LMFDB - Embedded newform 7623.2.a.cq.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$60.8699614608$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.3829849.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 6x^{4} + 15x^{3} - 5x^{2} - 3x + 1 $ x^6 - 2x^5 - 6x^4 + 15x^3 - 5x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.22391$ 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)

$f(q)$	$=$	$q+2.45784 q^{2} +4.04096 q^{4} -2.05098 q^{5} -1.00000 q^{7} +5.01636 q^{8} +O(q^{10})$	$q+2.45784 q^{2} +4.04096 q^{4} -2.05098 q^{5} -1.00000 q^{7} +5.01636 q^{8} -5.04096 q^{10} -0.206501 q^{13} -2.45784 q^{14} +4.24747 q^{16} +0.813723 q^{17} -6.28843 q^{19} -8.28792 q^{20} +5.11704 q^{23} -0.793499 q^{25} -0.507546 q^{26} -4.04096 q^{28} -3.87979 q^{29} +6.49493 q^{31} +0.406862 q^{32} +2.00000 q^{34} +2.05098 q^{35} +1.79350 q^{37} -15.4559 q^{38} -10.2884 q^{40} -1.82882 q^{41} -10.0819 q^{43} +12.5769 q^{46} -8.79547 q^{47} +1.00000 q^{49} -1.95029 q^{50} -0.834464 q^{52} -3.08686 q^{53} -5.01636 q^{56} -9.53590 q^{58} +2.66333 q^{59} -3.58700 q^{61} +15.9635 q^{62} -7.49493 q^{64} +0.423529 q^{65} +10.2884 q^{67} +3.28823 q^{68} +5.04096 q^{70} -15.5607 q^{71} -7.79350 q^{73} +4.40813 q^{74} -25.4113 q^{76} +12.1639 q^{79} -8.71145 q^{80} -4.49493 q^{82} -14.9484 q^{83} -1.66893 q^{85} -24.7797 q^{86} -5.72940 q^{89} +0.206501 q^{91} +20.6778 q^{92} -21.6178 q^{94} +12.8974 q^{95} -8.49493 q^{97} +2.45784 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4} - 6 q^{7}+O(q^{10}) $ 6 * q + 4 * q^4 - 6 * q^7	$ 6 q + 4 q^{4} - 6 q^{7} - 10 q^{10} - 4 q^{13} + 8 q^{16} - 2 q^{25} - 4 q^{28} + 4 q^{31} + 12 q^{34} + 8 q^{37} - 24 q^{40} - 20 q^{43} + 6 q^{49} + 18 q^{52} - 2 q^{58} - 16 q^{61} - 10 q^{64} + 24 q^{67} + 10 q^{70} - 44 q^{73} - 54 q^{76} - 8 q^{79} + 8 q^{82} + 36 q^{85} + 4 q^{91} - 34 q^{94} - 16 q^{97}+O(q^{100}) $ 6 * q + 4 * q^4 - 6 * q^7 - 10 * q^10 - 4 * q^13 + 8 * q^16 - 2 * q^25 - 4 * q^28 + 4 * q^31 + 12 * q^34 + 8 * q^37 - 24 * q^40 - 20 * q^43 + 6 * q^49 + 18 * q^52 - 2 * q^58 - 16 * q^61 - 10 * q^64 + 24 * q^67 + 10 * q^70 - 44 * q^73 - 54 * q^76 - 8 * q^79 + 8 * q^82 + 36 * q^85 + 4 * q^91 - 34 * 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 $

$2$	2.45784	1.73795	0.868977	−	0.494853i	$-0.164778\pi$
$2$	2.45784	1.73795	0.868977	+	0.494853i	$0.164778\pi$
$3$	0	0
$3$	0	0
$4$	4.04096	2.02048
$5$	−2.05098	−0.917224	−0.458612	−	0.888637i	$-0.651653\pi$
$5$	−2.05098	−0.917224	−0.458612	+	0.888637i	$0.651653\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	5.01636	1.77355
$9$	0	0
$10$	−5.04096	−1.59409
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.206501	−0.0572731	−0.0286365	−	0.999590i	$-0.509117\pi$
$13$	−0.206501	−0.0572731	−0.0286365	+	0.999590i	$0.509117\pi$
$14$	−2.45784	−0.656885
$15$	0	0
$16$	4.24747	1.06187
$17$	0.813723	0.197357	0.0986785	−	0.995119i	$-0.468538\pi$
$17$	0.813723	0.197357	0.0986785	+	0.995119i	$0.468538\pi$
$18$	0	0
$19$	−6.28843	−1.44266	−0.721332	−	0.692589i	$-0.756470\pi$
$19$	−6.28843	−1.44266	−0.721332	+	0.692589i	$0.756470\pi$
$20$	−8.28792	−1.85324
$21$	0	0
$22$	0	0
$23$	5.11704	1.06698	0.533489	−	0.845807i	$-0.320881\pi$
$23$	5.11704	1.06698	0.533489	+	0.845807i	$0.320881\pi$
$24$	0	0
$25$	−0.793499	−0.158700
$26$	−0.507546	−0.0995380
$27$	0	0
$28$	−4.04096	−0.763671
$29$	−3.87979	−0.720459	−0.360230	−	0.932864i	$-0.617302\pi$
$29$	−3.87979	−0.720459	−0.360230	+	0.932864i	$0.617302\pi$
$30$	0	0
$31$	6.49493	1.16652	0.583262	−	0.812284i	$-0.301776\pi$
$31$	6.49493	1.16652	0.583262	+	0.812284i	$0.301776\pi$
$32$	0.406862	0.0719237
$33$	0	0
$34$	2.00000	0.342997
$35$	2.05098	0.346678
$36$	0	0
$37$	1.79350	0.294849	0.147425	−	0.989073i	$-0.452902\pi$
$37$	1.79350	0.294849	0.147425	+	0.989073i	$0.452902\pi$
$38$	−15.4559	−2.50728
$39$	0	0
$40$	−10.2884	−1.62674
$41$	−1.82882	−0.285613	−0.142807	−	0.989751i	$-0.545613\pi$
$41$	−1.82882	−0.285613	−0.142807	+	0.989751i	$0.545613\pi$
$42$	0	0
$43$	−10.0819	−1.53748	−0.768740	−	0.639562i	$-0.779116\pi$
$43$	−10.0819	−1.53748	−0.768740	+	0.639562i	$0.779116\pi$
$44$	0	0
$45$	0	0
$46$	12.5769	1.85436
$47$	−8.79547	−1.28295	−0.641475	−	0.767144i	$-0.721677\pi$
$47$	−8.79547	−1.28295	−0.641475	+	0.767144i	$0.721677\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.95029	−0.275813
$51$	0	0
$52$	−0.834464	−0.115719
$53$	−3.08686	−0.424013	−0.212006	−	0.977268i	$-0.568000\pi$
$53$	−3.08686	−0.424013	−0.212006	+	0.977268i	$0.568000\pi$
$54$	0	0
$55$	0	0
$56$	−5.01636	−0.670339
$57$	0	0
$58$	−9.53590	−1.25212
$59$	2.66333	0.346736	0.173368	−	0.984857i	$-0.444535\pi$
$59$	2.66333	0.346736	0.173368	+	0.984857i	$0.444535\pi$
$60$	0	0
$61$	−3.58700	−0.459268	−0.229634	−	0.973277i	$-0.573753\pi$
$61$	−3.58700	−0.459268	−0.229634	+	0.973277i	$0.573753\pi$
$62$	15.9635	2.02736
$63$	0	0
$64$	−7.49493	−0.936866
$65$	0.423529	0.0525323
$66$	0	0
$67$	10.2884	1.25693	0.628466	−	0.777837i	$-0.283683\pi$
$67$	10.2884	1.25693	0.628466	+	0.777837i	$0.283683\pi$
$68$	3.28823	0.398756
$69$	0	0
$70$	5.04096	0.602511
$71$	−15.5607	−1.84672	−0.923361	−	0.383934i	$-0.874569\pi$
$71$	−15.5607	−1.84672	−0.923361	+	0.383934i	$0.874569\pi$
$72$	0	0
$73$	−7.79350	−0.912160	−0.456080	−	0.889939i	$-0.650747\pi$
$73$	−7.79350	−0.912160	−0.456080	+	0.889939i	$0.650747\pi$
$74$	4.40813	0.512435
$75$	0	0
$76$	−25.4113	−2.91488
$77$	0	0
$78$	0	0
$79$	12.1639	1.36854	0.684270	−	0.729228i	$-0.260121\pi$
$79$	12.1639	1.36854	0.684270	+	0.729228i	$0.260121\pi$
$80$	−8.71145	−0.973970
$81$	0	0
$82$	−4.49493	−0.496382
$83$	−14.9484	−1.64080	−0.820400	−	0.571791i	$-0.806249\pi$
$83$	−14.9484	−1.64080	−0.820400	+	0.571791i	$0.806249\pi$
$84$	0	0
$85$	−1.66893	−0.181021
$86$	−24.7797	−2.67207
$87$	0	0
$88$	0	0
$89$	−5.72940	−0.607315	−0.303657	−	0.952781i	$-0.598208\pi$
$89$	−5.72940	−0.607315	−0.303657	+	0.952781i	$0.598208\pi$
$90$	0	0
$91$	0.206501	0.0216472
$92$	20.6778	2.15581
$93$	0	0
$94$	−21.6178	−2.22971
$95$	12.8974	1.32325
$96$	0	0
$97$	−8.49493	−0.862530	−0.431265	−	0.902225i	$-0.641933\pi$
$97$	−8.49493	−0.862530	−0.431265	+	0.902225i	$0.641933\pi$
$98$	2.45784	0.248279
$99$	0	0
$100$	−3.20650	−0.320650
$101$	−10.6451	−1.05922	−0.529612	−	0.848240i	$-0.677663\pi$
$101$	−10.6451	−1.05922	−0.529612	+	0.848240i	$0.677663\pi$
$102$	0	0
$103$	−15.7509	−1.55198	−0.775989	−	0.630746i	$-0.782749\pi$
$103$	−15.7509	−1.55198	−0.775989	+	0.630746i	$0.782749\pi$
$104$	−1.03588	−0.101577
$105$	0	0
$106$	−7.58700	−0.736914
$107$	1.43862	0.139077	0.0695384	−	0.997579i	$-0.477847\pi$
$107$	1.43862	0.139077	0.0695384	+	0.997579i	$0.477847\pi$
$108$	0	0
$109$	2.49493	0.238971	0.119486	−	0.992836i	$-0.461875\pi$
$109$	2.49493	0.238971	0.119486	+	0.992836i	$0.461875\pi$
$110$	0	0
$111$	0	0
$112$	−4.24747	−0.401348
$113$	17.5909	1.65482	0.827408	−	0.561602i	$-0.189815\pi$
$113$	17.5909	1.65482	0.827408	+	0.561602i	$0.189815\pi$
$114$	0	0
$115$	−10.4949	−0.978657
$116$	−15.6781	−1.45567
$117$	0	0
$118$	6.54603	0.602611
$119$	−0.813723	−0.0745939
$120$	0	0
$121$	0	0
$122$	−8.81626	−0.798186
$123$	0	0
$124$	26.2458	2.35694
$125$	11.8823	1.06279
$126$	0	0
$127$	−10.9079	−0.967923	−0.483961	−	0.875089i	$-0.660802\pi$
$127$	−10.9079	−0.967923	−0.483961	+	0.875089i	$0.660802\pi$
$128$	−19.2350	−1.70015
$129$	0	0
$130$	1.04096	0.0912986
$131$	5.72940	0.500580	0.250290	−	0.968171i	$-0.419474\pi$
$131$	5.72940	0.500580	0.250290	+	0.968171i	$0.419474\pi$
$132$	0	0
$133$	6.28843	0.545276
$134$	25.2873	2.18449
$135$	0	0
$136$	4.08193	0.350023
$137$	−6.34175	−0.541813	−0.270906	−	0.962606i	$-0.587323\pi$
$137$	−6.34175	−0.541813	−0.270906	+	0.962606i	$0.587323\pi$
$138$	0	0
$139$	4.16386	0.353174	0.176587	−	0.984285i	$-0.443494\pi$
$139$	4.16386	0.353174	0.176587	+	0.984285i	$0.443494\pi$
$140$	8.28792	0.700457
$141$	0	0
$142$	−38.2458	−3.20952
$143$	0	0
$144$	0	0
$145$	7.95736	0.660823
$146$	−19.1552	−1.58529
$147$	0	0
$148$	7.24747	0.595738
$149$	16.1856	1.32598	0.662990	−	0.748628i	$-0.269287\pi$
$149$	16.1856	1.32598	0.662990	+	0.748628i	$0.269287\pi$
$150$	0	0
$151$	−1.66893	−0.135815	−0.0679077	−	0.997692i	$-0.521632\pi$
$151$	−1.66893	−0.135815	−0.0679077	+	0.997692i	$0.521632\pi$
$152$	−31.5450	−2.55864
$153$	0	0
$154$	0	0
$155$	−13.3209	−1.06996
$156$	0	0
$157$	−18.5769	−1.48259	−0.741297	−	0.671177i	$-0.765789\pi$
$157$	−18.5769	−1.48259	−0.741297	+	0.671177i	$0.765789\pi$
$158$	29.8968	2.37846
$159$	0	0
$160$	−0.834464	−0.0659701
$161$	−5.11704	−0.403280
$162$	0	0
$163$	10.2884	0.805852	0.402926	−	0.915233i	$-0.367993\pi$
$163$	10.2884	0.805852	0.402926	+	0.915233i	$0.367993\pi$
$164$	−7.39018	−0.577076
$165$	0	0
$166$	−36.7407	−2.85163
$167$	−23.5550	−1.82274	−0.911372	−	0.411585i	$-0.864975\pi$
$167$	−23.5550	−1.82274	−0.911372	+	0.411585i	$0.864975\pi$
$168$	0	0
$169$	−12.9574	−0.996720
$170$	−4.10195	−0.314605
$171$	0	0
$172$	−40.7407	−3.10645
$173$	5.93077	0.450908	0.225454	−	0.974254i	$-0.427613\pi$
$173$	5.93077	0.450908	0.225454	+	0.974254i	$0.427613\pi$
$174$	0	0
$175$	0.793499	0.0599829
$176$	0	0
$177$	0	0
$178$	−14.0819	−1.05549
$179$	16.4078	1.22638	0.613188	−	0.789937i	$-0.289887\pi$
$179$	16.4078	1.22638	0.613188	+	0.789937i	$0.289887\pi$
$180$	0	0
$181$	−3.50507	−0.260530	−0.130265	−	0.991479i	$-0.541583\pi$
$181$	−3.50507	−0.260530	−0.130265	+	0.991479i	$0.541583\pi$
$182$	0.507546	0.0376218
$183$	0	0
$184$	25.6689	1.89234
$185$	−3.67842	−0.270443
$186$	0	0
$187$	0	0
$188$	−35.5422	−2.59218
$189$	0	0
$190$	31.6998	2.29974
$191$	18.6476	1.34929	0.674647	−	0.738141i	$-0.264296\pi$
$191$	18.6476	1.34929	0.674647	+	0.738141i	$0.264296\pi$
$192$	0	0
$193$	−5.50507	−0.396264	−0.198132	−	0.980175i	$-0.563487\pi$
$193$	−5.50507	−0.396264	−0.198132	+	0.980175i	$0.563487\pi$
$194$	−20.8792	−1.49904
$195$	0	0
$196$	4.04096	0.288640
$197$	18.2366	1.29931	0.649653	−	0.760231i	$-0.274914\pi$
$197$	18.2366	1.29931	0.649653	+	0.760231i	$0.274914\pi$
$198$	0	0
$199$	−6.90793	−0.489690	−0.244845	−	0.969562i	$-0.578737\pi$
$199$	−6.90793	−0.489690	−0.244845	+	0.969562i	$0.578737\pi$
$200$	−3.98048	−0.281462
$201$	0	0
$202$	−26.1639	−1.84088
$203$	3.87979	0.272308
$204$	0	0
$205$	3.75086	0.261971
$206$	−38.7130	−2.69727
$207$	0	0
$208$	−0.877106	−0.0608164
$209$	0	0
$210$	0	0
$211$	−7.17400	−0.493878	−0.246939	−	0.969031i	$-0.579425\pi$
$211$	−7.17400	−0.493878	−0.246939	+	0.969031i	$0.579425\pi$
$212$	−12.4739	−0.856710
$213$	0	0
$214$	3.53590	0.241709
$215$	20.6778	1.41021
$216$	0	0
$217$	−6.49493	−0.440905
$218$	6.13214	0.415321
$219$	0	0
$220$	0	0
$221$	−0.168035	−0.0113032
$222$	0	0
$223$	8.16386	0.546692	0.273346	−	0.961916i	$-0.411870\pi$
$223$	8.16386	0.546692	0.273346	+	0.961916i	$0.411870\pi$
$224$	−0.406862	−0.0271846
$225$	0	0
$226$	43.2357	2.87599
$227$	−3.69921	−0.245525	−0.122763	−	0.992436i	$-0.539175\pi$
$227$	−3.69921	−0.245525	−0.122763	+	0.992436i	$0.539175\pi$
$228$	0	0
$229$	−20.2458	−1.33788	−0.668940	−	0.743317i	$-0.733252\pi$
$229$	−20.2458	−1.33788	−0.668940	+	0.743317i	$0.733252\pi$
$230$	−25.7948	−1.70086
$231$	0	0
$232$	−19.4624	−1.27777
$233$	13.6903	0.896885	0.448442	−	0.893812i	$-0.351979\pi$
$233$	13.6903	0.896885	0.448442	+	0.893812i	$0.351979\pi$
$234$	0	0
$235$	18.0393	1.17675
$236$	10.7624	0.700574
$237$	0	0
$238$	−2.00000	−0.129641
$239$	16.5966	1.07355	0.536773	−	0.843726i	$-0.319643\pi$
$239$	16.5966	1.07355	0.536773	+	0.843726i	$0.319643\pi$
$240$	0	0
$241$	11.9574	0.770241	0.385121	−	0.922866i	$-0.374160\pi$
$241$	11.9574	0.770241	0.385121	+	0.922866i	$0.374160\pi$
$242$	0	0
$243$	0	0
$244$	−14.4949	−0.927943
$245$	−2.05098	−0.131032
$246$	0	0
$247$	1.29857	0.0826259
$248$	32.5809	2.06889
$249$	0	0
$250$	29.2048	1.84708
$251$	5.54057	0.349718	0.174859	−	0.984594i	$-0.444053\pi$
$251$	5.54057	0.349718	0.174859	+	0.984594i	$0.444053\pi$
$252$	0	0
$253$	0	0
$254$	−26.8099	−1.68220
$255$	0	0
$256$	−32.2868	−2.01792
$257$	27.8458	1.73697	0.868487	−	0.495712i	$-0.165093\pi$
$257$	27.8458	1.73697	0.868487	+	0.495712i	$0.165093\pi$
$258$	0	0
$259$	−1.79350	−0.111443
$260$	1.71146	0.106141
$261$	0	0
$262$	14.0819	0.869984
$263$	−7.16802	−0.441999	−0.220999	−	0.975274i	$-0.570932\pi$
$263$	−7.16802	−0.441999	−0.220999	+	0.975274i	$0.570932\pi$
$264$	0	0
$265$	6.33107	0.388915
$266$	15.4559	0.947664
$267$	0	0
$268$	41.5752	2.53961
$269$	−11.9031	−0.725746	−0.362873	−	0.931839i	$-0.618204\pi$
$269$	−11.9031	−0.725746	−0.362873	+	0.931839i	$0.618204\pi$
$270$	0	0
$271$	−30.2884	−1.83989	−0.919946	−	0.392046i	$-0.871767\pi$
$271$	−30.2884	−1.83989	−0.919946	+	0.392046i	$0.871767\pi$
$272$	3.45626	0.209567
$273$	0	0
$274$	−15.5870	−0.941645
$275$	0	0
$276$	0	0
$277$	−28.5769	−1.71702	−0.858509	−	0.512799i	$-0.828609\pi$
$277$	−28.5769	−1.71702	−0.858509	+	0.512799i	$0.828609\pi$
$278$	10.2341	0.613800
$279$	0	0
$280$	10.2884	0.614851
$281$	4.72685	0.281980	0.140990	−	0.990011i	$-0.454971\pi$
$281$	4.72685	0.281980	0.140990	+	0.990011i	$0.454971\pi$
$282$	0	0
$283$	−10.4523	−0.621324	−0.310662	−	0.950520i	$-0.600551\pi$
$283$	−10.4523	−0.621324	−0.310662	+	0.950520i	$0.600551\pi$
$284$	−62.8804	−3.73127
$285$	0	0
$286$	0	0
$287$	1.82882	0.107952
$288$	0	0
$289$	−16.3379	−0.961050
$290$	19.5579	1.14848
$291$	0	0
$292$	−31.4933	−1.84300
$293$	32.1699	1.87939	0.939693	−	0.342018i	$-0.111110\pi$
$293$	32.1699	1.87939	0.939693	+	0.342018i	$0.111110\pi$
$294$	0	0
$295$	−5.46243	−0.318035
$296$	8.99683	0.522930
$297$	0	0
$298$	39.7817	2.30449
$299$	−1.05667	−0.0611091
$300$	0	0
$301$	10.0819	0.581113
$302$	−4.10195	−0.236041
$303$	0	0
$304$	−26.7099	−1.53192
$305$	7.35685	0.421252
$306$	0	0
$307$	24.3277	1.38846	0.694228	−	0.719755i	$-0.255746\pi$
$307$	24.3277	1.38846	0.694228	+	0.719755i	$0.255746\pi$
$308$	0	0
$309$	0	0
$310$	−32.7407	−1.85955
$311$	21.6929	1.23009	0.615045	−	0.788492i	$-0.289138\pi$
$311$	21.6929	1.23009	0.615045	+	0.788492i	$0.289138\pi$
$312$	0	0
$313$	−7.25592	−0.410129	−0.205065	−	0.978748i	$-0.565740\pi$
$313$	−7.25592	−0.410129	−0.205065	+	0.978748i	$0.565740\pi$
$314$	−45.6589	−2.57668
$315$	0	0
$316$	49.1537	2.76511
$317$	30.9119	1.73618	0.868092	−	0.496403i	$-0.165346\pi$
$317$	30.9119	1.73618	0.868092	+	0.496403i	$0.165346\pi$
$318$	0	0
$319$	0	0
$320$	15.3719	0.859317
$321$	0	0
$322$	−12.5769	−0.700881
$323$	−5.11704	−0.284720
$324$	0	0
$325$	0.163858	0.00908923
$326$	25.2873	1.40053
$327$	0	0
$328$	−9.17400	−0.506549
$329$	8.79547	0.484910
$330$	0	0
$331$	20.3277	1.11731	0.558656	−	0.829399i	$-0.311317\pi$
$331$	20.3277	1.11731	0.558656	+	0.829399i	$0.311317\pi$
$332$	−60.4059	−3.31521
$333$	0	0
$334$	−57.8944	−3.16784
$335$	−21.1013	−1.15289
$336$	0	0
$337$	26.8226	1.46112	0.730561	−	0.682847i	$-0.239258\pi$
$337$	26.8226	1.46112	0.730561	+	0.682847i	$0.239258\pi$
$338$	−31.8471	−1.73225
$339$	0	0
$340$	−6.74408	−0.365749
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−50.5746	−2.72680
$345$	0	0
$346$	14.5769	0.783657
$347$	−24.9478	−1.33927	−0.669633	−	0.742692i	$-0.733549\pi$
$347$	−24.9478	−1.33927	−0.669633	+	0.742692i	$0.733549\pi$
$348$	0	0
$349$	19.9574	1.06829	0.534146	−	0.845392i	$-0.320633\pi$
$349$	19.9574	1.06829	0.534146	+	0.845392i	$0.320633\pi$
$350$	1.95029	0.104247
$351$	0	0
$352$	0	0
$353$	−2.89803	−0.154247	−0.0771234	−	0.997022i	$-0.524574\pi$
$353$	−2.89803	−0.154247	−0.0771234	+	0.997022i	$0.524574\pi$
$354$	0	0
$355$	31.9147	1.69386
$356$	−23.1523	−1.22707
$357$	0	0
$358$	40.3277	2.13139
$359$	−26.2392	−1.38485	−0.692425	−	0.721490i	$-0.743458\pi$
$359$	−26.2392	−1.38485	−0.692425	+	0.721490i	$0.743458\pi$
$360$	0	0
$361$	20.5444	1.08128
$362$	−8.61489	−0.452789
$363$	0	0
$364$	0.834464	0.0437378
$365$	15.9843	0.836655
$366$	0	0
$367$	23.3379	1.21823	0.609113	−	0.793083i	$-0.291526\pi$
$367$	23.3379	1.21823	0.609113	+	0.793083i	$0.291526\pi$
$368$	21.7345	1.13299
$369$	0	0
$370$	−9.04096	−0.470017
$371$	3.08686	0.160262
$372$	0	0
$373$	2.08193	0.107798	0.0538991	−	0.998546i	$-0.482835\pi$
$373$	2.08193	0.107798	0.0538991	+	0.998546i	$0.482835\pi$
$374$	0	0
$375$	0	0
$376$	−44.1212	−2.27538
$377$	0.801181	0.0412629
$378$	0	0
$379$	−6.45229	−0.331432	−0.165716	−	0.986174i	$-0.552993\pi$
$379$	−6.45229	−0.331432	−0.165716	+	0.986174i	$0.552993\pi$
$380$	52.1180	2.67360
$381$	0	0
$382$	45.8328	2.34501
$383$	−6.97919	−0.356620	−0.178310	−	0.983974i	$-0.557063\pi$
$383$	−6.97919	−0.356620	−0.178310	+	0.983974i	$0.557063\pi$
$384$	0	0
$385$	0	0
$386$	−13.5306	−0.688688
$387$	0	0
$388$	−34.3277	−1.74273
$389$	−8.16232	−0.413846	−0.206923	−	0.978357i	$-0.566345\pi$
$389$	−8.16232	−0.413846	−0.206923	+	0.978357i	$0.566345\pi$
$390$	0	0
$391$	4.16386	0.210575
$392$	5.01636	0.253364
$393$	0	0
$394$	44.8226	2.25813
$395$	−24.9478	−1.25526
$396$	0	0
$397$	−27.4198	−1.37616	−0.688080	−	0.725635i	$-0.741546\pi$
$397$	−27.4198	−1.37616	−0.688080	+	0.725635i	$0.741546\pi$
$398$	−16.9786	−0.851059
$399$	0	0
$400$	−3.37036	−0.168518
$401$	−23.9327	−1.19514	−0.597571	−	0.801816i	$-0.703867\pi$
$401$	−23.9327	−1.19514	−0.597571	+	0.801816i	$0.703867\pi$
$402$	0	0
$403$	−1.34121	−0.0668104
$404$	−43.0164	−2.14014
$405$	0	0
$406$	9.53590	0.473259
$407$	0	0
$408$	0	0
$409$	−23.7509	−1.17440	−0.587202	−	0.809440i	$-0.699771\pi$
$409$	−23.7509	−1.17440	−0.587202	+	0.809440i	$0.699771\pi$
$410$	9.21899	0.455294
$411$	0	0
$412$	−63.6487	−3.13574
$413$	−2.66333	−0.131054
$414$	0	0
$415$	30.6588	1.50498
$416$	−0.0840174	−0.00411929
$417$	0	0
$418$	0	0
$419$	22.2844	1.08867	0.544333	−	0.838869i	$-0.316783\pi$
$419$	22.2844	1.08867	0.544333	+	0.838869i	$0.316783\pi$
$420$	0	0
$421$	9.38050	0.457177	0.228589	−	0.973523i	$-0.426589\pi$
$421$	9.38050	0.457177	0.228589	+	0.973523i	$0.426589\pi$
$422$	−17.6325	−0.858337
$423$	0	0
$424$	−15.4848	−0.752008
$425$	−0.645689	−0.0313205
$426$	0	0
$427$	3.58700	0.173587
$428$	5.81342	0.281002
$429$	0	0
$430$	50.8226	2.45089
$431$	−5.94331	−0.286279	−0.143140	−	0.989703i	$-0.545720\pi$
$431$	−5.94331	−0.286279	−0.143140	+	0.989703i	$0.545720\pi$
$432$	0	0
$433$	−5.42314	−0.260619	−0.130310	−	0.991473i	$-0.541597\pi$
$433$	−5.42314	−0.260619	−0.130310	+	0.991473i	$0.541597\pi$
$434$	−15.9635	−0.766272
$435$	0	0
$436$	10.0819	0.482837
$437$	−32.1782	−1.53929
$438$	0	0
$439$	26.0393	1.24279	0.621394	−	0.783499i	$-0.286567\pi$
$439$	26.0393	1.24279	0.621394	+	0.783499i	$0.286567\pi$
$440$	0	0
$441$	0	0
$442$	−0.413002	−0.0196445
$443$	7.35685	0.349534	0.174767	−	0.984610i	$-0.444083\pi$
$443$	7.35685	0.349534	0.174767	+	0.984610i	$0.444083\pi$
$444$	0	0
$445$	11.7509	0.557044
$446$	20.0654	0.950126
$447$	0	0
$448$	7.49493	0.354102
$449$	−11.4172	−0.538812	−0.269406	−	0.963027i	$-0.586827\pi$
$449$	−11.4172	−0.538812	−0.269406	+	0.963027i	$0.586827\pi$
$450$	0	0
$451$	0	0
$452$	71.0843	3.34353
$453$	0	0
$454$	−9.09207	−0.426712
$455$	−0.423529	−0.0198553
$456$	0	0
$457$	−11.3209	−0.529571	−0.264786	−	0.964307i	$-0.585301\pi$
$457$	−11.3209	−0.529571	−0.264786	+	0.964307i	$0.585301\pi$
$458$	−49.7609	−2.32517
$459$	0	0
$460$	−42.4096	−1.97736
$461$	−0.604106	−0.0281360	−0.0140680	−	0.999901i	$-0.504478\pi$
$461$	−0.604106	−0.0281360	−0.0140680	+	0.999901i	$0.504478\pi$
$462$	0	0
$463$	−34.8653	−1.62033	−0.810164	−	0.586204i	$-0.800622\pi$
$463$	−34.8653	−1.62033	−0.810164	+	0.586204i	$0.800622\pi$
$464$	−16.4793	−0.765031
$465$	0	0
$466$	33.6487	1.55874
$467$	−28.4582	−1.31689	−0.658443	−	0.752630i	$-0.728785\pi$
$467$	−28.4582	−1.31689	−0.658443	+	0.752630i	$0.728785\pi$
$468$	0	0
$469$	−10.2884	−0.475076
$470$	44.3376	2.04514
$471$	0	0
$472$	13.3602	0.614954
$473$	0	0
$474$	0	0
$475$	4.98986	0.228951
$476$	−3.28823	−0.150716
$477$	0	0
$478$	40.7918	1.86577
$479$	30.1064	1.37560	0.687798	−	0.725902i	$-0.258577\pi$
$479$	30.1064	1.37560	0.687798	+	0.725902i	$0.258577\pi$
$480$	0	0
$481$	−0.370359	−0.0168869
$482$	29.3892	1.33864
$483$	0	0
$484$	0	0
$485$	17.4229	0.791133
$486$	0	0
$487$	−33.1537	−1.50234	−0.751169	−	0.660110i	$-0.770510\pi$
$487$	−33.1537	−1.50234	−0.751169	+	0.660110i	$0.770510\pi$
$488$	−17.9937	−0.814535
$489$	0	0
$490$	−5.04096	−0.227728
$491$	−10.8257	−0.488555	−0.244277	−	0.969705i	$-0.578551\pi$
$491$	−10.8257	−0.488555	−0.244277	+	0.969705i	$0.578551\pi$
$492$	0	0
$493$	−3.15708	−0.142188
$494$	3.19167	0.143600
$495$	0	0
$496$	27.5870	1.23869
$497$	15.5607	0.697995
$498$	0	0
$499$	38.8653	1.73985	0.869925	−	0.493185i	$-0.164167\pi$
$499$	38.8653	1.73985	0.869925	+	0.493185i	$0.164167\pi$
$500$	48.0161	2.14734
$501$	0	0
$502$	13.6178	0.607793
$503$	10.4437	0.465662	0.232831	−	0.972517i	$-0.425201\pi$
$503$	10.4437	0.465662	0.232831	+	0.972517i	$0.425201\pi$
$504$	0	0
$505$	21.8328	0.971546
$506$	0	0
$507$	0	0
$508$	−44.0786	−1.95567
$509$	41.7999	1.85275	0.926374	−	0.376605i	$-0.122908\pi$
$509$	41.7999	1.85275	0.926374	+	0.376605i	$0.122908\pi$
$510$	0	0
$511$	7.79350	0.344764
$512$	−40.8855	−1.80690
$513$	0	0
$514$	68.4405	3.01878
$515$	32.3046	1.42351
$516$	0	0
$517$	0	0
$518$	−4.40813	−0.193682
$519$	0	0
$520$	2.12457	0.0931686
$521$	−27.8874	−1.22177	−0.610884	−	0.791720i	$-0.709186\pi$
$521$	−27.8874	−1.22177	−0.610884	+	0.791720i	$0.709186\pi$
$522$	0	0
$523$	30.6161	1.33875	0.669375	−	0.742924i	$-0.266562\pi$
$523$	30.6161	1.33875	0.669375	+	0.742924i	$0.266562\pi$
$524$	23.1523	1.01141
$525$	0	0
$526$	−17.6178	−0.768174
$527$	5.28508	0.230222
$528$	0	0
$529$	3.18413	0.138441
$530$	15.5607	0.675916
$531$	0	0
$532$	25.4113	1.10172
$533$	0.377652	0.0163579
$534$	0	0
$535$	−2.95058	−0.127565
$536$	51.6105	2.22923
$537$	0	0
$538$	−29.2559	−1.26131
$539$	0	0
$540$	0	0
$541$	16.7407	0.719740	0.359870	−	0.933003i	$-0.382821\pi$
$541$	16.7407	0.719740	0.359870	+	0.933003i	$0.382821\pi$
$542$	−74.4440	−3.19765
$543$	0	0
$544$	0.331073	0.0141946
$545$	−5.11704	−0.219190
$546$	0	0
$547$	−21.4029	−0.915120	−0.457560	−	0.889179i	$-0.651277\pi$
$547$	−21.4029	−0.915120	−0.457560	+	0.889179i	$0.651277\pi$
$548$	−25.6268	−1.09472
$549$	0	0
$550$	0	0
$551$	24.3978	1.03938
$552$	0	0
$553$	−12.1639	−0.517260
$554$	−70.2373	−2.98410
$555$	0	0
$556$	16.8260	0.713582
$557$	−7.53742	−0.319371	−0.159685	−	0.987168i	$-0.551048\pi$
$557$	−7.53742	−0.319371	−0.159685	+	0.987168i	$0.551048\pi$
$558$	0	0
$559$	2.08193	0.0880562
$560$	8.71145	0.368126
$561$	0	0
$562$	11.6178	0.490068
$563$	−14.9900	−0.631752	−0.315876	−	0.948800i	$-0.602298\pi$
$563$	−14.9900	−0.631752	−0.315876	+	0.948800i	$0.602298\pi$
$564$	0	0
$565$	−36.0786	−1.51784
$566$	−25.6900	−1.07983
$567$	0	0
$568$	−78.0583	−3.27525
$569$	−17.8339	−0.747635	−0.373818	−	0.927502i	$-0.621951\pi$
$569$	−17.8339	−0.747635	−0.373818	+	0.927502i	$0.621951\pi$
$570$	0	0
$571$	16.4130	0.686863	0.343431	−	0.939178i	$-0.388411\pi$
$571$	16.4130	0.686863	0.343431	+	0.939178i	$0.388411\pi$
$572$	0	0
$573$	0	0
$574$	4.49493	0.187615
$575$	−4.06037	−0.169329
$576$	0	0
$577$	25.2357	1.05057	0.525287	−	0.850925i	$-0.323958\pi$
$577$	25.2357	1.05057	0.525287	+	0.850925i	$0.323958\pi$
$578$	−40.1558	−1.67026
$579$	0	0
$580$	32.1554	1.33518
$581$	14.9484	0.620164
$582$	0	0
$583$	0	0
$584$	−39.0950	−1.61776
$585$	0	0
$586$	79.0684	3.26629
$587$	30.4883	1.25839	0.629194	−	0.777248i	$-0.283385\pi$
$587$	30.4883	1.25839	0.629194	+	0.777248i	$0.283385\pi$
$588$	0	0
$589$	−40.8429	−1.68290
$590$	−13.4258	−0.552730
$591$	0	0
$592$	7.61783	0.313091
$593$	36.2719	1.48951	0.744754	−	0.667340i	$-0.232567\pi$
$593$	36.2719	1.48951	0.744754	+	0.667340i	$0.232567\pi$
$594$	0	0
$595$	1.66893	0.0684193
$596$	65.4056	2.67912
$597$	0	0
$598$	−2.59714	−0.106205
$599$	−2.03018	−0.0829511	−0.0414755	−	0.999140i	$-0.513206\pi$
$599$	−2.03018	−0.0829511	−0.0414755	+	0.999140i	$0.513206\pi$
$600$	0	0
$601$	−28.7834	−1.17410	−0.587049	−	0.809551i	$-0.699710\pi$
$601$	−28.7834	−1.17410	−0.587049	+	0.809551i	$0.699710\pi$
$602$	24.7797	1.00995
$603$	0	0
$604$	−6.74408	−0.274413
$605$	0	0
$606$	0	0
$607$	−23.4422	−0.951488	−0.475744	−	0.879584i	$-0.657821\pi$
$607$	−23.4422	−0.951488	−0.475744	+	0.879584i	$0.657821\pi$
$608$	−2.55852	−0.103762
$609$	0	0
$610$	18.0819	0.732116
$611$	1.81627	0.0734785
$612$	0	0
$613$	−30.2458	−1.22162	−0.610808	−	0.791779i	$-0.709155\pi$
$613$	−30.2458	−1.22162	−0.610808	+	0.791779i	$0.709155\pi$
$614$	59.7936	2.41307
$615$	0	0
$616$	0	0
$617$	29.8968	1.20360	0.601800	−	0.798647i	$-0.294451\pi$
$617$	29.8968	1.20360	0.601800	+	0.798647i	$0.294451\pi$
$618$	0	0
$619$	23.0718	0.927334	0.463667	−	0.886010i	$-0.346533\pi$
$619$	23.0718	0.927334	0.463667	+	0.886010i	$0.346533\pi$
$620$	−53.8295	−2.16184
$621$	0	0
$622$	53.3176	2.13784
$623$	5.72940	0.229543
$624$	0	0
$625$	−20.4029	−0.816115
$626$	−17.8339	−0.712785
$627$	0	0
$628$	−75.0684	−2.99556
$629$	1.45941	0.0581906
$630$	0	0
$631$	−24.3277	−0.968471	−0.484236	−	0.874938i	$-0.660902\pi$
$631$	−24.3277	−0.968471	−0.484236	+	0.874938i	$0.660902\pi$
$632$	61.0183	2.42718
$633$	0	0
$634$	75.9764	3.01741
$635$	22.3719	0.887802
$636$	0	0
$637$	−0.206501	−0.00818187
$638$	0	0
$639$	0	0
$640$	39.4506	1.55942
$641$	−0.168035	−0.00663697	−0.00331849	−	0.999994i	$-0.501056\pi$
$641$	−0.168035	−0.00663697	−0.00331849	+	0.999994i	$0.501056\pi$
$642$	0	0
$643$	11.2357	0.443091	0.221545	−	0.975150i	$-0.428890\pi$
$643$	11.2357	0.443091	0.221545	+	0.975150i	$0.428890\pi$
$644$	−20.6778	−0.814819
$645$	0	0
$646$	−12.5769	−0.494830
$647$	33.4072	1.31337	0.656686	−	0.754164i	$-0.271958\pi$
$647$	33.4072	1.31337	0.656686	+	0.754164i	$0.271958\pi$
$648$	0	0
$649$	0	0
$650$	0.402737	0.0157967
$651$	0	0
$652$	41.5752	1.62821
$653$	4.94901	0.193670	0.0968348	−	0.995300i	$-0.469128\pi$
$653$	4.94901	0.193670	0.0968348	+	0.995300i	$0.469128\pi$
$654$	0	0
$655$	−11.7509	−0.459144
$656$	−7.76783	−0.303283
$657$	0	0
$658$	21.6178	0.842751
$659$	12.8974	0.502412	0.251206	−	0.967934i	$-0.419173\pi$
$659$	12.8974	0.502412	0.251206	+	0.967934i	$0.419173\pi$
$660$	0	0
$661$	25.8978	1.00731	0.503654	−	0.863906i	$-0.331989\pi$
$661$	25.8978	1.00731	0.503654	+	0.863906i	$0.331989\pi$
$662$	49.9622	1.94184
$663$	0	0
$664$	−74.9865	−2.91004
$665$	−12.8974	−0.500140
$666$	0	0
$667$	−19.8531	−0.768714
$668$	−95.1851	−3.68282
$669$	0	0
$670$	−51.8636	−2.00367
$671$	0	0
$672$	0	0
$673$	−47.0718	−1.81448	−0.907242	−	0.420609i	$-0.861817\pi$
$673$	−47.0718	−1.81448	−0.907242	+	0.420609i	$0.861817\pi$
$674$	65.9257	2.53936
$675$	0	0
$676$	−52.3602	−2.01385
$677$	5.97235	0.229536	0.114768	−	0.993392i	$-0.463388\pi$
$677$	5.97235	0.229536	0.114768	+	0.993392i	$0.463388\pi$
$678$	0	0
$679$	8.49493	0.326006
$680$	−8.37194	−0.321049
$681$	0	0
$682$	0	0
$683$	−4.26999	−0.163386	−0.0816932	−	0.996658i	$-0.526033\pi$
$683$	−4.26999	−0.163386	−0.0816932	+	0.996658i	$0.526033\pi$
$684$	0	0
$685$	13.0068	0.496964
$686$	−2.45784	−0.0938407
$687$	0	0
$688$	−42.8226	−1.63260
$689$	0.637440	0.0242845
$690$	0	0
$691$	−19.3379	−0.735647	−0.367823	−	0.929896i	$-0.619897\pi$
$691$	−19.3379	−0.735647	−0.367823	+	0.929896i	$0.619897\pi$
$692$	23.9660	0.911051
$693$	0	0
$694$	−61.3176	−2.32758
$695$	−8.53997	−0.323940
$696$	0	0
$697$	−1.48815	−0.0563677
$698$	49.0519	1.85664
$699$	0	0
$700$	3.20650	0.121194
$701$	22.3135	0.842769	0.421384	−	0.906882i	$-0.361544\pi$
$701$	22.3135	0.842769	0.421384	+	0.906882i	$0.361544\pi$
$702$	0	0
$703$	−11.2783	−0.425369
$704$	0	0
$705$	0	0
$706$	−7.12289	−0.268074
$707$	10.6451	0.400349
$708$	0	0
$709$	15.0325	0.564558	0.282279	−	0.959332i	$-0.408910\pi$
$709$	15.0325	0.564558	0.282279	+	0.959332i	$0.408910\pi$
$710$	78.4412	2.94384
$711$	0	0
$712$	−28.7407	−1.07710
$713$	33.2348	1.24465
$714$	0	0
$715$	0	0
$716$	66.3034	2.47787
$717$	0	0
$718$	−64.4916	−2.40680
$719$	−20.2958	−0.756907	−0.378454	−	0.925620i	$-0.623544\pi$
$719$	−20.2958	−0.756907	−0.378454	+	0.925620i	$0.623544\pi$
$720$	0	0
$721$	15.7509	0.586593
$722$	50.4947	1.87922
$723$	0	0
$724$	−14.1639	−0.526396
$725$	3.07861	0.114337
$726$	0	0
$727$	−26.9865	−1.00087	−0.500437	−	0.865773i	$-0.666827\pi$
$727$	−26.9865	−1.00087	−0.500437	+	0.865773i	$0.666827\pi$
$728$	1.03588	0.0383924
$729$	0	0
$730$	39.2868	1.45407
$731$	−8.20390	−0.303432
$732$	0	0
$733$	4.41300	0.162998	0.0814990	−	0.996673i	$-0.474029\pi$
$733$	4.41300	0.162998	0.0814990	+	0.996673i	$0.474029\pi$
$734$	57.3607	2.11722
$735$	0	0
$736$	2.08193	0.0767409
$737$	0	0
$738$	0	0
$739$	36.1808	1.33093	0.665466	−	0.746428i	$-0.268233\pi$
$739$	36.1808	1.33093	0.665466	+	0.746428i	$0.268233\pi$
$740$	−14.8644	−0.546425
$741$	0	0
$742$	7.58700	0.278527
$743$	−22.7703	−0.835363	−0.417682	−	0.908593i	$-0.637157\pi$
$743$	−22.7703	−0.835363	−0.417682	+	0.908593i	$0.637157\pi$
$744$	0	0
$745$	−33.1964	−1.21622
$746$	5.11704	0.187348
$747$	0	0
$748$	0	0
$749$	−1.43862	−0.0525661
$750$	0	0
$751$	−15.0291	−0.548421	−0.274211	−	0.961670i	$-0.588417\pi$
$751$	−15.0291	−0.548421	−0.274211	+	0.961670i	$0.588417\pi$
$752$	−37.3584	−1.36232
$753$	0	0
$754$	1.96917	0.0717130
$755$	3.42293	0.124573
$756$	0	0
$757$	23.6094	0.858097	0.429048	−	0.903281i	$-0.358849\pi$
$757$	23.6094	0.858097	0.429048	+	0.903281i	$0.358849\pi$
$758$	−15.8587	−0.576013
$759$	0	0
$760$	64.6981	2.34685
$761$	−50.9440	−1.84672	−0.923359	−	0.383938i	$-0.874568\pi$
$761$	−50.9440	−1.84672	−0.923359	+	0.383938i	$0.874568\pi$
$762$	0	0
$763$	−2.49493	−0.0903226
$764$	75.3543	2.72622
$765$	0	0
$766$	−17.1537	−0.619789
$767$	−0.549981	−0.0198586
$768$	0	0
$769$	−17.3602	−0.626026	−0.313013	−	0.949749i	$-0.601338\pi$
$769$	−17.3602	−0.626026	−0.313013	+	0.949749i	$0.601338\pi$
$770$	0	0
$771$	0	0
$772$	−22.2458	−0.800643
$773$	35.2442	1.26765	0.633824	−	0.773478i	$-0.281485\pi$
$773$	35.2442	1.26765	0.633824	+	0.773478i	$0.281485\pi$
$774$	0	0
$775$	−5.15372	−0.185127
$776$	−42.6136	−1.52974
$777$	0	0
$778$	−20.0617	−0.719245
$779$	11.5004	0.412044
$780$	0	0
$781$	0	0
$782$	10.2341	0.365970
$783$	0	0
$784$	4.24747	0.151695
$785$	38.1007	1.35987
$786$	0	0
$787$	−29.8754	−1.06494	−0.532472	−	0.846448i	$-0.678737\pi$
$787$	−29.8754	−1.06494	−0.532472	+	0.846448i	$0.678737\pi$
$788$	73.6935	2.62522
$789$	0	0
$790$	−61.3176	−2.18158
$791$	−17.5909	−0.625462
$792$	0	0
$793$	0.740719	0.0263037
$794$	−67.3934	−2.39170
$795$	0	0
$796$	−27.9147	−0.989411
$797$	12.2851	0.435159	0.217580	−	0.976043i	$-0.430184\pi$
$797$	12.2851	0.435159	0.217580	+	0.976043i	$0.430184\pi$
$798$	0	0
$799$	−7.15708	−0.253199
$800$	−0.322844	−0.0114143
$801$	0	0
$802$	−58.8226	−2.07710
$803$	0	0
$804$	0	0
$805$	10.4949	0.369898
$806$	−3.29648	−0.116113
$807$	0	0
$808$	−53.3995	−1.87859
$809$	−32.6350	−1.14739	−0.573693	−	0.819070i	$-0.694490\pi$
$809$	−32.6350	−1.14739	−0.573693	+	0.819070i	$0.694490\pi$
$810$	0	0
$811$	−13.8754	−0.487232	−0.243616	−	0.969872i	$-0.578334\pi$
$811$	−13.8754	−0.487232	−0.243616	+	0.969872i	$0.578334\pi$
$812$	15.6781	0.550193
$813$	0	0
$814$	0	0
$815$	−21.1013	−0.739147
$816$	0	0
$817$	63.3995	2.21807
$818$	−58.3757	−2.04106
$819$	0	0
$820$	15.1571	0.529308
$821$	8.42606	0.294072	0.147036	−	0.989131i	$-0.453027\pi$
$821$	8.42606	0.294072	0.147036	+	0.989131i	$0.453027\pi$
$822$	0	0
$823$	39.4422	1.37487	0.687433	−	0.726247i	$-0.258737\pi$
$823$	39.4422	1.37487	0.687433	+	0.726247i	$0.258737\pi$
$824$	−79.0120	−2.75251
$825$	0	0
$826$	−6.54603	−0.227766
$827$	−0.591563	−0.0205707	−0.0102853	−	0.999947i	$-0.503274\pi$
$827$	−0.591563	−0.0205707	−0.0102853	+	0.999947i	$0.503274\pi$
$828$	0	0
$829$	23.5667	0.818506	0.409253	−	0.912421i	$-0.365789\pi$
$829$	23.5667	0.818506	0.409253	+	0.912421i	$0.365789\pi$
$830$	75.3543	2.61559
$831$	0	0
$832$	1.54771	0.0536572
$833$	0.813723	0.0281938
$834$	0	0
$835$	48.3108	1.67186
$836$	0	0
$837$	0	0
$838$	54.7715	1.89205
$839$	38.6507	1.33437	0.667185	−	0.744892i	$-0.267499\pi$
$839$	38.6507	1.33437	0.667185	+	0.744892i	$0.267499\pi$
$840$	0	0
$841$	−13.9472	−0.480939
$842$	23.0557	0.794553
$843$	0	0
$844$	−28.9899	−0.997872
$845$	26.5752	0.914216
$846$	0	0
$847$	0	0
$848$	−13.1113	−0.450245
$849$	0	0
$850$	−1.58700	−0.0544336
$851$	9.17741	0.314598
$852$	0	0
$853$	−38.3927	−1.31454	−0.657271	−	0.753654i	$-0.728289\pi$
$853$	−38.3927	−1.31454	−0.657271	+	0.753654i	$0.728289\pi$
$854$	8.81626	0.301686
$855$	0	0
$856$	7.21664	0.246660
$857$	−3.49785	−0.119484	−0.0597421	−	0.998214i	$-0.519028\pi$
$857$	−3.49785	−0.119484	−0.0597421	+	0.998214i	$0.519028\pi$
$858$	0	0
$859$	−31.1740	−1.06364	−0.531822	−	0.846856i	$-0.678492\pi$
$859$	−31.1740	−1.06364	−0.531822	+	0.846856i	$0.678492\pi$
$860$	83.5582	2.84931
$861$	0	0
$862$	−14.6077	−0.497540
$863$	30.9119	1.05225	0.526126	−	0.850406i	$-0.323644\pi$
$863$	30.9119	1.05225	0.526126	+	0.850406i	$0.323644\pi$
$864$	0	0
$865$	−12.1639	−0.413584
$866$	−13.3292	−0.452944
$867$	0	0
$868$	−26.2458	−0.890840
$869$	0	0
$870$	0	0
$871$	−2.12457	−0.0719884
$872$	12.5155	0.423827
$873$	0	0
$874$	−79.0887	−2.67522
$875$	−11.8823	−0.401696
$876$	0	0
$877$	−26.5118	−0.895242	−0.447621	−	0.894223i	$-0.647729\pi$
$877$	−26.5118	−0.895242	−0.447621	+	0.894223i	$0.647729\pi$
$878$	64.0003	2.15991
$879$	0	0
$880$	0	0
$881$	−11.0604	−0.372633	−0.186316	−	0.982490i	$-0.559655\pi$
$881$	−11.0604	−0.372633	−0.186316	+	0.982490i	$0.559655\pi$
$882$	0	0
$883$	17.8754	0.601556	0.300778	−	0.953694i	$-0.402754\pi$
$883$	17.8754	0.601556	0.300778	+	0.953694i	$0.402754\pi$
$884$	−0.679023	−0.0228380
$885$	0	0
$886$	18.0819	0.607474
$887$	−24.7382	−0.830626	−0.415313	−	0.909679i	$-0.636328\pi$
$887$	−24.7382	−0.830626	−0.415313	+	0.909679i	$0.636328\pi$
$888$	0	0
$889$	10.9079	0.365840
$890$	28.8817	0.968117
$891$	0	0
$892$	32.9899	1.10458
$893$	55.3097	1.85087
$894$	0	0
$895$	−33.6520	−1.12486
$896$	19.2350	0.642598
$897$	0	0
$898$	−28.0617	−0.936430
$899$	−25.1990	−0.840433
$900$	0	0
$901$	−2.51185	−0.0836818
$902$	0	0
$903$	0	0
$904$	88.2424	2.93490
$905$	7.18881	0.238964
$906$	0	0
$907$	25.3176	0.840656	0.420328	−	0.907372i	$-0.361915\pi$
$907$	25.3176	0.840656	0.420328	+	0.907372i	$0.361915\pi$
$908$	−14.9484	−0.496080
$909$	0	0
$910$	−1.04096	−0.0345076
$911$	48.7124	1.61391	0.806957	−	0.590610i	$-0.201113\pi$
$911$	48.7124	1.61391	0.806957	+	0.590610i	$0.201113\pi$
$912$	0	0
$913$	0	0
$914$	−27.8250	−0.920370
$915$	0	0
$916$	−81.8125	−2.70316
$917$	−5.72940	−0.189201
$918$	0	0
$919$	−37.0068	−1.22074	−0.610371	−	0.792116i	$-0.708979\pi$
$919$	−37.0068	−1.22074	−0.610371	+	0.792116i	$0.708979\pi$
$920$	−52.6463	−1.73570
$921$	0	0
$922$	−1.48479	−0.0488991
$923$	3.21331	0.105767
$924$	0	0
$925$	−1.42314	−0.0467925
$926$	−85.6932	−2.81605
$927$	0	0
$928$	−1.57854	−0.0518181
$929$	−13.4682	−0.441877	−0.220938	−	0.975288i	$-0.570912\pi$
$929$	−13.4682	−0.441877	−0.220938	+	0.975288i	$0.570912\pi$
$930$	0	0
$931$	−6.28843	−0.206095
$932$	55.3222	1.81214
$933$	0	0
$934$	−69.9455	−2.28869
$935$	0	0
$936$	0	0
$937$	54.7204	1.78764	0.893819	−	0.448427i	$-0.148016\pi$
$937$	54.7204	1.78764	0.893819	+	0.448427i	$0.148016\pi$
$938$	−25.2873	−0.825659
$939$	0	0
$940$	72.8961	2.37761
$941$	32.9919	1.07551	0.537753	−	0.843103i	$-0.319273\pi$
$941$	32.9919	1.07551	0.537753	+	0.843103i	$0.319273\pi$
$942$	0	0
$943$	−9.35813	−0.304743
$944$	11.3124	0.368187
$945$	0	0
$946$	0	0
$947$	20.5513	0.667829	0.333914	−	0.942603i	$-0.391630\pi$
$947$	20.5513	0.667829	0.333914	+	0.942603i	$0.391630\pi$
$948$	0	0
$949$	1.60937	0.0522422
$950$	12.2643	0.397905
$951$	0	0
$952$	−4.08193	−0.132296
$953$	55.9138	1.81123	0.905613	−	0.424106i	$-0.139412\pi$
$953$	55.9138	1.81123	0.905613	+	0.424106i	$0.139412\pi$
$954$	0	0
$955$	−38.2458	−1.23760
$956$	67.0664	2.16908
$957$	0	0
$958$	73.9966	2.39072
$959$	6.34175	0.204786
$960$	0	0
$961$	11.1841	0.360778
$962$	−0.910283	−0.0293487
$963$	0	0
$964$	48.3193	1.55626
$965$	11.2908	0.363462
$966$	0	0
$967$	−28.9046	−0.929509	−0.464754	−	0.885440i	$-0.653857\pi$
$967$	−28.9046	−0.929509	−0.464754	+	0.885440i	$0.653857\pi$
$968$	0	0
$969$	0	0
$970$	42.8226	1.37495
$971$	36.6621	1.17654	0.588271	−	0.808664i	$-0.299809\pi$
$971$	36.6621	1.17654	0.588271	+	0.808664i	$0.299809\pi$
$972$	0	0
$973$	−4.16386	−0.133487
$974$	−81.4865	−2.61099
$975$	0	0
$976$	−15.2357	−0.487681
$977$	14.5457	0.465357	0.232678	−	0.972554i	$-0.425251\pi$
$977$	14.5457	0.465357	0.232678	+	0.972554i	$0.425251\pi$
$978$	0	0
$979$	0	0
$980$	−8.28792	−0.264748
$981$	0	0
$982$	−26.6077	−0.849085
$983$	17.6325	0.562390	0.281195	−	0.959651i	$-0.409269\pi$
$983$	17.6325	0.562390	0.281195	+	0.959651i	$0.409269\pi$
$984$	0	0
$985$	−37.4029	−1.19175
$986$	−7.75958	−0.247115
$987$	0	0
$988$	5.24747	0.166944
$989$	−51.5897	−1.64046
$990$	0	0
$991$	15.1144	0.480126	0.240063	−	0.970757i	$-0.422832\pi$
$991$	15.1144	0.480126	0.240063	+	0.970757i	$0.422832\pi$
$992$	2.64254	0.0839007
$993$	0	0
$994$	38.2458	1.21308
$995$	14.1680	0.449156
$996$	0	0
$997$	24.5769	0.778357	0.389178	−	0.921162i	$-0.372759\pi$
$997$	24.5769	0.778357	0.389178	+	0.921162i	$0.372759\pi$
$998$	95.5246	3.02378
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cq.1.6	yes	6
3.2	odd	2	inner	7623.2.a.cq.1.1	✓	6
11.10	odd	2		7623.2.a.cr.1.1	yes	6
33.32	even	2		7623.2.a.cr.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7623.2.a.cq.1.1	✓	6	3.2	odd	2	inner
7623.2.a.cq.1.6	yes	6	1.1	even	1	trivial
7623.2.a.cr.1.1	yes	6	11.10	odd	2
7623.2.a.cr.1.6	yes	6	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)

\(f(q)\)	\(=\)	\(q+2.45784 q^{2} +4.04096 q^{4} -2.05098 q^{5} -1.00000 q^{7} +5.01636 q^{8} +O(q^{10})\)	\(q+2.45784 q^{2} +4.04096 q^{4} -2.05098 q^{5} -1.00000 q^{7} +5.01636 q^{8} -5.04096 q^{10} -0.206501 q^{13} -2.45784 q^{14} +4.24747 q^{16} +0.813723 q^{17} -6.28843 q^{19} -8.28792 q^{20} +5.11704 q^{23} -0.793499 q^{25} -0.507546 q^{26} -4.04096 q^{28} -3.87979 q^{29} +6.49493 q^{31} +0.406862 q^{32} +2.00000 q^{34} +2.05098 q^{35} +1.79350 q^{37} -15.4559 q^{38} -10.2884 q^{40} -1.82882 q^{41} -10.0819 q^{43} +12.5769 q^{46} -8.79547 q^{47} +1.00000 q^{49} -1.95029 q^{50} -0.834464 q^{52} -3.08686 q^{53} -5.01636 q^{56} -9.53590 q^{58} +2.66333 q^{59} -3.58700 q^{61} +15.9635 q^{62} -7.49493 q^{64} +0.423529 q^{65} +10.2884 q^{67} +3.28823 q^{68} +5.04096 q^{70} -15.5607 q^{71} -7.79350 q^{73} +4.40813 q^{74} -25.4113 q^{76} +12.1639 q^{79} -8.71145 q^{80} -4.49493 q^{82} -14.9484 q^{83} -1.66893 q^{85} -24.7797 q^{86} -5.72940 q^{89} +0.206501 q^{91} +20.6778 q^{92} -21.6178 q^{94} +12.8974 q^{95} -8.49493 q^{97} +2.45784 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} - 6 q^{7}+O(q^{10}) \) 6 * q + 4 * q^4 - 6 * q^7	\( 6 q + 4 q^{4} - 6 q^{7} - 10 q^{10} - 4 q^{13} + 8 q^{16} - 2 q^{25} - 4 q^{28} + 4 q^{31} + 12 q^{34} + 8 q^{37} - 24 q^{40} - 20 q^{43} + 6 q^{49} + 18 q^{52} - 2 q^{58} - 16 q^{61} - 10 q^{64} + 24 q^{67} + 10 q^{70} - 44 q^{73} - 54 q^{76} - 8 q^{79} + 8 q^{82} + 36 q^{85} + 4 q^{91} - 34 q^{94} - 16 q^{97}+O(q^{100}) \) 6 * q + 4 * q^4 - 6 * q^7 - 10 * q^10 - 4 * q^13 + 8 * q^16 - 2 * q^25 - 4 * q^28 + 4 * q^31 + 12 * q^34 + 8 * q^37 - 24 * q^40 - 20 * q^43 + 6 * q^49 + 18 * q^52 - 2 * q^58 - 16 * q^61 - 10 * q^64 + 24 * q^67 + 10 * q^70 - 44 * q^73 - 54 * q^76 - 8 * q^79 + 8 * q^82 + 36 * q^85 + 4 * q^91 - 34 * q^94 - 16 * q^97

Introduction

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