LMFDB - Embedded newform 7623.2.a.s.1.1

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:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q+2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} +O(q^{10})$

$q+2.00000 q^{2} +2.00000 q^{4} +3.00000 q^{5} +1.00000 q^{7} +6.00000 q^{10} -2.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} +3.00000 q^{17} +4.00000 q^{19} +6.00000 q^{20} -2.00000 q^{23} +4.00000 q^{25} -4.00000 q^{26} +2.00000 q^{28} +8.00000 q^{29} +2.00000 q^{31} -8.00000 q^{32} +6.00000 q^{34} +3.00000 q^{35} +10.0000 q^{37} +8.00000 q^{38} +2.00000 q^{41} -9.00000 q^{43} -4.00000 q^{46} -9.00000 q^{47} +1.00000 q^{49} +8.00000 q^{50} -4.00000 q^{52} +8.00000 q^{53} +16.0000 q^{58} +3.00000 q^{59} +10.0000 q^{61} +4.00000 q^{62} -8.00000 q^{64} -6.00000 q^{65} +13.0000 q^{67} +6.00000 q^{68} +6.00000 q^{70} -14.0000 q^{71} +10.0000 q^{73} +20.0000 q^{74} +8.00000 q^{76} +4.00000 q^{79} -12.0000 q^{80} +4.00000 q^{82} +1.00000 q^{83} +9.00000 q^{85} -18.0000 q^{86} +9.00000 q^{89} -2.00000 q^{91} -4.00000 q^{92} -18.0000 q^{94} +12.0000 q^{95} -16.0000 q^{97} +2.00000 q^{98} +O(q^{100})$

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	6.00000	1.89737
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	6.00000	1.34164
$21$	0	0
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	2.00000	0.377964
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	6.00000	1.02899
$35$	3.00000	0.507093
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	0	0
$45$	0	0
$46$	−4.00000	−0.589768
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	8.00000	1.13137
$51$	0	0
$52$	−4.00000	−0.554700
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	16.0000	2.10090
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−6.00000	−0.744208
$66$	0	0
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	6.00000	0.717137
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	20.0000	2.32495
$75$	0	0
$76$	8.00000	0.917663
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	−12.0000	−1.34164
$81$	0	0
$82$	4.00000	0.441726
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	0	0
$85$	9.00000	0.976187
$86$	−18.0000	−1.94099
$87$	0	0
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−18.0000	−1.85656
$95$	12.0000	1.23117
$96$	0	0
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	2.00000	0.202031
$99$	0	0
$100$	8.00000	0.800000
$101$	17.0000	1.69156	0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000	1.69156	0.845782	+	0.533529i	$0.179135\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	0	0
$106$	16.0000	1.55406
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	0	0
$109$	−13.0000	−1.24517	−0.622587	−	0.782551i	$-0.713918\pi$
$109$	−13.0000	−1.24517	−0.622587	+	0.782551i	$0.713918\pi$
$110$	0	0
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	16.0000	1.48556
$117$	0	0
$118$	6.00000	0.552345
$119$	3.00000	0.275010
$120$	0	0
$121$	0	0
$122$	20.0000	1.81071
$123$	0	0
$124$	4.00000	0.359211
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	0	0
$129$	0	0
$130$	−12.0000	−1.05247
$131$	13.0000	1.13582	0.567908	−	0.823092i	$-0.307753\pi$
$131$	13.0000	1.13582	0.567908	+	0.823092i	$0.307753\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	26.0000	2.24606
$135$	0	0
$136$	0	0
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	6.00000	0.507093
$141$	0	0
$142$	−28.0000	−2.34971
$143$	0	0
$144$	0	0
$145$	24.0000	1.99309
$146$	20.0000	1.65521
$147$	0	0
$148$	20.0000	1.64399
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	−8.00000	−0.638470	−0.319235	−	0.947676i	$-0.603426\pi$
$157$	−8.00000	−0.638470	−0.319235	+	0.947676i	$0.603426\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	−24.0000	−1.89737
$161$	−2.00000	−0.157622
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	2.00000	0.155230
$167$	−3.00000	−0.232147	−0.116073	−	0.993241i	$-0.537031\pi$
$167$	−3.00000	−0.232147	−0.116073	+	0.993241i	$0.537031\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	18.0000	1.38054
$171$	0	0
$172$	−18.0000	−1.37249
$173$	25.0000	1.90071	0.950357	−	0.311160i	$-0.100718\pi$
$173$	25.0000	1.90071	0.950357	+	0.311160i	$0.100718\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	0	0
$178$	18.0000	1.34916
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	0	0
$185$	30.0000	2.20564
$186$	0	0
$187$	0	0
$188$	−18.0000	−1.31278
$189$	0	0
$190$	24.0000	1.74114
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	23.0000	1.65558	0.827788	−	0.561041i	$-0.189599\pi$
$193$	23.0000	1.65558	0.827788	+	0.561041i	$0.189599\pi$
$194$	−32.0000	−2.29747
$195$	0	0
$196$	2.00000	0.142857
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	0	0
$202$	34.0000	2.39223
$203$	8.00000	0.561490
$204$	0	0
$205$	6.00000	0.419058
$206$	28.0000	1.95085
$207$	0	0
$208$	8.00000	0.554700
$209$	0	0
$210$	0	0
$211$	−7.00000	−0.481900	−0.240950	−	0.970538i	$-0.577459\pi$
$211$	−7.00000	−0.481900	−0.240950	+	0.970538i	$0.577459\pi$
$212$	16.0000	1.09888
$213$	0	0
$214$	−20.0000	−1.36717
$215$	−27.0000	−1.84138
$216$	0	0
$217$	2.00000	0.135769
$218$	−26.0000	−1.76094
$219$	0	0
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	−8.00000	−0.534522
$225$	0	0
$226$	−12.0000	−0.798228
$227$	7.00000	0.464606	0.232303	−	0.972643i	$-0.425374\pi$
$227$	7.00000	0.464606	0.232303	+	0.972643i	$0.425374\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	−12.0000	−0.791257
$231$	0	0
$232$	0	0
$233$	−4.00000	−0.262049	−0.131024	−	0.991379i	$-0.541827\pi$
$233$	−4.00000	−0.262049	−0.131024	+	0.991379i	$0.541827\pi$
$234$	0	0
$235$	−27.0000	−1.76129
$236$	6.00000	0.390567
$237$	0	0
$238$	6.00000	0.388922
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	0	0
$243$	0	0
$244$	20.0000	1.28037
$245$	3.00000	0.191663
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	0	0
$250$	−6.00000	−0.379473
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	−26.0000	−1.63139
$255$	0	0
$256$	16.0000	1.00000
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	−12.0000	−0.744208
$261$	0	0
$262$	26.0000	1.60629
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	8.00000	0.490511
$267$	0	0
$268$	26.0000	1.58820
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−12.0000	−0.727607
$273$	0	0
$274$	−16.0000	−0.966595
$275$	0	0
$276$	0	0
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−28.0000	−1.66149
$285$	0	0
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	−8.00000	−0.470588
$290$	48.0000	2.81866
$291$	0	0
$292$	20.0000	1.17041
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	9.00000	0.524000
$296$	0	0
$297$	0	0
$298$	−24.0000	−1.39028
$299$	4.00000	0.231326
$300$	0	0
$301$	−9.00000	−0.518751
$302$	10.0000	0.575435
$303$	0	0
$304$	−16.0000	−0.917663
$305$	30.0000	1.71780
$306$	0	0
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	0	0
$310$	12.0000	0.681554
$311$	25.0000	1.41762	0.708810	−	0.705399i	$-0.249232\pi$
$311$	25.0000	1.41762	0.708810	+	0.705399i	$0.249232\pi$
$312$	0	0
$313$	−24.0000	−1.35656	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	−24.0000	−1.35656	−0.678280	+	0.734803i	$0.737274\pi$
$314$	−16.0000	−0.902932
$315$	0	0
$316$	8.00000	0.450035
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	0	0
$320$	−24.0000	−1.34164
$321$	0	0
$322$	−4.00000	−0.222911
$323$	12.0000	0.667698
$324$	0	0
$325$	−8.00000	−0.443760
$326$	8.00000	0.443079
$327$	0	0
$328$	0	0
$329$	−9.00000	−0.496186
$330$	0	0
$331$	3.00000	0.164895	0.0824475	−	0.996595i	$-0.473726\pi$
$331$	3.00000	0.164895	0.0824475	+	0.996595i	$0.473726\pi$
$332$	2.00000	0.109764
$333$	0	0
$334$	−6.00000	−0.328305
$335$	39.0000	2.13080
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	−18.0000	−0.979071
$339$	0	0
$340$	18.0000	0.976187
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	50.0000	2.68802
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	8.00000	0.427618
$351$	0	0
$352$	0	0
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	0	0
$355$	−42.0000	−2.22913
$356$	18.0000	0.953998
$357$	0	0
$358$	−40.0000	−2.11407
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−32.0000	−1.68188
$363$	0	0
$364$	−4.00000	−0.209657
$365$	30.0000	1.57027
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	8.00000	0.417029
$369$	0	0
$370$	60.0000	3.11925
$371$	8.00000	0.415339
$372$	0	0
$373$	−21.0000	−1.08734	−0.543669	−	0.839299i	$-0.682965\pi$
$373$	−21.0000	−1.08734	−0.543669	+	0.839299i	$0.682965\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−16.0000	−0.824042
$378$	0	0
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	24.0000	1.23117
$381$	0	0
$382$	40.0000	2.04658
$383$	−19.0000	−0.970855	−0.485427	−	0.874277i	$-0.661336\pi$
$383$	−19.0000	−0.970855	−0.485427	+	0.874277i	$0.661336\pi$
$384$	0	0
$385$	0	0
$386$	46.0000	2.34134
$387$	0	0
$388$	−32.0000	−1.62455
$389$	−4.00000	−0.202808	−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000	−0.202808	−0.101404	+	0.994845i	$0.532333\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	0	0
$393$	0	0
$394$	44.0000	2.21669
$395$	12.0000	0.603786
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−28.0000	−1.40351
$399$	0	0
$400$	−16.0000	−0.800000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	34.0000	1.69156
$405$	0	0
$406$	16.0000	0.794067
$407$	0	0
$408$	0	0
$409$	−28.0000	−1.38451	−0.692255	−	0.721653i	$-0.743383\pi$
$409$	−28.0000	−1.38451	−0.692255	+	0.721653i	$0.743383\pi$
$410$	12.0000	0.592638
$411$	0	0
$412$	28.0000	1.37946
$413$	3.00000	0.147620
$414$	0	0
$415$	3.00000	0.147264
$416$	16.0000	0.784465
$417$	0	0
$418$	0	0
$419$	−19.0000	−0.928211	−0.464105	−	0.885780i	$-0.653624\pi$
$419$	−19.0000	−0.928211	−0.464105	+	0.885780i	$0.653624\pi$
$420$	0	0
$421$	33.0000	1.60832	0.804161	−	0.594412i	$-0.202615\pi$
$421$	33.0000	1.60832	0.804161	+	0.594412i	$0.202615\pi$
$422$	−14.0000	−0.681509
$423$	0	0
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	10.0000	0.483934
$428$	−20.0000	−0.966736
$429$	0	0
$430$	−54.0000	−2.60411
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	0	0
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−26.0000	−1.24517
$437$	−8.00000	−0.382692
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	0	0
$442$	−12.0000	−0.570782
$443$	38.0000	1.80543	0.902717	−	0.430234i	$-0.141569\pi$
$443$	38.0000	1.80543	0.902717	+	0.430234i	$0.141569\pi$
$444$	0	0
$445$	27.0000	1.27992
$446$	−48.0000	−2.27287
$447$	0	0
$448$	−8.00000	−0.377964
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	−12.0000	−0.564433
$453$	0	0
$454$	14.0000	0.657053
$455$	−6.00000	−0.281284
$456$	0	0
$457$	−3.00000	−0.140334	−0.0701670	−	0.997535i	$-0.522353\pi$
$457$	−3.00000	−0.140334	−0.0701670	+	0.997535i	$0.522353\pi$
$458$	−8.00000	−0.373815
$459$	0	0
$460$	−12.0000	−0.559503
$461$	−15.0000	−0.698620	−0.349310	−	0.937007i	$-0.613584\pi$
$461$	−15.0000	−0.698620	−0.349310	+	0.937007i	$0.613584\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−32.0000	−1.48556
$465$	0	0
$466$	−8.00000	−0.370593
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	0	0
$469$	13.0000	0.600284
$470$	−54.0000	−2.49083
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	16.0000	0.734130
$476$	6.00000	0.275010
$477$	0	0
$478$	−12.0000	−0.548867
$479$	29.0000	1.32504	0.662522	−	0.749043i	$-0.269486\pi$
$479$	29.0000	1.32504	0.662522	+	0.749043i	$0.269486\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	−56.0000	−2.55073
$483$	0	0
$484$	0	0
$485$	−48.0000	−2.17957
$486$	0	0
$487$	37.0000	1.67663	0.838315	−	0.545186i	$-0.183541\pi$
$487$	37.0000	1.67663	0.838315	+	0.545186i	$0.183541\pi$
$488$	0	0
$489$	0	0
$490$	6.00000	0.271052
$491$	42.0000	1.89543	0.947717	−	0.319113i	$-0.103385\pi$
$491$	42.0000	1.89543	0.947717	+	0.319113i	$0.103385\pi$
$492$	0	0
$493$	24.0000	1.08091
$494$	−16.0000	−0.719874
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−14.0000	−0.627986
$498$	0	0
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	−6.00000	−0.268328
$501$	0	0
$502$	−24.0000	−1.07117
$503$	11.0000	0.490466	0.245233	−	0.969464i	$-0.421136\pi$
$503$	11.0000	0.490466	0.245233	+	0.969464i	$0.421136\pi$
$504$	0	0
$505$	51.0000	2.26947
$506$	0	0
$507$	0	0
$508$	−26.0000	−1.15356
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	32.0000	1.41421
$513$	0	0
$514$	−14.0000	−0.617514
$515$	42.0000	1.85074
$516$	0	0
$517$	0	0
$518$	20.0000	0.878750
$519$	0	0
$520$	0	0
$521$	−1.00000	−0.0438108	−0.0219054	−	0.999760i	$-0.506973\pi$
$521$	−1.00000	−0.0438108	−0.0219054	+	0.999760i	$0.506973\pi$
$522$	0	0
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	26.0000	1.13582
$525$	0	0
$526$	−36.0000	−1.56967
$527$	6.00000	0.261364
$528$	0	0
$529$	−19.0000	−0.826087
$530$	48.0000	2.08499
$531$	0	0
$532$	8.00000	0.346844
$533$	−4.00000	−0.173259
$534$	0	0
$535$	−30.0000	−1.29701
$536$	0	0
$537$	0	0
$538$	28.0000	1.20717
$539$	0	0
$540$	0	0
$541$	33.0000	1.41878	0.709390	−	0.704816i	$-0.248970\pi$
$541$	33.0000	1.41878	0.709390	+	0.704816i	$0.248970\pi$
$542$	−40.0000	−1.71815
$543$	0	0
$544$	−24.0000	−1.02899
$545$	−39.0000	−1.67058
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−16.0000	−0.683486
$549$	0	0
$550$	0	0
$551$	32.0000	1.36325
$552$	0	0
$553$	4.00000	0.170097
$554$	34.0000	1.44452
$555$	0	0
$556$	4.00000	0.169638
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	18.0000	0.761319
$560$	−12.0000	−0.507093
$561$	0	0
$562$	24.0000	1.01238
$563$	21.0000	0.885044	0.442522	−	0.896758i	$-0.354084\pi$
$563$	21.0000	0.885044	0.442522	+	0.896758i	$0.354084\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	−8.00000	−0.336265
$567$	0	0
$568$	0	0
$569$	−40.0000	−1.67689	−0.838444	−	0.544988i	$-0.816534\pi$
$569$	−40.0000	−1.67689	−0.838444	+	0.544988i	$0.816534\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	0	0
$574$	4.00000	0.166957
$575$	−8.00000	−0.333623
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	−16.0000	−0.665512
$579$	0	0
$580$	48.0000	1.99309
$581$	1.00000	0.0414870
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	18.0000	0.743573
$587$	−39.0000	−1.60970	−0.804851	−	0.593477i	$-0.797755\pi$
$587$	−39.0000	−1.60970	−0.804851	+	0.593477i	$0.797755\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	18.0000	0.741048
$591$	0	0
$592$	−40.0000	−1.64399
$593$	11.0000	0.451716	0.225858	−	0.974160i	$-0.427481\pi$
$593$	11.0000	0.451716	0.225858	+	0.974160i	$0.427481\pi$
$594$	0	0
$595$	9.00000	0.368964
$596$	−24.0000	−0.983078
$597$	0	0
$598$	8.00000	0.327144
$599$	−34.0000	−1.38920	−0.694601	−	0.719395i	$-0.744419\pi$
$599$	−34.0000	−1.38920	−0.694601	+	0.719395i	$0.744419\pi$
$600$	0	0
$601$	6.00000	0.244745	0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000	0.244745	0.122373	+	0.992484i	$0.460950\pi$
$602$	−18.0000	−0.733625
$603$	0	0
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	−32.0000	−1.29777
$609$	0	0
$610$	60.0000	2.42933
$611$	18.0000	0.728202
$612$	0	0
$613$	29.0000	1.17130	0.585649	−	0.810564i	$-0.300840\pi$
$613$	29.0000	1.17130	0.585649	+	0.810564i	$0.300840\pi$
$614$	−24.0000	−0.968561
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−18.0000	−0.723481	−0.361741	−	0.932279i	$-0.617817\pi$
$619$	−18.0000	−0.723481	−0.361741	+	0.932279i	$0.617817\pi$
$620$	12.0000	0.481932
$621$	0	0
$622$	50.0000	2.00482
$623$	9.00000	0.360577
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−48.0000	−1.91847
$627$	0	0
$628$	−16.0000	−0.638470
$629$	30.0000	1.19618
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	0	0
$633$	0	0
$634$	12.0000	0.476581
$635$	−39.0000	−1.54767
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	24.0000	0.944267
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	0	0
$649$	0	0
$650$	−16.0000	−0.627572
$651$	0	0
$652$	8.00000	0.313304
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	0	0
$655$	39.0000	1.52386
$656$	−8.00000	−0.312348
$657$	0	0
$658$	−18.0000	−0.701713
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	−40.0000	−1.55582	−0.777910	−	0.628376i	$-0.783720\pi$
$661$	−40.0000	−1.55582	−0.777910	+	0.628376i	$0.783720\pi$
$662$	6.00000	0.233197
$663$	0	0
$664$	0	0
$665$	12.0000	0.465340
$666$	0	0
$667$	−16.0000	−0.619522
$668$	−6.00000	−0.232147
$669$	0	0
$670$	78.0000	3.01340
$671$	0	0
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	−4.00000	−0.154074
$675$	0	0
$676$	−18.0000	−0.692308
$677$	9.00000	0.345898	0.172949	−	0.984931i	$-0.444670\pi$
$677$	9.00000	0.345898	0.172949	+	0.984931i	$0.444670\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	0	0
$681$	0	0
$682$	0	0
$683$	34.0000	1.30097	0.650487	−	0.759517i	$-0.274565\pi$
$683$	34.0000	1.30097	0.650487	+	0.759517i	$0.274565\pi$
$684$	0	0
$685$	−24.0000	−0.916993
$686$	2.00000	0.0763604
$687$	0	0
$688$	36.0000	1.37249
$689$	−16.0000	−0.609551
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	50.0000	1.90071
$693$	0	0
$694$	32.0000	1.21470
$695$	6.00000	0.227593
$696$	0	0
$697$	6.00000	0.227266
$698$	−12.0000	−0.454207
$699$	0	0
$700$	8.00000	0.302372
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	0	0
$703$	40.0000	1.50863
$704$	0	0
$705$	0	0
$706$	−60.0000	−2.25813
$707$	17.0000	0.639351
$708$	0	0
$709$	−9.00000	−0.338002	−0.169001	−	0.985616i	$-0.554054\pi$
$709$	−9.00000	−0.338002	−0.169001	+	0.985616i	$0.554054\pi$
$710$	−84.0000	−3.15246
$711$	0	0
$712$	0	0
$713$	−4.00000	−0.149801
$714$	0	0
$715$	0	0
$716$	−40.0000	−1.49487
$717$	0	0
$718$	−36.0000	−1.34351
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	14.0000	0.521387
$722$	−6.00000	−0.223297
$723$	0	0
$724$	−32.0000	−1.18927
$725$	32.0000	1.18845
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	0	0
$730$	60.0000	2.22070
$731$	−27.0000	−0.998631
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	16.0000	0.589768
$737$	0	0
$738$	0	0
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	60.0000	2.20564
$741$	0	0
$742$	16.0000	0.587378
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	0	0
$745$	−36.0000	−1.31894
$746$	−42.0000	−1.53773
$747$	0	0
$748$	0	0
$749$	−10.0000	−0.365392
$750$	0	0
$751$	−37.0000	−1.35015	−0.675075	−	0.737749i	$-0.735889\pi$
$751$	−37.0000	−1.35015	−0.675075	+	0.737749i	$0.735889\pi$
$752$	36.0000	1.31278
$753$	0	0
$754$	−32.0000	−1.16537
$755$	15.0000	0.545906
$756$	0	0
$757$	−9.00000	−0.327111	−0.163555	−	0.986534i	$-0.552296\pi$
$757$	−9.00000	−0.327111	−0.163555	+	0.986534i	$0.552296\pi$
$758$	−10.0000	−0.363216
$759$	0	0
$760$	0	0
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	0	0
$763$	−13.0000	−0.470632
$764$	40.0000	1.44715
$765$	0	0
$766$	−38.0000	−1.37300
$767$	−6.00000	−0.216647
$768$	0	0
$769$	−24.0000	−0.865462	−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000	−0.865462	−0.432731	+	0.901523i	$0.642450\pi$
$770$	0	0
$771$	0	0
$772$	46.0000	1.65558
$773$	−15.0000	−0.539513	−0.269756	−	0.962929i	$-0.586943\pi$
$773$	−15.0000	−0.539513	−0.269756	+	0.962929i	$0.586943\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	0	0
$778$	−8.00000	−0.286814
$779$	8.00000	0.286630
$780$	0	0
$781$	0	0
$782$	−12.0000	−0.429119
$783$	0	0
$784$	−4.00000	−0.142857
$785$	−24.0000	−0.856597
$786$	0	0
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	44.0000	1.56744
$789$	0	0
$790$	24.0000	0.853882
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−20.0000	−0.710221
$794$	−36.0000	−1.27759
$795$	0	0
$796$	−28.0000	−0.992434
$797$	−11.0000	−0.389640	−0.194820	−	0.980839i	$-0.562412\pi$
$797$	−11.0000	−0.389640	−0.194820	+	0.980839i	$0.562412\pi$
$798$	0	0
$799$	−27.0000	−0.955191
$800$	−32.0000	−1.13137
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−6.00000	−0.211472
$806$	−8.00000	−0.281788
$807$	0	0
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−52.0000	−1.82597	−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000	−1.82597	−0.912983	+	0.407997i	$0.866228\pi$
$812$	16.0000	0.561490
$813$	0	0
$814$	0	0
$815$	12.0000	0.420342
$816$	0	0
$817$	−36.0000	−1.25948
$818$	−56.0000	−1.95799
$819$	0	0
$820$	12.0000	0.419058
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	0	0
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	0	0
$825$	0	0
$826$	6.00000	0.208767
$827$	38.0000	1.32139	0.660695	−	0.750655i	$-0.270262\pi$
$827$	38.0000	1.32139	0.660695	+	0.750655i	$0.270262\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	6.00000	0.208263
$831$	0	0
$832$	16.0000	0.554700
$833$	3.00000	0.103944
$834$	0	0
$835$	−9.00000	−0.311458
$836$	0	0
$837$	0	0
$838$	−38.0000	−1.31269
$839$	−51.0000	−1.76072	−0.880358	−	0.474310i	$-0.842698\pi$
$839$	−51.0000	−1.76072	−0.880358	+	0.474310i	$0.842698\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	66.0000	2.27451
$843$	0	0
$844$	−14.0000	−0.481900
$845$	−27.0000	−0.928828
$846$	0	0
$847$	0	0
$848$	−32.0000	−1.09888
$849$	0	0
$850$	24.0000	0.823193
$851$	−20.0000	−0.685591
$852$	0	0
$853$	32.0000	1.09566	0.547830	−	0.836590i	$-0.315454\pi$
$853$	32.0000	1.09566	0.547830	+	0.836590i	$0.315454\pi$
$854$	20.0000	0.684386
$855$	0	0
$856$	0	0
$857$	−31.0000	−1.05894	−0.529470	−	0.848329i	$-0.677609\pi$
$857$	−31.0000	−1.05894	−0.529470	+	0.848329i	$0.677609\pi$
$858$	0	0
$859$	18.0000	0.614152	0.307076	−	0.951685i	$-0.400649\pi$
$859$	18.0000	0.614152	0.307076	+	0.951685i	$0.400649\pi$
$860$	−54.0000	−1.84138
$861$	0	0
$862$	−8.00000	−0.272481
$863$	26.0000	0.885050	0.442525	−	0.896756i	$-0.354083\pi$
$863$	26.0000	0.885050	0.442525	+	0.896756i	$0.354083\pi$
$864$	0	0
$865$	75.0000	2.55008
$866$	8.00000	0.271851
$867$	0	0
$868$	4.00000	0.135769
$869$	0	0
$870$	0	0
$871$	−26.0000	−0.880976
$872$	0	0
$873$	0	0
$874$	−16.0000	−0.541208
$875$	−3.00000	−0.101419
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	−52.0000	−1.75491
$879$	0	0
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	0	0
$883$	−21.0000	−0.706706	−0.353353	−	0.935490i	$-0.614959\pi$
$883$	−21.0000	−0.706706	−0.353353	+	0.935490i	$0.614959\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	76.0000	2.55327
$887$	−55.0000	−1.84672	−0.923360	−	0.383936i	$-0.874568\pi$
$887$	−55.0000	−1.84672	−0.923360	+	0.383936i	$0.874568\pi$
$888$	0	0
$889$	−13.0000	−0.436006
$890$	54.0000	1.81008
$891$	0	0
$892$	−48.0000	−1.60716
$893$	−36.0000	−1.20469
$894$	0	0
$895$	−60.0000	−2.00558
$896$	0	0
$897$	0	0
$898$	−12.0000	−0.400445
$899$	16.0000	0.533630
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−48.0000	−1.59557
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	14.0000	0.464606
$909$	0	0
$910$	−12.0000	−0.397796
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	0	0
$913$	0	0
$914$	−6.00000	−0.198462
$915$	0	0
$916$	−8.00000	−0.264327
$917$	13.0000	0.429298
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	0	0
$922$	−30.0000	−0.987997
$923$	28.0000	0.921631
$924$	0	0
$925$	40.0000	1.31519
$926$	32.0000	1.05159
$927$	0	0
$928$	−64.0000	−2.10090
$929$	−3.00000	−0.0984268	−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000	−0.0984268	−0.0492134	+	0.998788i	$0.515671\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	−8.00000	−0.262049
$933$	0	0
$934$	8.00000	0.261768
$935$	0	0
$936$	0	0
$937$	16.0000	0.522697	0.261349	−	0.965244i	$-0.415833\pi$
$937$	16.0000	0.522697	0.261349	+	0.965244i	$0.415833\pi$
$938$	26.0000	0.848930
$939$	0	0
$940$	−54.0000	−1.76129
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	−4.00000	−0.130258
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	30.0000	0.974869	0.487435	−	0.873160i	$-0.337933\pi$
$947$	30.0000	0.974869	0.487435	+	0.873160i	$0.337933\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	32.0000	1.03822
$951$	0	0
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	0	0
$955$	60.0000	1.94155
$956$	−12.0000	−0.388108
$957$	0	0
$958$	58.0000	1.87389
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−40.0000	−1.28965
$963$	0	0
$964$	−56.0000	−1.80364
$965$	69.0000	2.22119
$966$	0	0
$967$	−1.00000	−0.0321578	−0.0160789	−	0.999871i	$-0.505118\pi$
$967$	−1.00000	−0.0321578	−0.0160789	+	0.999871i	$0.505118\pi$
$968$	0	0
$969$	0	0
$970$	−96.0000	−3.08237
$971$	−37.0000	−1.18739	−0.593693	−	0.804691i	$-0.702331\pi$
$971$	−37.0000	−1.18739	−0.593693	+	0.804691i	$0.702331\pi$
$972$	0	0
$973$	2.00000	0.0641171
$974$	74.0000	2.37111
$975$	0	0
$976$	−40.0000	−1.28037
$977$	14.0000	0.447900	0.223950	−	0.974601i	$-0.428105\pi$
$977$	14.0000	0.447900	0.223950	+	0.974601i	$0.428105\pi$
$978$	0	0
$979$	0	0
$980$	6.00000	0.191663
$981$	0	0
$982$	84.0000	2.68055
$983$	8.00000	0.255160	0.127580	−	0.991828i	$-0.459279\pi$
$983$	8.00000	0.255160	0.127580	+	0.991828i	$0.459279\pi$
$984$	0	0
$985$	66.0000	2.10293
$986$	48.0000	1.52863
$987$	0	0
$988$	−16.0000	−0.509028
$989$	18.0000	0.572367
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	−16.0000	−0.508001
$993$	0	0
$994$	−28.0000	−0.888106
$995$	−42.0000	−1.33149
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	−10.0000	−0.316544
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.s.1.1		1
3.2	odd	2		2541.2.a.a.1.1	✓	1
11.10	odd	2		7623.2.a.d.1.1		1
33.32	even	2		2541.2.a.k.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.a.1.1	✓	1	3.2	odd	2
2541.2.a.k.1.1	yes	1	33.32	even	2
7623.2.a.d.1.1		1	11.10	odd	2
7623.2.a.s.1.1		1	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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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