LMFDB - Embedded newform 7623.2.a.cy.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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 $ x^10 - 19x^8 - x^7 + 124x^6 + 6x^5 - 316x^4 + 17x^3 + 253x^2 - 70*x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.112481$ 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)

$f(q)$	$=$	$q-0.112481 q^{2} -1.98735 q^{4} -1.06131 q^{5} +1.00000 q^{7} +0.448501 q^{8} +O(q^{10})$	$q-0.112481 q^{2} -1.98735 q^{4} -1.06131 q^{5} +1.00000 q^{7} +0.448501 q^{8} +0.119378 q^{10} +5.84183 q^{13} -0.112481 q^{14} +3.92425 q^{16} +2.80535 q^{17} +3.56160 q^{19} +2.10920 q^{20} -4.72213 q^{23} -3.87361 q^{25} -0.657096 q^{26} -1.98735 q^{28} +8.59581 q^{29} +1.74159 q^{31} -1.33841 q^{32} -0.315549 q^{34} -1.06131 q^{35} +4.74526 q^{37} -0.400613 q^{38} -0.476001 q^{40} +6.13922 q^{41} -5.25083 q^{43} +0.531150 q^{46} +8.19149 q^{47} +1.00000 q^{49} +0.435708 q^{50} -11.6098 q^{52} -14.3391 q^{53} +0.448501 q^{56} -0.966866 q^{58} -3.90570 q^{59} +14.7518 q^{61} -0.195896 q^{62} -7.69795 q^{64} -6.20002 q^{65} +8.29079 q^{67} -5.57521 q^{68} +0.119378 q^{70} -11.6370 q^{71} +4.20049 q^{73} -0.533752 q^{74} -7.07815 q^{76} -6.80421 q^{79} -4.16486 q^{80} -0.690546 q^{82} +1.16511 q^{83} -2.97736 q^{85} +0.590619 q^{86} -2.90784 q^{89} +5.84183 q^{91} +9.38451 q^{92} -0.921388 q^{94} -3.77998 q^{95} -4.85294 q^{97} -0.112481 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8}+O(q^{10}) $ 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8	$ 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8} - 6 q^{10} + 6 q^{13} + 38 q^{16} - 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} + 18 q^{28} + 14 q^{29} + 26 q^{31} + 41 q^{32} + 21 q^{34} - 5 q^{35} + 24 q^{37} - 8 q^{38} - 5 q^{40} - 19 q^{41} - 6 q^{43} - q^{46} - 15 q^{47} + 10 q^{49} + q^{50} - 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} - 11 q^{62} + 53 q^{64} + 29 q^{65} + 38 q^{67} - 87 q^{68} - 6 q^{70} - 26 q^{71} - q^{73} + 39 q^{74} - 2 q^{76} + 5 q^{79} - 6 q^{80} + 5 q^{82} - 6 q^{83} - q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} + 42 q^{95} + 24 q^{97}+O(q^{100}) $ 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8 - 6 * q^10 + 6 * q^13 + 38 * q^16 - 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 + 18 * q^28 + 14 * q^29 + 26 * q^31 + 41 * q^32 + 21 * q^34 - 5 * q^35 + 24 * q^37 - 8 * q^38 - 5 * q^40 - 19 * q^41 - 6 * q^43 - q^46 - 15 * q^47 + 10 * q^49 + q^50 - 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 - 11 * q^62 + 53 * q^64 + 29 * q^65 + 38 * q^67 - 87 * q^68 - 6 * q^70 - 26 * q^71 - q^73 + 39 * q^74 - 2 * q^76 + 5 * q^79 - 6 * q^80 + 5 * q^82 - 6 * q^83 - q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 + 42 * q^95 + 24 * q^97

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 $

$2$	−0.112481	−0.0795362	−0.0397681	−	0.999209i	$-0.512662\pi$
$2$	−0.112481	−0.0795362	−0.0397681	+	0.999209i	$0.512662\pi$
$3$	0	0
$3$	0	0
$4$	−1.98735	−0.993674
$5$	−1.06131	−0.474634	−0.237317	−	0.971432i	$-0.576268\pi$
$5$	−1.06131	−0.474634	−0.237317	+	0.971432i	$0.576268\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.448501	0.158569
$9$	0	0
$10$	0.119378	0.0377506
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.84183	1.62023	0.810117	−	0.586269i	$-0.199404\pi$
$13$	5.84183	1.62023	0.810117	+	0.586269i	$0.199404\pi$
$14$	−0.112481	−0.0300618
$15$	0	0
$16$	3.92425	0.981062
$17$	2.80535	0.680397	0.340199	−	0.940354i	$-0.389506\pi$
$17$	2.80535	0.680397	0.340199	+	0.940354i	$0.389506\pi$
$18$	0	0
$19$	3.56160	0.817088	0.408544	−	0.912739i	$-0.366037\pi$
$19$	3.56160	0.817088	0.408544	+	0.912739i	$0.366037\pi$
$20$	2.10920	0.471631
$21$	0	0
$22$	0	0
$23$	−4.72213	−0.984631	−0.492316	−	0.870417i	$-0.663849\pi$
$23$	−4.72213	−0.984631	−0.492316	+	0.870417i	$0.663849\pi$
$24$	0	0
$25$	−3.87361	−0.774723
$26$	−0.657096	−0.128867
$27$	0	0
$28$	−1.98735	−0.375573
$29$	8.59581	1.59620	0.798100	−	0.602524i	$-0.205838\pi$
$29$	8.59581	1.59620	0.798100	+	0.602524i	$0.205838\pi$
$30$	0	0
$31$	1.74159	0.312798	0.156399	−	0.987694i	$-0.450011\pi$
$31$	1.74159	0.312798	0.156399	+	0.987694i	$0.450011\pi$
$32$	−1.33841	−0.236599
$33$	0	0
$34$	−0.315549	−0.0541162
$35$	−1.06131	−0.179395
$36$	0	0
$37$	4.74526	0.780116	0.390058	−	0.920790i	$-0.372455\pi$
$37$	4.74526	0.780116	0.390058	+	0.920790i	$0.372455\pi$
$38$	−0.400613	−0.0649880
$39$	0	0
$40$	−0.476001	−0.0752623
$41$	6.13922	0.958784	0.479392	−	0.877601i	$-0.340857\pi$
$41$	6.13922	0.958784	0.479392	+	0.877601i	$0.340857\pi$
$42$	0	0
$43$	−5.25083	−0.800744	−0.400372	−	0.916353i	$-0.631119\pi$
$43$	−5.25083	−0.800744	−0.400372	+	0.916353i	$0.631119\pi$
$44$	0	0
$45$	0	0
$46$	0.531150	0.0783138
$47$	8.19149	1.19485	0.597426	−	0.801924i	$-0.296190\pi$
$47$	8.19149	1.19485	0.597426	+	0.801924i	$0.296190\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.435708	0.0616185
$51$	0	0
$52$	−11.6098	−1.60998
$53$	−14.3391	−1.96963	−0.984815	−	0.173609i	$-0.944457\pi$
$53$	−14.3391	−1.96963	−0.984815	+	0.173609i	$0.944457\pi$
$54$	0	0
$55$	0	0
$56$	0.448501	0.0599335
$57$	0	0
$58$	−0.966866	−0.126956
$59$	−3.90570	−0.508479	−0.254239	−	0.967141i	$-0.581825\pi$
$59$	−3.90570	−0.508479	−0.254239	+	0.967141i	$0.581825\pi$
$60$	0	0
$61$	14.7518	1.88878	0.944390	−	0.328827i	$-0.106653\pi$
$61$	14.7518	1.88878	0.944390	+	0.328827i	$0.106653\pi$
$62$	−0.195896	−0.0248788
$63$	0	0
$64$	−7.69795	−0.962244
$65$	−6.20002	−0.769018
$66$	0	0
$67$	8.29079	1.01288	0.506441	−	0.862275i	$-0.330961\pi$
$67$	8.29079	1.01288	0.506441	+	0.862275i	$0.330961\pi$
$68$	−5.57521	−0.676093
$69$	0	0
$70$	0.119378	0.0142684
$71$	−11.6370	−1.38106	−0.690531	−	0.723303i	$-0.742623\pi$
$71$	−11.6370	−1.38106	−0.690531	+	0.723303i	$0.742623\pi$
$72$	0	0
$73$	4.20049	0.491630	0.245815	−	0.969317i	$-0.420944\pi$
$73$	4.20049	0.491630	0.245815	+	0.969317i	$0.420944\pi$
$74$	−0.533752	−0.0620474
$75$	0	0
$76$	−7.07815	−0.811919
$77$	0	0
$78$	0	0
$79$	−6.80421	−0.765533	−0.382767	−	0.923845i	$-0.625029\pi$
$79$	−6.80421	−0.765533	−0.382767	+	0.923845i	$0.625029\pi$
$80$	−4.16486	−0.465645
$81$	0	0
$82$	−0.690546	−0.0762580
$83$	1.16511	0.127887	0.0639437	−	0.997954i	$-0.479632\pi$
$83$	1.16511	0.127887	0.0639437	+	0.997954i	$0.479632\pi$
$84$	0	0
$85$	−2.97736	−0.322940
$86$	0.590619	0.0636881
$87$	0	0
$88$	0	0
$89$	−2.90784	−0.308230	−0.154115	−	0.988053i	$-0.549253\pi$
$89$	−2.90784	−0.308230	−0.154115	+	0.988053i	$0.549253\pi$
$90$	0	0
$91$	5.84183	0.612391
$92$	9.38451	0.978402
$93$	0	0
$94$	−0.921388	−0.0950340
$95$	−3.77998	−0.387818
$96$	0	0
$97$	−4.85294	−0.492741	−0.246371	−	0.969176i	$-0.579238\pi$
$97$	−4.85294	−0.492741	−0.246371	+	0.969176i	$0.579238\pi$
$98$	−0.112481	−0.0113623
$99$	0	0
$100$	7.69822	0.769822
$101$	−9.29858	−0.925243	−0.462622	−	0.886556i	$-0.653091\pi$
$101$	−9.29858	−0.925243	−0.462622	+	0.886556i	$0.653091\pi$
$102$	0	0
$103$	12.9880	1.27974	0.639871	−	0.768482i	$-0.278988\pi$
$103$	12.9880	1.27974	0.639871	+	0.768482i	$0.278988\pi$
$104$	2.62007	0.256919
$105$	0	0
$106$	1.61288	0.156657
$107$	−1.85275	−0.179112	−0.0895558	−	0.995982i	$-0.528545\pi$
$107$	−1.85275	−0.179112	−0.0895558	+	0.995982i	$0.528545\pi$
$108$	0	0
$109$	4.65704	0.446063	0.223032	−	0.974811i	$-0.428405\pi$
$109$	4.65704	0.446063	0.223032	+	0.974811i	$0.428405\pi$
$110$	0	0
$111$	0	0
$112$	3.92425	0.370807
$113$	−10.8478	−1.02048	−0.510238	−	0.860033i	$-0.670443\pi$
$113$	−10.8478	−1.02048	−0.510238	+	0.860033i	$0.670443\pi$
$114$	0	0
$115$	5.01166	0.467339
$116$	−17.0829	−1.58610
$117$	0	0
$118$	0.439317	0.0404425
$119$	2.80535	0.257166
$120$	0	0
$121$	0	0
$122$	−1.65930	−0.150226
$123$	0	0
$124$	−3.46114	−0.310820
$125$	9.41769	0.842344
$126$	0	0
$127$	7.62096	0.676251	0.338125	−	0.941101i	$-0.390207\pi$
$127$	7.62096	0.676251	0.338125	+	0.941101i	$0.390207\pi$
$128$	3.54269	0.313132
$129$	0	0
$130$	0.697385	0.0611647
$131$	5.66648	0.495083	0.247541	−	0.968877i	$-0.420377\pi$
$131$	5.66648	0.495083	0.247541	+	0.968877i	$0.420377\pi$
$132$	0	0
$133$	3.56160	0.308830
$134$	−0.932557	−0.0805607
$135$	0	0
$136$	1.25820	0.107890
$137$	−16.4003	−1.40117	−0.700585	−	0.713569i	$-0.747078\pi$
$137$	−16.4003	−1.40117	−0.700585	+	0.713569i	$0.747078\pi$
$138$	0	0
$139$	−18.9801	−1.60987	−0.804934	−	0.593364i	$-0.797799\pi$
$139$	−18.9801	−1.60987	−0.804934	+	0.593364i	$0.797799\pi$
$140$	2.10920	0.178260
$141$	0	0
$142$	1.30895	0.109844
$143$	0	0
$144$	0	0
$145$	−9.12285	−0.757611
$146$	−0.472475	−0.0391023
$147$	0	0
$148$	−9.43048	−0.775181
$149$	10.7491	0.880601	0.440301	−	0.897850i	$-0.354872\pi$
$149$	10.7491	0.880601	0.440301	+	0.897850i	$0.354872\pi$
$150$	0	0
$151$	−7.18148	−0.584420	−0.292210	−	0.956354i	$-0.594391\pi$
$151$	−7.18148	−0.584420	−0.292210	+	0.956354i	$0.594391\pi$
$152$	1.59738	0.129565
$153$	0	0
$154$	0	0
$155$	−1.84837	−0.148465
$156$	0	0
$157$	2.24993	0.179564	0.0897821	−	0.995961i	$-0.471383\pi$
$157$	2.24993	0.179564	0.0897821	+	0.995961i	$0.471383\pi$
$158$	0.765345	0.0608876
$159$	0	0
$160$	1.42047	0.112298
$161$	−4.72213	−0.372156
$162$	0	0
$163$	−15.9040	−1.24570	−0.622850	−	0.782341i	$-0.714025\pi$
$163$	−15.9040	−1.24570	−0.622850	+	0.782341i	$0.714025\pi$
$164$	−12.2008	−0.952719
$165$	0	0
$166$	−0.131053	−0.0101717
$167$	−6.98786	−0.540737	−0.270368	−	0.962757i	$-0.587146\pi$
$167$	−6.98786	−0.540737	−0.270368	+	0.962757i	$0.587146\pi$
$168$	0	0
$169$	21.1270	1.62516
$170$	0.334896	0.0256854
$171$	0	0
$172$	10.4352	0.795679
$173$	20.8297	1.58365	0.791826	−	0.610747i	$-0.209131\pi$
$173$	20.8297	1.58365	0.791826	+	0.610747i	$0.209131\pi$
$174$	0	0
$175$	−3.87361	−0.292818
$176$	0	0
$177$	0	0
$178$	0.327077	0.0245154
$179$	13.4636	1.00631	0.503157	−	0.864195i	$-0.332172\pi$
$179$	13.4636	1.00631	0.503157	+	0.864195i	$0.332172\pi$
$180$	0	0
$181$	−0.103986	−0.00772924	−0.00386462	−	0.999993i	$-0.501230\pi$
$181$	−0.103986	−0.00772924	−0.00386462	+	0.999993i	$0.501230\pi$
$182$	−0.657096	−0.0487072
$183$	0	0
$184$	−2.11788	−0.156132
$185$	−5.03621	−0.370269
$186$	0	0
$187$	0	0
$188$	−16.2794	−1.18729
$189$	0	0
$190$	0.425176	0.0308455
$191$	−13.7323	−0.993633	−0.496817	−	0.867856i	$-0.665498\pi$
$191$	−13.7323	−0.993633	−0.496817	+	0.867856i	$0.665498\pi$
$192$	0	0
$193$	12.5508	0.903424	0.451712	−	0.892164i	$-0.350813\pi$
$193$	12.5508	0.903424	0.451712	+	0.892164i	$0.350813\pi$
$194$	0.545864	0.0391908
$195$	0	0
$196$	−1.98735	−0.141953
$197$	2.10513	0.149984	0.0749920	−	0.997184i	$-0.476107\pi$
$197$	2.10513	0.149984	0.0749920	+	0.997184i	$0.476107\pi$
$198$	0	0
$199$	−6.57256	−0.465916	−0.232958	−	0.972487i	$-0.574840\pi$
$199$	−6.57256	−0.465916	−0.232958	+	0.972487i	$0.574840\pi$
$200$	−1.73732	−0.122847
$201$	0	0
$202$	1.04591	0.0735903
$203$	8.59581	0.603307
$204$	0	0
$205$	−6.51564	−0.455072
$206$	−1.46090	−0.101786
$207$	0	0
$208$	22.9248	1.58955
$209$	0	0
$210$	0	0
$211$	−6.99883	−0.481819	−0.240910	−	0.970548i	$-0.577446\pi$
$211$	−6.99883	−0.481819	−0.240910	+	0.970548i	$0.577446\pi$
$212$	28.4968	1.95717
$213$	0	0
$214$	0.208399	0.0142459
$215$	5.57278	0.380060
$216$	0	0
$217$	1.74159	0.118227
$218$	−0.523829	−0.0354782
$219$	0	0
$220$	0	0
$221$	16.3884	1.10240
$222$	0	0
$223$	18.5592	1.24281	0.621406	−	0.783488i	$-0.286562\pi$
$223$	18.5592	1.24281	0.621406	+	0.783488i	$0.286562\pi$
$224$	−1.33841	−0.0894260
$225$	0	0
$226$	1.22017	0.0811647
$227$	9.07063	0.602039	0.301019	−	0.953618i	$-0.402673\pi$
$227$	9.07063	0.602039	0.301019	+	0.953618i	$0.402673\pi$
$228$	0	0
$229$	27.1538	1.79437	0.897186	−	0.441653i	$-0.145608\pi$
$229$	27.1538	1.79437	0.897186	+	0.441653i	$0.145608\pi$
$230$	−0.563717	−0.0371704
$231$	0	0
$232$	3.85523	0.253108
$233$	−1.57052	−0.102888	−0.0514439	−	0.998676i	$-0.516382\pi$
$233$	−1.57052	−0.102888	−0.0514439	+	0.998676i	$0.516382\pi$
$234$	0	0
$235$	−8.69375	−0.567118
$236$	7.76199	0.505262
$237$	0	0
$238$	−0.315549	−0.0204540
$239$	−15.3004	−0.989702	−0.494851	−	0.868978i	$-0.664777\pi$
$239$	−15.3004	−0.989702	−0.494851	+	0.868978i	$0.664777\pi$
$240$	0	0
$241$	−21.1065	−1.35959	−0.679796	−	0.733401i	$-0.737932\pi$
$241$	−21.1065	−1.35959	−0.679796	+	0.733401i	$0.737932\pi$
$242$	0	0
$243$	0	0
$244$	−29.3171	−1.87683
$245$	−1.06131	−0.0678049
$246$	0	0
$247$	20.8063	1.32387
$248$	0.781104	0.0496002
$249$	0	0
$250$	−1.05931	−0.0669968
$251$	−2.70154	−0.170519	−0.0852597	−	0.996359i	$-0.527172\pi$
$251$	−2.70154	−0.170519	−0.0852597	+	0.996359i	$0.527172\pi$
$252$	0	0
$253$	0	0
$254$	−0.857214	−0.0537864
$255$	0	0
$256$	14.9974	0.937339
$257$	28.4992	1.77773	0.888865	−	0.458169i	$-0.151495\pi$
$257$	28.4992	1.77773	0.888865	+	0.458169i	$0.151495\pi$
$258$	0	0
$259$	4.74526	0.294856
$260$	12.3216	0.764153
$261$	0	0
$262$	−0.637372	−0.0393770
$263$	−15.9889	−0.985917	−0.492959	−	0.870053i	$-0.664085\pi$
$263$	−15.9889	−0.985917	−0.492959	+	0.870053i	$0.664085\pi$
$264$	0	0
$265$	15.2183	0.934853
$266$	−0.400613	−0.0245632
$267$	0	0
$268$	−16.4767	−1.00647
$269$	13.9351	0.849638	0.424819	−	0.905278i	$-0.360338\pi$
$269$	13.9351	0.849638	0.424819	+	0.905278i	$0.360338\pi$
$270$	0	0
$271$	17.0917	1.03825	0.519125	−	0.854698i	$-0.326258\pi$
$271$	17.0917	1.03825	0.519125	+	0.854698i	$0.326258\pi$
$272$	11.0089	0.667512
$273$	0	0
$274$	1.84472	0.111444
$275$	0	0
$276$	0	0
$277$	29.1940	1.75410	0.877048	−	0.480402i	$-0.159509\pi$
$277$	29.1940	1.75410	0.877048	+	0.480402i	$0.159509\pi$
$278$	2.13490	0.128043
$279$	0	0
$280$	−0.476001	−0.0284465
$281$	14.4070	0.859449	0.429724	−	0.902960i	$-0.358611\pi$
$281$	14.4070	0.859449	0.429724	+	0.902960i	$0.358611\pi$
$282$	0	0
$283$	−0.457299	−0.0271836	−0.0135918	−	0.999908i	$-0.504327\pi$
$283$	−0.457299	−0.0271836	−0.0135918	+	0.999908i	$0.504327\pi$
$284$	23.1268	1.37233
$285$	0	0
$286$	0	0
$287$	6.13922	0.362386
$288$	0	0
$289$	−9.13001	−0.537060
$290$	1.02615	0.0602575
$291$	0	0
$292$	−8.34783	−0.488520
$293$	23.4778	1.37159	0.685793	−	0.727797i	$-0.259456\pi$
$293$	23.4778	1.37159	0.685793	+	0.727797i	$0.259456\pi$
$294$	0	0
$295$	4.14517	0.241341
$296$	2.12825	0.123702
$297$	0	0
$298$	−1.20907	−0.0700396
$299$	−27.5859	−1.59533
$300$	0	0
$301$	−5.25083	−0.302653
$302$	0.807781	0.0464825
$303$	0	0
$304$	13.9766	0.801614
$305$	−15.6563	−0.896479
$306$	0	0
$307$	−6.02778	−0.344023	−0.172012	−	0.985095i	$-0.555027\pi$
$307$	−6.02778	−0.344023	−0.172012	+	0.985095i	$0.555027\pi$
$308$	0	0
$309$	0	0
$310$	0.207907	0.0118083
$311$	27.0571	1.53427	0.767134	−	0.641487i	$-0.221682\pi$
$311$	27.0571	1.53427	0.767134	+	0.641487i	$0.221682\pi$
$312$	0	0
$313$	−3.09281	−0.174816	−0.0874080	−	0.996173i	$-0.527858\pi$
$313$	−3.09281	−0.174816	−0.0874080	+	0.996173i	$0.527858\pi$
$314$	−0.253075	−0.0142818
$315$	0	0
$316$	13.5223	0.760690
$317$	26.3521	1.48008	0.740042	−	0.672561i	$-0.234806\pi$
$317$	26.3521	1.48008	0.740042	+	0.672561i	$0.234806\pi$
$318$	0	0
$319$	0	0
$320$	8.16994	0.456714
$321$	0	0
$322$	0.531150	0.0295998
$323$	9.99154	0.555944
$324$	0	0
$325$	−22.6290	−1.25523
$326$	1.78890	0.0990782
$327$	0	0
$328$	2.75345	0.152034
$329$	8.19149	0.451612
$330$	0	0
$331$	10.1728	0.559150	0.279575	−	0.960124i	$-0.409806\pi$
$331$	10.1728	0.559150	0.279575	+	0.960124i	$0.409806\pi$
$332$	−2.31548	−0.127078
$333$	0	0
$334$	0.786002	0.0430081
$335$	−8.79913	−0.480748
$336$	0	0
$337$	0.483631	0.0263451	0.0131725	−	0.999913i	$-0.495807\pi$
$337$	0.483631	0.0263451	0.0131725	+	0.999913i	$0.495807\pi$
$338$	−2.37639	−0.129259
$339$	0	0
$340$	5.91704	0.320897
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−2.35500	−0.126973
$345$	0	0
$346$	−2.34295	−0.125958
$347$	2.19027	0.117580	0.0587900	−	0.998270i	$-0.481276\pi$
$347$	2.19027	0.117580	0.0587900	+	0.998270i	$0.481276\pi$
$348$	0	0
$349$	4.11130	0.220073	0.110036	−	0.993928i	$-0.464903\pi$
$349$	4.11130	0.220073	0.110036	+	0.993928i	$0.464903\pi$
$350$	0.435708	0.0232896
$351$	0	0
$352$	0	0
$353$	36.0161	1.91694	0.958471	−	0.285191i	$-0.0920571\pi$
$353$	36.0161	1.91694	0.958471	+	0.285191i	$0.0920571\pi$
$354$	0	0
$355$	12.3506	0.655499
$356$	5.77888	0.306280
$357$	0	0
$358$	−1.51440	−0.0800383
$359$	28.3376	1.49560	0.747801	−	0.663923i	$-0.231110\pi$
$359$	28.3376	1.49560	0.747801	+	0.663923i	$0.231110\pi$
$360$	0	0
$361$	−6.31498	−0.332367
$362$	0.0116965	0.000614754 0
$363$	0	0
$364$	−11.6098	−0.608517
$365$	−4.45804	−0.233344
$366$	0	0
$367$	5.71592	0.298369	0.149184	−	0.988809i	$-0.452335\pi$
$367$	5.71592	0.298369	0.149184	+	0.988809i	$0.452335\pi$
$368$	−18.5308	−0.965984
$369$	0	0
$370$	0.566478	0.0294498
$371$	−14.3391	−0.744450
$372$	0	0
$373$	0.428982	0.0222119	0.0111059	−	0.999938i	$-0.496465\pi$
$373$	0.428982	0.0222119	0.0111059	+	0.999938i	$0.496465\pi$
$374$	0	0
$375$	0	0
$376$	3.67390	0.189467
$377$	50.2153	2.58622
$378$	0	0
$379$	6.21110	0.319043	0.159521	−	0.987194i	$-0.449005\pi$
$379$	6.21110	0.319043	0.159521	+	0.987194i	$0.449005\pi$
$380$	7.51213	0.385364
$381$	0	0
$382$	1.54462	0.0790298
$383$	−36.9787	−1.88952	−0.944762	−	0.327759i	$-0.893707\pi$
$383$	−36.9787	−1.88952	−0.944762	+	0.327759i	$0.893707\pi$
$384$	0	0
$385$	0	0
$386$	−1.41172	−0.0718549
$387$	0	0
$388$	9.64448	0.489624
$389$	14.4235	0.731301	0.365650	−	0.930752i	$-0.380847\pi$
$389$	14.4235	0.731301	0.365650	+	0.930752i	$0.380847\pi$
$390$	0	0
$391$	−13.2472	−0.669940
$392$	0.448501	0.0226527
$393$	0	0
$394$	−0.236787	−0.0119292
$395$	7.22140	0.363348
$396$	0	0
$397$	−16.6802	−0.837156	−0.418578	−	0.908181i	$-0.637471\pi$
$397$	−16.6802	−0.837156	−0.418578	+	0.908181i	$0.637471\pi$
$398$	0.739288	0.0370572
$399$	0	0
$400$	−15.2010	−0.760051
$401$	−0.429298	−0.0214381	−0.0107191	−	0.999943i	$-0.503412\pi$
$401$	−0.429298	−0.0214381	−0.0107191	+	0.999943i	$0.503412\pi$
$402$	0	0
$403$	10.1741	0.506806
$404$	18.4795	0.919390
$405$	0	0
$406$	−0.966866	−0.0479847
$407$	0	0
$408$	0	0
$409$	−2.53087	−0.125144	−0.0625718	−	0.998040i	$-0.519930\pi$
$409$	−2.53087	−0.125144	−0.0625718	+	0.998040i	$0.519930\pi$
$410$	0.732886	0.0361947
$411$	0	0
$412$	−25.8116	−1.27165
$413$	−3.90570	−0.192187
$414$	0	0
$415$	−1.23655	−0.0606997
$416$	−7.81875	−0.383346
$417$	0	0
$418$	0	0
$419$	19.2313	0.939508	0.469754	−	0.882797i	$-0.344343\pi$
$419$	19.2313	0.939508	0.469754	+	0.882797i	$0.344343\pi$
$420$	0	0
$421$	−16.8316	−0.820322	−0.410161	−	0.912013i	$-0.634527\pi$
$421$	−16.8316	−0.820322	−0.410161	+	0.912013i	$0.634527\pi$
$422$	0.787236	0.0383221
$423$	0	0
$424$	−6.43111	−0.312322
$425$	−10.8668	−0.527119
$426$	0	0
$427$	14.7518	0.713892
$428$	3.68205	0.177979
$429$	0	0
$430$	−0.626832	−0.0302285
$431$	−13.1978	−0.635713	−0.317857	−	0.948139i	$-0.602963\pi$
$431$	−13.1978	−0.635713	−0.317857	+	0.948139i	$0.602963\pi$
$432$	0	0
$433$	12.4390	0.597779	0.298890	−	0.954288i	$-0.403384\pi$
$433$	12.4390	0.597779	0.298890	+	0.954288i	$0.403384\pi$
$434$	−0.195896	−0.00940330
$435$	0	0
$436$	−9.25516	−0.443242
$437$	−16.8183	−0.804530
$438$	0	0
$439$	−2.80796	−0.134017	−0.0670084	−	0.997752i	$-0.521345\pi$
$439$	−2.80796	−0.134017	−0.0670084	+	0.997752i	$0.521345\pi$
$440$	0	0
$441$	0	0
$442$	−1.84338	−0.0876808
$443$	−12.2025	−0.579758	−0.289879	−	0.957063i	$-0.593615\pi$
$443$	−12.2025	−0.579758	−0.289879	+	0.957063i	$0.593615\pi$
$444$	0	0
$445$	3.08613	0.146297
$446$	−2.08755	−0.0988485
$447$	0	0
$448$	−7.69795	−0.363694
$449$	4.53613	0.214073	0.107037	−	0.994255i	$-0.465864\pi$
$449$	4.53613	0.214073	0.107037	+	0.994255i	$0.465864\pi$
$450$	0	0
$451$	0	0
$452$	21.5584	1.01402
$453$	0	0
$454$	−1.02027	−0.0478839
$455$	−6.20002	−0.290661
$456$	0	0
$457$	−35.9900	−1.68354	−0.841772	−	0.539834i	$-0.818487\pi$
$457$	−35.9900	−1.68354	−0.841772	+	0.539834i	$0.818487\pi$
$458$	−3.05429	−0.142717
$459$	0	0
$460$	−9.95991	−0.464383
$461$	−3.38164	−0.157499	−0.0787493	−	0.996894i	$-0.525093\pi$
$461$	−3.38164	−0.157499	−0.0787493	+	0.996894i	$0.525093\pi$
$462$	0	0
$463$	0.189385	0.00880145	0.00440073	−	0.999990i	$-0.498599\pi$
$463$	0.189385	0.00880145	0.00440073	+	0.999990i	$0.498599\pi$
$464$	33.7321	1.56597
$465$	0	0
$466$	0.176653	0.00818331
$467$	9.82028	0.454428	0.227214	−	0.973845i	$-0.427038\pi$
$467$	9.82028	0.454428	0.227214	+	0.973845i	$0.427038\pi$
$468$	0	0
$469$	8.29079	0.382833
$470$	0.977882	0.0451063
$471$	0	0
$472$	−1.75171	−0.0806291
$473$	0	0
$474$	0	0
$475$	−13.7963	−0.633016
$476$	−5.57521	−0.255539
$477$	0	0
$478$	1.72101	0.0787171
$479$	−6.10821	−0.279091	−0.139546	−	0.990216i	$-0.544564\pi$
$479$	−6.10821	−0.279091	−0.139546	+	0.990216i	$0.544564\pi$
$480$	0	0
$481$	27.7210	1.26397
$482$	2.37409	0.108137
$483$	0	0
$484$	0	0
$485$	5.15049	0.233872
$486$	0	0
$487$	3.41377	0.154693	0.0773464	−	0.997004i	$-0.475355\pi$
$487$	3.41377	0.154693	0.0773464	+	0.997004i	$0.475355\pi$
$488$	6.61622	0.299502
$489$	0	0
$490$	0.119378	0.00539294
$491$	31.9564	1.44217	0.721087	−	0.692845i	$-0.243643\pi$
$491$	31.9564	1.44217	0.721087	+	0.692845i	$0.243643\pi$
$492$	0	0
$493$	24.1142	1.08605
$494$	−2.34032	−0.105296
$495$	0	0
$496$	6.83442	0.306875
$497$	−11.6370	−0.521992
$498$	0	0
$499$	21.9134	0.980979	0.490489	−	0.871447i	$-0.336818\pi$
$499$	21.9134	0.980979	0.490489	+	0.871447i	$0.336818\pi$
$500$	−18.7162	−0.837015
$501$	0	0
$502$	0.303872	0.0135625
$503$	−32.9708	−1.47010	−0.735049	−	0.678014i	$-0.762841\pi$
$503$	−32.9708	−1.47010	−0.735049	+	0.678014i	$0.762841\pi$
$504$	0	0
$505$	9.86871	0.439152
$506$	0	0
$507$	0	0
$508$	−15.1455	−0.671973
$509$	−26.7351	−1.18501	−0.592506	−	0.805566i	$-0.701861\pi$
$509$	−26.7351	−1.18501	−0.592506	+	0.805566i	$0.701861\pi$
$510$	0	0
$511$	4.20049	0.185819
$512$	−8.77230	−0.387685
$513$	0	0
$514$	−3.20562	−0.141394
$515$	−13.7843	−0.607409
$516$	0	0
$517$	0	0
$518$	−0.533752	−0.0234517
$519$	0	0
$520$	−2.78072	−0.121943
$521$	−11.4012	−0.499496	−0.249748	−	0.968311i	$-0.580348\pi$
$521$	−11.4012	−0.499496	−0.249748	+	0.968311i	$0.580348\pi$
$522$	0	0
$523$	15.0700	0.658966	0.329483	−	0.944161i	$-0.393125\pi$
$523$	15.0700	0.658966	0.329483	+	0.944161i	$0.393125\pi$
$524$	−11.2613	−0.491951
$525$	0	0
$526$	1.79845	0.0784161
$527$	4.88576	0.212827
$528$	0	0
$529$	−0.701531	−0.0305014
$530$	−1.71177	−0.0743546
$531$	0	0
$532$	−7.07815	−0.306877
$533$	35.8643	1.55345
$534$	0	0
$535$	1.96634	0.0850125
$536$	3.71843	0.160612
$537$	0	0
$538$	−1.56744	−0.0675769
$539$	0	0
$540$	0	0
$541$	−26.8970	−1.15639	−0.578196	−	0.815898i	$-0.696243\pi$
$541$	−26.8970	−1.15639	−0.578196	+	0.815898i	$0.696243\pi$
$542$	−1.92250	−0.0825784
$543$	0	0
$544$	−3.75470	−0.160981
$545$	−4.94258	−0.211717
$546$	0	0
$547$	10.0770	0.430860	0.215430	−	0.976519i	$-0.430885\pi$
$547$	10.0770	0.430860	0.215430	+	0.976519i	$0.430885\pi$
$548$	32.5931	1.39231
$549$	0	0
$550$	0	0
$551$	30.6148	1.30424
$552$	0	0
$553$	−6.80421	−0.289344
$554$	−3.28377	−0.139514
$555$	0	0
$556$	37.7200	1.59968
$557$	−30.8543	−1.30734	−0.653669	−	0.756781i	$-0.726771\pi$
$557$	−30.8543	−1.30734	−0.653669	+	0.756781i	$0.726771\pi$
$558$	0	0
$559$	−30.6745	−1.29739
$560$	−4.16486	−0.175997
$561$	0	0
$562$	−1.62051	−0.0683572
$563$	−14.0259	−0.591122	−0.295561	−	0.955324i	$-0.595507\pi$
$563$	−14.0259	−0.591122	−0.295561	+	0.955324i	$0.595507\pi$
$564$	0	0
$565$	11.5129	0.484353
$566$	0.0514375	0.00216208
$567$	0	0
$568$	−5.21923	−0.218994
$569$	−28.5547	−1.19708	−0.598538	−	0.801094i	$-0.704251\pi$
$569$	−28.5547	−1.19708	−0.598538	+	0.801094i	$0.704251\pi$
$570$	0	0
$571$	−18.0629	−0.755907	−0.377953	−	0.925825i	$-0.623372\pi$
$571$	−18.0629	−0.755907	−0.377953	+	0.925825i	$0.623372\pi$
$572$	0	0
$573$	0	0
$574$	−0.690546	−0.0288228
$575$	18.2917	0.762816
$576$	0	0
$577$	−35.7797	−1.48953	−0.744764	−	0.667328i	$-0.767438\pi$
$577$	−35.7797	−1.48953	−0.744764	+	0.667328i	$0.767438\pi$
$578$	1.02695	0.0427157
$579$	0	0
$580$	18.1303	0.752819
$581$	1.16511	0.0483369
$582$	0	0
$583$	0	0
$584$	1.88392	0.0779573
$585$	0	0
$586$	−2.64080	−0.109091
$587$	16.8458	0.695301	0.347651	−	0.937624i	$-0.386980\pi$
$587$	16.8458	0.695301	0.347651	+	0.937624i	$0.386980\pi$
$588$	0	0
$589$	6.20285	0.255584
$590$	−0.466254	−0.0191954
$591$	0	0
$592$	18.6216	0.765342
$593$	36.1560	1.48475	0.742375	−	0.669985i	$-0.233699\pi$
$593$	36.1560	1.48475	0.742375	+	0.669985i	$0.233699\pi$
$594$	0	0
$595$	−2.97736	−0.122060
$596$	−21.3622	−0.875030
$597$	0	0
$598$	3.10289	0.126887
$599$	34.5880	1.41323	0.706614	−	0.707599i	$-0.250222\pi$
$599$	34.5880	1.41323	0.706614	+	0.707599i	$0.250222\pi$
$600$	0	0
$601$	38.8660	1.58538	0.792688	−	0.609627i	$-0.208681\pi$
$601$	38.8660	1.58538	0.792688	+	0.609627i	$0.208681\pi$
$602$	0.590619	0.0240718
$603$	0	0
$604$	14.2721	0.580723
$605$	0	0
$606$	0	0
$607$	21.3067	0.864813	0.432406	−	0.901679i	$-0.357665\pi$
$607$	21.3067	0.864813	0.432406	+	0.901679i	$0.357665\pi$
$608$	−4.76687	−0.193322
$609$	0	0
$610$	1.76104	0.0713025
$611$	47.8534	1.93594
$612$	0	0
$613$	0.456270	0.0184286	0.00921429	−	0.999958i	$-0.497067\pi$
$613$	0.456270	0.0184286	0.00921429	+	0.999958i	$0.497067\pi$
$614$	0.678011	0.0273623
$615$	0	0
$616$	0	0
$617$	18.2733	0.735656	0.367828	−	0.929894i	$-0.380101\pi$
$617$	18.2733	0.735656	0.367828	+	0.929894i	$0.380101\pi$
$618$	0	0
$619$	31.0436	1.24775	0.623875	−	0.781525i	$-0.285558\pi$
$619$	31.0436	1.24775	0.623875	+	0.781525i	$0.285558\pi$
$620$	3.67336	0.147526
$621$	0	0
$622$	−3.04341	−0.122030
$623$	−2.90784	−0.116500
$624$	0	0
$625$	9.37294	0.374918
$626$	0.347883	0.0139042
$627$	0	0
$628$	−4.47140	−0.178428
$629$	13.3121	0.530788
$630$	0	0
$631$	−29.8472	−1.18820	−0.594098	−	0.804393i	$-0.702491\pi$
$631$	−29.8472	−1.18820	−0.594098	+	0.804393i	$0.702491\pi$
$632$	−3.05170	−0.121390
$633$	0	0
$634$	−2.96412	−0.117720
$635$	−8.08823	−0.320972
$636$	0	0
$637$	5.84183	0.231462
$638$	0	0
$639$	0	0
$640$	−3.75990	−0.148623
$641$	3.64805	0.144089	0.0720447	−	0.997401i	$-0.477048\pi$
$641$	3.64805	0.144089	0.0720447	+	0.997401i	$0.477048\pi$
$642$	0	0
$643$	34.5924	1.36419	0.682095	−	0.731263i	$-0.261069\pi$
$643$	34.5924	1.36419	0.682095	+	0.731263i	$0.261069\pi$
$644$	9.38451	0.369801
$645$	0	0
$646$	−1.12386	−0.0442177
$647$	−14.3914	−0.565784	−0.282892	−	0.959152i	$-0.591294\pi$
$647$	−14.3914	−0.565784	−0.282892	+	0.959152i	$0.591294\pi$
$648$	0	0
$649$	0	0
$650$	2.54534	0.0998363
$651$	0	0
$652$	31.6068	1.23782
$653$	23.4343	0.917056	0.458528	−	0.888680i	$-0.348377\pi$
$653$	23.4343	0.917056	0.458528	+	0.888680i	$0.348377\pi$
$654$	0	0
$655$	−6.01392	−0.234983
$656$	24.0918	0.940627
$657$	0	0
$658$	−0.921388	−0.0359195
$659$	38.6951	1.50735	0.753674	−	0.657248i	$-0.228280\pi$
$659$	38.6951	1.50735	0.753674	+	0.657248i	$0.228280\pi$
$660$	0	0
$661$	−30.3882	−1.18196	−0.590982	−	0.806684i	$-0.701260\pi$
$661$	−30.3882	−1.18196	−0.590982	+	0.806684i	$0.701260\pi$
$662$	−1.14425	−0.0444726
$663$	0	0
$664$	0.522553	0.0202790
$665$	−3.77998	−0.146581
$666$	0	0
$667$	−40.5905	−1.57167
$668$	13.8873	0.537316
$669$	0	0
$670$	0.989736	0.0382368
$671$	0	0
$672$	0	0
$673$	−18.1911	−0.701217	−0.350609	−	0.936522i	$-0.614025\pi$
$673$	−18.1911	−0.701217	−0.350609	+	0.936522i	$0.614025\pi$
$674$	−0.0543994	−0.00209539
$675$	0	0
$676$	−41.9867	−1.61487
$677$	−5.83832	−0.224385	−0.112192	−	0.993687i	$-0.535787\pi$
$677$	−5.83832	−0.224385	−0.112192	+	0.993687i	$0.535787\pi$
$678$	0	0
$679$	−4.85294	−0.186239
$680$	−1.33535	−0.0512083
$681$	0	0
$682$	0	0
$683$	−3.17118	−0.121342	−0.0606710	−	0.998158i	$-0.519324\pi$
$683$	−3.17118	−0.121342	−0.0606710	+	0.998158i	$0.519324\pi$
$684$	0	0
$685$	17.4058	0.665043
$686$	−0.112481	−0.00429455
$687$	0	0
$688$	−20.6056	−0.785580
$689$	−83.7668	−3.19126
$690$	0	0
$691$	−23.9093	−0.909554	−0.454777	−	0.890605i	$-0.650281\pi$
$691$	−23.9093	−0.909554	−0.454777	+	0.890605i	$0.650281\pi$
$692$	−41.3958	−1.57363
$693$	0	0
$694$	−0.246364	−0.00935186
$695$	20.1438	0.764098
$696$	0	0
$697$	17.2226	0.652354
$698$	−0.462443	−0.0175037
$699$	0	0
$700$	7.69822	0.290965
$701$	41.3167	1.56051	0.780254	−	0.625462i	$-0.215090\pi$
$701$	41.3167	1.56051	0.780254	+	0.625462i	$0.215090\pi$
$702$	0	0
$703$	16.9007	0.637423
$704$	0	0
$705$	0	0
$706$	−4.05113	−0.152466
$707$	−9.29858	−0.349709
$708$	0	0
$709$	−49.4717	−1.85795	−0.928975	−	0.370143i	$-0.879309\pi$
$709$	−49.4717	−1.85795	−0.928975	+	0.370143i	$0.879309\pi$
$710$	−1.38920	−0.0521359
$711$	0	0
$712$	−1.30417	−0.0488758
$713$	−8.22400	−0.307991
$714$	0	0
$715$	0	0
$716$	−26.7568	−0.999948
$717$	0	0
$718$	−3.18745	−0.118954
$719$	18.6951	0.697209	0.348605	−	0.937270i	$-0.386656\pi$
$719$	18.6951	0.697209	0.348605	+	0.937270i	$0.386656\pi$
$720$	0	0
$721$	12.9880	0.483697
$722$	0.710316	0.0264352
$723$	0	0
$724$	0.206657	0.00768034
$725$	−33.2968	−1.23661
$726$	0	0
$727$	−26.9216	−0.998468	−0.499234	−	0.866467i	$-0.666385\pi$
$727$	−26.9216	−0.998468	−0.499234	+	0.866467i	$0.666385\pi$
$728$	2.62007	0.0971063
$729$	0	0
$730$	0.501445	0.0185593
$731$	−14.7304	−0.544824
$732$	0	0
$733$	30.5427	1.12812	0.564061	−	0.825733i	$-0.309238\pi$
$733$	30.5427	1.12812	0.564061	+	0.825733i	$0.309238\pi$
$734$	−0.642933	−0.0237311
$735$	0	0
$736$	6.32012	0.232963
$737$	0	0
$738$	0	0
$739$	12.7624	0.469472	0.234736	−	0.972059i	$-0.424577\pi$
$739$	12.7624	0.469472	0.234736	+	0.972059i	$0.424577\pi$
$740$	10.0087	0.367927
$741$	0	0
$742$	1.61288	0.0592107
$743$	31.0657	1.13969	0.569844	−	0.821753i	$-0.307004\pi$
$743$	31.0657	1.13969	0.569844	+	0.821753i	$0.307004\pi$
$744$	0	0
$745$	−11.4082	−0.417963
$746$	−0.0482524	−0.00176665
$747$	0	0
$748$	0	0
$749$	−1.85275	−0.0676978
$750$	0	0
$751$	−12.6379	−0.461164	−0.230582	−	0.973053i	$-0.574063\pi$
$751$	−12.6379	−0.461164	−0.230582	+	0.973053i	$0.574063\pi$
$752$	32.1455	1.17222
$753$	0	0
$754$	−5.64827	−0.205698
$755$	7.62180	0.277386
$756$	0	0
$757$	−13.6857	−0.497415	−0.248708	−	0.968579i	$-0.580006\pi$
$757$	−13.6857	−0.497415	−0.248708	+	0.968579i	$0.580006\pi$
$758$	−0.698631	−0.0253754
$759$	0	0
$760$	−1.69533	−0.0614959
$761$	24.1444	0.875233	0.437617	−	0.899162i	$-0.355823\pi$
$761$	24.1444	0.875233	0.437617	+	0.899162i	$0.355823\pi$
$762$	0	0
$763$	4.65704	0.168596
$764$	27.2908	0.987348
$765$	0	0
$766$	4.15940	0.150285
$767$	−22.8165	−0.823854
$768$	0	0
$769$	−3.51499	−0.126754	−0.0633770	−	0.997990i	$-0.520187\pi$
$769$	−3.51499	−0.126754	−0.0633770	+	0.997990i	$0.520187\pi$
$770$	0	0
$771$	0	0
$772$	−24.9428	−0.897709
$773$	−8.07699	−0.290509	−0.145255	−	0.989394i	$-0.546400\pi$
$773$	−8.07699	−0.290509	−0.145255	+	0.989394i	$0.546400\pi$
$774$	0	0
$775$	−6.74624	−0.242332
$776$	−2.17655	−0.0781336
$777$	0	0
$778$	−1.62237	−0.0581648
$779$	21.8655	0.783411
$780$	0	0
$781$	0	0
$782$	1.49006	0.0532845
$783$	0	0
$784$	3.92425	0.140152
$785$	−2.38788	−0.0852273
$786$	0	0
$787$	5.32791	0.189920	0.0949598	−	0.995481i	$-0.469728\pi$
$787$	5.32791	0.189920	0.0949598	+	0.995481i	$0.469728\pi$
$788$	−4.18362	−0.149035
$789$	0	0
$790$	−0.812271	−0.0288993
$791$	−10.8478	−0.385704
$792$	0	0
$793$	86.1778	3.06026
$794$	1.87621	0.0665842
$795$	0	0
$796$	13.0620	0.462969
$797$	−44.3138	−1.56968	−0.784838	−	0.619701i	$-0.787254\pi$
$797$	−44.3138	−1.56968	−0.784838	+	0.619701i	$0.787254\pi$
$798$	0	0
$799$	22.9800	0.812974
$800$	5.18447	0.183299
$801$	0	0
$802$	0.0482880	0.00170511
$803$	0	0
$804$	0	0
$805$	5.01166	0.176638
$806$	−1.14439	−0.0403094
$807$	0	0
$808$	−4.17043	−0.146715
$809$	10.8499	0.381462	0.190731	−	0.981642i	$-0.438914\pi$
$809$	10.8499	0.381462	0.190731	+	0.981642i	$0.438914\pi$
$810$	0	0
$811$	−36.0883	−1.26723	−0.633615	−	0.773648i	$-0.718430\pi$
$811$	−36.0883	−1.26723	−0.633615	+	0.773648i	$0.718430\pi$
$812$	−17.0829	−0.599491
$813$	0	0
$814$	0	0
$815$	16.8792	0.591252
$816$	0	0
$817$	−18.7014	−0.654278
$818$	0.284675	0.00995343
$819$	0	0
$820$	12.9488	0.452193
$821$	52.1407	1.81972	0.909861	−	0.414913i	$-0.136188\pi$
$821$	52.1407	1.81972	0.909861	+	0.414913i	$0.136188\pi$
$822$	0	0
$823$	26.1122	0.910213	0.455106	−	0.890437i	$-0.349601\pi$
$823$	26.1122	0.910213	0.455106	+	0.890437i	$0.349601\pi$
$824$	5.82512	0.202928
$825$	0	0
$826$	0.439317	0.0152858
$827$	7.68971	0.267397	0.133699	−	0.991022i	$-0.457315\pi$
$827$	7.68971	0.267397	0.133699	+	0.991022i	$0.457315\pi$
$828$	0	0
$829$	−25.3084	−0.878997	−0.439498	−	0.898243i	$-0.644844\pi$
$829$	−25.3084	−0.878997	−0.439498	+	0.898243i	$0.644844\pi$
$830$	0.139088	0.00482782
$831$	0	0
$832$	−44.9701	−1.55906
$833$	2.80535	0.0971996
$834$	0	0
$835$	7.41632	0.256652
$836$	0	0
$837$	0	0
$838$	−2.16315	−0.0747249
$839$	7.10737	0.245374	0.122687	−	0.992445i	$-0.460849\pi$
$839$	7.10737	0.245374	0.122687	+	0.992445i	$0.460849\pi$
$840$	0	0
$841$	44.8879	1.54786
$842$	1.89324	0.0652452
$843$	0	0
$844$	13.9091	0.478771
$845$	−22.4224	−0.771354
$846$	0	0
$847$	0	0
$848$	−56.2703	−1.93233
$849$	0	0
$850$	1.22231	0.0419250
$851$	−22.4077	−0.768126
$852$	0	0
$853$	−29.8933	−1.02353	−0.511764	−	0.859126i	$-0.671008\pi$
$853$	−29.8933	−1.02353	−0.511764	+	0.859126i	$0.671008\pi$
$854$	−1.65930	−0.0567802
$855$	0	0
$856$	−0.830959	−0.0284016
$857$	−31.9946	−1.09291	−0.546457	−	0.837487i	$-0.684024\pi$
$857$	−31.9946	−1.09291	−0.546457	+	0.837487i	$0.684024\pi$
$858$	0	0
$859$	−18.8113	−0.641835	−0.320917	−	0.947107i	$-0.603991\pi$
$859$	−18.8113	−0.641835	−0.320917	+	0.947107i	$0.603991\pi$
$860$	−11.0751	−0.377656
$861$	0	0
$862$	1.48450	0.0505622
$863$	−19.1665	−0.652435	−0.326217	−	0.945295i	$-0.605774\pi$
$863$	−19.1665	−0.652435	−0.326217	+	0.945295i	$0.605774\pi$
$864$	0	0
$865$	−22.1068	−0.751655
$866$	−1.39915	−0.0475451
$867$	0	0
$868$	−3.46114	−0.117479
$869$	0	0
$870$	0	0
$871$	48.4334	1.64110
$872$	2.08869	0.0707319
$873$	0	0
$874$	1.89175	0.0639892
$875$	9.41769	0.318376
$876$	0	0
$877$	20.4286	0.689825	0.344912	−	0.938635i	$-0.387909\pi$
$877$	20.4286	0.689825	0.344912	+	0.938635i	$0.387909\pi$
$878$	0.315843	0.0106592
$879$	0	0
$880$	0	0
$881$	−29.2279	−0.984712	−0.492356	−	0.870394i	$-0.663864\pi$
$881$	−29.2279	−0.984712	−0.492356	+	0.870394i	$0.663864\pi$
$882$	0	0
$883$	−29.5803	−0.995456	−0.497728	−	0.867333i	$-0.665832\pi$
$883$	−29.5803	−0.995456	−0.497728	+	0.867333i	$0.665832\pi$
$884$	−32.5694	−1.09543
$885$	0	0
$886$	1.37255	0.0461117
$887$	56.1787	1.88630	0.943148	−	0.332373i	$-0.107849\pi$
$887$	56.1787	1.88630	0.943148	+	0.332373i	$0.107849\pi$
$888$	0	0
$889$	7.62096	0.255599
$890$	−0.347131	−0.0116359
$891$	0	0
$892$	−36.8835	−1.23495
$893$	29.1749	0.976299
$894$	0	0
$895$	−14.2891	−0.477631
$896$	3.54269	0.118353
$897$	0	0
$898$	−0.510229	−0.0170265
$899$	14.9704	0.499289
$900$	0	0
$901$	−40.2262	−1.34013
$902$	0	0
$903$	0	0
$904$	−4.86526	−0.161816
$905$	0.110362	0.00366856
$906$	0	0
$907$	−37.3467	−1.24008	−0.620038	−	0.784572i	$-0.712883\pi$
$907$	−37.3467	−1.24008	−0.620038	+	0.784572i	$0.712883\pi$
$908$	−18.0265	−0.598230
$909$	0	0
$910$	0.697385	0.0231181
$911$	17.4355	0.577664	0.288832	−	0.957380i	$-0.406733\pi$
$911$	17.4355	0.577664	0.288832	+	0.957380i	$0.406733\pi$
$912$	0	0
$913$	0	0
$914$	4.04820	0.133903
$915$	0	0
$916$	−53.9640	−1.78302
$917$	5.66648	0.187124
$918$	0	0
$919$	50.9223	1.67977	0.839885	−	0.542764i	$-0.182622\pi$
$919$	50.9223	1.67977	0.839885	+	0.542764i	$0.182622\pi$
$920$	2.24774	0.0741056
$921$	0	0
$922$	0.380371	0.0125268
$923$	−67.9816	−2.23764
$924$	0	0
$925$	−18.3813	−0.604373
$926$	−0.0213022	−0.000700034 0
$927$	0	0
$928$	−11.5047	−0.377660
$929$	57.2057	1.87686	0.938428	−	0.345474i	$-0.112282\pi$
$929$	57.2057	1.87686	0.938428	+	0.345474i	$0.112282\pi$
$930$	0	0
$931$	3.56160	0.116727
$932$	3.12116	0.102237
$933$	0	0
$934$	−1.10460	−0.0361435
$935$	0	0
$936$	0	0
$937$	27.1139	0.885773	0.442887	−	0.896578i	$-0.353954\pi$
$937$	27.1139	0.885773	0.442887	+	0.896578i	$0.353954\pi$
$938$	−0.932557	−0.0304491
$939$	0	0
$940$	17.2775	0.563530
$941$	−4.91368	−0.160181	−0.0800907	−	0.996788i	$-0.525521\pi$
$941$	−4.91368	−0.160181	−0.0800907	+	0.996788i	$0.525521\pi$
$942$	0	0
$943$	−28.9901	−0.944049
$944$	−15.3269	−0.498849
$945$	0	0
$946$	0	0
$947$	−26.1937	−0.851181	−0.425591	−	0.904916i	$-0.639934\pi$
$947$	−26.1937	−0.851181	−0.425591	+	0.904916i	$0.639934\pi$
$948$	0	0
$949$	24.5385	0.796555
$950$	1.55182	0.0503477
$951$	0	0
$952$	1.25820	0.0407786
$953$	−10.0177	−0.324505	−0.162252	−	0.986749i	$-0.551876\pi$
$953$	−10.0177	−0.324505	−0.162252	+	0.986749i	$0.551876\pi$
$954$	0	0
$955$	14.5743	0.471612
$956$	30.4073	0.983441
$957$	0	0
$958$	0.687058	0.0221978
$959$	−16.4003	−0.529593
$960$	0	0
$961$	−27.9669	−0.902157
$962$	−3.11809	−0.100531
$963$	0	0
$964$	41.9461	1.35099
$965$	−13.3203	−0.428796
$966$	0	0
$967$	6.98983	0.224778	0.112389	−	0.993664i	$-0.464150\pi$
$967$	6.98983	0.224778	0.112389	+	0.993664i	$0.464150\pi$
$968$	0	0
$969$	0	0
$970$	−0.579333	−0.0186013
$971$	−58.1274	−1.86540	−0.932699	−	0.360656i	$-0.882553\pi$
$971$	−58.1274	−1.86540	−0.932699	+	0.360656i	$0.882553\pi$
$972$	0	0
$973$	−18.9801	−0.608473
$974$	−0.383985	−0.0123037
$975$	0	0
$976$	57.8899	1.85301
$977$	3.78047	0.120948	0.0604739	−	0.998170i	$-0.480739\pi$
$977$	3.78047	0.120948	0.0604739	+	0.998170i	$0.480739\pi$
$978$	0	0
$979$	0	0
$980$	2.10920	0.0673759
$981$	0	0
$982$	−3.59449	−0.114705
$983$	−10.2712	−0.327602	−0.163801	−	0.986493i	$-0.552375\pi$
$983$	−10.2712	−0.327602	−0.163801	+	0.986493i	$0.552375\pi$
$984$	0	0
$985$	−2.23420	−0.0711875
$986$	−2.71240	−0.0863803
$987$	0	0
$988$	−41.3493	−1.31550
$989$	24.7951	0.788438
$990$	0	0
$991$	−20.5225	−0.651919	−0.325960	−	0.945384i	$-0.605687\pi$
$991$	−20.5225	−0.651919	−0.325960	+	0.945384i	$0.605687\pi$
$992$	−2.33095	−0.0740078
$993$	0	0
$994$	1.30895	0.0415173
$995$	6.97555	0.221140
$996$	0	0
$997$	−7.37003	−0.233411	−0.116706	−	0.993167i	$-0.537233\pi$
$997$	−7.37003	−0.233411	−0.116706	+	0.993167i	$0.537233\pi$
$998$	−2.46484	−0.0780233
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cy.1.5		10
3.2	odd	2		2541.2.a.br.1.6		10
11.7	odd	10		693.2.m.j.379.3		20
11.8	odd	10		693.2.m.j.64.3		20
11.10	odd	2		7623.2.a.cx.1.6		10
33.8	even	10		231.2.j.g.64.3	✓	20
33.29	even	10		231.2.j.g.148.3	yes	20
33.32	even	2		2541.2.a.bq.1.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.g.64.3	✓	20	33.8	even	10
231.2.j.g.148.3	yes	20	33.29	even	10
693.2.m.j.64.3		20	11.8	odd	10
693.2.m.j.379.3		20	11.7	odd	10
2541.2.a.bq.1.5		10	33.32	even	2
2541.2.a.br.1.6		10	3.2	odd	2
7623.2.a.cx.1.6		10	11.10	odd	2
7623.2.a.cy.1.5		10	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)

\(f(q)\)	\(=\)	\(q-0.112481 q^{2} -1.98735 q^{4} -1.06131 q^{5} +1.00000 q^{7} +0.448501 q^{8} +O(q^{10})\)	\(q-0.112481 q^{2} -1.98735 q^{4} -1.06131 q^{5} +1.00000 q^{7} +0.448501 q^{8} +0.119378 q^{10} +5.84183 q^{13} -0.112481 q^{14} +3.92425 q^{16} +2.80535 q^{17} +3.56160 q^{19} +2.10920 q^{20} -4.72213 q^{23} -3.87361 q^{25} -0.657096 q^{26} -1.98735 q^{28} +8.59581 q^{29} +1.74159 q^{31} -1.33841 q^{32} -0.315549 q^{34} -1.06131 q^{35} +4.74526 q^{37} -0.400613 q^{38} -0.476001 q^{40} +6.13922 q^{41} -5.25083 q^{43} +0.531150 q^{46} +8.19149 q^{47} +1.00000 q^{49} +0.435708 q^{50} -11.6098 q^{52} -14.3391 q^{53} +0.448501 q^{56} -0.966866 q^{58} -3.90570 q^{59} +14.7518 q^{61} -0.195896 q^{62} -7.69795 q^{64} -6.20002 q^{65} +8.29079 q^{67} -5.57521 q^{68} +0.119378 q^{70} -11.6370 q^{71} +4.20049 q^{73} -0.533752 q^{74} -7.07815 q^{76} -6.80421 q^{79} -4.16486 q^{80} -0.690546 q^{82} +1.16511 q^{83} -2.97736 q^{85} +0.590619 q^{86} -2.90784 q^{89} +5.84183 q^{91} +9.38451 q^{92} -0.921388 q^{94} -3.77998 q^{95} -4.85294 q^{97} -0.112481 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8}+O(q^{10}) \) 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8	\( 10 q + 18 q^{4} - 5 q^{5} + 10 q^{7} + 3 q^{8} - 6 q^{10} + 6 q^{13} + 38 q^{16} - 8 q^{17} - 7 q^{20} + 31 q^{25} - q^{26} + 18 q^{28} + 14 q^{29} + 26 q^{31} + 41 q^{32} + 21 q^{34} - 5 q^{35} + 24 q^{37} - 8 q^{38} - 5 q^{40} - 19 q^{41} - 6 q^{43} - q^{46} - 15 q^{47} + 10 q^{49} + q^{50} - 25 q^{52} + q^{53} + 3 q^{56} + 11 q^{58} - 23 q^{59} - 11 q^{62} + 53 q^{64} + 29 q^{65} + 38 q^{67} - 87 q^{68} - 6 q^{70} - 26 q^{71} - q^{73} + 39 q^{74} - 2 q^{76} + 5 q^{79} - 6 q^{80} + 5 q^{82} - 6 q^{83} - q^{85} + 41 q^{86} + 9 q^{89} + 6 q^{91} + 48 q^{92} + 42 q^{95} + 24 q^{97}+O(q^{100}) \) 10 * q + 18 * q^4 - 5 * q^5 + 10 * q^7 + 3 * q^8 - 6 * q^10 + 6 * q^13 + 38 * q^16 - 8 * q^17 - 7 * q^20 + 31 * q^25 - q^26 + 18 * q^28 + 14 * q^29 + 26 * q^31 + 41 * q^32 + 21 * q^34 - 5 * q^35 + 24 * q^37 - 8 * q^38 - 5 * q^40 - 19 * q^41 - 6 * q^43 - q^46 - 15 * q^47 + 10 * q^49 + q^50 - 25 * q^52 + q^53 + 3 * q^56 + 11 * q^58 - 23 * q^59 - 11 * q^62 + 53 * q^64 + 29 * q^65 + 38 * q^67 - 87 * q^68 - 6 * q^70 - 26 * q^71 - q^73 + 39 * q^74 - 2 * q^76 + 5 * q^79 - 6 * q^80 + 5 * q^82 - 6 * q^83 - q^85 + 41 * q^86 + 9 * q^89 + 6 * q^91 + 48 * q^92 + 42 * q^95 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more