LMFDB - Embedded newform 7623.2.a.cj.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:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 9 $ x^4 - 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.57794$ 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)

$f(q)$	$=$	$q+1.41421 q^{2} -2.64575 q^{5} -1.00000 q^{7} -2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} -2.64575 q^{5} -1.00000 q^{7} -2.82843 q^{8} -3.74166 q^{10} -1.74166 q^{13} -1.41421 q^{14} -4.00000 q^{16} +0.182676 q^{17} +1.74166 q^{19} +6.70572 q^{23} +2.00000 q^{25} -2.46308 q^{26} -6.70572 q^{29} -9.48331 q^{31} +0.258343 q^{34} +2.64575 q^{35} -11.4833 q^{37} +2.46308 q^{38} +7.48331 q^{40} -5.65685 q^{41} +1.00000 q^{43} +9.48331 q^{46} -0.182676 q^{47} +1.00000 q^{49} +2.82843 q^{50} -7.75458 q^{53} +2.82843 q^{56} -9.48331 q^{58} +3.01110 q^{59} +5.74166 q^{61} -13.4114 q^{62} +8.00000 q^{64} +4.60799 q^{65} +8.48331 q^{67} +3.74166 q^{70} -1.41421 q^{71} +8.25834 q^{73} -16.2399 q^{74} -1.48331 q^{79} +10.5830 q^{80} -8.00000 q^{82} +5.83953 q^{83} -0.483315 q^{85} +1.41421 q^{86} +11.1310 q^{89} +1.74166 q^{91} -0.258343 q^{94} -4.60799 q^{95} +15.2250 q^{97} +1.41421 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^7	$ 4 q - 4 q^{7} + 8 q^{13} - 16 q^{16} - 8 q^{19} + 8 q^{25} - 8 q^{31} + 16 q^{34} - 16 q^{37} + 4 q^{43} + 8 q^{46} + 4 q^{49} - 8 q^{58} + 8 q^{61} + 32 q^{64} + 4 q^{67} + 48 q^{73} + 24 q^{79} - 32 q^{82} + 28 q^{85} - 8 q^{91} - 16 q^{94} + 16 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 + 8 * q^13 - 16 * q^16 - 8 * q^19 + 8 * q^25 - 8 * q^31 + 16 * q^34 - 16 * q^37 + 4 * q^43 + 8 * q^46 + 4 * q^49 - 8 * q^58 + 8 * q^61 + 32 * q^64 + 4 * q^67 + 48 * q^73 + 24 * q^79 - 32 * q^82 + 28 * q^85 - 8 * q^91 - 16 * q^94 + 16 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.64575	−1.18322	−0.591608	−	0.806226i	$-0.701507\pi$
$5$	−2.64575	−1.18322	−0.591608	+	0.806226i	$0.701507\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.82843	−1.00000
$9$	0	0
$10$	−3.74166	−1.18322
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.74166	−0.483049	−0.241524	−	0.970395i	$-0.577647\pi$
$13$	−1.74166	−0.483049	−0.241524	+	0.970395i	$0.577647\pi$
$14$	−1.41421	−0.377964
$15$	0	0
$16$	−4.00000	−1.00000
$17$	0.182676	0.0443054	0.0221527	−	0.999755i	$-0.492948\pi$
$17$	0.182676	0.0443054	0.0221527	+	0.999755i	$0.492948\pi$
$18$	0	0
$19$	1.74166	0.399564	0.199782	−	0.979840i	$-0.435977\pi$
$19$	1.74166	0.399564	0.199782	+	0.979840i	$0.435977\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.70572	1.39824	0.699119	−	0.715005i	$-0.253576\pi$
$23$	6.70572	1.39824	0.699119	+	0.715005i	$0.253576\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	−2.46308	−0.483049
$27$	0	0
$28$	0	0
$29$	−6.70572	−1.24522	−0.622610	−	0.782532i	$-0.713928\pi$
$29$	−6.70572	−1.24522	−0.622610	+	0.782532i	$0.713928\pi$
$30$	0	0
$31$	−9.48331	−1.70325	−0.851627	−	0.524149i	$-0.824384\pi$
$31$	−9.48331	−1.70325	−0.851627	+	0.524149i	$0.824384\pi$
$32$	0	0
$33$	0	0
$34$	0.258343	0.0443054
$35$	2.64575	0.447214
$36$	0	0
$37$	−11.4833	−1.88785	−0.943923	−	0.330167i	$-0.892895\pi$
$37$	−11.4833	−1.88785	−0.943923	+	0.330167i	$0.892895\pi$
$38$	2.46308	0.399564
$39$	0	0
$40$	7.48331	1.18322
$41$	−5.65685	−0.883452	−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685	−0.883452	−0.441726	+	0.897150i	$0.645634\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	0	0
$46$	9.48331	1.39824
$47$	−0.182676	−0.0266460	−0.0133230	−	0.999911i	$-0.504241\pi$
$47$	−0.182676	−0.0266460	−0.0133230	+	0.999911i	$0.504241\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.82843	0.400000
$51$	0	0
$52$	0	0
$53$	−7.75458	−1.06517	−0.532587	−	0.846376i	$-0.678780\pi$
$53$	−7.75458	−1.06517	−0.532587	+	0.846376i	$0.678780\pi$
$54$	0	0
$55$	0	0
$56$	2.82843	0.377964
$57$	0	0
$58$	−9.48331	−1.24522
$59$	3.01110	0.392012	0.196006	−	0.980603i	$-0.437203\pi$
$59$	3.01110	0.392012	0.196006	+	0.980603i	$0.437203\pi$
$60$	0	0
$61$	5.74166	0.735144	0.367572	−	0.929995i	$-0.380189\pi$
$61$	5.74166	0.735144	0.367572	+	0.929995i	$0.380189\pi$
$62$	−13.4114	−1.70325
$63$	0	0
$64$	8.00000	1.00000
$65$	4.60799	0.571551
$66$	0	0
$67$	8.48331	1.03640	0.518201	−	0.855259i	$-0.326602\pi$
$67$	8.48331	1.03640	0.518201	+	0.855259i	$0.326602\pi$
$68$	0	0
$69$	0	0
$70$	3.74166	0.447214
$71$	−1.41421	−0.167836	−0.0839181	−	0.996473i	$-0.526743\pi$
$71$	−1.41421	−0.167836	−0.0839181	+	0.996473i	$0.526743\pi$
$72$	0	0
$73$	8.25834	0.966566	0.483283	−	0.875464i	$-0.339444\pi$
$73$	8.25834	0.966566	0.483283	+	0.875464i	$0.339444\pi$
$74$	−16.2399	−1.88785
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.48331	−0.166886	−0.0834430	−	0.996513i	$-0.526592\pi$
$79$	−1.48331	−0.166886	−0.0834430	+	0.996513i	$0.526592\pi$
$80$	10.5830	1.18322
$81$	0	0
$82$	−8.00000	−0.883452
$83$	5.83953	0.640972	0.320486	−	0.947253i	$-0.396154\pi$
$83$	5.83953	0.640972	0.320486	+	0.947253i	$0.396154\pi$
$84$	0	0
$85$	−0.483315	−0.0524228
$86$	1.41421	0.152499
$87$	0	0
$88$	0	0
$89$	11.1310	1.17989	0.589944	−	0.807444i	$-0.299150\pi$
$89$	11.1310	1.17989	0.589944	+	0.807444i	$0.299150\pi$
$90$	0	0
$91$	1.74166	0.182575
$92$	0	0
$93$	0	0
$94$	−0.258343	−0.0266460
$95$	−4.60799	−0.472770
$96$	0	0
$97$	15.2250	1.54586	0.772931	−	0.634490i	$-0.218790\pi$
$97$	15.2250	1.54586	0.772931	+	0.634490i	$0.218790\pi$
$98$	1.41421	0.142857
$99$	0	0
$100$	0	0
$101$	−13.9595	−1.38902	−0.694509	−	0.719484i	$-0.744378\pi$
$101$	−13.9595	−1.38902	−0.694509	+	0.719484i	$0.744378\pi$
$102$	0	0
$103$	9.22497	0.908964	0.454482	−	0.890756i	$-0.349824\pi$
$103$	9.22497	0.908964	0.454482	+	0.890756i	$0.349824\pi$
$104$	4.92615	0.483049
$105$	0	0
$106$	−10.9666	−1.06517
$107$	1.04886	0.101397	0.0506987	−	0.998714i	$-0.483855\pi$
$107$	1.04886	0.101397	0.0506987	+	0.998714i	$0.483855\pi$
$108$	0	0
$109$	−1.51669	−0.145272	−0.0726360	−	0.997359i	$-0.523141\pi$
$109$	−1.51669	−0.145272	−0.0726360	+	0.997359i	$0.523141\pi$
$110$	0	0
$111$	0	0
$112$	4.00000	0.377964
$113$	−5.29150	−0.497783	−0.248891	−	0.968531i	$-0.580066\pi$
$113$	−5.29150	−0.497783	−0.248891	+	0.968531i	$0.580066\pi$
$114$	0	0
$115$	−17.7417	−1.65442
$116$	0	0
$117$	0	0
$118$	4.25834	0.392012
$119$	−0.182676	−0.0167459
$120$	0	0
$121$	0	0
$122$	8.11993	0.735144
$123$	0	0
$124$	0	0
$125$	7.93725	0.709930
$126$	0	0
$127$	1.51669	0.134584	0.0672920	−	0.997733i	$-0.478564\pi$
$127$	1.51669	0.134584	0.0672920	+	0.997733i	$0.478564\pi$
$128$	11.3137	1.00000
$129$	0	0
$130$	6.51669	0.571551
$131$	13.9595	1.21964	0.609822	−	0.792539i	$-0.291241\pi$
$131$	13.9595	1.21964	0.609822	+	0.792539i	$0.291241\pi$
$132$	0	0
$133$	−1.74166	−0.151021
$134$	11.9972	1.03640
$135$	0	0
$136$	−0.516685	−0.0443054
$137$	−22.9456	−1.96037	−0.980186	−	0.198077i	$-0.936530\pi$
$137$	−22.9456	−1.96037	−0.980186	+	0.198077i	$0.936530\pi$
$138$	0	0
$139$	3.74166	0.317363	0.158682	−	0.987330i	$-0.449276\pi$
$139$	3.74166	0.317363	0.158682	+	0.987330i	$0.449276\pi$
$140$	0	0
$141$	0	0
$142$	−2.00000	−0.167836
$143$	0	0
$144$	0	0
$145$	17.7417	1.47336
$146$	11.6791	0.966566
$147$	0	0
$148$	0	0
$149$	2.14492	0.175718	0.0878592	−	0.996133i	$-0.471997\pi$
$149$	2.14492	0.175718	0.0878592	+	0.996133i	$0.471997\pi$
$150$	0	0
$151$	22.4833	1.82967	0.914833	−	0.403832i	$-0.132322\pi$
$151$	22.4833	1.82967	0.914833	+	0.403832i	$0.132322\pi$
$152$	−4.92615	−0.399564
$153$	0	0
$154$	0	0
$155$	25.0905	2.01532
$156$	0	0
$157$	16.9666	1.35408	0.677042	−	0.735944i	$-0.263261\pi$
$157$	16.9666	1.35408	0.677042	+	0.735944i	$0.263261\pi$
$158$	−2.09772	−0.166886
$159$	0	0
$160$	0	0
$161$	−6.70572	−0.528484
$162$	0	0
$163$	20.4499	1.60176	0.800882	−	0.598823i	$-0.204365\pi$
$163$	20.4499	1.60176	0.800882	+	0.598823i	$0.204365\pi$
$164$	0	0
$165$	0	0
$166$	8.25834	0.640972
$167$	−13.9595	−1.08022	−0.540108	−	0.841596i	$-0.681617\pi$
$167$	−13.9595	−1.08022	−0.540108	+	0.841596i	$0.681617\pi$
$168$	0	0
$169$	−9.96663	−0.766664
$170$	−0.683510	−0.0524228
$171$	0	0
$172$	0	0
$173$	−10.7657	−0.818500	−0.409250	−	0.912422i	$-0.634210\pi$
$173$	−10.7657	−0.818500	−0.409250	+	0.912422i	$0.634210\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	15.7417	1.17989
$179$	14.1421	1.05703	0.528516	−	0.848923i	$-0.322748\pi$
$179$	14.1421	1.05703	0.528516	+	0.848923i	$0.322748\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	2.46308	0.182575
$183$	0	0
$184$	−18.9666	−1.39824
$185$	30.3820	2.23373
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−6.51669	−0.472770
$191$	−1.41421	−0.102329	−0.0511645	−	0.998690i	$-0.516293\pi$
$191$	−1.41421	−0.102329	−0.0511645	+	0.998690i	$0.516293\pi$
$192$	0	0
$193$	1.51669	0.109173	0.0545867	−	0.998509i	$-0.482616\pi$
$193$	1.51669	0.109173	0.0545867	+	0.998509i	$0.482616\pi$
$194$	21.5314	1.54586
$195$	0	0
$196$	0	0
$197$	11.6791	0.832099	0.416049	−	0.909342i	$-0.363414\pi$
$197$	11.6791	0.832099	0.416049	+	0.909342i	$0.363414\pi$
$198$	0	0
$199$	9.74166	0.690568	0.345284	−	0.938498i	$-0.387783\pi$
$199$	9.74166	0.690568	0.345284	+	0.938498i	$0.387783\pi$
$200$	−5.65685	−0.400000
$201$	0	0
$202$	−19.7417	−1.38902
$203$	6.70572	0.470649
$204$	0	0
$205$	14.9666	1.04531
$206$	13.0461	0.908964
$207$	0	0
$208$	6.96663	0.483049
$209$	0	0
$210$	0	0
$211$	24.4833	1.68550	0.842750	−	0.538304i	$-0.180935\pi$
$211$	24.4833	1.68550	0.842750	+	0.538304i	$0.180935\pi$
$212$	0	0
$213$	0	0
$214$	1.48331	0.101397
$215$	−2.64575	−0.180439
$216$	0	0
$217$	9.48331	0.643769
$218$	−2.14492	−0.145272
$219$	0	0
$220$	0	0
$221$	−0.318159	−0.0214017
$222$	0	0
$223$	2.51669	0.168530	0.0842649	−	0.996443i	$-0.473146\pi$
$223$	2.51669	0.168530	0.0842649	+	0.996443i	$0.473146\pi$
$224$	0	0
$225$	0	0
$226$	−7.48331	−0.497783
$227$	11.8617	0.787291	0.393646	−	0.919262i	$-0.371214\pi$
$227$	11.8617	0.787291	0.393646	+	0.919262i	$0.371214\pi$
$228$	0	0
$229$	−1.48331	−0.0980202	−0.0490101	−	0.998798i	$-0.515607\pi$
$229$	−1.48331	−0.0980202	−0.0490101	+	0.998798i	$0.515607\pi$
$230$	−25.0905	−1.65442
$231$	0	0
$232$	18.9666	1.24522
$233$	−14.8256	−0.971260	−0.485630	−	0.874164i	$-0.661410\pi$
$233$	−14.8256	−0.971260	−0.485630	+	0.874164i	$0.661410\pi$
$234$	0	0
$235$	0.483315	0.0315280
$236$	0	0
$237$	0	0
$238$	−0.258343	−0.0167459
$239$	2.46308	0.159323	0.0796616	−	0.996822i	$-0.474616\pi$
$239$	2.46308	0.159323	0.0796616	+	0.996822i	$0.474616\pi$
$240$	0	0
$241$	19.2250	1.23839	0.619195	−	0.785238i	$-0.287459\pi$
$241$	19.2250	1.23839	0.619195	+	0.785238i	$0.287459\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.64575	−0.169031
$246$	0	0
$247$	−3.03337	−0.193009
$248$	26.8229	1.70325
$249$	0	0
$250$	11.2250	0.709930
$251$	19.0683	1.20358	0.601790	−	0.798655i	$-0.294455\pi$
$251$	19.0683	1.20358	0.601790	+	0.798655i	$0.294455\pi$
$252$	0	0
$253$	0	0
$254$	2.14492	0.134584
$255$	0	0
$256$	0	0
$257$	16.4225	1.02441	0.512205	−	0.858863i	$-0.328829\pi$
$257$	16.4225	1.02441	0.512205	+	0.858863i	$0.328829\pi$
$258$	0	0
$259$	11.4833	0.713538
$260$	0	0
$261$	0	0
$262$	19.7417	1.21964
$263$	11.6319	0.717252	0.358626	−	0.933481i	$-0.383245\pi$
$263$	11.6319	0.717252	0.358626	+	0.933481i	$0.383245\pi$
$264$	0	0
$265$	20.5167	1.26033
$266$	−2.46308	−0.151021
$267$	0	0
$268$	0	0
$269$	7.38923	0.450529	0.225265	−	0.974298i	$-0.427675\pi$
$269$	7.38923	0.450529	0.225265	+	0.974298i	$0.427675\pi$
$270$	0	0
$271$	−20.9666	−1.27363	−0.636816	−	0.771016i	$-0.719749\pi$
$271$	−20.9666	−1.27363	−0.636816	+	0.771016i	$0.719749\pi$
$272$	−0.730703	−0.0443054
$273$	0	0
$274$	−32.4499	−1.96037
$275$	0	0
$276$	0	0
$277$	−9.96663	−0.598837	−0.299418	−	0.954122i	$-0.596793\pi$
$277$	−9.96663	−0.598837	−0.299418	+	0.954122i	$0.596793\pi$
$278$	5.29150	0.317363
$279$	0	0
$280$	−7.48331	−0.447214
$281$	4.92615	0.293870	0.146935	−	0.989146i	$-0.453059\pi$
$281$	4.92615	0.293870	0.146935	+	0.989146i	$0.453059\pi$
$282$	0	0
$283$	20.9666	1.24634	0.623168	−	0.782088i	$-0.285845\pi$
$283$	20.9666	1.24634	0.623168	+	0.782088i	$0.285845\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.65685	0.333914
$288$	0	0
$289$	−16.9666	−0.998037
$290$	25.0905	1.47336
$291$	0	0
$292$	0	0
$293$	−22.8101	−1.33258	−0.666290	−	0.745693i	$-0.732119\pi$
$293$	−22.8101	−1.33258	−0.666290	+	0.745693i	$0.732119\pi$
$294$	0	0
$295$	−7.96663	−0.463835
$296$	32.4797	1.88785
$297$	0	0
$298$	3.03337	0.175718
$299$	−11.6791	−0.675417
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	31.7962	1.82967
$303$	0	0
$304$	−6.96663	−0.399564
$305$	−15.1910	−0.869834
$306$	0	0
$307$	−30.7083	−1.75261	−0.876307	−	0.481753i	$-0.840000\pi$
$307$	−30.7083	−1.75261	−0.876307	+	0.481753i	$0.840000\pi$
$308$	0	0
$309$	0	0
$310$	35.4833	2.01532
$311$	−32.2970	−1.83140	−0.915699	−	0.401866i	$-0.868362\pi$
$311$	−32.2970	−1.83140	−0.915699	+	0.401866i	$0.868362\pi$
$312$	0	0
$313$	−19.4833	−1.10126	−0.550631	−	0.834749i	$-0.685613\pi$
$313$	−19.4833	−1.10126	−0.550631	+	0.834749i	$0.685613\pi$
$314$	23.9944	1.35408
$315$	0	0
$316$	0	0
$317$	−1.04886	−0.0589100	−0.0294550	−	0.999566i	$-0.509377\pi$
$317$	−1.04886	−0.0589100	−0.0294550	+	0.999566i	$0.509377\pi$
$318$	0	0
$319$	0	0
$320$	−21.1660	−1.18322
$321$	0	0
$322$	−9.48331	−0.528484
$323$	0.318159	0.0177028
$324$	0	0
$325$	−3.48331	−0.193220
$326$	28.9206	1.60176
$327$	0	0
$328$	16.0000	0.883452
$329$	0.182676	0.0100712
$330$	0	0
$331$	5.51669	0.303224	0.151612	−	0.988440i	$-0.451554\pi$
$331$	5.51669	0.303224	0.151612	+	0.988440i	$0.451554\pi$
$332$	0	0
$333$	0	0
$334$	−19.7417	−1.08022
$335$	−22.4447	−1.22629
$336$	0	0
$337$	6.96663	0.379496	0.189748	−	0.981833i	$-0.439233\pi$
$337$	6.96663	0.379496	0.189748	+	0.981833i	$0.439233\pi$
$338$	−14.0949	−0.766664
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−2.82843	−0.152499
$345$	0	0
$346$	−15.2250	−0.818500
$347$	−30.3348	−1.62846	−0.814229	−	0.580544i	$-0.802840\pi$
$347$	−30.3348	−1.62846	−0.814229	+	0.580544i	$0.802840\pi$
$348$	0	0
$349$	18.5167	0.991175	0.495588	−	0.868558i	$-0.334953\pi$
$349$	18.5167	0.991175	0.495588	+	0.868558i	$0.334953\pi$
$350$	−2.82843	−0.151186
$351$	0	0
$352$	0	0
$353$	−33.5758	−1.78706	−0.893529	−	0.449005i	$-0.851778\pi$
$353$	−33.5758	−1.78706	−0.893529	+	0.449005i	$0.851778\pi$
$354$	0	0
$355$	3.74166	0.198587
$356$	0	0
$357$	0	0
$358$	20.0000	1.05703
$359$	28.2371	1.49030	0.745148	−	0.666899i	$-0.232379\pi$
$359$	28.2371	1.49030	0.745148	+	0.666899i	$0.232379\pi$
$360$	0	0
$361$	−15.9666	−0.840349
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−21.8495	−1.14366
$366$	0	0
$367$	−5.22497	−0.272741	−0.136371	−	0.990658i	$-0.543544\pi$
$367$	−5.22497	−0.272741	−0.136371	+	0.990658i	$0.543544\pi$
$368$	−26.8229	−1.39824
$369$	0	0
$370$	42.9666	2.23373
$371$	7.75458	0.402598
$372$	0	0
$373$	−25.4499	−1.31775	−0.658874	−	0.752253i	$-0.728967\pi$
$373$	−25.4499	−1.31775	−0.658874	+	0.752253i	$0.728967\pi$
$374$	0	0
$375$	0	0
$376$	0.516685	0.0266460
$377$	11.6791	0.601502
$378$	0	0
$379$	−19.5167	−1.00250	−0.501252	−	0.865301i	$-0.667127\pi$
$379$	−19.5167	−1.00250	−0.501252	+	0.865301i	$0.667127\pi$
$380$	0	0
$381$	0	0
$382$	−2.00000	−0.102329
$383$	15.6918	0.801815	0.400908	−	0.916119i	$-0.368695\pi$
$383$	15.6918	0.801815	0.400908	+	0.916119i	$0.368695\pi$
$384$	0	0
$385$	0	0
$386$	2.14492	0.109173
$387$	0	0
$388$	0	0
$389$	−22.9928	−1.16578	−0.582890	−	0.812551i	$-0.698078\pi$
$389$	−22.9928	−1.16578	−0.582890	+	0.812551i	$0.698078\pi$
$390$	0	0
$391$	1.22497	0.0619495
$392$	−2.82843	−0.142857
$393$	0	0
$394$	16.5167	0.832099
$395$	3.92448	0.197462
$396$	0	0
$397$	−13.4833	−0.676708	−0.338354	−	0.941019i	$-0.609870\pi$
$397$	−13.4833	−0.676708	−0.338354	+	0.941019i	$0.609870\pi$
$398$	13.7768	0.690568
$399$	0	0
$400$	−8.00000	−0.400000
$401$	−2.46308	−0.123000	−0.0615001	−	0.998107i	$-0.519588\pi$
$401$	−2.46308	−0.123000	−0.0615001	+	0.998107i	$0.519588\pi$
$402$	0	0
$403$	16.5167	0.822755
$404$	0	0
$405$	0	0
$406$	9.48331	0.470649
$407$	0	0
$408$	0	0
$409$	13.2250	0.653933	0.326966	−	0.945036i	$-0.393974\pi$
$409$	13.2250	0.653933	0.326966	+	0.945036i	$0.393974\pi$
$410$	21.1660	1.04531
$411$	0	0
$412$	0	0
$413$	−3.01110	−0.148167
$414$	0	0
$415$	−15.4499	−0.758408
$416$	0	0
$417$	0	0
$418$	0	0
$419$	18.1549	0.886925	0.443463	−	0.896293i	$-0.353750\pi$
$419$	18.1549	0.886925	0.443463	+	0.896293i	$0.353750\pi$
$420$	0	0
$421$	−22.4833	−1.09577	−0.547885	−	0.836554i	$-0.684567\pi$
$421$	−22.4833	−1.09577	−0.547885	+	0.836554i	$0.684567\pi$
$422$	34.6246	1.68550
$423$	0	0
$424$	21.9333	1.06517
$425$	0.365352	0.0177222
$426$	0	0
$427$	−5.74166	−0.277858
$428$	0	0
$429$	0	0
$430$	−3.74166	−0.180439
$431$	15.8745	0.764648	0.382324	−	0.924028i	$-0.375124\pi$
$431$	15.8745	0.764648	0.382324	+	0.924028i	$0.375124\pi$
$432$	0	0
$433$	5.74166	0.275926	0.137963	−	0.990437i	$-0.455944\pi$
$433$	5.74166	0.275926	0.137963	+	0.990437i	$0.455944\pi$
$434$	13.4114	0.643769
$435$	0	0
$436$	0	0
$437$	11.6791	0.558685
$438$	0	0
$439$	4.51669	0.215570	0.107785	−	0.994174i	$-0.465624\pi$
$439$	4.51669	0.215570	0.107785	+	0.994174i	$0.465624\pi$
$440$	0	0
$441$	0	0
$442$	−0.449944	−0.0214017
$443$	−37.8184	−1.79681	−0.898404	−	0.439171i	$-0.855272\pi$
$443$	−37.8184	−1.79681	−0.898404	+	0.439171i	$0.855272\pi$
$444$	0	0
$445$	−29.4499	−1.39606
$446$	3.55913	0.168530
$447$	0	0
$448$	−8.00000	−0.377964
$449$	−21.5314	−1.01613	−0.508064	−	0.861319i	$-0.669639\pi$
$449$	−21.5314	−1.01613	−0.508064	+	0.861319i	$0.669639\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	16.7750	0.787291
$455$	−4.60799	−0.216026
$456$	0	0
$457$	23.0000	1.07589	0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000	1.07589	0.537947	+	0.842978i	$0.319200\pi$
$458$	−2.09772	−0.0980202
$459$	0	0
$460$	0	0
$461$	−29.8340	−1.38951	−0.694753	−	0.719248i	$-0.744486\pi$
$461$	−29.8340	−1.38951	−0.694753	+	0.719248i	$0.744486\pi$
$462$	0	0
$463$	−2.51669	−0.116960	−0.0584801	−	0.998289i	$-0.518625\pi$
$463$	−2.51669	−0.116960	−0.0584801	+	0.998289i	$0.518625\pi$
$464$	26.8229	1.24522
$465$	0	0
$466$	−20.9666	−0.971260
$467$	−32.1144	−1.48608	−0.743038	−	0.669249i	$-0.766616\pi$
$467$	−32.1144	−1.48608	−0.743038	+	0.669249i	$0.766616\pi$
$468$	0	0
$469$	−8.48331	−0.391723
$470$	0.683510	0.0315280
$471$	0	0
$472$	−8.51669	−0.392012
$473$	0	0
$474$	0	0
$475$	3.48331	0.159825
$476$	0	0
$477$	0	0
$478$	3.48331	0.159323
$479$	39.0500	1.78424	0.892119	−	0.451801i	$-0.149218\pi$
$479$	39.0500	1.78424	0.892119	+	0.451801i	$0.149218\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	27.1882	1.23839
$483$	0	0
$484$	0	0
$485$	−40.2815	−1.82909
$486$	0	0
$487$	20.4833	0.928188	0.464094	−	0.885786i	$-0.346380\pi$
$487$	20.4833	0.928188	0.464094	+	0.885786i	$0.346380\pi$
$488$	−16.2399	−0.735144
$489$	0	0
$490$	−3.74166	−0.169031
$491$	−29.6985	−1.34027	−0.670137	−	0.742237i	$-0.733765\pi$
$491$	−29.6985	−1.34027	−0.670137	+	0.742237i	$0.733765\pi$
$492$	0	0
$493$	−1.22497	−0.0551700
$494$	−4.28983	−0.193009
$495$	0	0
$496$	37.9333	1.70325
$497$	1.41421	0.0634361
$498$	0	0
$499$	11.9666	0.535700	0.267850	−	0.963461i	$-0.413687\pi$
$499$	11.9666	0.535700	0.267850	+	0.963461i	$0.413687\pi$
$500$	0	0
$501$	0	0
$502$	26.9666	1.20358
$503$	34.3948	1.53359	0.766793	−	0.641894i	$-0.221851\pi$
$503$	34.3948	1.53359	0.766793	+	0.641894i	$0.221851\pi$
$504$	0	0
$505$	36.9333	1.64351
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−27.7362	−1.22939	−0.614694	−	0.788766i	$-0.710720\pi$
$509$	−27.7362	−1.22939	−0.614694	+	0.788766i	$0.710720\pi$
$510$	0	0
$511$	−8.25834	−0.365328
$512$	−22.6274	−1.00000
$513$	0	0
$514$	23.2250	1.02441
$515$	−24.4070	−1.07550
$516$	0	0
$517$	0	0
$518$	16.2399	0.713538
$519$	0	0
$520$	−13.0334	−0.571551
$521$	29.8340	1.30705	0.653525	−	0.756905i	$-0.273289\pi$
$521$	29.8340	1.30705	0.653525	+	0.756905i	$0.273289\pi$
$522$	0	0
$523$	15.2250	0.665742	0.332871	−	0.942972i	$-0.391983\pi$
$523$	15.2250	0.665742	0.332871	+	0.942972i	$0.391983\pi$
$524$	0	0
$525$	0	0
$526$	16.4499	0.717252
$527$	−1.73237	−0.0754633
$528$	0	0
$529$	21.9666	0.955071
$530$	29.0150	1.26033
$531$	0	0
$532$	0	0
$533$	9.85230	0.426751
$534$	0	0
$535$	−2.77503	−0.119975
$536$	−23.9944	−1.03640
$537$	0	0
$538$	10.4499	0.450529
$539$	0	0
$540$	0	0
$541$	15.0000	0.644900	0.322450	−	0.946586i	$-0.395494\pi$
$541$	15.0000	0.644900	0.322450	+	0.946586i	$0.395494\pi$
$542$	−29.6513	−1.27363
$543$	0	0
$544$	0	0
$545$	4.01277	0.171888
$546$	0	0
$547$	−10.0000	−0.427569	−0.213785	−	0.976881i	$-0.568579\pi$
$547$	−10.0000	−0.427569	−0.213785	+	0.976881i	$0.568579\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−11.6791	−0.497545
$552$	0	0
$553$	1.48331	0.0630770
$554$	−14.0949	−0.598837
$555$	0	0
$556$	0	0
$557$	−15.2382	−0.645663	−0.322831	−	0.946456i	$-0.604635\pi$
$557$	−15.2382	−0.645663	−0.322831	+	0.946456i	$0.604635\pi$
$558$	0	0
$559$	−1.74166	−0.0736643
$560$	−10.5830	−0.447214
$561$	0	0
$562$	6.96663	0.293870
$563$	13.2288	0.557526	0.278763	−	0.960360i	$-0.410076\pi$
$563$	13.2288	0.557526	0.278763	+	0.960360i	$0.410076\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	29.6513	1.24634
$567$	0	0
$568$	4.00000	0.167836
$569$	18.3848	0.770730	0.385365	−	0.922764i	$-0.374076\pi$
$569$	18.3848	0.770730	0.385365	+	0.922764i	$0.374076\pi$
$570$	0	0
$571$	−2.96663	−0.124150	−0.0620748	−	0.998072i	$-0.519772\pi$
$571$	−2.96663	−0.124150	−0.0620748	+	0.998072i	$0.519772\pi$
$572$	0	0
$573$	0	0
$574$	8.00000	0.333914
$575$	13.4114	0.559295
$576$	0	0
$577$	18.7083	0.778836	0.389418	−	0.921061i	$-0.372676\pi$
$577$	18.7083	0.778836	0.389418	+	0.921061i	$0.372676\pi$
$578$	−23.9944	−0.998037
$579$	0	0
$580$	0	0
$581$	−5.83953	−0.242265
$582$	0	0
$583$	0	0
$584$	−23.3581	−0.966566
$585$	0	0
$586$	−32.2583	−1.33258
$587$	−35.4908	−1.46486	−0.732431	−	0.680841i	$-0.761615\pi$
$587$	−35.4908	−1.46486	−0.732431	+	0.680841i	$0.761615\pi$
$588$	0	0
$589$	−16.5167	−0.680558
$590$	−11.2665	−0.463835
$591$	0	0
$592$	45.9333	1.88785
$593$	39.6863	1.62972	0.814860	−	0.579658i	$-0.196814\pi$
$593$	39.6863	1.62972	0.814860	+	0.579658i	$0.196814\pi$
$594$	0	0
$595$	0.483315	0.0198140
$596$	0	0
$597$	0	0
$598$	−16.5167	−0.675417
$599$	−12.6807	−0.518121	−0.259060	−	0.965861i	$-0.583413\pi$
$599$	−12.6807	−0.518121	−0.259060	+	0.965861i	$0.583413\pi$
$600$	0	0
$601$	14.7750	0.602686	0.301343	−	0.953516i	$-0.402565\pi$
$601$	14.7750	0.602686	0.301343	+	0.953516i	$0.402565\pi$
$602$	−1.41421	−0.0576390
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28.2583	1.14697	0.573485	−	0.819216i	$-0.305591\pi$
$607$	28.2583	1.14697	0.573485	+	0.819216i	$0.305591\pi$
$608$	0	0
$609$	0	0
$610$	−21.4833	−0.869834
$611$	0.318159	0.0128713
$612$	0	0
$613$	11.5167	0.465155	0.232577	−	0.972578i	$-0.425284\pi$
$613$	11.5167	0.465155	0.232577	+	0.972578i	$0.425284\pi$
$614$	−43.4281	−1.75261
$615$	0	0
$616$	0	0
$617$	19.0683	0.767660	0.383830	−	0.923404i	$-0.374605\pi$
$617$	19.0683	0.767660	0.383830	+	0.923404i	$0.374605\pi$
$618$	0	0
$619$	−10.4499	−0.420019	−0.210009	−	0.977699i	$-0.567349\pi$
$619$	−10.4499	−0.420019	−0.210009	+	0.977699i	$0.567349\pi$
$620$	0	0
$621$	0	0
$622$	−45.6749	−1.83140
$623$	−11.1310	−0.445955
$624$	0	0
$625$	−31.0000	−1.24000
$626$	−27.5536	−1.10126
$627$	0	0
$628$	0	0
$629$	−2.09772	−0.0836417
$630$	0	0
$631$	19.4499	0.774290	0.387145	−	0.922019i	$-0.373461\pi$
$631$	19.4499	0.774290	0.387145	+	0.922019i	$0.373461\pi$
$632$	4.19545	0.166886
$633$	0	0
$634$	−1.48331	−0.0589100
$635$	−4.01277	−0.159242
$636$	0	0
$637$	−1.74166	−0.0690070
$638$	0	0
$639$	0	0
$640$	−29.9333	−1.18322
$641$	31.0655	1.22701	0.613507	−	0.789689i	$-0.289758\pi$
$641$	31.0655	1.22701	0.613507	+	0.789689i	$0.289758\pi$
$642$	0	0
$643$	4.44994	0.175489	0.0877443	−	0.996143i	$-0.472034\pi$
$643$	4.44994	0.175489	0.0877443	+	0.996143i	$0.472034\pi$
$644$	0	0
$645$	0	0
$646$	0.449944	0.0177028
$647$	42.6091	1.67514	0.837568	−	0.546333i	$-0.183977\pi$
$647$	42.6091	1.67514	0.837568	+	0.546333i	$0.183977\pi$
$648$	0	0
$649$	0	0
$650$	−4.92615	−0.193220
$651$	0	0
$652$	0	0
$653$	−25.0433	−0.980020	−0.490010	−	0.871717i	$-0.663007\pi$
$653$	−25.0433	−0.980020	−0.490010	+	0.871717i	$0.663007\pi$
$654$	0	0
$655$	−36.9333	−1.44310
$656$	22.6274	0.883452
$657$	0	0
$658$	0.258343	0.0100712
$659$	−32.5269	−1.26707	−0.633534	−	0.773715i	$-0.718396\pi$
$659$	−32.5269	−1.26707	−0.633534	+	0.773715i	$0.718396\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	7.80177	0.303224
$663$	0	0
$664$	−16.5167	−0.640972
$665$	4.60799	0.178690
$666$	0	0
$667$	−44.9666	−1.74111
$668$	0	0
$669$	0	0
$670$	−31.7417	−1.22629
$671$	0	0
$672$	0	0
$673$	39.4499	1.52068	0.760342	−	0.649523i	$-0.225031\pi$
$673$	39.4499	1.52068	0.760342	+	0.649523i	$0.225031\pi$
$674$	9.85230	0.379496
$675$	0	0
$676$	0	0
$677$	−32.2970	−1.24128	−0.620638	−	0.784097i	$-0.713126\pi$
$677$	−32.2970	−1.24128	−0.620638	+	0.784097i	$0.713126\pi$
$678$	0	0
$679$	−15.2250	−0.584281
$680$	1.36702	0.0524228
$681$	0	0
$682$	0	0
$683$	−14.1421	−0.541134	−0.270567	−	0.962701i	$-0.587211\pi$
$683$	−14.1421	−0.541134	−0.270567	+	0.962701i	$0.587211\pi$
$684$	0	0
$685$	60.7083	2.31954
$686$	−1.41421	−0.0539949
$687$	0	0
$688$	−4.00000	−0.152499
$689$	13.5058	0.514531
$690$	0	0
$691$	34.4499	1.31054	0.655269	−	0.755396i	$-0.272555\pi$
$691$	34.4499	1.31054	0.655269	+	0.755396i	$0.272555\pi$
$692$	0	0
$693$	0	0
$694$	−42.8999	−1.62846
$695$	−9.89949	−0.375509
$696$	0	0
$697$	−1.03337	−0.0391417
$698$	26.1865	0.991175
$699$	0	0
$700$	0	0
$701$	26.1394	0.987270	0.493635	−	0.869669i	$-0.335668\pi$
$701$	26.1394	0.987270	0.493635	+	0.869669i	$0.335668\pi$
$702$	0	0
$703$	−20.0000	−0.754314
$704$	0	0
$705$	0	0
$706$	−47.4833	−1.78706
$707$	13.9595	0.525000
$708$	0	0
$709$	−34.4166	−1.29254	−0.646271	−	0.763108i	$-0.723672\pi$
$709$	−34.4166	−1.29254	−0.646271	+	0.763108i	$0.723672\pi$
$710$	5.29150	0.198587
$711$	0	0
$712$	−31.4833	−1.17989
$713$	−63.5924	−2.38155
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	39.9333	1.49030
$719$	−2.82843	−0.105483	−0.0527413	−	0.998608i	$-0.516796\pi$
$719$	−2.82843	−0.105483	−0.0527413	+	0.998608i	$0.516796\pi$
$720$	0	0
$721$	−9.22497	−0.343556
$722$	−22.5802	−0.840349
$723$	0	0
$724$	0	0
$725$	−13.4114	−0.498088
$726$	0	0
$727$	−42.1916	−1.56480	−0.782400	−	0.622776i	$-0.786005\pi$
$727$	−42.1916	−1.56480	−0.782400	+	0.622776i	$0.786005\pi$
$728$	−4.92615	−0.182575
$729$	0	0
$730$	−30.8999	−1.14366
$731$	0.182676	0.00675651
$732$	0	0
$733$	−38.7083	−1.42972	−0.714862	−	0.699266i	$-0.753510\pi$
$733$	−38.7083	−1.42972	−0.714862	+	0.699266i	$0.753510\pi$
$734$	−7.38923	−0.272741
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	31.9333	1.17468	0.587342	−	0.809339i	$-0.300174\pi$
$739$	31.9333	1.17468	0.587342	+	0.809339i	$0.300174\pi$
$740$	0	0
$741$	0	0
$742$	10.9666	0.402598
$743$	−21.5786	−0.791640	−0.395820	−	0.918328i	$-0.629540\pi$
$743$	−21.5786	−0.791640	−0.395820	+	0.918328i	$0.629540\pi$
$744$	0	0
$745$	−5.67492	−0.207913
$746$	−35.9917	−1.31775
$747$	0	0
$748$	0	0
$749$	−1.04886	−0.0383246
$750$	0	0
$751$	−2.03337	−0.0741987	−0.0370994	−	0.999312i	$-0.511812\pi$
$751$	−2.03337	−0.0741987	−0.0370994	+	0.999312i	$0.511812\pi$
$752$	0.730703	0.0266460
$753$	0	0
$754$	16.5167	0.601502
$755$	−59.4853	−2.16489
$756$	0	0
$757$	−9.51669	−0.345890	−0.172945	−	0.984932i	$-0.555328\pi$
$757$	−9.51669	−0.345890	−0.172945	+	0.984932i	$0.555328\pi$
$758$	−27.6008	−1.00250
$759$	0	0
$760$	13.0334	0.472770
$761$	−15.0555	−0.545762	−0.272881	−	0.962048i	$-0.587977\pi$
$761$	−15.0555	−0.545762	−0.272881	+	0.962048i	$0.587977\pi$
$762$	0	0
$763$	1.51669	0.0549077
$764$	0	0
$765$	0	0
$766$	22.1916	0.801815
$767$	−5.24431	−0.189361
$768$	0	0
$769$	48.4499	1.74715	0.873575	−	0.486690i	$-0.161796\pi$
$769$	48.4499	1.74715	0.873575	+	0.486690i	$0.161796\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.76401	−0.351187	−0.175594	−	0.984463i	$-0.556184\pi$
$773$	−9.76401	−0.351187	−0.175594	+	0.984463i	$0.556184\pi$
$774$	0	0
$775$	−18.9666	−0.681301
$776$	−43.0627	−1.54586
$777$	0	0
$778$	−32.5167	−1.16578
$779$	−9.85230	−0.352995
$780$	0	0
$781$	0	0
$782$	1.73237	0.0619495
$783$	0	0
$784$	−4.00000	−0.142857
$785$	−44.8895	−1.60217
$786$	0	0
$787$	52.4499	1.86964	0.934819	−	0.355124i	$-0.115561\pi$
$787$	52.4499	1.86964	0.934819	+	0.355124i	$0.115561\pi$
$788$	0	0
$789$	0	0
$790$	5.55006	0.197462
$791$	5.29150	0.188144
$792$	0	0
$793$	−10.0000	−0.355110
$794$	−19.0683	−0.676708
$795$	0	0
$796$	0	0
$797$	53.8284	1.90670	0.953350	−	0.301867i	$-0.0976099\pi$
$797$	53.8284	1.90670	0.953350	+	0.301867i	$0.0976099\pi$
$798$	0	0
$799$	−0.0333705	−0.00118056
$800$	0	0
$801$	0	0
$802$	−3.48331	−0.123000
$803$	0	0
$804$	0	0
$805$	17.7417	0.625311
$806$	23.3581	0.822755
$807$	0	0
$808$	39.4833	1.38902
$809$	−42.7446	−1.50282	−0.751409	−	0.659836i	$-0.770626\pi$
$809$	−42.7446	−1.50282	−0.751409	+	0.659836i	$0.770626\pi$
$810$	0	0
$811$	49.2250	1.72852	0.864261	−	0.503043i	$-0.167786\pi$
$811$	49.2250	1.72852	0.864261	+	0.503043i	$0.167786\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−54.1055	−1.89523
$816$	0	0
$817$	1.74166	0.0609329
$818$	18.7029	0.653933
$819$	0	0
$820$	0	0
$821$	33.9411	1.18455	0.592277	−	0.805735i	$-0.298229\pi$
$821$	33.9411	1.18455	0.592277	+	0.805735i	$0.298229\pi$
$822$	0	0
$823$	3.93326	0.137105	0.0685524	−	0.997648i	$-0.478162\pi$
$823$	3.93326	0.137105	0.0685524	+	0.997648i	$0.478162\pi$
$824$	−26.0922	−0.908964
$825$	0	0
$826$	−4.25834	−0.148167
$827$	52.5969	1.82897	0.914486	−	0.404617i	$-0.132595\pi$
$827$	52.5969	1.82897	0.914486	+	0.404617i	$0.132595\pi$
$828$	0	0
$829$	27.2250	0.945562	0.472781	−	0.881180i	$-0.343250\pi$
$829$	27.2250	0.945562	0.472781	+	0.881180i	$0.343250\pi$
$830$	−21.8495	−0.758408
$831$	0	0
$832$	−13.9333	−0.483049
$833$	0.182676	0.00632934
$834$	0	0
$835$	36.9333	1.27813
$836$	0	0
$837$	0	0
$838$	25.6749	0.886925
$839$	−6.20488	−0.214216	−0.107108	−	0.994247i	$-0.534159\pi$
$839$	−6.20488	−0.214216	−0.107108	+	0.994247i	$0.534159\pi$
$840$	0	0
$841$	15.9666	0.550573
$842$	−31.7962	−1.09577
$843$	0	0
$844$	0	0
$845$	26.3692	0.907129
$846$	0	0
$847$	0	0
$848$	31.0183	1.06517
$849$	0	0
$850$	0.516685	0.0177222
$851$	−77.0038	−2.63966
$852$	0	0
$853$	−9.03337	−0.309297	−0.154648	−	0.987970i	$-0.549424\pi$
$853$	−9.03337	−0.309297	−0.154648	+	0.987970i	$0.549424\pi$
$854$	−8.11993	−0.277858
$855$	0	0
$856$	−2.96663	−0.101397
$857$	24.1771	0.825874	0.412937	−	0.910759i	$-0.364503\pi$
$857$	24.1771	0.825874	0.412937	+	0.910759i	$0.364503\pi$
$858$	0	0
$859$	14.4499	0.493026	0.246513	−	0.969140i	$-0.420715\pi$
$859$	14.4499	0.493026	0.246513	+	0.969140i	$0.420715\pi$
$860$	0	0
$861$	0	0
$862$	22.4499	0.764648
$863$	−13.0461	−0.444094	−0.222047	−	0.975036i	$-0.571274\pi$
$863$	−13.0461	−0.444094	−0.222047	+	0.975036i	$0.571274\pi$
$864$	0	0
$865$	28.4833	0.968462
$866$	8.11993	0.275926
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−14.7750	−0.500633
$872$	4.28983	0.145272
$873$	0	0
$874$	16.5167	0.558685
$875$	−7.93725	−0.268328
$876$	0	0
$877$	40.8999	1.38109	0.690546	−	0.723289i	$-0.257371\pi$
$877$	40.8999	1.38109	0.690546	+	0.723289i	$0.257371\pi$
$878$	6.38756	0.215570
$879$	0	0
$880$	0	0
$881$	22.2621	0.750028	0.375014	−	0.927019i	$-0.377638\pi$
$881$	22.2621	0.750028	0.375014	+	0.927019i	$0.377638\pi$
$882$	0	0
$883$	−22.4166	−0.754378	−0.377189	−	0.926136i	$-0.623109\pi$
$883$	−22.4166	−0.754378	−0.377189	+	0.926136i	$0.623109\pi$
$884$	0	0
$885$	0	0
$886$	−53.4833	−1.79681
$887$	18.1549	0.609582	0.304791	−	0.952419i	$-0.401413\pi$
$887$	18.1549	0.609582	0.304791	+	0.952419i	$0.401413\pi$
$888$	0	0
$889$	−1.51669	−0.0508680
$890$	−41.6485	−1.39606
$891$	0	0
$892$	0	0
$893$	−0.318159	−0.0106468
$894$	0	0
$895$	−37.4166	−1.25070
$896$	−11.3137	−0.377964
$897$	0	0
$898$	−30.4499	−1.01613
$899$	63.5924	2.12093
$900$	0	0
$901$	−1.41657	−0.0471929
$902$	0	0
$903$	0	0
$904$	14.9666	0.497783
$905$	0	0
$906$	0	0
$907$	13.4833	0.447706	0.223853	−	0.974623i	$-0.428136\pi$
$907$	13.4833	0.447706	0.223853	+	0.974623i	$0.428136\pi$
$908$	0	0
$909$	0	0
$910$	−6.51669	−0.216026
$911$	−17.7013	−0.586469	−0.293235	−	0.956041i	$-0.594732\pi$
$911$	−17.7013	−0.586469	−0.293235	+	0.956041i	$0.594732\pi$
$912$	0	0
$913$	0	0
$914$	32.5269	1.07589
$915$	0	0
$916$	0	0
$917$	−13.9595	−0.460982
$918$	0	0
$919$	56.9666	1.87916	0.939578	−	0.342335i	$-0.111218\pi$
$919$	56.9666	1.87916	0.939578	+	0.342335i	$0.111218\pi$
$920$	50.1810	1.65442
$921$	0	0
$922$	−42.1916	−1.38951
$923$	2.46308	0.0810731
$924$	0	0
$925$	−22.9666	−0.755138
$926$	−3.55913	−0.116960
$927$	0	0
$928$	0	0
$929$	−0.548027	−0.0179802	−0.00899010	−	0.999960i	$-0.502862\pi$
$929$	−0.548027	−0.0179802	−0.00899010	+	0.999960i	$0.502862\pi$
$930$	0	0
$931$	1.74166	0.0570805
$932$	0	0
$933$	0	0
$934$	−45.4166	−1.48608
$935$	0	0
$936$	0	0
$937$	−28.2583	−0.923160	−0.461580	−	0.887099i	$-0.652717\pi$
$937$	−28.2583	−0.923160	−0.461580	+	0.887099i	$0.652717\pi$
$938$	−11.9972	−0.391723
$939$	0	0
$940$	0	0
$941$	−17.2415	−0.562058	−0.281029	−	0.959699i	$-0.590676\pi$
$941$	−17.2415	−0.562058	−0.281029	+	0.959699i	$0.590676\pi$
$942$	0	0
$943$	−37.9333	−1.23528
$944$	−12.0444	−0.392012
$945$	0	0
$946$	0	0
$947$	−12.7751	−0.415135	−0.207568	−	0.978221i	$-0.566555\pi$
$947$	−12.7751	−0.415135	−0.207568	+	0.978221i	$0.566555\pi$
$948$	0	0
$949$	−14.3832	−0.466899
$950$	4.92615	0.159825
$951$	0	0
$952$	0.516685	0.0167459
$953$	−39.9633	−1.29454	−0.647270	−	0.762261i	$-0.724089\pi$
$953$	−39.9633	−1.29454	−0.647270	+	0.762261i	$0.724089\pi$
$954$	0	0
$955$	3.74166	0.121077
$956$	0	0
$957$	0	0
$958$	55.2250	1.78424
$959$	22.9456	0.740951
$960$	0	0
$961$	58.9333	1.90107
$962$	28.2843	0.911922
$963$	0	0
$964$	0	0
$965$	−4.01277	−0.129176
$966$	0	0
$967$	28.9333	0.930431	0.465215	−	0.885197i	$-0.345977\pi$
$967$	28.9333	0.930431	0.465215	+	0.885197i	$0.345977\pi$
$968$	0	0
$969$	0	0
$970$	−56.9666	−1.82909
$971$	32.2970	1.03646	0.518231	−	0.855241i	$-0.326591\pi$
$971$	32.2970	1.03646	0.518231	+	0.855241i	$0.326591\pi$
$972$	0	0
$973$	−3.74166	−0.119952
$974$	28.9678	0.928188
$975$	0	0
$976$	−22.9666	−0.735144
$977$	−22.5802	−0.722405	−0.361203	−	0.932487i	$-0.617634\pi$
$977$	−22.5802	−0.722405	−0.361203	+	0.932487i	$0.617634\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−42.0000	−1.34027
$983$	2.82843	0.0902128	0.0451064	−	0.998982i	$-0.485637\pi$
$983$	2.82843	0.0902128	0.0451064	+	0.998982i	$0.485637\pi$
$984$	0	0
$985$	−30.8999	−0.984552
$986$	−1.73237	−0.0551700
$987$	0	0
$988$	0	0
$989$	6.70572	0.213229
$990$	0	0
$991$	−44.4499	−1.41200	−0.706000	−	0.708212i	$-0.749502\pi$
$991$	−44.4499	−1.41200	−0.706000	+	0.708212i	$0.749502\pi$
$992$	0	0
$993$	0	0
$994$	2.00000	0.0634361
$995$	−25.7740	−0.817091
$996$	0	0
$997$	18.9666	0.600679	0.300340	−	0.953832i	$-0.402900\pi$
$997$	18.9666	0.600679	0.300340	+	0.953832i	$0.402900\pi$
$998$	16.9234	0.535700
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cj.1.3	yes	4
3.2	odd	2	inner	7623.2.a.cj.1.2	✓	4
11.10	odd	2		7623.2.a.ck.1.1	yes	4
33.32	even	2		7623.2.a.ck.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7623.2.a.cj.1.2	✓	4	3.2	odd	2	inner
7623.2.a.cj.1.3	yes	4	1.1	even	1	trivial
7623.2.a.ck.1.1	yes	4	11.10	odd	2
7623.2.a.ck.1.4	yes	4	33.32	even	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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -2.64575 q^{5} -1.00000 q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} -2.64575 q^{5} -1.00000 q^{7} -2.82843 q^{8} -3.74166 q^{10} -1.74166 q^{13} -1.41421 q^{14} -4.00000 q^{16} +0.182676 q^{17} +1.74166 q^{19} +6.70572 q^{23} +2.00000 q^{25} -2.46308 q^{26} -6.70572 q^{29} -9.48331 q^{31} +0.258343 q^{34} +2.64575 q^{35} -11.4833 q^{37} +2.46308 q^{38} +7.48331 q^{40} -5.65685 q^{41} +1.00000 q^{43} +9.48331 q^{46} -0.182676 q^{47} +1.00000 q^{49} +2.82843 q^{50} -7.75458 q^{53} +2.82843 q^{56} -9.48331 q^{58} +3.01110 q^{59} +5.74166 q^{61} -13.4114 q^{62} +8.00000 q^{64} +4.60799 q^{65} +8.48331 q^{67} +3.74166 q^{70} -1.41421 q^{71} +8.25834 q^{73} -16.2399 q^{74} -1.48331 q^{79} +10.5830 q^{80} -8.00000 q^{82} +5.83953 q^{83} -0.483315 q^{85} +1.41421 q^{86} +11.1310 q^{89} +1.74166 q^{91} -0.258343 q^{94} -4.60799 q^{95} +15.2250 q^{97} +1.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^7	\( 4 q - 4 q^{7} + 8 q^{13} - 16 q^{16} - 8 q^{19} + 8 q^{25} - 8 q^{31} + 16 q^{34} - 16 q^{37} + 4 q^{43} + 8 q^{46} + 4 q^{49} - 8 q^{58} + 8 q^{61} + 32 q^{64} + 4 q^{67} + 48 q^{73} + 24 q^{79} - 32 q^{82} + 28 q^{85} - 8 q^{91} - 16 q^{94} + 16 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 + 8 * q^13 - 16 * q^16 - 8 * q^19 + 8 * q^25 - 8 * q^31 + 16 * q^34 - 16 * q^37 + 4 * q^43 + 8 * q^46 + 4 * q^49 - 8 * q^58 + 8 * q^61 + 32 * q^64 + 4 * q^67 + 48 * q^73 + 24 * q^79 - 32 * q^82 + 28 * q^85 - 8 * q^91 - 16 * q^94 + 16 * q^97

Introduction

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