LMFDB - Embedded newform 2240.2.a.j.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,-1,0,1,0,1,0,-2,0,-5,0,-1] 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 280)
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)

$f(q)$

$=$

$q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} -2.00000 q^{9} -5.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} +3.00000 q^{17} -6.00000 q^{19} -1.00000 q^{21} +6.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} +9.00000 q^{29} +5.00000 q^{33} +1.00000 q^{35} -6.00000 q^{37} +1.00000 q^{39} +8.00000 q^{41} +6.00000 q^{43} -2.00000 q^{45} -3.00000 q^{47} +1.00000 q^{49} -3.00000 q^{51} +12.0000 q^{53} -5.00000 q^{55} +6.00000 q^{57} +8.00000 q^{59} +4.00000 q^{61} -2.00000 q^{63} -1.00000 q^{65} -4.00000 q^{67} -6.00000 q^{69} -8.00000 q^{71} +10.0000 q^{73} -1.00000 q^{75} -5.00000 q^{77} +3.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +3.00000 q^{85} -9.00000 q^{87} -16.0000 q^{89} -1.00000 q^{91} -6.00000 q^{95} +7.00000 q^{97} +10.0000 q^{99} +O(q^{100})$

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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	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
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	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
$19$	−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
$21$	−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
$25$	1.00000	0.200000
$26$	0	0
$27$	5.00000	0.962250
$28$	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
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	−5.00000	−0.674200
$56$	0	0
$57$	6.00000	0.794719
$58$	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
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−5.00000	−0.569803
$78$	0	0
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	−9.00000	−0.964901
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	0	0
$99$	10.0000	1.00504
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	9.00000	0.886796	0.443398	−	0.896325i	$-0.353773\pi$
$103$	9.00000	0.886796	0.443398	+	0.896325i	$0.353773\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	−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
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	−8.00000	−0.721336
$124$	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
$129$	−6.00000	−0.528271
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	5.00000	0.430331
$136$	0	0
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	0	0
$139$	18.0000	1.52674	0.763370	−	0.645961i	$-0.223543\pi$
$139$	18.0000	1.52674	0.763370	+	0.645961i	$0.223543\pi$
$140$	0	0
$141$	3.00000	0.252646
$142$	0	0
$143$	5.00000	0.418121
$144$	0	0
$145$	9.00000	0.747409
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	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
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	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
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	5.00000	0.389249
$166$	0	0
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	12.0000	0.917663
$172$	0	0
$173$	−19.0000	−1.44454	−0.722272	−	0.691609i	$-0.756902\pi$
$173$	−19.0000	−1.44454	−0.722272	+	0.691609i	$0.756902\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	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
$183$	−4.00000	−0.295689
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−15.0000	−1.09691
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	−11.0000	−0.795932	−0.397966	−	0.917400i	$-0.630284\pi$
$191$	−11.0000	−0.795932	−0.397966	+	0.917400i	$0.630284\pi$
$192$	0	0
$193$	8.00000	0.575853	0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000	0.575853	0.287926	+	0.957653i	$0.407034\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	12.0000	0.854965	0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000	0.854965	0.427482	+	0.904024i	$0.359401\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	9.00000	0.631676
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	−12.0000	−0.834058
$208$	0	0
$209$	30.0000	2.07514
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−3.00000	−0.201802
$222$	0	0
$223$	25.0000	1.67412	0.837062	−	0.547108i	$-0.184271\pi$
$223$	25.0000	1.67412	0.837062	+	0.547108i	$0.184271\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	−13.0000	−0.862840	−0.431420	−	0.902151i	$-0.641987\pi$
$227$	−13.0000	−0.862840	−0.431420	+	0.902151i	$0.641987\pi$
$228$	0	0
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	5.00000	0.328976
$232$	0	0
$233$	8.00000	0.524097	0.262049	−	0.965055i	$-0.415602\pi$
$233$	8.00000	0.524097	0.262049	+	0.965055i	$0.415602\pi$
$234$	0	0
$235$	−3.00000	−0.195698
$236$	0	0
$237$	−3.00000	−0.194871
$238$	0	0
$239$	7.00000	0.452792	0.226396	−	0.974035i	$-0.427306\pi$
$239$	7.00000	0.452792	0.226396	+	0.974035i	$0.427306\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	14.0000	0.883672	0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000	0.883672	0.441836	+	0.897096i	$0.354327\pi$
$252$	0	0
$253$	−30.0000	−1.88608
$254$	0	0
$255$	−3.00000	−0.187867
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	−18.0000	−1.11417
$262$	0	0
$263$	−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
$265$	12.0000	0.737154
$266$	0	0
$267$	16.0000	0.979184
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	−5.00000	−0.301511
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−13.0000	−0.775515	−0.387757	−	0.921761i	$-0.626750\pi$
$281$	−13.0000	−0.775515	−0.387757	+	0.921761i	$0.626750\pi$
$282$	0	0
$283$	29.0000	1.72387	0.861936	−	0.507018i	$-0.169252\pi$
$283$	29.0000	1.72387	0.861936	+	0.507018i	$0.169252\pi$
$284$	0	0
$285$	6.00000	0.355409
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−7.00000	−0.410347
$292$	0	0
$293$	1.00000	0.0584206	0.0292103	−	0.999573i	$-0.490701\pi$
$293$	1.00000	0.0584206	0.0292103	+	0.999573i	$0.490701\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	−25.0000	−1.45065
$298$	0	0
$299$	−6.00000	−0.346989
$300$	0	0
$301$	6.00000	0.345834
$302$	0	0
$303$	−14.0000	−0.804279
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	−27.0000	−1.54097	−0.770486	−	0.637457i	$-0.779986\pi$
$307$	−27.0000	−1.54097	−0.770486	+	0.637457i	$0.779986\pi$
$308$	0	0
$309$	−9.00000	−0.511992
$310$	0	0
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	0	0
$313$	29.0000	1.63918	0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000	1.63918	0.819588	+	0.572953i	$0.194202\pi$
$314$	0	0
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	0	0
$319$	−45.0000	−2.51952
$320$	0	0
$321$	−2.00000	−0.111629
$322$	0	0
$323$	−18.0000	−1.00155
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	−11.0000	−0.608301
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	12.0000	0.657596
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−6.00000	−0.323029
$346$	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
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−33.0000	−1.75641	−0.878206	−	0.478282i	$-0.841260\pi$
$353$	−33.0000	−1.75641	−0.878206	+	0.478282i	$0.841260\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	−3.00000	−0.158777
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−14.0000	−0.734809
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	29.0000	1.51379	0.756894	−	0.653538i	$-0.226716\pi$
$367$	29.0000	1.51379	0.756894	+	0.653538i	$0.226716\pi$
$368$	0	0
$369$	−16.0000	−0.832927
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−9.00000	−0.463524
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	−5.00000	−0.254824
$386$	0	0
$387$	−12.0000	−0.609994
$388$	0	0
$389$	−25.0000	−1.26755	−0.633775	−	0.773517i	$-0.718496\pi$
$389$	−25.0000	−1.26755	−0.633775	+	0.773517i	$0.718496\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	6.00000	0.302660
$394$	0	0
$395$	3.00000	0.150946
$396$	0	0
$397$	29.0000	1.45547	0.727734	−	0.685859i	$-0.240573\pi$
$397$	29.0000	1.45547	0.727734	+	0.685859i	$0.240573\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	9.00000	0.449439	0.224719	−	0.974424i	$-0.427853\pi$
$401$	9.00000	0.449439	0.224719	+	0.974424i	$0.427853\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	30.0000	1.48704
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	16.0000	0.789222
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	−18.0000	−0.881464
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	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
$425$	3.00000	0.145521
$426$	0	0
$427$	4.00000	0.193574
$428$	0	0
$429$	−5.00000	−0.241402
$430$	0	0
$431$	−23.0000	−1.10787	−0.553936	−	0.832560i	$-0.686875\pi$
$431$	−23.0000	−1.10787	−0.553936	+	0.832560i	$0.686875\pi$
$432$	0	0
$433$	−30.0000	−1.44171	−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000	−1.44171	−0.720854	+	0.693087i	$0.756250\pi$
$434$	0	0
$435$	−9.00000	−0.431517
$436$	0	0
$437$	−36.0000	−1.72211
$438$	0	0
$439$	34.0000	1.62273	0.811366	−	0.584539i	$-0.198725\pi$
$439$	34.0000	1.62273	0.811366	+	0.584539i	$0.198725\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	30.0000	1.42534	0.712672	−	0.701498i	$-0.247485\pi$
$443$	30.0000	1.42534	0.712672	+	0.701498i	$0.247485\pi$
$444$	0	0
$445$	−16.0000	−0.758473
$446$	0	0
$447$	−14.0000	−0.662177
$448$	0	0
$449$	−33.0000	−1.55737	−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000	−1.55737	−0.778683	+	0.627417i	$0.784112\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	0	0
$453$	−19.0000	−0.892698
$454$	0	0
$455$	−1.00000	−0.0468807
$456$	0	0
$457$	−28.0000	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$457$	−28.0000	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$458$	0	0
$459$	15.0000	0.700140
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.0000	1.52706	0.763529	−	0.645774i	$-0.223465\pi$
$467$	33.0000	1.52706	0.763529	+	0.645774i	$0.223465\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	0	0
$473$	−30.0000	−1.37940
$474$	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
$483$	−6.00000	−0.273009
$484$	0	0
$485$	7.00000	0.317854
$486$	0	0
$487$	−26.0000	−1.17817	−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000	−1.17817	−0.589086	+	0.808070i	$0.700512\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	0	0
$491$	33.0000	1.48927	0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000	1.48927	0.744635	+	0.667472i	$0.232624\pi$
$492$	0	0
$493$	27.0000	1.21602
$494$	0	0
$495$	10.0000	0.449467
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	0	0
$501$	−9.00000	−0.402090
$502$	0	0
$503$	31.0000	1.38222	0.691111	−	0.722749i	$-0.257122\pi$
$503$	31.0000	1.38222	0.691111	+	0.722749i	$0.257122\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	12.0000	0.532939
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	−30.0000	−1.32453
$514$	0	0
$515$	9.00000	0.396587
$516$	0	0
$517$	15.0000	0.659699
$518$	0	0
$519$	19.0000	0.834007
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−16.0000	−0.694341
$532$	0	0
$533$	−8.00000	−0.346518
$534$	0	0
$535$	2.00000	0.0864675
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	−5.00000	−0.215365
$540$	0	0
$541$	9.00000	0.386940	0.193470	−	0.981106i	$-0.438026\pi$
$541$	9.00000	0.386940	0.193470	+	0.981106i	$0.438026\pi$
$542$	0	0
$543$	−20.0000	−0.858282
$544$	0	0
$545$	11.0000	0.471188
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	0	0
$549$	−8.00000	−0.341432
$550$	0	0
$551$	−54.0000	−2.30048
$552$	0	0
$553$	3.00000	0.127573
$554$	0	0
$555$	6.00000	0.254686
$556$	0	0
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	−6.00000	−0.253773
$560$	0	0
$561$	15.0000	0.633300
$562$	0	0
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	0	0
$573$	11.0000	0.459532
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	17.0000	0.707719	0.353860	−	0.935299i	$-0.384869\pi$
$577$	17.0000	0.707719	0.353860	+	0.935299i	$0.384869\pi$
$578$	0	0
$579$	−8.00000	−0.332469
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	−60.0000	−2.48495
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−12.0000	−0.493614
$592$	0	0
$593$	−7.00000	−0.287456	−0.143728	−	0.989617i	$-0.545909\pi$
$593$	−7.00000	−0.287456	−0.143728	+	0.989617i	$0.545909\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	0	0
$597$	0	0
$598$	0	0
$599$	33.0000	1.34834	0.674172	−	0.738575i	$-0.264501\pi$
$599$	33.0000	1.34834	0.674172	+	0.738575i	$0.264501\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	14.0000	0.569181
$606$	0	0
$607$	−17.0000	−0.690009	−0.345004	−	0.938601i	$-0.612123\pi$
$607$	−17.0000	−0.690009	−0.345004	+	0.938601i	$0.612123\pi$
$608$	0	0
$609$	−9.00000	−0.364698
$610$	0	0
$611$	3.00000	0.121367
$612$	0	0
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	0	0
$615$	−8.00000	−0.322591
$616$	0	0
$617$	10.0000	0.402585	0.201292	−	0.979531i	$-0.435486\pi$
$617$	10.0000	0.402585	0.201292	+	0.979531i	$0.435486\pi$
$618$	0	0
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	0	0
$621$	30.0000	1.20386
$622$	0	0
$623$	−16.0000	−0.641026
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−30.0000	−1.19808
$628$	0	0
$629$	−18.0000	−0.717707
$630$	0	0
$631$	9.00000	0.358284	0.179142	−	0.983823i	$-0.442668\pi$
$631$	9.00000	0.358284	0.179142	+	0.983823i	$0.442668\pi$
$632$	0	0
$633$	−13.0000	−0.516704
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	16.0000	0.632950
$640$	0	0
$641$	34.0000	1.34292	0.671460	−	0.741041i	$-0.265668\pi$
$641$	34.0000	1.34292	0.671460	+	0.741041i	$0.265668\pi$
$642$	0	0
$643$	47.0000	1.85350	0.926750	−	0.375680i	$-0.122591\pi$
$643$	47.0000	1.85350	0.926750	+	0.375680i	$0.122591\pi$
$644$	0	0
$645$	−6.00000	−0.236250
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−40.0000	−1.57014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	0	0
$657$	−20.0000	−0.780274
$658$	0	0
$659$	−25.0000	−0.973862	−0.486931	−	0.873441i	$-0.661884\pi$
$659$	−25.0000	−0.973862	−0.486931	+	0.873441i	$0.661884\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	0	0
$663$	3.00000	0.116510
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	54.0000	2.09089
$668$	0	0
$669$	−25.0000	−0.966556
$670$	0	0
$671$	−20.0000	−0.772091
$672$	0	0
$673$	32.0000	1.23351	0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000	1.23351	0.616755	+	0.787155i	$0.288447\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	−33.0000	−1.26829	−0.634147	−	0.773213i	$-0.718648\pi$
$677$	−33.0000	−1.26829	−0.634147	+	0.773213i	$0.718648\pi$
$678$	0	0
$679$	7.00000	0.268635
$680$	0	0
$681$	13.0000	0.498161
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	−16.0000	−0.611329
$686$	0	0
$687$	16.0000	0.610438
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	10.0000	0.379869
$694$	0	0
$695$	18.0000	0.682779
$696$	0	0
$697$	24.0000	0.909065
$698$	0	0
$699$	−8.00000	−0.302588
$700$	0	0
$701$	21.0000	0.793159	0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000	0.793159	0.396580	+	0.918000i	$0.370197\pi$
$702$	0	0
$703$	36.0000	1.35777
$704$	0	0
$705$	3.00000	0.112987
$706$	0	0
$707$	14.0000	0.526524
$708$	0	0
$709$	41.0000	1.53979	0.769894	−	0.638172i	$-0.220309\pi$
$709$	41.0000	1.53979	0.769894	+	0.638172i	$0.220309\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	0	0
$714$	0	0
$715$	5.00000	0.186989
$716$	0	0
$717$	−7.00000	−0.261420
$718$	0	0
$719$	−50.0000	−1.86469	−0.932343	−	0.361576i	$-0.882239\pi$
$719$	−50.0000	−1.86469	−0.932343	+	0.361576i	$0.882239\pi$
$720$	0	0
$721$	9.00000	0.335178
$722$	0	0
$723$	−18.0000	−0.669427
$724$	0	0
$725$	9.00000	0.334252
$726$	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
$729$	13.0000	0.481481
$730$	0	0
$731$	18.0000	0.665754
$732$	0	0
$733$	−5.00000	−0.184679	−0.0923396	−	0.995728i	$-0.529435\pi$
$733$	−5.00000	−0.184679	−0.0923396	+	0.995728i	$0.529435\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	20.0000	0.736709
$738$	0	0
$739$	−37.0000	−1.36107	−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−37.0000	−1.36107	−0.680534	+	0.732717i	$0.738252\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	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
$745$	14.0000	0.512920
$746$	0	0
$747$	24.0000	0.878114
$748$	0	0
$749$	2.00000	0.0730784
$750$	0	0
$751$	35.0000	1.27717	0.638584	−	0.769552i	$-0.279520\pi$
$751$	35.0000	1.27717	0.638584	+	0.769552i	$0.279520\pi$
$752$	0	0
$753$	−14.0000	−0.510188
$754$	0	0
$755$	19.0000	0.691481
$756$	0	0
$757$	−8.00000	−0.290765	−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000	−0.290765	−0.145382	+	0.989376i	$0.546441\pi$
$758$	0	0
$759$	30.0000	1.08893
$760$	0	0
$761$	−46.0000	−1.66750	−0.833749	−	0.552143i	$-0.813810\pi$
$761$	−46.0000	−1.66750	−0.833749	+	0.552143i	$0.813810\pi$
$762$	0	0
$763$	11.0000	0.398227
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	0	0
$773$	−1.00000	−0.0359675	−0.0179838	−	0.999838i	$-0.505725\pi$
$773$	−1.00000	−0.0359675	−0.0179838	+	0.999838i	$0.505725\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.00000	0.215249
$778$	0	0
$779$	−48.0000	−1.71978
$780$	0	0
$781$	40.0000	1.43131
$782$	0	0
$783$	45.0000	1.60817
$784$	0	0
$785$	2.00000	0.0713831
$786$	0	0
$787$	11.0000	0.392108	0.196054	−	0.980593i	$-0.437187\pi$
$787$	11.0000	0.392108	0.196054	+	0.980593i	$0.437187\pi$
$788$	0	0
$789$	18.0000	0.640817
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	−12.0000	−0.425596
$796$	0	0
$797$	−35.0000	−1.23976	−0.619882	−	0.784695i	$-0.712819\pi$
$797$	−35.0000	−1.23976	−0.619882	+	0.784695i	$0.712819\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	0	0
$801$	32.0000	1.13066
$802$	0	0
$803$	−50.0000	−1.76446
$804$	0	0
$805$	6.00000	0.211472
$806$	0	0
$807$	10.0000	0.352017
$808$	0	0
$809$	−23.0000	−0.808637	−0.404318	−	0.914618i	$-0.632491\pi$
$809$	−23.0000	−0.808637	−0.404318	+	0.914618i	$0.632491\pi$
$810$	0	0
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	0	0
$813$	4.00000	0.140286
$814$	0	0
$815$	6.00000	0.210171
$816$	0	0
$817$	−36.0000	−1.25948
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	7.00000	0.244302	0.122151	−	0.992512i	$-0.461021\pi$
$821$	7.00000	0.244302	0.122151	+	0.992512i	$0.461021\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	5.00000	0.174078
$826$	0	0
$827$	2.00000	0.0695468	0.0347734	−	0.999395i	$-0.488929\pi$
$827$	2.00000	0.0695468	0.0347734	+	0.999395i	$0.488929\pi$
$828$	0	0
$829$	−16.0000	−0.555703	−0.277851	−	0.960624i	$-0.589622\pi$
$829$	−16.0000	−0.555703	−0.277851	+	0.960624i	$0.589622\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	3.00000	0.103944
$834$	0	0
$835$	9.00000	0.311458
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26.0000	0.897620	0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000	0.897620	0.448810	+	0.893627i	$0.351848\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	0	0
$843$	13.0000	0.447744
$844$	0	0
$845$	−12.0000	−0.412813
$846$	0	0
$847$	14.0000	0.481046
$848$	0	0
$849$	−29.0000	−0.995277
$850$	0	0
$851$	−36.0000	−1.23406
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	12.0000	0.410391
$856$	0	0
$857$	−14.0000	−0.478231	−0.239115	−	0.970991i	$-0.576857\pi$
$857$	−14.0000	−0.478231	−0.239115	+	0.970991i	$0.576857\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	−20.0000	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$863$	−20.0000	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$864$	0	0
$865$	−19.0000	−0.646019
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−15.0000	−0.508840
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	0	0
$879$	−1.00000	−0.0337292
$880$	0	0
$881$	−16.0000	−0.539054	−0.269527	−	0.962993i	$-0.586867\pi$
$881$	−16.0000	−0.539054	−0.269527	+	0.962993i	$0.586867\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	−8.00000	−0.268917
$886$	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
$891$	−5.00000	−0.167506
$892$	0	0
$893$	18.0000	0.602347
$894$	0	0
$895$	4.00000	0.133705
$896$	0	0
$897$	6.00000	0.200334
$898$	0	0
$899$	0	0
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	−6.00000	−0.199667
$904$	0	0
$905$	20.0000	0.664822
$906$	0	0
$907$	−30.0000	−0.996134	−0.498067	−	0.867139i	$-0.665957\pi$
$907$	−30.0000	−0.996134	−0.498067	+	0.867139i	$0.665957\pi$
$908$	0	0
$909$	−28.0000	−0.928701
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	60.0000	1.98571
$914$	0	0
$915$	−4.00000	−0.132236
$916$	0	0
$917$	−6.00000	−0.198137
$918$	0	0
$919$	−25.0000	−0.824674	−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000	−0.824674	−0.412337	+	0.911031i	$0.635287\pi$
$920$	0	0
$921$	27.0000	0.889680
$922$	0	0
$923$	8.00000	0.263323
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	−18.0000	−0.591198
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	−14.0000	−0.458339
$934$	0	0
$935$	−15.0000	−0.490552
$936$	0	0
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	0	0
$939$	−29.0000	−0.946379
$940$	0	0
$941$	−4.00000	−0.130396	−0.0651981	−	0.997872i	$-0.520768\pi$
$941$	−4.00000	−0.130396	−0.0651981	+	0.997872i	$0.520768\pi$
$942$	0	0
$943$	48.0000	1.56310
$944$	0	0
$945$	5.00000	0.162650
$946$	0	0
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	30.0000	0.972817
$952$	0	0
$953$	20.0000	0.647864	0.323932	−	0.946080i	$-0.394995\pi$
$953$	20.0000	0.647864	0.323932	+	0.946080i	$0.394995\pi$
$954$	0	0
$955$	−11.0000	−0.355952
$956$	0	0
$957$	45.0000	1.45464
$958$	0	0
$959$	−16.0000	−0.516667
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−4.00000	−0.128898
$964$	0	0
$965$	8.00000	0.257529
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	0	0
$969$	18.0000	0.578243
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	18.0000	0.577054
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	80.0000	2.55681
$980$	0	0
$981$	−22.0000	−0.702406
$982$	0	0
$983$	−9.00000	−0.287055	−0.143528	−	0.989646i	$-0.545845\pi$
$983$	−9.00000	−0.287055	−0.143528	+	0.989646i	$0.545845\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	3.00000	0.0954911
$988$	0	0
$989$	36.0000	1.14473
$990$	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
$993$	20.0000	0.634681
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.00000	0.0316703	0.0158352	−	0.999875i	$-0.494959\pi$
$997$	1.00000	0.0316703	0.0158352	+	0.999875i	$0.494959\pi$
$998$	0	0
$999$	−30.0000	−0.949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.a.j.1.1		1
4.3	odd	2		2240.2.a.v.1.1		1
8.3	odd	2		280.2.a.b.1.1	✓	1
8.5	even	2		560.2.a.e.1.1		1
24.5	odd	2		5040.2.a.be.1.1		1
24.11	even	2		2520.2.a.p.1.1		1
40.3	even	4		1400.2.g.e.449.1		2
40.13	odd	4		2800.2.g.m.449.2		2
40.19	odd	2		1400.2.a.k.1.1		1
40.27	even	4		1400.2.g.e.449.2		2
40.29	even	2		2800.2.a.i.1.1		1
40.37	odd	4		2800.2.g.m.449.1		2
56.3	even	6		1960.2.q.e.961.1		2
56.11	odd	6		1960.2.q.m.961.1		2
56.13	odd	2		3920.2.a.r.1.1		1
56.19	even	6		1960.2.q.e.361.1		2
56.27	even	2		1960.2.a.k.1.1		1
56.51	odd	6		1960.2.q.m.361.1		2
280.139	even	2		9800.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.a.b.1.1	✓	1	8.3	odd	2
560.2.a.e.1.1		1	8.5	even	2
1400.2.a.k.1.1		1	40.19	odd	2
1400.2.g.e.449.1		2	40.3	even	4
1400.2.g.e.449.2		2	40.27	even	4
1960.2.a.k.1.1		1	56.27	even	2
1960.2.q.e.361.1		2	56.19	even	6
1960.2.q.e.961.1		2	56.3	even	6
1960.2.q.m.361.1		2	56.51	odd	6
1960.2.q.m.961.1		2	56.11	odd	6
2240.2.a.j.1.1		1	1.1	even	1	trivial
2240.2.a.v.1.1		1	4.3	odd	2
2520.2.a.p.1.1		1	24.11	even	2
2800.2.a.i.1.1		1	40.29	even	2
2800.2.g.m.449.1		2	40.37	odd	4
2800.2.g.m.449.2		2	40.13	odd	4
3920.2.a.r.1.1		1	56.13	odd	2
5040.2.a.be.1.1		1	24.5	odd	2
9800.2.a.n.1.1		1	280.139	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