LMFDB - Embedded newform 7623.2.a.cz.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:	$1$
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	$-0.691588\pi$
$5$	−2.53213	−1.13240	−0.566202	+	0.824267i	$0.691588\pi$
$6$	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	$-0.790024\pi$
$13$	−5.69822	−1.58040	−0.790201	+	0.612848i	$0.790024\pi$
$14$	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	$-0.743128\pi$
$19$	−6.02990	−1.38335	−0.691677	+	0.722207i	$0.743128\pi$
$20$	−6.07805	−1.35909
$21$	0	0
$22$	0	0
$23$	−1.25785	−0.262279	−0.131140	−	0.991364i	$-0.541864\pi$
$23$	−1.25785	−0.262279	−0.131140	+	0.991364i	$0.541864\pi$
$24$	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	$-0.374287\pi$
$43$	5.04597	0.769504	0.384752	+	0.923020i	$0.374287\pi$
$44$	0	0
$45$	0	0
$46$	2.63859	0.389039
$47$	−1.03023	−0.150274	−0.0751371	−	0.997173i	$-0.523939\pi$
$47$	−1.03023	−0.150274	−0.0751371	+	0.997173i	$0.523939\pi$
$48$	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	$-0.329288\pi$
$53$	7.43975	1.02193	0.510964	+	0.859602i	$0.329288\pi$
$54$	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	$-0.359690\pi$
$59$	6.55446	0.853318	0.426659	+	0.904413i	$0.359690\pi$
$60$	0	0
$61$	−3.76506	−0.482066	−0.241033	−	0.970517i	$-0.577486\pi$
$61$	−3.76506	−0.482066	−0.241033	+	0.970517i	$0.577486\pi$
$62$	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	$-0.110191\pi$
$71$	15.8526	1.88136	0.940678	+	0.339301i	$0.110191\pi$
$72$	0	0
$73$	−6.41645	−0.750988	−0.375494	−	0.926825i	$-0.622527\pi$
$73$	−6.41645	−0.750988	−0.375494	+	0.926825i	$0.622527\pi$
$74$	−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	$-0.707928\pi$
$79$	−10.8036	−1.21550	−0.607751	+	0.794128i	$0.707928\pi$
$80$	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	$-0.806835\pi$
$89$	−15.4991	−1.64290	−0.821451	+	0.570279i	$0.806835\pi$
$90$	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	$-0.589708\pi$
$109$	−5.80710	−0.556220	−0.278110	+	0.960549i	$0.589708\pi$
$110$	0	0
$111$	0	0
$112$	3.03896	0.287155
$113$	13.2820	1.24947	0.624733	−	0.780839i	$-0.285208\pi$
$113$	13.2820	1.24947	0.624733	+	0.780839i	$0.285208\pi$
$114$	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	$-0.414197\pi$
$127$	6.00223	0.532611	0.266306	+	0.963889i	$0.414197\pi$
$128$	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	$-0.740048\pi$
$137$	−16.0274	−1.36931	−0.684657	+	0.728866i	$0.740048\pi$
$138$	0	0
$139$	−8.31663	−0.705407	−0.352704	−	0.935735i	$-0.614738\pi$
$139$	−8.31663	−0.705407	−0.352704	+	0.935735i	$0.614738\pi$
$140$	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	$-0.275929\pi$
$151$	15.9065	1.29445	0.647227	+	0.762298i	$0.275929\pi$
$152$	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	$-0.392337\pi$
$179$	8.87894	0.663643	0.331821	+	0.943342i	$0.392337\pi$
$180$	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	$-0.257638\pi$
$191$	19.0702	1.37987	0.689937	+	0.723869i	$0.257638\pi$
$192$	0	0
$193$	−4.47300	−0.321973	−0.160987	−	0.986957i	$-0.551468\pi$
$193$	−4.47300	−0.321973	−0.160987	+	0.986957i	$0.551468\pi$
$194$	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	$-0.634326\pi$
$211$	−11.8991	−0.819170	−0.409585	+	0.912272i	$0.634326\pi$
$212$	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	$-0.578018\pi$
$241$	−7.53402	−0.485309	−0.242655	+	0.970113i	$0.578018\pi$
$242$	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	$-0.486729\pi$
$251$	1.32063	0.0833572	0.0416786	+	0.999131i	$0.486729\pi$
$252$	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	$-0.502383\pi$
$257$	−0.240009	−0.0149713	−0.00748567	+	0.999972i	$0.502383\pi$
$258$	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	$-0.296319\pi$
$269$	19.5864	1.19420	0.597102	+	0.802165i	$0.296319\pi$
$270$	0	0
$271$	20.2033	1.22727	0.613633	−	0.789592i	$-0.289708\pi$
$271$	20.2033	1.22727	0.613633	+	0.789592i	$0.289708\pi$
$272$	−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	$-0.219461\pi$
$277$	25.6837	1.54318	0.771592	+	0.636118i	$0.219461\pi$
$278$	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	$-0.463168\pi$
$283$	3.88445	0.230907	0.115453	+	0.993313i	$0.463168\pi$
$284$	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	$-0.597920\pi$
$307$	−10.6108	−0.605592	−0.302796	+	0.953055i	$0.597920\pi$
$308$	0	0
$309$	0	0
$310$	−9.83004	−0.558309
$311$	−25.6921	−1.45686	−0.728432	−	0.685118i	$-0.759751\pi$
$311$	−25.6921	−1.45686	−0.728432	+	0.685118i	$0.759751\pi$
$312$	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	$-0.212972\pi$
$317$	27.9316	1.56880	0.784398	+	0.620258i	$0.212972\pi$
$318$	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	$-0.340457\pi$
$337$	17.6414	0.960990	0.480495	+	0.876997i	$0.340457\pi$
$338$	−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	$-0.724980\pi$
$349$	−24.2636	−1.29880	−0.649401	+	0.760447i	$0.724980\pi$
$350$	2.96131	0.158289
$351$	0	0
$352$	0	0
$353$	−35.1150	−1.86898	−0.934491	−	0.355986i	$-0.884145\pi$
$353$	−35.1150	−1.86898	−0.934491	+	0.355986i	$0.884145\pi$
$354$	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	$-0.722785\pi$
$373$	−24.8809	−1.28828	−0.644142	+	0.764906i	$0.722785\pi$
$374$	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	$-0.572650\pi$
$383$	−8.85605	−0.452523	−0.226261	+	0.974067i	$0.572650\pi$
$384$	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	$-0.520503\pi$
$389$	−2.53908	−0.128736	−0.0643682	+	0.997926i	$0.520503\pi$
$390$	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	$-0.251599\pi$
$401$	28.1770	1.40709	0.703546	+	0.710650i	$0.251599\pi$
$402$	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	$-0.369479\pi$
$409$	16.1243	0.797297	0.398648	+	0.917104i	$0.369479\pi$
$410$	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	$-0.446834\pi$
$419$	6.80618	0.332503	0.166252	+	0.986083i	$0.446834\pi$
$420$	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	$-0.236944\pi$
$439$	30.8212	1.47102	0.735508	+	0.677516i	$0.236944\pi$
$440$	0	0
$441$	0	0
$442$	11.9114	0.566568
$443$	−13.1717	−0.625806	−0.312903	−	0.949785i	$-0.601302\pi$
$443$	−13.1717	−0.625806	−0.312903	+	0.949785i	$0.601302\pi$
$444$	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	$-0.397584\pi$
$449$	13.4014	0.632452	0.316226	+	0.948684i	$0.397584\pi$
$450$	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	$-0.538828\pi$
$457$	−5.20237	−0.243356	−0.121678	+	0.992570i	$0.538828\pi$
$458$	−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	$-0.560052\pi$
$467$	−8.10568	−0.375086	−0.187543	+	0.982256i	$0.560052\pi$
$468$	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	$-0.349617\pi$
$509$	20.5334	0.910126	0.455063	+	0.890459i	$0.349617\pi$
$510$	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	$-0.381791\pi$
$521$	16.5662	0.725777	0.362888	+	0.931833i	$0.381791\pi$
$522$	0	0
$523$	−10.4119	−0.455282	−0.227641	−	0.973745i	$-0.573101\pi$
$523$	−10.4119	−0.455282	−0.227641	+	0.973745i	$0.573101\pi$
$524$	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	$-0.167412\pi$
$541$	40.2319	1.72970	0.864852	+	0.502027i	$0.167412\pi$
$542$	−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	$-0.428363\pi$
$547$	10.4385	0.446320	0.223160	+	0.974782i	$0.428363\pi$
$548$	−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	$-0.857620\pi$
$571$	−43.0895	−1.80324	−0.901619	+	0.432532i	$0.857620\pi$
$572$	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	$-0.585035\pi$
$587$	−12.7914	−0.527958	−0.263979	+	0.964528i	$0.585035\pi$
$588$	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	$-0.621739\pi$
$599$	−18.2676	−0.746395	−0.373198	+	0.927752i	$0.621739\pi$
$600$	0	0
$601$	3.22883	0.131707	0.0658534	−	0.997829i	$-0.479023\pi$
$601$	3.22883	0.131707	0.0658534	+	0.997829i	$0.479023\pi$
$602$	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	$-0.700231\pi$
$607$	−28.9919	−1.17675	−0.588373	+	0.808590i	$0.700231\pi$
$608$	−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	$-0.343651\pi$
$613$	23.3561	0.943345	0.471672	+	0.881774i	$0.343651\pi$
$614$	22.2584	0.898275
$615$	0	0
$616$	0	0
$617$	−33.0024	−1.32863	−0.664313	−	0.747454i	$-0.731276\pi$
$617$	−33.0024	−1.32863	−0.664313	+	0.747454i	$0.731276\pi$
$618$	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	$-0.628901\pi$
$641$	−19.9494	−0.787954	−0.393977	+	0.919120i	$0.628901\pi$
$642$	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	$-0.875451\pi$
$647$	−47.0275	−1.84884	−0.924421	+	0.381373i	$0.875451\pi$
$648$	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	$-0.697445\pi$
$653$	−29.7075	−1.16254	−0.581272	+	0.813709i	$0.697445\pi$
$654$	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	$-0.627841\pi$
$673$	−20.2823	−0.781826	−0.390913	+	0.920428i	$0.627841\pi$
$674$	−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	$-0.579555\pi$
$683$	−12.9279	−0.494674	−0.247337	+	0.968930i	$0.579555\pi$
$684$	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	$-0.329036\pi$
$719$	27.4387	1.02329	0.511646	+	0.859196i	$0.329036\pi$
$720$	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	$-0.687872\pi$
$733$	−30.1355	−1.11308	−0.556541	+	0.830820i	$0.687872\pi$
$734$	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	$-0.417546\pi$
$739$	13.9266	0.512298	0.256149	+	0.966637i	$0.417546\pi$
$740$	−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	$-0.385284\pi$
$769$	19.5580	0.705280	0.352640	+	0.935759i	$0.385284\pi$
$770$	0	0
$771$	0	0
$772$	−10.7368	−0.386428
$773$	−1.14034	−0.0410153	−0.0205076	−	0.999790i	$-0.506528\pi$
$773$	−1.14034	−0.0410153	−0.0205076	+	0.999790i	$0.506528\pi$
$774$	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	$-0.419658\pi$
$787$	14.0116	0.499458	0.249729	+	0.968316i	$0.419658\pi$
$788$	−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	$-0.462556\pi$
$797$	6.62662	0.234727	0.117364	+	0.993089i	$0.462556\pi$
$798$	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	$-0.609931\pi$
$811$	−19.2816	−0.677068	−0.338534	+	0.940954i	$0.609931\pi$
$812$	−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	$-0.377976\pi$
$839$	21.6679	0.748059	0.374030	+	0.927417i	$0.377976\pi$
$840$	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	$-0.572356\pi$
$853$	−13.1638	−0.450721	−0.225360	+	0.974275i	$0.572356\pi$
$854$	−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	$-0.357359\pi$
$863$	25.4564	0.866545	0.433273	+	0.901263i	$0.357359\pi$
$864$	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	$-0.716948\pi$
$877$	−37.3143	−1.26002	−0.630008	+	0.776589i	$0.716948\pi$
$878$	−64.6538	−2.18196
$879$	0	0
$880$	0	0
$881$	1.73357	0.0584055	0.0292027	−	0.999574i	$-0.490703\pi$
$881$	1.73357	0.0584055	0.0292027	+	0.999574i	$0.490703\pi$
$882$	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	$-0.446337\pi$
$911$	10.1287	0.335579	0.167789	+	0.985823i	$0.446337\pi$
$912$	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	$-0.366261\pi$
$919$	24.7309	0.815799	0.407899	+	0.913027i	$0.366261\pi$
$920$	−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	$-0.809984\pi$
$929$	−50.4163	−1.65411	−0.827053	+	0.562124i	$0.809984\pi$
$930$	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	$-0.770982\pi$
$937$	−46.0471	−1.50429	−0.752147	+	0.658995i	$0.770982\pi$
$938$	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	$-0.565680\pi$
$947$	−12.6097	−0.409759	−0.204880	+	0.978787i	$0.565680\pi$
$948$	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	$-0.322544\pi$
$967$	32.9041	1.05812	0.529062	+	0.848583i	$0.322544\pi$
$968$	0	0
$969$	0	0
$970$	−10.9366	−0.351153
$971$	−30.0232	−0.963491	−0.481746	−	0.876311i	$-0.659997\pi$
$971$	−30.0232	−0.963491	−0.481746	+	0.876311i	$0.659997\pi$
$972$	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	$-0.383362\pi$
$977$	22.3977	0.716566	0.358283	+	0.933613i	$0.383362\pi$
$978$	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	$-0.480100\pi$
$983$	3.91774	0.124956	0.0624782	+	0.998046i	$0.480100\pi$
$984$	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	$-0.503410\pi$
$997$	−0.676566	−0.0214271	−0.0107135	+	0.999943i	$0.503410\pi$
$998$	−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.cz.1.3	✓	12
3.2	odd	2	inner	7623.2.a.cz.1.10	yes	12
11.10	odd	2		7623.2.a.da.1.10	yes	12
33.32	even	2		7623.2.a.da.1.3	yes	12

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

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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