LMFDB - Embedded newform 7623.2.a.cx.1.10

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 $ x^10 - 19x^8 - x^7 + 124x^6 + 6x^5 - 316x^4 + 17x^3 + 253x^2 - 70*x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.65195$ 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)

$f(q)$	$=$	$q+2.65195 q^{2} +5.03286 q^{4} -1.71311 q^{5} -1.00000 q^{7} +8.04301 q^{8} -4.54308 q^{10} +1.51161 q^{13} -2.65195 q^{14} +11.2640 q^{16} +2.39386 q^{17} +6.33123 q^{19} -8.62183 q^{20} +5.05885 q^{23} -2.06526 q^{25} +4.00873 q^{26} -5.03286 q^{28} -8.34554 q^{29} +6.03882 q^{31} +13.7855 q^{32} +6.34841 q^{34} +1.71311 q^{35} -7.47381 q^{37} +16.7901 q^{38} -13.7785 q^{40} +5.03457 q^{41} +2.02506 q^{43} +13.4158 q^{46} +5.53026 q^{47} +1.00000 q^{49} -5.47698 q^{50} +7.60775 q^{52} -6.23728 q^{53} -8.04301 q^{56} -22.1320 q^{58} -11.1742 q^{59} +13.8096 q^{61} +16.0147 q^{62} +14.0306 q^{64} -2.58956 q^{65} +13.6173 q^{67} +12.0480 q^{68} +4.54308 q^{70} +14.3329 q^{71} -1.43030 q^{73} -19.8202 q^{74} +31.8642 q^{76} +6.71734 q^{79} -19.2964 q^{80} +13.3515 q^{82} -5.86977 q^{83} -4.10094 q^{85} +5.37036 q^{86} -16.4241 q^{89} -1.51161 q^{91} +25.4605 q^{92} +14.6660 q^{94} -10.8461 q^{95} +12.0516 q^{97} +2.65195 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 18 q^{4} - 5 q^{5} - 10 q^{7} - 3 q^{8} + 6 q^{10} - 6 q^{13} + 38 q^{16} + 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} - 18 q^{28} - 14 q^{29} + 26 q^{31} - 41 q^{32} + 21 q^{34} + 5 q^{35} + 24 q^{37}+ \cdots + 24 q^{97}+O(q^{100}) $ 10 * q + 18 * q^4 - 5 * q^5 - 10 * q^7 - 3 * q^8 + 6 * q^10 - 6 * q^13 + 38 * q^16 + 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 - 18 * q^28 - 14 * q^29 + 26 * q^31 - 41 * q^32 + 21 * q^34 + 5 * q^35 + 24 * q^37 - 8 * q^38 + 5 * q^40 + 19 * q^41 + 6 * q^43 + q^46 - 15 * q^47 + 10 * q^49 - q^50 + 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 + 11 * q^62 + 53 * q^64 - 29 * q^65 + 38 * q^67 + 87 * q^68 - 6 * q^70 - 26 * q^71 + q^73 - 39 * q^74 + 2 * q^76 - 5 * q^79 - 6 * q^80 + 5 * q^82 + 6 * q^83 + q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 - 42 * q^95 + 24 * q^97

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 $

$2$	2.65195	1.87521	0.937607	−	0.347696i	$-0.113036\pi$
$2$	2.65195	1.87521	0.937607	+	0.347696i	$0.113036\pi$
$3$	0	0
$3$	0	0
$4$	5.03286	2.51643
$5$	−1.71311	−0.766125	−0.383063	−	0.923722i	$-0.625131\pi$
$5$	−1.71311	−0.766125	−0.383063	+	0.923722i	$0.625131\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	8.04301	2.84363
$9$	0	0
$10$	−4.54308	−1.43665
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.51161	0.419246	0.209623	−	0.977782i	$-0.432776\pi$
$13$	1.51161	0.419246	0.209623	+	0.977782i	$0.432776\pi$
$14$	−2.65195	−0.708765
$15$	0	0
$16$	11.2640	2.81599
$17$	2.39386	0.580597	0.290298	−	0.956936i	$-0.406245\pi$
$17$	2.39386	0.580597	0.290298	+	0.956936i	$0.406245\pi$
$18$	0	0
$19$	6.33123	1.45248	0.726242	−	0.687439i	$-0.241265\pi$
$19$	6.33123	1.45248	0.726242	+	0.687439i	$0.241265\pi$
$20$	−8.62183	−1.92790
$21$	0	0
$22$	0	0
$23$	5.05885	1.05484	0.527422	−	0.849604i	$-0.323159\pi$
$23$	5.05885	1.05484	0.527422	+	0.849604i	$0.323159\pi$
$24$	0	0
$25$	−2.06526	−0.413052
$26$	4.00873	0.786177
$27$	0	0
$28$	−5.03286	−0.951121
$29$	−8.34554	−1.54973	−0.774864	−	0.632128i	$-0.782182\pi$
$29$	−8.34554	−1.54973	−0.774864	+	0.632128i	$0.782182\pi$
$30$	0	0
$31$	6.03882	1.08460	0.542302	−	0.840184i	$-0.317553\pi$
$31$	6.03882	1.08460	0.542302	+	0.840184i	$0.317553\pi$
$32$	13.7855	2.43696
$33$	0	0
$34$	6.34841	1.08874
$35$	1.71311	0.289568
$36$	0	0
$37$	−7.47381	−1.22869	−0.614343	−	0.789039i	$-0.710579\pi$
$37$	−7.47381	−1.22869	−0.614343	+	0.789039i	$0.710579\pi$
$38$	16.7901	2.72372
$39$	0	0
$40$	−13.7785	−2.17858
$41$	5.03457	0.786269	0.393134	−	0.919481i	$-0.371391\pi$
$41$	5.03457	0.786269	0.393134	+	0.919481i	$0.371391\pi$
$42$	0	0
$43$	2.02506	0.308818	0.154409	−	0.988007i	$-0.450653\pi$
$43$	2.02506	0.308818	0.154409	+	0.988007i	$0.450653\pi$
$44$	0	0
$45$	0	0
$46$	13.4158	1.97806
$47$	5.53026	0.806672	0.403336	−	0.915052i	$-0.367851\pi$
$47$	5.53026	0.806672	0.403336	+	0.915052i	$0.367851\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−5.47698	−0.774562
$51$	0	0
$52$	7.60775	1.05500
$53$	−6.23728	−0.856757	−0.428378	−	0.903599i	$-0.640915\pi$
$53$	−6.23728	−0.856757	−0.428378	+	0.903599i	$0.640915\pi$
$54$	0	0
$55$	0	0
$56$	−8.04301	−1.07479
$57$	0	0
$58$	−22.1320	−2.90607
$59$	−11.1742	−1.45476	−0.727381	−	0.686234i	$-0.759263\pi$
$59$	−11.1742	−1.45476	−0.727381	+	0.686234i	$0.759263\pi$
$60$	0	0
$61$	13.8096	1.76814	0.884072	−	0.467351i	$-0.154791\pi$
$61$	13.8096	1.76814	0.884072	+	0.467351i	$0.154791\pi$
$62$	16.0147	2.03387
$63$	0	0
$64$	14.0306	1.75382
$65$	−2.58956	−0.321195
$66$	0	0
$67$	13.6173	1.66362	0.831811	−	0.555059i	$-0.187304\pi$
$67$	13.6173	1.66362	0.831811	+	0.555059i	$0.187304\pi$
$68$	12.0480	1.46103
$69$	0	0
$70$	4.54308	0.543002
$71$	14.3329	1.70100	0.850501	−	0.525974i	$-0.176299\pi$
$71$	14.3329	1.70100	0.850501	+	0.525974i	$0.176299\pi$
$72$	0	0
$73$	−1.43030	−0.167404	−0.0837022	−	0.996491i	$-0.526674\pi$
$73$	−1.43030	−0.167404	−0.0837022	+	0.996491i	$0.526674\pi$
$74$	−19.8202	−2.30405
$75$	0	0
$76$	31.8642	3.65508
$77$	0	0
$78$	0	0
$79$	6.71734	0.755760	0.377880	−	0.925855i	$-0.376653\pi$
$79$	6.71734	0.755760	0.377880	+	0.925855i	$0.376653\pi$
$80$	−19.2964	−2.15740
$81$	0	0
$82$	13.3515	1.47442
$83$	−5.86977	−0.644291	−0.322145	−	0.946690i	$-0.604404\pi$
$83$	−5.86977	−0.644291	−0.322145	+	0.946690i	$0.604404\pi$
$84$	0	0
$85$	−4.10094	−0.444810
$86$	5.37036	0.579101
$87$	0	0
$88$	0	0
$89$	−16.4241	−1.74095	−0.870473	−	0.492215i	$-0.836187\pi$
$89$	−16.4241	−1.74095	−0.870473	+	0.492215i	$0.836187\pi$
$90$	0	0
$91$	−1.51161	−0.158460
$92$	25.4605	2.65444
$93$	0	0
$94$	14.6660	1.51268
$95$	−10.8461	−1.11279
$96$	0	0
$97$	12.0516	1.22365	0.611827	−	0.790992i	$-0.290435\pi$
$97$	12.0516	1.22365	0.611827	+	0.790992i	$0.290435\pi$
$98$	2.65195	0.267888
$99$	0	0
$100$	−10.3942	−1.03942
$101$	5.34845	0.532190	0.266095	−	0.963947i	$-0.414266\pi$
$101$	5.34845	0.532190	0.266095	+	0.963947i	$0.414266\pi$
$102$	0	0
$103$	9.37773	0.924015	0.462008	−	0.886876i	$-0.347129\pi$
$103$	9.37773	0.924015	0.462008	+	0.886876i	$0.347129\pi$
$104$	12.1579	1.19218
$105$	0	0
$106$	−16.5410	−1.60660
$107$	12.1711	1.17662	0.588311	−	0.808635i	$-0.299793\pi$
$107$	12.1711	1.17662	0.588311	+	0.808635i	$0.299793\pi$
$108$	0	0
$109$	−6.31429	−0.604799	−0.302399	−	0.953181i	$-0.597788\pi$
$109$	−6.31429	−0.604799	−0.302399	+	0.953181i	$0.597788\pi$
$110$	0	0
$111$	0	0
$112$	−11.2640	−1.06434
$113$	−3.36767	−0.316804	−0.158402	−	0.987375i	$-0.550634\pi$
$113$	−3.36767	−0.316804	−0.158402	+	0.987375i	$0.550634\pi$
$114$	0	0
$115$	−8.66636	−0.808142
$116$	−42.0020	−3.89978
$117$	0	0
$118$	−29.6336	−2.72799
$119$	−2.39386	−0.219445
$120$	0	0
$121$	0	0
$122$	36.6225	3.31565
$123$	0	0
$124$	30.3925	2.72933
$125$	12.1036	1.08257
$126$	0	0
$127$	−1.11374	−0.0988282	−0.0494141	−	0.998778i	$-0.515735\pi$
$127$	−1.11374	−0.0988282	−0.0494141	+	0.998778i	$0.515735\pi$
$128$	9.63748	0.851841
$129$	0	0
$130$	−6.86739	−0.602310
$131$	14.3843	1.25676	0.628379	−	0.777907i	$-0.283719\pi$
$131$	14.3843	1.25676	0.628379	+	0.777907i	$0.283719\pi$
$132$	0	0
$133$	−6.33123	−0.548988
$134$	36.1125	3.11965
$135$	0	0
$136$	19.2539	1.65100
$137$	−2.72791	−0.233061	−0.116530	−	0.993187i	$-0.537177\pi$
$137$	−2.72791	−0.233061	−0.116530	+	0.993187i	$0.537177\pi$
$138$	0	0
$139$	0.353144	0.0299533	0.0149766	−	0.999888i	$-0.495233\pi$
$139$	0.353144	0.0299533	0.0149766	+	0.999888i	$0.495233\pi$
$140$	8.62183	0.728678
$141$	0	0
$142$	38.0102	3.18974
$143$	0	0
$144$	0	0
$145$	14.2968	1.18729
$146$	−3.79310	−0.313919
$147$	0	0
$148$	−37.6146	−3.09190
$149$	0.511399	0.0418954	0.0209477	−	0.999781i	$-0.493332\pi$
$149$	0.511399	0.0418954	0.0209477	+	0.999781i	$0.493332\pi$
$150$	0	0
$151$	0.977763	0.0795693	0.0397846	−	0.999208i	$-0.487333\pi$
$151$	0.977763	0.0795693	0.0397846	+	0.999208i	$0.487333\pi$
$152$	50.9222	4.13033
$153$	0	0
$154$	0	0
$155$	−10.3452	−0.830942
$156$	0	0
$157$	−3.28963	−0.262541	−0.131270	−	0.991347i	$-0.541906\pi$
$157$	−3.28963	−0.262541	−0.131270	+	0.991347i	$0.541906\pi$
$158$	17.8141	1.41721
$159$	0	0
$160$	−23.6160	−1.86701
$161$	−5.05885	−0.398693
$162$	0	0
$163$	3.48225	0.272751	0.136375	−	0.990657i	$-0.456455\pi$
$163$	3.48225	0.272751	0.136375	+	0.990657i	$0.456455\pi$
$164$	25.3383	1.97859
$165$	0	0
$166$	−15.5664	−1.20818
$167$	4.57477	0.354006	0.177003	−	0.984210i	$-0.443360\pi$
$167$	4.57477	0.354006	0.177003	+	0.984210i	$0.443360\pi$
$168$	0	0
$169$	−10.7150	−0.824232
$170$	−10.8755	−0.834114
$171$	0	0
$172$	10.1918	0.777120
$173$	−5.72474	−0.435244	−0.217622	−	0.976033i	$-0.569830\pi$
$173$	−5.72474	−0.435244	−0.217622	+	0.976033i	$0.569830\pi$
$174$	0	0
$175$	2.06526	0.156119
$176$	0	0
$177$	0	0
$178$	−43.5559	−3.26465
$179$	−2.95701	−0.221017	−0.110509	−	0.993875i	$-0.535248\pi$
$179$	−2.95701	−0.221017	−0.110509	+	0.993875i	$0.535248\pi$
$180$	0	0
$181$	−9.40004	−0.698699	−0.349350	−	0.936992i	$-0.613597\pi$
$181$	−9.40004	−0.698699	−0.349350	+	0.936992i	$0.613597\pi$
$182$	−4.00873	−0.297147
$183$	0	0
$184$	40.6884	2.99959
$185$	12.8034	0.941328
$186$	0	0
$187$	0	0
$188$	27.8330	2.02993
$189$	0	0
$190$	−28.7633	−2.08671
$191$	9.20911	0.666348	0.333174	−	0.942865i	$-0.391880\pi$
$191$	9.20911	0.666348	0.333174	+	0.942865i	$0.391880\pi$
$192$	0	0
$193$	−1.49221	−0.107412	−0.0537059	−	0.998557i	$-0.517103\pi$
$193$	−1.49221	−0.107412	−0.0537059	+	0.998557i	$0.517103\pi$
$194$	31.9603	2.29461
$195$	0	0
$196$	5.03286	0.359490
$197$	−3.37718	−0.240614	−0.120307	−	0.992737i	$-0.538388\pi$
$197$	−3.37718	−0.240614	−0.120307	+	0.992737i	$0.538388\pi$
$198$	0	0
$199$	27.8114	1.97150	0.985750	−	0.168215i	$-0.0538002\pi$
$199$	27.8114	1.97150	0.985750	+	0.168215i	$0.0538002\pi$
$200$	−16.6109	−1.17457
$201$	0	0
$202$	14.1838	0.997971
$203$	8.34554	0.585742
$204$	0	0
$205$	−8.62477	−0.602380
$206$	24.8693	1.73273
$207$	0	0
$208$	17.0268	1.18059
$209$	0	0
$210$	0	0
$211$	−24.0068	−1.65270	−0.826350	−	0.563157i	$-0.809587\pi$
$211$	−24.0068	−1.65270	−0.826350	+	0.563157i	$0.809587\pi$
$212$	−31.3914	−2.15597
$213$	0	0
$214$	32.2771	2.20642
$215$	−3.46914	−0.236593
$216$	0	0
$217$	−6.03882	−0.409942
$218$	−16.7452	−1.13413
$219$	0	0
$220$	0	0
$221$	3.61860	0.243413
$222$	0	0
$223$	22.8178	1.52799	0.763997	−	0.645220i	$-0.223234\pi$
$223$	22.8178	1.52799	0.763997	+	0.645220i	$0.223234\pi$
$224$	−13.7855	−0.921083
$225$	0	0
$226$	−8.93091	−0.594075
$227$	−14.4047	−0.956071	−0.478035	−	0.878341i	$-0.658651\pi$
$227$	−14.4047	−0.956071	−0.478035	+	0.878341i	$0.658651\pi$
$228$	0	0
$229$	−3.79367	−0.250693	−0.125346	−	0.992113i	$-0.540004\pi$
$229$	−3.79367	−0.250693	−0.125346	+	0.992113i	$0.540004\pi$
$230$	−22.9828	−1.51544
$231$	0	0
$232$	−67.1233	−4.40686
$233$	−27.4844	−1.80056	−0.900282	−	0.435307i	$-0.856640\pi$
$233$	−27.4844	−1.80056	−0.900282	+	0.435307i	$0.856640\pi$
$234$	0	0
$235$	−9.47394	−0.618011
$236$	−56.2384	−3.66081
$237$	0	0
$238$	−6.34841	−0.411507
$239$	−11.2335	−0.726636	−0.363318	−	0.931665i	$-0.618356\pi$
$239$	−11.2335	−0.726636	−0.363318	+	0.931665i	$0.618356\pi$
$240$	0	0
$241$	14.7496	0.950105	0.475052	−	0.879958i	$-0.342429\pi$
$241$	14.7496	0.950105	0.475052	+	0.879958i	$0.342429\pi$
$242$	0	0
$243$	0	0
$244$	69.5020	4.44941
$245$	−1.71311	−0.109446
$246$	0	0
$247$	9.57039	0.608949
$248$	48.5703	3.08422
$249$	0	0
$250$	32.0981	2.03006
$251$	−7.61933	−0.480928	−0.240464	−	0.970658i	$-0.577300\pi$
$251$	−7.61933	−0.480928	−0.240464	+	0.970658i	$0.577300\pi$
$252$	0	0
$253$	0	0
$254$	−2.95358	−0.185324
$255$	0	0
$256$	−2.50303	−0.156440
$257$	0.289382	0.0180512	0.00902559	−	0.999959i	$-0.497127\pi$
$257$	0.289382	0.0180512	0.00902559	+	0.999959i	$0.497127\pi$
$258$	0	0
$259$	7.47381	0.464400
$260$	−13.0329	−0.808265
$261$	0	0
$262$	38.1464	2.35669
$263$	−12.3251	−0.759998	−0.379999	−	0.924987i	$-0.624076\pi$
$263$	−12.3251	−0.759998	−0.379999	+	0.924987i	$0.624076\pi$
$264$	0	0
$265$	10.6851	0.656383
$266$	−16.7901	−1.02947
$267$	0	0
$268$	68.5341	4.18639
$269$	−12.6511	−0.771349	−0.385674	−	0.922635i	$-0.626031\pi$
$269$	−12.6511	−0.771349	−0.385674	+	0.922635i	$0.626031\pi$
$270$	0	0
$271$	10.7502	0.653029	0.326515	−	0.945192i	$-0.394126\pi$
$271$	10.7502	0.653029	0.326515	+	0.945192i	$0.394126\pi$
$272$	26.9644	1.63496
$273$	0	0
$274$	−7.23428	−0.437039
$275$	0	0
$276$	0	0
$277$	−6.08896	−0.365850	−0.182925	−	0.983127i	$-0.558557\pi$
$277$	−6.08896	−0.365850	−0.182925	+	0.983127i	$0.558557\pi$
$278$	0.936521	0.0561688
$279$	0	0
$280$	13.7785	0.823425
$281$	8.61127	0.513705	0.256853	−	0.966451i	$-0.417314\pi$
$281$	8.61127	0.513705	0.256853	+	0.966451i	$0.417314\pi$
$282$	0	0
$283$	3.19557	0.189957	0.0949785	−	0.995479i	$-0.469722\pi$
$283$	3.19557	0.189957	0.0949785	+	0.995479i	$0.469722\pi$
$284$	72.1354	4.28045
$285$	0	0
$286$	0	0
$287$	−5.03457	−0.297182
$288$	0	0
$289$	−11.2694	−0.662907
$290$	37.9145	2.22642
$291$	0	0
$292$	−7.19852	−0.421262
$293$	8.55173	0.499597	0.249799	−	0.968298i	$-0.419636\pi$
$293$	8.55173	0.499597	0.249799	+	0.968298i	$0.419636\pi$
$294$	0	0
$295$	19.1427	1.11453
$296$	−60.1119	−3.49393
$297$	0	0
$298$	1.35621	0.0785629
$299$	7.64703	0.442239
$300$	0	0
$301$	−2.02506	−0.116722
$302$	2.59298	0.149209
$303$	0	0
$304$	71.3148	4.09018
$305$	−23.6574	−1.35462
$306$	0	0
$307$	−3.49789	−0.199635	−0.0998176	−	0.995006i	$-0.531826\pi$
$307$	−3.49789	−0.199635	−0.0998176	+	0.995006i	$0.531826\pi$
$308$	0	0
$309$	0	0
$310$	−27.4349	−1.55820
$311$	3.69038	0.209262	0.104631	−	0.994511i	$-0.466634\pi$
$311$	3.69038	0.209262	0.104631	+	0.994511i	$0.466634\pi$
$312$	0	0
$313$	−26.2642	−1.48454	−0.742270	−	0.670101i	$-0.766251\pi$
$313$	−26.2642	−1.48454	−0.742270	+	0.670101i	$0.766251\pi$
$314$	−8.72394	−0.492320
$315$	0	0
$316$	33.8074	1.90182
$317$	−13.0802	−0.734657	−0.367328	−	0.930091i	$-0.619727\pi$
$317$	−13.0802	−0.734657	−0.367328	+	0.930091i	$0.619727\pi$
$318$	0	0
$319$	0	0
$320$	−24.0359	−1.34365
$321$	0	0
$322$	−13.4158	−0.747636
$323$	15.1561	0.843308
$324$	0	0
$325$	−3.12188	−0.173171
$326$	9.23477	0.511466
$327$	0	0
$328$	40.4931	2.23586
$329$	−5.53026	−0.304893
$330$	0	0
$331$	7.93676	0.436244	0.218122	−	0.975922i	$-0.430007\pi$
$331$	7.93676	0.436244	0.218122	+	0.975922i	$0.430007\pi$
$332$	−29.5417	−1.62131
$333$	0	0
$334$	12.1321	0.663838
$335$	−23.3280	−1.27454
$336$	0	0
$337$	10.0006	0.544765	0.272382	−	0.962189i	$-0.412188\pi$
$337$	10.0006	0.544765	0.272382	+	0.962189i	$0.412188\pi$
$338$	−28.4157	−1.54561
$339$	0	0
$340$	−20.6395	−1.11933
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	16.2876	0.878166
$345$	0	0
$346$	−15.1818	−0.816176
$347$	5.40853	0.290345	0.145173	−	0.989406i	$-0.453626\pi$
$347$	5.40853	0.290345	0.145173	+	0.989406i	$0.453626\pi$
$348$	0	0
$349$	−28.5311	−1.52723	−0.763617	−	0.645670i	$-0.776578\pi$
$349$	−28.5311	−1.52723	−0.763617	+	0.645670i	$0.776578\pi$
$350$	5.47698	0.292757
$351$	0	0
$352$	0	0
$353$	19.2226	1.02312	0.511559	−	0.859248i	$-0.329068\pi$
$353$	19.2226	1.02312	0.511559	+	0.859248i	$0.329068\pi$
$354$	0	0
$355$	−24.5538	−1.30318
$356$	−82.6600	−4.38097
$357$	0	0
$358$	−7.84185	−0.414455
$359$	−22.3528	−1.17974	−0.589868	−	0.807500i	$-0.700820\pi$
$359$	−22.3528	−1.17974	−0.589868	+	0.807500i	$0.700820\pi$
$360$	0	0
$361$	21.0845	1.10971
$362$	−24.9285	−1.31021
$363$	0	0
$364$	−7.60775	−0.398754
$365$	2.45027	0.128253
$366$	0	0
$367$	−23.4448	−1.22381	−0.611905	−	0.790931i	$-0.709597\pi$
$367$	−23.4448	−1.22381	−0.611905	+	0.790931i	$0.709597\pi$
$368$	56.9827	2.97043
$369$	0	0
$370$	33.9541	1.76519
$371$	6.23728	0.323824
$372$	0	0
$373$	−33.4536	−1.73216	−0.866082	−	0.499903i	$-0.833369\pi$
$373$	−33.4536	−1.73216	−0.866082	+	0.499903i	$0.833369\pi$
$374$	0	0
$375$	0	0
$376$	44.4799	2.29388
$377$	−12.6152	−0.649718
$378$	0	0
$379$	−18.3796	−0.944094	−0.472047	−	0.881573i	$-0.656485\pi$
$379$	−18.3796	−0.944094	−0.472047	+	0.881573i	$0.656485\pi$
$380$	−54.5868	−2.80025
$381$	0	0
$382$	24.4221	1.24954
$383$	−21.3403	−1.09044	−0.545220	−	0.838293i	$-0.683554\pi$
$383$	−21.3403	−1.09044	−0.545220	+	0.838293i	$0.683554\pi$
$384$	0	0
$385$	0	0
$386$	−3.95728	−0.201420
$387$	0	0
$388$	60.6540	3.07924
$389$	−15.5214	−0.786969	−0.393484	−	0.919331i	$-0.628730\pi$
$389$	−15.5214	−0.786969	−0.393484	+	0.919331i	$0.628730\pi$
$390$	0	0
$391$	12.1102	0.612439
$392$	8.04301	0.406233
$393$	0	0
$394$	−8.95614	−0.451204
$395$	−11.5075	−0.579007
$396$	0	0
$397$	21.4777	1.07794	0.538968	−	0.842326i	$-0.318815\pi$
$397$	21.4777	1.07794	0.538968	+	0.842326i	$0.318815\pi$
$398$	73.7547	3.69699
$399$	0	0
$400$	−23.2630	−1.16315
$401$	−8.88106	−0.443499	−0.221749	−	0.975104i	$-0.571177\pi$
$401$	−8.88106	−0.443499	−0.221749	+	0.975104i	$0.571177\pi$
$402$	0	0
$403$	9.12837	0.454717
$404$	26.9180	1.33922
$405$	0	0
$406$	22.1320	1.09839
$407$	0	0
$408$	0	0
$409$	5.45776	0.269869	0.134934	−	0.990855i	$-0.456918\pi$
$409$	5.45776	0.269869	0.134934	+	0.990855i	$0.456918\pi$
$410$	−22.8725	−1.12959
$411$	0	0
$412$	47.1968	2.32522
$413$	11.1742	0.549848
$414$	0	0
$415$	10.0555	0.493607
$416$	20.8384	1.02169
$417$	0	0
$418$	0	0
$419$	−19.9805	−0.976114	−0.488057	−	0.872812i	$-0.662294\pi$
$419$	−19.9805	−0.976114	−0.488057	+	0.872812i	$0.662294\pi$
$420$	0	0
$421$	−28.3122	−1.37985	−0.689926	−	0.723880i	$-0.742357\pi$
$421$	−28.3122	−1.37985	−0.689926	+	0.723880i	$0.742357\pi$
$422$	−63.6651	−3.09917
$423$	0	0
$424$	−50.1665	−2.43630
$425$	−4.94395	−0.239817
$426$	0	0
$427$	−13.8096	−0.668296
$428$	61.2553	2.96089
$429$	0	0
$430$	−9.20000	−0.443664
$431$	−31.7647	−1.53005	−0.765026	−	0.643999i	$-0.777274\pi$
$431$	−31.7647	−1.53005	−0.765026	+	0.643999i	$0.777274\pi$
$432$	0	0
$433$	−13.1661	−0.632725	−0.316362	−	0.948638i	$-0.602462\pi$
$433$	−13.1661	−0.632725	−0.316362	+	0.948638i	$0.602462\pi$
$434$	−16.0147	−0.768729
$435$	0	0
$436$	−31.7789	−1.52193
$437$	32.0288	1.53214
$438$	0	0
$439$	−9.81161	−0.468283	−0.234141	−	0.972203i	$-0.575228\pi$
$439$	−9.81161	−0.468283	−0.234141	+	0.972203i	$0.575228\pi$
$440$	0	0
$441$	0	0
$442$	9.59636	0.456452
$443$	−2.48514	−0.118072	−0.0590362	−	0.998256i	$-0.518803\pi$
$443$	−2.48514	−0.118072	−0.0590362	+	0.998256i	$0.518803\pi$
$444$	0	0
$445$	28.1362	1.33378
$446$	60.5118	2.86532
$447$	0	0
$448$	−14.0306	−0.662883
$449$	15.6097	0.736667	0.368334	−	0.929694i	$-0.379928\pi$
$449$	15.6097	0.736667	0.368334	+	0.929694i	$0.379928\pi$
$450$	0	0
$451$	0	0
$452$	−16.9490	−0.797215
$453$	0	0
$454$	−38.2005	−1.79284
$455$	2.58956	0.121400
$456$	0	0
$457$	13.6652	0.639233	0.319616	−	0.947547i	$-0.396446\pi$
$457$	13.6652	0.639233	0.319616	+	0.947547i	$0.396446\pi$
$458$	−10.0606	−0.470103
$459$	0	0
$460$	−43.6166	−2.03363
$461$	−18.6545	−0.868825	−0.434412	−	0.900714i	$-0.643044\pi$
$461$	−18.6545	−0.868825	−0.434412	+	0.900714i	$0.643044\pi$
$462$	0	0
$463$	−18.5697	−0.863006	−0.431503	−	0.902111i	$-0.642017\pi$
$463$	−18.5697	−0.863006	−0.431503	+	0.902111i	$0.642017\pi$
$464$	−94.0039	−4.36402
$465$	0	0
$466$	−72.8874	−3.37644
$467$	−34.9579	−1.61766	−0.808828	−	0.588045i	$-0.799898\pi$
$467$	−34.9579	−1.61766	−0.808828	+	0.588045i	$0.799898\pi$
$468$	0	0
$469$	−13.6173	−0.628790
$470$	−25.1244	−1.15890
$471$	0	0
$472$	−89.8745	−4.13681
$473$	0	0
$474$	0	0
$475$	−13.0757	−0.599952
$476$	−12.0480	−0.552218
$477$	0	0
$478$	−29.7908	−1.36260
$479$	−40.5443	−1.85252	−0.926259	−	0.376888i	$-0.876994\pi$
$479$	−40.5443	−1.85252	−0.926259	+	0.376888i	$0.876994\pi$
$480$	0	0
$481$	−11.2975	−0.515123
$482$	39.1152	1.78165
$483$	0	0
$484$	0	0
$485$	−20.6457	−0.937472
$486$	0	0
$487$	13.5569	0.614320	0.307160	−	0.951658i	$-0.400621\pi$
$487$	13.5569	0.614320	0.307160	+	0.951658i	$0.400621\pi$
$488$	111.071	5.02795
$489$	0	0
$490$	−4.54308	−0.205236
$491$	23.4594	1.05871	0.529354	−	0.848401i	$-0.322435\pi$
$491$	23.4594	1.05871	0.529354	+	0.848401i	$0.322435\pi$
$492$	0	0
$493$	−19.9781	−0.899768
$494$	25.3802	1.14191
$495$	0	0
$496$	68.0211	3.05424
$497$	−14.3329	−0.642918
$498$	0	0
$499$	17.2876	0.773899	0.386949	−	0.922101i	$-0.373529\pi$
$499$	17.2876	0.773899	0.386949	+	0.922101i	$0.373529\pi$
$500$	60.9155	2.72422
$501$	0	0
$502$	−20.2061	−0.901843
$503$	−3.24592	−0.144728	−0.0723641	−	0.997378i	$-0.523054\pi$
$503$	−3.24592	−0.144728	−0.0723641	+	0.997378i	$0.523054\pi$
$504$	0	0
$505$	−9.16247	−0.407724
$506$	0	0
$507$	0	0
$508$	−5.60529	−0.248694
$509$	15.6005	0.691480	0.345740	−	0.938330i	$-0.387628\pi$
$509$	15.6005	0.691480	0.345740	+	0.938330i	$0.387628\pi$
$510$	0	0
$511$	1.43030	0.0632729
$512$	−25.9129	−1.14520
$513$	0	0
$514$	0.767429	0.0338498
$515$	−16.0651	−0.707911
$516$	0	0
$517$	0	0
$518$	19.8202	0.870850
$519$	0	0
$520$	−20.8278	−0.913361
$521$	4.34383	0.190307	0.0951534	−	0.995463i	$-0.469666\pi$
$521$	4.34383	0.190307	0.0951534	+	0.995463i	$0.469666\pi$
$522$	0	0
$523$	12.4657	0.545085	0.272543	−	0.962144i	$-0.412135\pi$
$523$	12.4657	0.545085	0.272543	+	0.962144i	$0.412135\pi$
$524$	72.3940	3.16254
$525$	0	0
$526$	−32.6856	−1.42516
$527$	14.4561	0.629718
$528$	0	0
$529$	2.59198	0.112695
$530$	28.3365	1.23086
$531$	0	0
$532$	−31.8642	−1.38149
$533$	7.61034	0.329640
$534$	0	0
$535$	−20.8504	−0.901439
$536$	109.524	4.73073
$537$	0	0
$538$	−33.5500	−1.44644
$539$	0	0
$540$	0	0
$541$	−12.5756	−0.540666	−0.270333	−	0.962767i	$-0.587134\pi$
$541$	−12.5756	−0.540666	−0.270333	+	0.962767i	$0.587134\pi$
$542$	28.5091	1.22457
$543$	0	0
$544$	33.0006	1.41489
$545$	10.8171	0.463352
$546$	0	0
$547$	−0.250803	−0.0107236	−0.00536178	−	0.999986i	$-0.501707\pi$
$547$	−0.250803	−0.0107236	−0.00536178	+	0.999986i	$0.501707\pi$
$548$	−13.7292	−0.586481
$549$	0	0
$550$	0	0
$551$	−52.8376	−2.25096
$552$	0	0
$553$	−6.71734	−0.285650
$554$	−16.1476	−0.686047
$555$	0	0
$556$	1.77732	0.0753753
$557$	−28.3165	−1.19981	−0.599904	−	0.800072i	$-0.704795\pi$
$557$	−28.3165	−1.19981	−0.599904	+	0.800072i	$0.704795\pi$
$558$	0	0
$559$	3.06111	0.129471
$560$	19.2964	0.815421
$561$	0	0
$562$	22.8367	0.963308
$563$	−16.2038	−0.682907	−0.341454	−	0.939899i	$-0.610919\pi$
$563$	−16.2038	−0.682907	−0.341454	+	0.939899i	$0.610919\pi$
$564$	0	0
$565$	5.76918	0.242711
$566$	8.47450	0.356210
$567$	0	0
$568$	115.280	4.83702
$569$	6.40589	0.268549	0.134274	−	0.990944i	$-0.457130\pi$
$569$	6.40589	0.268549	0.134274	+	0.990944i	$0.457130\pi$
$570$	0	0
$571$	25.9443	1.08574	0.542868	−	0.839818i	$-0.317338\pi$
$571$	25.9443	1.08574	0.542868	+	0.839818i	$0.317338\pi$
$572$	0	0
$573$	0	0
$574$	−13.3515	−0.557279
$575$	−10.4479	−0.435706
$576$	0	0
$577$	8.96220	0.373101	0.186551	−	0.982445i	$-0.440269\pi$
$577$	8.96220	0.373101	0.186551	+	0.982445i	$0.440269\pi$
$578$	−29.8860	−1.24309
$579$	0	0
$580$	71.9539	2.98772
$581$	5.86977	0.243519
$582$	0	0
$583$	0	0
$584$	−11.5039	−0.476037
$585$	0	0
$586$	22.6788	0.936852
$587$	−4.96137	−0.204778	−0.102389	−	0.994744i	$-0.532649\pi$
$587$	−4.96137	−0.204778	−0.102389	+	0.994744i	$0.532649\pi$
$588$	0	0
$589$	38.2332	1.57537
$590$	50.7655	2.08998
$591$	0	0
$592$	−84.1847	−3.45997
$593$	−30.7179	−1.26143	−0.630716	−	0.776013i	$-0.717239\pi$
$593$	−30.7179	−1.26143	−0.630716	+	0.776013i	$0.717239\pi$
$594$	0	0
$595$	4.10094	0.168122
$596$	2.57380	0.105427
$597$	0	0
$598$	20.2796	0.829294
$599$	32.7175	1.33680	0.668401	−	0.743801i	$-0.266979\pi$
$599$	32.7175	1.33680	0.668401	+	0.743801i	$0.266979\pi$
$600$	0	0
$601$	−40.7669	−1.66292	−0.831458	−	0.555587i	$-0.812493\pi$
$601$	−40.7669	−1.66292	−0.831458	+	0.555587i	$0.812493\pi$
$602$	−5.37036	−0.218880
$603$	0	0
$604$	4.92095	0.200230
$605$	0	0
$606$	0	0
$607$	−25.5265	−1.03609	−0.518043	−	0.855354i	$-0.673339\pi$
$607$	−25.5265	−1.03609	−0.518043	+	0.855354i	$0.673339\pi$
$608$	87.2792	3.53964
$609$	0	0
$610$	−62.7384	−2.54020
$611$	8.35963	0.338194
$612$	0	0
$613$	−37.0000	−1.49442	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	−37.0000	−1.49442	−0.747208	+	0.664590i	$0.768606\pi$
$614$	−9.27624	−0.374359
$615$	0	0
$616$	0	0
$617$	43.0676	1.73384	0.866918	−	0.498452i	$-0.166098\pi$
$617$	43.0676	1.73384	0.866918	+	0.498452i	$0.166098\pi$
$618$	0	0
$619$	14.5508	0.584847	0.292424	−	0.956289i	$-0.405538\pi$
$619$	14.5508	0.584847	0.292424	+	0.956289i	$0.405538\pi$
$620$	−52.0657	−2.09101
$621$	0	0
$622$	9.78672	0.392412
$623$	16.4241	0.658016
$624$	0	0
$625$	−10.4084	−0.416335
$626$	−69.6514	−2.78383
$627$	0	0
$628$	−16.5562	−0.660666
$629$	−17.8913	−0.713372
$630$	0	0
$631$	6.82874	0.271848	0.135924	−	0.990719i	$-0.456600\pi$
$631$	6.82874	0.271848	0.135924	+	0.990719i	$0.456600\pi$
$632$	54.0276	2.14910
$633$	0	0
$634$	−34.6881	−1.37764
$635$	1.90795	0.0757148
$636$	0	0
$637$	1.51161	0.0598924
$638$	0	0
$639$	0	0
$640$	−16.5100	−0.652616
$641$	11.4167	0.450934	0.225467	−	0.974251i	$-0.427609\pi$
$641$	11.4167	0.450934	0.225467	+	0.974251i	$0.427609\pi$
$642$	0	0
$643$	−36.2291	−1.42874	−0.714368	−	0.699770i	$-0.753286\pi$
$643$	−36.2291	−1.42874	−0.714368	+	0.699770i	$0.753286\pi$
$644$	−25.4605	−1.00328
$645$	0	0
$646$	40.1933	1.58138
$647$	24.1531	0.949557	0.474778	−	0.880105i	$-0.342528\pi$
$647$	24.1531	0.949557	0.474778	+	0.880105i	$0.342528\pi$
$648$	0	0
$649$	0	0
$650$	−8.27908	−0.324732
$651$	0	0
$652$	17.5257	0.686358
$653$	18.3457	0.717922	0.358961	−	0.933353i	$-0.383131\pi$
$653$	18.3457	0.717922	0.358961	+	0.933353i	$0.383131\pi$
$654$	0	0
$655$	−24.6418	−0.962834
$656$	56.7093	2.21413
$657$	0	0
$658$	−14.6660	−0.571740
$659$	9.32284	0.363166	0.181583	−	0.983376i	$-0.441878\pi$
$659$	9.32284	0.363166	0.181583	+	0.983376i	$0.441878\pi$
$660$	0	0
$661$	−9.30120	−0.361775	−0.180887	−	0.983504i	$-0.557897\pi$
$661$	−9.30120	−0.361775	−0.180887	+	0.983504i	$0.557897\pi$
$662$	21.0479	0.818051
$663$	0	0
$664$	−47.2106	−1.83213
$665$	10.8461	0.420593
$666$	0	0
$667$	−42.2189	−1.63472
$668$	23.0242	0.890832
$669$	0	0
$670$	−61.8647	−2.39004
$671$	0	0
$672$	0	0
$673$	35.9318	1.38507	0.692535	−	0.721385i	$-0.256494\pi$
$673$	35.9318	1.38507	0.692535	+	0.721385i	$0.256494\pi$
$674$	26.5210	1.02155
$675$	0	0
$676$	−53.9272	−2.07412
$677$	47.4738	1.82456	0.912282	−	0.409562i	$-0.134318\pi$
$677$	47.4738	1.82456	0.912282	+	0.409562i	$0.134318\pi$
$678$	0	0
$679$	−12.0516	−0.462498
$680$	−32.9839	−1.26488
$681$	0	0
$682$	0	0
$683$	30.1415	1.15333	0.576667	−	0.816980i	$-0.304353\pi$
$683$	30.1415	1.15333	0.576667	+	0.816980i	$0.304353\pi$
$684$	0	0
$685$	4.67320	0.178554
$686$	−2.65195	−0.101252
$687$	0	0
$688$	22.8102	0.869630
$689$	−9.42837	−0.359192
$690$	0	0
$691$	45.7782	1.74149	0.870743	−	0.491739i	$-0.163638\pi$
$691$	45.7782	1.74149	0.870743	+	0.491739i	$0.163638\pi$
$692$	−28.8118	−1.09526
$693$	0	0
$694$	14.3432	0.544459
$695$	−0.604973	−0.0229479
$696$	0	0
$697$	12.0521	0.456505
$698$	−75.6631	−2.86389
$699$	0	0
$700$	10.3942	0.392863
$701$	−12.4242	−0.469257	−0.234629	−	0.972085i	$-0.575387\pi$
$701$	−12.4242	−0.469257	−0.234629	+	0.972085i	$0.575387\pi$
$702$	0	0
$703$	−47.3184	−1.78465
$704$	0	0
$705$	0	0
$706$	50.9776	1.91857
$707$	−5.34845	−0.201149
$708$	0	0
$709$	−4.52276	−0.169856	−0.0849279	−	0.996387i	$-0.527066\pi$
$709$	−4.52276	−0.169856	−0.0849279	+	0.996387i	$0.527066\pi$
$710$	−65.1155	−2.44374
$711$	0	0
$712$	−132.099	−4.95061
$713$	30.5495	1.14409
$714$	0	0
$715$	0	0
$716$	−14.8822	−0.556174
$717$	0	0
$718$	−59.2786	−2.21226
$719$	8.32032	0.310296	0.155148	−	0.987891i	$-0.450415\pi$
$719$	8.32032	0.310296	0.155148	+	0.987891i	$0.450415\pi$
$720$	0	0
$721$	−9.37773	−0.349245
$722$	55.9152	2.08095
$723$	0	0
$724$	−47.3091	−1.75823
$725$	17.2357	0.640119
$726$	0	0
$727$	39.9612	1.48208	0.741040	−	0.671461i	$-0.234333\pi$
$727$	39.9612	1.48208	0.741040	+	0.671461i	$0.234333\pi$
$728$	−12.1579	−0.450603
$729$	0	0
$730$	6.49799	0.240501
$731$	4.84771	0.179299
$732$	0	0
$733$	−13.2877	−0.490793	−0.245396	−	0.969423i	$-0.578918\pi$
$733$	−13.2877	−0.490793	−0.245396	+	0.969423i	$0.578918\pi$
$734$	−62.1746	−2.29491
$735$	0	0
$736$	69.7388	2.57061
$737$	0	0
$738$	0	0
$739$	−6.54579	−0.240791	−0.120395	−	0.992726i	$-0.538416\pi$
$739$	−6.54579	−0.240791	−0.120395	+	0.992726i	$0.538416\pi$
$740$	64.4379	2.36879
$741$	0	0
$742$	16.5410	0.607239
$743$	33.2519	1.21989	0.609947	−	0.792442i	$-0.291191\pi$
$743$	33.2519	1.21989	0.609947	+	0.792442i	$0.291191\pi$
$744$	0	0
$745$	−0.876081	−0.0320971
$746$	−88.7175	−3.24818
$747$	0	0
$748$	0	0
$749$	−12.1711	−0.444721
$750$	0	0
$751$	−0.769312	−0.0280726	−0.0140363	−	0.999901i	$-0.504468\pi$
$751$	−0.769312	−0.0280726	−0.0140363	+	0.999901i	$0.504468\pi$
$752$	62.2927	2.27158
$753$	0	0
$754$	−33.4551	−1.21836
$755$	−1.67501	−0.0609600
$756$	0	0
$757$	−2.29424	−0.0833854	−0.0416927	−	0.999130i	$-0.513275\pi$
$757$	−2.29424	−0.0833854	−0.0416927	+	0.999130i	$0.513275\pi$
$758$	−48.7417	−1.77038
$759$	0	0
$760$	−87.2351	−3.16435
$761$	3.99916	0.144970	0.0724848	−	0.997370i	$-0.476907\pi$
$761$	3.99916	0.144970	0.0724848	+	0.997370i	$0.476907\pi$
$762$	0	0
$763$	6.31429	0.228592
$764$	46.3481	1.67682
$765$	0	0
$766$	−56.5936	−2.04481
$767$	−16.8911	−0.609904
$768$	0	0
$769$	48.3136	1.74223	0.871116	−	0.491077i	$-0.163396\pi$
$769$	48.3136	1.74223	0.871116	+	0.491077i	$0.163396\pi$
$770$	0	0
$771$	0	0
$772$	−7.51010	−0.270294
$773$	13.4380	0.483331	0.241666	−	0.970360i	$-0.422306\pi$
$773$	13.4380	0.483331	0.241666	+	0.970360i	$0.422306\pi$
$774$	0	0
$775$	−12.4717	−0.447998
$776$	96.9310	3.47962
$777$	0	0
$778$	−41.1622	−1.47574
$779$	31.8751	1.14204
$780$	0	0
$781$	0	0
$782$	32.1157	1.14845
$783$	0	0
$784$	11.2640	0.402284
$785$	5.63549	0.201139
$786$	0	0
$787$	−7.35759	−0.262270	−0.131135	−	0.991365i	$-0.541862\pi$
$787$	−7.35759	−0.262270	−0.131135	+	0.991365i	$0.541862\pi$
$788$	−16.9969	−0.605489
$789$	0	0
$790$	−30.5174	−1.08576
$791$	3.36767	0.119741
$792$	0	0
$793$	20.8749	0.741288
$794$	56.9579	2.02136
$795$	0	0
$796$	139.971	4.96114
$797$	4.82640	0.170960	0.0854799	−	0.996340i	$-0.472758\pi$
$797$	4.82640	0.170960	0.0854799	+	0.996340i	$0.472758\pi$
$798$	0	0
$799$	13.2387	0.468351
$800$	−28.4707	−1.00659
$801$	0	0
$802$	−23.5522	−0.831655
$803$	0	0
$804$	0	0
$805$	8.66636	0.305449
$806$	24.2080	0.852691
$807$	0	0
$808$	43.0176	1.51335
$809$	52.2852	1.83825	0.919125	−	0.393965i	$-0.128897\pi$
$809$	52.2852	1.83825	0.919125	+	0.393965i	$0.128897\pi$
$810$	0	0
$811$	−35.3236	−1.24038	−0.620189	−	0.784453i	$-0.712944\pi$
$811$	−35.3236	−1.24038	−0.620189	+	0.784453i	$0.712944\pi$
$812$	42.0020	1.47398
$813$	0	0
$814$	0	0
$815$	−5.96547	−0.208961
$816$	0	0
$817$	12.8211	0.448554
$818$	14.4737	0.506062
$819$	0	0
$820$	−43.4073	−1.51585
$821$	−30.4361	−1.06223	−0.531113	−	0.847301i	$-0.678226\pi$
$821$	−30.4361	−1.06223	−0.531113	+	0.847301i	$0.678226\pi$
$822$	0	0
$823$	45.8835	1.59940	0.799700	−	0.600400i	$-0.204992\pi$
$823$	45.8835	1.59940	0.799700	+	0.600400i	$0.204992\pi$
$824$	75.4251	2.62756
$825$	0	0
$826$	29.6336	1.03108
$827$	−12.2298	−0.425270	−0.212635	−	0.977132i	$-0.568205\pi$
$827$	−12.2298	−0.425270	−0.212635	+	0.977132i	$0.568205\pi$
$828$	0	0
$829$	19.7416	0.685655	0.342828	−	0.939398i	$-0.388615\pi$
$829$	19.7416	0.685655	0.342828	+	0.939398i	$0.388615\pi$
$830$	26.6668	0.925620
$831$	0	0
$832$	21.2088	0.735284
$833$	2.39386	0.0829424
$834$	0	0
$835$	−7.83707	−0.271213
$836$	0	0
$837$	0	0
$838$	−52.9875	−1.83042
$839$	−9.97411	−0.344345	−0.172172	−	0.985067i	$-0.555079\pi$
$839$	−9.97411	−0.344345	−0.172172	+	0.985067i	$0.555079\pi$
$840$	0	0
$841$	40.6481	1.40166
$842$	−75.0827	−2.58752
$843$	0	0
$844$	−120.823	−4.15890
$845$	18.3560	0.631465
$846$	0	0
$847$	0	0
$848$	−70.2565	−2.41262
$849$	0	0
$850$	−13.1111	−0.449708
$851$	−37.8089	−1.29607
$852$	0	0
$853$	−0.698961	−0.0239320	−0.0119660	−	0.999928i	$-0.503809\pi$
$853$	−0.698961	−0.0239320	−0.0119660	+	0.999928i	$0.503809\pi$
$854$	−36.6225	−1.25320
$855$	0	0
$856$	97.8920	3.34588
$857$	40.0835	1.36922	0.684612	−	0.728907i	$-0.259972\pi$
$857$	40.0835	1.36922	0.684612	+	0.728907i	$0.259972\pi$
$858$	0	0
$859$	−28.0045	−0.955503	−0.477751	−	0.878495i	$-0.658548\pi$
$859$	−28.0045	−0.955503	−0.477751	+	0.878495i	$0.658548\pi$
$860$	−17.4597	−0.595371
$861$	0	0
$862$	−84.2386	−2.86918
$863$	26.0484	0.886698	0.443349	−	0.896349i	$-0.353790\pi$
$863$	26.0484	0.886698	0.443349	+	0.896349i	$0.353790\pi$
$864$	0	0
$865$	9.80710	0.333452
$866$	−34.9160	−1.18649
$867$	0	0
$868$	−30.3925	−1.03159
$869$	0	0
$870$	0	0
$871$	20.5842	0.697468
$872$	−50.7858	−1.71983
$873$	0	0
$874$	84.9388	2.87310
$875$	−12.1036	−0.409175
$876$	0	0
$877$	−42.6421	−1.43992	−0.719961	−	0.694015i	$-0.755840\pi$
$877$	−42.6421	−1.43992	−0.719961	+	0.694015i	$0.755840\pi$
$878$	−26.0199	−0.878130
$879$	0	0
$880$	0	0
$881$	5.07985	0.171144	0.0855722	−	0.996332i	$-0.472728\pi$
$881$	5.07985	0.171144	0.0855722	+	0.996332i	$0.472728\pi$
$882$	0	0
$883$	49.0484	1.65061	0.825305	−	0.564687i	$-0.191003\pi$
$883$	49.0484	1.65061	0.825305	+	0.564687i	$0.191003\pi$
$884$	18.2119	0.612532
$885$	0	0
$886$	−6.59047	−0.221411
$887$	−53.9270	−1.81069	−0.905346	−	0.424675i	$-0.860388\pi$
$887$	−53.9270	−1.81069	−0.905346	+	0.424675i	$0.860388\pi$
$888$	0	0
$889$	1.11374	0.0373536
$890$	74.6159	2.50113
$891$	0	0
$892$	114.839	3.84509
$893$	35.0134	1.17168
$894$	0	0
$895$	5.06567	0.169327
$896$	−9.63748	−0.321965
$897$	0	0
$898$	41.3962	1.38141
$899$	−50.3972	−1.68084
$900$	0	0
$901$	−14.9312	−0.497430
$902$	0	0
$903$	0	0
$904$	−27.0862	−0.900874
$905$	16.1033	0.535291
$906$	0	0
$907$	25.7626	0.855435	0.427717	−	0.903913i	$-0.359318\pi$
$907$	25.7626	0.855435	0.427717	+	0.903913i	$0.359318\pi$
$908$	−72.4966	−2.40589
$909$	0	0
$910$	6.86739	0.227652
$911$	29.1872	0.967015	0.483507	−	0.875340i	$-0.339363\pi$
$911$	29.1872	0.967015	0.483507	+	0.875340i	$0.339363\pi$
$912$	0	0
$913$	0	0
$914$	36.2396	1.19870
$915$	0	0
$916$	−19.0930	−0.630851
$917$	−14.3843	−0.475010
$918$	0	0
$919$	−40.5202	−1.33664	−0.668320	−	0.743874i	$-0.732986\pi$
$919$	−40.5202	−1.33664	−0.668320	+	0.743874i	$0.732986\pi$
$920$	−69.7036	−2.29806
$921$	0	0
$922$	−49.4708	−1.62923
$923$	21.6658	0.713139
$924$	0	0
$925$	15.4354	0.507512
$926$	−49.2459	−1.61832
$927$	0	0
$928$	−115.048	−3.77662
$929$	−5.13344	−0.168423	−0.0842113	−	0.996448i	$-0.526837\pi$
$929$	−5.13344	−0.168423	−0.0842113	+	0.996448i	$0.526837\pi$
$930$	0	0
$931$	6.33123	0.207498
$932$	−138.325	−4.53099
$933$	0	0
$934$	−92.7066	−3.03345
$935$	0	0
$936$	0	0
$937$	−25.3284	−0.827443	−0.413722	−	0.910403i	$-0.635771\pi$
$937$	−25.3284	−0.827443	−0.413722	+	0.910403i	$0.635771\pi$
$938$	−36.1125	−1.17912
$939$	0	0
$940$	−47.6810	−1.55518
$941$	−21.2749	−0.693541	−0.346770	−	0.937950i	$-0.612722\pi$
$941$	−21.2749	−0.693541	−0.346770	+	0.937950i	$0.612722\pi$
$942$	0	0
$943$	25.4692	0.829390
$944$	−125.866	−4.09660
$945$	0	0
$946$	0	0
$947$	30.1418	0.979476	0.489738	−	0.871870i	$-0.337092\pi$
$947$	30.1418	0.979476	0.489738	+	0.871870i	$0.337092\pi$
$948$	0	0
$949$	−2.16207	−0.0701837
$950$	−34.6760	−1.12504
$951$	0	0
$952$	−19.2539	−0.624021
$953$	−8.93028	−0.289280	−0.144640	−	0.989484i	$-0.546202\pi$
$953$	−8.93028	−0.289280	−0.144640	+	0.989484i	$0.546202\pi$
$954$	0	0
$955$	−15.7762	−0.510506
$956$	−56.5367	−1.82853
$957$	0	0
$958$	−107.522	−3.47387
$959$	2.72791	0.0880887
$960$	0	0
$961$	5.46735	0.176366
$962$	−29.9605	−0.965965
$963$	0	0
$964$	74.2326	2.39087
$965$	2.55632	0.0822909
$966$	0	0
$967$	−38.3461	−1.23313	−0.616564	−	0.787305i	$-0.711476\pi$
$967$	−38.3461	−1.23313	−0.616564	+	0.787305i	$0.711476\pi$
$968$	0	0
$969$	0	0
$970$	−54.7514	−1.75796
$971$	−31.1508	−0.999676	−0.499838	−	0.866119i	$-0.666607\pi$
$971$	−31.1508	−0.999676	−0.499838	+	0.866119i	$0.666607\pi$
$972$	0	0
$973$	−0.353144	−0.0113213
$974$	35.9522	1.15198
$975$	0	0
$976$	155.551	4.97908
$977$	−29.7663	−0.952310	−0.476155	−	0.879361i	$-0.657970\pi$
$977$	−29.7663	−0.952310	−0.476155	+	0.879361i	$0.657970\pi$
$978$	0	0
$979$	0	0
$980$	−8.62183	−0.275414
$981$	0	0
$982$	62.2132	1.98530
$983$	39.9068	1.27283	0.636415	−	0.771347i	$-0.280417\pi$
$983$	39.9068	1.27283	0.636415	+	0.771347i	$0.280417\pi$
$984$	0	0
$985$	5.78548	0.184341
$986$	−52.9810	−1.68726
$987$	0	0
$988$	48.1664	1.53238
$989$	10.2445	0.325755
$990$	0	0
$991$	−10.7771	−0.342345	−0.171173	−	0.985241i	$-0.554756\pi$
$991$	−10.7771	−0.342345	−0.171173	+	0.985241i	$0.554756\pi$
$992$	83.2482	2.64313
$993$	0	0
$994$	−38.0102	−1.20561
$995$	−47.6440	−1.51042
$996$	0	0
$997$	1.13922	0.0360796	0.0180398	−	0.999837i	$-0.494257\pi$
$997$	1.13922	0.0360796	0.0180398	+	0.999837i	$0.494257\pi$
$998$	45.8459	1.45123
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cx.1.10		10
3.2	odd	2		2541.2.a.bq.1.1		10
11.5	even	5		693.2.m.j.190.1		20
11.9	even	5		693.2.m.j.631.1		20
11.10	odd	2		7623.2.a.cy.1.1		10
33.5	odd	10		231.2.j.g.190.5	yes	20
33.20	odd	10		231.2.j.g.169.5	✓	20
33.32	even	2		2541.2.a.br.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.g.169.5	✓	20	33.20	odd	10
231.2.j.g.190.5	yes	20	33.5	odd	10
693.2.m.j.190.1		20	11.5	even	5
693.2.m.j.631.1		20	11.9	even	5
2541.2.a.bq.1.1		10	3.2	odd	2
2541.2.a.br.1.10		10	33.32	even	2
7623.2.a.cx.1.10		10	1.1	even	1	trivial
7623.2.a.cy.1.1		10	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)

\(f(q)\)	\(=\)	\(q+2.65195 q^{2} +5.03286 q^{4} -1.71311 q^{5} -1.00000 q^{7} +8.04301 q^{8} -4.54308 q^{10} +1.51161 q^{13} -2.65195 q^{14} +11.2640 q^{16} +2.39386 q^{17} +6.33123 q^{19} -8.62183 q^{20} +5.05885 q^{23} -2.06526 q^{25} +4.00873 q^{26} -5.03286 q^{28} -8.34554 q^{29} +6.03882 q^{31} +13.7855 q^{32} +6.34841 q^{34} +1.71311 q^{35} -7.47381 q^{37} +16.7901 q^{38} -13.7785 q^{40} +5.03457 q^{41} +2.02506 q^{43} +13.4158 q^{46} +5.53026 q^{47} +1.00000 q^{49} -5.47698 q^{50} +7.60775 q^{52} -6.23728 q^{53} -8.04301 q^{56} -22.1320 q^{58} -11.1742 q^{59} +13.8096 q^{61} +16.0147 q^{62} +14.0306 q^{64} -2.58956 q^{65} +13.6173 q^{67} +12.0480 q^{68} +4.54308 q^{70} +14.3329 q^{71} -1.43030 q^{73} -19.8202 q^{74} +31.8642 q^{76} +6.71734 q^{79} -19.2964 q^{80} +13.3515 q^{82} -5.86977 q^{83} -4.10094 q^{85} +5.37036 q^{86} -16.4241 q^{89} -1.51161 q^{91} +25.4605 q^{92} +14.6660 q^{94} -10.8461 q^{95} +12.0516 q^{97} +2.65195 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 18 q^{4} - 5 q^{5} - 10 q^{7} - 3 q^{8} + 6 q^{10} - 6 q^{13} + 38 q^{16} + 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} - 18 q^{28} - 14 q^{29} + 26 q^{31} - 41 q^{32} + 21 q^{34} + 5 q^{35} + 24 q^{37}+ \cdots + 24 q^{97}+O(q^{100}) \) 10 * q + 18 * q^4 - 5 * q^5 - 10 * q^7 - 3 * q^8 + 6 * q^10 - 6 * q^13 + 38 * q^16 + 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 - 18 * q^28 - 14 * q^29 + 26 * q^31 - 41 * q^32 + 21 * q^34 + 5 * q^35 + 24 * q^37 - 8 * q^38 + 5 * q^40 + 19 * q^41 + 6 * q^43 + q^46 - 15 * q^47 + 10 * q^49 - q^50 + 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 + 11 * q^62 + 53 * q^64 - 29 * q^65 + 38 * q^67 + 87 * q^68 - 6 * q^70 - 26 * q^71 + q^73 - 39 * q^74 + 2 * q^76 - 5 * q^79 - 6 * q^80 + 5 * q^82 + 6 * q^83 + q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 - 42 * q^95 + 24 * q^97

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