LMFDB - Embedded newform 7623.2.a.ct.1.5

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			1.5
Root			$0.226211$ 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)

$f(q)$	$=$	$q-0.226211 q^{2} -1.94883 q^{4} +2.49552 q^{5} +1.00000 q^{7} +0.893270 q^{8} +O(q^{10})$	$q-0.226211 q^{2} -1.94883 q^{4} +2.49552 q^{5} +1.00000 q^{7} +0.893270 q^{8} -0.564516 q^{10} +5.13499 q^{13} -0.226211 q^{14} +3.69559 q^{16} +1.43752 q^{17} -6.06848 q^{19} -4.86335 q^{20} -7.08292 q^{23} +1.22764 q^{25} -1.16159 q^{26} -1.94883 q^{28} -6.51769 q^{29} +7.68895 q^{31} -2.62252 q^{32} -0.325184 q^{34} +2.49552 q^{35} -3.98432 q^{37} +1.37276 q^{38} +2.22918 q^{40} -6.74900 q^{41} -0.802299 q^{43} +1.60224 q^{46} -6.75222 q^{47} +1.00000 q^{49} -0.277706 q^{50} -10.0072 q^{52} -6.58167 q^{53} +0.893270 q^{56} +1.47437 q^{58} -2.87625 q^{59} -0.855342 q^{61} -1.73933 q^{62} -6.79793 q^{64} +12.8145 q^{65} -1.64668 q^{67} -2.80149 q^{68} -0.564516 q^{70} -4.52077 q^{71} -14.8479 q^{73} +0.901299 q^{74} +11.8264 q^{76} +2.45291 q^{79} +9.22243 q^{80} +1.52670 q^{82} -2.24780 q^{83} +3.58738 q^{85} +0.181489 q^{86} -1.73566 q^{89} +5.13499 q^{91} +13.8034 q^{92} +1.52743 q^{94} -15.1440 q^{95} -12.0776 q^{97} -0.226211 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.226211	−0.159956	−0.0799778	−	0.996797i	$-0.525485\pi$
$2$	−0.226211	−0.159956	−0.0799778	+	0.996797i	$0.525485\pi$
$3$	0	0
$3$	0	0
$4$	−1.94883	−0.974414
$5$	2.49552	1.11603	0.558016	−	0.829830i	$-0.311563\pi$
$5$	2.49552	1.11603	0.558016	+	0.829830i	$0.311563\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.893270	0.315819
$9$	0	0
$10$	−0.564516	−0.178516
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.13499	1.42419	0.712094	−	0.702084i	$-0.247747\pi$
$13$	5.13499	1.42419	0.712094	+	0.702084i	$0.247747\pi$
$14$	−0.226211	−0.0604575
$15$	0	0
$16$	3.69559	0.923897
$17$	1.43752	0.348651	0.174325	−	0.984688i	$-0.444226\pi$
$17$	1.43752	0.348651	0.174325	+	0.984688i	$0.444226\pi$
$18$	0	0
$19$	−6.06848	−1.39220	−0.696102	−	0.717943i	$-0.745084\pi$
$19$	−6.06848	−1.39220	−0.696102	+	0.717943i	$0.745084\pi$
$20$	−4.86335	−1.08748
$21$	0	0
$22$	0	0
$23$	−7.08292	−1.47689	−0.738446	−	0.674313i	$-0.764440\pi$
$23$	−7.08292	−1.47689	−0.738446	+	0.674313i	$0.764440\pi$
$24$	0	0
$25$	1.22764	0.245528
$26$	−1.16159	−0.227807
$27$	0	0
$28$	−1.94883	−0.368294
$29$	−6.51769	−1.21030	−0.605152	−	0.796110i	$-0.706888\pi$
$29$	−6.51769	−1.21030	−0.605152	+	0.796110i	$0.706888\pi$
$30$	0	0
$31$	7.68895	1.38098	0.690488	−	0.723344i	$-0.257396\pi$
$31$	7.68895	1.38098	0.690488	+	0.723344i	$0.257396\pi$
$32$	−2.62252	−0.463601
$33$	0	0
$34$	−0.325184	−0.0557687
$35$	2.49552	0.421820
$36$	0	0
$37$	−3.98432	−0.655019	−0.327509	−	0.944848i	$-0.606209\pi$
$37$	−3.98432	−0.655019	−0.327509	+	0.944848i	$0.606209\pi$
$38$	1.37276	0.222691
$39$	0	0
$40$	2.22918	0.352464
$41$	−6.74900	−1.05402	−0.527008	−	0.849860i	$-0.676686\pi$
$41$	−6.74900	−1.05402	−0.527008	+	0.849860i	$0.676686\pi$
$42$	0	0
$43$	−0.802299	−0.122349	−0.0611747	−	0.998127i	$-0.519485\pi$
$43$	−0.802299	−0.122349	−0.0611747	+	0.998127i	$0.519485\pi$
$44$	0	0
$45$	0	0
$46$	1.60224	0.236237
$47$	−6.75222	−0.984912	−0.492456	−	0.870337i	$-0.663901\pi$
$47$	−6.75222	−0.984912	−0.492456	+	0.870337i	$0.663901\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.277706	−0.0392735
$51$	0	0
$52$	−10.0072	−1.38775
$53$	−6.58167	−0.904062	−0.452031	−	0.892002i	$-0.649300\pi$
$53$	−6.58167	−0.904062	−0.452031	+	0.892002i	$0.649300\pi$
$54$	0	0
$55$	0	0
$56$	0.893270	0.119368
$57$	0	0
$58$	1.47437	0.193595
$59$	−2.87625	−0.374456	−0.187228	−	0.982316i	$-0.559950\pi$
$59$	−2.87625	−0.374456	−0.187228	+	0.982316i	$0.559950\pi$
$60$	0	0
$61$	−0.855342	−0.109515	−0.0547576	−	0.998500i	$-0.517439\pi$
$61$	−0.855342	−0.109515	−0.0547576	+	0.998500i	$0.517439\pi$
$62$	−1.73933	−0.220895
$63$	0	0
$64$	−6.79793	−0.849742
$65$	12.8145	1.58944
$66$	0	0
$67$	−1.64668	−0.201174	−0.100587	−	0.994928i	$-0.532072\pi$
$67$	−1.64668	−0.201174	−0.100587	+	0.994928i	$0.532072\pi$
$68$	−2.80149	−0.339730
$69$	0	0
$70$	−0.564516	−0.0674725
$71$	−4.52077	−0.536517	−0.268258	−	0.963347i	$-0.586448\pi$
$71$	−4.52077	−0.536517	−0.268258	+	0.963347i	$0.586448\pi$
$72$	0	0
$73$	−14.8479	−1.73782	−0.868910	−	0.494970i	$-0.835179\pi$
$73$	−14.8479	−1.73782	−0.868910	+	0.494970i	$0.835179\pi$
$74$	0.901299	0.104774
$75$	0	0
$76$	11.8264	1.35658
$77$	0	0
$78$	0	0
$79$	2.45291	0.275973	0.137987	−	0.990434i	$-0.455937\pi$
$79$	2.45291	0.275973	0.137987	+	0.990434i	$0.455937\pi$
$80$	9.22243	1.03110
$81$	0	0
$82$	1.52670	0.168596
$83$	−2.24780	−0.246728	−0.123364	−	0.992361i	$-0.539368\pi$
$83$	−2.24780	−0.246728	−0.123364	+	0.992361i	$0.539368\pi$
$84$	0	0
$85$	3.58738	0.389106
$86$	0.181489	0.0195705
$87$	0	0
$88$	0	0
$89$	−1.73566	−0.183980	−0.0919898	−	0.995760i	$-0.529323\pi$
$89$	−1.73566	−0.183980	−0.0919898	+	0.995760i	$0.529323\pi$
$90$	0	0
$91$	5.13499	0.538293
$92$	13.8034	1.43910
$93$	0	0
$94$	1.52743	0.157542
$95$	−15.1440	−1.55374
$96$	0	0
$97$	−12.0776	−1.22629	−0.613145	−	0.789970i	$-0.710096\pi$
$97$	−12.0776	−1.22629	−0.613145	+	0.789970i	$0.710096\pi$
$98$	−0.226211	−0.0228508
$99$	0	0
$100$	−2.39246	−0.239246
$101$	3.69338	0.367505	0.183753	−	0.982973i	$-0.441175\pi$
$101$	3.69338	0.367505	0.183753	+	0.982973i	$0.441175\pi$
$102$	0	0
$103$	−1.15156	−0.113467	−0.0567334	−	0.998389i	$-0.518069\pi$
$103$	−1.15156	−0.113467	−0.0567334	+	0.998389i	$0.518069\pi$
$104$	4.58693	0.449785
$105$	0	0
$106$	1.48885	0.144610
$107$	−1.16714	−0.112831	−0.0564157	−	0.998407i	$-0.517967\pi$
$107$	−1.16714	−0.112831	−0.0564157	+	0.998407i	$0.517967\pi$
$108$	0	0
$109$	9.30234	0.891003	0.445501	−	0.895281i	$-0.353025\pi$
$109$	9.30234	0.891003	0.445501	+	0.895281i	$0.353025\pi$
$110$	0	0
$111$	0	0
$112$	3.69559	0.349200
$113$	−3.29733	−0.310187	−0.155093	−	0.987900i	$-0.549568\pi$
$113$	−3.29733	−0.310187	−0.155093	+	0.987900i	$0.549568\pi$
$114$	0	0
$115$	−17.6756	−1.64826
$116$	12.7019	1.17934
$117$	0	0
$118$	0.650640	0.0598963
$119$	1.43752	0.131778
$120$	0	0
$121$	0	0
$122$	0.193488	0.0175176
$123$	0	0
$124$	−14.9844	−1.34564
$125$	−9.41402	−0.842015
$126$	0	0
$127$	−0.289205	−0.0256628	−0.0128314	−	0.999918i	$-0.504084\pi$
$127$	−0.289205	−0.0256628	−0.0128314	+	0.999918i	$0.504084\pi$
$128$	6.78282	0.599522
$129$	0	0
$130$	−2.89878	−0.254240
$131$	16.5059	1.44212	0.721062	−	0.692871i	$-0.243654\pi$
$131$	16.5059	1.44212	0.721062	+	0.692871i	$0.243654\pi$
$132$	0	0
$133$	−6.06848	−0.526204
$134$	0.372498	0.0321789
$135$	0	0
$136$	1.28410	0.110110
$137$	−9.32588	−0.796763	−0.398382	−	0.917220i	$-0.630428\pi$
$137$	−9.32588	−0.796763	−0.398382	+	0.917220i	$0.630428\pi$
$138$	0	0
$139$	4.82649	0.409377	0.204689	−	0.978827i	$-0.434382\pi$
$139$	4.82649	0.409377	0.204689	+	0.978827i	$0.434382\pi$
$140$	−4.86335	−0.411028
$141$	0	0
$142$	1.02265	0.0858189
$143$	0	0
$144$	0	0
$145$	−16.2650	−1.35074
$146$	3.35877	0.277974
$147$	0	0
$148$	7.76476	0.638260
$149$	−0.921915	−0.0755262	−0.0377631	−	0.999287i	$-0.512023\pi$
$149$	−0.921915	−0.0755262	−0.0377631	+	0.999287i	$0.512023\pi$
$150$	0	0
$151$	18.0964	1.47267	0.736333	−	0.676620i	$-0.236556\pi$
$151$	18.0964	1.47267	0.736333	+	0.676620i	$0.236556\pi$
$152$	−5.42079	−0.439684
$153$	0	0
$154$	0	0
$155$	19.1880	1.54121
$156$	0	0
$157$	−12.2733	−0.979518	−0.489759	−	0.871858i	$-0.662915\pi$
$157$	−12.2733	−0.979518	−0.489759	+	0.871858i	$0.662915\pi$
$158$	−0.554875	−0.0441435
$159$	0	0
$160$	−6.54457	−0.517394
$161$	−7.08292	−0.558212
$162$	0	0
$163$	8.09913	0.634373	0.317186	−	0.948363i	$-0.397262\pi$
$163$	8.09913	0.634373	0.317186	+	0.948363i	$0.397262\pi$
$164$	13.1526	1.02705
$165$	0	0
$166$	0.508478	0.0394655
$167$	13.0516	1.00996	0.504982	−	0.863130i	$-0.331499\pi$
$167$	13.0516	1.00996	0.504982	+	0.863130i	$0.331499\pi$
$168$	0	0
$169$	13.3681	1.02831
$170$	−0.811505	−0.0622396
$171$	0	0
$172$	1.56354	0.119219
$173$	−5.91219	−0.449495	−0.224748	−	0.974417i	$-0.572156\pi$
$173$	−5.91219	−0.449495	−0.224748	+	0.974417i	$0.572156\pi$
$174$	0	0
$175$	1.22764	0.0928007
$176$	0	0
$177$	0	0
$178$	0.392626	0.0294286
$179$	−4.33508	−0.324019	−0.162009	−	0.986789i	$-0.551798\pi$
$179$	−4.33508	−0.324019	−0.162009	+	0.986789i	$0.551798\pi$
$180$	0	0
$181$	10.8307	0.805040	0.402520	−	0.915411i	$-0.368134\pi$
$181$	10.8307	0.805040	0.402520	+	0.915411i	$0.368134\pi$
$182$	−1.16159	−0.0861029
$183$	0	0
$184$	−6.32696	−0.466430
$185$	−9.94297	−0.731022
$186$	0	0
$187$	0	0
$188$	13.1589	0.959713
$189$	0	0
$190$	3.42575	0.248530
$191$	−11.6556	−0.843370	−0.421685	−	0.906742i	$-0.638561\pi$
$191$	−11.6556	−0.843370	−0.421685	+	0.906742i	$0.638561\pi$
$192$	0	0
$193$	22.4454	1.61566	0.807829	−	0.589418i	$-0.200643\pi$
$193$	22.4454	1.61566	0.807829	+	0.589418i	$0.200643\pi$
$194$	2.73208	0.196152
$195$	0	0
$196$	−1.94883	−0.139202
$197$	−24.1022	−1.71721	−0.858604	−	0.512639i	$-0.828668\pi$
$197$	−24.1022	−1.71721	−0.858604	+	0.512639i	$0.828668\pi$
$198$	0	0
$199$	18.7205	1.32706	0.663531	−	0.748148i	$-0.269057\pi$
$199$	18.7205	1.32706	0.663531	+	0.748148i	$0.269057\pi$
$200$	1.09661	0.0775422
$201$	0	0
$202$	−0.835485	−0.0587845
$203$	−6.51769	−0.457452
$204$	0	0
$205$	−16.8423	−1.17632
$206$	0.260497	0.0181497
$207$	0	0
$208$	18.9768	1.31580
$209$	0	0
$210$	0	0
$211$	−7.56636	−0.520890	−0.260445	−	0.965489i	$-0.583869\pi$
$211$	−7.56636	−0.520890	−0.260445	+	0.965489i	$0.583869\pi$
$212$	12.8266	0.880931
$213$	0	0
$214$	0.264019	0.0180480
$215$	−2.00216	−0.136546
$216$	0	0
$217$	7.68895	0.521960
$218$	−2.10430	−0.142521
$219$	0	0
$220$	0	0
$221$	7.38167	0.496545
$222$	0	0
$223$	17.5244	1.17352	0.586760	−	0.809761i	$-0.300403\pi$
$223$	17.5244	1.17352	0.586760	+	0.809761i	$0.300403\pi$
$224$	−2.62252	−0.175225
$225$	0	0
$226$	0.745893	0.0496161
$227$	25.6773	1.70426	0.852130	−	0.523330i	$-0.175311\pi$
$227$	25.6773	1.70426	0.852130	+	0.523330i	$0.175311\pi$
$228$	0	0
$229$	−19.8369	−1.31086	−0.655429	−	0.755257i	$-0.727512\pi$
$229$	−19.8369	−1.31086	−0.655429	+	0.755257i	$0.727512\pi$
$230$	3.99842	0.263648
$231$	0	0
$232$	−5.82205	−0.382236
$233$	20.2146	1.32430	0.662151	−	0.749371i	$-0.269644\pi$
$233$	20.2146	1.32430	0.662151	+	0.749371i	$0.269644\pi$
$234$	0	0
$235$	−16.8503	−1.09919
$236$	5.60532	0.364875
$237$	0	0
$238$	−0.325184	−0.0210786
$239$	−17.1004	−1.10613	−0.553066	−	0.833137i	$-0.686542\pi$
$239$	−17.1004	−1.10613	−0.553066	+	0.833137i	$0.686542\pi$
$240$	0	0
$241$	−24.1529	−1.55582	−0.777912	−	0.628373i	$-0.783721\pi$
$241$	−24.1529	−1.55582	−0.777912	+	0.628373i	$0.783721\pi$
$242$	0	0
$243$	0	0
$244$	1.66691	0.106713
$245$	2.49552	0.159433
$246$	0	0
$247$	−31.1615	−1.98276
$248$	6.86831	0.436138
$249$	0	0
$250$	2.12956	0.134685
$251$	−11.8947	−0.750790	−0.375395	−	0.926865i	$-0.622493\pi$
$251$	−11.8947	−0.750790	−0.375395	+	0.926865i	$0.622493\pi$
$252$	0	0
$253$	0	0
$254$	0.0654215	0.00410491
$255$	0	0
$256$	12.0615	0.753845
$257$	−22.9609	−1.43226	−0.716131	−	0.697966i	$-0.754089\pi$
$257$	−22.9609	−1.43226	−0.716131	+	0.697966i	$0.754089\pi$
$258$	0	0
$259$	−3.98432	−0.247574
$260$	−24.9732	−1.54877
$261$	0	0
$262$	−3.73381	−0.230676
$263$	−1.93774	−0.119486	−0.0597432	−	0.998214i	$-0.519028\pi$
$263$	−1.93774	−0.119486	−0.0597432	+	0.998214i	$0.519028\pi$
$264$	0	0
$265$	−16.4247	−1.00896
$266$	1.37276	0.0841692
$267$	0	0
$268$	3.20910	0.196027
$269$	−7.18676	−0.438184	−0.219092	−	0.975704i	$-0.570310\pi$
$269$	−7.18676	−0.438184	−0.219092	+	0.975704i	$0.570310\pi$
$270$	0	0
$271$	1.19110	0.0723543	0.0361772	−	0.999345i	$-0.488482\pi$
$271$	1.19110	0.0723543	0.0361772	+	0.999345i	$0.488482\pi$
$272$	5.31250	0.322118
$273$	0	0
$274$	2.10962	0.127447
$275$	0	0
$276$	0	0
$277$	10.3402	0.621285	0.310642	−	0.950527i	$-0.399456\pi$
$277$	10.3402	0.621285	0.310642	+	0.950527i	$0.399456\pi$
$278$	−1.09181	−0.0654822
$279$	0	0
$280$	2.22918	0.133219
$281$	13.1513	0.784541	0.392271	−	0.919850i	$-0.371690\pi$
$281$	13.1513	0.784541	0.392271	+	0.919850i	$0.371690\pi$
$282$	0	0
$283$	0.300031	0.0178350	0.00891748	−	0.999960i	$-0.497161\pi$
$283$	0.300031	0.0178350	0.00891748	+	0.999960i	$0.497161\pi$
$284$	8.81021	0.522790
$285$	0	0
$286$	0	0
$287$	−6.74900	−0.398381
$288$	0	0
$289$	−14.9335	−0.878443
$290$	3.67934	0.216058
$291$	0	0
$292$	28.9361	1.69336
$293$	−16.1441	−0.943148	−0.471574	−	0.881826i	$-0.656314\pi$
$293$	−16.1441	−0.943148	−0.471574	+	0.881826i	$0.656314\pi$
$294$	0	0
$295$	−7.17775	−0.417905
$296$	−3.55908	−0.206867
$297$	0	0
$298$	0.208548	0.0120808
$299$	−36.3707	−2.10337
$300$	0	0
$301$	−0.802299	−0.0462437
$302$	−4.09361	−0.235561
$303$	0	0
$304$	−22.4266	−1.28625
$305$	−2.13453	−0.122223
$306$	0	0
$307$	−28.6376	−1.63443	−0.817217	−	0.576330i	$-0.804484\pi$
$307$	−28.6376	−1.63443	−0.817217	+	0.576330i	$0.804484\pi$
$308$	0	0
$309$	0	0
$310$	−4.34053	−0.246526
$311$	31.8228	1.80450	0.902252	−	0.431210i	$-0.141913\pi$
$311$	31.8228	1.80450	0.902252	+	0.431210i	$0.141913\pi$
$312$	0	0
$313$	0.0342232	0.00193441	0.000967206	−	1.00000i	$-0.499692\pi$
$313$	0.0342232	0.00193441	0.000967206		1.00000i	$0.499692\pi$
$314$	2.77636	0.156679
$315$	0	0
$316$	−4.78029	−0.268912
$317$	−22.3894	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$317$	−22.3894	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$318$	0	0
$319$	0	0
$320$	−16.9644	−0.948339
$321$	0	0
$322$	1.60224	0.0892892
$323$	−8.72359	−0.485393
$324$	0	0
$325$	6.30391	0.349678
$326$	−1.83211	−0.101471
$327$	0	0
$328$	−6.02868	−0.332878
$329$	−6.75222	−0.372262
$330$	0	0
$331$	10.7577	0.591297	0.295648	−	0.955297i	$-0.404464\pi$
$331$	10.7577	0.591297	0.295648	+	0.955297i	$0.404464\pi$
$332$	4.38057	0.240415
$333$	0	0
$334$	−2.95242	−0.161549
$335$	−4.10933	−0.224517
$336$	0	0
$337$	7.50492	0.408819	0.204410	−	0.978885i	$-0.434473\pi$
$337$	7.50492	0.408819	0.204410	+	0.978885i	$0.434473\pi$
$338$	−3.02401	−0.164485
$339$	0	0
$340$	−6.99118	−0.379150
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−0.716669	−0.0386402
$345$	0	0
$346$	1.33740	0.0718993
$347$	27.6818	1.48604	0.743019	−	0.669271i	$-0.233393\pi$
$347$	27.6818	1.48604	0.743019	+	0.669271i	$0.233393\pi$
$348$	0	0
$349$	−11.0605	−0.592055	−0.296028	−	0.955179i	$-0.595662\pi$
$349$	−11.0605	−0.592055	−0.296028	+	0.955179i	$0.595662\pi$
$350$	−0.277706	−0.0148440
$351$	0	0
$352$	0	0
$353$	−31.9202	−1.69894	−0.849469	−	0.527638i	$-0.823078\pi$
$353$	−31.9202	−1.69894	−0.849469	+	0.527638i	$0.823078\pi$
$354$	0	0
$355$	−11.2817	−0.598770
$356$	3.38251	0.179272
$357$	0	0
$358$	0.980644	0.0518286
$359$	−3.57826	−0.188854	−0.0944268	−	0.995532i	$-0.530102\pi$
$359$	−3.57826	−0.188854	−0.0944268	+	0.995532i	$0.530102\pi$
$360$	0	0
$361$	17.8264	0.938233
$362$	−2.45003	−0.128771
$363$	0	0
$364$	−10.0072	−0.524520
$365$	−37.0534	−1.93946
$366$	0	0
$367$	2.29397	0.119744	0.0598720	−	0.998206i	$-0.480931\pi$
$367$	2.29397	0.119744	0.0598720	+	0.998206i	$0.480931\pi$
$368$	−26.1756	−1.36450
$369$	0	0
$370$	2.24921	0.116931
$371$	−6.58167	−0.341703
$372$	0	0
$373$	7.96856	0.412596	0.206298	−	0.978489i	$-0.433858\pi$
$373$	7.96856	0.412596	0.206298	+	0.978489i	$0.433858\pi$
$374$	0	0
$375$	0	0
$376$	−6.03155	−0.311054
$377$	−33.4682	−1.72370
$378$	0	0
$379$	−11.6212	−0.596941	−0.298470	−	0.954419i	$-0.596476\pi$
$379$	−11.6212	−0.596941	−0.298470	+	0.954419i	$0.596476\pi$
$380$	29.5131	1.51399
$381$	0	0
$382$	2.63663	0.134902
$383$	−12.5785	−0.642729	−0.321364	−	0.946956i	$-0.604141\pi$
$383$	−12.5785	−0.642729	−0.321364	+	0.946956i	$0.604141\pi$
$384$	0	0
$385$	0	0
$386$	−5.07741	−0.258433
$387$	0	0
$388$	23.5371	1.19491
$389$	0.438294	0.0222224	0.0111112	−	0.999938i	$-0.496463\pi$
$389$	0.438294	0.0222224	0.0111112	+	0.999938i	$0.496463\pi$
$390$	0	0
$391$	−10.1819	−0.514920
$392$	0.893270	0.0451169
$393$	0	0
$394$	5.45218	0.274677
$395$	6.12128	0.307995
$396$	0	0
$397$	16.8147	0.843905	0.421952	−	0.906618i	$-0.361345\pi$
$397$	16.8147	0.843905	0.421952	+	0.906618i	$0.361345\pi$
$398$	−4.23480	−0.212271
$399$	0	0
$400$	4.53685	0.226842
$401$	−36.8609	−1.84074	−0.920372	−	0.391043i	$-0.872114\pi$
$401$	−36.8609	−1.84074	−0.920372	+	0.391043i	$0.872114\pi$
$402$	0	0
$403$	39.4827	1.96677
$404$	−7.19776	−0.358102
$405$	0	0
$406$	1.47437	0.0731720
$407$	0	0
$408$	0	0
$409$	16.2739	0.804692	0.402346	−	0.915488i	$-0.368195\pi$
$409$	16.2739	0.804692	0.402346	+	0.915488i	$0.368195\pi$
$410$	3.80992	0.188158
$411$	0	0
$412$	2.24420	0.110564
$413$	−2.87625	−0.141531
$414$	0	0
$415$	−5.60944	−0.275356
$416$	−13.4666	−0.660255
$417$	0	0
$418$	0	0
$419$	−5.56352	−0.271796	−0.135898	−	0.990723i	$-0.543392\pi$
$419$	−5.56352	−0.271796	−0.135898	+	0.990723i	$0.543392\pi$
$420$	0	0
$421$	21.4914	1.04743	0.523713	−	0.851895i	$-0.324546\pi$
$421$	21.4914	1.04743	0.523713	+	0.851895i	$0.324546\pi$
$422$	1.71160	0.0833192
$423$	0	0
$424$	−5.87921	−0.285520
$425$	1.76476	0.0856035
$426$	0	0
$427$	−0.855342	−0.0413929
$428$	2.27455	0.109944
$429$	0	0
$430$	0.452910	0.0218413
$431$	27.7188	1.33517	0.667583	−	0.744536i	$-0.267329\pi$
$431$	27.7188	1.33517	0.667583	+	0.744536i	$0.267329\pi$
$432$	0	0
$433$	9.28812	0.446358	0.223179	−	0.974777i	$-0.428356\pi$
$433$	9.28812	0.446358	0.223179	+	0.974777i	$0.428356\pi$
$434$	−1.73933	−0.0834904
$435$	0	0
$436$	−18.1287	−0.868206
$437$	42.9826	2.05613
$438$	0	0
$439$	−14.7118	−0.702156	−0.351078	−	0.936346i	$-0.614185\pi$
$439$	−14.7118	−0.702156	−0.351078	+	0.936346i	$0.614185\pi$
$440$	0	0
$441$	0	0
$442$	−1.66982	−0.0794251
$443$	−2.44345	−0.116092	−0.0580459	−	0.998314i	$-0.518487\pi$
$443$	−2.44345	−0.116092	−0.0580459	+	0.998314i	$0.518487\pi$
$444$	0	0
$445$	−4.33138	−0.205327
$446$	−3.96421	−0.187711
$447$	0	0
$448$	−6.79793	−0.321172
$449$	−4.76935	−0.225080	−0.112540	−	0.993647i	$-0.535899\pi$
$449$	−4.76935	−0.225080	−0.112540	+	0.993647i	$0.535899\pi$
$450$	0	0
$451$	0	0
$452$	6.42593	0.302250
$453$	0	0
$454$	−5.80849	−0.272606
$455$	12.8145	0.600752
$456$	0	0
$457$	31.0875	1.45421	0.727106	−	0.686525i	$-0.240865\pi$
$457$	31.0875	1.45421	0.727106	+	0.686525i	$0.240865\pi$
$458$	4.48733	0.209679
$459$	0	0
$460$	34.4467	1.60609
$461$	−29.7215	−1.38427	−0.692134	−	0.721769i	$-0.743329\pi$
$461$	−29.7215	−1.38427	−0.692134	+	0.721769i	$0.743329\pi$
$462$	0	0
$463$	−25.4553	−1.18301	−0.591505	−	0.806302i	$-0.701466\pi$
$463$	−25.4553	−1.18301	−0.591505	+	0.806302i	$0.701466\pi$
$464$	−24.0867	−1.11820
$465$	0	0
$466$	−4.57277	−0.211829
$467$	3.18491	0.147380	0.0736901	−	0.997281i	$-0.476522\pi$
$467$	3.18491	0.147380	0.0736901	+	0.997281i	$0.476522\pi$
$468$	0	0
$469$	−1.64668	−0.0760366
$470$	3.81173	0.175822
$471$	0	0
$472$	−2.56927	−0.118260
$473$	0	0
$474$	0	0
$475$	−7.44990	−0.341825
$476$	−2.80149	−0.128406
$477$	0	0
$478$	3.86830	0.176932
$479$	−3.23354	−0.147744	−0.0738720	−	0.997268i	$-0.523536\pi$
$479$	−3.23354	−0.147744	−0.0738720	+	0.997268i	$0.523536\pi$
$480$	0	0
$481$	−20.4594	−0.932870
$482$	5.46366	0.248863
$483$	0	0
$484$	0	0
$485$	−30.1398	−1.36858
$486$	0	0
$487$	9.87096	0.447296	0.223648	−	0.974670i	$-0.428203\pi$
$487$	9.87096	0.447296	0.223648	+	0.974670i	$0.428203\pi$
$488$	−0.764051	−0.0345870
$489$	0	0
$490$	−0.564516	−0.0255022
$491$	−4.22600	−0.190717	−0.0953584	−	0.995443i	$-0.530400\pi$
$491$	−4.22600	−0.190717	−0.0953584	+	0.995443i	$0.530400\pi$
$492$	0	0
$493$	−9.36933	−0.421974
$494$	7.04910	0.317154
$495$	0	0
$496$	28.4152	1.27588
$497$	−4.52077	−0.202784
$498$	0	0
$499$	−20.3953	−0.913018	−0.456509	−	0.889719i	$-0.650900\pi$
$499$	−20.3953	−0.913018	−0.456509	+	0.889719i	$0.650900\pi$
$500$	18.3463	0.820472
$501$	0	0
$502$	2.69073	0.120093
$503$	23.9593	1.06829	0.534145	−	0.845393i	$-0.320634\pi$
$503$	23.9593	1.06829	0.534145	+	0.845393i	$0.320634\pi$
$504$	0	0
$505$	9.21692	0.410147
$506$	0	0
$507$	0	0
$508$	0.563611	0.0250062
$509$	−3.69341	−0.163708	−0.0818538	−	0.996644i	$-0.526084\pi$
$509$	−3.69341	−0.163708	−0.0818538	+	0.996644i	$0.526084\pi$
$510$	0	0
$511$	−14.8479	−0.656834
$512$	−16.2941	−0.720104
$513$	0	0
$514$	5.19402	0.229098
$515$	−2.87375	−0.126633
$516$	0	0
$517$	0	0
$518$	0.901299	0.0396008
$519$	0	0
$520$	11.4468	0.501975
$521$	0.238270	0.0104388	0.00521939	−	0.999986i	$-0.498339\pi$
$521$	0.238270	0.0104388	0.00521939	+	0.999986i	$0.498339\pi$
$522$	0	0
$523$	−21.9914	−0.961618	−0.480809	−	0.876825i	$-0.659657\pi$
$523$	−21.9914	−0.961618	−0.480809	+	0.876825i	$0.659657\pi$
$524$	−32.1671	−1.40523
$525$	0	0
$526$	0.438339	0.0191125
$527$	11.0531	0.481479
$528$	0	0
$529$	27.1678	1.18121
$530$	3.71546	0.161389
$531$	0	0
$532$	11.8264	0.512740
$533$	−34.6560	−1.50112
$534$	0	0
$535$	−2.91262	−0.125923
$536$	−1.47093	−0.0635345
$537$	0	0
$538$	1.62573	0.0700900
$539$	0	0
$540$	0	0
$541$	28.6309	1.23094	0.615470	−	0.788161i	$-0.288966\pi$
$541$	28.6309	1.23094	0.615470	+	0.788161i	$0.288966\pi$
$542$	−0.269441	−0.0115735
$543$	0	0
$544$	−3.76994	−0.161635
$545$	23.2142	0.994388
$546$	0	0
$547$	6.40847	0.274007	0.137003	−	0.990571i	$-0.456253\pi$
$547$	6.40847	0.274007	0.137003	+	0.990571i	$0.456253\pi$
$548$	18.1745	0.776378
$549$	0	0
$550$	0	0
$551$	39.5524	1.68499
$552$	0	0
$553$	2.45291	0.104308
$554$	−2.33908	−0.0993780
$555$	0	0
$556$	−9.40600	−0.398903
$557$	22.5193	0.954172	0.477086	−	0.878857i	$-0.341693\pi$
$557$	22.5193	0.954172	0.477086	+	0.878857i	$0.341693\pi$
$558$	0	0
$559$	−4.11979	−0.174249
$560$	9.22243	0.389719
$561$	0	0
$562$	−2.97498	−0.125492
$563$	29.8267	1.25705	0.628523	−	0.777791i	$-0.283660\pi$
$563$	29.8267	1.25705	0.628523	+	0.777791i	$0.283660\pi$
$564$	0	0
$565$	−8.22856	−0.346178
$566$	−0.0678703	−0.00285280
$567$	0	0
$568$	−4.03827	−0.169442
$569$	−29.2764	−1.22733	−0.613665	−	0.789567i	$-0.710305\pi$
$569$	−29.2764	−1.22733	−0.613665	+	0.789567i	$0.710305\pi$
$570$	0	0
$571$	37.9252	1.58712	0.793559	−	0.608493i	$-0.208226\pi$
$571$	37.9252	1.58712	0.793559	+	0.608493i	$0.208226\pi$
$572$	0	0
$573$	0	0
$574$	1.52670	0.0637232
$575$	−8.69527	−0.362618
$576$	0	0
$577$	8.77263	0.365209	0.182605	−	0.983186i	$-0.441547\pi$
$577$	8.77263	0.365209	0.182605	+	0.983186i	$0.441547\pi$
$578$	3.37813	0.140512
$579$	0	0
$580$	31.6978	1.31618
$581$	−2.24780	−0.0932544
$582$	0	0
$583$	0	0
$584$	−13.2632	−0.548836
$585$	0	0
$586$	3.65198	0.150862
$587$	−9.17011	−0.378491	−0.189246	−	0.981930i	$-0.560604\pi$
$587$	−9.17011	−0.378491	−0.189246	+	0.981930i	$0.560604\pi$
$588$	0	0
$589$	−46.6602	−1.92260
$590$	1.62369	0.0668462
$591$	0	0
$592$	−14.7244	−0.605170
$593$	−7.25596	−0.297967	−0.148983	−	0.988840i	$-0.547600\pi$
$593$	−7.25596	−0.297967	−0.148983	+	0.988840i	$0.547600\pi$
$594$	0	0
$595$	3.58738	0.147068
$596$	1.79665	0.0735938
$597$	0	0
$598$	8.22747	0.336446
$599$	20.0181	0.817916	0.408958	−	0.912553i	$-0.365892\pi$
$599$	20.0181	0.817916	0.408958	+	0.912553i	$0.365892\pi$
$600$	0	0
$601$	−11.2706	−0.459736	−0.229868	−	0.973222i	$-0.573829\pi$
$601$	−11.2706	−0.459736	−0.229868	+	0.973222i	$0.573829\pi$
$602$	0.181489	0.00739694
$603$	0	0
$604$	−35.2668	−1.43499
$605$	0	0
$606$	0	0
$607$	−13.5260	−0.549005	−0.274502	−	0.961586i	$-0.588513\pi$
$607$	−13.5260	−0.549005	−0.274502	+	0.961586i	$0.588513\pi$
$608$	15.9147	0.645427
$609$	0	0
$610$	0.482854	0.0195502
$611$	−34.6725	−1.40270
$612$	0	0
$613$	−30.7968	−1.24387	−0.621935	−	0.783069i	$-0.713653\pi$
$613$	−30.7968	−1.24387	−0.621935	+	0.783069i	$0.713653\pi$
$614$	6.47815	0.261437
$615$	0	0
$616$	0	0
$617$	−23.6896	−0.953707	−0.476853	−	0.878983i	$-0.658223\pi$
$617$	−23.6896	−0.953707	−0.476853	+	0.878983i	$0.658223\pi$
$618$	0	0
$619$	−32.4878	−1.30579	−0.652897	−	0.757446i	$-0.726447\pi$
$619$	−32.4878	−1.30579	−0.652897	+	0.757446i	$0.726447\pi$
$620$	−37.3940	−1.50178
$621$	0	0
$622$	−7.19867	−0.288640
$623$	−1.73566	−0.0695378
$624$	0	0
$625$	−29.6311	−1.18524
$626$	−0.00774169	−0.000309420 0
$627$	0	0
$628$	23.9186	0.954456
$629$	−5.72756	−0.228373
$630$	0	0
$631$	−15.1333	−0.602448	−0.301224	−	0.953553i	$-0.597395\pi$
$631$	−15.1333	−0.602448	−0.301224	+	0.953553i	$0.597395\pi$
$632$	2.19111	0.0871575
$633$	0	0
$634$	5.06474	0.201147
$635$	−0.721718	−0.0286405
$636$	0	0
$637$	5.13499	0.203456
$638$	0	0
$639$	0	0
$640$	16.9267	0.669086
$641$	16.5502	0.653695	0.326848	−	0.945077i	$-0.394014\pi$
$641$	16.5502	0.653695	0.326848	+	0.945077i	$0.394014\pi$
$642$	0	0
$643$	1.99506	0.0786773	0.0393387	−	0.999226i	$-0.487475\pi$
$643$	1.99506	0.0786773	0.0393387	+	0.999226i	$0.487475\pi$
$644$	13.8034	0.543930
$645$	0	0
$646$	1.97337	0.0776414
$647$	−40.4517	−1.59032	−0.795160	−	0.606399i	$-0.792613\pi$
$647$	−40.4517	−1.59032	−0.795160	+	0.606399i	$0.792613\pi$
$648$	0	0
$649$	0	0
$650$	−1.42602	−0.0559329
$651$	0	0
$652$	−15.7838	−0.618142
$653$	41.4856	1.62346	0.811729	−	0.584034i	$-0.198527\pi$
$653$	41.4856	1.62346	0.811729	+	0.584034i	$0.198527\pi$
$654$	0	0
$655$	41.1908	1.60946
$656$	−24.9415	−0.973803
$657$	0	0
$658$	1.52743	0.0595454
$659$	51.1359	1.99197	0.995985	−	0.0895158i	$-0.0285320\pi$
$659$	51.1359	1.99197	0.995985	+	0.0895158i	$0.0285320\pi$
$660$	0	0
$661$	−42.8840	−1.66800	−0.833998	−	0.551768i	$-0.813954\pi$
$661$	−42.8840	−1.66800	−0.833998	+	0.551768i	$0.813954\pi$
$662$	−2.43351	−0.0945812
$663$	0	0
$664$	−2.00789	−0.0779213
$665$	−15.1440	−0.587260
$666$	0	0
$667$	46.1643	1.78749
$668$	−25.4353	−0.984122
$669$	0	0
$670$	0.929577	0.0359127
$671$	0	0
$672$	0	0
$673$	−25.0072	−0.963958	−0.481979	−	0.876183i	$-0.660082\pi$
$673$	−25.0072	−0.963958	−0.481979	+	0.876183i	$0.660082\pi$
$674$	−1.69770	−0.0653929
$675$	0	0
$676$	−26.0521	−1.00200
$677$	9.91890	0.381214	0.190607	−	0.981666i	$-0.438954\pi$
$677$	9.91890	0.381214	0.190607	+	0.981666i	$0.438954\pi$
$678$	0	0
$679$	−12.0776	−0.463494
$680$	3.20450	0.122887
$681$	0	0
$682$	0	0
$683$	39.8980	1.52666	0.763328	−	0.646011i	$-0.223564\pi$
$683$	39.8980	1.52666	0.763328	+	0.646011i	$0.223564\pi$
$684$	0	0
$685$	−23.2729	−0.889214
$686$	−0.226211	−0.00863679
$687$	0	0
$688$	−2.96497	−0.113038
$689$	−33.7968	−1.28756
$690$	0	0
$691$	−6.61401	−0.251609	−0.125804	−	0.992055i	$-0.540151\pi$
$691$	−6.61401	−0.251609	−0.125804	+	0.992055i	$0.540151\pi$
$692$	11.5218	0.437995
$693$	0	0
$694$	−6.26194	−0.237700
$695$	12.0446	0.456878
$696$	0	0
$697$	−9.70185	−0.367484
$698$	2.50201	0.0947025
$699$	0	0
$700$	−2.39246	−0.0904264
$701$	−14.6016	−0.551495	−0.275748	−	0.961230i	$-0.588925\pi$
$701$	−14.6016	−0.551495	−0.275748	+	0.961230i	$0.588925\pi$
$702$	0	0
$703$	24.1788	0.911920
$704$	0	0
$705$	0	0
$706$	7.22070	0.271755
$707$	3.69338	0.138904
$708$	0	0
$709$	−4.14406	−0.155633	−0.0778167	−	0.996968i	$-0.524795\pi$
$709$	−4.14406	−0.155633	−0.0778167	+	0.996968i	$0.524795\pi$
$710$	2.55205	0.0957766
$711$	0	0
$712$	−1.55041	−0.0581042
$713$	−54.4602	−2.03955
$714$	0	0
$715$	0	0
$716$	8.44832	0.315729
$717$	0	0
$718$	0.809444	0.0302082
$719$	−17.0223	−0.634823	−0.317412	−	0.948288i	$-0.602814\pi$
$719$	−17.0223	−0.634823	−0.317412	+	0.948288i	$0.602814\pi$
$720$	0	0
$721$	−1.15156	−0.0428864
$722$	−4.03254	−0.150076
$723$	0	0
$724$	−21.1072	−0.784442
$725$	−8.00136	−0.297163
$726$	0	0
$727$	−21.6199	−0.801837	−0.400918	−	0.916114i	$-0.631309\pi$
$727$	−21.6199	−0.801837	−0.400918	+	0.916114i	$0.631309\pi$
$728$	4.58693	0.170003
$729$	0	0
$730$	8.38190	0.310228
$731$	−1.15332	−0.0426572
$732$	0	0
$733$	−48.2326	−1.78151	−0.890755	−	0.454484i	$-0.849824\pi$
$733$	−48.2326	−1.78151	−0.890755	+	0.454484i	$0.849824\pi$
$734$	−0.518921	−0.0191537
$735$	0	0
$736$	18.5751	0.684688
$737$	0	0
$738$	0	0
$739$	−8.16347	−0.300298	−0.150149	−	0.988663i	$-0.547975\pi$
$739$	−8.16347	−0.300298	−0.150149	+	0.988663i	$0.547975\pi$
$740$	19.3772	0.712318
$741$	0	0
$742$	1.48885	0.0546574
$743$	−19.5612	−0.717633	−0.358816	−	0.933408i	$-0.616820\pi$
$743$	−19.5612	−0.717633	−0.358816	+	0.933408i	$0.616820\pi$
$744$	0	0
$745$	−2.30066	−0.0842897
$746$	−1.80258	−0.0659971
$747$	0	0
$748$	0	0
$749$	−1.16714	−0.0426462
$750$	0	0
$751$	1.11630	0.0407343	0.0203671	−	0.999793i	$-0.493516\pi$
$751$	1.11630	0.0407343	0.0203671	+	0.999793i	$0.493516\pi$
$752$	−24.9534	−0.909958
$753$	0	0
$754$	7.57089	0.275716
$755$	45.1600	1.64354
$756$	0	0
$757$	−26.5773	−0.965969	−0.482984	−	0.875629i	$-0.660447\pi$
$757$	−26.5773	−0.965969	−0.482984	+	0.875629i	$0.660447\pi$
$758$	2.62885	0.0954840
$759$	0	0
$760$	−13.5277	−0.490701
$761$	−6.30003	−0.228376	−0.114188	−	0.993459i	$-0.536427\pi$
$761$	−6.30003	−0.228376	−0.114188	+	0.993459i	$0.536427\pi$
$762$	0	0
$763$	9.30234	0.336767
$764$	22.7148	0.821792
$765$	0	0
$766$	2.84539	0.102808
$767$	−14.7695	−0.533296
$768$	0	0
$769$	13.1916	0.475700	0.237850	−	0.971302i	$-0.423557\pi$
$769$	13.1916	0.475700	0.237850	+	0.971302i	$0.423557\pi$
$770$	0	0
$771$	0	0
$772$	−43.7423	−1.57432
$773$	−45.8178	−1.64795	−0.823976	−	0.566625i	$-0.808249\pi$
$773$	−45.8178	−1.64795	−0.823976	+	0.566625i	$0.808249\pi$
$774$	0	0
$775$	9.43925	0.339068
$776$	−10.7885	−0.387285
$777$	0	0
$778$	−0.0991470	−0.00355459
$779$	40.9562	1.46741
$780$	0	0
$781$	0	0
$782$	2.30326	0.0823643
$783$	0	0
$784$	3.69559	0.131985
$785$	−30.6284	−1.09317
$786$	0	0
$787$	−18.8479	−0.671856	−0.335928	−	0.941888i	$-0.609050\pi$
$787$	−18.8479	−0.671856	−0.335928	+	0.941888i	$0.609050\pi$
$788$	46.9710	1.67327
$789$	0	0
$790$	−1.38470	−0.0492656
$791$	−3.29733	−0.117240
$792$	0	0
$793$	−4.39217	−0.155970
$794$	−3.80367	−0.134987
$795$	0	0
$796$	−36.4831	−1.29311
$797$	28.1613	0.997526	0.498763	−	0.866738i	$-0.333788\pi$
$797$	28.1613	0.997526	0.498763	+	0.866738i	$0.333788\pi$
$798$	0	0
$799$	−9.70648	−0.343391
$800$	−3.21951	−0.113827
$801$	0	0
$802$	8.33835	0.294437
$803$	0	0
$804$	0	0
$805$	−17.6756	−0.622983
$806$	−8.93143	−0.314596
$807$	0	0
$808$	3.29919	0.116065
$809$	6.54552	0.230128	0.115064	−	0.993358i	$-0.463293\pi$
$809$	6.54552	0.230128	0.115064	+	0.993358i	$0.463293\pi$
$810$	0	0
$811$	18.8046	0.660320	0.330160	−	0.943925i	$-0.392897\pi$
$811$	18.8046	0.660320	0.330160	+	0.943925i	$0.392897\pi$
$812$	12.7019	0.445748
$813$	0	0
$814$	0	0
$815$	20.2116	0.707980
$816$	0	0
$817$	4.86873	0.170335
$818$	−3.68134	−0.128715
$819$	0	0
$820$	32.8227	1.14622
$821$	11.5880	0.404424	0.202212	−	0.979342i	$-0.435187\pi$
$821$	11.5880	0.404424	0.202212	+	0.979342i	$0.435187\pi$
$822$	0	0
$823$	12.3464	0.430368	0.215184	−	0.976574i	$-0.430965\pi$
$823$	12.3464	0.430368	0.215184	+	0.976574i	$0.430965\pi$
$824$	−1.02866	−0.0358349
$825$	0	0
$826$	0.650640	0.0226387
$827$	4.49579	0.156334	0.0781669	−	0.996940i	$-0.475093\pi$
$827$	4.49579	0.156334	0.0781669	+	0.996940i	$0.475093\pi$
$828$	0	0
$829$	−19.8629	−0.689866	−0.344933	−	0.938627i	$-0.612098\pi$
$829$	−19.8629	−0.689866	−0.344933	+	0.938627i	$0.612098\pi$
$830$	1.26892	0.0440448
$831$	0	0
$832$	−34.9073	−1.21019
$833$	1.43752	0.0498073
$834$	0	0
$835$	32.5706	1.12715
$836$	0	0
$837$	0	0
$838$	1.25853	0.0434753
$839$	43.8528	1.51397	0.756984	−	0.653434i	$-0.226672\pi$
$839$	43.8528	1.51397	0.756984	+	0.653434i	$0.226672\pi$
$840$	0	0
$841$	13.4802	0.464836
$842$	−4.86160	−0.167542
$843$	0	0
$844$	14.7455	0.507562
$845$	33.3604	1.14763
$846$	0	0
$847$	0	0
$848$	−24.3232	−0.835261
$849$	0	0
$850$	−0.399209	−0.0136928
$851$	28.2207	0.967392
$852$	0	0
$853$	39.1407	1.34015	0.670076	−	0.742293i	$-0.266262\pi$
$853$	39.1407	1.34015	0.670076	+	0.742293i	$0.266262\pi$
$854$	0.193488	0.00662102
$855$	0	0
$856$	−1.04257	−0.0356342
$857$	−35.0524	−1.19737	−0.598684	−	0.800986i	$-0.704309\pi$
$857$	−35.0524	−1.19737	−0.598684	+	0.800986i	$0.704309\pi$
$858$	0	0
$859$	−32.5206	−1.10959	−0.554794	−	0.831988i	$-0.687203\pi$
$859$	−32.5206	−1.10959	−0.554794	+	0.831988i	$0.687203\pi$
$860$	3.90186	0.133052
$861$	0	0
$862$	−6.27030	−0.213567
$863$	−12.6940	−0.432107	−0.216054	−	0.976381i	$-0.569319\pi$
$863$	−12.6940	−0.432107	−0.216054	+	0.976381i	$0.569319\pi$
$864$	0	0
$865$	−14.7540	−0.501651
$866$	−2.10108	−0.0713975
$867$	0	0
$868$	−14.9844	−0.508605
$869$	0	0
$870$	0	0
$871$	−8.45568	−0.286510
$872$	8.30950	0.281395
$873$	0	0
$874$	−9.72314	−0.328890
$875$	−9.41402	−0.318252
$876$	0	0
$877$	−28.1340	−0.950017	−0.475008	−	0.879981i	$-0.657555\pi$
$877$	−28.1340	−0.950017	−0.475008	+	0.879981i	$0.657555\pi$
$878$	3.32798	0.112314
$879$	0	0
$880$	0	0
$881$	36.7964	1.23970	0.619850	−	0.784720i	$-0.287193\pi$
$881$	36.7964	1.23970	0.619850	+	0.784720i	$0.287193\pi$
$882$	0	0
$883$	−2.28419	−0.0768692	−0.0384346	−	0.999261i	$-0.512237\pi$
$883$	−2.28419	−0.0768692	−0.0384346	+	0.999261i	$0.512237\pi$
$884$	−14.3856	−0.483840
$885$	0	0
$886$	0.552736	0.0185695
$887$	−42.2676	−1.41921	−0.709604	−	0.704600i	$-0.751126\pi$
$887$	−42.2676	−1.41921	−0.709604	+	0.704600i	$0.751126\pi$
$888$	0	0
$889$	−0.289205	−0.00969963
$890$	0.979808	0.0328432
$891$	0	0
$892$	−34.1520	−1.14349
$893$	40.9757	1.37120
$894$	0	0
$895$	−10.8183	−0.361616
$896$	6.78282	0.226598
$897$	0	0
$898$	1.07888	0.0360027
$899$	−50.1142	−1.67140
$900$	0	0
$901$	−9.46132	−0.315202
$902$	0	0
$903$	0	0
$904$	−2.94540	−0.0979627
$905$	27.0283	0.898451
$906$	0	0
$907$	39.5286	1.31252	0.656262	−	0.754533i	$-0.272136\pi$
$907$	39.5286	1.31252	0.656262	+	0.754533i	$0.272136\pi$
$908$	−50.0406	−1.66065
$909$	0	0
$910$	−2.89878	−0.0960936
$911$	35.2296	1.16721	0.583604	−	0.812039i	$-0.301642\pi$
$911$	35.2296	1.16721	0.583604	+	0.812039i	$0.301642\pi$
$912$	0	0
$913$	0	0
$914$	−7.03234	−0.232609
$915$	0	0
$916$	38.6587	1.27732
$917$	16.5059	0.545072
$918$	0	0
$919$	−39.6105	−1.30663	−0.653314	−	0.757087i	$-0.726622\pi$
$919$	−39.6105	−1.30663	−0.653314	+	0.757087i	$0.726622\pi$
$920$	−15.7891	−0.520551
$921$	0	0
$922$	6.72334	0.221421
$923$	−23.2141	−0.764101
$924$	0	0
$925$	−4.89131	−0.160825
$926$	5.75828	0.189229
$927$	0	0
$928$	17.0928	0.561098
$929$	−2.61657	−0.0858469	−0.0429234	−	0.999078i	$-0.513667\pi$
$929$	−2.61657	−0.0858469	−0.0429234	+	0.999078i	$0.513667\pi$
$930$	0	0
$931$	−6.06848	−0.198886
$932$	−39.3947	−1.29042
$933$	0	0
$934$	−0.720463	−0.0235743
$935$	0	0
$936$	0	0
$937$	53.8401	1.75888	0.879439	−	0.476011i	$-0.157918\pi$
$937$	53.8401	1.75888	0.879439	+	0.476011i	$0.157918\pi$
$938$	0.372498	0.0121625
$939$	0	0
$940$	32.8384	1.07107
$941$	−29.6476	−0.966484	−0.483242	−	0.875487i	$-0.660541\pi$
$941$	−29.6476	−0.966484	−0.483242	+	0.875487i	$0.660541\pi$
$942$	0	0
$943$	47.8026	1.55667
$944$	−10.6294	−0.345959
$945$	0	0
$946$	0	0
$947$	15.7861	0.512980	0.256490	−	0.966547i	$-0.417434\pi$
$947$	15.7861	0.512980	0.256490	+	0.966547i	$0.417434\pi$
$948$	0	0
$949$	−76.2440	−2.47498
$950$	1.68525	0.0546768
$951$	0	0
$952$	1.28410	0.0416178
$953$	35.1119	1.13739	0.568693	−	0.822550i	$-0.307449\pi$
$953$	35.1119	1.13739	0.568693	+	0.822550i	$0.307449\pi$
$954$	0	0
$955$	−29.0869	−0.941229
$956$	33.3257	1.07783
$957$	0	0
$958$	0.731463	0.0236325
$959$	−9.32588	−0.301148
$960$	0	0
$961$	28.1200	0.907096
$962$	4.62816	0.149218
$963$	0	0
$964$	47.0698	1.51602
$965$	56.0131	1.80313
$966$	0	0
$967$	−49.2820	−1.58480	−0.792401	−	0.610001i	$-0.791169\pi$
$967$	−49.2820	−1.58480	−0.792401	+	0.610001i	$0.791169\pi$
$968$	0	0
$969$	0	0
$970$	6.81797	0.218912
$971$	−37.3424	−1.19838	−0.599188	−	0.800608i	$-0.704510\pi$
$971$	−37.3424	−1.19838	−0.599188	+	0.800608i	$0.704510\pi$
$972$	0	0
$973$	4.82649	0.154730
$974$	−2.23292	−0.0715475
$975$	0	0
$976$	−3.16099	−0.101181
$977$	−40.0485	−1.28126	−0.640632	−	0.767848i	$-0.721328\pi$
$977$	−40.0485	−1.28126	−0.640632	+	0.767848i	$0.721328\pi$
$978$	0	0
$979$	0	0
$980$	−4.86335	−0.155354
$981$	0	0
$982$	0.955970	0.0305062
$983$	52.0414	1.65986	0.829932	−	0.557864i	$-0.188379\pi$
$983$	52.0414	1.65986	0.829932	+	0.557864i	$0.188379\pi$
$984$	0	0
$985$	−60.1475	−1.91646
$986$	2.11945	0.0674970
$987$	0	0
$988$	60.7285	1.93203
$989$	5.68262	0.180697
$990$	0	0
$991$	45.4828	1.44481	0.722404	−	0.691471i	$-0.243037\pi$
$991$	45.4828	1.44481	0.722404	+	0.691471i	$0.243037\pi$
$992$	−20.1645	−0.640222
$993$	0	0
$994$	1.02265	0.0324365
$995$	46.7175	1.48104
$996$	0	0
$997$	27.9594	0.885482	0.442741	−	0.896650i	$-0.354006\pi$
$997$	27.9594	0.885482	0.442741	+	0.896650i	$0.354006\pi$
$998$	4.61364	0.146042
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.ct.1.5		8
3.2	odd	2		847.2.a.p.1.4		8
11.5	even	5		693.2.m.i.190.2		16
11.9	even	5		693.2.m.i.631.2		16
11.10	odd	2		7623.2.a.cw.1.4		8
21.20	even	2		5929.2.a.bt.1.4		8
33.2	even	10		847.2.f.x.323.2		16
33.5	odd	10		77.2.f.b.36.3	yes	16
33.8	even	10		847.2.f.v.372.3		16
33.14	odd	10		847.2.f.w.372.2		16
33.17	even	10		847.2.f.x.729.2		16
33.20	odd	10		77.2.f.b.15.3	✓	16
33.26	odd	10		847.2.f.w.148.2		16
33.29	even	10		847.2.f.v.148.3		16
33.32	even	2		847.2.a.o.1.5		8
231.5	even	30		539.2.q.f.410.2		32
231.20	even	10		539.2.f.e.246.3		16
231.38	even	30		539.2.q.f.520.3		32
231.53	odd	30		539.2.q.g.422.2		32
231.86	odd	30		539.2.q.g.312.3		32
231.104	even	10		539.2.f.e.344.3		16
231.137	odd	30		539.2.q.g.520.3		32
231.152	even	30		539.2.q.f.312.3		32
231.170	odd	30		539.2.q.g.410.2		32
231.185	even	30		539.2.q.f.422.2		32
231.230	odd	2		5929.2.a.bs.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.b.15.3	✓	16	33.20	odd	10
77.2.f.b.36.3	yes	16	33.5	odd	10
539.2.f.e.246.3		16	231.20	even	10
539.2.f.e.344.3		16	231.104	even	10
539.2.q.f.312.3		32	231.152	even	30
539.2.q.f.410.2		32	231.5	even	30
539.2.q.f.422.2		32	231.185	even	30
539.2.q.f.520.3		32	231.38	even	30
539.2.q.g.312.3		32	231.86	odd	30
539.2.q.g.410.2		32	231.170	odd	30
539.2.q.g.422.2		32	231.53	odd	30
539.2.q.g.520.3		32	231.137	odd	30
693.2.m.i.190.2		16	11.5	even	5
693.2.m.i.631.2		16	11.9	even	5
847.2.a.o.1.5		8	33.32	even	2
847.2.a.p.1.4		8	3.2	odd	2
847.2.f.v.148.3		16	33.29	even	10
847.2.f.v.372.3		16	33.8	even	10
847.2.f.w.148.2		16	33.26	odd	10
847.2.f.w.372.2		16	33.14	odd	10
847.2.f.x.323.2		16	33.2	even	10
847.2.f.x.729.2		16	33.17	even	10
5929.2.a.bs.1.5		8	231.230	odd	2
5929.2.a.bt.1.4		8	21.20	even	2
7623.2.a.ct.1.5		8	1.1	even	1	trivial
7623.2.a.cw.1.4		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)

\(f(q)\)	\(=\)	\(q-0.226211 q^{2} -1.94883 q^{4} +2.49552 q^{5} +1.00000 q^{7} +0.893270 q^{8} +O(q^{10})\)	\(q-0.226211 q^{2} -1.94883 q^{4} +2.49552 q^{5} +1.00000 q^{7} +0.893270 q^{8} -0.564516 q^{10} +5.13499 q^{13} -0.226211 q^{14} +3.69559 q^{16} +1.43752 q^{17} -6.06848 q^{19} -4.86335 q^{20} -7.08292 q^{23} +1.22764 q^{25} -1.16159 q^{26} -1.94883 q^{28} -6.51769 q^{29} +7.68895 q^{31} -2.62252 q^{32} -0.325184 q^{34} +2.49552 q^{35} -3.98432 q^{37} +1.37276 q^{38} +2.22918 q^{40} -6.74900 q^{41} -0.802299 q^{43} +1.60224 q^{46} -6.75222 q^{47} +1.00000 q^{49} -0.277706 q^{50} -10.0072 q^{52} -6.58167 q^{53} +0.893270 q^{56} +1.47437 q^{58} -2.87625 q^{59} -0.855342 q^{61} -1.73933 q^{62} -6.79793 q^{64} +12.8145 q^{65} -1.64668 q^{67} -2.80149 q^{68} -0.564516 q^{70} -4.52077 q^{71} -14.8479 q^{73} +0.901299 q^{74} +11.8264 q^{76} +2.45291 q^{79} +9.22243 q^{80} +1.52670 q^{82} -2.24780 q^{83} +3.58738 q^{85} +0.181489 q^{86} -1.73566 q^{89} +5.13499 q^{91} +13.8034 q^{92} +1.52743 q^{94} -15.1440 q^{95} -12.0776 q^{97} -0.226211 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7}+O(q^{10}) \) 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

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