LMFDB - Embedded newform 7623.2.a.co.1.4

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:	$4$
Coefficient field:	4.4.2525.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 5 $ x^4 - 2x^3 - 4x^2 + 5*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.46673$ 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)

$f(q)$	$=$	$q+2.46673 q^{2} +4.08477 q^{4} +3.46673 q^{5} +1.00000 q^{7} +5.14256 q^{8} +O(q^{10})$	$q+2.46673 q^{2} +4.08477 q^{4} +3.46673 q^{5} +1.00000 q^{7} +5.14256 q^{8} +8.55150 q^{10} +0.653752 q^{13} +2.46673 q^{14} +4.51578 q^{16} -1.13715 q^{17} +6.07602 q^{19} +14.1608 q^{20} +6.66708 q^{23} +7.01823 q^{25} +1.61263 q^{26} +4.08477 q^{28} -4.57357 q^{29} +2.79631 q^{31} +0.854102 q^{32} -2.80505 q^{34} +3.46673 q^{35} -0.439758 q^{37} +14.9879 q^{38} +17.8279 q^{40} -5.90315 q^{41} -8.70820 q^{43} +16.4459 q^{46} +0.604703 q^{47} +1.00000 q^{49} +17.3121 q^{50} +2.67042 q^{52} -9.82247 q^{53} +5.14256 q^{56} -11.2818 q^{58} -1.69406 q^{59} -6.85818 q^{61} +6.89775 q^{62} -6.92472 q^{64} +2.26638 q^{65} -6.17828 q^{67} -4.64501 q^{68} +8.55150 q^{70} +5.41687 q^{71} -6.70198 q^{73} -1.08477 q^{74} +24.8191 q^{76} -2.65375 q^{79} +15.6550 q^{80} -14.5615 q^{82} +6.69658 q^{83} -3.94221 q^{85} -21.4808 q^{86} +0.698213 q^{89} +0.653752 q^{91} +27.2335 q^{92} +1.49164 q^{94} +21.0639 q^{95} -14.8587 q^{97} +2.46673 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{7} + 9 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^7 + 9 * q^8	$ 4 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{7} + 9 q^{8} + 14 q^{10} + 2 q^{14} - 4 q^{16} - 3 q^{17} - 3 q^{19} + 17 q^{20} + 8 q^{23} + 12 q^{26} + 4 q^{28} + 3 q^{29} - 3 q^{31} - 10 q^{32} - 12 q^{34} + 6 q^{35} - 7 q^{37} + 20 q^{38} + 13 q^{40} + 4 q^{41} - 8 q^{43} + 3 q^{46} + 14 q^{47} + 4 q^{49} + 33 q^{50} + 17 q^{52} + 9 q^{53} + 9 q^{56} + 3 q^{58} + 25 q^{59} + 19 q^{61} + 10 q^{62} + 3 q^{64} + 12 q^{65} - 15 q^{67} - q^{68} + 14 q^{70} + 7 q^{71} + 11 q^{73} + 8 q^{74} + 26 q^{76} - 8 q^{79} + 4 q^{80} + 3 q^{82} - q^{83} - 15 q^{85} - 4 q^{86} + 17 q^{89} + 17 q^{92} + 20 q^{94} + 17 q^{95} - 15 q^{97} + 2 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^7 + 9 * q^8 + 14 * q^10 + 2 * q^14 - 4 * q^16 - 3 * q^17 - 3 * q^19 + 17 * q^20 + 8 * q^23 + 12 * q^26 + 4 * q^28 + 3 * q^29 - 3 * q^31 - 10 * q^32 - 12 * q^34 + 6 * q^35 - 7 * q^37 + 20 * q^38 + 13 * q^40 + 4 * q^41 - 8 * q^43 + 3 * q^46 + 14 * q^47 + 4 * q^49 + 33 * q^50 + 17 * q^52 + 9 * q^53 + 9 * q^56 + 3 * q^58 + 25 * q^59 + 19 * q^61 + 10 * q^62 + 3 * q^64 + 12 * q^65 - 15 * q^67 - q^68 + 14 * q^70 + 7 * q^71 + 11 * q^73 + 8 * q^74 + 26 * q^76 - 8 * q^79 + 4 * q^80 + 3 * q^82 - q^83 - 15 * q^85 - 4 * q^86 + 17 * q^89 + 17 * q^92 + 20 * q^94 + 17 * q^95 - 15 * q^97 + 2 * q^98

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 $

$2$	2.46673	1.74424	0.872121	−	0.489290i	$-0.162744\pi$
$2$	2.46673	1.74424	0.872121	+	0.489290i	$0.162744\pi$
$3$	0	0
$3$	0	0
$4$	4.08477	2.04238
$5$	3.46673	1.55037	0.775185	−	0.631735i	$-0.217657\pi$
$5$	3.46673	1.55037	0.775185	+	0.631735i	$0.217657\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	5.14256	1.81817
$9$	0	0
$10$	8.55150	2.70422
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.653752	0.181318	0.0906590	−	0.995882i	$-0.471103\pi$
$13$	0.653752	0.181318	0.0906590	+	0.995882i	$0.471103\pi$
$14$	2.46673	0.659262
$15$	0	0
$16$	4.51578	1.12894
$17$	−1.13715	−0.275800	−0.137900	−	0.990446i	$-0.544035\pi$
$17$	−1.13715	−0.275800	−0.137900	+	0.990446i	$0.544035\pi$
$18$	0	0
$19$	6.07602	1.39393	0.696967	−	0.717103i	$-0.254532\pi$
$19$	6.07602	1.39393	0.696967	+	0.717103i	$0.254532\pi$
$20$	14.1608	3.16645
$21$	0	0
$22$	0	0
$23$	6.66708	1.39018	0.695091	−	0.718921i	$-0.255364\pi$
$23$	6.66708	1.39018	0.695091	+	0.718921i	$0.255364\pi$
$24$	0	0
$25$	7.01823	1.40365
$26$	1.61263	0.316263
$27$	0	0
$28$	4.08477	0.771948
$29$	−4.57357	−0.849291	−0.424646	−	0.905360i	$-0.639601\pi$
$29$	−4.57357	−0.849291	−0.424646	+	0.905360i	$0.639601\pi$
$30$	0	0
$31$	2.79631	0.502232	0.251116	−	0.967957i	$-0.419202\pi$
$31$	2.79631	0.502232	0.251116	+	0.967957i	$0.419202\pi$
$32$	0.854102	0.150985
$33$	0	0
$34$	−2.80505	−0.481063
$35$	3.46673	0.585985
$36$	0	0
$37$	−0.439758	−0.0722958	−0.0361479	−	0.999346i	$-0.511509\pi$
$37$	−0.439758	−0.0722958	−0.0361479	+	0.999346i	$0.511509\pi$
$38$	14.9879	2.43136
$39$	0	0
$40$	17.8279	2.81883
$41$	−5.90315	−0.921917	−0.460959	−	0.887422i	$-0.652494\pi$
$41$	−5.90315	−0.921917	−0.460959	+	0.887422i	$0.652494\pi$
$42$	0	0
$43$	−8.70820	−1.32799	−0.663994	−	0.747738i	$-0.731140\pi$
$43$	−8.70820	−1.32799	−0.663994	+	0.747738i	$0.731140\pi$
$44$	0	0
$45$	0	0
$46$	16.4459	2.42482
$47$	0.604703	0.0882051	0.0441025	−	0.999027i	$-0.485957\pi$
$47$	0.604703	0.0882051	0.0441025	+	0.999027i	$0.485957\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	17.3121	2.44830
$51$	0	0
$52$	2.67042	0.370321
$53$	−9.82247	−1.34922	−0.674610	−	0.738175i	$-0.735688\pi$
$53$	−9.82247	−1.34922	−0.674610	+	0.738175i	$0.735688\pi$
$54$	0	0
$55$	0	0
$56$	5.14256	0.687203
$57$	0	0
$58$	−11.2818	−1.48137
$59$	−1.69406	−0.220547	−0.110274	−	0.993901i	$-0.535173\pi$
$59$	−1.69406	−0.220547	−0.110274	+	0.993901i	$0.535173\pi$
$60$	0	0
$61$	−6.85818	−0.878100	−0.439050	−	0.898463i	$-0.644685\pi$
$61$	−6.85818	−0.878100	−0.439050	+	0.898463i	$0.644685\pi$
$62$	6.89775	0.876015
$63$	0	0
$64$	−6.92472	−0.865590
$65$	2.26638	0.281110
$66$	0	0
$67$	−6.17828	−0.754797	−0.377398	−	0.926051i	$-0.623181\pi$
$67$	−6.17828	−0.754797	−0.377398	+	0.926051i	$0.623181\pi$
$68$	−4.64501	−0.563290
$69$	0	0
$70$	8.55150	1.02210
$71$	5.41687	0.642864	0.321432	−	0.946933i	$-0.395836\pi$
$71$	5.41687	0.642864	0.321432	+	0.946933i	$0.395836\pi$
$72$	0	0
$73$	−6.70198	−0.784408	−0.392204	−	0.919878i	$-0.628287\pi$
$73$	−6.70198	−0.784408	−0.392204	+	0.919878i	$0.628287\pi$
$74$	−1.08477	−0.126101
$75$	0	0
$76$	24.8191	2.84695
$77$	0	0
$78$	0	0
$79$	−2.65375	−0.298570	−0.149285	−	0.988794i	$-0.547697\pi$
$79$	−2.65375	−0.298570	−0.149285	+	0.988794i	$0.547697\pi$
$80$	15.6550	1.75028
$81$	0	0
$82$	−14.5615	−1.60805
$83$	6.69658	0.735045	0.367522	−	0.930015i	$-0.380206\pi$
$83$	6.69658	0.735045	0.367522	+	0.930015i	$0.380206\pi$
$84$	0	0
$85$	−3.94221	−0.427592
$86$	−21.4808	−2.31633
$87$	0	0
$88$	0	0
$89$	0.698213	0.0740105	0.0370052	−	0.999315i	$-0.488218\pi$
$89$	0.698213	0.0740105	0.0370052	+	0.999315i	$0.488218\pi$
$90$	0	0
$91$	0.653752	0.0685318
$92$	27.2335	2.83929
$93$	0	0
$94$	1.49164	0.153851
$95$	21.0639	2.16111
$96$	0	0
$97$	−14.8587	−1.50867	−0.754336	−	0.656489i	$-0.772041\pi$
$97$	−14.8587	−1.50867	−0.754336	+	0.656489i	$0.772041\pi$
$98$	2.46673	0.249178
$99$	0	0
$100$	28.6678	2.86678
$101$	8.61959	0.857682	0.428841	−	0.903380i	$-0.358922\pi$
$101$	8.61959	0.857682	0.428841	+	0.903380i	$0.358922\pi$
$102$	0	0
$103$	−0.932958	−0.0919271	−0.0459636	−	0.998943i	$-0.514636\pi$
$103$	−0.932958	−0.0919271	−0.0459636	+	0.998943i	$0.514636\pi$
$104$	3.36196	0.329667
$105$	0	0
$106$	−24.2294	−2.35337
$107$	6.62212	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$107$	6.62212	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$108$	0	0
$109$	4.12507	0.395110	0.197555	−	0.980292i	$-0.436700\pi$
$109$	4.12507	0.395110	0.197555	+	0.980292i	$0.436700\pi$
$110$	0	0
$111$	0	0
$112$	4.51578	0.426701
$113$	18.8258	1.77098	0.885491	−	0.464656i	$-0.153822\pi$
$113$	18.8258	1.77098	0.885491	+	0.464656i	$0.153822\pi$
$114$	0	0
$115$	23.1130	2.15530
$116$	−18.6820	−1.73458
$117$	0	0
$118$	−4.17878	−0.384688
$119$	−1.13715	−0.104243
$120$	0	0
$121$	0	0
$122$	−16.9173	−1.53162
$123$	0	0
$124$	11.4223	1.02575
$125$	6.99666	0.625800
$126$	0	0
$127$	7.96635	0.706899	0.353449	−	0.935454i	$-0.385009\pi$
$127$	7.96635	0.706899	0.353449	+	0.935454i	$0.385009\pi$
$128$	−18.7896	−1.66078
$129$	0	0
$130$	5.59056	0.490324
$131$	−4.80505	−0.419819	−0.209910	−	0.977721i	$-0.567317\pi$
$131$	−4.80505	−0.419819	−0.209910	+	0.977721i	$0.567317\pi$
$132$	0	0
$133$	6.07602	0.526858
$134$	−15.2401	−1.31655
$135$	0	0
$136$	−5.84788	−0.501452
$137$	21.8777	1.86914	0.934571	−	0.355778i	$-0.115784\pi$
$137$	21.8777	1.86914	0.934571	+	0.355778i	$0.115784\pi$
$138$	0	0
$139$	19.8137	1.68058	0.840289	−	0.542139i	$-0.182385\pi$
$139$	19.8137	1.68058	0.840289	+	0.542139i	$0.182385\pi$
$140$	14.1608	1.19680
$141$	0	0
$142$	13.3620	1.12131
$143$	0	0
$144$	0	0
$145$	−15.8553	−1.31672
$146$	−16.5320	−1.36820
$147$	0	0
$148$	−1.79631	−0.147656
$149$	3.16211	0.259050	0.129525	−	0.991576i	$-0.458655\pi$
$149$	3.16211	0.259050	0.129525	+	0.991576i	$0.458655\pi$
$150$	0	0
$151$	−8.92806	−0.726555	−0.363278	−	0.931681i	$-0.618342\pi$
$151$	−8.92806	−0.726555	−0.363278	+	0.931681i	$0.618342\pi$
$152$	31.2463	2.53441
$153$	0	0
$154$	0	0
$155$	9.69406	0.778645
$156$	0	0
$157$	1.13968	0.0909561	0.0454780	−	0.998965i	$-0.485519\pi$
$157$	1.13968	0.0909561	0.0454780	+	0.998965i	$0.485519\pi$
$158$	−6.54609	−0.520779
$159$	0	0
$160$	2.96094	0.234083
$161$	6.66708	0.525440
$162$	0	0
$163$	−5.06728	−0.396900	−0.198450	−	0.980111i	$-0.563591\pi$
$163$	−5.06728	−0.396900	−0.198450	+	0.980111i	$0.563591\pi$
$164$	−24.1130	−1.88291
$165$	0	0
$166$	16.5187	1.28210
$167$	−19.7069	−1.52496	−0.762482	−	0.647009i	$-0.776019\pi$
$167$	−19.7069	−1.52496	−0.762482	+	0.647009i	$0.776019\pi$
$168$	0	0
$169$	−12.5726	−0.967124
$170$	−9.72437	−0.745825
$171$	0	0
$172$	−35.5710	−2.71226
$173$	−14.3948	−1.09442	−0.547208	−	0.836997i	$-0.684309\pi$
$173$	−14.3948	−1.09442	−0.547208	+	0.836997i	$0.684309\pi$
$174$	0	0
$175$	7.01823	0.530528
$176$	0	0
$177$	0	0
$178$	1.72230	0.129092
$179$	4.66420	0.348619	0.174309	−	0.984691i	$-0.444231\pi$
$179$	4.66420	0.348619	0.174309	+	0.984691i	$0.444231\pi$
$180$	0	0
$181$	9.90805	0.736459	0.368230	−	0.929735i	$-0.379964\pi$
$181$	9.90805	0.736459	0.368230	+	0.929735i	$0.379964\pi$
$182$	1.61263	0.119536
$183$	0	0
$184$	34.2859	2.52759
$185$	−1.52452	−0.112085
$186$	0	0
$187$	0	0
$188$	2.47007	0.180148
$189$	0	0
$190$	51.9591	3.76951
$191$	−9.94295	−0.719447	−0.359723	−	0.933059i	$-0.617129\pi$
$191$	−9.94295	−0.719447	−0.359723	+	0.933059i	$0.617129\pi$
$192$	0	0
$193$	−3.70665	−0.266810	−0.133405	−	0.991062i	$-0.542591\pi$
$193$	−3.70665	−0.266810	−0.133405	+	0.991062i	$0.542591\pi$
$194$	−36.6524	−2.63149
$195$	0	0
$196$	4.08477	0.291769
$197$	5.91982	0.421770	0.210885	−	0.977511i	$-0.432365\pi$
$197$	5.91982	0.421770	0.210885	+	0.977511i	$0.432365\pi$
$198$	0	0
$199$	11.4842	0.814095	0.407047	−	0.913407i	$-0.366558\pi$
$199$	11.4842	0.814095	0.407047	+	0.913407i	$0.366558\pi$
$200$	36.0917	2.55207
$201$	0	0
$202$	21.2622	1.49600
$203$	−4.57357	−0.321002
$204$	0	0
$205$	−20.4646	−1.42931
$206$	−2.30136	−0.160343
$207$	0	0
$208$	2.95220	0.204698
$209$	0	0
$210$	0	0
$211$	−8.72487	−0.600645	−0.300323	−	0.953838i	$-0.597094\pi$
$211$	−8.72487	−0.600645	−0.300323	+	0.953838i	$0.597094\pi$
$212$	−40.1225	−2.75562
$213$	0	0
$214$	16.3350	1.11664
$215$	−30.1890	−2.05887
$216$	0	0
$217$	2.79631	0.189826
$218$	10.1754	0.689168
$219$	0	0
$220$	0	0
$221$	−0.743416	−0.0500076
$222$	0	0
$223$	10.3328	0.691938	0.345969	−	0.938246i	$-0.387550\pi$
$223$	10.3328	0.691938	0.345969	+	0.938246i	$0.387550\pi$
$224$	0.854102	0.0570671
$225$	0	0
$226$	46.4382	3.08902
$227$	−13.2690	−0.880691	−0.440346	−	0.897828i	$-0.645144\pi$
$227$	−13.2690	−0.880691	−0.440346	+	0.897828i	$0.645144\pi$
$228$	0	0
$229$	−2.49623	−0.164955	−0.0824777	−	0.996593i	$-0.526283\pi$
$229$	−2.49623	−0.164955	−0.0824777	+	0.996593i	$0.526283\pi$
$230$	57.0135	3.75936
$231$	0	0
$232$	−23.5199	−1.54415
$233$	−9.60389	−0.629171	−0.314586	−	0.949229i	$-0.601866\pi$
$233$	−9.60389	−0.629171	−0.314586	+	0.949229i	$0.601866\pi$
$234$	0	0
$235$	2.09634	0.136750
$236$	−6.91982	−0.450442
$237$	0	0
$238$	−2.80505	−0.181825
$239$	5.56327	0.359858	0.179929	−	0.983680i	$-0.442413\pi$
$239$	5.56327	0.359858	0.179929	+	0.983680i	$0.442413\pi$
$240$	0	0
$241$	−15.0208	−0.967572	−0.483786	−	0.875186i	$-0.660739\pi$
$241$	−15.0208	−0.967572	−0.483786	+	0.875186i	$0.660739\pi$
$242$	0	0
$243$	0	0
$244$	−28.0141	−1.79342
$245$	3.46673	0.221481
$246$	0	0
$247$	3.97221	0.252746
$248$	14.3802	0.913143
$249$	0	0
$250$	17.2589	1.09155
$251$	−27.6131	−1.74292	−0.871460	−	0.490466i	$-0.836827\pi$
$251$	−27.6131	−1.74292	−0.871460	+	0.490466i	$0.836827\pi$
$252$	0	0
$253$	0	0
$254$	19.6508	1.23300
$255$	0	0
$256$	−32.4995	−2.03122
$257$	28.4680	1.77578	0.887892	−	0.460052i	$-0.152169\pi$
$257$	28.4680	1.77578	0.887892	+	0.460052i	$0.152169\pi$
$258$	0	0
$259$	−0.439758	−0.0273253
$260$	9.25764	0.574134
$261$	0	0
$262$	−11.8528	−0.732267
$263$	14.1803	0.874397	0.437199	−	0.899365i	$-0.355971\pi$
$263$	14.1803	0.874397	0.437199	+	0.899365i	$0.355971\pi$
$264$	0	0
$265$	−34.0519	−2.09179
$266$	14.9879	0.918968
$267$	0	0
$268$	−25.2368	−1.54158
$269$	24.1937	1.47511	0.737557	−	0.675285i	$-0.235979\pi$
$269$	24.1937	1.47511	0.737557	+	0.675285i	$0.235979\pi$
$270$	0	0
$271$	−7.44975	−0.452540	−0.226270	−	0.974065i	$-0.572653\pi$
$271$	−7.44975	−0.452540	−0.226270	+	0.974065i	$0.572653\pi$
$272$	−5.13514	−0.311363
$273$	0	0
$274$	53.9665	3.26024
$275$	0	0
$276$	0	0
$277$	19.1890	1.15296	0.576478	−	0.817113i	$-0.304427\pi$
$277$	19.1890	1.15296	0.576478	+	0.817113i	$0.304427\pi$
$278$	48.8751	2.93134
$279$	0	0
$280$	17.8279	1.06542
$281$	−1.90063	−0.113382	−0.0566910	−	0.998392i	$-0.518055\pi$
$281$	−1.90063	−0.113382	−0.0566910	+	0.998392i	$0.518055\pi$
$282$	0	0
$283$	−7.25178	−0.431073	−0.215537	−	0.976496i	$-0.569150\pi$
$283$	−7.25178	−0.431073	−0.215537	+	0.976496i	$0.569150\pi$
$284$	22.1266	1.31297
$285$	0	0
$286$	0	0
$287$	−5.90315	−0.348452
$288$	0	0
$289$	−15.7069	−0.923934
$290$	−39.1109	−2.29667
$291$	0	0
$292$	−27.3760	−1.60206
$293$	−3.27097	−0.191092	−0.0955460	−	0.995425i	$-0.530460\pi$
$293$	−3.27097	−0.191092	−0.0955460	+	0.995425i	$0.530460\pi$
$294$	0	0
$295$	−5.87284	−0.341930
$296$	−2.26148	−0.131446
$297$	0	0
$298$	7.80008	0.451846
$299$	4.35862	0.252065
$300$	0	0
$301$	−8.70820	−0.501933
$302$	−22.0231	−1.26729
$303$	0	0
$304$	27.4380	1.57368
$305$	−23.7755	−1.36138
$306$	0	0
$307$	−31.6121	−1.80420	−0.902099	−	0.431530i	$-0.857974\pi$
$307$	−31.6121	−1.80420	−0.902099	+	0.431530i	$0.857974\pi$
$308$	0	0
$309$	0	0
$310$	23.9126	1.35815
$311$	9.03829	0.512514	0.256257	−	0.966609i	$-0.417511\pi$
$311$	9.03829	0.512514	0.256257	+	0.966609i	$0.417511\pi$
$312$	0	0
$313$	14.4990	0.819534	0.409767	−	0.912190i	$-0.365610\pi$
$313$	14.4990	0.819534	0.409767	+	0.912190i	$0.365610\pi$
$314$	2.81128	0.158649
$315$	0	0
$316$	−10.8400	−0.609795
$317$	18.4174	1.03442	0.517211	−	0.855858i	$-0.326970\pi$
$317$	18.4174	1.03442	0.517211	+	0.855858i	$0.326970\pi$
$318$	0	0
$319$	0	0
$320$	−24.0061	−1.34198
$321$	0	0
$322$	16.4459	0.916494
$323$	−6.90937	−0.384448
$324$	0	0
$325$	4.58818	0.254506
$326$	−12.4996	−0.692290
$327$	0	0
$328$	−30.3573	−1.67620
$329$	0.604703	0.0333384
$330$	0	0
$331$	−6.47653	−0.355982	−0.177991	−	0.984032i	$-0.556960\pi$
$331$	−6.47653	−0.355982	−0.177991	+	0.984032i	$0.556960\pi$
$332$	27.3540	1.50124
$333$	0	0
$334$	−48.6116	−2.65991
$335$	−21.4184	−1.17021
$336$	0	0
$337$	−6.25682	−0.340831	−0.170415	−	0.985372i	$-0.554511\pi$
$337$	−6.25682	−0.340831	−0.170415	+	0.985372i	$0.554511\pi$
$338$	−31.0133	−1.68690
$339$	0	0
$340$	−16.1030	−0.873308
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−44.7824	−2.41451
$345$	0	0
$346$	−35.5081	−1.90893
$347$	22.7156	1.21944	0.609719	−	0.792617i	$-0.291282\pi$
$347$	22.7156	1.21944	0.609719	+	0.792617i	$0.291282\pi$
$348$	0	0
$349$	−29.6941	−1.58949	−0.794743	−	0.606946i	$-0.792394\pi$
$349$	−29.6941	−1.58949	−0.794743	+	0.606946i	$0.792394\pi$
$350$	17.3121	0.925370
$351$	0	0
$352$	0	0
$353$	6.82506	0.363262	0.181631	−	0.983367i	$-0.441862\pi$
$353$	6.82506	0.363262	0.181631	+	0.983367i	$0.441862\pi$
$354$	0	0
$355$	18.7788	0.996676
$356$	2.85204	0.151158
$357$	0	0
$358$	11.5053	0.608076
$359$	−7.25970	−0.383152	−0.191576	−	0.981478i	$-0.561360\pi$
$359$	−7.25970	−0.383152	−0.191576	+	0.981478i	$0.561360\pi$
$360$	0	0
$361$	17.9180	0.943055
$362$	24.4405	1.28456
$363$	0	0
$364$	2.67042	0.139968
$365$	−23.2340	−1.21612
$366$	0	0
$367$	−36.3366	−1.89676	−0.948378	−	0.317143i	$-0.897277\pi$
$367$	−36.3366	−1.89676	−0.948378	+	0.317143i	$0.897277\pi$
$368$	30.1071	1.56944
$369$	0	0
$370$	−3.76059	−0.195504
$371$	−9.82247	−0.509957
$372$	0	0
$373$	14.2913	0.739977	0.369989	−	0.929036i	$-0.379362\pi$
$373$	14.2913	0.739977	0.369989	+	0.929036i	$0.379362\pi$
$374$	0	0
$375$	0	0
$376$	3.10972	0.160372
$377$	−2.98998	−0.153992
$378$	0	0
$379$	−2.54528	−0.130742	−0.0653710	−	0.997861i	$-0.520823\pi$
$379$	−2.54528	−0.130742	−0.0653710	+	0.997861i	$0.520823\pi$
$380$	86.0413	4.41382
$381$	0	0
$382$	−24.5266	−1.25489
$383$	23.1367	1.18223	0.591115	−	0.806587i	$-0.298688\pi$
$383$	23.1367	1.18223	0.591115	+	0.806587i	$0.298688\pi$
$384$	0	0
$385$	0	0
$386$	−9.14330	−0.465382
$387$	0	0
$388$	−60.6943	−3.08128
$389$	−30.2615	−1.53432	−0.767158	−	0.641458i	$-0.778330\pi$
$389$	−30.2615	−1.53432	−0.767158	+	0.641458i	$0.778330\pi$
$390$	0	0
$391$	−7.58150	−0.383413
$392$	5.14256	0.259738
$393$	0	0
$394$	14.6026	0.735669
$395$	−9.19985	−0.462894
$396$	0	0
$397$	−22.6740	−1.13798	−0.568989	−	0.822345i	$-0.692665\pi$
$397$	−22.6740	−1.13798	−0.568989	+	0.822345i	$0.692665\pi$
$398$	28.3285	1.41998
$399$	0	0
$400$	31.6928	1.58464
$401$	16.2186	0.809917	0.404959	−	0.914335i	$-0.367286\pi$
$401$	16.2186	0.809917	0.404959	+	0.914335i	$0.367286\pi$
$402$	0	0
$403$	1.82809	0.0910637
$404$	35.2090	1.75171
$405$	0	0
$406$	−11.2818	−0.559905
$407$	0	0
$408$	0	0
$409$	35.0614	1.73368	0.866838	−	0.498590i	$-0.166149\pi$
$409$	35.0614	1.73368	0.866838	+	0.498590i	$0.166149\pi$
$410$	−50.4808	−2.49307
$411$	0	0
$412$	−3.81092	−0.187750
$413$	−1.69406	−0.0833590
$414$	0	0
$415$	23.2152	1.13959
$416$	0.558371	0.0273764
$417$	0	0
$418$	0	0
$419$	−28.2633	−1.38075	−0.690376	−	0.723451i	$-0.742555\pi$
$419$	−28.2633	−1.38075	−0.690376	+	0.723451i	$0.742555\pi$
$420$	0	0
$421$	−13.7947	−0.672311	−0.336156	−	0.941806i	$-0.609127\pi$
$421$	−13.7947	−0.672311	−0.336156	+	0.941806i	$0.609127\pi$
$422$	−21.5219	−1.04767
$423$	0	0
$424$	−50.5126	−2.45311
$425$	−7.98081	−0.387126
$426$	0	0
$427$	−6.85818	−0.331891
$428$	27.0498	1.30750
$429$	0	0
$430$	−74.4682	−3.59117
$431$	28.1700	1.35690	0.678451	−	0.734645i	$-0.262651\pi$
$431$	28.1700	1.35690	0.678451	+	0.734645i	$0.262651\pi$
$432$	0	0
$433$	14.1793	0.681415	0.340708	−	0.940169i	$-0.389333\pi$
$433$	14.1793	0.681415	0.340708	+	0.940169i	$0.389333\pi$
$434$	6.89775	0.331102
$435$	0	0
$436$	16.8499	0.806966
$437$	40.5093	1.93782
$438$	0	0
$439$	−28.0185	−1.33725	−0.668625	−	0.743599i	$-0.733117\pi$
$439$	−28.0185	−1.33725	−0.668625	+	0.743599i	$0.733117\pi$
$440$	0	0
$441$	0	0
$442$	−1.83381	−0.0872254
$443$	−17.3370	−0.823706	−0.411853	−	0.911250i	$-0.635118\pi$
$443$	−17.3370	−0.823706	−0.411853	+	0.911250i	$0.635118\pi$
$444$	0	0
$445$	2.42052	0.114744
$446$	25.4883	1.20691
$447$	0	0
$448$	−6.92472	−0.327162
$449$	29.5215	1.39321	0.696603	−	0.717457i	$-0.254694\pi$
$449$	29.5215	1.39321	0.696603	+	0.717457i	$0.254694\pi$
$450$	0	0
$451$	0	0
$452$	76.8990	3.61702
$453$	0	0
$454$	−32.7309	−1.53614
$455$	2.26638	0.106250
$456$	0	0
$457$	−9.69725	−0.453618	−0.226809	−	0.973939i	$-0.572829\pi$
$457$	−9.69725	−0.453618	−0.226809	+	0.973939i	$0.572829\pi$
$458$	−6.15752	−0.287722
$459$	0	0
$460$	94.4111	4.40194
$461$	−19.2216	−0.895240	−0.447620	−	0.894224i	$-0.647728\pi$
$461$	−19.2216	−0.895240	−0.447620	+	0.894224i	$0.647728\pi$
$462$	0	0
$463$	20.5327	0.954235	0.477117	−	0.878840i	$-0.341682\pi$
$463$	20.5327	0.954235	0.477117	+	0.878840i	$0.341682\pi$
$464$	−20.6532	−0.958803
$465$	0	0
$466$	−23.6902	−1.09743
$467$	−33.6714	−1.55813	−0.779064	−	0.626944i	$-0.784305\pi$
$467$	−33.6714	−1.55813	−0.779064	+	0.626944i	$0.784305\pi$
$468$	0	0
$469$	−6.17828	−0.285286
$470$	5.17112	0.238526
$471$	0	0
$472$	−8.71178	−0.400992
$473$	0	0
$474$	0	0
$475$	42.6429	1.95659
$476$	−4.64501	−0.212904
$477$	0	0
$478$	13.7231	0.627680
$479$	14.1708	0.647479	0.323740	−	0.946146i	$-0.395060\pi$
$479$	14.1708	0.647479	0.323740	+	0.946146i	$0.395060\pi$
$480$	0	0
$481$	−0.287493	−0.0131085
$482$	−37.0522	−1.68768
$483$	0	0
$484$	0	0
$485$	−51.5111	−2.33900
$486$	0	0
$487$	27.7415	1.25709	0.628543	−	0.777775i	$-0.283652\pi$
$487$	27.7415	1.25709	0.628543	+	0.777775i	$0.283652\pi$
$488$	−35.2686	−1.59653
$489$	0	0
$490$	8.55150	0.386317
$491$	−3.44295	−0.155378	−0.0776891	−	0.996978i	$-0.524754\pi$
$491$	−3.44295	−0.155378	−0.0776891	+	0.996978i	$0.524754\pi$
$492$	0	0
$493$	5.20086	0.234235
$494$	9.79837	0.440850
$495$	0	0
$496$	12.6275	0.566992
$497$	5.41687	0.242980
$498$	0	0
$499$	−2.58923	−0.115910	−0.0579550	−	0.998319i	$-0.518458\pi$
$499$	−2.58923	−0.115910	−0.0579550	+	0.998319i	$0.518458\pi$
$500$	28.5797	1.27812
$501$	0	0
$502$	−68.1140	−3.04008
$503$	−22.9026	−1.02118	−0.510589	−	0.859825i	$-0.670573\pi$
$503$	−22.9026	−1.02118	−0.510589	+	0.859825i	$0.670573\pi$
$504$	0	0
$505$	29.8818	1.32972
$506$	0	0
$507$	0	0
$508$	32.5407	1.44376
$509$	24.0985	1.06815	0.534074	−	0.845438i	$-0.320660\pi$
$509$	24.0985	1.06815	0.534074	+	0.845438i	$0.320660\pi$
$510$	0	0
$511$	−6.70198	−0.296478
$512$	−42.5884	−1.88216
$513$	0	0
$514$	70.2229	3.09740
$515$	−3.23432	−0.142521
$516$	0	0
$517$	0	0
$518$	−1.08477	−0.0476619
$519$	0	0
$520$	11.6550	0.511105
$521$	33.5057	1.46791	0.733956	−	0.679197i	$-0.237672\pi$
$521$	33.5057	1.46791	0.733956	+	0.679197i	$0.237672\pi$
$522$	0	0
$523$	31.1574	1.36242	0.681208	−	0.732090i	$-0.261455\pi$
$523$	31.1574	1.36242	0.681208	+	0.732090i	$0.261455\pi$
$524$	−19.6275	−0.857432
$525$	0	0
$526$	34.9791	1.52516
$527$	−3.17983	−0.138516
$528$	0	0
$529$	21.4500	0.932608
$530$	−83.9968	−3.64859
$531$	0	0
$532$	24.8191	1.07605
$533$	−3.85919	−0.167160
$534$	0	0
$535$	22.9571	0.992522
$536$	−31.7721	−1.37235
$537$	0	0
$538$	59.6793	2.57296
$539$	0	0
$540$	0	0
$541$	−22.6071	−0.971957	−0.485979	−	0.873971i	$-0.661537\pi$
$541$	−22.6071	−0.971957	−0.485979	+	0.873971i	$0.661537\pi$
$542$	−18.3765	−0.789340
$543$	0	0
$544$	−0.971245	−0.0416418
$545$	14.3005	0.612567
$546$	0	0
$547$	27.4442	1.17343	0.586715	−	0.809794i	$-0.300421\pi$
$547$	27.4442	1.17343	0.586715	+	0.809794i	$0.300421\pi$
$548$	89.3654	3.81750
$549$	0	0
$550$	0	0
$551$	−27.7891	−1.18386
$552$	0	0
$553$	−2.65375	−0.112849
$554$	47.3341	2.01103
$555$	0	0
$556$	80.9344	3.43238
$557$	−30.2504	−1.28175	−0.640875	−	0.767645i	$-0.721428\pi$
$557$	−30.2504	−1.28175	−0.640875	+	0.767645i	$0.721428\pi$
$558$	0	0
$559$	−5.69300	−0.240788
$560$	15.6550	0.661544
$561$	0	0
$562$	−4.68834	−0.197766
$563$	7.46234	0.314500	0.157250	−	0.987559i	$-0.449737\pi$
$563$	7.46234	0.314500	0.157250	+	0.987559i	$0.449737\pi$
$564$	0	0
$565$	65.2640	2.74568
$566$	−17.8882	−0.751896
$567$	0	0
$568$	27.8565	1.16883
$569$	35.6483	1.49446	0.747228	−	0.664568i	$-0.231384\pi$
$569$	35.6483	1.49446	0.747228	+	0.664568i	$0.231384\pi$
$570$	0	0
$571$	−25.8902	−1.08347	−0.541737	−	0.840548i	$-0.682233\pi$
$571$	−25.8902	−1.08347	−0.541737	+	0.840548i	$0.682233\pi$
$572$	0	0
$573$	0	0
$574$	−14.5615	−0.607785
$575$	46.7911	1.95132
$576$	0	0
$577$	8.16382	0.339864	0.169932	−	0.985456i	$-0.445645\pi$
$577$	8.16382	0.339864	0.169932	+	0.985456i	$0.445645\pi$
$578$	−38.7447	−1.61157
$579$	0	0
$580$	−64.7654	−2.68924
$581$	6.69658	0.277821
$582$	0	0
$583$	0	0
$584$	−34.4653	−1.42619
$585$	0	0
$586$	−8.06860	−0.333311
$587$	12.4634	0.514419	0.257210	−	0.966356i	$-0.417197\pi$
$587$	12.4634	0.514419	0.257210	+	0.966356i	$0.417197\pi$
$588$	0	0
$589$	16.9904	0.700079
$590$	−14.4867	−0.596409
$591$	0	0
$592$	−1.98585	−0.0816180
$593$	−23.6707	−0.972037	−0.486019	−	0.873948i	$-0.661551\pi$
$593$	−23.6707	−0.972037	−0.486019	+	0.873948i	$0.661551\pi$
$594$	0	0
$595$	−3.94221	−0.161615
$596$	12.9165	0.529080
$597$	0	0
$598$	10.7515	0.439663
$599$	−38.9809	−1.59272	−0.796358	−	0.604826i	$-0.793243\pi$
$599$	−38.9809	−1.59272	−0.796358	+	0.604826i	$0.793243\pi$
$600$	0	0
$601$	−30.5510	−1.24620	−0.623100	−	0.782142i	$-0.714127\pi$
$601$	−30.5510	−1.24620	−0.623100	+	0.782142i	$0.714127\pi$
$602$	−21.4808	−0.875492
$603$	0	0
$604$	−36.4690	−1.48390
$605$	0	0
$606$	0	0
$607$	−37.6173	−1.52684	−0.763420	−	0.645903i	$-0.776481\pi$
$607$	−37.6173	−1.52684	−0.763420	+	0.645903i	$0.776481\pi$
$608$	5.18954	0.210464
$609$	0	0
$610$	−58.6477	−2.37458
$611$	0.395326	0.0159932
$612$	0	0
$613$	−17.6272	−0.711956	−0.355978	−	0.934494i	$-0.615852\pi$
$613$	−17.6272	−0.711956	−0.355978	+	0.934494i	$0.615852\pi$
$614$	−77.9786	−3.14696
$615$	0	0
$616$	0	0
$617$	44.4849	1.79089	0.895447	−	0.445168i	$-0.146856\pi$
$617$	44.4849	1.79089	0.895447	+	0.445168i	$0.146856\pi$
$618$	0	0
$619$	−6.20424	−0.249369	−0.124685	−	0.992196i	$-0.539792\pi$
$619$	−6.20424	−0.249369	−0.124685	+	0.992196i	$0.539792\pi$
$620$	39.5979	1.59029
$621$	0	0
$622$	22.2950	0.893949
$623$	0.698213	0.0279733
$624$	0	0
$625$	−10.8356	−0.433424
$626$	35.7652	1.42947
$627$	0	0
$628$	4.65531	0.185767
$629$	0.500073	0.0199392
$630$	0	0
$631$	44.8057	1.78369	0.891844	−	0.452344i	$-0.149412\pi$
$631$	44.8057	1.78369	0.891844	+	0.452344i	$0.149412\pi$
$632$	−13.6471	−0.542851
$633$	0	0
$634$	45.4307	1.80428
$635$	27.6172	1.09595
$636$	0	0
$637$	0.653752	0.0259026
$638$	0	0
$639$	0	0
$640$	−65.1386	−2.57483
$641$	10.2756	0.405863	0.202932	−	0.979193i	$-0.434953\pi$
$641$	10.2756	0.405863	0.202932	+	0.979193i	$0.434953\pi$
$642$	0	0
$643$	−16.4446	−0.648511	−0.324255	−	0.945970i	$-0.605114\pi$
$643$	−16.4446	−0.648511	−0.324255	+	0.945970i	$0.605114\pi$
$644$	27.2335	1.07315
$645$	0	0
$646$	−17.0436	−0.670570
$647$	26.9686	1.06025	0.530123	−	0.847920i	$-0.322146\pi$
$647$	26.9686	1.06025	0.530123	+	0.847920i	$0.322146\pi$
$648$	0	0
$649$	0	0
$650$	11.3178	0.443921
$651$	0	0
$652$	−20.6986	−0.810621
$653$	−7.10602	−0.278080	−0.139040	−	0.990287i	$-0.544402\pi$
$653$	−7.10602	−0.278080	−0.139040	+	0.990287i	$0.544402\pi$
$654$	0	0
$655$	−16.6578	−0.650875
$656$	−26.6573	−1.04079
$657$	0	0
$658$	1.49164	0.0581502
$659$	−32.6279	−1.27100	−0.635502	−	0.772099i	$-0.719207\pi$
$659$	−32.6279	−1.27100	−0.635502	+	0.772099i	$0.719207\pi$
$660$	0	0
$661$	−33.8165	−1.31531	−0.657654	−	0.753320i	$-0.728451\pi$
$661$	−33.8165	−1.31531	−0.657654	+	0.753320i	$0.728451\pi$
$662$	−15.9759	−0.620919
$663$	0	0
$664$	34.4375	1.33644
$665$	21.0639	0.816824
$666$	0	0
$667$	−30.4924	−1.18067
$668$	−80.4980	−3.11456
$669$	0	0
$670$	−52.8335	−2.04114
$671$	0	0
$672$	0	0
$673$	−23.7496	−0.915478	−0.457739	−	0.889087i	$-0.651341\pi$
$673$	−23.7496	−0.915478	−0.457739	+	0.889087i	$0.651341\pi$
$674$	−15.4339	−0.594491
$675$	0	0
$676$	−51.3562	−1.97524
$677$	−45.6311	−1.75375	−0.876874	−	0.480721i	$-0.840375\pi$
$677$	−45.6311	−1.75375	−0.876874	+	0.480721i	$0.840375\pi$
$678$	0	0
$679$	−14.8587	−0.570224
$680$	−20.2730	−0.777435
$681$	0	0
$682$	0	0
$683$	28.8727	1.10478	0.552392	−	0.833585i	$-0.313715\pi$
$683$	28.8727	1.10478	0.552392	+	0.833585i	$0.313715\pi$
$684$	0	0
$685$	75.8442	2.89786
$686$	2.46673	0.0941803
$687$	0	0
$688$	−39.3243	−1.49923
$689$	−6.42145	−0.244638
$690$	0	0
$691$	−27.0635	−1.02954	−0.514772	−	0.857327i	$-0.672123\pi$
$691$	−27.0635	−1.02954	−0.514772	+	0.857327i	$0.672123\pi$
$692$	−58.7994	−2.23522
$693$	0	0
$694$	56.0334	2.12700
$695$	68.6889	2.60552
$696$	0	0
$697$	6.71279	0.254265
$698$	−73.2473	−2.77245
$699$	0	0
$700$	28.6678	1.08354
$701$	38.5156	1.45471	0.727357	−	0.686259i	$-0.240748\pi$
$701$	38.5156	1.45471	0.727357	+	0.686259i	$0.240748\pi$
$702$	0	0
$703$	−2.67198	−0.100776
$704$	0	0
$705$	0	0
$706$	16.8356	0.633616
$707$	8.61959	0.324173
$708$	0	0
$709$	−1.89464	−0.0711548	−0.0355774	−	0.999367i	$-0.511327\pi$
$709$	−1.89464	−0.0711548	−0.0355774	+	0.999367i	$0.511327\pi$
$710$	46.3223	1.73845
$711$	0	0
$712$	3.59060	0.134564
$713$	18.6432	0.698194
$714$	0	0
$715$	0	0
$716$	19.0522	0.712013
$717$	0	0
$718$	−17.9077	−0.668311
$719$	−14.5772	−0.543639	−0.271819	−	0.962348i	$-0.587625\pi$
$719$	−14.5772	−0.543639	−0.271819	+	0.962348i	$0.587625\pi$
$720$	0	0
$721$	−0.932958	−0.0347452
$722$	44.1990	1.64492
$723$	0	0
$724$	40.4721	1.50413
$725$	−32.0984	−1.19210
$726$	0	0
$727$	4.04780	0.150125	0.0750623	−	0.997179i	$-0.476084\pi$
$727$	4.04780	0.150125	0.0750623	+	0.997179i	$0.476084\pi$
$728$	3.36196	0.124602
$729$	0	0
$730$	−57.3120	−2.12121
$731$	9.90257	0.366260
$732$	0	0
$733$	23.5012	0.868036	0.434018	−	0.900904i	$-0.357095\pi$
$733$	23.5012	0.868036	0.434018	+	0.900904i	$0.357095\pi$
$734$	−89.6327	−3.30840
$735$	0	0
$736$	5.69437	0.209897
$737$	0	0
$738$	0	0
$739$	32.6786	1.20210	0.601051	−	0.799210i	$-0.294749\pi$
$739$	32.6786	1.20210	0.601051	+	0.799210i	$0.294749\pi$
$740$	−6.22732	−0.228921
$741$	0	0
$742$	−24.2294	−0.889489
$743$	−18.1058	−0.664237	−0.332119	−	0.943238i	$-0.607763\pi$
$743$	−18.1058	−0.664237	−0.332119	+	0.943238i	$0.607763\pi$
$744$	0	0
$745$	10.9622	0.401624
$746$	35.2529	1.29070
$747$	0	0
$748$	0	0
$749$	6.62212	0.241967
$750$	0	0
$751$	9.14357	0.333654	0.166827	−	0.985986i	$-0.446648\pi$
$751$	9.14357	0.333654	0.166827	+	0.985986i	$0.446648\pi$
$752$	2.73071	0.0995787
$753$	0	0
$754$	−7.37548	−0.268599
$755$	−30.9512	−1.12643
$756$	0	0
$757$	47.3509	1.72100	0.860499	−	0.509452i	$-0.170152\pi$
$757$	47.3509	1.72100	0.860499	+	0.509452i	$0.170152\pi$
$758$	−6.27851	−0.228046
$759$	0	0
$760$	108.323	3.92927
$761$	3.44828	0.125000	0.0625000	−	0.998045i	$-0.480093\pi$
$761$	3.44828	0.125000	0.0625000	+	0.998045i	$0.480093\pi$
$762$	0	0
$763$	4.12507	0.149338
$764$	−40.6146	−1.46939
$765$	0	0
$766$	57.0720	2.06210
$767$	−1.10749	−0.0399892
$768$	0	0
$769$	−34.9787	−1.26137	−0.630683	−	0.776041i	$-0.717225\pi$
$769$	−34.9787	−1.26137	−0.630683	+	0.776041i	$0.717225\pi$
$770$	0	0
$771$	0	0
$772$	−15.1408	−0.544928
$773$	−25.3794	−0.912832	−0.456416	−	0.889766i	$-0.650867\pi$
$773$	−25.3794	−0.912832	−0.456416	+	0.889766i	$0.650867\pi$
$774$	0	0
$775$	19.6251	0.704956
$776$	−76.4117	−2.74302
$777$	0	0
$778$	−74.6469	−2.67622
$779$	−35.8677	−1.28509
$780$	0	0
$781$	0	0
$782$	−18.7015	−0.668765
$783$	0	0
$784$	4.51578	0.161278
$785$	3.95095	0.141016
$786$	0	0
$787$	−29.7552	−1.06066	−0.530329	−	0.847792i	$-0.677932\pi$
$787$	−29.7552	−1.06066	−0.530329	+	0.847792i	$0.677932\pi$
$788$	24.1811	0.861415
$789$	0	0
$790$	−22.6936	−0.807400
$791$	18.8258	0.669369
$792$	0	0
$793$	−4.48355	−0.159215
$794$	−55.9308	−1.98491
$795$	0	0
$796$	46.9103	1.66269
$797$	6.24330	0.221149	0.110575	−	0.993868i	$-0.464731\pi$
$797$	6.24330	0.221149	0.110575	+	0.993868i	$0.464731\pi$
$798$	0	0
$799$	−0.687641	−0.0243270
$800$	5.99428	0.211930
$801$	0	0
$802$	40.0069	1.41269
$803$	0	0
$804$	0	0
$805$	23.1130	0.814626
$806$	4.50941	0.158837
$807$	0	0
$808$	44.3268	1.55941
$809$	39.1860	1.37771	0.688854	−	0.724900i	$-0.258114\pi$
$809$	39.1860	1.37771	0.688854	+	0.724900i	$0.258114\pi$
$810$	0	0
$811$	12.3809	0.434753	0.217376	−	0.976088i	$-0.430250\pi$
$811$	12.3809	0.434753	0.217376	+	0.976088i	$0.430250\pi$
$812$	−18.6820	−0.655609
$813$	0	0
$814$	0	0
$815$	−17.5669	−0.615341
$816$	0	0
$817$	−52.9112	−1.85113
$818$	86.4871	3.02395
$819$	0	0
$820$	−83.5933	−2.91920
$821$	47.6987	1.66470	0.832349	−	0.554252i	$-0.186996\pi$
$821$	47.6987	1.66470	0.832349	+	0.554252i	$0.186996\pi$
$822$	0	0
$823$	−14.5207	−0.506161	−0.253080	−	0.967445i	$-0.581444\pi$
$823$	−14.5207	−0.506161	−0.253080	+	0.967445i	$0.581444\pi$
$824$	−4.79779	−0.167139
$825$	0	0
$826$	−4.17878	−0.145398
$827$	−30.9372	−1.07579	−0.537896	−	0.843011i	$-0.680781\pi$
$827$	−30.9372	−1.07579	−0.537896	+	0.843011i	$0.680781\pi$
$828$	0	0
$829$	0.0858057	0.00298015	0.00149008	−	0.999999i	$-0.499526\pi$
$829$	0.0858057	0.00298015	0.00149008	+	0.999999i	$0.499526\pi$
$830$	57.2658	1.98772
$831$	0	0
$832$	−4.52705	−0.156947
$833$	−1.13715	−0.0394000
$834$	0	0
$835$	−68.3185	−2.36426
$836$	0	0
$837$	0	0
$838$	−69.7179	−2.40837
$839$	−10.6905	−0.369078	−0.184539	−	0.982825i	$-0.559079\pi$
$839$	−10.6905	−0.369078	−0.184539	+	0.982825i	$0.559079\pi$
$840$	0	0
$841$	−8.08244	−0.278705
$842$	−34.0278	−1.17267
$843$	0	0
$844$	−35.6391	−1.22675
$845$	−43.5859	−1.49940
$846$	0	0
$847$	0	0
$848$	−44.3561	−1.52319
$849$	0	0
$850$	−19.6865	−0.675242
$851$	−2.93190	−0.100504
$852$	0	0
$853$	−21.2003	−0.725884	−0.362942	−	0.931812i	$-0.618228\pi$
$853$	−21.2003	−0.725884	−0.362942	+	0.931812i	$0.618228\pi$
$854$	−16.9173	−0.578898
$855$	0	0
$856$	34.0546	1.16396
$857$	−9.45359	−0.322929	−0.161464	−	0.986879i	$-0.551622\pi$
$857$	−9.45359	−0.322929	−0.161464	+	0.986879i	$0.551622\pi$
$858$	0	0
$859$	−38.8261	−1.32473	−0.662365	−	0.749181i	$-0.730447\pi$
$859$	−38.8261	−1.32473	−0.662365	+	0.749181i	$0.730447\pi$
$860$	−123.315	−4.20501
$861$	0	0
$862$	69.4879	2.36677
$863$	37.6046	1.28008	0.640038	−	0.768343i	$-0.278919\pi$
$863$	37.6046	1.28008	0.640038	+	0.768343i	$0.278919\pi$
$864$	0	0
$865$	−49.9029	−1.69675
$866$	34.9766	1.18855
$867$	0	0
$868$	11.4223	0.387697
$869$	0	0
$870$	0	0
$871$	−4.03906	−0.136858
$872$	21.2134	0.718377
$873$	0	0
$874$	99.9257	3.38004
$875$	6.99666	0.236530
$876$	0	0
$877$	−22.0086	−0.743178	−0.371589	−	0.928397i	$-0.621187\pi$
$877$	−22.0086	−0.743178	−0.371589	+	0.928397i	$0.621187\pi$
$878$	−69.1142	−2.33249
$879$	0	0
$880$	0	0
$881$	6.92969	0.233467	0.116734	−	0.993163i	$-0.462758\pi$
$881$	6.92969	0.233467	0.116734	+	0.993163i	$0.462758\pi$
$882$	0	0
$883$	42.3388	1.42481	0.712407	−	0.701767i	$-0.247605\pi$
$883$	42.3388	1.42481	0.712407	+	0.701767i	$0.247605\pi$
$884$	−3.03668	−0.102135
$885$	0	0
$886$	−42.7657	−1.43674
$887$	8.05647	0.270510	0.135255	−	0.990811i	$-0.456815\pi$
$887$	8.05647	0.270510	0.135255	+	0.990811i	$0.456815\pi$
$888$	0	0
$889$	7.96635	0.267183
$890$	5.97077	0.200141
$891$	0	0
$892$	42.2072	1.41320
$893$	3.67419	0.122952
$894$	0	0
$895$	16.1695	0.540488
$896$	−18.7896	−0.627717
$897$	0	0
$898$	72.8216	2.43009
$899$	−12.7891	−0.426541
$900$	0	0
$901$	11.1697	0.372115
$902$	0	0
$903$	0	0
$904$	96.8128	3.21995
$905$	34.3485	1.14178
$906$	0	0
$907$	2.89429	0.0961033	0.0480517	−	0.998845i	$-0.484699\pi$
$907$	2.89429	0.0961033	0.0480517	+	0.998845i	$0.484699\pi$
$908$	−54.2006	−1.79871
$909$	0	0
$910$	5.59056	0.185325
$911$	20.2500	0.670912	0.335456	−	0.942056i	$-0.391110\pi$
$911$	20.2500	0.670912	0.335456	+	0.942056i	$0.391110\pi$
$912$	0	0
$913$	0	0
$914$	−23.9205	−0.791220
$915$	0	0
$916$	−10.1965	−0.336902
$917$	−4.80505	−0.158677
$918$	0	0
$919$	−18.7261	−0.617718	−0.308859	−	0.951108i	$-0.599947\pi$
$919$	−18.7261	−0.617718	−0.308859	+	0.951108i	$0.599947\pi$
$920$	118.860	3.91869
$921$	0	0
$922$	−47.4146	−1.56152
$923$	3.54128	0.116563
$924$	0	0
$925$	−3.08632	−0.101478
$926$	50.6486	1.66442
$927$	0	0
$928$	−3.90630	−0.128230
$929$	45.2701	1.48526	0.742631	−	0.669700i	$-0.233577\pi$
$929$	45.2701	1.48526	0.742631	+	0.669700i	$0.233577\pi$
$930$	0	0
$931$	6.07602	0.199134
$932$	−39.2296	−1.28501
$933$	0	0
$934$	−83.0584	−2.71775
$935$	0	0
$936$	0	0
$937$	−29.4599	−0.962413	−0.481207	−	0.876607i	$-0.659801\pi$
$937$	−29.4599	−0.962413	−0.481207	+	0.876607i	$0.659801\pi$
$938$	−15.2401	−0.497609
$939$	0	0
$940$	8.56308	0.279297
$941$	58.2346	1.89839	0.949197	−	0.314683i	$-0.101898\pi$
$941$	58.2346	1.89839	0.949197	+	0.314683i	$0.101898\pi$
$942$	0	0
$943$	−39.3568	−1.28163
$944$	−7.64998	−0.248986
$945$	0	0
$946$	0	0
$947$	−45.3642	−1.47414	−0.737069	−	0.675818i	$-0.763791\pi$
$947$	−45.3642	−1.47414	−0.737069	+	0.675818i	$0.763791\pi$
$948$	0	0
$949$	−4.38143	−0.142227
$950$	105.189	3.41277
$951$	0	0
$952$	−5.84788	−0.189531
$953$	−41.3375	−1.33905	−0.669527	−	0.742788i	$-0.733503\pi$
$953$	−41.3375	−1.33905	−0.669527	+	0.742788i	$0.733503\pi$
$954$	0	0
$955$	−34.4695	−1.11541
$956$	22.7247	0.734968
$957$	0	0
$958$	34.9555	1.12936
$959$	21.8777	0.706469
$960$	0	0
$961$	−23.1807	−0.747763
$962$	−0.709167	−0.0228645
$963$	0	0
$964$	−61.3563	−1.97615
$965$	−12.8499	−0.413654
$966$	0	0
$967$	6.52818	0.209932	0.104966	−	0.994476i	$-0.466527\pi$
$967$	6.52818	0.209932	0.104966	+	0.994476i	$0.466527\pi$
$968$	0	0
$969$	0	0
$970$	−127.064	−4.07978
$971$	−28.8587	−0.926119	−0.463060	−	0.886327i	$-0.653248\pi$
$971$	−28.8587	−0.926119	−0.463060	+	0.886327i	$0.653248\pi$
$972$	0	0
$973$	19.8137	0.635199
$974$	68.4308	2.19266
$975$	0	0
$976$	−30.9700	−0.991327
$977$	8.98453	0.287441	0.143720	−	0.989618i	$-0.454093\pi$
$977$	8.98453	0.287441	0.143720	+	0.989618i	$0.454093\pi$
$978$	0	0
$979$	0	0
$980$	14.1608	0.452350
$981$	0	0
$982$	−8.49284	−0.271017
$983$	−36.8819	−1.17635	−0.588175	−	0.808734i	$-0.700153\pi$
$983$	−36.8819	−1.17635	−0.588175	+	0.808734i	$0.700153\pi$
$984$	0	0
$985$	20.5224	0.653899
$986$	12.8291	0.408562
$987$	0	0
$988$	16.2255	0.516203
$989$	−58.0583	−1.84615
$990$	0	0
$991$	−2.98352	−0.0947746	−0.0473873	−	0.998877i	$-0.515089\pi$
$991$	−2.98352	−0.0947746	−0.0473873	+	0.998877i	$0.515089\pi$
$992$	2.38833	0.0758297
$993$	0	0
$994$	13.3620	0.423815
$995$	39.8127	1.26215
$996$	0	0
$997$	−62.8553	−1.99065	−0.995324	−	0.0965930i	$-0.969205\pi$
$997$	−62.8553	−1.99065	−0.995324	+	0.0965930i	$0.969205\pi$
$998$	−6.38694	−0.202175
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.co.1.4		4
3.2	odd	2		847.2.a.k.1.1		4
11.2	odd	10		693.2.m.g.631.2		8
11.6	odd	10		693.2.m.g.190.2		8
11.10	odd	2		7623.2.a.ch.1.1		4
21.20	even	2		5929.2.a.bb.1.1		4
33.2	even	10		77.2.f.a.15.1	✓	8
33.5	odd	10		847.2.f.q.729.2		8
33.8	even	10		847.2.f.p.372.2		8
33.14	odd	10		847.2.f.s.372.1		8
33.17	even	10		77.2.f.a.36.1	yes	8
33.20	odd	10		847.2.f.q.323.2		8
33.26	odd	10		847.2.f.s.148.1		8
33.29	even	10		847.2.f.p.148.2		8
33.32	even	2		847.2.a.l.1.4		4
231.2	even	30		539.2.q.c.312.1		16
231.17	odd	30		539.2.q.b.520.1		16
231.68	odd	30		539.2.q.b.312.1		16
231.83	odd	10		539.2.f.d.344.1		8
231.101	odd	30		539.2.q.b.422.2		16
231.116	even	30		539.2.q.c.520.1		16
231.149	even	30		539.2.q.c.410.2		16
231.167	odd	10		539.2.f.d.246.1		8
231.200	even	30		539.2.q.c.422.2		16
231.215	odd	30		539.2.q.b.410.2		16
231.230	odd	2		5929.2.a.bi.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.a.15.1	✓	8	33.2	even	10
77.2.f.a.36.1	yes	8	33.17	even	10
539.2.f.d.246.1		8	231.167	odd	10
539.2.f.d.344.1		8	231.83	odd	10
539.2.q.b.312.1		16	231.68	odd	30
539.2.q.b.410.2		16	231.215	odd	30
539.2.q.b.422.2		16	231.101	odd	30
539.2.q.b.520.1		16	231.17	odd	30
539.2.q.c.312.1		16	231.2	even	30
539.2.q.c.410.2		16	231.149	even	30
539.2.q.c.422.2		16	231.200	even	30
539.2.q.c.520.1		16	231.116	even	30
693.2.m.g.190.2		8	11.6	odd	10
693.2.m.g.631.2		8	11.2	odd	10
847.2.a.k.1.1		4	3.2	odd	2
847.2.a.l.1.4		4	33.32	even	2
847.2.f.p.148.2		8	33.29	even	10
847.2.f.p.372.2		8	33.8	even	10
847.2.f.q.323.2		8	33.20	odd	10
847.2.f.q.729.2		8	33.5	odd	10
847.2.f.s.148.1		8	33.26	odd	10
847.2.f.s.372.1		8	33.14	odd	10
5929.2.a.bb.1.1		4	21.20	even	2
5929.2.a.bi.1.4		4	231.230	odd	2
7623.2.a.ch.1.1		4	11.10	odd	2
7623.2.a.co.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.46673 q^{2} +4.08477 q^{4} +3.46673 q^{5} +1.00000 q^{7} +5.14256 q^{8} +O(q^{10})\)	\(q+2.46673 q^{2} +4.08477 q^{4} +3.46673 q^{5} +1.00000 q^{7} +5.14256 q^{8} +8.55150 q^{10} +0.653752 q^{13} +2.46673 q^{14} +4.51578 q^{16} -1.13715 q^{17} +6.07602 q^{19} +14.1608 q^{20} +6.66708 q^{23} +7.01823 q^{25} +1.61263 q^{26} +4.08477 q^{28} -4.57357 q^{29} +2.79631 q^{31} +0.854102 q^{32} -2.80505 q^{34} +3.46673 q^{35} -0.439758 q^{37} +14.9879 q^{38} +17.8279 q^{40} -5.90315 q^{41} -8.70820 q^{43} +16.4459 q^{46} +0.604703 q^{47} +1.00000 q^{49} +17.3121 q^{50} +2.67042 q^{52} -9.82247 q^{53} +5.14256 q^{56} -11.2818 q^{58} -1.69406 q^{59} -6.85818 q^{61} +6.89775 q^{62} -6.92472 q^{64} +2.26638 q^{65} -6.17828 q^{67} -4.64501 q^{68} +8.55150 q^{70} +5.41687 q^{71} -6.70198 q^{73} -1.08477 q^{74} +24.8191 q^{76} -2.65375 q^{79} +15.6550 q^{80} -14.5615 q^{82} +6.69658 q^{83} -3.94221 q^{85} -21.4808 q^{86} +0.698213 q^{89} +0.653752 q^{91} +27.2335 q^{92} +1.49164 q^{94} +21.0639 q^{95} -14.8587 q^{97} +2.46673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{7} + 9 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^7 + 9 * q^8	\( 4 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{7} + 9 q^{8} + 14 q^{10} + 2 q^{14} - 4 q^{16} - 3 q^{17} - 3 q^{19} + 17 q^{20} + 8 q^{23} + 12 q^{26} + 4 q^{28} + 3 q^{29} - 3 q^{31} - 10 q^{32} - 12 q^{34} + 6 q^{35} - 7 q^{37} + 20 q^{38} + 13 q^{40} + 4 q^{41} - 8 q^{43} + 3 q^{46} + 14 q^{47} + 4 q^{49} + 33 q^{50} + 17 q^{52} + 9 q^{53} + 9 q^{56} + 3 q^{58} + 25 q^{59} + 19 q^{61} + 10 q^{62} + 3 q^{64} + 12 q^{65} - 15 q^{67} - q^{68} + 14 q^{70} + 7 q^{71} + 11 q^{73} + 8 q^{74} + 26 q^{76} - 8 q^{79} + 4 q^{80} + 3 q^{82} - q^{83} - 15 q^{85} - 4 q^{86} + 17 q^{89} + 17 q^{92} + 20 q^{94} + 17 q^{95} - 15 q^{97} + 2 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^7 + 9 * q^8 + 14 * q^10 + 2 * q^14 - 4 * q^16 - 3 * q^17 - 3 * q^19 + 17 * q^20 + 8 * q^23 + 12 * q^26 + 4 * q^28 + 3 * q^29 - 3 * q^31 - 10 * q^32 - 12 * q^34 + 6 * q^35 - 7 * q^37 + 20 * q^38 + 13 * q^40 + 4 * q^41 - 8 * q^43 + 3 * q^46 + 14 * q^47 + 4 * q^49 + 33 * q^50 + 17 * q^52 + 9 * q^53 + 9 * q^56 + 3 * q^58 + 25 * q^59 + 19 * q^61 + 10 * q^62 + 3 * q^64 + 12 * q^65 - 15 * q^67 - q^68 + 14 * q^70 + 7 * q^71 + 11 * q^73 + 8 * q^74 + 26 * q^76 - 8 * q^79 + 4 * q^80 + 3 * q^82 - q^83 - 15 * q^85 - 4 * q^86 + 17 * q^89 + 17 * q^92 + 20 * q^94 + 17 * q^95 - 15 * q^97 + 2 * q^98

Introduction

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