LMFDB - Embedded newform 7623.2.a.dc.1.12

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26x^{14} + 268x^{12} - 1395x^{10} + 3876x^{8} - 5635x^{6} + 4042x^{4} - 1272x^{2} + 121 $ x^16 - 26x^14 + 268x^12 - 1395x^10 + 3876x^8 - 5635x^6 + 4042x^4 - 1272*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 693)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.31288$ 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)

$f(q)$	$=$	$q+1.31288 q^{2} -0.276342 q^{4} +3.45937 q^{5} +1.00000 q^{7} -2.98857 q^{8} +O(q^{10})$	$q+1.31288 q^{2} -0.276342 q^{4} +3.45937 q^{5} +1.00000 q^{7} -2.98857 q^{8} +4.54174 q^{10} -5.53810 q^{13} +1.31288 q^{14} -3.37095 q^{16} -4.96611 q^{17} +8.22969 q^{19} -0.955969 q^{20} +7.56119 q^{23} +6.96722 q^{25} -7.27087 q^{26} -0.276342 q^{28} +1.47011 q^{29} +2.41141 q^{31} +1.55148 q^{32} -6.51991 q^{34} +3.45937 q^{35} -2.83639 q^{37} +10.8046 q^{38} -10.3386 q^{40} -6.93310 q^{41} +5.62936 q^{43} +9.92695 q^{46} +0.0417127 q^{47} +1.00000 q^{49} +9.14714 q^{50} +1.53041 q^{52} +4.47145 q^{53} -2.98857 q^{56} +1.93008 q^{58} +2.99486 q^{59} -11.6684 q^{61} +3.16589 q^{62} +8.77881 q^{64} -19.1583 q^{65} +9.95852 q^{67} +1.37234 q^{68} +4.54174 q^{70} +6.99312 q^{71} +7.38320 q^{73} -3.72385 q^{74} -2.27421 q^{76} +16.2755 q^{79} -11.6614 q^{80} -9.10234 q^{82} +9.64377 q^{83} -17.1796 q^{85} +7.39068 q^{86} +7.94906 q^{89} -5.53810 q^{91} -2.08948 q^{92} +0.0547639 q^{94} +28.4695 q^{95} -2.35182 q^{97} +1.31288 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 20 q^{4} + 16 q^{7}+O(q^{10}) $ 16 * q + 20 * q^4 + 16 * q^7	$ 16 q + 20 q^{4} + 16 q^{7} - 6 q^{10} + 32 q^{16} + 10 q^{19} + 44 q^{25} + 20 q^{28} + 30 q^{31} + 12 q^{34} + 38 q^{37} - 68 q^{40} + 16 q^{43} + 80 q^{46} + 16 q^{49} + 2 q^{52} + 18 q^{58} - 28 q^{61} + 34 q^{64} + 52 q^{67} - 6 q^{70} - 14 q^{73} - 14 q^{76} + 54 q^{79} + 64 q^{82} + 30 q^{85} - 60 q^{94} + 108 q^{97}+O(q^{100}) $ 16 * q + 20 * q^4 + 16 * q^7 - 6 * q^10 + 32 * q^16 + 10 * q^19 + 44 * q^25 + 20 * q^28 + 30 * q^31 + 12 * q^34 + 38 * q^37 - 68 * q^40 + 16 * q^43 + 80 * q^46 + 16 * q^49 + 2 * q^52 + 18 * q^58 - 28 * q^61 + 34 * q^64 + 52 * q^67 - 6 * q^70 - 14 * q^73 - 14 * q^76 + 54 * q^79 + 64 * q^82 + 30 * q^85 - 60 * q^94 + 108 * 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 $

$2$	1.31288	0.928347	0.464174	−	0.885744i	$-0.346351\pi$
$2$	1.31288	0.928347	0.464174	+	0.885744i	$0.346351\pi$
$3$	0	0
$3$	0	0
$4$	−0.276342	−0.138171
$5$	3.45937	1.54708	0.773538	−	0.633750i	$-0.218485\pi$
$5$	3.45937	1.54708	0.773538	+	0.633750i	$0.218485\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.98857	−1.05662
$9$	0	0
$10$	4.54174	1.43622
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.53810	−1.53599	−0.767997	−	0.640454i	$-0.778746\pi$
$13$	−5.53810	−1.53599	−0.767997	+	0.640454i	$0.778746\pi$
$14$	1.31288	0.350882
$15$	0	0
$16$	−3.37095	−0.842738
$17$	−4.96611	−1.20446	−0.602229	−	0.798324i	$-0.705720\pi$
$17$	−4.96611	−1.20446	−0.602229	+	0.798324i	$0.705720\pi$
$18$	0	0
$19$	8.22969	1.88802	0.944010	−	0.329918i	$-0.107021\pi$
$19$	8.22969	1.88802	0.944010	+	0.329918i	$0.107021\pi$
$20$	−0.955969	−0.213761
$21$	0	0
$22$	0	0
$23$	7.56119	1.57662	0.788309	−	0.615280i	$-0.210957\pi$
$23$	7.56119	1.57662	0.788309	+	0.615280i	$0.210957\pi$
$24$	0	0
$25$	6.96722	1.39344
$26$	−7.27087	−1.42594
$27$	0	0
$28$	−0.276342	−0.0522238
$29$	1.47011	0.272993	0.136496	−	0.990641i	$-0.456416\pi$
$29$	1.47011	0.272993	0.136496	+	0.990641i	$0.456416\pi$
$30$	0	0
$31$	2.41141	0.433102	0.216551	−	0.976271i	$-0.430519\pi$
$31$	2.41141	0.433102	0.216551	+	0.976271i	$0.430519\pi$
$32$	1.55148	0.274265
$33$	0	0
$34$	−6.51991	−1.11815
$35$	3.45937	0.584740
$36$	0	0
$37$	−2.83639	−0.466300	−0.233150	−	0.972441i	$-0.574903\pi$
$37$	−2.83639	−0.466300	−0.233150	+	0.972441i	$0.574903\pi$
$38$	10.8046	1.75274
$39$	0	0
$40$	−10.3386	−1.63467
$41$	−6.93310	−1.08277	−0.541384	−	0.840775i	$-0.682100\pi$
$41$	−6.93310	−1.08277	−0.541384	+	0.840775i	$0.682100\pi$
$42$	0	0
$43$	5.62936	0.858470	0.429235	−	0.903193i	$-0.358783\pi$
$43$	5.62936	0.858470	0.429235	+	0.903193i	$0.358783\pi$
$44$	0	0
$45$	0	0
$46$	9.92695	1.46365
$47$	0.0417127	0.00608443	0.00304221	−	0.999995i	$-0.499032\pi$
$47$	0.0417127	0.00608443	0.00304221	+	0.999995i	$0.499032\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	9.14714	1.29360
$51$	0	0
$52$	1.53041	0.212230
$53$	4.47145	0.614201	0.307101	−	0.951677i	$-0.400641\pi$
$53$	4.47145	0.614201	0.307101	+	0.951677i	$0.400641\pi$
$54$	0	0
$55$	0	0
$56$	−2.98857	−0.399364
$57$	0	0
$58$	1.93008	0.253432
$59$	2.99486	0.389898	0.194949	−	0.980813i	$-0.437546\pi$
$59$	2.99486	0.389898	0.194949	+	0.980813i	$0.437546\pi$
$60$	0	0
$61$	−11.6684	−1.49399	−0.746995	−	0.664830i	$-0.768504\pi$
$61$	−11.6684	−1.49399	−0.746995	+	0.664830i	$0.768504\pi$
$62$	3.16589	0.402069
$63$	0	0
$64$	8.77881	1.09735
$65$	−19.1583	−2.37630
$66$	0	0
$67$	9.95852	1.21663	0.608314	−	0.793697i	$-0.291846\pi$
$67$	9.95852	1.21663	0.608314	+	0.793697i	$0.291846\pi$
$68$	1.37234	0.166421
$69$	0	0
$70$	4.54174	0.542842
$71$	6.99312	0.829931	0.414966	−	0.909837i	$-0.363794\pi$
$71$	6.99312	0.829931	0.414966	+	0.909837i	$0.363794\pi$
$72$	0	0
$73$	7.38320	0.864138	0.432069	−	0.901841i	$-0.357784\pi$
$73$	7.38320	0.864138	0.432069	+	0.901841i	$0.357784\pi$
$74$	−3.72385	−0.432888
$75$	0	0
$76$	−2.27421	−0.260870
$77$	0	0
$78$	0	0
$79$	16.2755	1.83114	0.915570	−	0.402159i	$-0.131740\pi$
$79$	16.2755	1.83114	0.915570	+	0.402159i	$0.131740\pi$
$80$	−11.6614	−1.30378
$81$	0	0
$82$	−9.10234	−1.00519
$83$	9.64377	1.05854	0.529271	−	0.848453i	$-0.322466\pi$
$83$	9.64377	1.05854	0.529271	+	0.848453i	$0.322466\pi$
$84$	0	0
$85$	−17.1796	−1.86339
$86$	7.39068	0.796958
$87$	0	0
$88$	0	0
$89$	7.94906	0.842599	0.421300	−	0.906922i	$-0.361574\pi$
$89$	7.94906	0.842599	0.421300	+	0.906922i	$0.361574\pi$
$90$	0	0
$91$	−5.53810	−0.580551
$92$	−2.08948	−0.217843
$93$	0	0
$94$	0.0547639	0.00564846
$95$	28.4695	2.92091
$96$	0	0
$97$	−2.35182	−0.238791	−0.119395	−	0.992847i	$-0.538096\pi$
$97$	−2.35182	−0.238791	−0.119395	+	0.992847i	$0.538096\pi$
$98$	1.31288	0.132621
$99$	0	0
$100$	−1.92534	−0.192534
$101$	13.8502	1.37814	0.689072	−	0.724693i	$-0.258018\pi$
$101$	13.8502	1.37814	0.689072	+	0.724693i	$0.258018\pi$
$102$	0	0
$103$	3.79732	0.374161	0.187081	−	0.982345i	$-0.440097\pi$
$103$	3.79732	0.374161	0.187081	+	0.982345i	$0.440097\pi$
$104$	16.5510	1.62296
$105$	0	0
$106$	5.87049	0.570192
$107$	1.34165	0.129702	0.0648512	−	0.997895i	$-0.479343\pi$
$107$	1.34165	0.129702	0.0648512	+	0.997895i	$0.479343\pi$
$108$	0	0
$109$	−17.5860	−1.68443	−0.842216	−	0.539140i	$-0.818749\pi$
$109$	−17.5860	−1.68443	−0.842216	+	0.539140i	$0.818749\pi$
$110$	0	0
$111$	0	0
$112$	−3.37095	−0.318525
$113$	12.7641	1.20075	0.600374	−	0.799719i	$-0.295018\pi$
$113$	12.7641	1.20075	0.600374	+	0.799719i	$0.295018\pi$
$114$	0	0
$115$	26.1569	2.43915
$116$	−0.406254	−0.0377197
$117$	0	0
$118$	3.93190	0.361961
$119$	−4.96611	−0.455242
$120$	0	0
$121$	0	0
$122$	−15.3193	−1.38694
$123$	0	0
$124$	−0.666374	−0.0598422
$125$	6.80535	0.608689
$126$	0	0
$127$	3.98767	0.353848	0.176924	−	0.984224i	$-0.443385\pi$
$127$	3.98767	0.353848	0.176924	+	0.984224i	$0.443385\pi$
$128$	8.42258	0.744458
$129$	0	0
$130$	−25.1526	−2.20603
$131$	2.02685	0.177087	0.0885433	−	0.996072i	$-0.471779\pi$
$131$	2.02685	0.177087	0.0885433	+	0.996072i	$0.471779\pi$
$132$	0	0
$133$	8.22969	0.713604
$134$	13.0744	1.12945
$135$	0	0
$136$	14.8415	1.27265
$137$	−15.2924	−1.30652	−0.653259	−	0.757135i	$-0.726599\pi$
$137$	−15.2924	−1.30652	−0.653259	+	0.757135i	$0.726599\pi$
$138$	0	0
$139$	−21.5228	−1.82554	−0.912772	−	0.408470i	$-0.866062\pi$
$139$	−21.5228	−1.82554	−0.912772	+	0.408470i	$0.866062\pi$
$140$	−0.955969	−0.0807941
$141$	0	0
$142$	9.18114	0.770464
$143$	0	0
$144$	0	0
$145$	5.08565	0.422340
$146$	9.69326	0.802220
$147$	0	0
$148$	0.783815	0.0644292
$149$	−1.21143	−0.0992440	−0.0496220	−	0.998768i	$-0.515802\pi$
$149$	−1.21143	−0.0992440	−0.0496220	+	0.998768i	$0.515802\pi$
$150$	0	0
$151$	9.71739	0.790790	0.395395	−	0.918511i	$-0.370608\pi$
$151$	9.71739	0.790790	0.395395	+	0.918511i	$0.370608\pi$
$152$	−24.5950	−1.99492
$153$	0	0
$154$	0	0
$155$	8.34195	0.670042
$156$	0	0
$157$	16.5385	1.31991	0.659957	−	0.751303i	$-0.270574\pi$
$157$	16.5385	1.31991	0.659957	+	0.751303i	$0.270574\pi$
$158$	21.3678	1.69993
$159$	0	0
$160$	5.36713	0.424309
$161$	7.56119	0.595905
$162$	0	0
$163$	−7.60022	−0.595295	−0.297648	−	0.954676i	$-0.596202\pi$
$163$	−7.60022	−0.595295	−0.297648	+	0.954676i	$0.596202\pi$
$164$	1.91591	0.149607
$165$	0	0
$166$	12.6611	0.982694
$167$	4.08561	0.316154	0.158077	−	0.987427i	$-0.449471\pi$
$167$	4.08561	0.316154	0.158077	+	0.987427i	$0.449471\pi$
$168$	0	0
$169$	17.6706	1.35928
$170$	−22.5548	−1.72987
$171$	0	0
$172$	−1.55563	−0.118616
$173$	−3.21279	−0.244264	−0.122132	−	0.992514i	$-0.538973\pi$
$173$	−3.21279	−0.244264	−0.122132	+	0.992514i	$0.538973\pi$
$174$	0	0
$175$	6.96722	0.526673
$176$	0	0
$177$	0	0
$178$	10.4362	0.782225
$179$	−0.207824	−0.0155335	−0.00776674	−	0.999970i	$-0.502472\pi$
$179$	−0.207824	−0.0155335	−0.00776674	+	0.999970i	$0.502472\pi$
$180$	0	0
$181$	−5.25519	−0.390615	−0.195307	−	0.980742i	$-0.562570\pi$
$181$	−5.25519	−0.390615	−0.195307	+	0.980742i	$0.562570\pi$
$182$	−7.27087	−0.538953
$183$	0	0
$184$	−22.5971	−1.66588
$185$	−9.81212	−0.721402
$186$	0	0
$187$	0	0
$188$	−0.0115270	−0.000840692 0
$189$	0	0
$190$	37.3771	2.71162
$191$	−2.70775	−0.195926	−0.0979630	−	0.995190i	$-0.531233\pi$
$191$	−2.70775	−0.195926	−0.0979630	+	0.995190i	$0.531233\pi$
$192$	0	0
$193$	−16.4899	−1.18697	−0.593484	−	0.804846i	$-0.702248\pi$
$193$	−16.4899	−1.18697	−0.593484	+	0.804846i	$0.702248\pi$
$194$	−3.08766	−0.221681
$195$	0	0
$196$	−0.276342	−0.0197387
$197$	3.21597	0.229128	0.114564	−	0.993416i	$-0.463453\pi$
$197$	3.21597	0.229128	0.114564	+	0.993416i	$0.463453\pi$
$198$	0	0
$199$	8.63547	0.612152	0.306076	−	0.952007i	$-0.400984\pi$
$199$	8.63547	0.612152	0.306076	+	0.952007i	$0.400984\pi$
$200$	−20.8220	−1.47234
$201$	0	0
$202$	18.1836	1.27940
$203$	1.47011	0.103182
$204$	0	0
$205$	−23.9842	−1.67513
$206$	4.98543	0.347351
$207$	0	0
$208$	18.6687	1.29444
$209$	0	0
$210$	0	0
$211$	4.91452	0.338329	0.169165	−	0.985588i	$-0.445893\pi$
$211$	4.91452	0.338329	0.169165	+	0.985588i	$0.445893\pi$
$212$	−1.23565	−0.0848649
$213$	0	0
$214$	1.76143	0.120409
$215$	19.4740	1.32812
$216$	0	0
$217$	2.41141	0.163697
$218$	−23.0883	−1.56374
$219$	0	0
$220$	0	0
$221$	27.5028	1.85004
$222$	0	0
$223$	−1.15583	−0.0774000	−0.0387000	−	0.999251i	$-0.512322\pi$
$223$	−1.15583	−0.0774000	−0.0387000	+	0.999251i	$0.512322\pi$
$224$	1.55148	0.103662
$225$	0	0
$226$	16.7578	1.11471
$227$	−2.62410	−0.174168	−0.0870839	−	0.996201i	$-0.527755\pi$
$227$	−2.62410	−0.174168	−0.0870839	+	0.996201i	$0.527755\pi$
$228$	0	0
$229$	−10.8283	−0.715555	−0.357777	−	0.933807i	$-0.616465\pi$
$229$	−10.8283	−0.715555	−0.357777	+	0.933807i	$0.616465\pi$
$230$	34.3410	2.26438
$231$	0	0
$232$	−4.39352	−0.288449
$233$	16.7906	1.09999	0.549995	−	0.835168i	$-0.314630\pi$
$233$	16.7906	1.09999	0.549995	+	0.835168i	$0.314630\pi$
$234$	0	0
$235$	0.144300	0.00941307
$236$	−0.827607	−0.0538726
$237$	0	0
$238$	−6.51991	−0.422623
$239$	12.8564	0.831613	0.415806	−	0.909453i	$-0.363499\pi$
$239$	12.8564	0.831613	0.415806	+	0.909453i	$0.363499\pi$
$240$	0	0
$241$	1.12325	0.0723548	0.0361774	−	0.999345i	$-0.488482\pi$
$241$	1.12325	0.0723548	0.0361774	+	0.999345i	$0.488482\pi$
$242$	0	0
$243$	0	0
$244$	3.22448	0.206426
$245$	3.45937	0.221011
$246$	0	0
$247$	−45.5769	−2.89999
$248$	−7.20666	−0.457623
$249$	0	0
$250$	8.93461	0.565075
$251$	24.8678	1.56964	0.784819	−	0.619725i	$-0.212756\pi$
$251$	24.8678	1.56964	0.784819	+	0.619725i	$0.212756\pi$
$252$	0	0
$253$	0	0
$254$	5.23533	0.328494
$255$	0	0
$256$	−6.49976	−0.406235
$257$	−12.9965	−0.810701	−0.405350	−	0.914161i	$-0.632850\pi$
$257$	−12.9965	−0.810701	−0.405350	+	0.914161i	$0.632850\pi$
$258$	0	0
$259$	−2.83639	−0.176245
$260$	5.29426	0.328336
$261$	0	0
$262$	2.66101	0.164398
$263$	−28.0645	−1.73053	−0.865266	−	0.501313i	$-0.832851\pi$
$263$	−28.0645	−1.73053	−0.865266	+	0.501313i	$0.832851\pi$
$264$	0	0
$265$	15.4684	0.950216
$266$	10.8046	0.662473
$267$	0	0
$268$	−2.75196	−0.168103
$269$	−30.4884	−1.85891	−0.929456	−	0.368933i	$-0.879723\pi$
$269$	−30.4884	−1.85891	−0.929456	+	0.368933i	$0.879723\pi$
$270$	0	0
$271$	−0.119754	−0.00727455	−0.00363727	−	0.999993i	$-0.501158\pi$
$271$	−0.119754	−0.00727455	−0.00363727	+	0.999993i	$0.501158\pi$
$272$	16.7405	1.01504
$273$	0	0
$274$	−20.0771	−1.21290
$275$	0	0
$276$	0	0
$277$	−5.63483	−0.338564	−0.169282	−	0.985568i	$-0.554145\pi$
$277$	−5.63483	−0.338564	−0.169282	+	0.985568i	$0.554145\pi$
$278$	−28.2569	−1.69474
$279$	0	0
$280$	−10.3386	−0.617847
$281$	4.16605	0.248526	0.124263	−	0.992249i	$-0.460343\pi$
$281$	4.16605	0.248526	0.124263	+	0.992249i	$0.460343\pi$
$282$	0	0
$283$	13.6746	0.812872	0.406436	−	0.913679i	$-0.366771\pi$
$283$	13.6746	0.812872	0.406436	+	0.913679i	$0.366771\pi$
$284$	−1.93250	−0.114673
$285$	0	0
$286$	0	0
$287$	−6.93310	−0.409248
$288$	0	0
$289$	7.66220	0.450718
$290$	6.67686	0.392079
$291$	0	0
$292$	−2.04029	−0.119399
$293$	−3.09005	−0.180523	−0.0902614	−	0.995918i	$-0.528770\pi$
$293$	−3.09005	−0.180523	−0.0902614	+	0.995918i	$0.528770\pi$
$294$	0	0
$295$	10.3603	0.603201
$296$	8.47675	0.492701
$297$	0	0
$298$	−1.59046	−0.0921329
$299$	−41.8747	−2.42167
$300$	0	0
$301$	5.62936	0.324471
$302$	12.7578	0.734128
$303$	0	0
$304$	−27.7419	−1.59111
$305$	−40.3654	−2.31132
$306$	0	0
$307$	5.33016	0.304208	0.152104	−	0.988364i	$-0.451395\pi$
$307$	5.33016	0.304208	0.152104	+	0.988364i	$0.451395\pi$
$308$	0	0
$309$	0	0
$310$	10.9520	0.622031
$311$	−7.55259	−0.428268	−0.214134	−	0.976804i	$-0.568693\pi$
$311$	−7.55259	−0.428268	−0.214134	+	0.976804i	$0.568693\pi$
$312$	0	0
$313$	4.84740	0.273991	0.136996	−	0.990572i	$-0.456255\pi$
$313$	4.84740	0.273991	0.136996	+	0.990572i	$0.456255\pi$
$314$	21.7131	1.22534
$315$	0	0
$316$	−4.49761	−0.253011
$317$	−31.0942	−1.74643	−0.873213	−	0.487339i	$-0.837967\pi$
$317$	−31.0942	−1.74643	−0.873213	+	0.487339i	$0.837967\pi$
$318$	0	0
$319$	0	0
$320$	30.3691	1.69769
$321$	0	0
$322$	9.92695	0.553207
$323$	−40.8695	−2.27404
$324$	0	0
$325$	−38.5852	−2.14032
$326$	−9.97819	−0.552641
$327$	0	0
$328$	20.7201	1.14407
$329$	0.0417127	0.00229970
$330$	0	0
$331$	−27.0887	−1.48893	−0.744465	−	0.667661i	$-0.767295\pi$
$331$	−27.0887	−1.48893	−0.744465	+	0.667661i	$0.767295\pi$
$332$	−2.66498	−0.146260
$333$	0	0
$334$	5.36392	0.293500
$335$	34.4502	1.88222
$336$	0	0
$337$	6.24394	0.340129	0.170065	−	0.985433i	$-0.445602\pi$
$337$	6.24394	0.340129	0.170065	+	0.985433i	$0.445602\pi$
$338$	23.1994	1.26188
$339$	0	0
$340$	4.74744	0.257466
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−16.8237	−0.907075
$345$	0	0
$346$	−4.21801	−0.226762
$347$	−1.55233	−0.0833332	−0.0416666	−	0.999132i	$-0.513267\pi$
$347$	−1.55233	−0.0833332	−0.0416666	+	0.999132i	$0.513267\pi$
$348$	0	0
$349$	−5.70726	−0.305503	−0.152751	−	0.988265i	$-0.548813\pi$
$349$	−5.70726	−0.305503	−0.152751	+	0.988265i	$0.548813\pi$
$350$	9.14714	0.488935
$351$	0	0
$352$	0	0
$353$	17.0387	0.906877	0.453438	−	0.891288i	$-0.350197\pi$
$353$	17.0387	0.906877	0.453438	+	0.891288i	$0.350197\pi$
$354$	0	0
$355$	24.1918	1.28397
$356$	−2.19666	−0.116423
$357$	0	0
$358$	−0.272848	−0.0144205
$359$	−16.5696	−0.874512	−0.437256	−	0.899337i	$-0.644050\pi$
$359$	−16.5696	−0.874512	−0.437256	+	0.899337i	$0.644050\pi$
$360$	0	0
$361$	48.7277	2.56462
$362$	−6.89944	−0.362626
$363$	0	0
$364$	1.53041	0.0802154
$365$	25.5412	1.33689
$366$	0	0
$367$	20.3623	1.06290	0.531450	−	0.847089i	$-0.321647\pi$
$367$	20.3623	1.06290	0.531450	+	0.847089i	$0.321647\pi$
$368$	−25.4884	−1.32867
$369$	0	0
$370$	−12.8822	−0.669711
$371$	4.47145	0.232146
$372$	0	0
$373$	13.5841	0.703356	0.351678	−	0.936121i	$-0.385611\pi$
$373$	13.5841	0.703356	0.351678	+	0.936121i	$0.385611\pi$
$374$	0	0
$375$	0	0
$376$	−0.124661	−0.00642892
$377$	−8.14163	−0.419315
$378$	0	0
$379$	12.7562	0.655244	0.327622	−	0.944809i	$-0.393753\pi$
$379$	12.7562	0.655244	0.327622	+	0.944809i	$0.393753\pi$
$380$	−7.86733	−0.403585
$381$	0	0
$382$	−3.55496	−0.181887
$383$	−11.7476	−0.600273	−0.300136	−	0.953896i	$-0.597032\pi$
$383$	−11.7476	−0.600273	−0.300136	+	0.953896i	$0.597032\pi$
$384$	0	0
$385$	0	0
$386$	−21.6493	−1.10192
$387$	0	0
$388$	0.649907	0.0329940
$389$	31.6905	1.60677	0.803385	−	0.595460i	$-0.203030\pi$
$389$	31.6905	1.60677	0.803385	+	0.595460i	$0.203030\pi$
$390$	0	0
$391$	−37.5497	−1.89897
$392$	−2.98857	−0.150945
$393$	0	0
$394$	4.22219	0.212711
$395$	56.3030	2.83291
$396$	0	0
$397$	24.2723	1.21819	0.609097	−	0.793096i	$-0.291532\pi$
$397$	24.2723	1.21819	0.609097	+	0.793096i	$0.291532\pi$
$398$	11.3374	0.568290
$399$	0	0
$400$	−23.4862	−1.17431
$401$	−7.94109	−0.396559	−0.198280	−	0.980146i	$-0.563535\pi$
$401$	−7.94109	−0.396559	−0.198280	+	0.980146i	$0.563535\pi$
$402$	0	0
$403$	−13.3546	−0.665242
$404$	−3.82739	−0.190420
$405$	0	0
$406$	1.93008	0.0957883
$407$	0	0
$408$	0	0
$409$	−1.58343	−0.0782956	−0.0391478	−	0.999233i	$-0.512464\pi$
$409$	−1.58343	−0.0782956	−0.0391478	+	0.999233i	$0.512464\pi$
$410$	−31.4884	−1.55510
$411$	0	0
$412$	−1.04936	−0.0516982
$413$	2.99486	0.147367
$414$	0	0
$415$	33.3613	1.63764
$416$	−8.59224	−0.421269
$417$	0	0
$418$	0	0
$419$	−18.0433	−0.881472	−0.440736	−	0.897637i	$-0.645282\pi$
$419$	−18.0433	−0.881472	−0.440736	+	0.897637i	$0.645282\pi$
$420$	0	0
$421$	−6.96477	−0.339442	−0.169721	−	0.985492i	$-0.554287\pi$
$421$	−6.96477	−0.339442	−0.169721	+	0.985492i	$0.554287\pi$
$422$	6.45218	0.314087
$423$	0	0
$424$	−13.3632	−0.648976
$425$	−34.6000	−1.67834
$426$	0	0
$427$	−11.6684	−0.564675
$428$	−0.370755	−0.0179211
$429$	0	0
$430$	25.5671	1.23295
$431$	−24.0781	−1.15980	−0.579901	−	0.814687i	$-0.696909\pi$
$431$	−24.0781	−1.15980	−0.579901	+	0.814687i	$0.696909\pi$
$432$	0	0
$433$	−24.2687	−1.16628	−0.583141	−	0.812371i	$-0.698176\pi$
$433$	−24.2687	−1.16628	−0.583141	+	0.812371i	$0.698176\pi$
$434$	3.16589	0.151968
$435$	0	0
$436$	4.85975	0.232740
$437$	62.2262	2.97668
$438$	0	0
$439$	−8.71332	−0.415864	−0.207932	−	0.978143i	$-0.566673\pi$
$439$	−8.71332	−0.415864	−0.207932	+	0.978143i	$0.566673\pi$
$440$	0	0
$441$	0	0
$442$	36.1079	1.71748
$443$	−25.3205	−1.20301	−0.601507	−	0.798868i	$-0.705433\pi$
$443$	−25.3205	−1.20301	−0.601507	+	0.798868i	$0.705433\pi$
$444$	0	0
$445$	27.4987	1.30357
$446$	−1.51747	−0.0718541
$447$	0	0
$448$	8.77881	0.414760
$449$	21.5816	1.01850	0.509250	−	0.860618i	$-0.329923\pi$
$449$	21.5816	1.01850	0.509250	+	0.860618i	$0.329923\pi$
$450$	0	0
$451$	0	0
$452$	−3.52727	−0.165909
$453$	0	0
$454$	−3.44514	−0.161688
$455$	−19.1583	−0.898157
$456$	0	0
$457$	29.5898	1.38415	0.692076	−	0.721824i	$-0.256696\pi$
$457$	29.5898	1.38415	0.692076	+	0.721824i	$0.256696\pi$
$458$	−14.2163	−0.664283
$459$	0	0
$460$	−7.22827	−0.337020
$461$	18.2856	0.851643	0.425821	−	0.904807i	$-0.359985\pi$
$461$	18.2856	0.851643	0.425821	+	0.904807i	$0.359985\pi$
$462$	0	0
$463$	21.5147	0.999875	0.499937	−	0.866062i	$-0.333356\pi$
$463$	21.5147	0.999875	0.499937	+	0.866062i	$0.333356\pi$
$464$	−4.95567	−0.230061
$465$	0	0
$466$	22.0441	1.02117
$467$	−33.0545	−1.52958	−0.764790	−	0.644280i	$-0.777157\pi$
$467$	−33.0545	−1.52958	−0.764790	+	0.644280i	$0.777157\pi$
$468$	0	0
$469$	9.95852	0.459842
$470$	0.189448	0.00873860
$471$	0	0
$472$	−8.95035	−0.411973
$473$	0	0
$474$	0	0
$475$	57.3381	2.63085
$476$	1.37234	0.0629013
$477$	0	0
$478$	16.8790	0.772026
$479$	8.22730	0.375915	0.187957	−	0.982177i	$-0.439813\pi$
$479$	8.22730	0.375915	0.187957	+	0.982177i	$0.439813\pi$
$480$	0	0
$481$	15.7082	0.716234
$482$	1.47469	0.0671704
$483$	0	0
$484$	0	0
$485$	−8.13580	−0.369428
$486$	0	0
$487$	16.7692	0.759884	0.379942	−	0.925010i	$-0.375944\pi$
$487$	16.7692	0.759884	0.379942	+	0.925010i	$0.375944\pi$
$488$	34.8719	1.57858
$489$	0	0
$490$	4.54174	0.205175
$491$	−21.7641	−0.982199	−0.491100	−	0.871103i	$-0.663405\pi$
$491$	−21.7641	−0.982199	−0.491100	+	0.871103i	$0.663405\pi$
$492$	0	0
$493$	−7.30072	−0.328808
$494$	−59.8370	−2.69219
$495$	0	0
$496$	−8.12874	−0.364991
$497$	6.99312	0.313684
$498$	0	0
$499$	27.4664	1.22956	0.614781	−	0.788698i	$-0.289244\pi$
$499$	27.4664	1.22956	0.614781	+	0.788698i	$0.289244\pi$
$500$	−1.88060	−0.0841032
$501$	0	0
$502$	32.6484	1.45717
$503$	−15.4162	−0.687375	−0.343688	−	0.939084i	$-0.611676\pi$
$503$	−15.4162	−0.687375	−0.343688	+	0.939084i	$0.611676\pi$
$504$	0	0
$505$	47.9129	2.13209
$506$	0	0
$507$	0	0
$508$	−1.10196	−0.0488916
$509$	24.5612	1.08866	0.544329	−	0.838872i	$-0.316784\pi$
$509$	24.5612	1.08866	0.544329	+	0.838872i	$0.316784\pi$
$510$	0	0
$511$	7.38320	0.326613
$512$	−25.3786	−1.12159
$513$	0	0
$514$	−17.0629	−0.752612
$515$	13.1363	0.578856
$516$	0	0
$517$	0	0
$518$	−3.72385	−0.163616
$519$	0	0
$520$	57.2560	2.51084
$521$	−12.1838	−0.533782	−0.266891	−	0.963727i	$-0.585996\pi$
$521$	−12.1838	−0.533782	−0.266891	+	0.963727i	$0.585996\pi$
$522$	0	0
$523$	29.8506	1.30527	0.652637	−	0.757671i	$-0.273663\pi$
$523$	29.8506	1.30527	0.652637	+	0.757671i	$0.273663\pi$
$524$	−0.560104	−0.0244682
$525$	0	0
$526$	−36.8454	−1.60653
$527$	−11.9753	−0.521653
$528$	0	0
$529$	34.1716	1.48572
$530$	20.3082	0.882131
$531$	0	0
$532$	−2.27421	−0.0985995
$533$	38.3963	1.66313
$534$	0	0
$535$	4.64127	0.200660
$536$	−29.7617	−1.28551
$537$	0	0
$538$	−40.0277	−1.72572
$539$	0	0
$540$	0	0
$541$	−33.5792	−1.44368	−0.721842	−	0.692058i	$-0.756704\pi$
$541$	−33.5792	−1.44368	−0.721842	+	0.692058i	$0.756704\pi$
$542$	−0.157223	−0.00675331
$543$	0	0
$544$	−7.70480	−0.330340
$545$	−60.8364	−2.60594
$546$	0	0
$547$	25.6065	1.09486	0.547428	−	0.836853i	$-0.315607\pi$
$547$	25.6065	1.09486	0.547428	+	0.836853i	$0.315607\pi$
$548$	4.22593	0.180523
$549$	0	0
$550$	0	0
$551$	12.0985	0.515415
$552$	0	0
$553$	16.2755	0.692106
$554$	−7.39787	−0.314305
$555$	0	0
$556$	5.94767	0.252237
$557$	5.85837	0.248227	0.124114	−	0.992268i	$-0.460391\pi$
$557$	5.85837	0.248227	0.124114	+	0.992268i	$0.460391\pi$
$558$	0	0
$559$	−31.1760	−1.31860
$560$	−11.6614	−0.492782
$561$	0	0
$562$	5.46953	0.230718
$563$	−6.17238	−0.260135	−0.130067	−	0.991505i	$-0.541519\pi$
$563$	−6.17238	−0.260135	−0.130067	+	0.991505i	$0.541519\pi$
$564$	0	0
$565$	44.1558	1.85765
$566$	17.9532	0.754628
$567$	0	0
$568$	−20.8994	−0.876920
$569$	−27.0150	−1.13253	−0.566264	−	0.824224i	$-0.691612\pi$
$569$	−27.0150	−1.13253	−0.566264	+	0.824224i	$0.691612\pi$
$570$	0	0
$571$	30.7018	1.28483	0.642416	−	0.766356i	$-0.277932\pi$
$571$	30.7018	1.28483	0.642416	+	0.766356i	$0.277932\pi$
$572$	0	0
$573$	0	0
$574$	−9.10234	−0.379925
$575$	52.6805	2.19693
$576$	0	0
$577$	−9.06613	−0.377428	−0.188714	−	0.982032i	$-0.560432\pi$
$577$	−9.06613	−0.377428	−0.188714	+	0.982032i	$0.560432\pi$
$578$	10.0596	0.418423
$579$	0	0
$580$	−1.40538	−0.0583552
$581$	9.64377	0.400091
$582$	0	0
$583$	0	0
$584$	−22.0652	−0.913064
$585$	0	0
$586$	−4.05687	−0.167588
$587$	−6.77919	−0.279807	−0.139904	−	0.990165i	$-0.544679\pi$
$587$	−6.77919	−0.279807	−0.139904	+	0.990165i	$0.544679\pi$
$588$	0	0
$589$	19.8451	0.817705
$590$	13.6019	0.559981
$591$	0	0
$592$	9.56134	0.392969
$593$	−32.2361	−1.32378	−0.661890	−	0.749601i	$-0.730245\pi$
$593$	−32.2361	−1.32378	−0.661890	+	0.749601i	$0.730245\pi$
$594$	0	0
$595$	−17.1796	−0.704294
$596$	0.334769	0.0137127
$597$	0	0
$598$	−54.9765	−2.24816
$599$	26.6954	1.09074	0.545372	−	0.838194i	$-0.316388\pi$
$599$	26.6954	1.09074	0.545372	+	0.838194i	$0.316388\pi$
$600$	0	0
$601$	4.26250	0.173871	0.0869354	−	0.996214i	$-0.472293\pi$
$601$	4.26250	0.173871	0.0869354	+	0.996214i	$0.472293\pi$
$602$	7.39068	0.301222
$603$	0	0
$604$	−2.68532	−0.109264
$605$	0	0
$606$	0	0
$607$	6.97851	0.283249	0.141625	−	0.989920i	$-0.454767\pi$
$607$	6.97851	0.283249	0.141625	+	0.989920i	$0.454767\pi$
$608$	12.7682	0.517817
$609$	0	0
$610$	−52.9950	−2.14570
$611$	−0.231009	−0.00934564
$612$	0	0
$613$	−25.3866	−1.02535	−0.512677	−	0.858582i	$-0.671346\pi$
$613$	−25.3866	−1.02535	−0.512677	+	0.858582i	$0.671346\pi$
$614$	6.99787	0.282411
$615$	0	0
$616$	0	0
$617$	−8.20576	−0.330351	−0.165176	−	0.986264i	$-0.552819\pi$
$617$	−8.20576	−0.330351	−0.165176	+	0.986264i	$0.552819\pi$
$618$	0	0
$619$	−4.24253	−0.170522	−0.0852609	−	0.996359i	$-0.527172\pi$
$619$	−4.24253	−0.170522	−0.0852609	+	0.996359i	$0.527172\pi$
$620$	−2.30523	−0.0925804
$621$	0	0
$622$	−9.91565	−0.397581
$623$	7.94906	0.318473
$624$	0	0
$625$	−11.2939	−0.451757
$626$	6.36406	0.254359
$627$	0	0
$628$	−4.57028	−0.182374
$629$	14.0858	0.561639
$630$	0	0
$631$	35.8595	1.42755	0.713773	−	0.700377i	$-0.246985\pi$
$631$	35.8595	1.42755	0.713773	+	0.700377i	$0.246985\pi$
$632$	−48.6405	−1.93482
$633$	0	0
$634$	−40.8230	−1.62129
$635$	13.7948	0.547430
$636$	0	0
$637$	−5.53810	−0.219428
$638$	0	0
$639$	0	0
$640$	29.1368	1.15173
$641$	30.8864	1.21994	0.609970	−	0.792424i	$-0.291181\pi$
$641$	30.8864	1.21994	0.609970	+	0.792424i	$0.291181\pi$
$642$	0	0
$643$	11.1381	0.439243	0.219621	−	0.975585i	$-0.429518\pi$
$643$	11.1381	0.439243	0.219621	+	0.975585i	$0.429518\pi$
$644$	−2.08948	−0.0823369
$645$	0	0
$646$	−53.6568	−2.11110
$647$	19.1705	0.753669	0.376835	−	0.926281i	$-0.377012\pi$
$647$	19.1705	0.753669	0.376835	+	0.926281i	$0.377012\pi$
$648$	0	0
$649$	0	0
$650$	−50.6578	−1.98696
$651$	0	0
$652$	2.10026	0.0822526
$653$	6.33344	0.247847	0.123923	−	0.992292i	$-0.460452\pi$
$653$	6.33344	0.247847	0.123923	+	0.992292i	$0.460452\pi$
$654$	0	0
$655$	7.01161	0.273966
$656$	23.3712	0.912490
$657$	0	0
$658$	0.0547639	0.00213492
$659$	−33.0256	−1.28650	−0.643248	−	0.765658i	$-0.722414\pi$
$659$	−33.0256	−1.28650	−0.643248	+	0.765658i	$0.722414\pi$
$660$	0	0
$661$	−20.1870	−0.785181	−0.392591	−	0.919713i	$-0.628421\pi$
$661$	−20.1870	−0.785181	−0.392591	+	0.919713i	$0.628421\pi$
$662$	−35.5643	−1.38224
$663$	0	0
$664$	−28.8211	−1.11847
$665$	28.4695	1.10400
$666$	0	0
$667$	11.1158	0.430405
$668$	−1.12903	−0.0436833
$669$	0	0
$670$	45.2290	1.74735
$671$	0	0
$672$	0	0
$673$	−23.2806	−0.897403	−0.448701	−	0.893682i	$-0.648113\pi$
$673$	−23.2806	−0.897403	−0.448701	+	0.893682i	$0.648113\pi$
$674$	8.19755	0.315758
$675$	0	0
$676$	−4.88313	−0.187813
$677$	−11.7124	−0.450143	−0.225072	−	0.974342i	$-0.572262\pi$
$677$	−11.7124	−0.450143	−0.225072	+	0.974342i	$0.572262\pi$
$678$	0	0
$679$	−2.35182	−0.0902545
$680$	51.3423	1.96889
$681$	0	0
$682$	0	0
$683$	20.9286	0.800811	0.400405	−	0.916338i	$-0.368869\pi$
$683$	20.9286	0.800811	0.400405	+	0.916338i	$0.368869\pi$
$684$	0	0
$685$	−52.9020	−2.02128
$686$	1.31288	0.0501260
$687$	0	0
$688$	−18.9763	−0.723465
$689$	−24.7634	−0.943409
$690$	0	0
$691$	5.06534	0.192695	0.0963473	−	0.995348i	$-0.469284\pi$
$691$	5.06534	0.192695	0.0963473	+	0.995348i	$0.469284\pi$
$692$	0.887829	0.0337502
$693$	0	0
$694$	−2.03802	−0.0773622
$695$	−74.4554	−2.82425
$696$	0	0
$697$	34.4305	1.30415
$698$	−7.49296	−0.283613
$699$	0	0
$700$	−1.92534	−0.0727709
$701$	−32.5293	−1.22861	−0.614307	−	0.789067i	$-0.710564\pi$
$701$	−32.5293	−1.22861	−0.614307	+	0.789067i	$0.710564\pi$
$702$	0	0
$703$	−23.3426	−0.880384
$704$	0	0
$705$	0	0
$706$	22.3698	0.841897
$707$	13.8502	0.520890
$708$	0	0
$709$	13.7164	0.515129	0.257564	−	0.966261i	$-0.417080\pi$
$709$	13.7164	0.515129	0.257564	+	0.966261i	$0.417080\pi$
$710$	31.7609	1.19197
$711$	0	0
$712$	−23.7563	−0.890306
$713$	18.2331	0.682836
$714$	0	0
$715$	0	0
$716$	0.0574305	0.00214628
$717$	0	0
$718$	−21.7540	−0.811851
$719$	−43.3822	−1.61788	−0.808941	−	0.587890i	$-0.799959\pi$
$719$	−43.3822	−1.61788	−0.808941	+	0.587890i	$0.799959\pi$
$720$	0	0
$721$	3.79732	0.141420
$722$	63.9737	2.38086
$723$	0	0
$724$	1.45223	0.0539717
$725$	10.2426	0.380400
$726$	0	0
$727$	−2.79407	−0.103626	−0.0518132	−	0.998657i	$-0.516500\pi$
$727$	−2.79407	−0.103626	−0.0518132	+	0.998657i	$0.516500\pi$
$728$	16.5510	0.613421
$729$	0	0
$730$	33.5325	1.24110
$731$	−27.9560	−1.03399
$732$	0	0
$733$	−13.5883	−0.501894	−0.250947	−	0.968001i	$-0.580742\pi$
$733$	−13.5883	−0.501894	−0.250947	+	0.968001i	$0.580742\pi$
$734$	26.7332	0.986741
$735$	0	0
$736$	11.7310	0.432411
$737$	0	0
$738$	0	0
$739$	23.3065	0.857342	0.428671	−	0.903461i	$-0.358982\pi$
$739$	23.3065	0.857342	0.428671	+	0.903461i	$0.358982\pi$
$740$	2.71150	0.0996769
$741$	0	0
$742$	5.87049	0.215512
$743$	−21.3263	−0.782387	−0.391193	−	0.920308i	$-0.627938\pi$
$743$	−21.3263	−0.782387	−0.391193	+	0.920308i	$0.627938\pi$
$744$	0	0
$745$	−4.19077	−0.153538
$746$	17.8343	0.652959
$747$	0	0
$748$	0	0
$749$	1.34165	0.0490229
$750$	0	0
$751$	8.57812	0.313020	0.156510	−	0.987676i	$-0.449976\pi$
$751$	8.57812	0.313020	0.156510	+	0.987676i	$0.449976\pi$
$752$	−0.140612	−0.00512758
$753$	0	0
$754$	−10.6890	−0.389270
$755$	33.6160	1.22341
$756$	0	0
$757$	−50.2751	−1.82728	−0.913639	−	0.406526i	$-0.866740\pi$
$757$	−50.2751	−1.82728	−0.913639	+	0.406526i	$0.866740\pi$
$758$	16.7474	0.608294
$759$	0	0
$760$	−85.0830	−3.08629
$761$	−10.5311	−0.381753	−0.190876	−	0.981614i	$-0.561133\pi$
$761$	−10.5311	−0.381753	−0.190876	+	0.981614i	$0.561133\pi$
$762$	0	0
$763$	−17.5860	−0.636655
$764$	0.748266	0.0270713
$765$	0	0
$766$	−15.4232	−0.557262
$767$	−16.5859	−0.598881
$768$	0	0
$769$	−49.1715	−1.77317	−0.886584	−	0.462567i	$-0.846929\pi$
$769$	−49.1715	−1.77317	−0.886584	+	0.462567i	$0.846929\pi$
$770$	0	0
$771$	0	0
$772$	4.55685	0.164005
$773$	15.4108	0.554289	0.277145	−	0.960828i	$-0.410612\pi$
$773$	15.4108	0.554289	0.277145	+	0.960828i	$0.410612\pi$
$774$	0	0
$775$	16.8008	0.603503
$776$	7.02857	0.252311
$777$	0	0
$778$	41.6058	1.49164
$779$	−57.0573	−2.04429
$780$	0	0
$781$	0	0
$782$	−49.2983	−1.76290
$783$	0	0
$784$	−3.37095	−0.120391
$785$	57.2127	2.04201
$786$	0	0
$787$	11.6189	0.414169	0.207084	−	0.978323i	$-0.433603\pi$
$787$	11.6189	0.414169	0.207084	+	0.978323i	$0.433603\pi$
$788$	−0.888709	−0.0316589
$789$	0	0
$790$	73.9192	2.62993
$791$	12.7641	0.453840
$792$	0	0
$793$	64.6210	2.29476
$794$	31.8667	1.13091
$795$	0	0
$796$	−2.38635	−0.0845818
$797$	8.66866	0.307060	0.153530	−	0.988144i	$-0.450936\pi$
$797$	8.66866	0.307060	0.153530	+	0.988144i	$0.450936\pi$
$798$	0	0
$799$	−0.207150	−0.00732843
$800$	10.8095	0.382173
$801$	0	0
$802$	−10.4257	−0.368145
$803$	0	0
$804$	0	0
$805$	26.1569	0.921911
$806$	−17.5331	−0.617576
$807$	0	0
$808$	−41.3922	−1.45617
$809$	−16.1800	−0.568857	−0.284429	−	0.958697i	$-0.591804\pi$
$809$	−16.1800	−0.568857	−0.284429	+	0.958697i	$0.591804\pi$
$810$	0	0
$811$	4.58424	0.160974	0.0804871	−	0.996756i	$-0.474352\pi$
$811$	4.58424	0.160974	0.0804871	+	0.996756i	$0.474352\pi$
$812$	−0.406254	−0.0142567
$813$	0	0
$814$	0	0
$815$	−26.2920	−0.920967
$816$	0	0
$817$	46.3279	1.62081
$818$	−2.07886	−0.0726855
$819$	0	0
$820$	6.62783	0.231454
$821$	−46.8787	−1.63608	−0.818040	−	0.575162i	$-0.804939\pi$
$821$	−46.8787	−1.63608	−0.818040	+	0.575162i	$0.804939\pi$
$822$	0	0
$823$	−0.814184	−0.0283807	−0.0141903	−	0.999899i	$-0.504517\pi$
$823$	−0.814184	−0.0283807	−0.0141903	+	0.999899i	$0.504517\pi$
$824$	−11.3485	−0.395345
$825$	0	0
$826$	3.93190	0.136808
$827$	−26.4971	−0.921395	−0.460698	−	0.887557i	$-0.652401\pi$
$827$	−26.4971	−0.921395	−0.460698	+	0.887557i	$0.652401\pi$
$828$	0	0
$829$	32.9292	1.14368	0.571839	−	0.820366i	$-0.306230\pi$
$829$	32.9292	1.14368	0.571839	+	0.820366i	$0.306230\pi$
$830$	43.7995	1.52030
$831$	0	0
$832$	−48.6179	−1.68552
$833$	−4.96611	−0.172065
$834$	0	0
$835$	14.1336	0.489114
$836$	0	0
$837$	0	0
$838$	−23.6887	−0.818312
$839$	31.4398	1.08542	0.542711	−	0.839919i	$-0.317398\pi$
$839$	31.4398	1.08542	0.542711	+	0.839919i	$0.317398\pi$
$840$	0	0
$841$	−26.8388	−0.925475
$842$	−9.14391	−0.315120
$843$	0	0
$844$	−1.35809	−0.0467473
$845$	61.1291	2.10290
$846$	0	0
$847$	0	0
$848$	−15.0730	−0.517610
$849$	0	0
$850$	−45.4256	−1.55809
$851$	−21.4465	−0.735177
$852$	0	0
$853$	−53.0965	−1.81799	−0.908995	−	0.416807i	$-0.863149\pi$
$853$	−53.0965	−1.81799	−0.908995	+	0.416807i	$0.863149\pi$
$854$	−15.3193	−0.524215
$855$	0	0
$856$	−4.00962	−0.137046
$857$	38.6964	1.32185	0.660923	−	0.750454i	$-0.270165\pi$
$857$	38.6964	1.32185	0.660923	+	0.750454i	$0.270165\pi$
$858$	0	0
$859$	−4.54428	−0.155049	−0.0775245	−	0.996990i	$-0.524702\pi$
$859$	−4.54428	−0.155049	−0.0775245	+	0.996990i	$0.524702\pi$
$860$	−5.38150	−0.183508
$861$	0	0
$862$	−31.6117	−1.07670
$863$	14.8810	0.506557	0.253278	−	0.967393i	$-0.418491\pi$
$863$	14.8810	0.506557	0.253278	+	0.967393i	$0.418491\pi$
$864$	0	0
$865$	−11.1142	−0.377895
$866$	−31.8620	−1.08271
$867$	0	0
$868$	−0.666374	−0.0226182
$869$	0	0
$870$	0	0
$871$	−55.1513	−1.86873
$872$	52.5569	1.77980
$873$	0	0
$874$	81.6957	2.76340
$875$	6.80535	0.230063
$876$	0	0
$877$	41.1431	1.38930	0.694652	−	0.719346i	$-0.255559\pi$
$877$	41.1431	1.38930	0.694652	+	0.719346i	$0.255559\pi$
$878$	−11.4396	−0.386067
$879$	0	0
$880$	0	0
$881$	−9.03118	−0.304268	−0.152134	−	0.988360i	$-0.548615\pi$
$881$	−9.03118	−0.304268	−0.152134	+	0.988360i	$0.548615\pi$
$882$	0	0
$883$	30.5916	1.02949	0.514745	−	0.857343i	$-0.327886\pi$
$883$	30.5916	1.02949	0.514745	+	0.857343i	$0.327886\pi$
$884$	−7.60019	−0.255622
$885$	0	0
$886$	−33.2428	−1.11681
$887$	29.7478	0.998833	0.499416	−	0.866362i	$-0.333548\pi$
$887$	29.7478	0.998833	0.499416	+	0.866362i	$0.333548\pi$
$888$	0	0
$889$	3.98767	0.133742
$890$	36.1026	1.21016
$891$	0	0
$892$	0.319404	0.0106945
$893$	0.343283	0.0114875
$894$	0	0
$895$	−0.718939	−0.0240315
$896$	8.42258	0.281379
$897$	0	0
$898$	28.3341	0.945523
$899$	3.54504	0.118234
$900$	0	0
$901$	−22.2057	−0.739779
$902$	0	0
$903$	0	0
$904$	−38.1465	−1.26873
$905$	−18.1796	−0.604311
$906$	0	0
$907$	39.1485	1.29991	0.649953	−	0.759975i	$-0.274789\pi$
$907$	39.1485	1.29991	0.649953	+	0.759975i	$0.274789\pi$
$908$	0.725150	0.0240650
$909$	0	0
$910$	−25.1526	−0.833801
$911$	−27.9931	−0.927454	−0.463727	−	0.885978i	$-0.653488\pi$
$911$	−27.9931	−0.927454	−0.463727	+	0.885978i	$0.653488\pi$
$912$	0	0
$913$	0	0
$914$	38.8479	1.28497
$915$	0	0
$916$	2.99232	0.0988690
$917$	2.02685	0.0669324
$918$	0	0
$919$	−38.4845	−1.26949	−0.634743	−	0.772723i	$-0.718894\pi$
$919$	−38.4845	−1.26949	−0.634743	+	0.772723i	$0.718894\pi$
$920$	−78.1718	−2.57725
$921$	0	0
$922$	24.0068	0.790621
$923$	−38.7287	−1.27477
$924$	0	0
$925$	−19.7618	−0.649763
$926$	28.2463	0.928231
$927$	0	0
$928$	2.28084	0.0748723
$929$	−47.2366	−1.54978	−0.774892	−	0.632094i	$-0.782196\pi$
$929$	−47.2366	−1.54978	−0.774892	+	0.632094i	$0.782196\pi$
$930$	0	0
$931$	8.22969	0.269717
$932$	−4.63996	−0.151987
$933$	0	0
$934$	−43.3966	−1.41998
$935$	0	0
$936$	0	0
$937$	−43.5825	−1.42378	−0.711888	−	0.702293i	$-0.752160\pi$
$937$	−43.5825	−1.42378	−0.711888	+	0.702293i	$0.752160\pi$
$938$	13.0744	0.426893
$939$	0	0
$940$	−0.0398761	−0.00130061
$941$	−33.8165	−1.10239	−0.551193	−	0.834378i	$-0.685827\pi$
$941$	−33.8165	−1.10239	−0.551193	+	0.834378i	$0.685827\pi$
$942$	0	0
$943$	−52.4225	−1.70711
$944$	−10.0955	−0.328582
$945$	0	0
$946$	0	0
$947$	3.79141	0.123204	0.0616021	−	0.998101i	$-0.480379\pi$
$947$	3.79141	0.123204	0.0616021	+	0.998101i	$0.480379\pi$
$948$	0	0
$949$	−40.8889	−1.32731
$950$	75.2781	2.44234
$951$	0	0
$952$	14.8415	0.481017
$953$	−3.16992	−0.102684	−0.0513419	−	0.998681i	$-0.516350\pi$
$953$	−3.16992	−0.102684	−0.0513419	+	0.998681i	$0.516350\pi$
$954$	0	0
$955$	−9.36711	−0.303113
$956$	−3.55277	−0.114905
$957$	0	0
$958$	10.8015	0.348980
$959$	−15.2924	−0.493817
$960$	0	0
$961$	−25.1851	−0.812423
$962$	20.6231	0.664914
$963$	0	0
$964$	−0.310401	−0.00999734
$965$	−57.0446	−1.83633
$966$	0	0
$967$	24.6865	0.793865	0.396932	−	0.917848i	$-0.370075\pi$
$967$	24.6865	0.793865	0.396932	+	0.917848i	$0.370075\pi$
$968$	0	0
$969$	0	0
$970$	−10.6813	−0.342957
$971$	−20.2070	−0.648473	−0.324237	−	0.945976i	$-0.605107\pi$
$971$	−20.2070	−0.648473	−0.324237	+	0.945976i	$0.605107\pi$
$972$	0	0
$973$	−21.5228	−0.689991
$974$	22.0159	0.705436
$975$	0	0
$976$	39.3337	1.25904
$977$	34.3098	1.09767	0.548834	−	0.835931i	$-0.315072\pi$
$977$	34.3098	1.09767	0.548834	+	0.835931i	$0.315072\pi$
$978$	0	0
$979$	0	0
$980$	−0.955969	−0.0305373
$981$	0	0
$982$	−28.5737	−0.911822
$983$	−33.0534	−1.05424	−0.527120	−	0.849791i	$-0.676728\pi$
$983$	−33.0534	−1.05424	−0.527120	+	0.849791i	$0.676728\pi$
$984$	0	0
$985$	11.1252	0.354479
$986$	−9.58498	−0.305248
$987$	0	0
$988$	12.5948	0.400694
$989$	42.5647	1.35348
$990$	0	0
$991$	39.3191	1.24901	0.624506	−	0.781020i	$-0.285300\pi$
$991$	39.3191	1.24901	0.624506	+	0.781020i	$0.285300\pi$
$992$	3.74124	0.118785
$993$	0	0
$994$	9.18114	0.291208
$995$	29.8733	0.947046
$996$	0	0
$997$	−2.41508	−0.0764865	−0.0382432	−	0.999268i	$-0.512176\pi$
$997$	−2.41508	−0.0764865	−0.0382432	+	0.999268i	$0.512176\pi$
$998$	36.0601	1.14146
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.dc.1.12		16
3.2	odd	2	inner	7623.2.a.dc.1.5		16
11.3	even	5		693.2.m.k.64.6	yes	32
11.4	even	5		693.2.m.k.379.6	yes	32
11.10	odd	2		7623.2.a.db.1.5		16
33.14	odd	10		693.2.m.k.64.3	✓	32
33.26	odd	10		693.2.m.k.379.3	yes	32
33.32	even	2		7623.2.a.db.1.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.m.k.64.3	✓	32	33.14	odd	10
693.2.m.k.64.6	yes	32	11.3	even	5
693.2.m.k.379.3	yes	32	33.26	odd	10
693.2.m.k.379.6	yes	32	11.4	even	5
7623.2.a.db.1.5		16	11.10	odd	2
7623.2.a.db.1.12		16	33.32	even	2
7623.2.a.dc.1.5		16	3.2	odd	2	inner
7623.2.a.dc.1.12		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.31288 q^{2} -0.276342 q^{4} +3.45937 q^{5} +1.00000 q^{7} -2.98857 q^{8} +O(q^{10})\)	\(q+1.31288 q^{2} -0.276342 q^{4} +3.45937 q^{5} +1.00000 q^{7} -2.98857 q^{8} +4.54174 q^{10} -5.53810 q^{13} +1.31288 q^{14} -3.37095 q^{16} -4.96611 q^{17} +8.22969 q^{19} -0.955969 q^{20} +7.56119 q^{23} +6.96722 q^{25} -7.27087 q^{26} -0.276342 q^{28} +1.47011 q^{29} +2.41141 q^{31} +1.55148 q^{32} -6.51991 q^{34} +3.45937 q^{35} -2.83639 q^{37} +10.8046 q^{38} -10.3386 q^{40} -6.93310 q^{41} +5.62936 q^{43} +9.92695 q^{46} +0.0417127 q^{47} +1.00000 q^{49} +9.14714 q^{50} +1.53041 q^{52} +4.47145 q^{53} -2.98857 q^{56} +1.93008 q^{58} +2.99486 q^{59} -11.6684 q^{61} +3.16589 q^{62} +8.77881 q^{64} -19.1583 q^{65} +9.95852 q^{67} +1.37234 q^{68} +4.54174 q^{70} +6.99312 q^{71} +7.38320 q^{73} -3.72385 q^{74} -2.27421 q^{76} +16.2755 q^{79} -11.6614 q^{80} -9.10234 q^{82} +9.64377 q^{83} -17.1796 q^{85} +7.39068 q^{86} +7.94906 q^{89} -5.53810 q^{91} -2.08948 q^{92} +0.0547639 q^{94} +28.4695 q^{95} -2.35182 q^{97} +1.31288 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 16 q^{7}+O(q^{10}) \) 16 * q + 20 * q^4 + 16 * q^7	\( 16 q + 20 q^{4} + 16 q^{7} - 6 q^{10} + 32 q^{16} + 10 q^{19} + 44 q^{25} + 20 q^{28} + 30 q^{31} + 12 q^{34} + 38 q^{37} - 68 q^{40} + 16 q^{43} + 80 q^{46} + 16 q^{49} + 2 q^{52} + 18 q^{58} - 28 q^{61} + 34 q^{64} + 52 q^{67} - 6 q^{70} - 14 q^{73} - 14 q^{76} + 54 q^{79} + 64 q^{82} + 30 q^{85} - 60 q^{94} + 108 q^{97}+O(q^{100}) \) 16 * q + 20 * q^4 + 16 * q^7 - 6 * q^10 + 32 * q^16 + 10 * q^19 + 44 * q^25 + 20 * q^28 + 30 * q^31 + 12 * q^34 + 38 * q^37 - 68 * q^40 + 16 * q^43 + 80 * q^46 + 16 * q^49 + 2 * q^52 + 18 * q^58 - 28 * q^61 + 34 * q^64 + 52 * q^67 - 6 * q^70 - 14 * q^73 - 14 * q^76 + 54 * q^79 + 64 * q^82 + 30 * q^85 - 60 * q^94 + 108 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more