LMFDB - Embedded newform 440.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [440,2,Mod(1,440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(440, 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("440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 440 = 2^{3} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	440.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:	$3.51341768894$
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	$\chi$	$=$	440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +6.00000 q^{9} -1.00000 q^{11} -6.00000 q^{13} +3.00000 q^{15} +3.00000 q^{17} -5.00000 q^{19} +3.00000 q^{21} -2.00000 q^{23} +1.00000 q^{25} +9.00000 q^{27} -5.00000 q^{29} +5.00000 q^{31} -3.00000 q^{33} +1.00000 q^{35} -1.00000 q^{37} -18.0000 q^{39} -2.00000 q^{41} +12.0000 q^{43} +6.00000 q^{45} -2.00000 q^{47} -6.00000 q^{49} +9.00000 q^{51} -13.0000 q^{53} -1.00000 q^{55} -15.0000 q^{57} +2.00000 q^{59} +1.00000 q^{61} +6.00000 q^{63} -6.00000 q^{65} +16.0000 q^{67} -6.00000 q^{69} +15.0000 q^{71} +10.0000 q^{73} +3.00000 q^{75} -1.00000 q^{77} +2.00000 q^{79} +9.00000 q^{81} -14.0000 q^{83} +3.00000 q^{85} -15.0000 q^{87} +9.00000 q^{89} -6.00000 q^{91} +15.0000 q^{93} -5.00000 q^{95} -16.0000 q^{97} -6.00000 q^{99} +O(q^{100})$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	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
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	9.00000	1.73205
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	−3.00000	−0.522233
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	0	0
$39$	−18.0000	−2.88231
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	9.00000	1.26025
$52$	0	0
$53$	−13.0000	−1.78569	−0.892844	−	0.450367i	$-0.851293\pi$
$53$	−13.0000	−1.78569	−0.892844	+	0.450367i	$0.851293\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−15.0000	−1.98680
$58$	0	0
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	16.0000	1.95471	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000	1.95471	0.977356	+	0.211604i	$0.0678686\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	15.0000	1.78017	0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000	1.78017	0.890086	+	0.455792i	$0.150644\pi$
$72$	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
$75$	3.00000	0.346410
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	−15.0000	−1.60817
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	15.0000	1.55543
$94$	0	0
$95$	−5.00000	−0.512989
$96$	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
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	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
$105$	3.00000	0.292770
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	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
$111$	−3.00000	−0.284747
$112$	0	0
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	0	0
$117$	−36.0000	−3.32820
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	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	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	36.0000	3.16962
$130$	0	0
$131$	7.00000	0.611593	0.305796	−	0.952097i	$-0.401077\pi$
$131$	7.00000	0.611593	0.305796	+	0.952097i	$0.401077\pi$
$132$	0	0
$133$	−5.00000	−0.433555
$134$	0	0
$135$	9.00000	0.774597
$136$	0	0
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	−5.00000	−0.415227
$146$	0	0
$147$	−18.0000	−1.48461
$148$	0	0
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	0	0
$153$	18.0000	1.45521
$154$	0	0
$155$	5.00000	0.401610
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	0	0
$159$	−39.0000	−3.09290
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	1.00000	0.0773823	0.0386912	−	0.999251i	$-0.487681\pi$
$167$	1.00000	0.0773823	0.0386912	+	0.999251i	$0.487681\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	−30.0000	−2.29416
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	6.00000	0.450988
$178$	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
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	3.00000	0.221766
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−3.00000	−0.219382
$188$	0	0
$189$	9.00000	0.654654
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	7.00000	0.503871	0.251936	−	0.967744i	$-0.418933\pi$
$193$	7.00000	0.503871	0.251936	+	0.967744i	$0.418933\pi$
$194$	0	0
$195$	−18.0000	−1.28901
$196$	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
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	0	0
$201$	48.0000	3.38566
$202$	0	0
$203$	−5.00000	−0.350931
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	−12.0000	−0.834058
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	0	0
$213$	45.0000	3.08335
$214$	0	0
$215$	12.0000	0.818393
$216$	0	0
$217$	5.00000	0.339422
$218$	0	0
$219$	30.0000	2.02721
$220$	0	0
$221$	−18.0000	−1.21081
$222$	0	0
$223$	26.0000	1.74109	0.870544	−	0.492090i	$-0.163767\pi$
$223$	26.0000	1.74109	0.870544	+	0.492090i	$0.163767\pi$
$224$	0	0
$225$	6.00000	0.400000
$226$	0	0
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	11.0000	0.720634	0.360317	−	0.932830i	$-0.382669\pi$
$233$	11.0000	0.720634	0.360317	+	0.932830i	$0.382669\pi$
$234$	0	0
$235$	−2.00000	−0.130466
$236$	0	0
$237$	6.00000	0.389742
$238$	0	0
$239$	−2.00000	−0.129369	−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000	−0.129369	−0.0646846	+	0.997906i	$0.520604\pi$
$240$	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
$243$	0	0
$244$	0	0
$245$	−6.00000	−0.383326
$246$	0	0
$247$	30.0000	1.90885
$248$	0	0
$249$	−42.0000	−2.66164
$250$	0	0
$251$	−10.0000	−0.631194	−0.315597	−	0.948893i	$-0.602205\pi$
$251$	−10.0000	−0.631194	−0.315597	+	0.948893i	$0.602205\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	9.00000	0.563602
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	0	0
$261$	−30.0000	−1.85695
$262$	0	0
$263$	−13.0000	−0.801614	−0.400807	−	0.916162i	$-0.631270\pi$
$263$	−13.0000	−0.801614	−0.400807	+	0.916162i	$0.631270\pi$
$264$	0	0
$265$	−13.0000	−0.798584
$266$	0	0
$267$	27.0000	1.65237
$268$	0	0
$269$	20.0000	1.21942	0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000	1.21942	0.609711	+	0.792624i	$0.291286\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	−18.0000	−1.08941
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	16.0000	0.961347	0.480673	−	0.876900i	$-0.340392\pi$
$277$	16.0000	0.961347	0.480673	+	0.876900i	$0.340392\pi$
$278$	0	0
$279$	30.0000	1.79605
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	0	0
$285$	−15.0000	−0.888523
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−48.0000	−2.81381
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	2.00000	0.116445
$296$	0	0
$297$	−9.00000	−0.522233
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	−30.0000	−1.72345
$304$	0	0
$305$	1.00000	0.0572598
$306$	0	0
$307$	−22.0000	−1.25561	−0.627803	−	0.778372i	$-0.716046\pi$
$307$	−22.0000	−1.25561	−0.627803	+	0.778372i	$0.716046\pi$
$308$	0	0
$309$	−48.0000	−2.73062
$310$	0	0
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	0	0
$313$	−30.0000	−1.69570	−0.847850	−	0.530236i	$-0.822103\pi$
$313$	−30.0000	−1.69570	−0.847850	+	0.530236i	$0.822103\pi$
$314$	0	0
$315$	6.00000	0.338062
$316$	0	0
$317$	−3.00000	−0.168497	−0.0842484	−	0.996445i	$-0.526849\pi$
$317$	−3.00000	−0.168497	−0.0842484	+	0.996445i	$0.526849\pi$
$318$	0	0
$319$	5.00000	0.279946
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−15.0000	−0.834622
$324$	0	0
$325$	−6.00000	−0.332820
$326$	0	0
$327$	30.0000	1.65900
$328$	0	0
$329$	−2.00000	−0.110264
$330$	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
$333$	−6.00000	−0.328798
$334$	0	0
$335$	16.0000	0.874173
$336$	0	0
$337$	29.0000	1.57973	0.789865	−	0.613280i	$-0.210150\pi$
$337$	29.0000	1.57973	0.789865	+	0.613280i	$0.210150\pi$
$338$	0	0
$339$	−48.0000	−2.60700
$340$	0	0
$341$	−5.00000	−0.270765
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	−6.00000	−0.323029
$346$	0	0
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	−54.0000	−2.88231
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	15.0000	0.796117
$356$	0	0
$357$	9.00000	0.476331
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	3.00000	0.157459
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	−13.0000	−0.674926
$372$	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
$375$	3.00000	0.154919
$376$	0	0
$377$	30.0000	1.54508
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	−24.0000	−1.22956
$382$	0	0
$383$	−14.0000	−0.715367	−0.357683	−	0.933843i	$-0.616433\pi$
$383$	−14.0000	−0.715367	−0.357683	+	0.933843i	$0.616433\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	72.0000	3.65997
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	0	0
$393$	21.0000	1.05931
$394$	0	0
$395$	2.00000	0.100631
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	−15.0000	−0.750939
$400$	0	0
$401$	−21.0000	−1.04869	−0.524345	−	0.851506i	$-0.675690\pi$
$401$	−21.0000	−1.04869	−0.524345	+	0.851506i	$0.675690\pi$
$402$	0	0
$403$	−30.0000	−1.49441
$404$	0	0
$405$	9.00000	0.447214
$406$	0	0
$407$	1.00000	0.0495682
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	36.0000	1.77575
$412$	0	0
$413$	2.00000	0.0984136
$414$	0	0
$415$	−14.0000	−0.687233
$416$	0	0
$417$	−12.0000	−0.587643
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−36.0000	−1.75453	−0.877266	−	0.480004i	$-0.840635\pi$
$421$	−36.0000	−1.75453	−0.877266	+	0.480004i	$0.840635\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	3.00000	0.145521
$426$	0	0
$427$	1.00000	0.0483934
$428$	0	0
$429$	18.0000	0.869048
$430$	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
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	0	0
$435$	−15.0000	−0.719195
$436$	0	0
$437$	10.0000	0.478365
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	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
$445$	9.00000	0.426641
$446$	0	0
$447$	−45.0000	−2.12843
$448$	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
$451$	2.00000	0.0941763
$452$	0	0
$453$	54.0000	2.53714
$454$	0	0
$455$	−6.00000	−0.281284
$456$	0	0
$457$	33.0000	1.54367	0.771837	−	0.635820i	$-0.219338\pi$
$457$	33.0000	1.54367	0.771837	+	0.635820i	$0.219338\pi$
$458$	0	0
$459$	27.0000	1.26025
$460$	0	0
$461$	5.00000	0.232873	0.116437	−	0.993198i	$-0.462853\pi$
$461$	5.00000	0.232873	0.116437	+	0.993198i	$0.462853\pi$
$462$	0	0
$463$	6.00000	0.278844	0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000	0.278844	0.139422	+	0.990233i	$0.455476\pi$
$464$	0	0
$465$	15.0000	0.695608
$466$	0	0
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	21.0000	0.967629
$472$	0	0
$473$	−12.0000	−0.551761
$474$	0	0
$475$	−5.00000	−0.229416
$476$	0	0
$477$	−78.0000	−3.57137
$478$	0	0
$479$	20.0000	0.913823	0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000	0.913823	0.456912	+	0.889512i	$0.348956\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	−6.00000	−0.273009
$484$	0	0
$485$	−16.0000	−0.726523
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	0	0
$489$	−3.00000	−0.135665
$490$	0	0
$491$	−13.0000	−0.586682	−0.293341	−	0.956008i	$-0.594767\pi$
$491$	−13.0000	−0.586682	−0.293341	+	0.956008i	$0.594767\pi$
$492$	0	0
$493$	−15.0000	−0.675566
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	15.0000	0.672842
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	0	0
$501$	3.00000	0.134030
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	69.0000	3.06440
$508$	0	0
$509$	8.00000	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$509$	8.00000	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	−45.0000	−1.98680
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	2.00000	0.0879599
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	3.00000	0.130931
$526$	0	0
$527$	15.0000	0.653410
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	12.0000	0.520756
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	15.0000	0.644900	0.322450	−	0.946586i	$-0.395494\pi$
$541$	15.0000	0.644900	0.322450	+	0.946586i	$0.395494\pi$
$542$	0	0
$543$	66.0000	2.83233
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	25.0000	1.06504
$552$	0	0
$553$	2.00000	0.0850487
$554$	0	0
$555$	−3.00000	−0.127343
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	−72.0000	−3.04528
$560$	0	0
$561$	−9.00000	−0.379980
$562$	0	0
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	0	0
$565$	−16.0000	−0.673125
$566$	0	0
$567$	9.00000	0.377964
$568$	0	0
$569$	36.0000	1.50920	0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000	1.50920	0.754599	+	0.656186i	$0.227831\pi$
$570$	0	0
$571$	−19.0000	−0.795125	−0.397563	−	0.917575i	$-0.630144\pi$
$571$	−19.0000	−0.795125	−0.397563	+	0.917575i	$0.630144\pi$
$572$	0	0
$573$	36.0000	1.50392
$574$	0	0
$575$	−2.00000	−0.0834058
$576$	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
$579$	21.0000	0.872730
$580$	0	0
$581$	−14.0000	−0.580818
$582$	0	0
$583$	13.0000	0.538405
$584$	0	0
$585$	−36.0000	−1.48842
$586$	0	0
$587$	−39.0000	−1.60970	−0.804851	−	0.593477i	$-0.797755\pi$
$587$	−39.0000	−1.60970	−0.804851	+	0.593477i	$0.797755\pi$
$588$	0	0
$589$	−25.0000	−1.03011
$590$	0	0
$591$	−72.0000	−2.96168
$592$	0	0
$593$	−22.0000	−0.903432	−0.451716	−	0.892162i	$-0.649188\pi$
$593$	−22.0000	−0.903432	−0.451716	+	0.892162i	$0.649188\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	0	0
$597$	−3.00000	−0.122782
$598$	0	0
$599$	−27.0000	−1.10319	−0.551595	−	0.834112i	$-0.685981\pi$
$599$	−27.0000	−1.10319	−0.551595	+	0.834112i	$0.685981\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	96.0000	3.90942
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−21.0000	−0.852364	−0.426182	−	0.904638i	$-0.640142\pi$
$607$	−21.0000	−0.852364	−0.426182	+	0.904638i	$0.640142\pi$
$608$	0	0
$609$	−15.0000	−0.607831
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	−18.0000	−0.722315
$622$	0	0
$623$	9.00000	0.360577
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	15.0000	0.599042
$628$	0	0
$629$	−3.00000	−0.119618
$630$	0	0
$631$	15.0000	0.597141	0.298570	−	0.954388i	$-0.403490\pi$
$631$	15.0000	0.597141	0.298570	+	0.954388i	$0.403490\pi$
$632$	0	0
$633$	−3.00000	−0.119239
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	36.0000	1.42637
$638$	0	0
$639$	90.0000	3.56034
$640$	0	0
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	0	0
$643$	7.00000	0.276053	0.138027	−	0.990429i	$-0.455924\pi$
$643$	7.00000	0.276053	0.138027	+	0.990429i	$0.455924\pi$
$644$	0	0
$645$	36.0000	1.41750
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	0	0
$649$	−2.00000	−0.0785069
$650$	0	0
$651$	15.0000	0.587896
$652$	0	0
$653$	−19.0000	−0.743527	−0.371764	−	0.928327i	$-0.621247\pi$
$653$	−19.0000	−0.743527	−0.371764	+	0.928327i	$0.621247\pi$
$654$	0	0
$655$	7.00000	0.273513
$656$	0	0
$657$	60.0000	2.34082
$658$	0	0
$659$	49.0000	1.90877	0.954384	−	0.298580i	$-0.0965131\pi$
$659$	49.0000	1.90877	0.954384	+	0.298580i	$0.0965131\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	0	0
$663$	−54.0000	−2.09719
$664$	0	0
$665$	−5.00000	−0.193892
$666$	0	0
$667$	10.0000	0.387202
$668$	0	0
$669$	78.0000	3.01565
$670$	0	0
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	0	0
$675$	9.00000	0.346410
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	0	0
$681$	54.0000	2.06928
$682$	0	0
$683$	−19.0000	−0.727015	−0.363507	−	0.931591i	$-0.618421\pi$
$683$	−19.0000	−0.727015	−0.363507	+	0.931591i	$0.618421\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	18.0000	0.686743
$688$	0	0
$689$	78.0000	2.97156
$690$	0	0
$691$	34.0000	1.29342	0.646710	−	0.762736i	$-0.276144\pi$
$691$	34.0000	1.29342	0.646710	+	0.762736i	$0.276144\pi$
$692$	0	0
$693$	−6.00000	−0.227921
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	−6.00000	−0.227266
$698$	0	0
$699$	33.0000	1.24817
$700$	0	0
$701$	−31.0000	−1.17085	−0.585427	−	0.810725i	$-0.699073\pi$
$701$	−31.0000	−1.17085	−0.585427	+	0.810725i	$0.699073\pi$
$702$	0	0
$703$	5.00000	0.188579
$704$	0	0
$705$	−6.00000	−0.225973
$706$	0	0
$707$	−10.0000	−0.376089
$708$	0	0
$709$	46.0000	1.72757	0.863783	−	0.503864i	$-0.168089\pi$
$709$	46.0000	1.72757	0.863783	+	0.503864i	$0.168089\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	−10.0000	−0.374503
$714$	0	0
$715$	6.00000	0.224387
$716$	0	0
$717$	−6.00000	−0.224074
$718$	0	0
$719$	1.00000	0.0372937	0.0186469	−	0.999826i	$-0.494064\pi$
$719$	1.00000	0.0372937	0.0186469	+	0.999826i	$0.494064\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	54.0000	2.00828
$724$	0	0
$725$	−5.00000	−0.185695
$726$	0	0
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	36.0000	1.33151
$732$	0	0
$733$	−32.0000	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$733$	−32.0000	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$734$	0	0
$735$	−18.0000	−0.663940
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	0	0
$741$	90.0000	3.30623
$742$	0	0
$743$	33.0000	1.21065	0.605326	−	0.795977i	$-0.293043\pi$
$743$	33.0000	1.21065	0.605326	+	0.795977i	$0.293043\pi$
$744$	0	0
$745$	−15.0000	−0.549557
$746$	0	0
$747$	−84.0000	−3.07340
$748$	0	0
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−7.00000	−0.255434	−0.127717	−	0.991811i	$-0.540765\pi$
$751$	−7.00000	−0.255434	−0.127717	+	0.991811i	$0.540765\pi$
$752$	0	0
$753$	−30.0000	−1.09326
$754$	0	0
$755$	18.0000	0.655087
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	6.00000	0.217786
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	10.0000	0.362024
$764$	0	0
$765$	18.0000	0.650791
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	−66.0000	−2.37693
$772$	0	0
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	0	0
$775$	5.00000	0.179605
$776$	0	0
$777$	−3.00000	−0.107624
$778$	0	0
$779$	10.0000	0.358287
$780$	0	0
$781$	−15.0000	−0.536742
$782$	0	0
$783$	−45.0000	−1.60817
$784$	0	0
$785$	7.00000	0.249841
$786$	0	0
$787$	−48.0000	−1.71102	−0.855508	−	0.517790i	$-0.826755\pi$
$787$	−48.0000	−1.71102	−0.855508	+	0.517790i	$0.826755\pi$
$788$	0	0
$789$	−39.0000	−1.38844
$790$	0	0
$791$	−16.0000	−0.568895
$792$	0	0
$793$	−6.00000	−0.213066
$794$	0	0
$795$	−39.0000	−1.38319
$796$	0	0
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	0	0
$801$	54.0000	1.90800
$802$	0	0
$803$	−10.0000	−0.352892
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	0	0
$807$	60.0000	2.11210
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	1.00000	0.0351147	0.0175574	−	0.999846i	$-0.494411\pi$
$811$	1.00000	0.0351147	0.0175574	+	0.999846i	$0.494411\pi$
$812$	0	0
$813$	72.0000	2.52515
$814$	0	0
$815$	−1.00000	−0.0350285
$816$	0	0
$817$	−60.0000	−2.09913
$818$	0	0
$819$	−36.0000	−1.25794
$820$	0	0
$821$	50.0000	1.74501	0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000	1.74501	0.872506	+	0.488603i	$0.162493\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	0	0
$825$	−3.00000	−0.104447
$826$	0	0
$827$	−52.0000	−1.80822	−0.904109	−	0.427303i	$-0.859464\pi$
$827$	−52.0000	−1.80822	−0.904109	+	0.427303i	$0.859464\pi$
$828$	0	0
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	48.0000	1.66510
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	1.00000	0.0346064
$836$	0	0
$837$	45.0000	1.55543
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	54.0000	1.85986
$844$	0	0
$845$	23.0000	0.791224
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	30.0000	1.02960
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	0	0
$855$	−30.0000	−1.02598
$856$	0	0
$857$	−21.0000	−0.717346	−0.358673	−	0.933463i	$-0.616771\pi$
$857$	−21.0000	−0.717346	−0.358673	+	0.933463i	$0.616771\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	−46.0000	−1.56586	−0.782929	−	0.622111i	$-0.786275\pi$
$863$	−46.0000	−1.56586	−0.782929	+	0.622111i	$0.786275\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	−24.0000	−0.815083
$868$	0	0
$869$	−2.00000	−0.0678454
$870$	0	0
$871$	−96.0000	−3.25284
$872$	0	0
$873$	−96.0000	−3.24911
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	0	0
$879$	18.0000	0.607125
$880$	0	0
$881$	50.0000	1.68454	0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000	1.68454	0.842271	+	0.539054i	$0.181218\pi$
$882$	0	0
$883$	5.00000	0.168263	0.0841317	−	0.996455i	$-0.473188\pi$
$883$	5.00000	0.168263	0.0841317	+	0.996455i	$0.473188\pi$
$884$	0	0
$885$	6.00000	0.201688
$886$	0	0
$887$	−40.0000	−1.34307	−0.671534	−	0.740973i	$-0.734364\pi$
$887$	−40.0000	−1.34307	−0.671534	+	0.740973i	$0.734364\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	−9.00000	−0.301511
$892$	0	0
$893$	10.0000	0.334637
$894$	0	0
$895$	4.00000	0.133705
$896$	0	0
$897$	36.0000	1.20201
$898$	0	0
$899$	−25.0000	−0.833797
$900$	0	0
$901$	−39.0000	−1.29928
$902$	0	0
$903$	36.0000	1.19800
$904$	0	0
$905$	22.0000	0.731305
$906$	0	0
$907$	−29.0000	−0.962929	−0.481465	−	0.876466i	$-0.659895\pi$
$907$	−29.0000	−0.962929	−0.481465	+	0.876466i	$0.659895\pi$
$908$	0	0
$909$	−60.0000	−1.99007
$910$	0	0
$911$	3.00000	0.0993944	0.0496972	−	0.998764i	$-0.484174\pi$
$911$	3.00000	0.0993944	0.0496972	+	0.998764i	$0.484174\pi$
$912$	0	0
$913$	14.0000	0.463332
$914$	0	0
$915$	3.00000	0.0991769
$916$	0	0
$917$	7.00000	0.231160
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	−66.0000	−2.17477
$922$	0	0
$923$	−90.0000	−2.96239
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	−96.0000	−3.15305
$928$	0	0
$929$	59.0000	1.93573	0.967864	−	0.251476i	$-0.0809159\pi$
$929$	59.0000	1.93573	0.967864	+	0.251476i	$0.0809159\pi$
$930$	0	0
$931$	30.0000	0.983210
$932$	0	0
$933$	63.0000	2.06253
$934$	0	0
$935$	−3.00000	−0.0981105
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	−90.0000	−2.93704
$940$	0	0
$941$	−41.0000	−1.33656	−0.668281	−	0.743909i	$-0.732970\pi$
$941$	−41.0000	−1.33656	−0.668281	+	0.743909i	$0.732970\pi$
$942$	0	0
$943$	4.00000	0.130258
$944$	0	0
$945$	9.00000	0.292770
$946$	0	0
$947$	41.0000	1.33232	0.666160	−	0.745808i	$-0.267937\pi$
$947$	41.0000	1.33232	0.666160	+	0.745808i	$0.267937\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	0	0
$951$	−9.00000	−0.291845
$952$	0	0
$953$	29.0000	0.939402	0.469701	−	0.882826i	$-0.344362\pi$
$953$	29.0000	0.939402	0.469701	+	0.882826i	$0.344362\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	0	0
$957$	15.0000	0.484881
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	−24.0000	−0.773389
$964$	0	0
$965$	7.00000	0.225338
$966$	0	0
$967$	55.0000	1.76868	0.884340	−	0.466843i	$-0.154609\pi$
$967$	55.0000	1.76868	0.884340	+	0.466843i	$0.154609\pi$
$968$	0	0
$969$	−45.0000	−1.44561
$970$	0	0
$971$	44.0000	1.41203	0.706014	−	0.708198i	$-0.250492\pi$
$971$	44.0000	1.41203	0.706014	+	0.708198i	$0.250492\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	0	0
$975$	−18.0000	−0.576461
$976$	0	0
$977$	−8.00000	−0.255943	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000	−0.255943	−0.127971	+	0.991778i	$0.540847\pi$
$978$	0	0
$979$	−9.00000	−0.287641
$980$	0	0
$981$	60.0000	1.91565
$982$	0	0
$983$	−14.0000	−0.446531	−0.223265	−	0.974758i	$-0.571672\pi$
$983$	−14.0000	−0.446531	−0.223265	+	0.974758i	$0.571672\pi$
$984$	0	0
$985$	−24.0000	−0.764704
$986$	0	0
$987$	−6.00000	−0.190982
$988$	0	0
$989$	−24.0000	−0.763156
$990$	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
$993$	−60.0000	−1.90404
$994$	0	0
$995$	−1.00000	−0.0317021
$996$	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
$999$	−9.00000	−0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	440.2.a.d.1.1	✓	1
3.2	odd	2		3960.2.a.f.1.1		1
4.3	odd	2		880.2.a.a.1.1		1
5.2	odd	4		2200.2.b.b.1849.1		2
5.3	odd	4		2200.2.b.b.1849.2		2
5.4	even	2		2200.2.a.a.1.1		1
8.3	odd	2		3520.2.a.bh.1.1		1
8.5	even	2		3520.2.a.a.1.1		1
11.10	odd	2		4840.2.a.i.1.1		1
12.11	even	2		7920.2.a.e.1.1		1
20.3	even	4		4400.2.b.a.4049.1		2
20.7	even	4		4400.2.b.a.4049.2		2
20.19	odd	2		4400.2.a.be.1.1		1
44.43	even	2		9680.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.a.d.1.1	✓	1	1.1	even	1	trivial
880.2.a.a.1.1		1	4.3	odd	2
2200.2.a.a.1.1		1	5.4	even	2
2200.2.b.b.1849.1		2	5.2	odd	4
2200.2.b.b.1849.2		2	5.3	odd	4
3520.2.a.a.1.1		1	8.5	even	2
3520.2.a.bh.1.1		1	8.3	odd	2
3960.2.a.f.1.1		1	3.2	odd	2
4400.2.a.be.1.1		1	20.19	odd	2
4400.2.b.a.4049.1		2	20.3	even	4
4400.2.b.a.4049.2		2	20.7	even	4
4840.2.a.i.1.1		1	11.10	odd	2
7920.2.a.e.1.1		1	12.11	even	2
9680.2.a.a.1.1		1	44.43	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:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.a (trivial)

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