LMFDB - Embedded newform 7623.2.a.ct.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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 $ x^8 - x^7 - 11x^6 + 10x^5 + 35x^4 - 30x^3 - 30x^2 + 30x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.3
Root			$1.11447$ 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)

$f(q)$	$=$	$q-1.11447 q^{2} -0.757964 q^{4} -3.45608 q^{5} +1.00000 q^{7} +3.07366 q^{8} +O(q^{10})$	$q-1.11447 q^{2} -0.757964 q^{4} -3.45608 q^{5} +1.00000 q^{7} +3.07366 q^{8} +3.85168 q^{10} -2.05965 q^{13} -1.11447 q^{14} -1.90956 q^{16} -1.93373 q^{17} +1.62296 q^{19} +2.61958 q^{20} +0.807136 q^{23} +6.94447 q^{25} +2.29541 q^{26} -0.757964 q^{28} -7.97368 q^{29} +0.788420 q^{31} -4.01918 q^{32} +2.15508 q^{34} -3.45608 q^{35} +10.0618 q^{37} -1.80873 q^{38} -10.6228 q^{40} +2.12613 q^{41} +3.08043 q^{43} -0.899526 q^{46} -7.56632 q^{47} +1.00000 q^{49} -7.73938 q^{50} +1.56114 q^{52} -10.8224 q^{53} +3.07366 q^{56} +8.88640 q^{58} +3.29664 q^{59} -1.07663 q^{61} -0.878667 q^{62} +8.29836 q^{64} +7.11832 q^{65} +2.40314 q^{67} +1.46570 q^{68} +3.85168 q^{70} +3.18859 q^{71} +1.22628 q^{73} -11.2135 q^{74} -1.23015 q^{76} -9.48182 q^{79} +6.59959 q^{80} -2.36950 q^{82} +16.0694 q^{83} +6.68312 q^{85} -3.43303 q^{86} +4.43830 q^{89} -2.05965 q^{91} -0.611780 q^{92} +8.43241 q^{94} -5.60908 q^{95} +6.46807 q^{97} -1.11447 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7}+O(q^{10}) $ 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7	$ 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7} + 6 q^{10} - 6 q^{13} - q^{14} + q^{16} + 5 q^{17} - 13 q^{19} - 23 q^{20} - 16 q^{23} + 16 q^{25} + 6 q^{26} + 7 q^{28} - 9 q^{29} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} + 7 q^{37} + 10 q^{38} + 5 q^{40} + 10 q^{41} - 4 q^{43} + 4 q^{46} - 16 q^{47} + 8 q^{49} - 6 q^{50} - 41 q^{52} - 37 q^{53} - 15 q^{58} - q^{59} + 19 q^{61} + 18 q^{62} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 6 q^{70} - 13 q^{71} - 25 q^{73} - 33 q^{74} + 26 q^{76} - 4 q^{80} - 13 q^{82} + 25 q^{83} + 3 q^{85} - 4 q^{86} - 37 q^{89} - 6 q^{91} - 35 q^{92} - 42 q^{94} - 21 q^{95} + 15 q^{97} - q^{98}+O(q^{100}) $ 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7 + 6 * q^10 - 6 * q^13 - q^14 + q^16 + 5 * q^17 - 13 * q^19 - 23 * q^20 - 16 * q^23 + 16 * q^25 + 6 * q^26 + 7 * q^28 - 9 * q^29 + 9 * q^31 - 16 * q^32 - 12 * q^34 - 10 * q^35 + 7 * q^37 + 10 * q^38 + 5 * q^40 + 10 * q^41 - 4 * q^43 + 4 * q^46 - 16 * q^47 + 8 * q^49 - 6 * q^50 - 41 * q^52 - 37 * q^53 - 15 * q^58 - q^59 + 19 * q^61 + 18 * q^62 - 4 * q^64 + 4 * q^65 - 19 * q^67 - 9 * q^68 + 6 * q^70 - 13 * q^71 - 25 * q^73 - 33 * q^74 + 26 * q^76 - 4 * q^80 - 13 * q^82 + 25 * q^83 + 3 * q^85 - 4 * q^86 - 37 * q^89 - 6 * q^91 - 35 * q^92 - 42 * q^94 - 21 * q^95 + 15 * q^97 - 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 $

$2$	−1.11447	−0.788047	−0.394023	−	0.919100i	$-0.628917\pi$
$2$	−1.11447	−0.788047	−0.394023	+	0.919100i	$0.628917\pi$
$3$	0	0
$3$	0	0
$4$	−0.757964	−0.378982
$5$	−3.45608	−1.54561	−0.772803	−	0.634647i	$-0.781146\pi$
$5$	−3.45608	−1.54561	−0.772803	+	0.634647i	$0.781146\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.07366	1.08670
$9$	0	0
$10$	3.85168	1.21801
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.05965	−0.571245	−0.285622	−	0.958342i	$-0.592200\pi$
$13$	−2.05965	−0.571245	−0.285622	+	0.958342i	$0.592200\pi$
$14$	−1.11447	−0.297854
$15$	0	0
$16$	−1.90956	−0.477390
$17$	−1.93373	−0.468998	−0.234499	−	0.972116i	$-0.575345\pi$
$17$	−1.93373	−0.468998	−0.234499	+	0.972116i	$0.575345\pi$
$18$	0	0
$19$	1.62296	0.372333	0.186166	−	0.982518i	$-0.440394\pi$
$19$	1.62296	0.372333	0.186166	+	0.982518i	$0.440394\pi$
$20$	2.61958	0.585757
$21$	0	0
$22$	0	0
$23$	0.807136	0.168299	0.0841497	−	0.996453i	$-0.473183\pi$
$23$	0.807136	0.168299	0.0841497	+	0.996453i	$0.473183\pi$
$24$	0	0
$25$	6.94447	1.38889
$26$	2.29541	0.450168
$27$	0	0
$28$	−0.757964	−0.143242
$29$	−7.97368	−1.48067	−0.740337	−	0.672235i	$-0.765334\pi$
$29$	−7.97368	−1.48067	−0.740337	+	0.672235i	$0.765334\pi$
$30$	0	0
$31$	0.788420	0.141604	0.0708022	−	0.997490i	$-0.477444\pi$
$31$	0.788420	0.141604	0.0708022	+	0.997490i	$0.477444\pi$
$32$	−4.01918	−0.710497
$33$	0	0
$34$	2.15508	0.369592
$35$	−3.45608	−0.584184
$36$	0	0
$37$	10.0618	1.65414	0.827072	−	0.562096i	$-0.190005\pi$
$37$	10.0618	1.65414	0.827072	+	0.562096i	$0.190005\pi$
$38$	−1.80873	−0.293415
$39$	0	0
$40$	−10.6228	−1.67961
$41$	2.12613	0.332046	0.166023	−	0.986122i	$-0.446907\pi$
$41$	2.12613	0.332046	0.166023	+	0.986122i	$0.446907\pi$
$42$	0	0
$43$	3.08043	0.469761	0.234880	−	0.972024i	$-0.424530\pi$
$43$	3.08043	0.469761	0.234880	+	0.972024i	$0.424530\pi$
$44$	0	0
$45$	0	0
$46$	−0.899526	−0.132628
$47$	−7.56632	−1.10366	−0.551831	−	0.833956i	$-0.686070\pi$
$47$	−7.56632	−1.10366	−0.551831	+	0.833956i	$0.686070\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−7.73938	−1.09451
$51$	0	0
$52$	1.56114	0.216492
$53$	−10.8224	−1.48658	−0.743289	−	0.668971i	$-0.766735\pi$
$53$	−10.8224	−1.48658	−0.743289	+	0.668971i	$0.766735\pi$
$54$	0	0
$55$	0	0
$56$	3.07366	0.410735
$57$	0	0
$58$	8.88640	1.16684
$59$	3.29664	0.429186	0.214593	−	0.976704i	$-0.431158\pi$
$59$	3.29664	0.429186	0.214593	+	0.976704i	$0.431158\pi$
$60$	0	0
$61$	−1.07663	−0.137848	−0.0689240	−	0.997622i	$-0.521957\pi$
$61$	−1.07663	−0.137848	−0.0689240	+	0.997622i	$0.521957\pi$
$62$	−0.878667	−0.111591
$63$	0	0
$64$	8.29836	1.03729
$65$	7.11832	0.882919
$66$	0	0
$67$	2.40314	0.293590	0.146795	−	0.989167i	$-0.453104\pi$
$67$	2.40314	0.293590	0.146795	+	0.989167i	$0.453104\pi$
$68$	1.46570	0.177742
$69$	0	0
$70$	3.85168	0.460364
$71$	3.18859	0.378417	0.189208	−	0.981937i	$-0.439408\pi$
$71$	3.18859	0.378417	0.189208	+	0.981937i	$0.439408\pi$
$72$	0	0
$73$	1.22628	0.143525	0.0717624	−	0.997422i	$-0.477138\pi$
$73$	1.22628	0.143525	0.0717624	+	0.997422i	$0.477138\pi$
$74$	−11.2135	−1.30354
$75$	0	0
$76$	−1.23015	−0.141107
$77$	0	0
$78$	0	0
$79$	−9.48182	−1.06679	−0.533394	−	0.845867i	$-0.679084\pi$
$79$	−9.48182	−1.06679	−0.533394	+	0.845867i	$0.679084\pi$
$80$	6.59959	0.737857
$81$	0	0
$82$	−2.36950	−0.261668
$83$	16.0694	1.76385	0.881923	−	0.471394i	$-0.156249\pi$
$83$	16.0694	1.76385	0.881923	+	0.471394i	$0.156249\pi$
$84$	0	0
$85$	6.68312	0.724886
$86$	−3.43303	−0.370193
$87$	0	0
$88$	0	0
$89$	4.43830	0.470459	0.235230	−	0.971940i	$-0.424416\pi$
$89$	4.43830	0.470459	0.235230	+	0.971940i	$0.424416\pi$
$90$	0	0
$91$	−2.05965	−0.215910
$92$	−0.611780	−0.0637825
$93$	0	0
$94$	8.43241	0.869737
$95$	−5.60908	−0.575479
$96$	0	0
$97$	6.46807	0.656733	0.328366	−	0.944550i	$-0.393502\pi$
$97$	6.46807	0.656733	0.328366	+	0.944550i	$0.393502\pi$
$98$	−1.11447	−0.112578
$99$	0	0
$100$	−5.26366	−0.526366
$101$	15.4449	1.53682	0.768412	−	0.639955i	$-0.221047\pi$
$101$	15.4449	1.53682	0.768412	+	0.639955i	$0.221047\pi$
$102$	0	0
$103$	8.90812	0.877743	0.438872	−	0.898550i	$-0.355378\pi$
$103$	8.90812	0.877743	0.438872	+	0.898550i	$0.355378\pi$
$104$	−6.33067	−0.620773
$105$	0	0
$106$	12.0613	1.17149
$107$	−3.51219	−0.339536	−0.169768	−	0.985484i	$-0.554302\pi$
$107$	−3.51219	−0.339536	−0.169768	+	0.985484i	$0.554302\pi$
$108$	0	0
$109$	3.87655	0.371306	0.185653	−	0.982615i	$-0.440560\pi$
$109$	3.87655	0.371306	0.185653	+	0.982615i	$0.440560\pi$
$110$	0	0
$111$	0	0
$112$	−1.90956	−0.180437
$113$	−10.6539	−1.00224	−0.501118	−	0.865379i	$-0.667078\pi$
$113$	−10.6539	−1.00224	−0.501118	+	0.865379i	$0.667078\pi$
$114$	0	0
$115$	−2.78952	−0.260124
$116$	6.04376	0.561149
$117$	0	0
$118$	−3.67399	−0.338218
$119$	−1.93373	−0.177265
$120$	0	0
$121$	0	0
$122$	1.19987	0.108631
$123$	0	0
$124$	−0.597594	−0.0536655
$125$	−6.72025	−0.601078
$126$	0	0
$127$	19.4509	1.72599	0.862994	−	0.505214i	$-0.168586\pi$
$127$	19.4509	1.72599	0.862994	+	0.505214i	$0.168586\pi$
$128$	−1.20989	−0.106941
$129$	0	0
$130$	−7.93313	−0.695782
$131$	−5.11284	−0.446711	−0.223355	−	0.974737i	$-0.571701\pi$
$131$	−5.11284	−0.446711	−0.223355	+	0.974737i	$0.571701\pi$
$132$	0	0
$133$	1.62296	0.140728
$134$	−2.67822	−0.231363
$135$	0	0
$136$	−5.94362	−0.509661
$137$	−9.10052	−0.777510	−0.388755	−	0.921341i	$-0.627095\pi$
$137$	−9.10052	−0.777510	−0.388755	+	0.921341i	$0.627095\pi$
$138$	0	0
$139$	−13.0166	−1.10405	−0.552026	−	0.833827i	$-0.686145\pi$
$139$	−13.0166	−1.10405	−0.552026	+	0.833827i	$0.686145\pi$
$140$	2.61958	0.221395
$141$	0	0
$142$	−3.55358	−0.298210
$143$	0	0
$144$	0	0
$145$	27.5577	2.28854
$146$	−1.36664	−0.113104
$147$	0	0
$148$	−7.62646	−0.626891
$149$	3.14650	0.257771	0.128886	−	0.991659i	$-0.458860\pi$
$149$	3.14650	0.257771	0.128886	+	0.991659i	$0.458860\pi$
$150$	0	0
$151$	2.86696	0.233310	0.116655	−	0.993172i	$-0.462783\pi$
$151$	2.86696	0.233310	0.116655	+	0.993172i	$0.462783\pi$
$152$	4.98843	0.404615
$153$	0	0
$154$	0	0
$155$	−2.72484	−0.218864
$156$	0	0
$157$	−21.4895	−1.71505	−0.857524	−	0.514443i	$-0.827999\pi$
$157$	−21.4895	−1.71505	−0.857524	+	0.514443i	$0.827999\pi$
$158$	10.5672	0.840679
$159$	0	0
$160$	13.8906	1.09815
$161$	0.807136	0.0636112
$162$	0	0
$163$	−8.22245	−0.644032	−0.322016	−	0.946734i	$-0.604361\pi$
$163$	−8.22245	−0.644032	−0.322016	+	0.946734i	$0.604361\pi$
$164$	−1.61153	−0.125840
$165$	0	0
$166$	−17.9088	−1.38999
$167$	21.7086	1.67986	0.839930	−	0.542695i	$-0.182596\pi$
$167$	21.7086	1.67986	0.839930	+	0.542695i	$0.182596\pi$
$168$	0	0
$169$	−8.75783	−0.673679
$170$	−7.44811	−0.571244
$171$	0	0
$172$	−2.33485	−0.178031
$173$	8.04563	0.611698	0.305849	−	0.952080i	$-0.401060\pi$
$173$	8.04563	0.611698	0.305849	+	0.952080i	$0.401060\pi$
$174$	0	0
$175$	6.94447	0.524953
$176$	0	0
$177$	0	0
$178$	−4.94634	−0.370744
$179$	3.62091	0.270640	0.135320	−	0.990802i	$-0.456794\pi$
$179$	3.62091	0.270640	0.135320	+	0.990802i	$0.456794\pi$
$180$	0	0
$181$	15.8179	1.17574	0.587868	−	0.808957i	$-0.299967\pi$
$181$	15.8179	1.17574	0.587868	+	0.808957i	$0.299967\pi$
$182$	2.29541	0.170147
$183$	0	0
$184$	2.48086	0.182891
$185$	−34.7743	−2.55665
$186$	0	0
$187$	0	0
$188$	5.73500	0.418268
$189$	0	0
$190$	6.25113	0.453504
$191$	−0.429081	−0.0310472	−0.0155236	−	0.999880i	$-0.504942\pi$
$191$	−0.429081	−0.0310472	−0.0155236	+	0.999880i	$0.504942\pi$
$192$	0	0
$193$	−15.1748	−1.09231	−0.546154	−	0.837685i	$-0.683909\pi$
$193$	−15.1748	−1.09231	−0.546154	+	0.837685i	$0.683909\pi$
$194$	−7.20845	−0.517536
$195$	0	0
$196$	−0.757964	−0.0541403
$197$	20.8082	1.48252	0.741262	−	0.671216i	$-0.234228\pi$
$197$	20.8082	1.48252	0.741262	+	0.671216i	$0.234228\pi$
$198$	0	0
$199$	8.44567	0.598698	0.299349	−	0.954144i	$-0.403231\pi$
$199$	8.44567	0.598698	0.299349	+	0.954144i	$0.403231\pi$
$200$	21.3449	1.50932
$201$	0	0
$202$	−17.2128	−1.21109
$203$	−7.97368	−0.559642
$204$	0	0
$205$	−7.34808	−0.513212
$206$	−9.92781	−0.691703
$207$	0	0
$208$	3.93303	0.272707
$209$	0	0
$210$	0	0
$211$	−9.85927	−0.678740	−0.339370	−	0.940653i	$-0.610214\pi$
$211$	−9.85927	−0.678740	−0.339370	+	0.940653i	$0.610214\pi$
$212$	8.20303	0.563386
$213$	0	0
$214$	3.91422	0.267570
$215$	−10.6462	−0.726065
$216$	0	0
$217$	0.788420	0.0535214
$218$	−4.32028	−0.292606
$219$	0	0
$220$	0	0
$221$	3.98281	0.267913
$222$	0	0
$223$	17.3959	1.16491	0.582457	−	0.812861i	$-0.302091\pi$
$223$	17.3959	1.16491	0.582457	+	0.812861i	$0.302091\pi$
$224$	−4.01918	−0.268542
$225$	0	0
$226$	11.8734	0.789809
$227$	12.6315	0.838381	0.419190	−	0.907898i	$-0.362314\pi$
$227$	12.6315	0.838381	0.419190	+	0.907898i	$0.362314\pi$
$228$	0	0
$229$	4.57341	0.302220	0.151110	−	0.988517i	$-0.451715\pi$
$229$	4.57341	0.302220	0.151110	+	0.988517i	$0.451715\pi$
$230$	3.10883	0.204990
$231$	0	0
$232$	−24.5084	−1.60905
$233$	−23.8569	−1.56292	−0.781458	−	0.623957i	$-0.785524\pi$
$233$	−23.8569	−1.56292	−0.781458	+	0.623957i	$0.785524\pi$
$234$	0	0
$235$	26.1498	1.70582
$236$	−2.49873	−0.162654
$237$	0	0
$238$	2.15508	0.139693
$239$	−8.83814	−0.571692	−0.285846	−	0.958276i	$-0.592275\pi$
$239$	−8.83814	−0.571692	−0.285846	+	0.958276i	$0.592275\pi$
$240$	0	0
$241$	18.9464	1.22045	0.610224	−	0.792229i	$-0.291079\pi$
$241$	18.9464	1.22045	0.610224	+	0.792229i	$0.291079\pi$
$242$	0	0
$243$	0	0
$244$	0.816045	0.0522419
$245$	−3.45608	−0.220801
$246$	0	0
$247$	−3.34273	−0.212693
$248$	2.42333	0.153882
$249$	0	0
$250$	7.48950	0.473678
$251$	2.86691	0.180958	0.0904790	−	0.995898i	$-0.471160\pi$
$251$	2.86691	0.180958	0.0904790	+	0.995898i	$0.471160\pi$
$252$	0	0
$253$	0	0
$254$	−21.6774	−1.36016
$255$	0	0
$256$	−15.2483	−0.953021
$257$	−22.4159	−1.39826	−0.699132	−	0.714993i	$-0.746430\pi$
$257$	−22.4159	−1.39826	−0.699132	+	0.714993i	$0.746430\pi$
$258$	0	0
$259$	10.0618	0.625208
$260$	−5.39543	−0.334611
$261$	0	0
$262$	5.69808	0.352029
$263$	−0.990706	−0.0610895	−0.0305448	−	0.999533i	$-0.509724\pi$
$263$	−0.990706	−0.0610895	−0.0305448	+	0.999533i	$0.509724\pi$
$264$	0	0
$265$	37.4032	2.29766
$266$	−1.80873	−0.110901
$267$	0	0
$268$	−1.82149	−0.111265
$269$	7.19036	0.438404	0.219202	−	0.975679i	$-0.429655\pi$
$269$	7.19036	0.438404	0.219202	+	0.975679i	$0.429655\pi$
$270$	0	0
$271$	−27.1643	−1.65011	−0.825056	−	0.565050i	$-0.808857\pi$
$271$	−27.1643	−1.65011	−0.825056	+	0.565050i	$0.808857\pi$
$272$	3.69257	0.223895
$273$	0	0
$274$	10.1422	0.612714
$275$	0	0
$276$	0	0
$277$	20.9856	1.26090	0.630451	−	0.776229i	$-0.282870\pi$
$277$	20.9856	1.26090	0.630451	+	0.776229i	$0.282870\pi$
$278$	14.5065	0.870045
$279$	0	0
$280$	−10.6228	−0.634834
$281$	28.1580	1.67977	0.839883	−	0.542768i	$-0.182624\pi$
$281$	28.1580	1.67977	0.839883	+	0.542768i	$0.182624\pi$
$282$	0	0
$283$	−26.1917	−1.55694	−0.778469	−	0.627684i	$-0.784003\pi$
$283$	−26.1917	−1.55694	−0.778469	+	0.627684i	$0.784003\pi$
$284$	−2.41684	−0.143413
$285$	0	0
$286$	0	0
$287$	2.12613	0.125502
$288$	0	0
$289$	−13.2607	−0.780041
$290$	−30.7121	−1.80348
$291$	0	0
$292$	−0.929473	−0.0543933
$293$	4.46385	0.260781	0.130391	−	0.991463i	$-0.458377\pi$
$293$	4.46385	0.260781	0.130391	+	0.991463i	$0.458377\pi$
$294$	0	0
$295$	−11.3934	−0.663351
$296$	30.9264	1.79756
$297$	0	0
$298$	−3.50667	−0.203136
$299$	−1.66242	−0.0961402
$300$	0	0
$301$	3.08043	0.177553
$302$	−3.19513	−0.183859
$303$	0	0
$304$	−3.09914	−0.177748
$305$	3.72091	0.213059
$306$	0	0
$307$	12.8841	0.735334	0.367667	−	0.929957i	$-0.380157\pi$
$307$	12.8841	0.735334	0.367667	+	0.929957i	$0.380157\pi$
$308$	0	0
$309$	0	0
$310$	3.03674	0.172475
$311$	−26.8199	−1.52081	−0.760407	−	0.649447i	$-0.775001\pi$
$311$	−26.8199	−1.52081	−0.760407	+	0.649447i	$0.775001\pi$
$312$	0	0
$313$	3.58869	0.202845	0.101422	−	0.994843i	$-0.467661\pi$
$313$	3.58869	0.202845	0.101422	+	0.994843i	$0.467661\pi$
$314$	23.9493	1.35154
$315$	0	0
$316$	7.18688	0.404294
$317$	16.9181	0.950213	0.475106	−	0.879928i	$-0.342410\pi$
$317$	16.9181	0.950213	0.475106	+	0.879928i	$0.342410\pi$
$318$	0	0
$319$	0	0
$320$	−28.6798	−1.60325
$321$	0	0
$322$	−0.899526	−0.0501286
$323$	−3.13836	−0.174623
$324$	0	0
$325$	−14.3032	−0.793399
$326$	9.16365	0.507528
$327$	0	0
$328$	6.53501	0.360836
$329$	−7.56632	−0.417145
$330$	0	0
$331$	−1.23826	−0.0680610	−0.0340305	−	0.999421i	$-0.510834\pi$
$331$	−1.23826	−0.0680610	−0.0340305	+	0.999421i	$0.510834\pi$
$332$	−12.1800	−0.668466
$333$	0	0
$334$	−24.1935	−1.32381
$335$	−8.30544	−0.453774
$336$	0	0
$337$	−20.4806	−1.11565	−0.557824	−	0.829959i	$-0.688364\pi$
$337$	−20.4806	−1.11565	−0.557824	+	0.829959i	$0.688364\pi$
$338$	9.76031	0.530891
$339$	0	0
$340$	−5.06556	−0.274719
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	9.46818	0.510490
$345$	0	0
$346$	−8.96659	−0.482047
$347$	−27.2699	−1.46392	−0.731961	−	0.681346i	$-0.761395\pi$
$347$	−27.2699	−1.46392	−0.731961	+	0.681346i	$0.761395\pi$
$348$	0	0
$349$	7.98434	0.427392	0.213696	−	0.976900i	$-0.431450\pi$
$349$	7.98434	0.427392	0.213696	+	0.976900i	$0.431450\pi$
$350$	−7.73938	−0.413688
$351$	0	0
$352$	0	0
$353$	−5.93472	−0.315873	−0.157937	−	0.987449i	$-0.550484\pi$
$353$	−5.93472	−0.315873	−0.157937	+	0.987449i	$0.550484\pi$
$354$	0	0
$355$	−11.0200	−0.584883
$356$	−3.36407	−0.178296
$357$	0	0
$358$	−4.03538	−0.213277
$359$	−28.4203	−1.49996	−0.749982	−	0.661458i	$-0.769938\pi$
$359$	−28.4203	−1.49996	−0.749982	+	0.661458i	$0.769938\pi$
$360$	0	0
$361$	−16.3660	−0.861368
$362$	−17.6285	−0.926535
$363$	0	0
$364$	1.56114	0.0818261
$365$	−4.23810	−0.221832
$366$	0	0
$367$	−30.4014	−1.58694	−0.793471	−	0.608608i	$-0.791728\pi$
$367$	−30.4014	−1.58694	−0.793471	+	0.608608i	$0.791728\pi$
$368$	−1.54128	−0.0803445
$369$	0	0
$370$	38.7547	2.01476
$371$	−10.8224	−0.561873
$372$	0	0
$373$	14.4226	0.746772	0.373386	−	0.927676i	$-0.378197\pi$
$373$	14.4226	0.746772	0.373386	+	0.927676i	$0.378197\pi$
$374$	0	0
$375$	0	0
$376$	−23.2563	−1.19935
$377$	16.4230	0.845828
$378$	0	0
$379$	−22.4072	−1.15098	−0.575490	−	0.817809i	$-0.695189\pi$
$379$	−22.4072	−1.15098	−0.575490	+	0.817809i	$0.695189\pi$
$380$	4.25148	0.218096
$381$	0	0
$382$	0.478196	0.0244667
$383$	−33.6785	−1.72089	−0.860446	−	0.509541i	$-0.829815\pi$
$383$	−33.6785	−1.72089	−0.860446	+	0.509541i	$0.829815\pi$
$384$	0	0
$385$	0	0
$386$	16.9118	0.860790
$387$	0	0
$388$	−4.90256	−0.248890
$389$	−2.42783	−0.123096	−0.0615479	−	0.998104i	$-0.519604\pi$
$389$	−2.42783	−0.123096	−0.0615479	+	0.998104i	$0.519604\pi$
$390$	0	0
$391$	−1.56078	−0.0789321
$392$	3.07366	0.155243
$393$	0	0
$394$	−23.1900	−1.16830
$395$	32.7699	1.64883
$396$	0	0
$397$	−5.89696	−0.295960	−0.147980	−	0.988990i	$-0.547277\pi$
$397$	−5.89696	−0.295960	−0.147980	+	0.988990i	$0.547277\pi$
$398$	−9.41242	−0.471802
$399$	0	0
$400$	−13.2609	−0.663045
$401$	−11.2396	−0.561278	−0.280639	−	0.959813i	$-0.590546\pi$
$401$	−11.2396	−0.561278	−0.280639	+	0.959813i	$0.590546\pi$
$402$	0	0
$403$	−1.62387	−0.0808908
$404$	−11.7067	−0.582429
$405$	0	0
$406$	8.88640	0.441025
$407$	0	0
$408$	0	0
$409$	29.3344	1.45049	0.725246	−	0.688490i	$-0.241726\pi$
$409$	29.3344	1.45049	0.725246	+	0.688490i	$0.241726\pi$
$410$	8.18919	0.404435
$411$	0	0
$412$	−6.75204	−0.332649
$413$	3.29664	0.162217
$414$	0	0
$415$	−55.5371	−2.72621
$416$	8.27811	0.405868
$417$	0	0
$418$	0	0
$419$	20.2858	0.991027	0.495514	−	0.868600i	$-0.334980\pi$
$419$	20.2858	0.991027	0.495514	+	0.868600i	$0.334980\pi$
$420$	0	0
$421$	−3.06003	−0.149137	−0.0745683	−	0.997216i	$-0.523758\pi$
$421$	−3.06003	−0.149137	−0.0745683	+	0.997216i	$0.523758\pi$
$422$	10.9878	0.534879
$423$	0	0
$424$	−33.2645	−1.61547
$425$	−13.4287	−0.651389
$426$	0	0
$427$	−1.07663	−0.0521017
$428$	2.66211	0.128678
$429$	0	0
$430$	11.8648	0.572173
$431$	−7.54197	−0.363284	−0.181642	−	0.983365i	$-0.558141\pi$
$431$	−7.54197	−0.363284	−0.181642	+	0.983365i	$0.558141\pi$
$432$	0	0
$433$	−32.1616	−1.54559	−0.772794	−	0.634657i	$-0.781141\pi$
$433$	−32.1616	−1.54559	−0.772794	+	0.634657i	$0.781141\pi$
$434$	−0.878667	−0.0421774
$435$	0	0
$436$	−2.93828	−0.140718
$437$	1.30995	0.0626633
$438$	0	0
$439$	−4.66725	−0.222756	−0.111378	−	0.993778i	$-0.535526\pi$
$439$	−4.66725	−0.222756	−0.111378	+	0.993778i	$0.535526\pi$
$440$	0	0
$441$	0	0
$442$	−4.43871	−0.211128
$443$	−17.2772	−0.820862	−0.410431	−	0.911892i	$-0.634622\pi$
$443$	−17.2772	−0.820862	−0.410431	+	0.911892i	$0.634622\pi$
$444$	0	0
$445$	−15.3391	−0.727144
$446$	−19.3871	−0.918007
$447$	0	0
$448$	8.29836	0.392061
$449$	16.7401	0.790013	0.395007	−	0.918678i	$-0.370742\pi$
$449$	16.7401	0.790013	0.395007	+	0.918678i	$0.370742\pi$
$450$	0	0
$451$	0	0
$452$	8.07529	0.379829
$453$	0	0
$454$	−14.0774	−0.660683
$455$	7.11832	0.333712
$456$	0	0
$457$	−21.6974	−1.01496	−0.507481	−	0.861663i	$-0.669423\pi$
$457$	−21.6974	−1.01496	−0.507481	+	0.861663i	$0.669423\pi$
$458$	−5.09692	−0.238163
$459$	0	0
$460$	2.11436	0.0985825
$461$	6.07778	0.283070	0.141535	−	0.989933i	$-0.454796\pi$
$461$	6.07778	0.283070	0.141535	+	0.989933i	$0.454796\pi$
$462$	0	0
$463$	−5.14719	−0.239210	−0.119605	−	0.992822i	$-0.538163\pi$
$463$	−5.14719	−0.239210	−0.119605	+	0.992822i	$0.538163\pi$
$464$	15.2262	0.706860
$465$	0	0
$466$	26.5877	1.23165
$467$	3.91927	0.181362	0.0906812	−	0.995880i	$-0.471096\pi$
$467$	3.91927	0.181362	0.0906812	+	0.995880i	$0.471096\pi$
$468$	0	0
$469$	2.40314	0.110967
$470$	−29.1431	−1.34427
$471$	0	0
$472$	10.1327	0.466397
$473$	0	0
$474$	0	0
$475$	11.2706	0.517131
$476$	1.46570	0.0671801
$477$	0	0
$478$	9.84982	0.450520
$479$	15.1129	0.690527	0.345263	−	0.938506i	$-0.387790\pi$
$479$	15.1129	0.690527	0.345263	+	0.938506i	$0.387790\pi$
$480$	0	0
$481$	−20.7237	−0.944922
$482$	−21.1152	−0.961770
$483$	0	0
$484$	0	0
$485$	−22.3541	−1.01505
$486$	0	0
$487$	25.0768	1.13634	0.568169	−	0.822912i	$-0.307652\pi$
$487$	25.0768	1.13634	0.568169	+	0.822912i	$0.307652\pi$
$488$	−3.30919	−0.149800
$489$	0	0
$490$	3.85168	0.174001
$491$	−37.2208	−1.67975	−0.839875	−	0.542780i	$-0.817372\pi$
$491$	−37.2208	−1.67975	−0.839875	+	0.542780i	$0.817372\pi$
$492$	0	0
$493$	15.4189	0.694434
$494$	3.72536	0.167612
$495$	0	0
$496$	−1.50554	−0.0676006
$497$	3.18859	0.143028
$498$	0	0
$499$	−31.8134	−1.42416	−0.712081	−	0.702098i	$-0.752247\pi$
$499$	−31.8134	−1.42416	−0.712081	+	0.702098i	$0.752247\pi$
$500$	5.09371	0.227798
$501$	0	0
$502$	−3.19508	−0.142603
$503$	19.2058	0.856346	0.428173	−	0.903697i	$-0.359157\pi$
$503$	19.2058	0.856346	0.428173	+	0.903697i	$0.359157\pi$
$504$	0	0
$505$	−53.3788	−2.37532
$506$	0	0
$507$	0	0
$508$	−14.7431	−0.654119
$509$	−2.51549	−0.111497	−0.0557485	−	0.998445i	$-0.517754\pi$
$509$	−2.51549	−0.111497	−0.0557485	+	0.998445i	$0.517754\pi$
$510$	0	0
$511$	1.22628	0.0542472
$512$	19.4135	0.857966
$513$	0	0
$514$	24.9817	1.10190
$515$	−30.7872	−1.35664
$516$	0	0
$517$	0	0
$518$	−11.2135	−0.492693
$519$	0	0
$520$	21.8793	0.959470
$521$	−14.8968	−0.652639	−0.326320	−	0.945260i	$-0.605809\pi$
$521$	−14.8968	−0.652639	−0.326320	+	0.945260i	$0.605809\pi$
$522$	0	0
$523$	−9.90502	−0.433116	−0.216558	−	0.976270i	$-0.569483\pi$
$523$	−9.90502	−0.433116	−0.216558	+	0.976270i	$0.569483\pi$
$524$	3.87535	0.169295
$525$	0	0
$526$	1.10411	0.0481414
$527$	−1.52459	−0.0664122
$528$	0	0
$529$	−22.3485	−0.971675
$530$	−41.6846	−1.81066
$531$	0	0
$532$	−1.23015	−0.0533336
$533$	−4.37910	−0.189680
$534$	0	0
$535$	12.1384	0.524789
$536$	7.38643	0.319045
$537$	0	0
$538$	−8.01342	−0.345483
$539$	0	0
$540$	0	0
$541$	−22.0084	−0.946214	−0.473107	−	0.881005i	$-0.656868\pi$
$541$	−22.0084	−0.946214	−0.473107	+	0.881005i	$0.656868\pi$
$542$	30.2737	1.30037
$543$	0	0
$544$	7.77199	0.333222
$545$	−13.3977	−0.573892
$546$	0	0
$547$	−10.8643	−0.464523	−0.232261	−	0.972653i	$-0.574612\pi$
$547$	−10.8643	−0.464523	−0.232261	+	0.972653i	$0.574612\pi$
$548$	6.89787	0.294662
$549$	0	0
$550$	0	0
$551$	−12.9410	−0.551303
$552$	0	0
$553$	−9.48182	−0.403208
$554$	−23.3878	−0.993650
$555$	0	0
$556$	9.86610	0.418416
$557$	−33.6126	−1.42421	−0.712106	−	0.702072i	$-0.752258\pi$
$557$	−33.6126	−1.42421	−0.712106	+	0.702072i	$0.752258\pi$
$558$	0	0
$559$	−6.34461	−0.268348
$560$	6.59959	0.278884
$561$	0	0
$562$	−31.3811	−1.32373
$563$	2.05348	0.0865440	0.0432720	−	0.999063i	$-0.486222\pi$
$563$	2.05348	0.0865440	0.0432720	+	0.999063i	$0.486222\pi$
$564$	0	0
$565$	36.8208	1.54906
$566$	29.1898	1.22694
$567$	0	0
$568$	9.80065	0.411226
$569$	−3.27416	−0.137260	−0.0686300	−	0.997642i	$-0.521863\pi$
$569$	−3.27416	−0.137260	−0.0686300	+	0.997642i	$0.521863\pi$
$570$	0	0
$571$	−43.8897	−1.83673	−0.918363	−	0.395738i	$-0.870489\pi$
$571$	−43.8897	−1.83673	−0.918363	+	0.395738i	$0.870489\pi$
$572$	0	0
$573$	0	0
$574$	−2.36950	−0.0989012
$575$	5.60513	0.233750
$576$	0	0
$577$	−43.8904	−1.82718	−0.913591	−	0.406635i	$-0.866702\pi$
$577$	−43.8904	−1.82718	−0.913591	+	0.406635i	$0.866702\pi$
$578$	14.7786	0.614709
$579$	0	0
$580$	−20.8877	−0.867315
$581$	16.0694	0.666671
$582$	0	0
$583$	0	0
$584$	3.76915	0.155969
$585$	0	0
$586$	−4.97481	−0.205508
$587$	−2.79166	−0.115224	−0.0576121	−	0.998339i	$-0.518349\pi$
$587$	−2.79166	−0.115224	−0.0576121	+	0.998339i	$0.518349\pi$
$588$	0	0
$589$	1.27957	0.0527239
$590$	12.6976	0.522752
$591$	0	0
$592$	−19.2136	−0.789673
$593$	−23.2526	−0.954871	−0.477435	−	0.878667i	$-0.658434\pi$
$593$	−23.2526	−0.954871	−0.477435	+	0.878667i	$0.658434\pi$
$594$	0	0
$595$	6.68312	0.273981
$596$	−2.38493	−0.0976907
$597$	0	0
$598$	1.85271	0.0757630
$599$	−10.4595	−0.427362	−0.213681	−	0.976904i	$-0.568545\pi$
$599$	−10.4595	−0.427362	−0.213681	+	0.976904i	$0.568545\pi$
$600$	0	0
$601$	22.3096	0.910029	0.455015	−	0.890484i	$-0.349634\pi$
$601$	22.3096	0.910029	0.455015	+	0.890484i	$0.349634\pi$
$602$	−3.43303	−0.139920
$603$	0	0
$604$	−2.17306	−0.0884204
$605$	0	0
$606$	0	0
$607$	18.9884	0.770716	0.385358	−	0.922767i	$-0.374078\pi$
$607$	18.9884	0.770716	0.385358	+	0.922767i	$0.374078\pi$
$608$	−6.52296	−0.264541
$609$	0	0
$610$	−4.14683	−0.167900
$611$	15.5840	0.630461
$612$	0	0
$613$	20.1617	0.814323	0.407161	−	0.913356i	$-0.366519\pi$
$613$	20.1617	0.814323	0.407161	+	0.913356i	$0.366519\pi$
$614$	−14.3589	−0.579478
$615$	0	0
$616$	0	0
$617$	−7.03919	−0.283387	−0.141694	−	0.989911i	$-0.545255\pi$
$617$	−7.03919	−0.283387	−0.141694	+	0.989911i	$0.545255\pi$
$618$	0	0
$619$	−31.0831	−1.24933	−0.624667	−	0.780891i	$-0.714765\pi$
$619$	−31.0831	−1.24933	−0.624667	+	0.780891i	$0.714765\pi$
$620$	2.06533	0.0829457
$621$	0	0
$622$	29.8898	1.19847
$623$	4.43830	0.177817
$624$	0	0
$625$	−11.4966	−0.459866
$626$	−3.99947	−0.159851
$627$	0	0
$628$	16.2883	0.649973
$629$	−19.4567	−0.775791
$630$	0	0
$631$	21.9720	0.874690	0.437345	−	0.899294i	$-0.355919\pi$
$631$	21.9720	0.874690	0.437345	+	0.899294i	$0.355919\pi$
$632$	−29.1439	−1.15928
$633$	0	0
$634$	−18.8546	−0.748812
$635$	−67.2238	−2.66770
$636$	0	0
$637$	−2.05965	−0.0816064
$638$	0	0
$639$	0	0
$640$	4.18149	0.165288
$641$	13.2909	0.524961	0.262480	−	0.964937i	$-0.415460\pi$
$641$	13.2909	0.524961	0.262480	+	0.964937i	$0.415460\pi$
$642$	0	0
$643$	14.6904	0.579333	0.289666	−	0.957128i	$-0.406456\pi$
$643$	14.6904	0.579333	0.289666	+	0.957128i	$0.406456\pi$
$644$	−0.611780	−0.0241075
$645$	0	0
$646$	3.49760	0.137611
$647$	6.14523	0.241594	0.120797	−	0.992677i	$-0.461455\pi$
$647$	6.14523	0.241594	0.120797	+	0.992677i	$0.461455\pi$
$648$	0	0
$649$	0	0
$650$	15.9404	0.625236
$651$	0	0
$652$	6.23232	0.244077
$653$	40.6691	1.59151	0.795753	−	0.605622i	$-0.207076\pi$
$653$	40.6691	1.59151	0.795753	+	0.605622i	$0.207076\pi$
$654$	0	0
$655$	17.6704	0.690438
$656$	−4.05998	−0.158516
$657$	0	0
$658$	8.43241	0.328730
$659$	−18.0090	−0.701531	−0.350765	−	0.936463i	$-0.614079\pi$
$659$	−18.0090	−0.701531	−0.350765	+	0.936463i	$0.614079\pi$
$660$	0	0
$661$	−17.1420	−0.666745	−0.333373	−	0.942795i	$-0.608187\pi$
$661$	−17.1420	−0.666745	−0.333373	+	0.942795i	$0.608187\pi$
$662$	1.38000	0.0536353
$663$	0	0
$664$	49.3919	1.91678
$665$	−5.60908	−0.217511
$666$	0	0
$667$	−6.43584	−0.249197
$668$	−16.4543	−0.636637
$669$	0	0
$670$	9.25613	0.357596
$671$	0	0
$672$	0	0
$673$	−23.1926	−0.894008	−0.447004	−	0.894532i	$-0.647509\pi$
$673$	−23.1926	−0.894008	−0.447004	+	0.894532i	$0.647509\pi$
$674$	22.8249	0.879183
$675$	0	0
$676$	6.63812	0.255312
$677$	27.5705	1.05962	0.529811	−	0.848116i	$-0.322263\pi$
$677$	27.5705	1.05962	0.529811	+	0.848116i	$0.322263\pi$
$678$	0	0
$679$	6.46807	0.248222
$680$	20.5416	0.787735
$681$	0	0
$682$	0	0
$683$	−21.9351	−0.839322	−0.419661	−	0.907681i	$-0.637851\pi$
$683$	−21.9351	−0.839322	−0.419661	+	0.907681i	$0.637851\pi$
$684$	0	0
$685$	31.4521	1.20172
$686$	−1.11447	−0.0425505
$687$	0	0
$688$	−5.88227	−0.224259
$689$	22.2905	0.849200
$690$	0	0
$691$	−25.6666	−0.976404	−0.488202	−	0.872731i	$-0.662347\pi$
$691$	−25.6666	−0.976404	−0.488202	+	0.872731i	$0.662347\pi$
$692$	−6.09830	−0.231823
$693$	0	0
$694$	30.3913	1.15364
$695$	44.9863	1.70643
$696$	0	0
$697$	−4.11137	−0.155729
$698$	−8.89828	−0.336805
$699$	0	0
$700$	−5.26366	−0.198948
$701$	18.4135	0.695467	0.347733	−	0.937593i	$-0.386951\pi$
$701$	18.4135	0.695467	0.347733	+	0.937593i	$0.386951\pi$
$702$	0	0
$703$	16.3298	0.615892
$704$	0	0
$705$	0	0
$706$	6.61404	0.248923
$707$	15.4449	0.580865
$708$	0	0
$709$	23.6621	0.888647	0.444323	−	0.895866i	$-0.353444\pi$
$709$	23.6621	0.888647	0.444323	+	0.895866i	$0.353444\pi$
$710$	12.2815	0.460915
$711$	0	0
$712$	13.6418	0.511249
$713$	0.636362	0.0238319
$714$	0	0
$715$	0	0
$716$	−2.74452	−0.102568
$717$	0	0
$718$	31.6734	1.18204
$719$	−11.1222	−0.414790	−0.207395	−	0.978257i	$-0.566498\pi$
$719$	−11.1222	−0.414790	−0.207395	+	0.978257i	$0.566498\pi$
$720$	0	0
$721$	8.90812	0.331756
$722$	18.2394	0.678799
$723$	0	0
$724$	−11.9894	−0.445583
$725$	−55.3730	−2.05650
$726$	0	0
$727$	42.4803	1.57551	0.787753	−	0.615991i	$-0.211244\pi$
$727$	42.4803	1.57551	0.787753	+	0.615991i	$0.211244\pi$
$728$	−6.33067	−0.234630
$729$	0	0
$730$	4.72323	0.174814
$731$	−5.95671	−0.220317
$732$	0	0
$733$	22.1884	0.819548	0.409774	−	0.912187i	$-0.365607\pi$
$733$	22.1884	0.819548	0.409774	+	0.912187i	$0.365607\pi$
$734$	33.8814	1.25058
$735$	0	0
$736$	−3.24402	−0.119576
$737$	0	0
$738$	0	0
$739$	29.4481	1.08327	0.541633	−	0.840615i	$-0.317806\pi$
$739$	29.4481	1.08327	0.541633	+	0.840615i	$0.317806\pi$
$740$	26.3576	0.968926
$741$	0	0
$742$	12.0613	0.442783
$743$	−16.9059	−0.620217	−0.310109	−	0.950701i	$-0.600365\pi$
$743$	−16.9059	−0.620217	−0.310109	+	0.950701i	$0.600365\pi$
$744$	0	0
$745$	−10.8745	−0.398412
$746$	−16.0735	−0.588491
$747$	0	0
$748$	0	0
$749$	−3.51219	−0.128333
$750$	0	0
$751$	−1.55147	−0.0566138	−0.0283069	−	0.999599i	$-0.509012\pi$
$751$	−1.55147	−0.0566138	−0.0283069	+	0.999599i	$0.509012\pi$
$752$	14.4484	0.526877
$753$	0	0
$754$	−18.3029	−0.666552
$755$	−9.90845	−0.360605
$756$	0	0
$757$	12.5467	0.456016	0.228008	−	0.973659i	$-0.426779\pi$
$757$	12.5467	0.456016	0.228008	+	0.973659i	$0.426779\pi$
$758$	24.9721	0.907027
$759$	0	0
$760$	−17.2404	−0.625375
$761$	9.03436	0.327495	0.163748	−	0.986502i	$-0.447642\pi$
$761$	9.03436	0.327495	0.163748	+	0.986502i	$0.447642\pi$
$762$	0	0
$763$	3.87655	0.140340
$764$	0.325228	0.0117663
$765$	0	0
$766$	37.5336	1.35614
$767$	−6.78993	−0.245170
$768$	0	0
$769$	−16.1383	−0.581963	−0.290981	−	0.956729i	$-0.593982\pi$
$769$	−16.1383	−0.581963	−0.290981	+	0.956729i	$0.593982\pi$
$770$	0	0
$771$	0	0
$772$	11.5020	0.413965
$773$	18.4134	0.662285	0.331143	−	0.943581i	$-0.392566\pi$
$773$	18.4134	0.662285	0.331143	+	0.943581i	$0.392566\pi$
$774$	0	0
$775$	5.47516	0.196674
$776$	19.8806	0.713673
$777$	0	0
$778$	2.70573	0.0970053
$779$	3.45063	0.123632
$780$	0	0
$781$	0	0
$782$	1.73944	0.0622022
$783$	0	0
$784$	−1.90956	−0.0681986
$785$	74.2694	2.65079
$786$	0	0
$787$	46.9870	1.67491	0.837453	−	0.546509i	$-0.184043\pi$
$787$	46.9870	1.67491	0.837453	+	0.546509i	$0.184043\pi$
$788$	−15.7719	−0.561850
$789$	0	0
$790$	−36.5210	−1.29936
$791$	−10.6539	−0.378810
$792$	0	0
$793$	2.21748	0.0787450
$794$	6.57197	0.233230
$795$	0	0
$796$	−6.40152	−0.226896
$797$	3.12454	0.110677	0.0553385	−	0.998468i	$-0.482376\pi$
$797$	3.12454	0.110677	0.0553385	+	0.998468i	$0.482376\pi$
$798$	0	0
$799$	14.6312	0.517615
$800$	−27.9111	−0.986805
$801$	0	0
$802$	12.5261	0.442314
$803$	0	0
$804$	0	0
$805$	−2.78952	−0.0983178
$806$	1.80975	0.0637457
$807$	0	0
$808$	47.4724	1.67007
$809$	32.7257	1.15057	0.575286	−	0.817952i	$-0.304891\pi$
$809$	32.7257	1.15057	0.575286	+	0.817952i	$0.304891\pi$
$810$	0	0
$811$	32.6613	1.14689	0.573447	−	0.819243i	$-0.305606\pi$
$811$	32.6613	1.14689	0.573447	+	0.819243i	$0.305606\pi$
$812$	6.04376	0.212095
$813$	0	0
$814$	0	0
$815$	28.4174	0.995419
$816$	0	0
$817$	4.99941	0.174907
$818$	−32.6922	−1.14306
$819$	0	0
$820$	5.56958	0.194498
$821$	7.43142	0.259358	0.129679	−	0.991556i	$-0.458605\pi$
$821$	7.43142	0.259358	0.129679	+	0.991556i	$0.458605\pi$
$822$	0	0
$823$	6.55866	0.228621	0.114310	−	0.993445i	$-0.463534\pi$
$823$	6.55866	0.228621	0.114310	+	0.993445i	$0.463534\pi$
$824$	27.3805	0.953846
$825$	0	0
$826$	−3.67399	−0.127835
$827$	−23.8538	−0.829479	−0.414739	−	0.909940i	$-0.636127\pi$
$827$	−23.8538	−0.829479	−0.414739	+	0.909940i	$0.636127\pi$
$828$	0	0
$829$	−26.1431	−0.907989	−0.453994	−	0.891005i	$-0.650001\pi$
$829$	−26.1431	−0.907989	−0.453994	+	0.891005i	$0.650001\pi$
$830$	61.8942	2.14838
$831$	0	0
$832$	−17.0917	−0.592549
$833$	−1.93373	−0.0669997
$834$	0	0
$835$	−75.0265	−2.59640
$836$	0	0
$837$	0	0
$838$	−22.6079	−0.780976
$839$	−34.2890	−1.18379	−0.591893	−	0.806016i	$-0.701619\pi$
$839$	−34.2890	−1.18379	−0.591893	+	0.806016i	$0.701619\pi$
$840$	0	0
$841$	34.5795	1.19240
$842$	3.41030	0.117527
$843$	0	0
$844$	7.47297	0.257230
$845$	30.2677	1.04124
$846$	0	0
$847$	0	0
$848$	20.6661	0.709678
$849$	0	0
$850$	14.9659	0.513325
$851$	8.12121	0.278392
$852$	0	0
$853$	21.3842	0.732181	0.366090	−	0.930579i	$-0.380696\pi$
$853$	21.3842	0.732181	0.366090	+	0.930579i	$0.380696\pi$
$854$	1.19987	0.0410585
$855$	0	0
$856$	−10.7953	−0.368975
$857$	−42.8697	−1.46440	−0.732200	−	0.681090i	$-0.761506\pi$
$857$	−42.8697	−1.46440	−0.732200	+	0.681090i	$0.761506\pi$
$858$	0	0
$859$	−30.3915	−1.03695	−0.518473	−	0.855094i	$-0.673499\pi$
$859$	−30.3915	−1.03695	−0.518473	+	0.855094i	$0.673499\pi$
$860$	8.06944	0.275165
$861$	0	0
$862$	8.40527	0.286285
$863$	−11.8184	−0.402304	−0.201152	−	0.979560i	$-0.564468\pi$
$863$	−11.8184	−0.402304	−0.201152	+	0.979560i	$0.564468\pi$
$864$	0	0
$865$	−27.8063	−0.945444
$866$	35.8430	1.21800
$867$	0	0
$868$	−0.597594	−0.0202837
$869$	0	0
$870$	0	0
$871$	−4.94963	−0.167712
$872$	11.9152	0.403499
$873$	0	0
$874$	−1.45989	−0.0493816
$875$	−6.72025	−0.227186
$876$	0	0
$877$	−9.20488	−0.310827	−0.155413	−	0.987850i	$-0.549671\pi$
$877$	−9.20488	−0.310827	−0.155413	+	0.987850i	$0.549671\pi$
$878$	5.20149	0.175542
$879$	0	0
$880$	0	0
$881$	41.9030	1.41175	0.705874	−	0.708338i	$-0.250555\pi$
$881$	41.9030	1.41175	0.705874	+	0.708338i	$0.250555\pi$
$882$	0	0
$883$	−16.0478	−0.540053	−0.270026	−	0.962853i	$-0.587032\pi$
$883$	−16.0478	−0.540053	−0.270026	+	0.962853i	$0.587032\pi$
$884$	−3.01883	−0.101534
$885$	0	0
$886$	19.2548	0.646878
$887$	33.8483	1.13652	0.568258	−	0.822850i	$-0.307618\pi$
$887$	33.8483	1.13652	0.568258	+	0.822850i	$0.307618\pi$
$888$	0	0
$889$	19.4509	0.652362
$890$	17.0949	0.573024
$891$	0	0
$892$	−13.1855	−0.441482
$893$	−12.2798	−0.410929
$894$	0	0
$895$	−12.5141	−0.418302
$896$	−1.20989	−0.0404197
$897$	0	0
$898$	−18.6563	−0.622568
$899$	−6.28660	−0.209670
$900$	0	0
$901$	20.9277	0.697202
$902$	0	0
$903$	0	0
$904$	−32.7465	−1.08913
$905$	−54.6679	−1.81722
$906$	0	0
$907$	−44.1013	−1.46436	−0.732181	−	0.681111i	$-0.761497\pi$
$907$	−44.1013	−1.46436	−0.732181	+	0.681111i	$0.761497\pi$
$908$	−9.57421	−0.317731
$909$	0	0
$910$	−7.93313	−0.262981
$911$	−49.5756	−1.64251	−0.821257	−	0.570558i	$-0.806727\pi$
$911$	−49.5756	−1.64251	−0.821257	+	0.570558i	$0.806727\pi$
$912$	0	0
$913$	0	0
$914$	24.1810	0.799837
$915$	0	0
$916$	−3.46648	−0.114536
$917$	−5.11284	−0.168841
$918$	0	0
$919$	−40.7926	−1.34563	−0.672813	−	0.739813i	$-0.734914\pi$
$919$	−40.7926	−1.34563	−0.672813	+	0.739813i	$0.734914\pi$
$920$	−8.57404	−0.282678
$921$	0	0
$922$	−6.77348	−0.223073
$923$	−6.56740	−0.216169
$924$	0	0
$925$	69.8737	2.29743
$926$	5.73637	0.188509
$927$	0	0
$928$	32.0476	1.05201
$929$	−41.3929	−1.35806	−0.679029	−	0.734112i	$-0.737599\pi$
$929$	−41.3929	−1.35806	−0.679029	+	0.734112i	$0.737599\pi$
$930$	0	0
$931$	1.62296	0.0531904
$932$	18.0827	0.592318
$933$	0	0
$934$	−4.36790	−0.142922
$935$	0	0
$936$	0	0
$937$	1.59644	0.0521534	0.0260767	−	0.999660i	$-0.491699\pi$
$937$	1.59644	0.0521534	0.0260767	+	0.999660i	$0.491699\pi$
$938$	−2.67822	−0.0874469
$939$	0	0
$940$	−19.8206	−0.646477
$941$	−7.10787	−0.231710	−0.115855	−	0.993266i	$-0.536961\pi$
$941$	−7.10787	−0.231710	−0.115855	+	0.993266i	$0.536961\pi$
$942$	0	0
$943$	1.71608	0.0558832
$944$	−6.29513	−0.204889
$945$	0	0
$946$	0	0
$947$	2.45986	0.0799347	0.0399674	−	0.999201i	$-0.487275\pi$
$947$	2.45986	0.0799347	0.0399674	+	0.999201i	$0.487275\pi$
$948$	0	0
$949$	−2.52570	−0.0819878
$950$	−12.5607	−0.407523
$951$	0	0
$952$	−5.94362	−0.192634
$953$	28.6000	0.926446	0.463223	−	0.886242i	$-0.346693\pi$
$953$	28.6000	0.926446	0.463223	+	0.886242i	$0.346693\pi$
$954$	0	0
$955$	1.48294	0.0479867
$956$	6.69900	0.216661
$957$	0	0
$958$	−16.8428	−0.544168
$959$	−9.10052	−0.293871
$960$	0	0
$961$	−30.3784	−0.979948
$962$	23.0959	0.744643
$963$	0	0
$964$	−14.3607	−0.462528
$965$	52.4454	1.68828
$966$	0	0
$967$	−0.213338	−0.00686047	−0.00343024	−	0.999994i	$-0.501092\pi$
$967$	−0.213338	−0.00686047	−0.00343024	+	0.999994i	$0.501092\pi$
$968$	0	0
$969$	0	0
$970$	24.9129	0.799907
$971$	0.828199	0.0265782	0.0132891	−	0.999912i	$-0.495770\pi$
$971$	0.828199	0.0265782	0.0132891	+	0.999912i	$0.495770\pi$
$972$	0	0
$973$	−13.0166	−0.417292
$974$	−27.9473	−0.895488
$975$	0	0
$976$	2.05589	0.0658073
$977$	−9.75714	−0.312159	−0.156079	−	0.987745i	$-0.549886\pi$
$977$	−9.75714	−0.312159	−0.156079	+	0.987745i	$0.549886\pi$
$978$	0	0
$979$	0	0
$980$	2.61958	0.0836795
$981$	0	0
$982$	41.4813	1.32372
$983$	−45.1198	−1.43910	−0.719548	−	0.694442i	$-0.755651\pi$
$983$	−45.1198	−1.43910	−0.719548	+	0.694442i	$0.755651\pi$
$984$	0	0
$985$	−71.9148	−2.29140
$986$	−17.1839	−0.547246
$987$	0	0
$988$	2.53367	0.0806069
$989$	2.48632	0.0790604
$990$	0	0
$991$	53.5405	1.70077	0.850384	−	0.526162i	$-0.176369\pi$
$991$	53.5405	1.70077	0.850384	+	0.526162i	$0.176369\pi$
$992$	−3.16880	−0.100609
$993$	0	0
$994$	−3.55358	−0.112713
$995$	−29.1889	−0.925351
$996$	0	0
$997$	−31.1607	−0.986869	−0.493435	−	0.869783i	$-0.664259\pi$
$997$	−31.1607	−0.986869	−0.493435	+	0.869783i	$0.664259\pi$
$998$	35.4549	1.12231
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.ct.1.3		8
3.2	odd	2		847.2.a.p.1.6		8
11.5	even	5		693.2.m.i.190.3		16
11.9	even	5		693.2.m.i.631.3		16
11.10	odd	2		7623.2.a.cw.1.6		8
21.20	even	2		5929.2.a.bt.1.6		8
33.2	even	10		847.2.f.x.323.3		16
33.5	odd	10		77.2.f.b.36.2	yes	16
33.8	even	10		847.2.f.v.372.2		16
33.14	odd	10		847.2.f.w.372.3		16
33.17	even	10		847.2.f.x.729.3		16
33.20	odd	10		77.2.f.b.15.2	✓	16
33.26	odd	10		847.2.f.w.148.3		16
33.29	even	10		847.2.f.v.148.2		16
33.32	even	2		847.2.a.o.1.3		8
231.5	even	30		539.2.q.f.410.3		32
231.20	even	10		539.2.f.e.246.2		16
231.38	even	30		539.2.q.f.520.2		32
231.53	odd	30		539.2.q.g.422.3		32
231.86	odd	30		539.2.q.g.312.2		32
231.104	even	10		539.2.f.e.344.2		16
231.137	odd	30		539.2.q.g.520.2		32
231.152	even	30		539.2.q.f.312.2		32
231.170	odd	30		539.2.q.g.410.3		32
231.185	even	30		539.2.q.f.422.3		32
231.230	odd	2		5929.2.a.bs.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.b.15.2	✓	16	33.20	odd	10
77.2.f.b.36.2	yes	16	33.5	odd	10
539.2.f.e.246.2		16	231.20	even	10
539.2.f.e.344.2		16	231.104	even	10
539.2.q.f.312.2		32	231.152	even	30
539.2.q.f.410.3		32	231.5	even	30
539.2.q.f.422.3		32	231.185	even	30
539.2.q.f.520.2		32	231.38	even	30
539.2.q.g.312.2		32	231.86	odd	30
539.2.q.g.410.3		32	231.170	odd	30
539.2.q.g.422.3		32	231.53	odd	30
539.2.q.g.520.2		32	231.137	odd	30
693.2.m.i.190.3		16	11.5	even	5
693.2.m.i.631.3		16	11.9	even	5
847.2.a.o.1.3		8	33.32	even	2
847.2.a.p.1.6		8	3.2	odd	2
847.2.f.v.148.2		16	33.29	even	10
847.2.f.v.372.2		16	33.8	even	10
847.2.f.w.148.3		16	33.26	odd	10
847.2.f.w.372.3		16	33.14	odd	10
847.2.f.x.323.3		16	33.2	even	10
847.2.f.x.729.3		16	33.17	even	10
5929.2.a.bs.1.3		8	231.230	odd	2
5929.2.a.bt.1.6		8	21.20	even	2
7623.2.a.ct.1.3		8	1.1	even	1	trivial
7623.2.a.cw.1.6		8	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)

\(f(q)\)	\(=\)	\(q-1.11447 q^{2} -0.757964 q^{4} -3.45608 q^{5} +1.00000 q^{7} +3.07366 q^{8} +O(q^{10})\)	\(q-1.11447 q^{2} -0.757964 q^{4} -3.45608 q^{5} +1.00000 q^{7} +3.07366 q^{8} +3.85168 q^{10} -2.05965 q^{13} -1.11447 q^{14} -1.90956 q^{16} -1.93373 q^{17} +1.62296 q^{19} +2.61958 q^{20} +0.807136 q^{23} +6.94447 q^{25} +2.29541 q^{26} -0.757964 q^{28} -7.97368 q^{29} +0.788420 q^{31} -4.01918 q^{32} +2.15508 q^{34} -3.45608 q^{35} +10.0618 q^{37} -1.80873 q^{38} -10.6228 q^{40} +2.12613 q^{41} +3.08043 q^{43} -0.899526 q^{46} -7.56632 q^{47} +1.00000 q^{49} -7.73938 q^{50} +1.56114 q^{52} -10.8224 q^{53} +3.07366 q^{56} +8.88640 q^{58} +3.29664 q^{59} -1.07663 q^{61} -0.878667 q^{62} +8.29836 q^{64} +7.11832 q^{65} +2.40314 q^{67} +1.46570 q^{68} +3.85168 q^{70} +3.18859 q^{71} +1.22628 q^{73} -11.2135 q^{74} -1.23015 q^{76} -9.48182 q^{79} +6.59959 q^{80} -2.36950 q^{82} +16.0694 q^{83} +6.68312 q^{85} -3.43303 q^{86} +4.43830 q^{89} -2.05965 q^{91} -0.611780 q^{92} +8.43241 q^{94} -5.60908 q^{95} +6.46807 q^{97} -1.11447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7}+O(q^{10}) \) 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7	\( 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7} + 6 q^{10} - 6 q^{13} - q^{14} + q^{16} + 5 q^{17} - 13 q^{19} - 23 q^{20} - 16 q^{23} + 16 q^{25} + 6 q^{26} + 7 q^{28} - 9 q^{29} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} + 7 q^{37} + 10 q^{38} + 5 q^{40} + 10 q^{41} - 4 q^{43} + 4 q^{46} - 16 q^{47} + 8 q^{49} - 6 q^{50} - 41 q^{52} - 37 q^{53} - 15 q^{58} - q^{59} + 19 q^{61} + 18 q^{62} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 6 q^{70} - 13 q^{71} - 25 q^{73} - 33 q^{74} + 26 q^{76} - 4 q^{80} - 13 q^{82} + 25 q^{83} + 3 q^{85} - 4 q^{86} - 37 q^{89} - 6 q^{91} - 35 q^{92} - 42 q^{94} - 21 q^{95} + 15 q^{97} - q^{98}+O(q^{100}) \) 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7 + 6 * q^10 - 6 * q^13 - q^14 + q^16 + 5 * q^17 - 13 * q^19 - 23 * q^20 - 16 * q^23 + 16 * q^25 + 6 * q^26 + 7 * q^28 - 9 * q^29 + 9 * q^31 - 16 * q^32 - 12 * q^34 - 10 * q^35 + 7 * q^37 + 10 * q^38 + 5 * q^40 + 10 * q^41 - 4 * q^43 + 4 * q^46 - 16 * q^47 + 8 * q^49 - 6 * q^50 - 41 * q^52 - 37 * q^53 - 15 * q^58 - q^59 + 19 * q^61 + 18 * q^62 - 4 * q^64 + 4 * q^65 - 19 * q^67 - 9 * q^68 + 6 * q^70 - 13 * q^71 - 25 * q^73 - 33 * q^74 + 26 * q^76 - 4 * q^80 - 13 * q^82 + 25 * q^83 + 3 * q^85 - 4 * q^86 - 37 * q^89 - 6 * q^91 - 35 * q^92 - 42 * q^94 - 21 * q^95 + 15 * q^97 - q^98

Introduction

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