LMFDB - Embedded newform 7623.2.a.da.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 22x^{10} + 181x^{8} - 692x^{6} + 1240x^{4} - 936x^{2} + 244 $ x^12 - 22x^10 + 181x^8 - 692x^6 + 1240x^4 - 936*x^2 + 244
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.09771$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.09771 q^{2} +2.40037 q^{4} +2.53213 q^{5} +1.00000 q^{7} -0.839859 q^{8} +O(q^{10})$	$q-2.09771 q^{2} +2.40037 q^{4} +2.53213 q^{5} +1.00000 q^{7} -0.839859 q^{8} -5.31167 q^{10} +5.69822 q^{13} -2.09771 q^{14} -3.03896 q^{16} +0.996504 q^{17} +6.02990 q^{19} +6.07805 q^{20} +1.25785 q^{23} +1.41169 q^{25} -11.9532 q^{26} +2.40037 q^{28} +1.33880 q^{29} -1.85065 q^{31} +8.05457 q^{32} -2.09037 q^{34} +2.53213 q^{35} +9.65942 q^{37} -12.6490 q^{38} -2.12663 q^{40} +10.0523 q^{41} -5.04597 q^{43} -2.63859 q^{46} +1.03023 q^{47} +1.00000 q^{49} -2.96131 q^{50} +13.6778 q^{52} -7.43975 q^{53} -0.839859 q^{56} -2.80840 q^{58} -6.55446 q^{59} +3.76506 q^{61} +3.88212 q^{62} -10.8182 q^{64} +14.4286 q^{65} +12.0485 q^{67} +2.39198 q^{68} -5.31167 q^{70} -15.8526 q^{71} +6.41645 q^{73} -20.2626 q^{74} +14.4740 q^{76} +10.8036 q^{79} -7.69506 q^{80} -21.0868 q^{82} -14.2491 q^{83} +2.52328 q^{85} +10.5850 q^{86} +15.4991 q^{89} +5.69822 q^{91} +3.01930 q^{92} -2.16112 q^{94} +15.2685 q^{95} -2.05898 q^{97} -2.09771 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 20 q^{4} + 12 q^{7}+O(q^{10}) $ 12 * q + 20 * q^4 + 12 * q^7	$ 12 q + 20 q^{4} + 12 q^{7} + 20 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 32 q^{25} + 20 q^{28} - 16 q^{31} - 24 q^{34} + 4 q^{37} + 48 q^{40} + 16 q^{43} + 24 q^{46} + 12 q^{49} + 96 q^{52} + 20 q^{58} + 44 q^{61} + 76 q^{64} + 20 q^{70} + 52 q^{73} - 8 q^{79} - 68 q^{82} + 72 q^{85} + 20 q^{91} + 20 q^{94} - 32 q^{97}+O(q^{100}) $ 12 * q + 20 * q^4 + 12 * q^7 + 20 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 32 * q^25 + 20 * q^28 - 16 * q^31 - 24 * q^34 + 4 * q^37 + 48 * q^40 + 16 * q^43 + 24 * q^46 + 12 * q^49 + 96 * q^52 + 20 * q^58 + 44 * q^61 + 76 * q^64 + 20 * q^70 + 52 * q^73 - 8 * q^79 - 68 * q^82 + 72 * q^85 + 20 * q^91 + 20 * q^94 - 32 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	−2.09771	−1.48330	−0.741651	−	0.670786i	$-0.765957\pi$
$2$	−2.09771	−1.48330	−0.741651	+	0.670786i	$0.765957\pi$
$3$	0	0
$3$	0	0
$4$	2.40037	1.20019
$5$	2.53213	1.13240	0.566202	−	0.824267i	$-0.308412\pi$
$5$	2.53213	1.13240	0.566202	+	0.824267i	$0.308412\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.839859	−0.296935
$9$	0	0
$10$	−5.31167	−1.67970
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.69822	1.58040	0.790201	−	0.612848i	$-0.209976\pi$
$13$	5.69822	1.58040	0.790201	+	0.612848i	$0.209976\pi$
$14$	−2.09771	−0.560635
$15$	0	0
$16$	−3.03896	−0.759741
$17$	0.996504	0.241688	0.120844	−	0.992672i	$-0.461440\pi$
$17$	0.996504	0.241688	0.120844	+	0.992672i	$0.461440\pi$
$18$	0	0
$19$	6.02990	1.38335	0.691677	−	0.722207i	$-0.256872\pi$
$19$	6.02990	1.38335	0.691677	+	0.722207i	$0.256872\pi$
$20$	6.07805	1.35909
$21$	0	0
$22$	0	0
$23$	1.25785	0.262279	0.131140	−	0.991364i	$-0.458136\pi$
$23$	1.25785	0.262279	0.131140	+	0.991364i	$0.458136\pi$
$24$	0	0
$25$	1.41169	0.282338
$26$	−11.9532	−2.34421
$27$	0	0
$28$	2.40037	0.453627
$29$	1.33880	0.248608	0.124304	−	0.992244i	$-0.460330\pi$
$29$	1.33880	0.248608	0.124304	+	0.992244i	$0.460330\pi$
$30$	0	0
$31$	−1.85065	−0.332387	−0.166193	−	0.986093i	$-0.553148\pi$
$31$	−1.85065	−0.332387	−0.166193	+	0.986093i	$0.553148\pi$
$32$	8.05457	1.42386
$33$	0	0
$34$	−2.09037	−0.358496
$35$	2.53213	0.428008
$36$	0	0
$37$	9.65942	1.58800	0.794000	−	0.607918i	$-0.207995\pi$
$37$	9.65942	1.58800	0.794000	+	0.607918i	$0.207995\pi$
$38$	−12.6490	−2.05193
$39$	0	0
$40$	−2.12663	−0.336250
$41$	10.0523	1.56991	0.784953	−	0.619555i	$-0.212687\pi$
$41$	10.0523	1.56991	0.784953	+	0.619555i	$0.212687\pi$
$42$	0	0
$43$	−5.04597	−0.769504	−0.384752	−	0.923020i	$-0.625713\pi$
$43$	−5.04597	−0.769504	−0.384752	+	0.923020i	$0.625713\pi$
$44$	0	0
$45$	0	0
$46$	−2.63859	−0.389039
$47$	1.03023	0.150274	0.0751371	−	0.997173i	$-0.476061\pi$
$47$	1.03023	0.150274	0.0751371	+	0.997173i	$0.476061\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.96131	−0.418793
$51$	0	0
$52$	13.6778	1.89677
$53$	−7.43975	−1.02193	−0.510964	−	0.859602i	$-0.670712\pi$
$53$	−7.43975	−1.02193	−0.510964	+	0.859602i	$0.670712\pi$
$54$	0	0
$55$	0	0
$56$	−0.839859	−0.112231
$57$	0	0
$58$	−2.80840	−0.368761
$59$	−6.55446	−0.853318	−0.426659	−	0.904413i	$-0.640310\pi$
$59$	−6.55446	−0.853318	−0.426659	+	0.904413i	$0.640310\pi$
$60$	0	0
$61$	3.76506	0.482066	0.241033	−	0.970517i	$-0.422514\pi$
$61$	3.76506	0.482066	0.241033	+	0.970517i	$0.422514\pi$
$62$	3.88212	0.493030
$63$	0	0
$64$	−10.8182	−1.35227
$65$	14.4286	1.78965
$66$	0	0
$67$	12.0485	1.47196	0.735978	−	0.677006i	$-0.236723\pi$
$67$	12.0485	1.47196	0.735978	+	0.677006i	$0.236723\pi$
$68$	2.39198	0.290070
$69$	0	0
$70$	−5.31167	−0.634866
$71$	−15.8526	−1.88136	−0.940678	−	0.339301i	$-0.889809\pi$
$71$	−15.8526	−1.88136	−0.940678	+	0.339301i	$0.889809\pi$
$72$	0	0
$73$	6.41645	0.750988	0.375494	−	0.926825i	$-0.377473\pi$
$73$	6.41645	0.750988	0.375494	+	0.926825i	$0.377473\pi$
$74$	−20.2626	−2.35548
$75$	0	0
$76$	14.4740	1.66028
$77$	0	0
$78$	0	0
$79$	10.8036	1.21550	0.607751	−	0.794128i	$-0.292072\pi$
$79$	10.8036	1.21550	0.607751	+	0.794128i	$0.292072\pi$
$80$	−7.69506	−0.860333
$81$	0	0
$82$	−21.0868	−2.32865
$83$	−14.2491	−1.56404	−0.782020	−	0.623253i	$-0.785811\pi$
$83$	−14.2491	−1.56404	−0.782020	+	0.623253i	$0.785811\pi$
$84$	0	0
$85$	2.52328	0.273688
$86$	10.5850	1.14141
$87$	0	0
$88$	0	0
$89$	15.4991	1.64290	0.821451	−	0.570279i	$-0.193165\pi$
$89$	15.4991	1.64290	0.821451	+	0.570279i	$0.193165\pi$
$90$	0	0
$91$	5.69822	0.597335
$92$	3.01930	0.314784
$93$	0	0
$94$	−2.16112	−0.222902
$95$	15.2685	1.56651
$96$	0	0
$97$	−2.05898	−0.209057	−0.104529	−	0.994522i	$-0.533333\pi$
$97$	−2.05898	−0.209057	−0.104529	+	0.994522i	$0.533333\pi$
$98$	−2.09771	−0.211900
$99$	0	0
$100$	3.38858	0.338858
$101$	4.79295	0.476916	0.238458	−	0.971153i	$-0.423358\pi$
$101$	4.79295	0.476916	0.238458	+	0.971153i	$0.423358\pi$
$102$	0	0
$103$	−7.54804	−0.743730	−0.371865	−	0.928287i	$-0.621282\pi$
$103$	−7.54804	−0.743730	−0.371865	+	0.928287i	$0.621282\pi$
$104$	−4.78570	−0.469276
$105$	0	0
$106$	15.6064	1.51583
$107$	0.498937	0.0482341	0.0241170	−	0.999709i	$-0.492323\pi$
$107$	0.498937	0.0482341	0.0241170	+	0.999709i	$0.492323\pi$
$108$	0	0
$109$	5.80710	0.556220	0.278110	−	0.960549i	$-0.410292\pi$
$109$	5.80710	0.556220	0.278110	+	0.960549i	$0.410292\pi$
$110$	0	0
$111$	0	0
$112$	−3.03896	−0.287155
$113$	−13.2820	−1.24947	−0.624733	−	0.780839i	$-0.714792\pi$
$113$	−13.2820	−1.24947	−0.624733	+	0.780839i	$0.714792\pi$
$114$	0	0
$115$	3.18503	0.297006
$116$	3.21361	0.298376
$117$	0	0
$118$	13.7493	1.26573
$119$	0.996504	0.0913494
$120$	0	0
$121$	0	0
$122$	−7.89798	−0.715049
$123$	0	0
$124$	−4.44225	−0.398926
$125$	−9.08607	−0.812683
$126$	0	0
$127$	−6.00223	−0.532611	−0.266306	−	0.963889i	$-0.585803\pi$
$127$	−6.00223	−0.532611	−0.266306	+	0.963889i	$0.585803\pi$
$128$	6.58425	0.581971
$129$	0	0
$130$	−30.2670	−2.65459
$131$	12.5395	1.09558	0.547792	−	0.836615i	$-0.315469\pi$
$131$	12.5395	1.09558	0.547792	+	0.836615i	$0.315469\pi$
$132$	0	0
$133$	6.02990	0.522858
$134$	−25.2742	−2.18335
$135$	0	0
$136$	−0.836923	−0.0717656
$137$	16.0274	1.36931	0.684657	−	0.728866i	$-0.259952\pi$
$137$	16.0274	1.36931	0.684657	+	0.728866i	$0.259952\pi$
$138$	0	0
$139$	8.31663	0.705407	0.352704	−	0.935735i	$-0.385262\pi$
$139$	8.31663	0.705407	0.352704	+	0.935735i	$0.385262\pi$
$140$	6.07805	0.513689
$141$	0	0
$142$	33.2541	2.79062
$143$	0	0
$144$	0	0
$145$	3.39001	0.281525
$146$	−13.4598	−1.11394
$147$	0	0
$148$	23.1862	1.90589
$149$	−9.91993	−0.812673	−0.406336	−	0.913724i	$-0.633194\pi$
$149$	−9.91993	−0.812673	−0.406336	+	0.913724i	$0.633194\pi$
$150$	0	0
$151$	−15.9065	−1.29445	−0.647227	−	0.762298i	$-0.724071\pi$
$151$	−15.9065	−1.29445	−0.647227	+	0.762298i	$0.724071\pi$
$152$	−5.06426	−0.410766
$153$	0	0
$154$	0	0
$155$	−4.68609	−0.376396
$156$	0	0
$157$	10.9327	0.872525	0.436262	−	0.899819i	$-0.356302\pi$
$157$	10.9327	0.872525	0.436262	+	0.899819i	$0.356302\pi$
$158$	−22.6628	−1.80296
$159$	0	0
$160$	20.3952	1.61238
$161$	1.25785	0.0991322
$162$	0	0
$163$	5.77018	0.451956	0.225978	−	0.974132i	$-0.427442\pi$
$163$	5.77018	0.451956	0.225978	+	0.974132i	$0.427442\pi$
$164$	24.1293	1.88418
$165$	0	0
$166$	29.8904	2.31995
$167$	−3.43598	−0.265884	−0.132942	−	0.991124i	$-0.542442\pi$
$167$	−3.43598	−0.265884	−0.132942	+	0.991124i	$0.542442\pi$
$168$	0	0
$169$	19.4697	1.49767
$170$	−5.29310	−0.405962
$171$	0	0
$172$	−12.1122	−0.923547
$173$	−16.6405	−1.26515	−0.632577	−	0.774498i	$-0.718003\pi$
$173$	−16.6405	−1.26515	−0.632577	+	0.774498i	$0.718003\pi$
$174$	0	0
$175$	1.41169	0.106714
$176$	0	0
$177$	0	0
$178$	−32.5126	−2.43692
$179$	−8.87894	−0.663643	−0.331821	−	0.943342i	$-0.607663\pi$
$179$	−8.87894	−0.663643	−0.331821	+	0.943342i	$0.607663\pi$
$180$	0	0
$181$	0.982518	0.0730300	0.0365150	−	0.999333i	$-0.488374\pi$
$181$	0.982518	0.0730300	0.0365150	+	0.999333i	$0.488374\pi$
$182$	−11.9532	−0.886029
$183$	0	0
$184$	−1.05641	−0.0778799
$185$	24.4589	1.79826
$186$	0	0
$187$	0	0
$188$	2.47293	0.180357
$189$	0	0
$190$	−32.0288	−2.32361
$191$	−19.0702	−1.37987	−0.689937	−	0.723869i	$-0.742362\pi$
$191$	−19.0702	−1.37987	−0.689937	+	0.723869i	$0.742362\pi$
$192$	0	0
$193$	4.47300	0.321973	0.160987	−	0.986957i	$-0.448532\pi$
$193$	4.47300	0.321973	0.160987	+	0.986957i	$0.448532\pi$
$194$	4.31913	0.310095
$195$	0	0
$196$	2.40037	0.171455
$197$	−9.54397	−0.679980	−0.339990	−	0.940429i	$-0.610424\pi$
$197$	−9.54397	−0.679980	−0.339990	+	0.940429i	$0.610424\pi$
$198$	0	0
$199$	22.7626	1.61360	0.806798	−	0.590828i	$-0.201199\pi$
$199$	22.7626	1.61360	0.806798	+	0.590828i	$0.201199\pi$
$200$	−1.18562	−0.0838361
$201$	0	0
$202$	−10.0542	−0.707410
$203$	1.33880	0.0939651
$204$	0	0
$205$	25.4538	1.77777
$206$	15.8336	1.10318
$207$	0	0
$208$	−17.3167	−1.20070
$209$	0	0
$210$	0	0
$211$	11.8991	0.819170	0.409585	−	0.912272i	$-0.365674\pi$
$211$	11.8991	0.819170	0.409585	+	0.912272i	$0.365674\pi$
$212$	−17.8582	−1.22650
$213$	0	0
$214$	−1.04662	−0.0715457
$215$	−12.7771	−0.871389
$216$	0	0
$217$	−1.85065	−0.125630
$218$	−12.1816	−0.825042
$219$	0	0
$220$	0	0
$221$	5.67830	0.381964
$222$	0	0
$223$	−13.9876	−0.936681	−0.468340	−	0.883548i	$-0.655148\pi$
$223$	−13.9876	−0.936681	−0.468340	+	0.883548i	$0.655148\pi$
$224$	8.05457	0.538169
$225$	0	0
$226$	27.8617	1.85334
$227$	19.1843	1.27331	0.636654	−	0.771149i	$-0.280318\pi$
$227$	19.1843	1.27331	0.636654	+	0.771149i	$0.280318\pi$
$228$	0	0
$229$	11.1268	0.735280	0.367640	−	0.929968i	$-0.380166\pi$
$229$	11.1268	0.735280	0.367640	+	0.929968i	$0.380166\pi$
$230$	−6.68127	−0.440550
$231$	0	0
$232$	−1.12440	−0.0738205
$233$	17.2671	1.13120	0.565602	−	0.824678i	$-0.308644\pi$
$233$	17.2671	1.13120	0.565602	+	0.824678i	$0.308644\pi$
$234$	0	0
$235$	2.60867	0.170171
$236$	−15.7331	−1.02414
$237$	0	0
$238$	−2.09037	−0.135499
$239$	5.85774	0.378906	0.189453	−	0.981890i	$-0.439329\pi$
$239$	5.85774	0.378906	0.189453	+	0.981890i	$0.439329\pi$
$240$	0	0
$241$	7.53402	0.485309	0.242655	−	0.970113i	$-0.421982\pi$
$241$	7.53402	0.485309	0.242655	+	0.970113i	$0.421982\pi$
$242$	0	0
$243$	0	0
$244$	9.03753	0.578568
$245$	2.53213	0.161772
$246$	0	0
$247$	34.3597	2.18625
$248$	1.55429	0.0986972
$249$	0	0
$250$	19.0599	1.20545
$251$	−1.32063	−0.0833572	−0.0416786	−	0.999131i	$-0.513271\pi$
$251$	−1.32063	−0.0833572	−0.0416786	+	0.999131i	$0.513271\pi$
$252$	0	0
$253$	0	0
$254$	12.5909	0.790024
$255$	0	0
$256$	7.82457	0.489036
$257$	0.240009	0.0149713	0.00748567	−	0.999972i	$-0.497617\pi$
$257$	0.240009	0.0149713	0.00748567	+	0.999972i	$0.497617\pi$
$258$	0	0
$259$	9.65942	0.600207
$260$	34.6341	2.14791
$261$	0	0
$262$	−26.3043	−1.62508
$263$	−25.8712	−1.59528	−0.797642	−	0.603131i	$-0.793920\pi$
$263$	−25.8712	−1.59528	−0.797642	+	0.603131i	$0.793920\pi$
$264$	0	0
$265$	−18.8384	−1.15724
$266$	−12.6490	−0.775557
$267$	0	0
$268$	28.9208	1.76662
$269$	−19.5864	−1.19420	−0.597102	−	0.802165i	$-0.703681\pi$
$269$	−19.5864	−1.19420	−0.597102	+	0.802165i	$0.703681\pi$
$270$	0	0
$271$	−20.2033	−1.22727	−0.613633	−	0.789592i	$-0.710292\pi$
$271$	−20.2033	−1.22727	−0.613633	+	0.789592i	$0.710292\pi$
$272$	−3.02834	−0.183620
$273$	0	0
$274$	−33.6208	−2.03111
$275$	0	0
$276$	0	0
$277$	−25.6837	−1.54318	−0.771592	−	0.636118i	$-0.780539\pi$
$277$	−25.6837	−1.54318	−0.771592	+	0.636118i	$0.780539\pi$
$278$	−17.4458	−1.04633
$279$	0	0
$280$	−2.12663	−0.127091
$281$	−28.3537	−1.69144	−0.845719	−	0.533628i	$-0.820828\pi$
$281$	−28.3537	−1.69144	−0.845719	+	0.533628i	$0.820828\pi$
$282$	0	0
$283$	−3.88445	−0.230907	−0.115453	−	0.993313i	$-0.536832\pi$
$283$	−3.88445	−0.230907	−0.115453	+	0.993313i	$0.536832\pi$
$284$	−38.0521	−2.25797
$285$	0	0
$286$	0	0
$287$	10.0523	0.593369
$288$	0	0
$289$	−16.0070	−0.941587
$290$	−7.11124	−0.417586
$291$	0	0
$292$	15.4018	0.901325
$293$	12.4915	0.729759	0.364879	−	0.931055i	$-0.381110\pi$
$293$	12.4915	0.729759	0.364879	+	0.931055i	$0.381110\pi$
$294$	0	0
$295$	−16.5968	−0.966301
$296$	−8.11255	−0.471533
$297$	0	0
$298$	20.8091	1.20544
$299$	7.16748	0.414506
$300$	0	0
$301$	−5.04597	−0.290845
$302$	33.3672	1.92007
$303$	0	0
$304$	−18.3246	−1.05099
$305$	9.53362	0.545893
$306$	0	0
$307$	10.6108	0.605592	0.302796	−	0.953055i	$-0.402080\pi$
$307$	10.6108	0.605592	0.302796	+	0.953055i	$0.402080\pi$
$308$	0	0
$309$	0	0
$310$	9.83004	0.558309
$311$	25.6921	1.45686	0.728432	−	0.685118i	$-0.240249\pi$
$311$	25.6921	1.45686	0.728432	+	0.685118i	$0.240249\pi$
$312$	0	0
$313$	−23.7556	−1.34274	−0.671372	−	0.741121i	$-0.734295\pi$
$313$	−23.7556	−1.34274	−0.671372	+	0.741121i	$0.734295\pi$
$314$	−22.9336	−1.29422
$315$	0	0
$316$	25.9327	1.45883
$317$	−27.9316	−1.56880	−0.784398	−	0.620258i	$-0.787028\pi$
$317$	−27.9316	−1.56880	−0.784398	+	0.620258i	$0.787028\pi$
$318$	0	0
$319$	0	0
$320$	−27.3931	−1.53132
$321$	0	0
$322$	−2.63859	−0.147043
$323$	6.00882	0.334340
$324$	0	0
$325$	8.04412	0.446208
$326$	−12.1042	−0.670387
$327$	0	0
$328$	−8.44252	−0.466160
$329$	1.03023	0.0567983
$330$	0	0
$331$	−10.6516	−0.585462	−0.292731	−	0.956195i	$-0.594564\pi$
$331$	−10.6516	−0.585462	−0.292731	+	0.956195i	$0.594564\pi$
$332$	−34.2031	−1.87714
$333$	0	0
$334$	7.20768	0.394386
$335$	30.5083	1.66685
$336$	0	0
$337$	−17.6414	−0.960990	−0.480495	−	0.876997i	$-0.659543\pi$
$337$	−17.6414	−0.960990	−0.480495	+	0.876997i	$0.659543\pi$
$338$	−40.8417	−2.22149
$339$	0	0
$340$	6.05681	0.328476
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	4.23791	0.228493
$345$	0	0
$346$	34.9069	1.87660
$347$	−31.7274	−1.70322	−0.851608	−	0.524180i	$-0.824372\pi$
$347$	−31.7274	−1.70322	−0.851608	+	0.524180i	$0.824372\pi$
$348$	0	0
$349$	24.2636	1.29880	0.649401	−	0.760447i	$-0.275020\pi$
$349$	24.2636	1.29880	0.649401	+	0.760447i	$0.275020\pi$
$350$	−2.96131	−0.158289
$351$	0	0
$352$	0	0
$353$	35.1150	1.86898	0.934491	−	0.355986i	$-0.115855\pi$
$353$	35.1150	1.86898	0.934491	+	0.355986i	$0.115855\pi$
$354$	0	0
$355$	−40.1408	−2.13045
$356$	37.2036	1.97179
$357$	0	0
$358$	18.6254	0.984383
$359$	−31.7004	−1.67308	−0.836541	−	0.547904i	$-0.815426\pi$
$359$	−31.7004	−1.67308	−0.836541	+	0.547904i	$0.815426\pi$
$360$	0	0
$361$	17.3597	0.913667
$362$	−2.06103	−0.108325
$363$	0	0
$364$	13.6778	0.716913
$365$	16.2473	0.850422
$366$	0	0
$367$	−6.48862	−0.338703	−0.169352	−	0.985556i	$-0.554167\pi$
$367$	−6.48862	−0.338703	−0.169352	+	0.985556i	$0.554167\pi$
$368$	−3.82255	−0.199264
$369$	0	0
$370$	−51.3077	−2.66736
$371$	−7.43975	−0.386253
$372$	0	0
$373$	24.8809	1.28828	0.644142	−	0.764906i	$-0.277215\pi$
$373$	24.8809	1.28828	0.644142	+	0.764906i	$0.277215\pi$
$374$	0	0
$375$	0	0
$376$	−0.865246	−0.0446217
$377$	7.62875	0.392901
$378$	0	0
$379$	−28.6289	−1.47057	−0.735284	−	0.677759i	$-0.762951\pi$
$379$	−28.6289	−1.47057	−0.735284	+	0.677759i	$0.762951\pi$
$380$	36.6500	1.88011
$381$	0	0
$382$	40.0038	2.04677
$383$	8.85605	0.452523	0.226261	−	0.974067i	$-0.427350\pi$
$383$	8.85605	0.452523	0.226261	+	0.974067i	$0.427350\pi$
$384$	0	0
$385$	0	0
$386$	−9.38303	−0.477584
$387$	0	0
$388$	−4.94230	−0.250908
$389$	2.53908	0.128736	0.0643682	−	0.997926i	$-0.479497\pi$
$389$	2.53908	0.128736	0.0643682	+	0.997926i	$0.479497\pi$
$390$	0	0
$391$	1.25345	0.0633897
$392$	−0.839859	−0.0424193
$393$	0	0
$394$	20.0205	1.00862
$395$	27.3562	1.37644
$396$	0	0
$397$	−26.9155	−1.35085	−0.675426	−	0.737428i	$-0.736040\pi$
$397$	−26.9155	−1.35085	−0.675426	+	0.737428i	$0.736040\pi$
$398$	−47.7492	−2.39345
$399$	0	0
$400$	−4.29008	−0.214504
$401$	−28.1770	−1.40709	−0.703546	−	0.710650i	$-0.748401\pi$
$401$	−28.1770	−1.40709	−0.703546	+	0.710650i	$0.748401\pi$
$402$	0	0
$403$	−10.5454	−0.525304
$404$	11.5048	0.572387
$405$	0	0
$406$	−2.80840	−0.139379
$407$	0	0
$408$	0	0
$409$	−16.1243	−0.797297	−0.398648	−	0.917104i	$-0.630521\pi$
$409$	−16.1243	−0.797297	−0.398648	+	0.917104i	$0.630521\pi$
$410$	−53.3945	−2.63697
$411$	0	0
$412$	−18.1181	−0.892614
$413$	−6.55446	−0.322524
$414$	0	0
$415$	−36.0806	−1.77113
$416$	45.8967	2.25027
$417$	0	0
$418$	0	0
$419$	−6.80618	−0.332503	−0.166252	−	0.986083i	$-0.553166\pi$
$419$	−6.80618	−0.332503	−0.166252	+	0.986083i	$0.553166\pi$
$420$	0	0
$421$	−33.7446	−1.64461	−0.822305	−	0.569047i	$-0.807312\pi$
$421$	−33.7446	−1.64461	−0.822305	+	0.569047i	$0.807312\pi$
$422$	−24.9609	−1.21508
$423$	0	0
$424$	6.24834	0.303446
$425$	1.40676	0.0682377
$426$	0	0
$427$	3.76506	0.182204
$428$	1.19763	0.0578898
$429$	0	0
$430$	26.8025	1.29253
$431$	30.0834	1.44907	0.724534	−	0.689239i	$-0.242055\pi$
$431$	30.0834	1.44907	0.724534	+	0.689239i	$0.242055\pi$
$432$	0	0
$433$	17.7479	0.852909	0.426455	−	0.904509i	$-0.359762\pi$
$433$	17.7479	0.852909	0.426455	+	0.904509i	$0.359762\pi$
$434$	3.88212	0.186348
$435$	0	0
$436$	13.9392	0.667566
$437$	7.58469	0.362825
$438$	0	0
$439$	−30.8212	−1.47102	−0.735508	−	0.677516i	$-0.763056\pi$
$439$	−30.8212	−1.47102	−0.735508	+	0.677516i	$0.763056\pi$
$440$	0	0
$441$	0	0
$442$	−11.9114	−0.566568
$443$	13.1717	0.625806	0.312903	−	0.949785i	$-0.398698\pi$
$443$	13.1717	0.625806	0.312903	+	0.949785i	$0.398698\pi$
$444$	0	0
$445$	39.2458	1.86043
$446$	29.3419	1.38938
$447$	0	0
$448$	−10.8182	−0.511111
$449$	−13.4014	−0.632452	−0.316226	−	0.948684i	$-0.602416\pi$
$449$	−13.4014	−0.632452	−0.316226	+	0.948684i	$0.602416\pi$
$450$	0	0
$451$	0	0
$452$	−31.8817	−1.49959
$453$	0	0
$454$	−40.2431	−1.88870
$455$	14.4286	0.676425
$456$	0	0
$457$	5.20237	0.243356	0.121678	−	0.992570i	$-0.461172\pi$
$457$	5.20237	0.243356	0.121678	+	0.992570i	$0.461172\pi$
$458$	−23.3408	−1.09064
$459$	0	0
$460$	7.64526	0.356462
$461$	28.2415	1.31534	0.657669	−	0.753307i	$-0.271543\pi$
$461$	28.2415	1.31534	0.657669	+	0.753307i	$0.271543\pi$
$462$	0	0
$463$	−1.83204	−0.0851423	−0.0425712	−	0.999093i	$-0.513555\pi$
$463$	−1.83204	−0.0851423	−0.0425712	+	0.999093i	$0.513555\pi$
$464$	−4.06855	−0.188878
$465$	0	0
$466$	−36.2213	−1.67792
$467$	8.10568	0.375086	0.187543	−	0.982256i	$-0.439948\pi$
$467$	8.10568	0.375086	0.187543	+	0.982256i	$0.439948\pi$
$468$	0	0
$469$	12.0485	0.556347
$470$	−5.47223	−0.252415
$471$	0	0
$472$	5.50482	0.253380
$473$	0	0
$474$	0	0
$475$	8.51235	0.390574
$476$	2.39198	0.109636
$477$	0	0
$478$	−12.2878	−0.562032
$479$	20.8348	0.951965	0.475982	−	0.879455i	$-0.342093\pi$
$479$	20.8348	0.951965	0.475982	+	0.879455i	$0.342093\pi$
$480$	0	0
$481$	55.0415	2.50968
$482$	−15.8042	−0.719860
$483$	0	0
$484$	0	0
$485$	−5.21360	−0.236737
$486$	0	0
$487$	−20.2679	−0.918426	−0.459213	−	0.888326i	$-0.651868\pi$
$487$	−20.2679	−0.918426	−0.459213	+	0.888326i	$0.651868\pi$
$488$	−3.16212	−0.143142
$489$	0	0
$490$	−5.31167	−0.239957
$491$	−36.0577	−1.62726	−0.813631	−	0.581382i	$-0.802512\pi$
$491$	−36.0577	−1.62726	−0.813631	+	0.581382i	$0.802512\pi$
$492$	0	0
$493$	1.33412	0.0600856
$494$	−72.0765	−3.24287
$495$	0	0
$496$	5.62406	0.252528
$497$	−15.8526	−0.711085
$498$	0	0
$499$	13.7325	0.614750	0.307375	−	0.951588i	$-0.400549\pi$
$499$	13.7325	0.614750	0.307375	+	0.951588i	$0.400549\pi$
$500$	−21.8099	−0.975370
$501$	0	0
$502$	2.77029	0.123644
$503$	20.0298	0.893083	0.446542	−	0.894763i	$-0.352655\pi$
$503$	20.0298	0.893083	0.446542	+	0.894763i	$0.352655\pi$
$504$	0	0
$505$	12.1364	0.540061
$506$	0	0
$507$	0	0
$508$	−14.4076	−0.639232
$509$	−20.5334	−0.910126	−0.455063	−	0.890459i	$-0.650383\pi$
$509$	−20.5334	−0.910126	−0.455063	+	0.890459i	$0.650383\pi$
$510$	0	0
$511$	6.41645	0.283847
$512$	−29.5821	−1.30736
$513$	0	0
$514$	−0.503468	−0.0222070
$515$	−19.1126	−0.842203
$516$	0	0
$517$	0	0
$518$	−20.2626	−0.890289
$519$	0	0
$520$	−12.1180	−0.531410
$521$	−16.5662	−0.725777	−0.362888	−	0.931833i	$-0.618209\pi$
$521$	−16.5662	−0.725777	−0.362888	+	0.931833i	$0.618209\pi$
$522$	0	0
$523$	10.4119	0.455282	0.227641	−	0.973745i	$-0.426899\pi$
$523$	10.4119	0.455282	0.227641	+	0.973745i	$0.426899\pi$
$524$	30.0995	1.31490
$525$	0	0
$526$	54.2701	2.36629
$527$	−1.84418	−0.0803338
$528$	0	0
$529$	−21.4178	−0.931210
$530$	39.5175	1.71653
$531$	0	0
$532$	14.4740	0.627527
$533$	57.2802	2.48108
$534$	0	0
$535$	1.26337	0.0546204
$536$	−10.1190	−0.437075
$537$	0	0
$538$	41.0865	1.77137
$539$	0	0
$540$	0	0
$541$	−40.2319	−1.72970	−0.864852	−	0.502027i	$-0.832588\pi$
$541$	−40.2319	−1.72970	−0.864852	+	0.502027i	$0.832588\pi$
$542$	42.3807	1.82041
$543$	0	0
$544$	8.02641	0.344130
$545$	14.7044	0.629865
$546$	0	0
$547$	−10.4385	−0.446320	−0.223160	−	0.974782i	$-0.571637\pi$
$547$	−10.4385	−0.446320	−0.223160	+	0.974782i	$0.571637\pi$
$548$	38.4717	1.64343
$549$	0	0
$550$	0	0
$551$	8.07280	0.343913
$552$	0	0
$553$	10.8036	0.459416
$554$	53.8768	2.28901
$555$	0	0
$556$	19.9630	0.846619
$557$	−8.50681	−0.360445	−0.180223	−	0.983626i	$-0.557682\pi$
$557$	−8.50681	−0.360445	−0.180223	+	0.983626i	$0.557682\pi$
$558$	0	0
$559$	−28.7531	−1.21612
$560$	−7.69506	−0.325175
$561$	0	0
$562$	59.4777	2.50891
$563$	45.2115	1.90544	0.952718	−	0.303855i	$-0.0982738\pi$
$563$	45.2115	1.90544	0.952718	+	0.303855i	$0.0982738\pi$
$564$	0	0
$565$	−33.6318	−1.41490
$566$	8.14844	0.342505
$567$	0	0
$568$	13.3139	0.558640
$569$	44.7713	1.87691	0.938454	−	0.345403i	$-0.112258\pi$
$569$	44.7713	1.87691	0.938454	+	0.345403i	$0.112258\pi$
$570$	0	0
$571$	43.0895	1.80324	0.901619	−	0.432532i	$-0.142380\pi$
$571$	43.0895	1.80324	0.901619	+	0.432532i	$0.142380\pi$
$572$	0	0
$573$	0	0
$574$	−21.0868	−0.880146
$575$	1.77569	0.0740515
$576$	0	0
$577$	−31.0429	−1.29233	−0.646166	−	0.763197i	$-0.723628\pi$
$577$	−31.0429	−1.29233	−0.646166	+	0.763197i	$0.723628\pi$
$578$	33.5779	1.39666
$579$	0	0
$580$	8.13727	0.337882
$581$	−14.2491	−0.591152
$582$	0	0
$583$	0	0
$584$	−5.38891	−0.222995
$585$	0	0
$586$	−26.2034	−1.08245
$587$	12.7914	0.527958	0.263979	−	0.964528i	$-0.414965\pi$
$587$	12.7914	0.527958	0.263979	+	0.964528i	$0.414965\pi$
$588$	0	0
$589$	−11.1592	−0.459808
$590$	34.8151	1.43332
$591$	0	0
$592$	−29.3546	−1.20647
$593$	−19.1567	−0.786669	−0.393335	−	0.919395i	$-0.628679\pi$
$593$	−19.1567	−0.786669	−0.393335	+	0.919395i	$0.628679\pi$
$594$	0	0
$595$	2.52328	0.103444
$596$	−23.8115	−0.975358
$597$	0	0
$598$	−15.0353	−0.614838
$599$	18.2676	0.746395	0.373198	−	0.927752i	$-0.378261\pi$
$599$	18.2676	0.746395	0.373198	+	0.927752i	$0.378261\pi$
$600$	0	0
$601$	−3.22883	−0.131707	−0.0658534	−	0.997829i	$-0.520977\pi$
$601$	−3.22883	−0.131707	−0.0658534	+	0.997829i	$0.520977\pi$
$602$	10.5850	0.431411
$603$	0	0
$604$	−38.1815	−1.55358
$605$	0	0
$606$	0	0
$607$	28.9919	1.17675	0.588373	−	0.808590i	$-0.299769\pi$
$607$	28.9919	1.17675	0.588373	+	0.808590i	$0.299769\pi$
$608$	48.5682	1.96970
$609$	0	0
$610$	−19.9987	−0.809725
$611$	5.87046	0.237493
$612$	0	0
$613$	−23.3561	−0.943345	−0.471672	−	0.881774i	$-0.656349\pi$
$613$	−23.3561	−0.943345	−0.471672	+	0.881774i	$0.656349\pi$
$614$	−22.2584	−0.898275
$615$	0	0
$616$	0	0
$617$	33.0024	1.32863	0.664313	−	0.747454i	$-0.268724\pi$
$617$	33.0024	1.32863	0.664313	+	0.747454i	$0.268724\pi$
$618$	0	0
$619$	−4.67984	−0.188099	−0.0940494	−	0.995568i	$-0.529981\pi$
$619$	−4.67984	−0.188099	−0.0940494	+	0.995568i	$0.529981\pi$
$620$	−11.2484	−0.451745
$621$	0	0
$622$	−53.8944	−2.16097
$623$	15.4991	0.620959
$624$	0	0
$625$	−30.0656	−1.20262
$626$	49.8322	1.99169
$627$	0	0
$628$	26.2425	1.04719
$629$	9.62566	0.383800
$630$	0	0
$631$	48.1057	1.91506	0.957530	−	0.288335i	$-0.0931017\pi$
$631$	48.1057	1.91506	0.957530	+	0.288335i	$0.0931017\pi$
$632$	−9.07351	−0.360925
$633$	0	0
$634$	58.5923	2.32700
$635$	−15.1984	−0.603131
$636$	0	0
$637$	5.69822	0.225772
$638$	0	0
$639$	0	0
$640$	16.6722	0.659026
$641$	19.9494	0.787954	0.393977	−	0.919120i	$-0.371099\pi$
$641$	19.9494	0.787954	0.393977	+	0.919120i	$0.371099\pi$
$642$	0	0
$643$	9.21828	0.363534	0.181767	−	0.983342i	$-0.441818\pi$
$643$	9.21828	0.363534	0.181767	+	0.983342i	$0.441818\pi$
$644$	3.01930	0.118977
$645$	0	0
$646$	−12.6047	−0.495927
$647$	47.0275	1.84884	0.924421	−	0.381373i	$-0.124549\pi$
$647$	47.0275	1.84884	0.924421	+	0.381373i	$0.124549\pi$
$648$	0	0
$649$	0	0
$650$	−16.8742	−0.661861
$651$	0	0
$652$	13.8506	0.542431
$653$	29.7075	1.16254	0.581272	−	0.813709i	$-0.302555\pi$
$653$	29.7075	1.16254	0.581272	+	0.813709i	$0.302555\pi$
$654$	0	0
$655$	31.7517	1.24064
$656$	−30.5486	−1.19272
$657$	0	0
$658$	−2.16112	−0.0842490
$659$	−11.4989	−0.447932	−0.223966	−	0.974597i	$-0.571900\pi$
$659$	−11.4989	−0.447932	−0.223966	+	0.974597i	$0.571900\pi$
$660$	0	0
$661$	−29.7497	−1.15713	−0.578565	−	0.815636i	$-0.696387\pi$
$661$	−29.7497	−1.15713	−0.578565	+	0.815636i	$0.696387\pi$
$662$	22.3438	0.868417
$663$	0	0
$664$	11.9672	0.464418
$665$	15.2685	0.592087
$666$	0	0
$667$	1.68400	0.0652048
$668$	−8.24762	−0.319110
$669$	0	0
$670$	−63.9975	−2.47244
$671$	0	0
$672$	0	0
$673$	20.2823	0.781826	0.390913	−	0.920428i	$-0.372159\pi$
$673$	20.2823	0.781826	0.390913	+	0.920428i	$0.372159\pi$
$674$	37.0065	1.42544
$675$	0	0
$676$	46.7344	1.79748
$677$	45.6199	1.75331	0.876657	−	0.481115i	$-0.159768\pi$
$677$	45.6199	1.75331	0.876657	+	0.481115i	$0.159768\pi$
$678$	0	0
$679$	−2.05898	−0.0790162
$680$	−2.11920	−0.0812676
$681$	0	0
$682$	0	0
$683$	12.9279	0.494674	0.247337	−	0.968930i	$-0.420445\pi$
$683$	12.9279	0.494674	0.247337	+	0.968930i	$0.420445\pi$
$684$	0	0
$685$	40.5835	1.55062
$686$	−2.09771	−0.0800908
$687$	0	0
$688$	15.3345	0.584623
$689$	−42.3933	−1.61506
$690$	0	0
$691$	22.2368	0.845928	0.422964	−	0.906147i	$-0.360990\pi$
$691$	22.2368	0.845928	0.422964	+	0.906147i	$0.360990\pi$
$692$	−39.9433	−1.51842
$693$	0	0
$694$	66.5547	2.52638
$695$	21.0588	0.798806
$696$	0	0
$697$	10.0172	0.379427
$698$	−50.8979	−1.92651
$699$	0	0
$700$	3.38858	0.128076
$701$	12.3613	0.466880	0.233440	−	0.972371i	$-0.425002\pi$
$701$	12.3613	0.466880	0.233440	+	0.972371i	$0.425002\pi$
$702$	0	0
$703$	58.2453	2.19676
$704$	0	0
$705$	0	0
$706$	−73.6609	−2.77227
$707$	4.79295	0.180257
$708$	0	0
$709$	46.7101	1.75423	0.877117	−	0.480277i	$-0.159464\pi$
$709$	46.7101	1.75423	0.877117	+	0.480277i	$0.159464\pi$
$710$	84.2037	3.16011
$711$	0	0
$712$	−13.0171	−0.487835
$713$	−2.32784	−0.0871781
$714$	0	0
$715$	0	0
$716$	−21.3127	−0.796494
$717$	0	0
$718$	66.4981	2.48169
$719$	−27.4387	−1.02329	−0.511646	−	0.859196i	$-0.670964\pi$
$719$	−27.4387	−1.02329	−0.511646	+	0.859196i	$0.670964\pi$
$720$	0	0
$721$	−7.54804	−0.281104
$722$	−36.4155	−1.35524
$723$	0	0
$724$	2.35841	0.0876495
$725$	1.88997	0.0701916
$726$	0	0
$727$	−34.8207	−1.29143	−0.645714	−	0.763579i	$-0.723440\pi$
$727$	−34.8207	−1.29143	−0.645714	+	0.763579i	$0.723440\pi$
$728$	−4.78570	−0.177370
$729$	0	0
$730$	−34.0820	−1.26143
$731$	−5.02834	−0.185980
$732$	0	0
$733$	30.1355	1.11308	0.556541	−	0.830820i	$-0.312128\pi$
$733$	30.1355	1.11308	0.556541	+	0.830820i	$0.312128\pi$
$734$	13.6112	0.502399
$735$	0	0
$736$	10.1314	0.373449
$737$	0	0
$738$	0	0
$739$	−13.9266	−0.512298	−0.256149	−	0.966637i	$-0.582454\pi$
$739$	−13.9266	−0.512298	−0.256149	+	0.966637i	$0.582454\pi$
$740$	58.7105	2.15824
$741$	0	0
$742$	15.6064	0.572929
$743$	−45.8625	−1.68253	−0.841267	−	0.540620i	$-0.818190\pi$
$743$	−45.8625	−1.68253	−0.841267	+	0.540620i	$0.818190\pi$
$744$	0	0
$745$	−25.1186	−0.920273
$746$	−52.1928	−1.91091
$747$	0	0
$748$	0	0
$749$	0.498937	0.0182308
$750$	0	0
$751$	21.6057	0.788402	0.394201	−	0.919024i	$-0.371021\pi$
$751$	21.6057	0.788402	0.394201	+	0.919024i	$0.371021\pi$
$752$	−3.13082	−0.114169
$753$	0	0
$754$	−16.0029	−0.582790
$755$	−40.2774	−1.46584
$756$	0	0
$757$	14.8957	0.541392	0.270696	−	0.962665i	$-0.412746\pi$
$757$	14.8957	0.541392	0.270696	+	0.962665i	$0.412746\pi$
$758$	60.0550	2.18130
$759$	0	0
$760$	−12.8234	−0.465153
$761$	−19.2120	−0.696435	−0.348217	−	0.937414i	$-0.613213\pi$
$761$	−19.2120	−0.696435	−0.348217	+	0.937414i	$0.613213\pi$
$762$	0	0
$763$	5.80710	0.210231
$764$	−45.7756	−1.65610
$765$	0	0
$766$	−18.5774	−0.671228
$767$	−37.3487	−1.34858
$768$	0	0
$769$	−19.5580	−0.705280	−0.352640	−	0.935759i	$-0.614716\pi$
$769$	−19.5580	−0.705280	−0.352640	+	0.935759i	$0.614716\pi$
$770$	0	0
$771$	0	0
$772$	10.7368	0.386428
$773$	1.14034	0.0410153	0.0205076	−	0.999790i	$-0.493472\pi$
$773$	1.14034	0.0410153	0.0205076	+	0.999790i	$0.493472\pi$
$774$	0	0
$775$	−2.61255	−0.0938455
$776$	1.72925	0.0620764
$777$	0	0
$778$	−5.32624	−0.190955
$779$	60.6144	2.17174
$780$	0	0
$781$	0	0
$782$	−2.62937	−0.0940261
$783$	0	0
$784$	−3.03896	−0.108534
$785$	27.6831	0.988051
$786$	0	0
$787$	−14.0116	−0.499458	−0.249729	−	0.968316i	$-0.580342\pi$
$787$	−14.0116	−0.499458	−0.249729	+	0.968316i	$0.580342\pi$
$788$	−22.9091	−0.816102
$789$	0	0
$790$	−57.3852	−2.04167
$791$	−13.2820	−0.472254
$792$	0	0
$793$	21.4541	0.761858
$794$	56.4609	2.00372
$795$	0	0
$796$	54.6386	1.93661
$797$	−6.62662	−0.234727	−0.117364	−	0.993089i	$-0.537444\pi$
$797$	−6.62662	−0.234727	−0.117364	+	0.993089i	$0.537444\pi$
$798$	0	0
$799$	1.02663	0.0363194
$800$	11.3706	0.402010
$801$	0	0
$802$	59.1070	2.08714
$803$	0	0
$804$	0	0
$805$	3.18503	0.112258
$806$	22.1212	0.779185
$807$	0	0
$808$	−4.02540	−0.141613
$809$	13.1145	0.461080	0.230540	−	0.973063i	$-0.425951\pi$
$809$	13.1145	0.461080	0.230540	+	0.973063i	$0.425951\pi$
$810$	0	0
$811$	19.2816	0.677068	0.338534	−	0.940954i	$-0.390069\pi$
$811$	19.2816	0.677068	0.338534	+	0.940954i	$0.390069\pi$
$812$	3.21361	0.112775
$813$	0	0
$814$	0	0
$815$	14.6109	0.511796
$816$	0	0
$817$	−30.4267	−1.06450
$818$	33.8241	1.18263
$819$	0	0
$820$	61.0985	2.13365
$821$	−23.7058	−0.827337	−0.413668	−	0.910428i	$-0.635753\pi$
$821$	−23.7058	−0.827337	−0.413668	+	0.910428i	$0.635753\pi$
$822$	0	0
$823$	−5.47655	−0.190900	−0.0954502	−	0.995434i	$-0.530429\pi$
$823$	−5.47655	−0.190900	−0.0954502	+	0.995434i	$0.530429\pi$
$824$	6.33929	0.220840
$825$	0	0
$826$	13.7493	0.478400
$827$	−23.6878	−0.823707	−0.411854	−	0.911250i	$-0.635118\pi$
$827$	−23.6878	−0.823707	−0.411854	+	0.911250i	$0.635118\pi$
$828$	0	0
$829$	37.6550	1.30781	0.653906	−	0.756576i	$-0.273129\pi$
$829$	37.6550	1.30781	0.653906	+	0.756576i	$0.273129\pi$
$830$	75.6864	2.62711
$831$	0	0
$832$	−61.6444	−2.13714
$833$	0.996504	0.0345268
$834$	0	0
$835$	−8.70035	−0.301088
$836$	0	0
$837$	0	0
$838$	14.2774	0.493203
$839$	−21.6679	−0.748059	−0.374030	−	0.927417i	$-0.622024\pi$
$839$	−21.6679	−0.748059	−0.374030	+	0.927417i	$0.622024\pi$
$840$	0	0
$841$	−27.2076	−0.938194
$842$	70.7862	2.43945
$843$	0	0
$844$	28.5623	0.983155
$845$	49.2998	1.69596
$846$	0	0
$847$	0	0
$848$	22.6091	0.776401
$849$	0	0
$850$	−2.95096	−0.101217
$851$	12.1501	0.416499
$852$	0	0
$853$	13.1638	0.450721	0.225360	−	0.974275i	$-0.427644\pi$
$853$	13.1638	0.450721	0.225360	+	0.974275i	$0.427644\pi$
$854$	−7.89798	−0.270263
$855$	0	0
$856$	−0.419037	−0.0143224
$857$	46.2746	1.58071	0.790355	−	0.612649i	$-0.209896\pi$
$857$	46.2746	1.58071	0.790355	+	0.612649i	$0.209896\pi$
$858$	0	0
$859$	−40.4096	−1.37876	−0.689379	−	0.724401i	$-0.742117\pi$
$859$	−40.4096	−1.37876	−0.689379	+	0.724401i	$0.742117\pi$
$860$	−30.6697	−1.04583
$861$	0	0
$862$	−63.1062	−2.14941
$863$	−25.4564	−0.866545	−0.433273	−	0.901263i	$-0.642641\pi$
$863$	−25.4564	−0.866545	−0.433273	+	0.901263i	$0.642641\pi$
$864$	0	0
$865$	−42.1359	−1.43266
$866$	−37.2299	−1.26512
$867$	0	0
$868$	−4.44225	−0.150780
$869$	0	0
$870$	0	0
$871$	68.6548	2.32628
$872$	−4.87715	−0.165161
$873$	0	0
$874$	−15.9104	−0.538179
$875$	−9.08607	−0.307165
$876$	0	0
$877$	37.3143	1.26002	0.630008	−	0.776589i	$-0.283052\pi$
$877$	37.3143	1.26002	0.630008	+	0.776589i	$0.283052\pi$
$878$	64.6538	2.18196
$879$	0	0
$880$	0	0
$881$	−1.73357	−0.0584055	−0.0292027	−	0.999574i	$-0.509297\pi$
$881$	−1.73357	−0.0584055	−0.0292027	+	0.999574i	$0.509297\pi$
$882$	0	0
$883$	33.0694	1.11287	0.556437	−	0.830890i	$-0.312168\pi$
$883$	33.0694	1.11287	0.556437	+	0.830890i	$0.312168\pi$
$884$	13.6300	0.458427
$885$	0	0
$886$	−27.6304	−0.928260
$887$	−27.4806	−0.922708	−0.461354	−	0.887216i	$-0.652636\pi$
$887$	−27.4806	−0.922708	−0.461354	+	0.887216i	$0.652636\pi$
$888$	0	0
$889$	−6.00223	−0.201308
$890$	−82.3261	−2.75958
$891$	0	0
$892$	−33.5755	−1.12419
$893$	6.21217	0.207882
$894$	0	0
$895$	−22.4826	−0.751512
$896$	6.58425	0.219964
$897$	0	0
$898$	28.1122	0.938117
$899$	−2.47764	−0.0826340
$900$	0	0
$901$	−7.41374	−0.246988
$902$	0	0
$903$	0	0
$904$	11.1550	0.371010
$905$	2.48786	0.0826994
$906$	0	0
$907$	−13.3271	−0.442521	−0.221260	−	0.975215i	$-0.571017\pi$
$907$	−13.3271	−0.442521	−0.221260	+	0.975215i	$0.571017\pi$
$908$	46.0495	1.52821
$909$	0	0
$910$	−30.2670	−1.00334
$911$	−10.1287	−0.335579	−0.167789	−	0.985823i	$-0.553663\pi$
$911$	−10.1287	−0.335579	−0.167789	+	0.985823i	$0.553663\pi$
$912$	0	0
$913$	0	0
$914$	−10.9130	−0.360971
$915$	0	0
$916$	26.7085	0.882472
$917$	12.5395	0.414092
$918$	0	0
$919$	−24.7309	−0.815799	−0.407899	−	0.913027i	$-0.633739\pi$
$919$	−24.7309	−0.815799	−0.407899	+	0.913027i	$0.633739\pi$
$920$	−2.67498	−0.0881915
$921$	0	0
$922$	−59.2424	−1.95104
$923$	−90.3315	−2.97330
$924$	0	0
$925$	13.6361	0.448353
$926$	3.84309	0.126292
$927$	0	0
$928$	10.7834	0.353983
$929$	50.4163	1.65411	0.827053	−	0.562124i	$-0.190016\pi$
$929$	50.4163	1.65411	0.827053	+	0.562124i	$0.190016\pi$
$930$	0	0
$931$	6.02990	0.197622
$932$	41.4474	1.35765
$933$	0	0
$934$	−17.0033	−0.556366
$935$	0	0
$936$	0	0
$937$	46.0471	1.50429	0.752147	−	0.658995i	$-0.229018\pi$
$937$	46.0471	1.50429	0.752147	+	0.658995i	$0.229018\pi$
$938$	−25.2742	−0.825230
$939$	0	0
$940$	6.26178	0.204237
$941$	41.8983	1.36584	0.682922	−	0.730491i	$-0.260709\pi$
$941$	41.8983	1.36584	0.682922	+	0.730491i	$0.260709\pi$
$942$	0	0
$943$	12.6443	0.411754
$944$	19.9188	0.648301
$945$	0	0
$946$	0	0
$947$	12.6097	0.409759	0.204880	−	0.978787i	$-0.434320\pi$
$947$	12.6097	0.409759	0.204880	+	0.978787i	$0.434320\pi$
$948$	0	0
$949$	36.5623	1.18686
$950$	−17.8564	−0.579339
$951$	0	0
$952$	−0.836923	−0.0271248
$953$	20.2815	0.656984	0.328492	−	0.944507i	$-0.393460\pi$
$953$	20.2815	0.656984	0.328492	+	0.944507i	$0.393460\pi$
$954$	0	0
$955$	−48.2884	−1.56257
$956$	14.0608	0.454757
$957$	0	0
$958$	−43.7052	−1.41205
$959$	16.0274	0.517552
$960$	0	0
$961$	−27.5751	−0.889519
$962$	−115.461	−3.72261
$963$	0	0
$964$	18.0844	0.582461
$965$	11.3262	0.364604
$966$	0	0
$967$	−32.9041	−1.05812	−0.529062	−	0.848583i	$-0.677456\pi$
$967$	−32.9041	−1.05812	−0.529062	+	0.848583i	$0.677456\pi$
$968$	0	0
$969$	0	0
$970$	10.9366	0.351153
$971$	30.0232	0.963491	0.481746	−	0.876311i	$-0.340003\pi$
$971$	30.0232	0.963491	0.481746	+	0.876311i	$0.340003\pi$
$972$	0	0
$973$	8.31663	0.266619
$974$	42.5161	1.36230
$975$	0	0
$976$	−11.4419	−0.366245
$977$	−22.3977	−0.716566	−0.358283	−	0.933613i	$-0.616638\pi$
$977$	−22.3977	−0.716566	−0.358283	+	0.933613i	$0.616638\pi$
$978$	0	0
$979$	0	0
$980$	6.07805	0.194156
$981$	0	0
$982$	75.6384	2.41372
$983$	−3.91774	−0.124956	−0.0624782	−	0.998046i	$-0.519900\pi$
$983$	−3.91774	−0.124956	−0.0624782	+	0.998046i	$0.519900\pi$
$984$	0	0
$985$	−24.1666	−0.770012
$986$	−2.79858	−0.0891250
$987$	0	0
$988$	82.4759	2.62391
$989$	−6.34706	−0.201825
$990$	0	0
$991$	−14.7632	−0.468970	−0.234485	−	0.972120i	$-0.575340\pi$
$991$	−14.7632	−0.468970	−0.234485	+	0.972120i	$0.575340\pi$
$992$	−14.9062	−0.473272
$993$	0	0
$994$	33.2541	1.05475
$995$	57.6378	1.82724
$996$	0	0
$997$	0.676566	0.0214271	0.0107135	−	0.999943i	$-0.496590\pi$
$997$	0.676566	0.0214271	0.0107135	+	0.999943i	$0.496590\pi$
$998$	−28.8067	−0.911860
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.da.1.3	yes	12
3.2	odd	2	inner	7623.2.a.da.1.10	yes	12
11.10	odd	2		7623.2.a.cz.1.10	yes	12
33.32	even	2		7623.2.a.cz.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7623.2.a.cz.1.3	✓	12	33.32	even	2
7623.2.a.cz.1.10	yes	12	11.10	odd	2
7623.2.a.da.1.3	yes	12	1.1	even	1	trivial
7623.2.a.da.1.10	yes	12	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q-2.09771 q^{2} +2.40037 q^{4} +2.53213 q^{5} +1.00000 q^{7} -0.839859 q^{8} +O(q^{10})\)	\(q-2.09771 q^{2} +2.40037 q^{4} +2.53213 q^{5} +1.00000 q^{7} -0.839859 q^{8} -5.31167 q^{10} +5.69822 q^{13} -2.09771 q^{14} -3.03896 q^{16} +0.996504 q^{17} +6.02990 q^{19} +6.07805 q^{20} +1.25785 q^{23} +1.41169 q^{25} -11.9532 q^{26} +2.40037 q^{28} +1.33880 q^{29} -1.85065 q^{31} +8.05457 q^{32} -2.09037 q^{34} +2.53213 q^{35} +9.65942 q^{37} -12.6490 q^{38} -2.12663 q^{40} +10.0523 q^{41} -5.04597 q^{43} -2.63859 q^{46} +1.03023 q^{47} +1.00000 q^{49} -2.96131 q^{50} +13.6778 q^{52} -7.43975 q^{53} -0.839859 q^{56} -2.80840 q^{58} -6.55446 q^{59} +3.76506 q^{61} +3.88212 q^{62} -10.8182 q^{64} +14.4286 q^{65} +12.0485 q^{67} +2.39198 q^{68} -5.31167 q^{70} -15.8526 q^{71} +6.41645 q^{73} -20.2626 q^{74} +14.4740 q^{76} +10.8036 q^{79} -7.69506 q^{80} -21.0868 q^{82} -14.2491 q^{83} +2.52328 q^{85} +10.5850 q^{86} +15.4991 q^{89} +5.69822 q^{91} +3.01930 q^{92} -2.16112 q^{94} +15.2685 q^{95} -2.05898 q^{97} -2.09771 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 20 q^{4} + 12 q^{7}+O(q^{10}) \) 12 * q + 20 * q^4 + 12 * q^7	\( 12 q + 20 q^{4} + 12 q^{7} + 20 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 32 q^{25} + 20 q^{28} - 16 q^{31} - 24 q^{34} + 4 q^{37} + 48 q^{40} + 16 q^{43} + 24 q^{46} + 12 q^{49} + 96 q^{52} + 20 q^{58} + 44 q^{61} + 76 q^{64} + 20 q^{70} + 52 q^{73} - 8 q^{79} - 68 q^{82} + 72 q^{85} + 20 q^{91} + 20 q^{94} - 32 q^{97}+O(q^{100}) \) 12 * q + 20 * q^4 + 12 * q^7 + 20 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 32 * q^25 + 20 * q^28 - 16 * q^31 - 24 * q^34 + 4 * q^37 + 48 * q^40 + 16 * q^43 + 24 * q^46 + 12 * q^49 + 96 * q^52 + 20 * q^58 + 44 * q^61 + 76 * q^64 + 20 * q^70 + 52 * q^73 - 8 * q^79 - 68 * q^82 + 72 * q^85 + 20 * q^91 + 20 * q^94 - 32 * q^97

Introduction

L-functions

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Database

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