.841197	+	0.540729i	$0.818149\pi$
$62$	−1.26824	−0.161066
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.7704	−1.58398
$66$	0	0
$67$	−10.4839	−1.28082	−0.640409	−	0.768034i	$-0.721235\pi$
$67$	−10.4839	−1.28082	−0.640409	+	0.768034i	$0.721235\pi$
$68$	−0.154095	−0.0186868
$69$	0	0
$70$	−19.4920	−2.32975
$71$	−14.2586	−1.69219	−0.846095	−	0.533032i	$-0.821052\pi$
$71$	−14.2586	−1.69219	−0.846095	+	0.533032i	$0.821052\pi$
$72$	0	0
$73$	6.20430	0.726158	0.363079	−	0.931758i	$-0.381725\pi$
$73$	6.20430	0.726158	0.363079	+	0.931758i	$0.381725\pi$
$74$	4.45795	0.518227
$75$	0	0
$76$	5.25031	0.602252
$77$	22.9552	2.61599
$78$	0	0
$79$	−2.91689	−0.328175	−0.164088	−	0.986446i	$-0.552468\pi$
$79$	−2.91689	−0.328175	−0.164088	+	0.986446i	$0.552468\pi$
$80$	3.83556	0.428828
$81$	0	0
$82$	2.21152	0.244222
$83$	2.33395	0.256185	0.128092	−	0.991762i	$-0.459115\pi$
$83$	2.33395	0.256185	0.128092	+	0.991762i	$0.459115\pi$
$84$	0	0
$85$	−0.591040	−0.0641073
$86$	2.63867	0.284535
$87$	0	0
$88$	−4.51702	−0.481516
$89$	5.50801	0.583847	0.291924	−	0.956442i	$-0.405705\pi$
$89$	5.50801	0.583847	0.291924	+	0.956442i	$0.405705\pi$
$90$	0	0
$91$	−16.9202	−1.77372
$92$	0	0
$93$	0	0
$94$	1.10906	0.114391
$95$	20.1378	2.06610
$96$	0	0
$97$	−13.1549	−1.33567	−0.667837	−	0.744307i	$-0.732780\pi$
$97$	−13.1549	−1.33567	−0.667837	+	0.744307i	$0.732780\pi$
$98$	−18.8261	−1.90172
$99$	0	0
$100$	9.71149	0.971149
$101$	6.51687	0.648453	0.324227	−	0.945979i	$-0.394896\pi$
$101$	6.51687	0.648453	0.324227	+	0.945979i	$0.394896\pi$
$102$	0	0
$103$	5.64524	0.556242	0.278121	−	0.960546i	$-0.410288\pi$
$103$	5.64524	0.556242	0.278121	+	0.960546i	$0.410288\pi$
$104$	3.32949	0.326483
$105$	0	0
$106$	−4.28457	−0.416154
$107$	6.79406	0.656807	0.328404	−	0.944538i	$-0.393489\pi$
$107$	6.79406	0.656807	0.328404	+	0.944538i	$0.393489\pi$
$108$	0	0
$109$	5.83285	0.558685	0.279343	−	0.960191i	$-0.409883\pi$
$109$	5.83285	0.558685	0.279343	+	0.960191i	$0.409883\pi$
$110$	−17.3253	−1.65190
$111$	0	0
$112$	5.08193	0.480198
$113$	14.6584	1.37895	0.689473	−	0.724311i	$-0.257842\pi$
$113$	14.6584	1.37895	0.689473	+	0.724311i	$0.257842\pi$
$114$	0	0
$115$	0	0
$116$	5.23379	0.485946
$117$	0	0
$118$	3.22790	0.297152
$119$	−0.783101	−0.0717868
$120$	0	0
$121$	9.40343	0.854858
$122$	13.1399	1.18963
$123$	0	0
$124$	1.26824	0.113891
$125$	18.0712	1.61634
$126$	0	0
$127$	−11.1093	−0.985787	−0.492893	−	0.870090i	$-0.664061\pi$
$127$	−11.1093	−0.985787	−0.492893	+	0.870090i	$0.664061\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	12.7704	1.12004
$131$	−2.08032	−0.181759	−0.0908793	−	0.995862i	$-0.528968\pi$
$131$	−2.08032	−0.181759	−0.0908793	+	0.995862i	$0.528968\pi$
$132$	0	0
$133$	26.6817	2.31360
$134$	10.4839	0.905675
$135$	0	0
$136$	0.154095	0.0132135
$137$	16.5855	1.41699	0.708496	−	0.705714i	$-0.249374\pi$
$137$	16.5855	1.41699	0.708496	+	0.705714i	$0.249374\pi$
$138$	0	0
$139$	4.74700	0.402635	0.201318	−	0.979526i	$-0.435478\pi$
$139$	4.74700	0.402635	0.201318	+	0.979526i	$0.435478\pi$
$140$	19.4920	1.64738
$141$	0	0
$142$	14.2586	1.19656
$143$	−15.0393	−1.25765
$144$	0	0
$145$	20.0745	1.66710
$146$	−6.20430	−0.513471
$147$	0	0
$148$	−4.45795	−0.366442
$149$	−17.0234	−1.39461	−0.697304	−	0.716775i	$-0.745617\pi$
$149$	−17.0234	−1.39461	−0.697304	+	0.716775i	$0.745617\pi$
$150$	0	0
$151$	−7.05024	−0.573741	−0.286870	−	0.957969i	$-0.592615\pi$
$151$	−7.05024	−0.573741	−0.286870	+	0.957969i	$0.592615\pi$
$152$	−5.25031	−0.425856
$153$	0	0
$154$	−22.9552	−1.84978
$155$	4.86439	0.390717
$156$	0	0
$157$	−6.11470	−0.488006	−0.244003	−	0.969774i	$-0.578461\pi$
$157$	−6.11470	−0.488006	−0.244003	+	0.969774i	$0.578461\pi$
$158$	2.91689	0.232055
$159$	0	0
$160$	−3.83556	−0.303227
$161$	0	0
$162$	0	0
$163$	−6.00203	−0.470116	−0.235058	−	0.971981i	$-0.575528\pi$
$163$	−6.00203	−0.470116	−0.235058	+	0.971981i	$0.575528\pi$
$164$	−2.21152	−0.172691
$165$	0	0
$166$	−2.33395	−0.181150
$167$	12.4878	0.966338	0.483169	−	0.875527i	$-0.339486\pi$
$167$	12.4878	0.966338	0.483169	+	0.875527i	$0.339486\pi$
$168$	0	0
$169$	−1.91453	−0.147271
$170$	0.591040	0.0453307
$171$	0	0
$172$	−2.63867	−0.201196
$173$	−3.79092	−0.288218	−0.144109	−	0.989562i	$-0.546032\pi$
$173$	−3.79092	−0.288218	−0.144109	+	0.989562i	$0.546032\pi$
$174$	0	0
$175$	49.3532	3.73075
$176$	4.51702	0.340483
$177$	0	0
$178$	−5.50801	−0.412842
$179$	0.453414	0.0338898	0.0169449	−	0.999856i	$-0.494606\pi$
$179$	0.453414	0.0338898	0.0169449	+	0.999856i	$0.494606\pi$
$180$	0	0
$181$	7.83553	0.582410	0.291205	−	0.956661i	$-0.405944\pi$
$181$	7.83553	0.582410	0.291205	+	0.956661i	$0.405944\pi$
$182$	16.9202	1.25421
$183$	0	0
$184$	0	0
$185$	−17.0987	−1.25712
$186$	0	0
$187$	−0.696050	−0.0509002
$188$	−1.10906	−0.0808865
$189$	0	0
$190$	−20.1378	−1.46095
$191$	−15.3660	−1.11184	−0.555921	−	0.831235i	$-0.687634\pi$
$191$	−15.3660	−1.11184	−0.555921	+	0.831235i	$0.687634\pi$
$192$	0	0
$193$	8.77768	0.631831	0.315915	−	0.948787i	$-0.397688\pi$
$193$	8.77768	0.631831	0.315915	+	0.948787i	$0.397688\pi$
$194$	13.1549	0.944464
$195$	0	0
$196$	18.8261	1.34472
$197$	8.01971	0.571381	0.285690	−	0.958322i	$-0.407777\pi$
$197$	8.01971	0.571381	0.285690	+	0.958322i	$0.407777\pi$
$198$	0	0
$199$	−6.20296	−0.439716	−0.219858	−	0.975532i	$-0.570559\pi$
$199$	−6.20296	−0.439716	−0.219858	+	0.975532i	$0.570559\pi$
$200$	−9.71149	−0.686706
$201$	0	0
$202$	−6.51687	−0.458526
$203$	26.5978	1.86680
$204$	0	0
$205$	−8.48242	−0.592438
$206$	−5.64524	−0.393322
$207$	0	0
$208$	−3.32949	−0.230858
$209$	23.7157	1.64045
$210$	0	0
$211$	−16.1371	−1.11092	−0.555461	−	0.831542i	$-0.687458\pi$
$211$	−16.1371	−1.11092	−0.555461	+	0.831542i	$0.687458\pi$
$212$	4.28457	0.294265
$213$	0	0
$214$	−6.79406	−0.464433
$215$	−10.1208	−0.690230
$216$	0	0
$217$	6.44510	0.437522
$218$	−5.83285	−0.395050
$219$	0	0
$220$	17.3253	1.16807
$221$	0.513057	0.0345120
$222$	0	0
$223$	−9.15732	−0.613219	−0.306610	−	0.951835i	$-0.599195\pi$
$223$	−9.15732	−0.613219	−0.306610	+	0.951835i	$0.599195\pi$
$224$	−5.08193	−0.339551
$225$	0	0
$226$	−14.6584	−0.975062
$227$	9.40435	0.624188	0.312094	−	0.950051i	$-0.398970\pi$
$227$	9.40435	0.624188	0.312094	+	0.950051i	$0.398970\pi$
$228$	0	0
$229$	−3.61058	−0.238594	−0.119297	−	0.992859i	$-0.538064\pi$
$229$	−3.61058	−0.238594	−0.119297	+	0.992859i	$0.538064\pi$
$230$	0	0
$231$	0	0
$232$	−5.23379	−0.343615
$233$	−0.558599	−0.0365950	−0.0182975	−	0.999833i	$-0.505825\pi$
$233$	−0.558599	−0.0365950	−0.0182975	+	0.999833i	$0.505825\pi$
$234$	0	0
$235$	−4.25386	−0.277491
$236$	−3.22790	−0.210118
$237$	0	0
$238$	0.783101	0.0507609
$239$	19.0720	1.23366	0.616832	−	0.787095i	$-0.288416\pi$
$239$	19.0720	1.23366	0.616832	+	0.787095i	$0.288416\pi$
$240$	0	0
$241$	−19.6475	−1.26561	−0.632805	−	0.774311i	$-0.718096\pi$
$241$	−19.6475	−1.26561	−0.632805	+	0.774311i	$0.718096\pi$
$242$	−9.40343	−0.604476
$243$	0	0
$244$	−13.1399	−0.841197
$245$	72.2084	4.61323
$246$	0	0
$247$	−17.4808	−1.11228
$248$	−1.26824	−0.0805331
$249$	0	0
$250$	−18.0712	−1.14292
$251$	−19.1101	−1.20622	−0.603111	−	0.797657i	$-0.706072\pi$
$251$	−19.1101	−1.20622	−0.603111	+	0.797657i	$0.706072\pi$
$252$	0	0
$253$	0	0
$254$	11.1093	0.697056
$255$	0	0
$256$	1.00000	0.0625000
$257$	25.8927	1.61514	0.807572	−	0.589769i	$-0.200781\pi$
$257$	25.8927	1.61514	0.807572	+	0.589769i	$0.200781\pi$
$258$	0	0
$259$	−22.6550	−1.40772
$260$	−12.7704	−0.791988
$261$	0	0
$262$	2.08032	0.128523
$263$	20.1945	1.24525	0.622623	−	0.782522i	$-0.286067\pi$
$263$	20.1945	1.24525	0.622623	+	0.782522i	$0.286067\pi$
$264$	0	0
$265$	16.4337	1.00951
$266$	−26.6817	−1.63596
$267$	0	0
$268$	−10.4839	−0.640409
$269$	−27.6781	−1.68756	−0.843780	−	0.536689i	$-0.819675\pi$
$269$	−27.6781	−1.68756	−0.843780	+	0.536689i	$0.819675\pi$
$270$	0	0
$271$	−9.58260	−0.582101	−0.291051	−	0.956708i	$-0.594005\pi$
$271$	−9.58260	−0.582101	−0.291051	+	0.956708i	$0.594005\pi$
$272$	−0.154095	−0.00934338
$273$	0	0
$274$	−16.5855	−1.00197
$275$	43.8670	2.64528
$276$	0	0
$277$	20.7038	1.24397	0.621986	−	0.783028i	$-0.286326\pi$
$277$	20.7038	1.24397	0.621986	+	0.783028i	$0.286326\pi$
$278$	−4.74700	−0.284706
$279$	0	0
$280$	−19.4920	−1.16487
$281$	−27.0825	−1.61561	−0.807803	−	0.589452i	$-0.799344\pi$
$281$	−27.0825	−1.61561	−0.807803	+	0.589452i	$0.799344\pi$
$282$	0	0
$283$	−18.1337	−1.07794	−0.538968	−	0.842326i	$-0.681186\pi$
$283$	−18.1337	−1.07794	−0.538968	+	0.842326i	$0.681186\pi$
$284$	−14.2586	−0.846095
$285$	0	0
$286$	15.0393	0.889295
$287$	−11.2388	−0.663407
$288$	0	0
$289$	−16.9763	−0.998603
$290$	−20.0745	−1.17882
$291$	0	0
$292$	6.20430	0.363079
$293$	22.2447	1.29955	0.649775	−	0.760126i	$-0.274863\pi$
$293$	22.2447	1.29955	0.649775	+	0.760126i	$0.274863\pi$
$294$	0	0
$295$	−12.3808	−0.720837
$296$	4.45795	0.259113
$297$	0	0
$298$	17.0234	0.986137
$299$	0	0
$300$	0	0
$301$	−13.4095	−0.772912
$302$	7.05024	0.405696
$303$	0	0
$304$	5.25031	0.301126
$305$	−50.3989	−2.88583
$306$	0	0
$307$	−11.0626	−0.631377	−0.315689	−	0.948863i	$-0.602236\pi$
$307$	−11.0626	−0.631377	−0.315689	+	0.948863i	$0.602236\pi$
$308$	22.9552	1.30799
$309$	0	0
$310$	−4.86439	−0.276279
$311$	9.31522	0.528218	0.264109	−	0.964493i	$-0.414922\pi$
$311$	9.31522	0.528218	0.264109	+	0.964493i	$0.414922\pi$
$312$	0	0
$313$	−24.9866	−1.41233	−0.706163	−	0.708049i	$-0.749576\pi$
$313$	−24.9866	−1.41233	−0.706163	+	0.708049i	$0.749576\pi$
$314$	6.11470	0.345073
$315$	0	0
$316$	−2.91689	−0.164088
$317$	−17.8811	−1.00430	−0.502152	−	0.864779i	$-0.667458\pi$
$317$	−17.8811	−1.00430	−0.502152	+	0.864779i	$0.667458\pi$
$318$	0	0
$319$	23.6411	1.32365
$320$	3.83556	0.214414
$321$	0	0
$322$	0	0
$323$	−0.809046	−0.0450165
$324$	0	0
$325$	−32.3343	−1.79358
$326$	6.00203	0.332422
$327$	0	0
$328$	2.21152	0.122111
$329$	−5.63617	−0.310732
$330$	0	0
$331$	−4.40749	−0.242258	−0.121129	−	0.992637i	$-0.538651\pi$
$331$	−4.40749	−0.242258	−0.121129	+	0.992637i	$0.538651\pi$
$332$	2.33395	0.128092
$333$	0	0
$334$	−12.4878	−0.683304
$335$	−40.2118	−2.19700
$336$	0	0
$337$	13.5323	0.737153	0.368577	−	0.929597i	$-0.379845\pi$
$337$	13.5323	0.737153	0.368577	+	0.929597i	$0.379845\pi$
$338$	1.91453	0.104136
$339$	0	0
$340$	−0.591040	−0.0320537
$341$	5.72865	0.310224
$342$	0	0
$343$	60.0993	3.24506
$344$	2.63867	0.142267
$345$	0	0
$346$	3.79092	0.203801
$347$	18.5971	0.998345	0.499173	−	0.866503i	$-0.333637\pi$
$347$	18.5971	0.998345	0.499173	+	0.866503i	$0.333637\pi$
$348$	0	0
$349$	0.177691	0.00951160	0.00475580	−	0.999989i	$-0.498486\pi$
$349$	0.177691	0.00951160	0.00475580	+	0.999989i	$0.498486\pi$
$350$	−49.3532	−2.63804
$351$	0	0
$352$	−4.51702	−0.240758
$353$	−13.5394	−0.720632	−0.360316	−	0.932830i	$-0.617331\pi$
$353$	−13.5394	−0.720632	−0.360316	+	0.932830i	$0.617331\pi$
$354$	0	0
$355$	−54.6898	−2.90264
$356$	5.50801	0.291924
$357$	0	0
$358$	−0.453414	−0.0239637
$359$	16.6948	0.881117	0.440558	−	0.897724i	$-0.354781\pi$
$359$	16.6948	0.881117	0.440558	+	0.897724i	$0.354781\pi$
$360$	0	0
$361$	8.56572	0.450827
$362$	−7.83553	−0.411826
$363$	0	0
$364$	−16.9202	−0.886861
$365$	23.7969	1.24559
$366$	0	0
$367$	28.8815	1.50760	0.753800	−	0.657104i	$-0.228219\pi$
$367$	28.8815	1.50760	0.753800	+	0.657104i	$0.228219\pi$
$368$	0	0
$369$	0	0
$370$	17.0987	0.888921
$371$	21.7739	1.13044
$372$	0	0
$373$	−31.5934	−1.63585	−0.817923	−	0.575327i	$-0.804875\pi$
$373$	−31.5934	−1.63585	−0.817923	+	0.575327i	$0.804875\pi$
$374$	0.696050	0.0359919
$375$	0	0
$376$	1.10906	0.0571954
$377$	−17.4258	−0.897477
$378$	0	0
$379$	−14.6640	−0.753237	−0.376618	−	0.926369i	$-0.622913\pi$
$379$	−14.6640	−0.753237	−0.376618	+	0.926369i	$0.622913\pi$
$380$	20.1378	1.03305
$381$	0	0
$382$	15.3660	0.786190
$383$	−6.99286	−0.357318	−0.178659	−	0.983911i	$-0.557176\pi$
$383$	−6.99286	−0.357318	−0.178659	+	0.983911i	$0.557176\pi$
$384$	0	0
$385$	88.0459	4.48723
$386$	−8.77768	−0.446772
$387$	0	0
$388$	−13.1549	−0.667837
$389$	−26.8167	−1.35966	−0.679830	−	0.733370i	$-0.737946\pi$
$389$	−26.8167	−1.35966	−0.679830	+	0.733370i	$0.737946\pi$
$390$	0	0
$391$	0	0
$392$	−18.8261	−0.950860
$393$	0	0
$394$	−8.01971	−0.404027
$395$	−11.1879	−0.562924
$396$	0	0
$397$	−7.49361	−0.376093	−0.188047	−	0.982160i	$-0.560216\pi$
$397$	−7.49361	−0.376093	−0.188047	+	0.982160i	$0.560216\pi$
$398$	6.20296	0.310926
$399$	0	0
$400$	9.71149	0.485575
$401$	−32.4616	−1.62106	−0.810528	−	0.585699i	$-0.800820\pi$
$401$	−32.4616	−1.62106	−0.810528	+	0.585699i	$0.800820\pi$
$402$	0	0
$403$	−4.22258	−0.210341
$404$	6.51687	0.324227
$405$	0	0
$406$	−26.5978	−1.32003
$407$	−20.1367	−0.998137
$408$	0	0
$409$	−12.4518	−0.615703	−0.307852	−	0.951434i	$-0.599610\pi$
$409$	−12.4518	−0.615703	−0.307852	+	0.951434i	$0.599610\pi$
$410$	8.48242	0.418917
$411$	0	0
$412$	5.64524	0.278121
$413$	−16.4040	−0.807186
$414$	0	0
$415$	8.95201	0.439437
$416$	3.32949	0.163241
$417$	0	0
$418$	−23.7157	−1.15997
$419$	−37.6239	−1.83805	−0.919025	−	0.394199i	$-0.871022\pi$
$419$	−37.6239	−1.83805	−0.919025	+	0.394199i	$0.871022\pi$
$420$	0	0
$421$	29.4657	1.43607	0.718035	−	0.696007i	$-0.245042\pi$
$421$	29.4657	1.43607	0.718035	+	0.696007i	$0.245042\pi$
$422$	16.1371	0.785541
$423$	0	0
$424$	−4.28457	−0.208077
$425$	−1.49649	−0.0725906
$426$	0	0
$427$	−66.7762	−3.23153
$428$	6.79406	0.328404
$429$	0	0
$430$	10.1208	0.488066
$431$	−26.4006	−1.27167	−0.635836	−	0.771824i	$-0.719345\pi$
$431$	−26.4006	−1.27167	−0.635836	+	0.771824i	$0.719345\pi$
$432$	0	0
$433$	−4.67389	−0.224613	−0.112306	−	0.993674i	$-0.535824\pi$
$433$	−4.67389	−0.224613	−0.112306	+	0.993674i	$0.535824\pi$
$434$	−6.44510	−0.309374
$435$	0	0
$436$	5.83285	0.279343
$437$	0	0
$438$	0	0
$439$	−18.9955	−0.906605	−0.453303	−	0.891357i	$-0.649754\pi$
$439$	−18.9955	−0.906605	−0.453303	+	0.891357i	$0.649754\pi$
$440$	−17.3253	−0.825950
$441$	0	0
$442$	−0.513057	−0.0244036
$443$	6.95701	0.330537	0.165269	−	0.986249i	$-0.447151\pi$
$443$	6.95701	0.330537	0.165269	+	0.986249i	$0.447151\pi$
$444$	0	0
$445$	21.1263	1.00148
$446$	9.15732	0.433612
$447$	0	0
$448$	5.08193	0.240099
$449$	19.2356	0.907784	0.453892	−	0.891057i	$-0.350035\pi$
$449$	19.2356	0.907784	0.453892	+	0.891057i	$0.350035\pi$
$450$	0	0
$451$	−9.98949	−0.470387
$452$	14.6584	0.689473
$453$	0	0
$454$	−9.40435	−0.441368
$455$	−64.8985	−3.04249
$456$	0	0
$457$	−14.8469	−0.694507	−0.347253	−	0.937771i	$-0.612886\pi$
$457$	−14.8469	−0.694507	−0.347253	+	0.937771i	$0.612886\pi$
$458$	3.61058	0.168711
$459$	0	0
$460$	0	0
$461$	−5.15752	−0.240209	−0.120105	−	0.992761i	$-0.538323\pi$
$461$	−5.15752	−0.240209	−0.120105	+	0.992761i	$0.538323\pi$
$462$	0	0
$463$	−18.4681	−0.858284	−0.429142	−	0.903237i	$-0.641184\pi$
$463$	−18.4681	−0.858284	−0.429142	+	0.903237i	$0.641184\pi$
$464$	5.23379	0.242973
$465$	0	0
$466$	0.558599	0.0258766
$467$	33.2249	1.53747	0.768733	−	0.639570i	$-0.220888\pi$
$467$	33.2249	1.53747	0.768733	+	0.639570i	$0.220888\pi$
$468$	0	0
$469$	−53.2787	−2.46018
$470$	4.25386	0.196216
$471$	0	0
$472$	3.22790	0.148576
$473$	−11.9189	−0.548031
$474$	0	0
$475$	50.9883	2.33950
$476$	−0.783101	−0.0358934
$477$	0	0
$478$	−19.0720	−0.872332
$479$	20.1872	0.922375	0.461187	−	0.887303i	$-0.347424\pi$
$479$	20.1872	0.922375	0.461187	+	0.887303i	$0.347424\pi$
$480$	0	0
$481$	14.8427	0.676769
$482$	19.6475	0.894921
$483$	0	0
$484$	9.40343	0.427429
$485$	−50.4562	−2.29110
$486$	0	0
$487$	−1.03341	−0.0468284	−0.0234142	−	0.999726i	$-0.507454\pi$
$487$	−1.03341	−0.0468284	−0.0234142	+	0.999726i	$0.507454\pi$
$488$	13.1399	0.594816
$489$	0	0
$490$	−72.2084	−3.26204
$491$	8.77567	0.396040	0.198020	−	0.980198i	$-0.436549\pi$
$491$	8.77567	0.396040	0.198020	+	0.980198i	$0.436549\pi$
$492$	0	0
$493$	−0.806502	−0.0363230
$494$	17.4808	0.786499
$495$	0	0
$496$	1.26824	0.0569455
$497$	−72.4615	−3.25034
$498$	0	0
$499$	32.9855	1.47663	0.738316	−	0.674455i	$-0.235621\pi$
$499$	32.9855	1.47663	0.738316	+	0.674455i	$0.235621\pi$
$500$	18.0712	0.808168
$501$	0	0
$502$	19.1101	0.852927
$503$	43.2295	1.92751	0.963754	−	0.266794i	$-0.0859642\pi$
$503$	43.2295	1.92751	0.963754	+	0.266794i	$0.0859642\pi$
$504$	0	0
$505$	24.9958	1.11230
$506$	0	0
$507$	0	0
$508$	−11.1093	−0.492893
$509$	23.1012	1.02394	0.511972	−	0.859002i	$-0.328915\pi$
$509$	23.1012	1.02394	0.511972	+	0.859002i	$0.328915\pi$
$510$	0	0
$511$	31.5298	1.39480
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−25.8927	−1.14208
$515$	21.6526	0.954129
$516$	0	0
$517$	−5.00964	−0.220324
$518$	22.6550	0.995405
$519$	0	0
$520$	12.7704	0.560020
$521$	15.6277	0.684660	0.342330	−	0.939580i	$-0.388784\pi$
$521$	15.6277	0.684660	0.342330	+	0.939580i	$0.388784\pi$
$522$	0	0
$523$	−21.8417	−0.955070	−0.477535	−	0.878613i	$-0.658470\pi$
$523$	−21.8417	−0.955070	−0.477535	+	0.878613i	$0.658470\pi$
$524$	−2.08032	−0.0908793
$525$	0	0
$526$	−20.1945	−0.880521
$527$	−0.195429	−0.00851302
$528$	0	0
$529$	0	0
$530$	−16.4337	−0.713834
$531$	0	0
$532$	26.6817	1.15680
$533$	7.36324	0.318937
$534$	0	0
$535$	26.0590	1.12663
$536$	10.4839	0.452837
$537$	0	0
$538$	27.6781	1.19329
$539$	85.0376	3.66283
$540$	0	0
$541$	27.2889	1.17324	0.586620	−	0.809862i	$-0.300458\pi$
$541$	27.2889	1.17324	0.586620	+	0.809862i	$0.300458\pi$
$542$	9.58260	0.411608
$543$	0	0
$544$	0.154095	0.00660677
$545$	22.3722	0.958320
$546$	0	0
$547$	−4.17954	−0.178704	−0.0893521	−	0.996000i	$-0.528480\pi$
$547$	−4.17954	−0.178704	−0.0893521	+	0.996000i	$0.528480\pi$
$548$	16.5855	0.708496
$549$	0	0
$550$	−43.8670	−1.87049
$551$	27.4790	1.17065
$552$	0	0
$553$	−14.8234	−0.630356
$554$	−20.7038	−0.879621
$555$	0	0
$556$	4.74700	0.201318
$557$	24.6304	1.04362	0.521812	−	0.853060i	$-0.325256\pi$
$557$	24.6304	1.04362	0.521812	+	0.853060i	$0.325256\pi$
$558$	0	0
$559$	8.78540	0.371583
$560$	19.4920	0.823689
$561$	0	0
$562$	27.0825	1.14241
$563$	7.92169	0.333859	0.166930	−	0.985969i	$-0.446615\pi$
$563$	7.92169	0.333859	0.166930	+	0.985969i	$0.446615\pi$
$564$	0	0
$565$	56.2231	2.36532
$566$	18.1337	0.762216
$567$	0	0
$568$	14.2586	0.598280
$569$	−28.0564	−1.17618	−0.588092	−	0.808794i	$-0.700121\pi$
$569$	−28.0564	−1.17618	−0.588092	+	0.808794i	$0.700121\pi$
$570$	0	0
$571$	−33.4816	−1.40116	−0.700582	−	0.713572i	$-0.747076\pi$
$571$	−33.4816	−1.40116	−0.700582	+	0.713572i	$0.747076\pi$
$572$	−15.0393	−0.628826
$573$	0	0
$574$	11.2388	0.469099
$575$	0	0
$576$	0	0
$577$	26.8994	1.11984	0.559919	−	0.828548i	$-0.310832\pi$
$577$	26.8994	1.11984	0.559919	+	0.828548i	$0.310832\pi$
$578$	16.9763	0.706119
$579$	0	0
$580$	20.0745	0.833549
$581$	11.8610	0.492077
$582$	0	0
$583$	19.3535	0.801538
$584$	−6.20430	−0.256736
$585$	0	0
$586$	−22.2447	−0.918921
$587$	27.8283	1.14860	0.574298	−	0.818647i	$-0.305275\pi$
$587$	27.8283	1.14860	0.574298	+	0.818647i	$0.305275\pi$
$588$	0	0
$589$	6.65863	0.274364
$590$	12.3808	0.509709
$591$	0	0
$592$	−4.45795	−0.183221
$593$	−25.8705	−1.06237	−0.531187	−	0.847255i	$-0.678254\pi$
$593$	−25.8705	−1.06237	−0.531187	+	0.847255i	$0.678254\pi$
$594$	0	0
$595$	−3.00363	−0.123137
$596$	−17.0234	−0.697304
$597$	0	0
$598$	0	0
$599$	22.3667	0.913878	0.456939	−	0.889498i	$-0.348946\pi$
$599$	22.3667	0.913878	0.456939	+	0.889498i	$0.348946\pi$
$600$	0	0
$601$	−27.9093	−1.13844	−0.569221	−	0.822184i	$-0.692755\pi$
$601$	−27.9093	−1.13844	−0.569221	+	0.822184i	$0.692755\pi$
$602$	13.4095	0.546532
$603$	0	0
$604$	−7.05024	−0.286870
$605$	36.0674	1.46635
$606$	0	0
$607$	−11.1033	−0.450670	−0.225335	−	0.974281i	$-0.572348\pi$
$607$	−11.1033	−0.450670	−0.225335	+	0.974281i	$0.572348\pi$
$608$	−5.25031	−0.212928
$609$	0	0
$610$	50.3989	2.04059
$611$	3.69260	0.149387
$612$	0	0
$613$	2.48067	0.100193	0.0500966	−	0.998744i	$-0.484047\pi$
$613$	2.48067	0.100193	0.0500966	+	0.998744i	$0.484047\pi$
$614$	11.0626	0.446451
$615$	0	0
$616$	−22.9552	−0.924891
$617$	−18.2825	−0.736025	−0.368012	−	0.929821i	$-0.619962\pi$
$617$	−18.2825	−0.736025	−0.368012	+	0.929821i	$0.619962\pi$
$618$	0	0
$619$	−32.6082	−1.31064	−0.655318	−	0.755353i	$-0.727465\pi$
$619$	−32.6082	−1.31064	−0.655318	+	0.755353i	$0.727465\pi$
$620$	4.86439	0.195359
$621$	0	0
$622$	−9.31522	−0.373506
$623$	27.9913	1.12145
$624$	0	0
$625$	20.7556	0.830225
$626$	24.9866	0.998666
$627$	0	0
$628$	−6.11470	−0.244003
$629$	0.686949	0.0273904
$630$	0	0
$631$	11.2094	0.446240	0.223120	−	0.974791i	$-0.428376\pi$
$631$	11.2094	0.446240	0.223120	+	0.974791i	$0.428376\pi$
$632$	2.91689	0.116028
$633$	0	0
$634$	17.8811	0.710150
$635$	−42.6102	−1.69093
$636$	0	0
$637$	−62.6811	−2.48352
$638$	−23.6411	−0.935961
$639$	0	0
$640$	−3.83556	−0.151614
$641$	5.46480	0.215847	0.107923	−	0.994159i	$-0.465580\pi$
$641$	5.46480	0.215847	0.107923	+	0.994159i	$0.465580\pi$
$642$	0	0
$643$	−13.2191	−0.521310	−0.260655	−	0.965432i	$-0.583939\pi$
$643$	−13.2191	−0.521310	−0.260655	+	0.965432i	$0.583939\pi$
$644$	0	0
$645$	0	0
$646$	0.809046	0.0318315
$647$	−18.6775	−0.734289	−0.367144	−	0.930164i	$-0.619664\pi$
$647$	−18.6775	−0.734289	−0.367144	+	0.930164i	$0.619664\pi$
$648$	0	0
$649$	−14.5805	−0.572333
$650$	32.3343	1.26825
$651$	0	0
$652$	−6.00203	−0.235058
$653$	7.33246	0.286941	0.143471	−	0.989655i	$-0.454174\pi$
$653$	7.33246	0.286941	0.143471	+	0.989655i	$0.454174\pi$
$654$	0	0
$655$	−7.97919	−0.311773
$656$	−2.21152	−0.0863455
$657$	0	0
$658$	5.63617	0.219721
$659$	−44.6806	−1.74051	−0.870255	−	0.492602i	$-0.836046\pi$
$659$	−44.6806	−1.74051	−0.870255	+	0.492602i	$0.836046\pi$
$660$	0	0
$661$	−20.9982	−0.816736	−0.408368	−	0.912817i	$-0.633902\pi$
$661$	−20.9982	−0.816736	−0.408368	+	0.912817i	$0.633902\pi$
$662$	4.40749	0.171302
$663$	0	0
$664$	−2.33395	−0.0905750
$665$	102.339	3.96854
$666$	0	0
$667$	0	0
$668$	12.4878	0.483169
$669$	0	0
$670$	40.2118	1.55352
$671$	−59.3532	−2.29131
$672$	0	0
$673$	28.0107	1.07973	0.539867	−	0.841750i	$-0.318474\pi$
$673$	28.0107	1.07973	0.539867	+	0.841750i	$0.318474\pi$
$674$	−13.5323	−0.521246
$675$	0	0
$676$	−1.91453	−0.0736356
$677$	−43.2019	−1.66038	−0.830192	−	0.557477i	$-0.811769\pi$
$677$	−43.2019	−1.66038	−0.830192	+	0.557477i	$0.811769\pi$
$678$	0	0
$679$	−66.8522	−2.56555
$680$	0.591040	0.0226654
$681$	0	0
$682$	−5.72865	−0.219361
$683$	23.1421	0.885507	0.442754	−	0.896643i	$-0.354002\pi$
$683$	23.1421	0.885507	0.442754	+	0.896643i	$0.354002\pi$
$684$	0	0
$685$	63.6145	2.43059
$686$	−60.0993	−2.29460
$687$	0	0
$688$	−2.63867	−0.100598
$689$	−14.2654	−0.543469
$690$	0	0
$691$	31.0298	1.18043	0.590215	−	0.807246i	$-0.299043\pi$
$691$	31.0298	1.18043	0.590215	+	0.807246i	$0.299043\pi$
$692$	−3.79092	−0.144109
$693$	0	0
$694$	−18.5971	−0.705937
$695$	18.2074	0.690646
$696$	0	0
$697$	0.340785	0.0129081
$698$	−0.177691	−0.00672572
$699$	0	0
$700$	49.3532	1.86537
$701$	−13.3405	−0.503863	−0.251931	−	0.967745i	$-0.581066\pi$
$701$	−13.3405	−0.503863	−0.251931	+	0.967745i	$0.581066\pi$
$702$	0	0
$703$	−23.4056	−0.882760
$704$	4.51702	0.170241
$705$	0	0
$706$	13.5394	0.509564
$707$	33.1183	1.24554
$708$	0	0
$709$	43.0894	1.61826	0.809128	−	0.587632i	$-0.199940\pi$
$709$	43.0894	1.61826	0.809128	+	0.587632i	$0.199940\pi$
$710$	54.6898	2.05247
$711$	0	0
$712$	−5.50801	−0.206421
$713$	0	0
$714$	0	0
$715$	−57.6842	−2.15727
$716$	0.453414	0.0169449
$717$	0	0
$718$	−16.6948	−0.623043
$719$	−20.4222	−0.761618	−0.380809	−	0.924654i	$-0.624354\pi$
$719$	−20.4222	−0.761618	−0.380809	+	0.924654i	$0.624354\pi$
$720$	0	0
$721$	28.6887	1.06842
$722$	−8.56572	−0.318783
$723$	0	0
$724$	7.83553	0.291205
$725$	50.8280	1.88770
$726$	0	0
$727$	−14.8691	−0.551463	−0.275732	−	0.961235i	$-0.588920\pi$
$727$	−14.8691	−0.551463	−0.275732	+	0.961235i	$0.588920\pi$
$728$	16.9202	0.627105
$729$	0	0
$730$	−23.7969	−0.880764
$731$	0.406605	0.0150388
$732$	0	0
$733$	26.4036	0.975241	0.487620	−	0.873056i	$-0.337865\pi$
$733$	26.4036	0.975241	0.487620	+	0.873056i	$0.337865\pi$
$734$	−28.8815	−1.06603
$735$	0	0
$736$	0	0
$737$	−47.3561	−1.74439
$738$	0	0
$739$	−8.98205	−0.330410	−0.165205	−	0.986259i	$-0.552829\pi$
$739$	−8.98205	−0.330410	−0.165205	+	0.986259i	$0.552829\pi$
$740$	−17.0987	−0.628562
$741$	0	0
$742$	−21.7739	−0.799345
$743$	−2.40734	−0.0883169	−0.0441584	−	0.999025i	$-0.514061\pi$
$743$	−2.40734	−0.0883169	−0.0441584	+	0.999025i	$0.514061\pi$
$744$	0	0
$745$	−65.2941	−2.39219
$746$	31.5934	1.15672
$747$	0	0
$748$	−0.696050	−0.0254501
$749$	34.5270	1.26159
$750$	0	0
$751$	28.7899	1.05056	0.525279	−	0.850930i	$-0.323961\pi$
$751$	28.7899	1.05056	0.525279	+	0.850930i	$0.323961\pi$
$752$	−1.10906	−0.0404433
$753$	0	0
$754$	17.4258	0.634612
$755$	−27.0416	−0.984145
$756$	0	0
$757$	33.9406	1.23359	0.616796	−	0.787123i	$-0.288430\pi$
$757$	33.9406	1.23359	0.616796	+	0.787123i	$0.288430\pi$
$758$	14.6640	0.532619
$759$	0	0
$760$	−20.1378	−0.730476
$761$	33.7946	1.22505	0.612526	−	0.790450i	$-0.290153\pi$
$761$	33.7946	1.22505	0.612526	+	0.790450i	$0.290153\pi$
$762$	0	0
$763$	29.6421	1.07312
$764$	−15.3660	−0.555921
$765$	0	0
$766$	6.99286	0.252662
$767$	10.7472	0.388060
$768$	0	0
$769$	−52.8036	−1.90415	−0.952073	−	0.305872i	$-0.901052\pi$
$769$	−52.8036	−1.90415	−0.952073	+	0.305872i	$0.901052\pi$
$770$	−88.0459	−3.17295
$771$	0	0
$772$	8.77768	0.315915
$773$	−1.46268	−0.0526089	−0.0263045	−	0.999654i	$-0.508374\pi$
$773$	−1.46268	−0.0526089	−0.0263045	+	0.999654i	$0.508374\pi$
$774$	0	0
$775$	12.3165	0.442421
$776$	13.1549	0.472232
$777$	0	0
$778$	26.8167	0.961425
$779$	−11.6112	−0.416014
$780$	0	0
$781$	−64.4065	−2.30465
$782$	0	0
$783$	0	0
$784$	18.8261	0.672359
$785$	−23.4533	−0.837084
$786$	0	0
$787$	17.3652	0.619002	0.309501	−	0.950899i	$-0.399838\pi$
$787$	17.3652	0.619002	0.309501	+	0.950899i	$0.399838\pi$
$788$	8.01971	0.285690
$789$	0	0
$790$	11.1879	0.398047
$791$	74.4930	2.64867
$792$	0	0
$793$	43.7492	1.55358
$794$	7.49361	0.265938
$795$	0	0
$796$	−6.20296	−0.219858
$797$	52.5218	1.86042	0.930209	−	0.367029i	$-0.119625\pi$
$797$	52.5218	1.86042	0.930209	+	0.367029i	$0.119625\pi$
$798$	0	0
$799$	0.170901	0.00604603
$800$	−9.71149	−0.343353
$801$	0	0
$802$	32.4616	1.14626
$803$	28.0249	0.988978
$804$	0	0
$805$	0	0
$806$	4.22258	0.148734
$807$	0	0
$808$	−6.51687	−0.229263
$809$	−36.6261	−1.28770	−0.643852	−	0.765150i	$-0.722665\pi$
$809$	−36.6261	−1.28770	−0.643852	+	0.765150i	$0.722665\pi$
$810$	0	0
$811$	39.6571	1.39255	0.696274	−	0.717776i	$-0.254840\pi$
$811$	39.6571	1.39255	0.696274	+	0.717776i	$0.254840\pi$
$812$	26.5978	0.933400
$813$	0	0
$814$	20.1367	0.705789
$815$	−23.0211	−0.806395
$816$	0	0
$817$	−13.8538	−0.484683
$818$	12.4518	0.435368
$819$	0	0
$820$	−8.48242	−0.296219
$821$	−30.7368	−1.07272	−0.536362	−	0.843988i	$-0.680202\pi$
$821$	−30.7368	−1.07272	−0.536362	+	0.843988i	$0.680202\pi$
$822$	0	0
$823$	16.1882	0.564286	0.282143	−	0.959372i	$-0.408955\pi$
$823$	16.1882	0.564286	0.282143	+	0.959372i	$0.408955\pi$
$824$	−5.64524	−0.196661
$825$	0	0
$826$	16.4040	0.570767
$827$	−53.7834	−1.87023	−0.935115	−	0.354344i	$-0.884704\pi$
$827$	−53.7834	−1.87023	−0.935115	+	0.354344i	$0.884704\pi$
$828$	0	0
$829$	6.11983	0.212550	0.106275	−	0.994337i	$-0.466108\pi$
$829$	6.11983	0.212550	0.106275	+	0.994337i	$0.466108\pi$
$830$	−8.95201	−0.310729
$831$	0	0
$832$	−3.32949	−0.115429
$833$	−2.90100	−0.100514
$834$	0	0
$835$	47.8978	1.65757
$836$	23.7157	0.820225
$837$	0	0
$838$	37.6239	1.29970
$839$	32.1106	1.10858	0.554290	−	0.832323i	$-0.312990\pi$
$839$	32.1106	1.10858	0.554290	+	0.832323i	$0.312990\pi$
$840$	0	0
$841$	−1.60740	−0.0554274
$842$	−29.4657	−1.01545
$843$	0	0
$844$	−16.1371	−0.555461
$845$	−7.34327	−0.252616
$846$	0	0
$847$	47.7876	1.64200
$848$	4.28457	0.147133
$849$	0	0
$850$	1.49649	0.0513293
$851$	0	0
$852$	0	0
$853$	24.9558	0.854469	0.427234	−	0.904141i	$-0.359488\pi$
$853$	24.9558	0.854469	0.427234	+	0.904141i	$0.359488\pi$
$854$	66.7762	2.28503
$855$	0	0
$856$	−6.79406	−0.232216
$857$	−45.2936	−1.54720	−0.773599	−	0.633675i	$-0.781546\pi$
$857$	−45.2936	−1.54720	−0.773599	+	0.633675i	$0.781546\pi$
$858$	0	0
$859$	−56.5676	−1.93006	−0.965031	−	0.262136i	$-0.915573\pi$
$859$	−56.5676	−1.93006	−0.965031	+	0.262136i	$0.915573\pi$
$860$	−10.1208	−0.345115
$861$	0	0
$862$	26.4006	0.899208
$863$	11.1251	0.378701	0.189351	−	0.981910i	$-0.439362\pi$
$863$	11.1251	0.378701	0.189351	+	0.981910i	$0.439362\pi$
$864$	0	0
$865$	−14.5403	−0.494384
$866$	4.67389	0.158825
$867$	0	0
$868$	6.44510	0.218761
$869$	−13.1756	−0.446953
$870$	0	0
$871$	34.9061	1.18275
$872$	−5.83285	−0.197525
$873$	0	0
$874$	0	0
$875$	91.8366	3.10464
$876$	0	0
$877$	−40.0892	−1.35372	−0.676858	−	0.736114i	$-0.736659\pi$
$877$	−40.0892	−1.35372	−0.676858	+	0.736114i	$0.736659\pi$
$878$	18.9955	0.641067
$879$	0	0
$880$	17.3253	0.584035
$881$	44.2762	1.49170	0.745851	−	0.666113i	$-0.232043\pi$
$881$	44.2762	1.49170	0.745851	+	0.666113i	$0.232043\pi$
$882$	0	0
$883$	45.7486	1.53956	0.769781	−	0.638308i	$-0.220365\pi$
$883$	45.7486	1.53956	0.769781	+	0.638308i	$0.220365\pi$
$884$	0.513057	0.0172560
$885$	0	0
$886$	−6.95701	−0.233725
$887$	−1.14721	−0.0385195	−0.0192597	−	0.999815i	$-0.506131\pi$
$887$	−1.14721	−0.0385195	−0.0192597	+	0.999815i	$0.506131\pi$
$888$	0	0
$889$	−56.4565	−1.89349
$890$	−21.1263	−0.708154
$891$	0	0
$892$	−9.15732	−0.306610
$893$	−5.82291	−0.194856
$894$	0	0
$895$	1.73910	0.0581316
$896$	−5.08193	−0.169776
$897$	0	0
$898$	−19.2356	−0.641900
$899$	6.63769	0.221379
$900$	0	0
$901$	−0.660231	−0.0219955
$902$	9.98949	0.332614
$903$	0	0
$904$	−14.6584	−0.487531
$905$	30.0536	0.999016
$906$	0	0
$907$	1.31302	0.0435980	0.0217990	−	0.999762i	$-0.493061\pi$
$907$	1.31302	0.0435980	0.0217990	+	0.999762i	$0.493061\pi$
$908$	9.40435	0.312094
$909$	0	0
$910$	64.8985	2.15136
$911$	30.7937	1.02024	0.510120	−	0.860103i	$-0.329601\pi$
$911$	30.7937	1.02024	0.510120	+	0.860103i	$0.329601\pi$
$912$	0	0
$913$	10.5425	0.348906
$914$	14.8469	0.491090
$915$	0	0
$916$	−3.61058	−0.119297
$917$	−10.5721	−0.349120
$918$	0	0
$919$	−13.9421	−0.459906	−0.229953	−	0.973202i	$-0.573857\pi$
$919$	−13.9421	−0.459906	−0.229953	+	0.973202i	$0.573857\pi$
$920$	0	0
$921$	0	0
$922$	5.15752	0.169854
$923$	47.4740	1.56262
$924$	0	0
$925$	−43.2934	−1.42348
$926$	18.4681	0.606898
$927$	0	0
$928$	−5.23379	−0.171808
$929$	21.2473	0.697100	0.348550	−	0.937290i	$-0.386674\pi$
$929$	21.2473	0.697100	0.348550	+	0.937290i	$0.386674\pi$
$930$	0	0
$931$	98.8426	3.23944
$932$	−0.558599	−0.0182975
$933$	0	0
$934$	−33.2249	−1.08715
$935$	−2.66974	−0.0873098
$936$	0	0
$937$	5.87293	0.191860	0.0959302	−	0.995388i	$-0.469417\pi$
$937$	5.87293	0.191860	0.0959302	+	0.995388i	$0.469417\pi$
$938$	53.2787	1.73961
$939$	0	0
$940$	−4.25386	−0.138746
$941$	24.8884	0.811337	0.405669	−	0.914020i	$-0.367039\pi$
$941$	24.8884	0.811337	0.405669	+	0.914020i	$0.367039\pi$
$942$	0	0
$943$	0	0
$944$	−3.22790	−0.105059
$945$	0	0
$946$	11.9189	0.387517
$947$	49.0721	1.59463	0.797314	−	0.603564i	$-0.206253\pi$
$947$	49.0721	1.59463	0.797314	+	0.603564i	$0.206253\pi$
$948$	0	0
$949$	−20.6571	−0.670558
$950$	−50.9883	−1.65428
$951$	0	0
$952$	0.783101	0.0253804
$953$	28.0932	0.910027	0.455014	−	0.890484i	$-0.349634\pi$
$953$	28.0932	0.910027	0.455014	+	0.890484i	$0.349634\pi$
$954$	0	0
$955$	−58.9370	−1.90716
$956$	19.0720	0.616832
$957$	0	0
$958$	−20.1872	−0.652218
$959$	84.2863	2.72175
$960$	0	0
$961$	−29.3916	−0.948115
$962$	−14.8427	−0.478548
$963$	0	0
$964$	−19.6475	−0.632805
$965$	33.6673	1.08379
$966$	0	0
$967$	−25.6391	−0.824498	−0.412249	−	0.911071i	$-0.635257\pi$
$967$	−25.6391	−0.824498	−0.412249	+	0.911071i	$0.635257\pi$
$968$	−9.40343	−0.302238
$969$	0	0
$970$	50.4562	1.62005
$971$	−8.44711	−0.271081	−0.135540	−	0.990772i	$-0.543277\pi$
$971$	−8.44711	−0.271081	−0.135540	+	0.990772i	$0.543277\pi$
$972$	0	0
$973$	24.1240	0.773378
$974$	1.03341	0.0331126
$975$	0	0
$976$	−13.1399	−0.420599
$977$	37.8257	1.21015	0.605076	−	0.796168i	$-0.293143\pi$
$977$	37.8257	1.21015	0.605076	+	0.796168i	$0.293143\pi$
$978$	0	0
$979$	24.8798	0.795160
$980$	72.2084	2.30661
$981$	0	0
$982$	−8.77567	−0.280043
$983$	0.780538	0.0248953	0.0124477	−	0.999923i	$-0.496038\pi$
$983$	0.780538	0.0248953	0.0124477	+	0.999923i	$0.496038\pi$
$984$	0	0
$985$	30.7600	0.980096
$986$	0.806502	0.0256843
$987$	0	0
$988$	−17.4808	−0.556139
$989$	0	0
$990$	0	0
$991$	39.0952	1.24190	0.620950	−	0.783850i	$-0.286747\pi$
$991$	39.0952	1.24190	0.620950	+	0.783850i	$0.286747\pi$
$992$	−1.26824	−0.0402666
$993$	0	0
$994$	72.4615	2.29834
$995$	−23.7918	−0.754251
$996$	0	0
$997$	−40.9119	−1.29569	−0.647846	−	0.761771i	$-0.724330\pi$
$997$	−40.9119	−1.29569	−0.647846	+	0.761771i	$0.724330\pi$
$998$	−32.9855	−1.04414
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9522.2.a.ch.1.9		10
3.2	odd	2		9522.2.a.ci.1.2		10
23.5	odd	22		414.2.i.h.163.2	yes	20
23.14	odd	22		414.2.i.h.127.2	yes	20
23.22	odd	2		9522.2.a.cg.1.2		10
69.5	even	22		414.2.i.g.163.1	yes	20
69.14	even	22		414.2.i.g.127.1	✓	20
69.68	even	2		9522.2.a.cj.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.2.i.g.127.1	✓	20	69.14	even	22
414.2.i.g.163.1	yes	20	69.5	even	22
414.2.i.h.127.2	yes	20	23.14	odd	22
414.2.i.h.163.2	yes	20	23.5	odd	22
9522.2.a.cg.1.2		10	23.22	odd	2
9522.2.a.ch.1.9		10	1.1	even	1	trivial
9522.2.a.ci.1.2		10	3.2	odd	2
9522.2.a.cj.1.9		10	69.68	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.83556 q^{5} +5.08193 q^{7} -1.00000 q^{8} -3.83556 q^{10} +4.51702 q^{11} -3.32949 q^{13} -5.08193 q^{14} +1.00000 q^{16} -0.154095 q^{17} +5.25031 q^{19} +3.83556 q^{20} -4.51702 q^{22} +9.71149 q^{25} +3.32949 q^{26} +5.08193 q^{28} +5.23379 q^{29} +1.26824 q^{31} -1.00000 q^{32} +0.154095 q^{34} +19.4920 q^{35} -4.45795 q^{37} -5.25031 q^{38} -3.83556 q^{40} -2.21152 q^{41} -2.63867 q^{43} +4.51702 q^{44} -1.10906 q^{47} +18.8261 q^{49} -9.71149 q^{50} -3.32949 q^{52} +4.28457 q^{53} +17.3253 q^{55} -5.08193 q^{56} -5.23379 q^{58} -3.22790 q^{59} -13.1399 q^{61} -1.26824 q^{62} +1.00000 q^{64} -12.7704 q^{65} -10.4839 q^{67} -0.154095 q^{68} -19.4920 q^{70} -14.2586 q^{71} +6.20430 q^{73} +4.45795 q^{74} +5.25031 q^{76} +22.9552 q^{77} -2.91689 q^{79} +3.83556 q^{80} +2.21152 q^{82} +2.33395 q^{83} -0.591040 q^{85} +2.63867 q^{86} -4.51702 q^{88} +5.50801 q^{89} -16.9202 q^{91} +1.10906 q^{94} +20.1378 q^{95} -13.1549 q^{97} -18.8261 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{2} + 10 q^{4} + 10 q^{5} + 2 q^{7} - 10 q^{8} - 10 q^{10} + 12 q^{11} - 2 q^{14} + 10 q^{16} + 24 q^{17} - 8 q^{19} + 10 q^{20} - 12 q^{22} + 8 q^{25} + 2 q^{28} - 4 q^{29} + 18 q^{31} - 10 q^{32}+ \cdots - 36 q^{98}+O(q^{100}) \) 10 * q - 10 * q^2 + 10 * q^4 + 10 * q^5 + 2 * q^7 - 10 * q^8 - 10 * q^10 + 12 * q^11 - 2 * q^14 + 10 * q^16 + 24 * q^17 - 8 * q^19 + 10 * q^20 - 12 * q^22 + 8 * q^25 + 2 * q^28 - 4 * q^29 + 18 * q^31 - 10 * q^32 - 24 * q^34 + 24 * q^35 - 12 * q^37 + 8 * q^38 - 10 * q^40 - 28 * q^41 - 8 * q^43 + 12 * q^44 + 16 * q^47 + 36 * q^49 - 8 * q^50 + 34 * q^53 + 30 * q^55 - 2 * q^56 + 4 * q^58 + 22 * q^59 - 30 * q^61 - 18 * q^62 + 10 * q^64 + 36 * q^65 - 18 * q^67 + 24 * q^68 - 24 * q^70 + 28 * q^71 - 20 * q^73 + 12 * q^74 - 8 * q^76 + 20 * q^77 - 2 * q^79 + 10 * q^80 + 28 * q^82 + 44 * q^83 + 16 * q^85 + 8 * q^86 - 12 * q^88 + 44 * q^89 - 22 * q^91 - 16 * q^94 + 10 * q^95 - 18 * q^97 - 36 * q^98

Introduction

L-functions

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Database

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