LMFDB - Embedded newform 2541.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2541,2,Mod(1,2541)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2541, 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("2541.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2541 = 3 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2541.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:	$20.2899871536$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2541.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} -2.00000 q^{12} +6.00000 q^{13} +2.00000 q^{14} -1.00000 q^{15} -4.00000 q^{16} +7.00000 q^{17} -2.00000 q^{18} +8.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} +6.00000 q^{23} -4.00000 q^{25} -12.0000 q^{26} -1.00000 q^{27} -2.00000 q^{28} -4.00000 q^{29} +2.00000 q^{30} +2.00000 q^{31} +8.00000 q^{32} -14.0000 q^{34} -1.00000 q^{35} +2.00000 q^{36} -6.00000 q^{37} -16.0000 q^{38} -6.00000 q^{39} +2.00000 q^{41} -2.00000 q^{42} +1.00000 q^{43} +1.00000 q^{45} -12.0000 q^{46} +13.0000 q^{47} +4.00000 q^{48} +1.00000 q^{49} +8.00000 q^{50} -7.00000 q^{51} +12.0000 q^{52} +2.00000 q^{54} -8.00000 q^{57} +8.00000 q^{58} +1.00000 q^{59} -2.00000 q^{60} +10.0000 q^{61} -4.00000 q^{62} -1.00000 q^{63} -8.00000 q^{64} +6.00000 q^{65} -3.00000 q^{67} +14.0000 q^{68} -6.00000 q^{69} +2.00000 q^{70} -6.00000 q^{71} -14.0000 q^{73} +12.0000 q^{74} +4.00000 q^{75} +16.0000 q^{76} +12.0000 q^{78} -4.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} +5.00000 q^{83} +2.00000 q^{84} +7.00000 q^{85} -2.00000 q^{86} +4.00000 q^{87} +3.00000 q^{89} -2.00000 q^{90} -6.00000 q^{91} +12.0000 q^{92} -2.00000 q^{93} -26.0000 q^{94} +8.00000 q^{95} -8.00000 q^{96} -4.00000 q^{97} -2.00000 q^{98} +O(q^{100})\)

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	$-0.750000\pi$
$2$	−2.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.00000	1.00000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	2.00000	0.816497
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	0	0
$11$	0	0
$12$	−2.00000	−0.577350
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	2.00000	0.534522
$15$	−1.00000	−0.258199
$16$	−4.00000	−1.00000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	−2.00000	−0.471405
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	2.00000	0.447214
$21$	1.00000	0.218218
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−12.0000	−2.35339
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	2.00000	0.365148
$31$	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
$34$	−14.0000	−2.40098
$35$	−1.00000	−0.169031
$36$	2.00000	0.333333
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	−16.0000	−2.59554
$39$	−6.00000	−0.960769
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−2.00000	−0.308607
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	−12.0000	−1.76930
$47$	13.0000	1.89624	0.948122	−	0.317905i	$-0.102979\pi$
$47$	13.0000	1.89624	0.948122	+	0.317905i	$0.102979\pi$
$48$	4.00000	0.577350
$49$	1.00000	0.142857
$50$	8.00000	1.13137
$51$	−7.00000	−0.980196
$52$	12.0000	1.66410
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	2.00000	0.272166
$55$	0	0
$56$	0	0
$57$	−8.00000	−1.05963
$58$	8.00000	1.05045
$59$	1.00000	0.130189	0.0650945	−	0.997879i	$-0.479265\pi$
$59$	1.00000	0.130189	0.0650945	+	0.997879i	$0.479265\pi$
$60$	−2.00000	−0.258199
$61$	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
$63$	−1.00000	−0.125988
$64$	−8.00000	−1.00000
$65$	6.00000	0.744208
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	14.0000	1.69775
$69$	−6.00000	−0.722315
$70$	2.00000	0.239046
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	12.0000	1.39497
$75$	4.00000	0.461880
$76$	16.0000	1.83533
$77$	0	0
$78$	12.0000	1.35873
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−4.00000	−0.447214
$81$	1.00000	0.111111
$82$	−4.00000	−0.441726
$83$	5.00000	0.548821	0.274411	−	0.961613i	$-0.411517\pi$
$83$	5.00000	0.548821	0.274411	+	0.961613i	$0.411517\pi$
$84$	2.00000	0.218218
$85$	7.00000	0.759257
$86$	−2.00000	−0.215666
$87$	4.00000	0.428845
$88$	0	0
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	−2.00000	−0.210819
$91$	−6.00000	−0.628971
$92$	12.0000	1.25109
$93$	−2.00000	−0.207390
$94$	−26.0000	−2.68170
$95$	8.00000	0.820783
$96$	−8.00000	−0.816497
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	−2.00000	−0.202031
$99$	0	0
$100$	−8.00000	−0.800000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	14.0000	1.38621
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−2.00000	−0.192450
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	4.00000	0.377964
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	16.0000	1.49854
$115$	6.00000	0.559503
$116$	−8.00000	−0.742781
$117$	6.00000	0.554700
$118$	−2.00000	−0.184115
$119$	−7.00000	−0.641689
$120$	0	0
$121$	0	0
$122$	−20.0000	−1.81071
$123$	−2.00000	−0.180334
$124$	4.00000	0.359211
$125$	−9.00000	−0.804984
$126$	2.00000	0.178174
$127$	−11.0000	−0.976092	−0.488046	−	0.872818i	$-0.662290\pi$
$127$	−11.0000	−0.976092	−0.488046	+	0.872818i	$0.662290\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	−12.0000	−1.05247
$131$	17.0000	1.48530	0.742648	−	0.669681i	$-0.233569\pi$
$131$	17.0000	1.48530	0.742648	+	0.669681i	$0.233569\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	6.00000	0.518321
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	12.0000	1.02151
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	−2.00000	−0.169031
$141$	−13.0000	−1.09480
$142$	12.0000	1.00702
$143$	0	0
$144$	−4.00000	−0.333333
$145$	−4.00000	−0.332182
$146$	28.0000	2.31730
$147$	−1.00000	−0.0824786
$148$	−12.0000	−0.986394
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	−8.00000	−0.653197
$151$	19.0000	1.54620	0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000	1.54620	0.773099	+	0.634285i	$0.218706\pi$
$152$	0	0
$153$	7.00000	0.565916
$154$	0	0
$155$	2.00000	0.160644
$156$	−12.0000	−0.960769
$157$	−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
$160$	8.00000	0.632456
$161$	−6.00000	−0.472866
$162$	−2.00000	−0.157135
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	−10.0000	−0.776151
$167$	17.0000	1.31550	0.657750	−	0.753237i	$-0.271508\pi$
$167$	17.0000	1.31550	0.657750	+	0.753237i	$0.271508\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	−14.0000	−1.07375
$171$	8.00000	0.611775
$172$	2.00000	0.152499
$173$	13.0000	0.988372	0.494186	−	0.869356i	$-0.335466\pi$
$173$	13.0000	0.988372	0.494186	+	0.869356i	$0.335466\pi$
$174$	−8.00000	−0.606478
$175$	4.00000	0.302372
$176$	0	0
$177$	−1.00000	−0.0751646
$178$	−6.00000	−0.449719
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	2.00000	0.149071
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	12.0000	0.889499
$183$	−10.0000	−0.739221
$184$	0	0
$185$	−6.00000	−0.441129
$186$	4.00000	0.293294
$187$	0	0
$188$	26.0000	1.89624
$189$	1.00000	0.0727393
$190$	−16.0000	−1.16076
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	8.00000	0.577350
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	8.00000	0.574367
$195$	−6.00000	−0.429669
$196$	2.00000	0.142857
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	3.00000	0.211604
$202$	6.00000	0.422159
$203$	4.00000	0.280745
$204$	−14.0000	−0.980196
$205$	2.00000	0.139686
$206$	12.0000	0.836080
$207$	6.00000	0.417029
$208$	−24.0000	−1.66410
$209$	0	0
$210$	−2.00000	−0.138013
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	12.0000	0.820303
$215$	1.00000	0.0681994
$216$	0	0
$217$	−2.00000	−0.135769
$218$	38.0000	2.57368
$219$	14.0000	0.946032
$220$	0	0
$221$	42.0000	2.82523
$222$	−12.0000	−0.805387
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−8.00000	−0.534522
$225$	−4.00000	−0.266667
$226$	36.0000	2.39468
$227$	3.00000	0.199117	0.0995585	−	0.995032i	$-0.468257\pi$
$227$	3.00000	0.199117	0.0995585	+	0.995032i	$0.468257\pi$
$228$	−16.0000	−1.05963
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	−12.0000	−0.791257
$231$	0	0
$232$	0	0
$233$	−8.00000	−0.524097	−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000	−0.524097	−0.262049	+	0.965055i	$0.584398\pi$
$234$	−12.0000	−0.784465
$235$	13.0000	0.848026
$236$	2.00000	0.130189
$237$	4.00000	0.259828
$238$	14.0000	0.907485
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	4.00000	0.258199
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	20.0000	1.28037
$245$	1.00000	0.0638877
$246$	4.00000	0.255031
$247$	48.0000	3.05417
$248$	0	0
$249$	−5.00000	−0.316862
$250$	18.0000	1.13842
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	22.0000	1.38040
$255$	−7.00000	−0.438357
$256$	16.0000	1.00000
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	2.00000	0.124515
$259$	6.00000	0.372822
$260$	12.0000	0.744208
$261$	−4.00000	−0.247594
$262$	−34.0000	−2.10053
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	0	0
$266$	16.0000	0.981023
$267$	−3.00000	−0.183597
$268$	−6.00000	−0.366508
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	2.00000	0.121716
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−28.0000	−1.69775
$273$	6.00000	0.363137
$274$	−24.0000	−1.44989
$275$	0	0
$276$	−12.0000	−0.722315
$277$	23.0000	1.38194	0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000	1.38194	0.690968	+	0.722885i	$0.257185\pi$
$278$	−4.00000	−0.239904
$279$	2.00000	0.119737
$280$	0	0
$281$	−28.0000	−1.67034	−0.835170	−	0.549992i	$-0.814631\pi$
$281$	−28.0000	−1.67034	−0.835170	+	0.549992i	$0.814631\pi$
$282$	26.0000	1.54828
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	−12.0000	−0.712069
$285$	−8.00000	−0.473879
$286$	0	0
$287$	−2.00000	−0.118056
$288$	8.00000	0.471405
$289$	32.0000	1.88235
$290$	8.00000	0.469776
$291$	4.00000	0.234484
$292$	−28.0000	−1.63858
$293$	−11.0000	−0.642627	−0.321313	−	0.946973i	$-0.604124\pi$
$293$	−11.0000	−0.642627	−0.321313	+	0.946973i	$0.604124\pi$
$294$	2.00000	0.116642
$295$	1.00000	0.0582223
$296$	0	0
$297$	0	0
$298$	−32.0000	−1.85371
$299$	36.0000	2.08193
$300$	8.00000	0.461880
$301$	−1.00000	−0.0576390
$302$	−38.0000	−2.18665
$303$	3.00000	0.172345
$304$	−32.0000	−1.83533
$305$	10.0000	0.572598
$306$	−14.0000	−0.800327
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	0	0
$309$	6.00000	0.341328
$310$	−4.00000	−0.227185
$311$	−13.0000	−0.737162	−0.368581	−	0.929596i	$-0.620156\pi$
$311$	−13.0000	−0.737162	−0.368581	+	0.929596i	$0.620156\pi$
$312$	0	0
$313$	24.0000	1.35656	0.678280	−	0.734803i	$-0.262726\pi$
$313$	24.0000	1.35656	0.678280	+	0.734803i	$0.262726\pi$
$314$	16.0000	0.902932
$315$	−1.00000	−0.0563436
$316$	−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
$320$	−8.00000	−0.447214
$321$	6.00000	0.334887
$322$	12.0000	0.668734
$323$	56.0000	3.11592
$324$	2.00000	0.111111
$325$	−24.0000	−1.33128
$326$	24.0000	1.32924
$327$	19.0000	1.05070
$328$	0	0
$329$	−13.0000	−0.716713
$330$	0	0
$331$	19.0000	1.04433	0.522167	−	0.852843i	$-0.325124\pi$
$331$	19.0000	1.04433	0.522167	+	0.852843i	$0.325124\pi$
$332$	10.0000	0.548821
$333$	−6.00000	−0.328798
$334$	−34.0000	−1.86040
$335$	−3.00000	−0.163908
$336$	−4.00000	−0.218218
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	−46.0000	−2.50207
$339$	18.0000	0.977626
$340$	14.0000	0.759257
$341$	0	0
$342$	−16.0000	−0.865181
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−6.00000	−0.323029
$346$	−26.0000	−1.39777
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	8.00000	0.428845
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	−8.00000	−0.427618
$351$	−6.00000	−0.320256
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	2.00000	0.106299
$355$	−6.00000	−0.318447
$356$	6.00000	0.317999
$357$	7.00000	0.370479
$358$	−24.0000	−1.26844
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	−16.0000	−0.840941
$363$	0	0
$364$	−12.0000	−0.628971
$365$	−14.0000	−0.732793
$366$	20.0000	1.04542
$367$	−20.0000	−1.04399	−0.521996	−	0.852948i	$-0.674812\pi$
$367$	−20.0000	−1.04399	−0.521996	+	0.852948i	$0.674812\pi$
$368$	−24.0000	−1.25109
$369$	2.00000	0.104116
$370$	12.0000	0.623850
$371$	0	0
$372$	−4.00000	−0.207390
$373$	5.00000	0.258890	0.129445	−	0.991587i	$-0.458680\pi$
$373$	5.00000	0.258890	0.129445	+	0.991587i	$0.458680\pi$
$374$	0	0
$375$	9.00000	0.464758
$376$	0	0
$377$	−24.0000	−1.23606
$378$	−2.00000	−0.102869
$379$	−5.00000	−0.256833	−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000	−0.256833	−0.128416	+	0.991720i	$0.540989\pi$
$380$	16.0000	0.820783
$381$	11.0000	0.563547
$382$	0	0
$383$	23.0000	1.17525	0.587623	−	0.809135i	$-0.300064\pi$
$383$	23.0000	1.17525	0.587623	+	0.809135i	$0.300064\pi$
$384$	0	0
$385$	0	0
$386$	46.0000	2.34134
$387$	1.00000	0.0508329
$388$	−8.00000	−0.406138
$389$	−36.0000	−1.82527	−0.912636	−	0.408773i	$-0.865957\pi$
$389$	−36.0000	−1.82527	−0.912636	+	0.408773i	$0.865957\pi$
$390$	12.0000	0.607644
$391$	42.0000	2.12403
$392$	0	0
$393$	−17.0000	−0.857537
$394$	20.0000	1.00759
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	4.00000	0.200502
$399$	8.00000	0.400501
$400$	16.0000	0.800000
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	−6.00000	−0.299253
$403$	12.0000	0.597763
$404$	−6.00000	−0.298511
$405$	1.00000	0.0496904
$406$	−8.00000	−0.397033
$407$	0	0
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	−4.00000	−0.197546
$411$	−12.0000	−0.591916
$412$	−12.0000	−0.591198
$413$	−1.00000	−0.0492068
$414$	−12.0000	−0.589768
$415$	5.00000	0.245440
$416$	48.0000	2.35339
$417$	−2.00000	−0.0979404
$418$	0	0
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	2.00000	0.0975900
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	−30.0000	−1.46038
$423$	13.0000	0.632082
$424$	0	0
$425$	−28.0000	−1.35820
$426$	−12.0000	−0.581402
$427$	−10.0000	−0.483934
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	4.00000	0.192450
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	4.00000	0.192006
$435$	4.00000	0.191785
$436$	−38.0000	−1.81987
$437$	48.0000	2.29615
$438$	−28.0000	−1.33789
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−84.0000	−3.99547
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	12.0000	0.569495
$445$	3.00000	0.142214
$446$	0	0
$447$	−16.0000	−0.756774
$448$	8.00000	0.377964
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	8.00000	0.377124
$451$	0	0
$452$	−36.0000	−1.69330
$453$	−19.0000	−0.892698
$454$	−6.00000	−0.281594
$455$	−6.00000	−0.281284
$456$	0	0
$457$	−29.0000	−1.35656	−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000	−1.35656	−0.678281	+	0.734802i	$0.737275\pi$
$458$	40.0000	1.86908
$459$	−7.00000	−0.326732
$460$	12.0000	0.559503
$461$	37.0000	1.72326	0.861631	−	0.507535i	$-0.169443\pi$
$461$	37.0000	1.72326	0.861631	+	0.507535i	$0.169443\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	16.0000	0.742781
$465$	−2.00000	−0.0927478
$466$	16.0000	0.741186
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	12.0000	0.554700
$469$	3.00000	0.138527
$470$	−26.0000	−1.19929
$471$	8.00000	0.368621
$472$	0	0
$473$	0	0
$474$	−8.00000	−0.367452
$475$	−32.0000	−1.46826
$476$	−14.0000	−0.641689
$477$	0	0
$478$	−36.0000	−1.64660
$479$	−39.0000	−1.78196	−0.890978	−	0.454047i	$-0.849980\pi$
$479$	−39.0000	−1.78196	−0.890978	+	0.454047i	$0.849980\pi$
$480$	−8.00000	−0.365148
$481$	−36.0000	−1.64146
$482$	16.0000	0.728780
$483$	6.00000	0.273009
$484$	0	0
$485$	−4.00000	−0.181631
$486$	2.00000	0.0907218
$487$	−19.0000	−0.860972	−0.430486	−	0.902597i	$-0.641658\pi$
$487$	−19.0000	−0.860972	−0.430486	+	0.902597i	$0.641658\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	−2.00000	−0.0903508
$491$	−26.0000	−1.17336	−0.586682	−	0.809818i	$-0.699566\pi$
$491$	−26.0000	−1.17336	−0.586682	+	0.809818i	$0.699566\pi$
$492$	−4.00000	−0.180334
$493$	−28.0000	−1.26106
$494$	−96.0000	−4.31924
$495$	0	0
$496$	−8.00000	−0.359211
$497$	6.00000	0.269137
$498$	10.0000	0.448111
$499$	11.0000	0.492428	0.246214	−	0.969216i	$-0.420813\pi$
$499$	11.0000	0.492428	0.246214	+	0.969216i	$0.420813\pi$
$500$	−18.0000	−0.804984
$501$	−17.0000	−0.759504
$502$	−24.0000	−1.07117
$503$	15.0000	0.668817	0.334408	−	0.942428i	$-0.391463\pi$
$503$	15.0000	0.668817	0.334408	+	0.942428i	$0.391463\pi$
$504$	0	0
$505$	−3.00000	−0.133498
$506$	0	0
$507$	−23.0000	−1.02147
$508$	−22.0000	−0.976092
$509$	21.0000	0.930809	0.465404	−	0.885098i	$-0.345909\pi$
$509$	21.0000	0.930809	0.465404	+	0.885098i	$0.345909\pi$
$510$	14.0000	0.619930
$511$	14.0000	0.619324
$512$	−32.0000	−1.41421
$513$	−8.00000	−0.353209
$514$	−6.00000	−0.264649
$515$	−6.00000	−0.264392
$516$	−2.00000	−0.0880451
$517$	0	0
$518$	−12.0000	−0.527250
$519$	−13.0000	−0.570637
$520$	0	0
$521$	5.00000	0.219054	0.109527	−	0.993984i	$-0.465066\pi$
$521$	5.00000	0.219054	0.109527	+	0.993984i	$0.465066\pi$
$522$	8.00000	0.350150
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	34.0000	1.48530
$525$	−4.00000	−0.174574
$526$	12.0000	0.523225
$527$	14.0000	0.609850
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	1.00000	0.0433963
$532$	−16.0000	−0.693688
$533$	12.0000	0.519778
$534$	6.00000	0.259645
$535$	−6.00000	−0.259403
$536$	0	0
$537$	−12.0000	−0.517838
$538$	−4.00000	−0.172452
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	7.00000	0.300954	0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000	0.300954	0.150477	+	0.988614i	$0.451919\pi$
$542$	−40.0000	−1.71815
$543$	−8.00000	−0.343313
$544$	56.0000	2.40098
$545$	−19.0000	−0.813871
$546$	−12.0000	−0.513553
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	24.0000	1.02523
$549$	10.0000	0.426790
$550$	0	0
$551$	−32.0000	−1.36325
$552$	0	0
$553$	4.00000	0.170097
$554$	−46.0000	−1.95435
$555$	6.00000	0.254686
$556$	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	$0.540571\pi$
$558$	−4.00000	−0.169334
$559$	6.00000	0.253773
$560$	4.00000	0.169031
$561$	0	0
$562$	56.0000	2.36222
$563$	−39.0000	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$563$	−39.0000	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$564$	−26.0000	−1.09480
$565$	−18.0000	−0.757266
$566$	40.0000	1.68133
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	16.0000	0.670166
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	0	0
$574$	4.00000	0.166957
$575$	−24.0000	−1.00087
$576$	−8.00000	−0.333333
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−64.0000	−2.66205
$579$	23.0000	0.955847
$580$	−8.00000	−0.332182
$581$	−5.00000	−0.207435
$582$	−8.00000	−0.331611
$583$	0	0
$584$	0	0
$585$	6.00000	0.248069
$586$	22.0000	0.908812
$587$	−21.0000	−0.866763	−0.433381	−	0.901211i	$-0.642680\pi$
$587$	−21.0000	−0.866763	−0.433381	+	0.901211i	$0.642680\pi$
$588$	−2.00000	−0.0824786
$589$	16.0000	0.659269
$590$	−2.00000	−0.0823387
$591$	10.0000	0.411345
$592$	24.0000	0.986394
$593$	23.0000	0.944497	0.472248	−	0.881466i	$-0.343443\pi$
$593$	23.0000	0.944497	0.472248	+	0.881466i	$0.343443\pi$
$594$	0	0
$595$	−7.00000	−0.286972
$596$	32.0000	1.31077
$597$	2.00000	0.0818546
$598$	−72.0000	−2.94430
$599$	−38.0000	−1.55264	−0.776319	−	0.630340i	$-0.782915\pi$
$599$	−38.0000	−1.55264	−0.776319	+	0.630340i	$0.782915\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	2.00000	0.0815139
$603$	−3.00000	−0.122169
$604$	38.0000	1.54620
$605$	0	0
$606$	−6.00000	−0.243733
$607$	6.00000	0.243532	0.121766	−	0.992559i	$-0.461144\pi$
$607$	6.00000	0.243532	0.121766	+	0.992559i	$0.461144\pi$
$608$	64.0000	2.59554
$609$	−4.00000	−0.162088
$610$	−20.0000	−0.809776
$611$	78.0000	3.15554
$612$	14.0000	0.565916
$613$	−5.00000	−0.201948	−0.100974	−	0.994889i	$-0.532196\pi$
$613$	−5.00000	−0.201948	−0.100974	+	0.994889i	$0.532196\pi$
$614$	16.0000	0.645707
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−12.0000	−0.482711
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	4.00000	0.160644
$621$	−6.00000	−0.240772
$622$	26.0000	1.04251
$623$	−3.00000	−0.120192
$624$	24.0000	0.960769
$625$	11.0000	0.440000
$626$	−48.0000	−1.91847
$627$	0	0
$628$	−16.0000	−0.638470
$629$	−42.0000	−1.67465
$630$	2.00000	0.0796819
$631$	31.0000	1.23409	0.617045	−	0.786928i	$-0.288330\pi$
$631$	31.0000	1.23409	0.617045	+	0.786928i	$0.288330\pi$
$632$	0	0
$633$	−15.0000	−0.596196
$634$	−12.0000	−0.476581
$635$	−11.0000	−0.436522
$636$	0	0
$637$	6.00000	0.237729
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	−12.0000	−0.473602
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	−12.0000	−0.472866
$645$	−1.00000	−0.0393750
$646$	−112.000	−4.40658
$647$	−25.0000	−0.982851	−0.491426	−	0.870919i	$-0.663524\pi$
$647$	−25.0000	−0.982851	−0.491426	+	0.870919i	$0.663524\pi$
$648$	0	0
$649$	0	0
$650$	48.0000	1.88271
$651$	2.00000	0.0783862
$652$	−24.0000	−0.939913
$653$	12.0000	0.469596	0.234798	−	0.972044i	$-0.424557\pi$
$653$	12.0000	0.469596	0.234798	+	0.972044i	$0.424557\pi$
$654$	−38.0000	−1.48592
$655$	17.0000	0.664245
$656$	−8.00000	−0.312348
$657$	−14.0000	−0.546192
$658$	26.0000	1.01359
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	−38.0000	−1.47691
$663$	−42.0000	−1.63114
$664$	0	0
$665$	−8.00000	−0.310227
$666$	12.0000	0.464991
$667$	−24.0000	−0.929284
$668$	34.0000	1.31550
$669$	0	0
$670$	6.00000	0.231800
$671$	0	0
$672$	8.00000	0.308607
$673$	−33.0000	−1.27206	−0.636028	−	0.771666i	$-0.719424\pi$
$673$	−33.0000	−1.27206	−0.636028	+	0.771666i	$0.719424\pi$
$674$	−20.0000	−0.770371
$675$	4.00000	0.153960
$676$	46.0000	1.76923
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	−36.0000	−1.38257
$679$	4.00000	0.153506
$680$	0	0
$681$	−3.00000	−0.114960
$682$	0	0
$683$	−18.0000	−0.688751	−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000	−0.688751	−0.344375	+	0.938832i	$0.611909\pi$
$684$	16.0000	0.611775
$685$	12.0000	0.458496
$686$	2.00000	0.0763604
$687$	20.0000	0.763048
$688$	−4.00000	−0.152499
$689$	0	0
$690$	12.0000	0.456832
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	26.0000	0.988372
$693$	0	0
$694$	−40.0000	−1.51838
$695$	2.00000	0.0758643
$696$	0	0
$697$	14.0000	0.530288
$698$	52.0000	1.96823
$699$	8.00000	0.302588
$700$	8.00000	0.302372
$701$	−48.0000	−1.81293	−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000	−1.81293	−0.906467	+	0.422276i	$0.861231\pi$
$702$	12.0000	0.452911
$703$	−48.0000	−1.81035
$704$	0	0
$705$	−13.0000	−0.489608
$706$	−28.0000	−1.05379
$707$	3.00000	0.112827
$708$	−2.00000	−0.0751646
$709$	−49.0000	−1.84023	−0.920117	−	0.391644i	$-0.871906\pi$
$709$	−49.0000	−1.84023	−0.920117	+	0.391644i	$0.871906\pi$
$710$	12.0000	0.450352
$711$	−4.00000	−0.150012
$712$	0	0
$713$	12.0000	0.449404
$714$	−14.0000	−0.523937
$715$	0	0
$716$	24.0000	0.896922
$717$	−18.0000	−0.672222
$718$	−36.0000	−1.34351
$719$	−52.0000	−1.93927	−0.969636	−	0.244551i	$-0.921359\pi$
$719$	−52.0000	−1.93927	−0.969636	+	0.244551i	$0.921359\pi$
$720$	−4.00000	−0.149071
$721$	6.00000	0.223452
$722$	−90.0000	−3.34945
$723$	8.00000	0.297523
$724$	16.0000	0.594635
$725$	16.0000	0.594225
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	28.0000	1.03633
$731$	7.00000	0.258904
$732$	−20.0000	−0.739221
$733$	−6.00000	−0.221615	−0.110808	−	0.993842i	$-0.535344\pi$
$733$	−6.00000	−0.221615	−0.110808	+	0.993842i	$0.535344\pi$
$734$	40.0000	1.47643
$735$	−1.00000	−0.0368856
$736$	48.0000	1.76930
$737$	0	0
$738$	−4.00000	−0.147242
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−12.0000	−0.441129
$741$	−48.0000	−1.76332
$742$	0	0
$743$	−30.0000	−1.10059	−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000	−1.10059	−0.550297	+	0.834969i	$0.685485\pi$
$744$	0	0
$745$	16.0000	0.586195
$746$	−10.0000	−0.366126
$747$	5.00000	0.182940
$748$	0	0
$749$	6.00000	0.219235
$750$	−18.0000	−0.657267
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	−52.0000	−1.89624
$753$	−12.0000	−0.437304
$754$	48.0000	1.74806
$755$	19.0000	0.691481
$756$	2.00000	0.0727393
$757$	23.0000	0.835949	0.417975	−	0.908459i	$-0.362740\pi$
$757$	23.0000	0.835949	0.417975	+	0.908459i	$0.362740\pi$
$758$	10.0000	0.363216
$759$	0	0
$760$	0	0
$761$	17.0000	0.616250	0.308125	−	0.951346i	$-0.400299\pi$
$761$	17.0000	0.616250	0.308125	+	0.951346i	$0.400299\pi$
$762$	−22.0000	−0.796976
$763$	19.0000	0.687846
$764$	0	0
$765$	7.00000	0.253086
$766$	−46.0000	−1.66205
$767$	6.00000	0.216647
$768$	−16.0000	−0.577350
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	−3.00000	−0.108042
$772$	−46.0000	−1.65558
$773$	−21.0000	−0.755318	−0.377659	−	0.925945i	$-0.623271\pi$
$773$	−21.0000	−0.755318	−0.377659	+	0.925945i	$0.623271\pi$
$774$	−2.00000	−0.0718885
$775$	−8.00000	−0.287368
$776$	0	0
$777$	−6.00000	−0.215249
$778$	72.0000	2.58133
$779$	16.0000	0.573259
$780$	−12.0000	−0.429669
$781$	0	0
$782$	−84.0000	−3.00383
$783$	4.00000	0.142948
$784$	−4.00000	−0.142857
$785$	−8.00000	−0.285532
$786$	34.0000	1.21274
$787$	−42.0000	−1.49714	−0.748569	−	0.663057i	$-0.769259\pi$
$787$	−42.0000	−1.49714	−0.748569	+	0.663057i	$0.769259\pi$
$788$	−20.0000	−0.712470
$789$	6.00000	0.213606
$790$	8.00000	0.284627
$791$	18.0000	0.640006
$792$	0	0
$793$	60.0000	2.13066
$794$	20.0000	0.709773
$795$	0	0
$796$	−4.00000	−0.141776
$797$	15.0000	0.531327	0.265664	−	0.964066i	$-0.414409\pi$
$797$	15.0000	0.531327	0.265664	+	0.964066i	$0.414409\pi$
$798$	−16.0000	−0.566394
$799$	91.0000	3.21935
$800$	−32.0000	−1.13137
$801$	3.00000	0.106000
$802$	16.0000	0.564980
$803$	0	0
$804$	6.00000	0.211604
$805$	−6.00000	−0.211472
$806$	−24.0000	−0.845364
$807$	−2.00000	−0.0704033
$808$	0	0
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	−2.00000	−0.0702728
$811$	−32.0000	−1.12367	−0.561836	−	0.827249i	$-0.689905\pi$
$811$	−32.0000	−1.12367	−0.561836	+	0.827249i	$0.689905\pi$
$812$	8.00000	0.280745
$813$	−20.0000	−0.701431
$814$	0	0
$815$	−12.0000	−0.420342
$816$	28.0000	0.980196
$817$	8.00000	0.279885
$818$	−64.0000	−2.23771
$819$	−6.00000	−0.209657
$820$	4.00000	0.139686
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	24.0000	0.837096
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	0	0
$826$	2.00000	0.0695889
$827$	50.0000	1.73867	0.869335	−	0.494223i	$-0.164547\pi$
$827$	50.0000	1.73867	0.869335	+	0.494223i	$0.164547\pi$
$828$	12.0000	0.417029
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	−10.0000	−0.347105
$831$	−23.0000	−0.797861
$832$	−48.0000	−1.66410
$833$	7.00000	0.242536
$834$	4.00000	0.138509
$835$	17.0000	0.588309
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	18.0000	0.621800
$839$	39.0000	1.34643	0.673215	−	0.739447i	$-0.264913\pi$
$839$	39.0000	1.34643	0.673215	+	0.739447i	$0.264913\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	30.0000	1.03387
$843$	28.0000	0.964371
$844$	30.0000	1.03264
$845$	23.0000	0.791224
$846$	−26.0000	−0.893898
$847$	0	0
$848$	0	0
$849$	20.0000	0.686398
$850$	56.0000	1.92078
$851$	−36.0000	−1.23406
$852$	12.0000	0.411113
$853$	−32.0000	−1.09566	−0.547830	−	0.836590i	$-0.684546\pi$
$853$	−32.0000	−1.09566	−0.547830	+	0.836590i	$0.684546\pi$
$854$	20.0000	0.684386
$855$	8.00000	0.273594
$856$	0	0
$857$	5.00000	0.170797	0.0853984	−	0.996347i	$-0.472784\pi$
$857$	5.00000	0.170797	0.0853984	+	0.996347i	$0.472784\pi$
$858$	0	0
$859$	10.0000	0.341196	0.170598	−	0.985341i	$-0.445430\pi$
$859$	10.0000	0.341196	0.170598	+	0.985341i	$0.445430\pi$
$860$	2.00000	0.0681994
$861$	2.00000	0.0681598
$862$	24.0000	0.817443
$863$	−34.0000	−1.15737	−0.578687	−	0.815550i	$-0.696435\pi$
$863$	−34.0000	−1.15737	−0.578687	+	0.815550i	$0.696435\pi$
$864$	−8.00000	−0.272166
$865$	13.0000	0.442013
$866$	0	0
$867$	−32.0000	−1.08678
$868$	−4.00000	−0.135769
$869$	0	0
$870$	−8.00000	−0.271225
$871$	−18.0000	−0.609907
$872$	0	0
$873$	−4.00000	−0.135379
$874$	−96.0000	−3.24725
$875$	9.00000	0.304256
$876$	28.0000	0.946032
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−20.0000	−0.674967
$879$	11.0000	0.371021
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	−2.00000	−0.0673435
$883$	27.0000	0.908622	0.454311	−	0.890843i	$-0.349885\pi$
$883$	27.0000	0.908622	0.454311	+	0.890843i	$0.349885\pi$
$884$	84.0000	2.82523
$885$	−1.00000	−0.0336146
$886$	36.0000	1.20944
$887$	53.0000	1.77957	0.889783	−	0.456384i	$-0.150856\pi$
$887$	53.0000	1.77957	0.889783	+	0.456384i	$0.150856\pi$
$888$	0	0
$889$	11.0000	0.368928
$890$	−6.00000	−0.201120
$891$	0	0
$892$	0	0
$893$	104.000	3.48023
$894$	32.0000	1.07024
$895$	12.0000	0.401116
$896$	0	0
$897$	−36.0000	−1.20201
$898$	52.0000	1.73526
$899$	−8.00000	−0.266815
$900$	−8.00000	−0.266667
$901$	0	0
$902$	0	0
$903$	1.00000	0.0332779
$904$	0	0
$905$	8.00000	0.265929
$906$	38.0000	1.26247
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	6.00000	0.199117
$909$	−3.00000	−0.0995037
$910$	12.0000	0.397796
$911$	−14.0000	−0.463841	−0.231920	−	0.972735i	$-0.574501\pi$
$911$	−14.0000	−0.463841	−0.231920	+	0.972735i	$0.574501\pi$
$912$	32.0000	1.05963
$913$	0	0
$914$	58.0000	1.91847
$915$	−10.0000	−0.330590
$916$	−40.0000	−1.32164
$917$	−17.0000	−0.561389
$918$	14.0000	0.462069
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	−74.0000	−2.43706
$923$	−36.0000	−1.18495
$924$	0	0
$925$	24.0000	0.789115
$926$	−32.0000	−1.05159
$927$	−6.00000	−0.197066
$928$	−32.0000	−1.05045
$929$	−9.00000	−0.295280	−0.147640	−	0.989041i	$-0.547168\pi$
$929$	−9.00000	−0.295280	−0.147640	+	0.989041i	$0.547168\pi$
$930$	4.00000	0.131165
$931$	8.00000	0.262189
$932$	−16.0000	−0.524097
$933$	13.0000	0.425601
$934$	24.0000	0.785304
$935$	0	0
$936$	0	0
$937$	4.00000	0.130674	0.0653372	−	0.997863i	$-0.479188\pi$
$937$	4.00000	0.130674	0.0653372	+	0.997863i	$0.479188\pi$
$938$	−6.00000	−0.195907
$939$	−24.0000	−0.783210
$940$	26.0000	0.848026
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	−16.0000	−0.521308
$943$	12.0000	0.390774
$944$	−4.00000	−0.130189
$945$	1.00000	0.0325300
$946$	0	0
$947$	−30.0000	−0.974869	−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000	−0.974869	−0.487435	+	0.873160i	$0.662067\pi$
$948$	8.00000	0.259828
$949$	−84.0000	−2.72676
$950$	64.0000	2.07643
$951$	−6.00000	−0.194563
$952$	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
$955$	0	0
$956$	36.0000	1.16432
$957$	0	0
$958$	78.0000	2.52007
$959$	−12.0000	−0.387500
$960$	8.00000	0.258199
$961$	−27.0000	−0.870968
$962$	72.0000	2.32137
$963$	−6.00000	−0.193347
$964$	−16.0000	−0.515325
$965$	−23.0000	−0.740396
$966$	−12.0000	−0.386094
$967$	1.00000	0.0321578	0.0160789	−	0.999871i	$-0.494882\pi$
$967$	1.00000	0.0321578	0.0160789	+	0.999871i	$0.494882\pi$
$968$	0	0
$969$	−56.0000	−1.79898
$970$	8.00000	0.256865
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	−2.00000	−0.0641500
$973$	−2.00000	−0.0641171
$974$	38.0000	1.21760
$975$	24.0000	0.768615
$976$	−40.0000	−1.28037
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	−24.0000	−0.767435
$979$	0	0
$980$	2.00000	0.0638877
$981$	−19.0000	−0.606623
$982$	52.0000	1.65939
$983$	56.0000	1.78612	0.893061	−	0.449935i	$-0.148553\pi$
$983$	56.0000	1.78612	0.893061	+	0.449935i	$0.148553\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	56.0000	1.78340
$987$	13.0000	0.413795
$988$	96.0000	3.05417
$989$	6.00000	0.190789
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	16.0000	0.508001
$993$	−19.0000	−0.602947
$994$	−12.0000	−0.380617
$995$	−2.00000	−0.0634043
$996$	−10.0000	−0.316862
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−22.0000	−0.696398
$999$	6.00000	0.189832

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.b.1.1	✓	1	1.1	even	1	trivial
2541.2.a.l.1.1	yes	1	11.10	odd	2
7623.2.a.a.1.1		1	33.32	even	2
7623.2.a.q.1.1		1	3.2	odd	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:	\( N \)	\(=\)	\( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2541.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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