LMFDB - Embedded newform 2240.2.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2240,2,Mod(1,2240)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2240.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,1,0,-1,0,6,0,5,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.8864900528$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2240.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +6.00000 q^{9} +5.00000 q^{11} +3.00000 q^{13} -3.00000 q^{15} -1.00000 q^{17} -6.00000 q^{19} +3.00000 q^{21} +6.00000 q^{23} +1.00000 q^{25} -9.00000 q^{27} +9.00000 q^{29} -4.00000 q^{31} -15.0000 q^{33} -1.00000 q^{35} -2.00000 q^{37} -9.00000 q^{39} -4.00000 q^{41} -10.0000 q^{43} +6.00000 q^{45} -1.00000 q^{47} +1.00000 q^{49} +3.00000 q^{51} -4.00000 q^{53} +5.00000 q^{55} +18.0000 q^{57} +8.00000 q^{59} +8.00000 q^{61} -6.00000 q^{63} +3.00000 q^{65} -12.0000 q^{67} -18.0000 q^{69} +8.00000 q^{71} +2.00000 q^{73} -3.00000 q^{75} -5.00000 q^{77} +13.0000 q^{79} +9.00000 q^{81} +4.00000 q^{83} -1.00000 q^{85} -27.0000 q^{87} +4.00000 q^{89} -3.00000 q^{91} +12.0000 q^{93} -6.00000 q^{95} -13.0000 q^{97} +30.0000 q^{99} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	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
$25$	1.00000	0.200000
$26$	0	0
$27$	−9.00000	−1.73205
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−15.0000	−2.61116
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−9.00000	−1.44115
$40$	0	0
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	0	0
$47$	−1.00000	−0.145865	−0.0729325	−	0.997337i	$-0.523236\pi$
$47$	−1.00000	−0.145865	−0.0729325	+	0.997337i	$0.523236\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	5.00000	0.674200
$56$	0	0
$57$	18.0000	2.38416
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	−6.00000	−0.755929
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	−18.0000	−2.16695
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	−5.00000	−0.569803
$78$	0	0
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	−27.0000	−2.89470
$88$	0	0
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	12.0000	1.24434
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	0	0
$99$	30.0000	3.01511
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	19.0000	1.87213	0.936063	−	0.351833i	$-0.114441\pi$
$103$	19.0000	1.87213	0.936063	+	0.351833i	$0.114441\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	3.00000	0.287348	0.143674	−	0.989625i	$-0.454108\pi$
$109$	3.00000	0.287348	0.143674	+	0.989625i	$0.454108\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	0	0
$117$	18.0000	1.66410
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	12.0000	1.08200
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	30.0000	2.64135
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	−9.00000	−0.774597
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	0	0
$143$	15.0000	1.25436
$144$	0	0
$145$	9.00000	0.747409
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	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
$153$	−6.00000	−0.485071
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	12.0000	0.951662
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	0	0
$165$	−15.0000	−1.16775
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−36.0000	−2.75299
$172$	0	0
$173$	1.00000	0.0760286	0.0380143	−	0.999277i	$-0.487897\pi$
$173$	1.00000	0.0760286	0.0380143	+	0.999277i	$0.487897\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−24.0000	−1.80395
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	−24.0000	−1.77413
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−5.00000	−0.365636
$188$	0	0
$189$	9.00000	0.654654
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	−9.00000	−0.644503
$196$	0	0
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	36.0000	2.53924
$202$	0	0
$203$	−9.00000	−0.631676
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	36.0000	2.50217
$208$	0	0
$209$	−30.0000	−2.07514
$210$	0	0
$211$	11.0000	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$212$	0	0
$213$	−24.0000	−1.64445
$214$	0	0
$215$	−10.0000	−0.681994
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−3.00000	−0.201802
$222$	0	0
$223$	−5.00000	−0.334825	−0.167412	−	0.985887i	$-0.553541\pi$
$223$	−5.00000	−0.334825	−0.167412	+	0.985887i	$0.553541\pi$
$224$	0	0
$225$	6.00000	0.400000
$226$	0	0
$227$	1.00000	0.0663723	0.0331862	−	0.999449i	$-0.489435\pi$
$227$	1.00000	0.0663723	0.0331862	+	0.999449i	$0.489435\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	15.0000	0.986928
$232$	0	0
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	−1.00000	−0.0652328
$236$	0	0
$237$	−39.0000	−2.53332
$238$	0	0
$239$	1.00000	0.0646846	0.0323423	−	0.999477i	$-0.489703\pi$
$239$	1.00000	0.0646846	0.0323423	+	0.999477i	$0.489703\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−18.0000	−1.14531
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	0	0
$253$	30.0000	1.88608
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	54.0000	3.34252
$262$	0	0
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	−12.0000	−0.734388
$268$	0	0
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	9.00000	0.544705
$274$	0	0
$275$	5.00000	0.301511
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	0	0
$279$	−24.0000	−1.43684
$280$	0	0
$281$	11.0000	0.656205	0.328102	−	0.944642i	$-0.393591\pi$
$281$	11.0000	0.656205	0.328102	+	0.944642i	$0.393591\pi$
$282$	0	0
$283$	31.0000	1.84276	0.921379	−	0.388664i	$-0.127063\pi$
$283$	31.0000	1.84276	0.921379	+	0.388664i	$0.127063\pi$
$284$	0	0
$285$	18.0000	1.06623
$286$	0	0
$287$	4.00000	0.236113
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	39.0000	2.28622
$292$	0	0
$293$	5.00000	0.292103	0.146052	−	0.989277i	$-0.453343\pi$
$293$	5.00000	0.292103	0.146052	+	0.989277i	$0.453343\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	−45.0000	−2.61116
$298$	0	0
$299$	18.0000	1.04097
$300$	0	0
$301$	10.0000	0.576390
$302$	0	0
$303$	18.0000	1.03407
$304$	0	0
$305$	8.00000	0.458079
$306$	0	0
$307$	23.0000	1.31268	0.656340	−	0.754466i	$-0.272104\pi$
$307$	23.0000	1.31268	0.656340	+	0.754466i	$0.272104\pi$
$308$	0	0
$309$	−57.0000	−3.24262
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	−7.00000	−0.395663	−0.197832	−	0.980236i	$-0.563390\pi$
$313$	−7.00000	−0.395663	−0.197832	+	0.980236i	$0.563390\pi$
$314$	0	0
$315$	−6.00000	−0.338062
$316$	0	0
$317$	26.0000	1.46031	0.730153	−	0.683284i	$-0.239449\pi$
$317$	26.0000	1.46031	0.730153	+	0.683284i	$0.239449\pi$
$318$	0	0
$319$	45.0000	2.51952
$320$	0	0
$321$	−18.0000	−1.00466
$322$	0	0
$323$	6.00000	0.333849
$324$	0	0
$325$	3.00000	0.166410
$326$	0	0
$327$	−9.00000	−0.497701
$328$	0	0
$329$	1.00000	0.0551318
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	−12.0000	−0.657596
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	−42.0000	−2.28113
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−18.0000	−0.969087
$346$	0	0
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−27.0000	−1.44115
$352$	0	0
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	0	0
$357$	−3.00000	−0.158777
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−42.0000	−2.20443
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	0	0
$369$	−24.0000	−1.24939
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	27.0000	1.39057
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	24.0000	1.22956
$382$	0	0
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	−5.00000	−0.254824
$386$	0	0
$387$	−60.0000	−3.04997
$388$	0	0
$389$	31.0000	1.57176	0.785881	−	0.618378i	$-0.212210\pi$
$389$	31.0000	1.57176	0.785881	+	0.618378i	$0.212210\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	0	0
$393$	−30.0000	−1.51330
$394$	0	0
$395$	13.0000	0.654101
$396$	0	0
$397$	−7.00000	−0.351320	−0.175660	−	0.984451i	$-0.556206\pi$
$397$	−7.00000	−0.351320	−0.175660	+	0.984451i	$0.556206\pi$
$398$	0	0
$399$	−18.0000	−0.901127
$400$	0	0
$401$	25.0000	1.24844	0.624220	−	0.781248i	$-0.285417\pi$
$401$	25.0000	1.24844	0.624220	+	0.781248i	$0.285417\pi$
$402$	0	0
$403$	−12.0000	−0.597763
$404$	0	0
$405$	9.00000	0.447214
$406$	0	0
$407$	−10.0000	−0.495682
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	36.0000	1.77575
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	−42.0000	−2.05675
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	19.0000	0.926003	0.463002	−	0.886357i	$-0.346772\pi$
$421$	19.0000	0.926003	0.463002	+	0.886357i	$0.346772\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−8.00000	−0.387147
$428$	0	0
$429$	−45.0000	−2.17262
$430$	0	0
$431$	−9.00000	−0.433515	−0.216757	−	0.976226i	$-0.569548\pi$
$431$	−9.00000	−0.433515	−0.216757	+	0.976226i	$0.569548\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	−27.0000	−1.29455
$436$	0	0
$437$	−36.0000	−1.72211
$438$	0	0
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	14.0000	0.665160	0.332580	−	0.943075i	$-0.392081\pi$
$443$	14.0000	0.665160	0.332580	+	0.943075i	$0.392081\pi$
$444$	0	0
$445$	4.00000	0.189618
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	−9.00000	−0.424736	−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000	−0.424736	−0.212368	+	0.977190i	$0.568118\pi$
$450$	0	0
$451$	−20.0000	−0.941763
$452$	0	0
$453$	−15.0000	−0.704761
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	16.0000	0.748448	0.374224	−	0.927338i	$-0.377909\pi$
$457$	16.0000	0.748448	0.374224	+	0.927338i	$0.377909\pi$
$458$	0	0
$459$	9.00000	0.420084
$460$	0	0
$461$	−40.0000	−1.86299	−0.931493	−	0.363760i	$-0.881493\pi$
$461$	−40.0000	−1.86299	−0.931493	+	0.363760i	$0.881493\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	12.0000	0.556487
$466$	0	0
$467$	−5.00000	−0.231372	−0.115686	−	0.993286i	$-0.536907\pi$
$467$	−5.00000	−0.231372	−0.115686	+	0.993286i	$0.536907\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	−6.00000	−0.276465
$472$	0	0
$473$	−50.0000	−2.29900
$474$	0	0
$475$	−6.00000	−0.275299
$476$	0	0
$477$	−24.0000	−1.09888
$478$	0	0
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	18.0000	0.819028
$484$	0	0
$485$	−13.0000	−0.590300
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	0	0
$489$	−30.0000	−1.35665
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	0	0
$495$	30.0000	1.34840
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	9.00000	0.402895	0.201448	−	0.979499i	$-0.435435\pi$
$499$	9.00000	0.402895	0.201448	+	0.979499i	$0.435435\pi$
$500$	0	0
$501$	−9.00000	−0.402090
$502$	0	0
$503$	−3.00000	−0.133763	−0.0668817	−	0.997761i	$-0.521305\pi$
$503$	−3.00000	−0.133763	−0.0668817	+	0.997761i	$0.521305\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	0	0
$513$	54.0000	2.38416
$514$	0	0
$515$	19.0000	0.837240
$516$	0	0
$517$	−5.00000	−0.219900
$518$	0	0
$519$	−3.00000	−0.131685
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	3.00000	0.130931
$526$	0	0
$527$	4.00000	0.174243
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	48.0000	2.08302
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	−36.0000	−1.55351
$538$	0	0
$539$	5.00000	0.215365
$540$	0	0
$541$	−39.0000	−1.67674	−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000	−1.67674	−0.838370	+	0.545101i	$0.816491\pi$
$542$	0	0
$543$	−60.0000	−2.57485
$544$	0	0
$545$	3.00000	0.128506
$546$	0	0
$547$	24.0000	1.02617	0.513083	−	0.858339i	$-0.328503\pi$
$547$	24.0000	1.02617	0.513083	+	0.858339i	$0.328503\pi$
$548$	0	0
$549$	48.0000	2.04859
$550$	0	0
$551$	−54.0000	−2.30048
$552$	0	0
$553$	−13.0000	−0.552816
$554$	0	0
$555$	6.00000	0.254686
$556$	0	0
$557$	−40.0000	−1.69485	−0.847427	−	0.530912i	$-0.821850\pi$
$557$	−40.0000	−1.69485	−0.847427	+	0.530912i	$0.821850\pi$
$558$	0	0
$559$	−30.0000	−1.26886
$560$	0	0
$561$	15.0000	0.633300
$562$	0	0
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	0	0
$567$	−9.00000	−0.377964
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	−9.00000	−0.375980
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	13.0000	0.541197	0.270599	−	0.962692i	$-0.412778\pi$
$577$	13.0000	0.541197	0.270599	+	0.962692i	$0.412778\pi$
$578$	0	0
$579$	12.0000	0.498703
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	−20.0000	−0.828315
$584$	0	0
$585$	18.0000	0.744208
$586$	0	0
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	24.0000	0.987228
$592$	0	0
$593$	−27.0000	−1.10876	−0.554379	−	0.832265i	$-0.687044\pi$
$593$	−27.0000	−1.10876	−0.554379	+	0.832265i	$0.687044\pi$
$594$	0	0
$595$	1.00000	0.0409960
$596$	0	0
$597$	−24.0000	−0.982255
$598$	0	0
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	−72.0000	−2.93207
$604$	0	0
$605$	14.0000	0.569181
$606$	0	0
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	0	0
$609$	27.0000	1.09410
$610$	0	0
$611$	−3.00000	−0.121367
$612$	0	0
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	−46.0000	−1.85189	−0.925945	−	0.377658i	$-0.876729\pi$
$617$	−46.0000	−1.85189	−0.925945	+	0.377658i	$0.876729\pi$
$618$	0	0
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−54.0000	−2.16695
$622$	0	0
$623$	−4.00000	−0.160257
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	90.0000	3.59425
$628$	0	0
$629$	2.00000	0.0797452
$630$	0	0
$631$	47.0000	1.87104	0.935520	−	0.353273i	$-0.114931\pi$
$631$	47.0000	1.87104	0.935520	+	0.353273i	$0.114931\pi$
$632$	0	0
$633$	−33.0000	−1.31163
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	3.00000	0.118864
$638$	0	0
$639$	48.0000	1.89885
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	0	0
$645$	30.0000	1.18125
$646$	0	0
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	40.0000	1.57014
$650$	0	0
$651$	−12.0000	−0.470317
$652$	0	0
$653$	−2.00000	−0.0782660	−0.0391330	−	0.999234i	$-0.512460\pi$
$653$	−2.00000	−0.0782660	−0.0391330	+	0.999234i	$0.512460\pi$
$654$	0	0
$655$	10.0000	0.390732
$656$	0	0
$657$	12.0000	0.468165
$658$	0	0
$659$	−31.0000	−1.20759	−0.603794	−	0.797140i	$-0.706345\pi$
$659$	−31.0000	−1.20759	−0.603794	+	0.797140i	$0.706345\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	0	0
$663$	9.00000	0.349531
$664$	0	0
$665$	6.00000	0.232670
$666$	0	0
$667$	54.0000	2.09089
$668$	0	0
$669$	15.0000	0.579934
$670$	0	0
$671$	40.0000	1.54418
$672$	0	0
$673$	4.00000	0.154189	0.0770943	−	0.997024i	$-0.475436\pi$
$673$	4.00000	0.154189	0.0770943	+	0.997024i	$0.475436\pi$
$674$	0	0
$675$	−9.00000	−0.346410
$676$	0	0
$677$	27.0000	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$678$	0	0
$679$	13.0000	0.498894
$680$	0	0
$681$	−3.00000	−0.114960
$682$	0	0
$683$	−8.00000	−0.306111	−0.153056	−	0.988218i	$-0.548911\pi$
$683$	−8.00000	−0.306111	−0.153056	+	0.988218i	$0.548911\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	12.0000	0.457829
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	−30.0000	−1.13961
$694$	0	0
$695$	14.0000	0.531050
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	−72.0000	−2.72329
$700$	0	0
$701$	5.00000	0.188847	0.0944237	−	0.995532i	$-0.469899\pi$
$701$	5.00000	0.188847	0.0944237	+	0.995532i	$0.469899\pi$
$702$	0	0
$703$	12.0000	0.452589
$704$	0	0
$705$	3.00000	0.112987
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	−31.0000	−1.16423	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−31.0000	−1.16423	−0.582115	+	0.813107i	$0.697775\pi$
$710$	0	0
$711$	78.0000	2.92523
$712$	0	0
$713$	−24.0000	−0.898807
$714$	0	0
$715$	15.0000	0.560968
$716$	0	0
$717$	−3.00000	−0.112037
$718$	0	0
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	−19.0000	−0.707597
$722$	0	0
$723$	78.0000	2.90085
$724$	0	0
$725$	9.00000	0.334252
$726$	0	0
$727$	24.0000	0.890111	0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000	0.890111	0.445055	+	0.895503i	$0.353184\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	10.0000	0.369863
$732$	0	0
$733$	47.0000	1.73598	0.867992	−	0.496578i	$-0.165410\pi$
$733$	47.0000	1.73598	0.867992	+	0.496578i	$0.165410\pi$
$734$	0	0
$735$	−3.00000	−0.110657
$736$	0	0
$737$	−60.0000	−2.21013
$738$	0	0
$739$	5.00000	0.183928	0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000	0.183928	0.0919640	+	0.995762i	$0.470686\pi$
$740$	0	0
$741$	54.0000	1.98374
$742$	0	0
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	24.0000	0.878114
$748$	0	0
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−19.0000	−0.693320	−0.346660	−	0.937991i	$-0.612684\pi$
$751$	−19.0000	−0.693320	−0.346660	+	0.937991i	$0.612684\pi$
$752$	0	0
$753$	90.0000	3.27978
$754$	0	0
$755$	5.00000	0.181969
$756$	0	0
$757$	32.0000	1.16306	0.581530	−	0.813525i	$-0.302454\pi$
$757$	32.0000	1.16306	0.581530	+	0.813525i	$0.302454\pi$
$758$	0	0
$759$	−90.0000	−3.26679
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	−3.00000	−0.108607
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	24.0000	0.866590
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	−42.0000	−1.51259
$772$	0	0
$773$	−45.0000	−1.61854	−0.809269	−	0.587439i	$-0.800136\pi$
$773$	−45.0000	−1.61854	−0.809269	+	0.587439i	$0.800136\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	−6.00000	−0.215249
$778$	0	0
$779$	24.0000	0.859889
$780$	0	0
$781$	40.0000	1.43131
$782$	0	0
$783$	−81.0000	−2.89470
$784$	0	0
$785$	2.00000	0.0713831
$786$	0	0
$787$	−31.0000	−1.10503	−0.552515	−	0.833503i	$-0.686332\pi$
$787$	−31.0000	−1.10503	−0.552515	+	0.833503i	$0.686332\pi$
$788$	0	0
$789$	−6.00000	−0.213606
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	24.0000	0.852265
$794$	0	0
$795$	12.0000	0.425596
$796$	0	0
$797$	33.0000	1.16892	0.584460	−	0.811423i	$-0.301306\pi$
$797$	33.0000	1.16892	0.584460	+	0.811423i	$0.301306\pi$
$798$	0	0
$799$	1.00000	0.0353775
$800$	0	0
$801$	24.0000	0.847998
$802$	0	0
$803$	10.0000	0.352892
$804$	0	0
$805$	−6.00000	−0.211472
$806$	0	0
$807$	54.0000	1.90089
$808$	0	0
$809$	9.00000	0.316423	0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000	0.316423	0.158212	+	0.987405i	$0.449427\pi$
$810$	0	0
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	10.0000	0.350285
$816$	0	0
$817$	60.0000	2.09913
$818$	0	0
$819$	−18.0000	−0.628971
$820$	0	0
$821$	39.0000	1.36111	0.680555	−	0.732697i	$-0.261739\pi$
$821$	39.0000	1.36111	0.680555	+	0.732697i	$0.261739\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	0	0
$825$	−15.0000	−0.522233
$826$	0	0
$827$	−50.0000	−1.73867	−0.869335	−	0.494223i	$-0.835453\pi$
$827$	−50.0000	−1.73867	−0.869335	+	0.494223i	$0.835453\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	3.00000	0.103819
$836$	0	0
$837$	36.0000	1.24434
$838$	0	0
$839$	2.00000	0.0690477	0.0345238	−	0.999404i	$-0.489009\pi$
$839$	2.00000	0.0690477	0.0345238	+	0.999404i	$0.489009\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	−33.0000	−1.13658
$844$	0	0
$845$	−4.00000	−0.137604
$846$	0	0
$847$	−14.0000	−0.481046
$848$	0	0
$849$	−93.0000	−3.19175
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	−36.0000	−1.23117
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	0	0
$863$	16.0000	0.544646	0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000	0.544646	0.272323	+	0.962206i	$0.412208\pi$
$864$	0	0
$865$	1.00000	0.0340010
$866$	0	0
$867$	48.0000	1.63017
$868$	0	0
$869$	65.0000	2.20497
$870$	0	0
$871$	−36.0000	−1.21981
$872$	0	0
$873$	−78.0000	−2.63990
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−26.0000	−0.877958	−0.438979	−	0.898497i	$-0.644660\pi$
$877$	−26.0000	−0.877958	−0.438979	+	0.898497i	$0.644660\pi$
$878$	0	0
$879$	−15.0000	−0.505937
$880$	0	0
$881$	32.0000	1.07811	0.539054	−	0.842271i	$-0.318782\pi$
$881$	32.0000	1.07811	0.539054	+	0.842271i	$0.318782\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	0	0
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	45.0000	1.50756
$892$	0	0
$893$	6.00000	0.200782
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	−54.0000	−1.80301
$898$	0	0
$899$	−36.0000	−1.20067
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	−30.0000	−0.998337
$904$	0	0
$905$	20.0000	0.664822
$906$	0	0
$907$	−14.0000	−0.464862	−0.232431	−	0.972613i	$-0.574668\pi$
$907$	−14.0000	−0.464862	−0.232431	+	0.972613i	$0.574668\pi$
$908$	0	0
$909$	−36.0000	−1.19404
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	20.0000	0.661903
$914$	0	0
$915$	−24.0000	−0.793416
$916$	0	0
$917$	−10.0000	−0.330229
$918$	0	0
$919$	−31.0000	−1.02260	−0.511298	−	0.859404i	$-0.670835\pi$
$919$	−31.0000	−1.02260	−0.511298	+	0.859404i	$0.670835\pi$
$920$	0	0
$921$	−69.0000	−2.27363
$922$	0	0
$923$	24.0000	0.789970
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	114.000	3.74425
$928$	0	0
$929$	28.0000	0.918650	0.459325	−	0.888268i	$-0.348091\pi$
$929$	28.0000	0.918650	0.459325	+	0.888268i	$0.348091\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	54.0000	1.76788
$934$	0	0
$935$	−5.00000	−0.163517
$936$	0	0
$937$	−37.0000	−1.20874	−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000	−1.20874	−0.604369	+	0.796705i	$0.706575\pi$
$938$	0	0
$939$	21.0000	0.685309
$940$	0	0
$941$	−32.0000	−1.04317	−0.521585	−	0.853199i	$-0.674659\pi$
$941$	−32.0000	−1.04317	−0.521585	+	0.853199i	$0.674659\pi$
$942$	0	0
$943$	−24.0000	−0.781548
$944$	0	0
$945$	9.00000	0.292770
$946$	0	0
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	−78.0000	−2.52932
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	0	0
$955$	3.00000	0.0970777
$956$	0	0
$957$	−135.000	−4.36393
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	36.0000	1.16008
$964$	0	0
$965$	−4.00000	−0.128765
$966$	0	0
$967$	−2.00000	−0.0643157	−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000	−0.0643157	−0.0321578	+	0.999483i	$0.510238\pi$
$968$	0	0
$969$	−18.0000	−0.578243
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	−14.0000	−0.448819
$974$	0	0
$975$	−9.00000	−0.288231
$976$	0	0
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	20.0000	0.639203
$980$	0	0
$981$	18.0000	0.574696
$982$	0	0
$983$	−43.0000	−1.37149	−0.685744	−	0.727843i	$-0.740523\pi$
$983$	−43.0000	−1.37149	−0.685744	+	0.727843i	$0.740523\pi$
$984$	0	0
$985$	−8.00000	−0.254901
$986$	0	0
$987$	−3.00000	−0.0954911
$988$	0	0
$989$	−60.0000	−1.90789
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	84.0000	2.66566
$994$	0	0
$995$	8.00000	0.253617
$996$	0	0
$997$	−11.0000	−0.348373	−0.174187	−	0.984713i	$-0.555730\pi$
$997$	−11.0000	−0.348373	−0.174187	+	0.984713i	$0.555730\pi$
$998$	0	0
$999$	18.0000	0.569495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.a.c.1.1		1
4.3	odd	2		2240.2.a.bb.1.1		1
8.3	odd	2		560.2.a.a.1.1		1
8.5	even	2		140.2.a.b.1.1	✓	1
24.5	odd	2		1260.2.a.h.1.1		1
24.11	even	2		5040.2.a.bd.1.1		1
40.3	even	4		2800.2.g.c.449.1		2
40.13	odd	4		700.2.e.a.449.2		2
40.19	odd	2		2800.2.a.be.1.1		1
40.27	even	4		2800.2.g.c.449.2		2
40.29	even	2		700.2.a.b.1.1		1
40.37	odd	4		700.2.e.a.449.1		2
56.5	odd	6		980.2.i.j.361.1		2
56.13	odd	2		980.2.a.b.1.1		1
56.27	even	2		3920.2.a.bl.1.1		1
56.37	even	6		980.2.i.b.361.1		2
56.45	odd	6		980.2.i.j.961.1		2
56.53	even	6		980.2.i.b.961.1		2
120.29	odd	2		6300.2.a.bf.1.1		1
120.53	even	4		6300.2.k.p.6049.2		2
120.77	even	4		6300.2.k.p.6049.1		2
168.125	even	2		8820.2.a.n.1.1		1
280.13	even	4		4900.2.e.a.2549.1		2
280.69	odd	2		4900.2.a.u.1.1		1
280.237	even	4		4900.2.e.a.2549.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.a.b.1.1	✓	1	8.5	even	2
560.2.a.a.1.1		1	8.3	odd	2
700.2.a.b.1.1		1	40.29	even	2
700.2.e.a.449.1		2	40.37	odd	4
700.2.e.a.449.2		2	40.13	odd	4
980.2.a.b.1.1		1	56.13	odd	2
980.2.i.b.361.1		2	56.37	even	6
980.2.i.b.961.1		2	56.53	even	6
980.2.i.j.361.1		2	56.5	odd	6
980.2.i.j.961.1		2	56.45	odd	6
1260.2.a.h.1.1		1	24.5	odd	2
2240.2.a.c.1.1		1	1.1	even	1	trivial
2240.2.a.bb.1.1		1	4.3	odd	2
2800.2.a.be.1.1		1	40.19	odd	2
2800.2.g.c.449.1		2	40.3	even	4
2800.2.g.c.449.2		2	40.27	even	4
3920.2.a.bl.1.1		1	56.27	even	2
4900.2.a.u.1.1		1	280.69	odd	2
4900.2.e.a.2549.1		2	280.13	even	4
4900.2.e.a.2549.2		2	280.237	even	4
5040.2.a.bd.1.1		1	24.11	even	2
6300.2.a.bf.1.1		1	120.29	odd	2
6300.2.k.p.6049.1		2	120.77	even	4
6300.2.k.p.6049.2		2	120.53	even	4
8820.2.a.n.1.1		1	168.125	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more