LMFDB - Embedded newform 224.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,2,Mod(1,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("224.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	224.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:	$1.78864900528$
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	$\chi$	$=$	224.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+2.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} -4.00000 q^{13} -2.00000 q^{17} +6.00000 q^{19} +2.00000 q^{21} -8.00000 q^{23} -5.00000 q^{25} -4.00000 q^{27} +2.00000 q^{29} +4.00000 q^{31} +8.00000 q^{33} +10.0000 q^{37} -8.00000 q^{39} -10.0000 q^{41} -4.00000 q^{43} -4.00000 q^{47} +1.00000 q^{49} -4.00000 q^{51} -2.00000 q^{53} +12.0000 q^{57} -10.0000 q^{59} -8.00000 q^{61} +1.00000 q^{63} +8.00000 q^{67} -16.0000 q^{69} -6.00000 q^{73} -10.0000 q^{75} +4.00000 q^{77} +16.0000 q^{79} -11.0000 q^{81} -2.00000 q^{83} +4.00000 q^{87} +18.0000 q^{89} -4.00000 q^{91} +8.00000 q^{93} -2.00000 q^{97} +4.00000 q^{99} +O(q^{100})$

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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	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
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	8.00000	1.39262
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	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
$45$	0	0
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.0000	1.58944
$58$	0	0
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	−16.0000	−1.92617
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	−10.0000	−1.15470
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	20.0000	1.89832
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−20.0000	−1.80334
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−16.0000	−1.33799
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.00000	0.164957
$148$	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
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	0	0
$159$	−4.00000	−0.317221
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	−12.0000	−0.912343	−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000	−0.912343	−0.456172	+	0.889892i	$0.650780\pi$
$174$	0	0
$175$	−5.00000	−0.377964
$176$	0	0
$177$	−20.0000	−1.50329
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	0	0
$183$	−16.0000	−1.18275
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	24.0000	1.66011
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	−12.0000	−0.810885
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	−24.0000	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	22.0000	1.46019	0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000	1.46019	0.730096	+	0.683345i	$0.239475\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	32.0000	2.07862
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	−10.0000	−0.641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−24.0000	−1.52708
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	0	0
$253$	−32.0000	−2.01182
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	36.0000	2.20316
$268$	0	0
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	−20.0000	−1.20605
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−4.00000	−0.234484
$292$	0	0
$293$	8.00000	0.467365	0.233682	−	0.972313i	$-0.424922\pi$
$293$	8.00000	0.467365	0.233682	+	0.972313i	$0.424922\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−16.0000	−0.928414
$298$	0	0
$299$	32.0000	1.85061
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−34.0000	−1.90963	−0.954815	−	0.297200i	$-0.903947\pi$
$317$	−34.0000	−1.90963	−0.954815	+	0.297200i	$0.903947\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	32.0000	1.78607
$322$	0	0
$323$	−12.0000	−0.667698
$324$	0	0
$325$	20.0000	1.10940
$326$	0	0
$327$	20.0000	1.10600
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	16.0000	0.866449
$342$	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
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.00000	−0.412021
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	32.0000	1.63941
$382$	0	0
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−28.0000	−1.40528	−0.702640	−	0.711546i	$-0.747995\pi$
$397$	−28.0000	−1.40528	−0.702640	+	0.711546i	$0.747995\pi$
$398$	0	0
$399$	12.0000	0.600751
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	0	0
$405$	0	0
$406$	0	0
$407$	40.0000	1.98273
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	−28.0000	−1.38114
$412$	0	0
$413$	−10.0000	−0.492068
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	10.0000	0.485071
$426$	0	0
$427$	−8.00000	−0.387147
$428$	0	0
$429$	−32.0000	−1.54497
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−48.0000	−2.29615
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	12.0000	0.567581
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	0	0
$453$	32.0000	1.50349
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	28.0000	1.30409	0.652045	−	0.758180i	$-0.273911\pi$
$461$	28.0000	1.30409	0.652045	+	0.758180i	$0.273911\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.0000	−0.832941	−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000	−0.832941	−0.416470	+	0.909149i	$0.636733\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	24.0000	1.10586
$472$	0	0
$473$	−16.0000	−0.735681
$474$	0	0
$475$	−30.0000	−1.37649
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	0	0
$483$	−16.0000	−0.728025
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	0	0
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	−40.0000	−1.78707
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.00000	0.266469
$508$	0	0
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	−24.0000	−1.05963
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−16.0000	−0.703679
$518$	0	0
$519$	−24.0000	−1.05348
$520$	0	0
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	0	0
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	0	0
$525$	−10.0000	−0.436436
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−10.0000	−0.433963
$532$	0	0
$533$	40.0000	1.73259
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	−24.0000	−1.02994
$544$	0	0
$545$	0	0
$546$	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
$551$	12.0000	0.511217
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	−16.0000	−0.675521
$562$	0	0
$563$	−46.0000	−1.93867	−0.969334	−	0.245745i	$-0.920967\pi$
$563$	−46.0000	−1.93867	−0.969334	+	0.245745i	$0.920967\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−11.0000	−0.461957
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	40.0000	1.66812
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	0	0
$579$	28.0000	1.16364
$580$	0	0
$581$	−2.00000	−0.0829740
$582$	0	0
$583$	−8.00000	−0.331326
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.0000	0.742940	0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000	0.742940	0.371470	+	0.928445i	$0.378854\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−48.0000	−1.94826	−0.974130	−	0.225989i	$-0.927439\pi$
$607$	−48.0000	−1.94826	−0.974130	+	0.225989i	$0.927439\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	22.0000	0.884255	0.442127	−	0.896952i	$-0.354224\pi$
$619$	22.0000	0.884255	0.442127	+	0.896952i	$0.354224\pi$
$620$	0	0
$621$	32.0000	1.28412
$622$	0	0
$623$	18.0000	0.721155
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	48.0000	1.91694
$628$	0	0
$629$	−20.0000	−0.797452
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	−16.0000	−0.635943
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−34.0000	−1.34292	−0.671460	−	0.741041i	$-0.734332\pi$
$641$	−34.0000	−1.34292	−0.671460	+	0.741041i	$0.734332\pi$
$642$	0	0
$643$	34.0000	1.34083	0.670415	−	0.741987i	$-0.266116\pi$
$643$	34.0000	1.34083	0.670415	+	0.741987i	$0.266116\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	−40.0000	−1.57014
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	−38.0000	−1.48705	−0.743527	−	0.668705i	$-0.766849\pi$
$653$	−38.0000	−1.48705	−0.743527	+	0.668705i	$0.766849\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	24.0000	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$662$	0	0
$663$	16.0000	0.621389
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−16.0000	−0.619522
$668$	0	0
$669$	−48.0000	−1.85579
$670$	0	0
$671$	−32.0000	−1.23535
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	0	0
$675$	20.0000	0.769800
$676$	0	0
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	44.0000	1.68608
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−40.0000	−1.52610
$688$	0	0
$689$	8.00000	0.304776
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	0	0
$693$	4.00000	0.151947
$694$	0	0
$695$	0	0
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	−12.0000	−0.453882
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	60.0000	2.26294
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−48.0000	−1.79259
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	28.0000	1.04133
$724$	0	0
$725$	−10.0000	−0.371391
$726$	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
$729$	13.0000	0.481481
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	8.00000	0.295487	0.147743	−	0.989026i	$-0.452799\pi$
$733$	8.00000	0.295487	0.147743	+	0.989026i	$0.452799\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.0000	1.17874
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	−48.0000	−1.76332
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.00000	−0.0731762
$748$	0	0
$749$	16.0000	0.584627
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	0	0
$759$	−64.0000	−2.32305
$760$	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
$763$	10.0000	0.362024
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40.0000	1.44432
$768$	0	0
$769$	6.00000	0.216366	0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000	0.216366	0.108183	+	0.994131i	$0.465497\pi$
$770$	0	0
$771$	4.00000	0.144056
$772$	0	0
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	20.0000	0.717496
$778$	0	0
$779$	−60.0000	−2.14972
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−8.00000	−0.285897
$784$	0	0
$785$	0	0
$786$	0	0
$787$	46.0000	1.63972	0.819861	−	0.572562i	$-0.194050\pi$
$787$	46.0000	1.63972	0.819861	+	0.572562i	$0.194050\pi$
$788$	0	0
$789$	32.0000	1.13923
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	32.0000	1.13635
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.0000	0.708436	0.354218	−	0.935163i	$-0.384747\pi$
$797$	20.0000	0.708436	0.354218	+	0.935163i	$0.384747\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	−24.0000	−0.846942
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−24.0000	−0.844840
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−34.0000	−1.19390	−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000	−1.19390	−0.596951	+	0.802278i	$0.703621\pi$
$812$	0	0
$813$	32.0000	1.12229
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−24.0000	−0.839654
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	0	0
$825$	−40.0000	−1.39262
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	24.0000	0.833554	0.416777	−	0.909009i	$-0.363160\pi$
$829$	24.0000	0.833554	0.416777	+	0.909009i	$0.363160\pi$
$830$	0	0
$831$	28.0000	0.971309
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−16.0000	−0.553041
$838$	0	0
$839$	52.0000	1.79524	0.897620	−	0.440771i	$-0.145295\pi$
$839$	52.0000	1.79524	0.897620	+	0.440771i	$0.145295\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−12.0000	−0.413302
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.00000	0.171802
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	−80.0000	−2.74236
$852$	0	0
$853$	−36.0000	−1.23262	−0.616308	−	0.787505i	$-0.711372\pi$
$853$	−36.0000	−1.23262	−0.616308	+	0.787505i	$0.711372\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	0	0
$861$	−20.0000	−0.681598
$862$	0	0
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−26.0000	−0.883006
$868$	0	0
$869$	64.0000	2.17105
$870$	0	0
$871$	−32.0000	−1.08428
$872$	0	0
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	0	0
$879$	16.0000	0.539667
$880$	0	0
$881$	−38.0000	−1.28025	−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000	−1.28025	−0.640126	+	0.768270i	$0.721118\pi$
$882$	0	0
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	−44.0000	−1.47406
$892$	0	0
$893$	−24.0000	−0.803129
$894$	0	0
$895$	0	0
$896$	0	0
$897$	64.0000	2.13690
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.00000	0.198137
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	0	0
$921$	−20.0000	−0.659022
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−50.0000	−1.64399
$926$	0	0
$927$	4.00000	0.131377
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	48.0000	1.57145
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	−52.0000	−1.69696
$940$	0	0
$941$	−40.0000	−1.30396	−0.651981	−	0.758235i	$-0.726062\pi$
$941$	−40.0000	−1.30396	−0.651981	+	0.758235i	$0.726062\pi$
$942$	0	0
$943$	80.0000	2.60516
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	−68.0000	−2.20505
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16.0000	0.517207
$958$	0	0
$959$	−14.0000	−0.452084
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	16.0000	0.515593
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	0	0
$975$	40.0000	1.28103
$976$	0	0
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	0	0
$979$	72.0000	2.30113
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−8.00000	−0.254643
$988$	0	0
$989$	32.0000	1.01754
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	24.0000	0.761617
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	0	0
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.2.a.b.1.1	yes	1
3.2	odd	2		2016.2.a.g.1.1		1
4.3	odd	2		224.2.a.a.1.1	✓	1
5.4	even	2		5600.2.a.c.1.1		1
7.2	even	3		1568.2.i.b.1537.1		2
7.3	odd	6		1568.2.i.j.961.1		2
7.4	even	3		1568.2.i.b.961.1		2
7.5	odd	6		1568.2.i.j.1537.1		2
7.6	odd	2		1568.2.a.b.1.1		1
8.3	odd	2		448.2.a.f.1.1		1
8.5	even	2		448.2.a.b.1.1		1
12.11	even	2		2016.2.a.e.1.1		1
16.3	odd	4		1792.2.b.f.897.1		2
16.5	even	4		1792.2.b.b.897.1		2
16.11	odd	4		1792.2.b.f.897.2		2
16.13	even	4		1792.2.b.b.897.2		2
20.19	odd	2		5600.2.a.t.1.1		1
24.5	odd	2		4032.2.a.z.1.1		1
24.11	even	2		4032.2.a.p.1.1		1
28.3	even	6		1568.2.i.c.961.1		2
28.11	odd	6		1568.2.i.k.961.1		2
28.19	even	6		1568.2.i.c.1537.1		2
28.23	odd	6		1568.2.i.k.1537.1		2
28.27	even	2		1568.2.a.h.1.1		1
56.13	odd	2		3136.2.a.y.1.1		1
56.27	even	2		3136.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.a.a.1.1	✓	1	4.3	odd	2
224.2.a.b.1.1	yes	1	1.1	even	1	trivial
448.2.a.b.1.1		1	8.5	even	2
448.2.a.f.1.1		1	8.3	odd	2
1568.2.a.b.1.1		1	7.6	odd	2
1568.2.a.h.1.1		1	28.27	even	2
1568.2.i.b.961.1		2	7.4	even	3
1568.2.i.b.1537.1		2	7.2	even	3
1568.2.i.c.961.1		2	28.3	even	6
1568.2.i.c.1537.1		2	28.19	even	6
1568.2.i.j.961.1		2	7.3	odd	6
1568.2.i.j.1537.1		2	7.5	odd	6
1568.2.i.k.961.1		2	28.11	odd	6
1568.2.i.k.1537.1		2	28.23	odd	6
1792.2.b.b.897.1		2	16.5	even	4
1792.2.b.b.897.2		2	16.13	even	4
1792.2.b.f.897.1		2	16.3	odd	4
1792.2.b.f.897.2		2	16.11	odd	4
2016.2.a.e.1.1		1	12.11	even	2
2016.2.a.g.1.1		1	3.2	odd	2
3136.2.a.f.1.1		1	56.27	even	2
3136.2.a.y.1.1		1	56.13	odd	2
4032.2.a.p.1.1		1	24.11	even	2
4032.2.a.z.1.1		1	24.5	odd	2
5600.2.a.c.1.1		1	5.4	even	2
5600.2.a.t.1.1		1	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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