LMFDB - Embedded newform 420.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [420,2,Mod(1,420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(420, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("420.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	420.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:	$3.35371688489$
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	$\chi$	$=$	420.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +6.00000 q^{11} -4.00000 q^{13} -1.00000 q^{15} +6.00000 q^{17} +2.00000 q^{19} +1.00000 q^{21} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -10.0000 q^{31} +6.00000 q^{33} -1.00000 q^{35} +2.00000 q^{37} -4.00000 q^{39} -6.00000 q^{41} -4.00000 q^{43} -1.00000 q^{45} +1.00000 q^{49} +6.00000 q^{51} -12.0000 q^{53} -6.00000 q^{55} +2.00000 q^{57} +14.0000 q^{61} +1.00000 q^{63} +4.00000 q^{65} -4.00000 q^{67} +6.00000 q^{71} -4.00000 q^{73} +1.00000 q^{75} +6.00000 q^{77} -16.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -6.00000 q^{85} +6.00000 q^{87} +6.00000 q^{89} -4.00000 q^{91} -10.0000 q^{93} -2.00000 q^{95} -16.0000 q^{97} +6.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	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
$9$	1.00000	0.333333
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.00000	0.840168
$52$	0	0
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	4.00000	0.496139
$66$	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
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	6.00000	0.683763
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	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
$85$	−6.00000	−0.650791
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	−10.0000	−1.03695
$94$	0	0
$95$	−2.00000	−0.205196
$96$	0	0
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	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
$139$	−22.0000	−1.86602	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−22.0000	−1.86602	−0.933008	+	0.359856i	$0.882826\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−24.0000	−2.00698
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	10.0000	0.803219
$156$	0	0
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	0	0
$159$	−12.0000	−0.951662
$160$	0	0
$161$	0	0
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	36.0000	2.63258
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	−10.0000	−0.678844
$218$	0	0
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−6.00000	−0.375735
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	0	0
$275$	6.00000	0.361814
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	0	0
$279$	−10.0000	−0.598684
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	−6.00000	−0.354169
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	−16.0000	−0.937937
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.00000	0.348155
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	−14.0000	−0.801638
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	−12.0000	−0.673987	−0.336994	−	0.941507i	$-0.609410\pi$
$317$	−12.0000	−0.673987	−0.336994	+	0.941507i	$0.609410\pi$
$318$	0	0
$319$	36.0000	2.01561
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.0000	0.667698
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	2.00000	0.110600
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−60.0000	−3.24918
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	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
$351$	−4.00000	−0.213504
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	0	0
$357$	6.00000	0.317554
$358$	0	0
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	25.0000	1.31216
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	16.0000	0.805047
$396$	0	0
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	40.0000	1.99254
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	12.0000	0.594818
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	−22.0000	−1.07734
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	14.0000	0.677507
$428$	0	0
$429$	−24.0000	−1.15873
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	18.0000	0.851371
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−36.0000	−1.69517
$452$	0	0
$453$	−4.00000	−0.187936
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−34.0000	−1.59045	−0.795226	−	0.606313i	$-0.792648\pi$
$457$	−34.0000	−1.59045	−0.795226	+	0.606313i	$0.792648\pi$
$458$	0	0
$459$	6.00000	0.280056
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	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
$465$	10.0000	0.463739
$466$	0	0
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	8.00000	0.368621
$472$	0	0
$473$	−24.0000	−1.10352
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.0000	0.726523
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	6.00000	0.269137
$498$	0	0
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	−4.00000	−0.176950
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	16.0000	0.705044
$516$	0	0
$517$	0	0
$518$	0	0
$519$	6.00000	0.263371
$520$	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
$523$	32.0000	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−60.0000	−2.61364
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	24.0000	1.03956
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.00000	0.258919
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	14.0000	0.597505
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	−16.0000	−0.680389
$554$	0	0
$555$	−2.00000	−0.0848953
$556$	0	0
$557$	12.0000	0.508456	0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000	0.508456	0.254228	+	0.967144i	$0.418179\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	36.0000	1.51992
$562$	0	0
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	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
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	0	0
$575$	0	0
$576$	0	0
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	0	0
$579$	26.0000	1.08052
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	−72.0000	−2.98194
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	0	0
$591$	−24.0000	−0.987228
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	0	0
$597$	−10.0000	−0.409273
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	−25.0000	−1.01639
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	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
$621$	0	0
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	12.0000	0.479234
$628$	0	0
$629$	12.0000	0.478471
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	0	0
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−10.0000	−0.391931
$652$	0	0
$653$	48.0000	1.87839	0.939193	−	0.343391i	$-0.111576\pi$
$653$	48.0000	1.87839	0.939193	+	0.343391i	$0.111576\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	42.0000	1.63609	0.818044	−	0.575156i	$-0.195059\pi$
$659$	42.0000	1.63609	0.818044	+	0.575156i	$0.195059\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−4.00000	−0.154649
$670$	0	0
$671$	84.0000	3.24278
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	48.0000	1.82865
$690$	0	0
$691$	−34.0000	−1.29342	−0.646710	−	0.762736i	$-0.723856\pi$
$691$	−34.0000	−1.29342	−0.646710	+	0.762736i	$0.723856\pi$
$692$	0	0
$693$	6.00000	0.227921
$694$	0	0
$695$	22.0000	0.834508
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	−24.0000	−0.907763
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	0	0
$713$	0	0
$714$	0	0
$715$	24.0000	0.897549
$716$	0	0
$717$	6.00000	0.224074
$718$	0	0
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	−10.0000	−0.371904
$724$	0	0
$725$	6.00000	0.222834
$726$	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
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	20.0000	0.738717	0.369358	−	0.929287i	$-0.379577\pi$
$733$	20.0000	0.738717	0.369358	+	0.929287i	$0.379577\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	24.0000	0.874609
$754$	0	0
$755$	4.00000	0.145575
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	0	0
$768$	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
$771$	18.0000	0.648254
$772$	0	0
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	−8.00000	−0.285532
$786$	0	0
$787$	−16.0000	−0.570338	−0.285169	−	0.958477i	$-0.592050\pi$
$787$	−16.0000	−0.570338	−0.285169	+	0.958477i	$0.592050\pi$
$788$	0	0
$789$	−12.0000	−0.427211
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−56.0000	−1.98862
$794$	0	0
$795$	12.0000	0.425596
$796$	0	0
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	−24.0000	−0.846942
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.00000	0.211210
$808$	0	0
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−46.0000	−1.61528	−0.807639	−	0.589677i	$-0.799255\pi$
$811$	−46.0000	−1.61528	−0.807639	+	0.589677i	$0.799255\pi$
$812$	0	0
$813$	2.00000	0.0701431
$814$	0	0
$815$	−20.0000	−0.700569
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	6.00000	0.208893
$826$	0	0
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−10.0000	−0.345651
$838$	0	0
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	6.00000	0.206651
$844$	0	0
$845$	−3.00000	−0.103203
$846$	0	0
$847$	25.0000	0.859010
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−4.00000	−0.136957	−0.0684787	−	0.997653i	$-0.521815\pi$
$853$	−4.00000	−0.136957	−0.0684787	+	0.997653i	$0.521815\pi$
$854$	0	0
$855$	−2.00000	−0.0683986
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	19.0000	0.645274
$868$	0	0
$869$	−96.0000	−3.25658
$870$	0	0
$871$	16.0000	0.542139
$872$	0	0
$873$	−16.0000	−0.541518
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	0	0
$879$	−18.0000	−0.607125
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	6.00000	0.201008
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−60.0000	−2.00111
$900$	0	0
$901$	−72.0000	−2.39867
$902$	0	0
$903$	−4.00000	−0.133112
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	−72.0000	−2.38285
$914$	0	0
$915$	−14.0000	−0.462826
$916$	0	0
$917$	−12.0000	−0.396275
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	0	0
$923$	−24.0000	−0.789970
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	−24.0000	−0.785725
$934$	0	0
$935$	−36.0000	−1.17733
$936$	0	0
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	0	0
$939$	8.00000	0.261070
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	0	0
$949$	16.0000	0.519382
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	48.0000	1.55487	0.777436	−	0.628962i	$-0.216520\pi$
$953$	48.0000	1.55487	0.777436	+	0.628962i	$0.216520\pi$
$954$	0	0
$955$	−18.0000	−0.582466
$956$	0	0
$957$	36.0000	1.16371
$958$	0	0
$959$	−12.0000	−0.387500
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−26.0000	−0.836970
$966$	0	0
$967$	56.0000	1.80084	0.900419	−	0.435023i	$-0.143260\pi$
$967$	56.0000	1.80084	0.900419	+	0.435023i	$0.143260\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−22.0000	−0.705288
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	48.0000	1.53566	0.767828	−	0.640656i	$-0.221338\pi$
$977$	48.0000	1.53566	0.767828	+	0.640656i	$0.221338\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	10.0000	0.317021
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.a.c.1.1	✓	1
3.2	odd	2		1260.2.a.i.1.1		1
4.3	odd	2		1680.2.a.a.1.1		1
5.2	odd	4		2100.2.k.j.1849.1		2
5.3	odd	4		2100.2.k.j.1849.2		2
5.4	even	2		2100.2.a.d.1.1		1
7.2	even	3		2940.2.q.e.361.1		2
7.3	odd	6		2940.2.q.i.961.1		2
7.4	even	3		2940.2.q.e.961.1		2
7.5	odd	6		2940.2.q.i.361.1		2
7.6	odd	2		2940.2.a.f.1.1		1
8.3	odd	2		6720.2.a.ch.1.1		1
8.5	even	2		6720.2.a.x.1.1		1
12.11	even	2		5040.2.a.bc.1.1		1
15.2	even	4		6300.2.k.a.6049.2		2
15.8	even	4		6300.2.k.a.6049.1		2
15.14	odd	2		6300.2.a.a.1.1		1
20.19	odd	2		8400.2.a.cj.1.1		1
21.20	even	2		8820.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.a.c.1.1	✓	1	1.1	even	1	trivial
1260.2.a.i.1.1		1	3.2	odd	2
1680.2.a.a.1.1		1	4.3	odd	2
2100.2.a.d.1.1		1	5.4	even	2
2100.2.k.j.1849.1		2	5.2	odd	4
2100.2.k.j.1849.2		2	5.3	odd	4
2940.2.a.f.1.1		1	7.6	odd	2
2940.2.q.e.361.1		2	7.2	even	3
2940.2.q.e.961.1		2	7.4	even	3
2940.2.q.i.361.1		2	7.5	odd	6
2940.2.q.i.961.1		2	7.3	odd	6
5040.2.a.bc.1.1		1	12.11	even	2
6300.2.a.a.1.1		1	15.14	odd	2
6300.2.k.a.6049.1		2	15.8	even	4
6300.2.k.a.6049.2		2	15.2	even	4
6720.2.a.x.1.1		1	8.5	even	2
6720.2.a.ch.1.1		1	8.3	odd	2
8400.2.a.cj.1.1		1	20.19	odd	2
8820.2.a.b.1.1		1	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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