LMFDB - Embedded newform 675.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(1,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	675.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:	$5.38990213644$
Analytic rank:	$1$
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	$\chi$	$=$	675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} -1.00000 q^{4} +3.00000 q^{8} -5.00000 q^{11} +5.00000 q^{13} -1.00000 q^{16} -4.00000 q^{17} -2.00000 q^{19} +5.00000 q^{22} +3.00000 q^{23} -5.00000 q^{26} -10.0000 q^{29} +6.00000 q^{31} -5.00000 q^{32} +4.00000 q^{34} -5.00000 q^{37} +2.00000 q^{38} -10.0000 q^{41} -10.0000 q^{43} +5.00000 q^{44} -3.00000 q^{46} +5.00000 q^{47} -7.00000 q^{49} -5.00000 q^{52} +2.00000 q^{53} +10.0000 q^{58} -5.00000 q^{59} -11.0000 q^{61} -6.00000 q^{62} +7.00000 q^{64} +4.00000 q^{68} +5.00000 q^{71} -10.0000 q^{73} +5.00000 q^{74} +2.00000 q^{76} +12.0000 q^{79} +10.0000 q^{82} -12.0000 q^{83} +10.0000 q^{86} -15.0000 q^{88} -3.00000 q^{92} -5.00000 q^{94} -5.00000 q^{97} +7.00000 q^{98} +O(q^{100})$

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 $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	5.00000	1.06600
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	0	0
$26$	−5.00000	−0.980581
$27$	0	0
$28$	0	0
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	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
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	5.00000	0.753778
$45$	0	0
$46$	−3.00000	−0.442326
$47$	5.00000	0.729325	0.364662	−	0.931140i	$-0.381184\pi$
$47$	5.00000	0.729325	0.364662	+	0.931140i	$0.381184\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	−5.00000	−0.693375
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	10.0000	1.31306
$59$	−5.00000	−0.650945	−0.325472	−	0.945552i	$-0.605523\pi$
$59$	−5.00000	−0.650945	−0.325472	+	0.945552i	$0.605523\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	0	0
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	5.00000	0.581238
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	0	0
$82$	10.0000	1.10432
$83$	−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
$85$	0	0
$86$	10.0000	1.07833
$87$	0	0
$88$	−15.0000	−1.59901
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	−3.00000	−0.312772
$93$	0	0
$94$	−5.00000	−0.515711
$95$	0	0
$96$	0	0
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	7.00000	0.707107
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	15.0000	1.47087
$105$	0	0
$106$	−2.00000	−0.194257
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	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
$111$	0	0
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	0	0
$116$	10.0000	0.928477
$117$	0	0
$118$	5.00000	0.460287
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	11.0000	0.995893
$123$	0	0
$124$	−6.00000	−0.538816
$125$	0	0
$126$	0	0
$127$	−10.0000	−0.887357	−0.443678	−	0.896186i	$-0.646327\pi$
$127$	−10.0000	−0.887357	−0.443678	+	0.896186i	$0.646327\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	−12.0000	−1.02899
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	−5.00000	−0.419591
$143$	−25.0000	−2.09061
$144$	0	0
$145$	0	0
$146$	10.0000	0.827606
$147$	0	0
$148$	5.00000	0.410997
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−12.0000	−0.954669
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	12.0000	0.931381
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	10.0000	0.762493
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	0	0
$176$	5.00000	0.376889
$177$	0	0
$178$	0	0
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	0	0
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	0	0
$183$	0	0
$184$	9.00000	0.663489
$185$	0	0
$186$	0	0
$187$	20.0000	1.46254
$188$	−5.00000	−0.364662
$189$	0	0
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	5.00000	0.358979
$195$	0	0
$196$	7.00000	0.500000
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−10.0000	−0.696733
$207$	0	0
$208$	−5.00000	−0.346688
$209$	10.0000	0.691714
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	−2.00000	−0.137361
$213$	0	0
$214$	−15.0000	−1.02538
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−2.00000	−0.135457
$219$	0	0
$220$	0	0
$221$	−20.0000	−1.34535
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	0	0
$226$	10.0000	0.665190
$227$	−11.0000	−0.730096	−0.365048	−	0.930989i	$-0.618947\pi$
$227$	−11.0000	−0.730096	−0.365048	+	0.930989i	$0.618947\pi$
$228$	0	0
$229$	3.00000	0.198246	0.0991228	−	0.995075i	$-0.468396\pi$
$229$	3.00000	0.198246	0.0991228	+	0.995075i	$0.468396\pi$
$230$	0	0
$231$	0	0
$232$	−30.0000	−1.96960
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	5.00000	0.325472
$237$	0	0
$238$	0	0
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	0	0
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	−14.0000	−0.899954
$243$	0	0
$244$	11.0000	0.704203
$245$	0	0
$246$	0	0
$247$	−10.0000	−0.636285
$248$	18.0000	1.14300
$249$	0	0
$250$	0	0
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	0	0
$253$	−15.0000	−0.943042
$254$	10.0000	0.627456
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−15.0000	−0.926703
$263$	−5.00000	−0.308313	−0.154157	−	0.988046i	$-0.549266\pi$
$263$	−5.00000	−0.308313	−0.154157	+	0.988046i	$0.549266\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	30.0000	1.80253	0.901263	−	0.433273i	$-0.142641\pi$
$277$	30.0000	1.80253	0.901263	+	0.433273i	$0.142641\pi$
$278$	16.0000	0.959616
$279$	0	0
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	−30.0000	−1.78331	−0.891657	−	0.452711i	$-0.850457\pi$
$283$	−30.0000	−1.78331	−0.891657	+	0.452711i	$0.850457\pi$
$284$	−5.00000	−0.296695
$285$	0	0
$286$	25.0000	1.47828
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	10.0000	0.585206
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	0	0
$296$	−15.0000	−0.871857
$297$	0	0
$298$	10.0000	0.579284
$299$	15.0000	0.867472
$300$	0	0
$301$	0	0
$302$	2.00000	0.115087
$303$	0	0
$304$	2.00000	0.114708
$305$	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
$309$	0	0
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−12.0000	−0.675053
$317$	−26.0000	−1.46031	−0.730153	−	0.683284i	$-0.760551\pi$
$317$	−26.0000	−1.46031	−0.730153	+	0.683284i	$0.760551\pi$
$318$	0	0
$319$	50.0000	2.79946
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	0	0
$326$	−10.0000	−0.553849
$327$	0	0
$328$	−30.0000	−1.65647
$329$	0	0
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−9.00000	−0.492458
$335$	0	0
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	0	0
$341$	−30.0000	−1.62459
$342$	0	0
$343$	0	0
$344$	−30.0000	−1.61749
$345$	0	0
$346$	−12.0000	−0.645124
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	0	0
$352$	25.0000	1.33250
$353$	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
$355$	0	0
$356$	0	0
$357$	0	0
$358$	5.00000	0.264258
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−11.0000	−0.578147
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	−3.00000	−0.156386
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	−20.0000	−1.03418
$375$	0	0
$376$	15.0000	0.773566
$377$	−50.0000	−2.57513
$378$	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
$381$	0	0
$382$	−20.0000	−1.02329
$383$	−3.00000	−0.153293	−0.0766464	−	0.997058i	$-0.524421\pi$
$383$	−3.00000	−0.153293	−0.0766464	+	0.997058i	$0.524421\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	5.00000	0.253837
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	−21.0000	−1.06066
$393$	0	0
$394$	−24.0000	−1.20910
$395$	0	0
$396$	0	0
$397$	−25.0000	−1.25471	−0.627357	−	0.778732i	$-0.715863\pi$
$397$	−25.0000	−1.25471	−0.627357	+	0.778732i	$0.715863\pi$
$398$	14.0000	0.701757
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	30.0000	1.49441
$404$	0	0
$405$	0	0
$406$	0	0
$407$	25.0000	1.23920
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	0	0
$411$	0	0
$412$	−10.0000	−0.492665
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−25.0000	−1.22573
$417$	0	0
$418$	−10.0000	−0.489116
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	31.0000	1.51085	0.755424	−	0.655237i	$-0.227431\pi$
$421$	31.0000	1.51085	0.755424	+	0.655237i	$0.227431\pi$
$422$	16.0000	0.778868
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−15.0000	−0.725052
$429$	0	0
$430$	0	0
$431$	15.0000	0.722525	0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000	0.722525	0.361262	+	0.932464i	$0.382346\pi$
$432$	0	0
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	20.0000	0.951303
$443$	−3.00000	−0.142534	−0.0712672	−	0.997457i	$-0.522704\pi$
$443$	−3.00000	−0.142534	−0.0712672	+	0.997457i	$0.522704\pi$
$444$	0	0
$445$	0	0
$446$	−10.0000	−0.473514
$447$	0	0
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	50.0000	2.35441
$452$	10.0000	0.470360
$453$	0	0
$454$	11.0000	0.516256
$455$	0	0
$456$	0	0
$457$	35.0000	1.63723	0.818615	−	0.574342i	$-0.194742\pi$
$457$	35.0000	1.63723	0.818615	+	0.574342i	$0.194742\pi$
$458$	−3.00000	−0.140181
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−30.0000	−1.39422	−0.697109	−	0.716965i	$-0.745531\pi$
$463$	−30.0000	−1.39422	−0.697109	+	0.716965i	$0.745531\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	0	0
$467$	−11.0000	−0.509019	−0.254510	−	0.967070i	$-0.581914\pi$
$467$	−11.0000	−0.509019	−0.254510	+	0.967070i	$0.581914\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−15.0000	−0.690431
$473$	50.0000	2.29900
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	5.00000	0.228695
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−25.0000	−1.13990
$482$	11.0000	0.501036
$483$	0	0
$484$	−14.0000	−0.636364
$485$	0	0
$486$	0	0
$487$	40.0000	1.81257	0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000	1.81257	0.906287	+	0.422664i	$0.138905\pi$
$488$	−33.0000	−1.49384
$489$	0	0
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	40.0000	1.80151
$494$	10.0000	0.449921
$495$	0	0
$496$	−6.00000	−0.269408
$497$	0	0
$498$	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
$501$	0	0
$502$	−15.0000	−0.669483
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	0	0
$505$	0	0
$506$	15.0000	0.666831
$507$	0	0
$508$	10.0000	0.443678
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	0	0
$512$	11.0000	0.486136
$513$	0	0
$514$	24.0000	1.05859
$515$	0	0
$516$	0	0
$517$	−25.0000	−1.09950
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.0000	0.876216	0.438108	−	0.898922i	$-0.355649\pi$
$521$	20.0000	0.876216	0.438108	+	0.898922i	$0.355649\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−15.0000	−0.655278
$525$	0	0
$526$	5.00000	0.218010
$527$	−24.0000	−1.04546
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−50.0000	−2.16574
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	20.0000	0.862261
$539$	35.0000	1.50756
$540$	0	0
$541$	−33.0000	−1.41878	−0.709390	−	0.704816i	$-0.751030\pi$
$541$	−33.0000	−1.41878	−0.709390	+	0.704816i	$0.751030\pi$
$542$	−4.00000	−0.171815
$543$	0	0
$544$	20.0000	0.857493
$545$	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	$0.359370\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	0	0
$551$	20.0000	0.852029
$552$	0	0
$553$	0	0
$554$	−30.0000	−1.27458
$555$	0	0
$556$	16.0000	0.678551
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	−50.0000	−2.11477
$560$	0	0
$561$	0	0
$562$	30.0000	1.26547
$563$	−45.0000	−1.89652	−0.948262	−	0.317489i	$-0.897160\pi$
$563$	−45.0000	−1.89652	−0.948262	+	0.317489i	$0.897160\pi$
$564$	0	0
$565$	0	0
$566$	30.0000	1.26099
$567$	0	0
$568$	15.0000	0.629386
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	14.0000	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$572$	25.0000	1.04530
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−35.0000	−1.45707	−0.728535	−	0.685009i	$-0.759798\pi$
$577$	−35.0000	−1.45707	−0.728535	+	0.685009i	$0.759798\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−10.0000	−0.414158
$584$	−30.0000	−1.24141
$585$	0	0
$586$	−12.0000	−0.495715
$587$	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
$589$	−12.0000	−0.494451
$590$	0	0
$591$	0	0
$592$	5.00000	0.205499
$593$	28.0000	1.14982	0.574911	−	0.818216i	$-0.305037\pi$
$593$	28.0000	1.14982	0.574911	+	0.818216i	$0.305037\pi$
$594$	0	0
$595$	0	0
$596$	10.0000	0.409616
$597$	0	0
$598$	−15.0000	−0.613396
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	11.0000	0.448699	0.224350	−	0.974509i	$-0.427974\pi$
$601$	11.0000	0.448699	0.224350	+	0.974509i	$0.427974\pi$
$602$	0	0
$603$	0	0
$604$	2.00000	0.0813788
$605$	0	0
$606$	0	0
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	10.0000	0.405554
$609$	0	0
$610$	0	0
$611$	25.0000	1.01139
$612$	0	0
$613$	−25.0000	−1.00974	−0.504870	−	0.863195i	$-0.668460\pi$
$613$	−25.0000	−1.00974	−0.504870	+	0.863195i	$0.668460\pi$
$614$	10.0000	0.403567
$615$	0	0
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	0	0
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	0	0
$621$	0	0
$622$	15.0000	0.601445
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−10.0000	−0.399680
$627$	0	0
$628$	−10.0000	−0.399043
$629$	20.0000	0.797452
$630$	0	0
$631$	34.0000	1.35352	0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000	1.35352	0.676759	+	0.736204i	$0.263384\pi$
$632$	36.0000	1.43200
$633$	0	0
$634$	26.0000	1.03259
$635$	0	0
$636$	0	0
$637$	−35.0000	−1.38675
$638$	−50.0000	−1.97952
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−30.0000	−1.18308	−0.591542	−	0.806274i	$-0.701481\pi$
$643$	−30.0000	−1.18308	−0.591542	+	0.806274i	$0.701481\pi$
$644$	0	0
$645$	0	0
$646$	−8.00000	−0.314756
$647$	29.0000	1.14011	0.570054	−	0.821607i	$-0.306922\pi$
$647$	29.0000	1.14011	0.570054	+	0.821607i	$0.306922\pi$
$648$	0	0
$649$	25.0000	0.981336
$650$	0	0
$651$	0	0
$652$	−10.0000	−0.391630
$653$	−8.00000	−0.313064	−0.156532	−	0.987673i	$-0.550031\pi$
$653$	−8.00000	−0.313064	−0.156532	+	0.987673i	$0.550031\pi$
$654$	0	0
$655$	0	0
$656$	10.0000	0.390434
$657$	0	0
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	13.0000	0.505641	0.252821	−	0.967513i	$-0.418642\pi$
$661$	13.0000	0.505641	0.252821	+	0.967513i	$0.418642\pi$
$662$	18.0000	0.699590
$663$	0	0
$664$	−36.0000	−1.39707
$665$	0	0
$666$	0	0
$667$	−30.0000	−1.16160
$668$	−9.00000	−0.348220
$669$	0	0
$670$	0	0
$671$	55.0000	2.12325
$672$	0	0
$673$	−15.0000	−0.578208	−0.289104	−	0.957298i	$-0.593357\pi$
$673$	−15.0000	−0.578208	−0.289104	+	0.957298i	$0.593357\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	−12.0000	−0.461538
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	30.0000	1.14876
$683$	−33.0000	−1.26271	−0.631355	−	0.775494i	$-0.717501\pi$
$683$	−33.0000	−1.26271	−0.631355	+	0.775494i	$0.717501\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	10.0000	0.381246
$689$	10.0000	0.380970
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	−12.0000	−0.456172
$693$	0	0
$694$	−20.0000	−0.759190
$695$	0	0
$696$	0	0
$697$	40.0000	1.51511
$698$	2.00000	0.0757011
$699$	0	0
$700$	0	0
$701$	−40.0000	−1.51078	−0.755390	−	0.655276i	$-0.772552\pi$
$701$	−40.0000	−1.51078	−0.755390	+	0.655276i	$0.772552\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	−35.0000	−1.31911
$705$	0	0
$706$	−30.0000	−1.12906
$707$	0	0
$708$	0	0
$709$	19.0000	0.713560	0.356780	−	0.934188i	$-0.383875\pi$
$709$	19.0000	0.713560	0.356780	+	0.934188i	$0.383875\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	18.0000	0.674105
$714$	0	0
$715$	0	0
$716$	5.00000	0.186859
$717$	0	0
$718$	−15.0000	−0.559795
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	0	0
$722$	15.0000	0.558242
$723$	0	0
$724$	−11.0000	−0.408812
$725$	0	0
$726$	0	0
$727$	−20.0000	−0.741759	−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000	−0.741759	−0.370879	+	0.928681i	$0.620944\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	40.0000	1.47945
$732$	0	0
$733$	25.0000	0.923396	0.461698	−	0.887037i	$-0.347240\pi$
$733$	25.0000	0.923396	0.461698	+	0.887037i	$0.347240\pi$
$734$	0	0
$735$	0	0
$736$	−15.0000	−0.552907
$737$	0	0
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−25.0000	−0.917161	−0.458581	−	0.888653i	$-0.651642\pi$
$743$	−25.0000	−0.917161	−0.458581	+	0.888653i	$0.651642\pi$
$744$	0	0
$745$	0	0
$746$	10.0000	0.366126
$747$	0	0
$748$	−20.0000	−0.731272
$749$	0	0
$750$	0	0
$751$	−44.0000	−1.60558	−0.802791	−	0.596260i	$-0.796653\pi$
$751$	−44.0000	−1.60558	−0.802791	+	0.596260i	$0.796653\pi$
$752$	−5.00000	−0.182331
$753$	0	0
$754$	50.0000	1.82089
$755$	0	0
$756$	0	0
$757$	5.00000	0.181728	0.0908640	−	0.995863i	$-0.471037\pi$
$757$	5.00000	0.181728	0.0908640	+	0.995863i	$0.471037\pi$
$758$	2.00000	0.0726433
$759$	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
$763$	0	0
$764$	−20.0000	−0.723575
$765$	0	0
$766$	3.00000	0.108394
$767$	−25.0000	−0.902698
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	0	0
$772$	−10.0000	−0.359908
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	0	0
$776$	−15.0000	−0.538469
$777$	0	0
$778$	30.0000	1.07555
$779$	20.0000	0.716574
$780$	0	0
$781$	−25.0000	−0.894570
$782$	12.0000	0.429119
$783$	0	0
$784$	7.00000	0.250000
$785$	0	0
$786$	0	0
$787$	−40.0000	−1.42585	−0.712923	−	0.701242i	$-0.752629\pi$
$787$	−40.0000	−1.42585	−0.712923	+	0.701242i	$0.752629\pi$
$788$	−24.0000	−0.854965
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−55.0000	−1.95311
$794$	25.0000	0.887217
$795$	0	0
$796$	14.0000	0.496217
$797$	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
$799$	−20.0000	−0.707549
$800$	0	0
$801$	0	0
$802$	0	0
$803$	50.0000	1.76446
$804$	0	0
$805$	0	0
$806$	−30.0000	−1.05670
$807$	0	0
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	−12.0000	−0.421377	−0.210688	−	0.977553i	$-0.567571\pi$
$811$	−12.0000	−0.421377	−0.210688	+	0.977553i	$0.567571\pi$
$812$	0	0
$813$	0	0
$814$	−25.0000	−0.876250
$815$	0	0
$816$	0	0
$817$	20.0000	0.699711
$818$	7.00000	0.244749
$819$	0	0
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	10.0000	0.348578	0.174289	−	0.984695i	$-0.444237\pi$
$823$	10.0000	0.348578	0.174289	+	0.984695i	$0.444237\pi$
$824$	30.0000	1.04510
$825$	0	0
$826$	0	0
$827$	−19.0000	−0.660695	−0.330347	−	0.943859i	$-0.607166\pi$
$827$	−19.0000	−0.660695	−0.330347	+	0.943859i	$0.607166\pi$
$828$	0	0
$829$	−21.0000	−0.729360	−0.364680	−	0.931133i	$-0.618822\pi$
$829$	−21.0000	−0.729360	−0.364680	+	0.931133i	$0.618822\pi$
$830$	0	0
$831$	0	0
$832$	35.0000	1.21341
$833$	28.0000	0.970143
$834$	0	0
$835$	0	0
$836$	−10.0000	−0.345857
$837$	0	0
$838$	−20.0000	−0.690889
$839$	35.0000	1.20833	0.604167	−	0.796858i	$-0.293506\pi$
$839$	35.0000	1.20833	0.604167	+	0.796858i	$0.293506\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	−31.0000	−1.06833
$843$	0	0
$844$	16.0000	0.550743
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	0	0
$850$	0	0
$851$	−15.0000	−0.514193
$852$	0	0
$853$	−45.0000	−1.54077	−0.770385	−	0.637579i	$-0.779936\pi$
$853$	−45.0000	−1.54077	−0.770385	+	0.637579i	$0.779936\pi$
$854$	0	0
$855$	0	0
$856$	45.0000	1.53807
$857$	−4.00000	−0.136637	−0.0683187	−	0.997664i	$-0.521763\pi$
$857$	−4.00000	−0.136637	−0.0683187	+	0.997664i	$0.521763\pi$
$858$	0	0
$859$	22.0000	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$860$	0	0
$861$	0	0
$862$	−15.0000	−0.510902
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	0	0
$865$	0	0
$866$	−5.00000	−0.169907
$867$	0	0
$868$	0	0
$869$	−60.0000	−2.03536
$870$	0	0
$871$	0	0
$872$	6.00000	0.203186
$873$	0	0
$874$	6.00000	0.202953
$875$	0	0
$876$	0	0
$877$	15.0000	0.506514	0.253257	−	0.967399i	$-0.418498\pi$
$877$	15.0000	0.506514	0.253257	+	0.967399i	$0.418498\pi$
$878$	8.00000	0.269987
$879$	0	0
$880$	0	0
$881$	−20.0000	−0.673817	−0.336909	−	0.941537i	$-0.609381\pi$
$881$	−20.0000	−0.673817	−0.336909	+	0.941537i	$0.609381\pi$
$882$	0	0
$883$	40.0000	1.34611	0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000	1.34611	0.673054	+	0.739594i	$0.264982\pi$
$884$	20.0000	0.672673
$885$	0	0
$886$	3.00000	0.100787
$887$	−9.00000	−0.302190	−0.151095	−	0.988519i	$-0.548280\pi$
$887$	−9.00000	−0.302190	−0.151095	+	0.988519i	$0.548280\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−10.0000	−0.334825
$893$	−10.0000	−0.334637
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−10.0000	−0.333704
$899$	−60.0000	−2.00111
$900$	0	0
$901$	−8.00000	−0.266519
$902$	−50.0000	−1.66482
$903$	0	0
$904$	−30.0000	−0.997785
$905$	0	0
$906$	0	0
$907$	−30.0000	−0.996134	−0.498067	−	0.867139i	$-0.665957\pi$
$907$	−30.0000	−0.996134	−0.498067	+	0.867139i	$0.665957\pi$
$908$	11.0000	0.365048
$909$	0	0
$910$	0	0
$911$	−5.00000	−0.165657	−0.0828287	−	0.996564i	$-0.526395\pi$
$911$	−5.00000	−0.165657	−0.0828287	+	0.996564i	$0.526395\pi$
$912$	0	0
$913$	60.0000	1.98571
$914$	−35.0000	−1.15770
$915$	0	0
$916$	−3.00000	−0.0991228
$917$	0	0
$918$	0	0
$919$	−26.0000	−0.857661	−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000	−0.857661	−0.428830	+	0.903385i	$0.641074\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	25.0000	0.822885
$924$	0	0
$925$	0	0
$926$	30.0000	0.985861
$927$	0	0
$928$	50.0000	1.64133
$929$	50.0000	1.64045	0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000	1.64045	0.820223	+	0.572043i	$0.193849\pi$
$930$	0	0
$931$	14.0000	0.458831
$932$	0	0
$933$	0	0
$934$	11.0000	0.359931
$935$	0	0
$936$	0	0
$937$	−15.0000	−0.490029	−0.245014	−	0.969519i	$-0.578793\pi$
$937$	−15.0000	−0.490029	−0.245014	+	0.969519i	$0.578793\pi$
$938$	0	0
$939$	0	0
$940$	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
$943$	−30.0000	−0.976934
$944$	5.00000	0.162736
$945$	0	0
$946$	−50.0000	−1.62564
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	0	0
$949$	−50.0000	−1.62307
$950$	0	0
$951$	0	0
$952$	0	0
$953$	20.0000	0.647864	0.323932	−	0.946080i	$-0.394995\pi$
$953$	20.0000	0.647864	0.323932	+	0.946080i	$0.394995\pi$
$954$	0	0
$955$	0	0
$956$	5.00000	0.161712
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	25.0000	0.806032
$963$	0	0
$964$	11.0000	0.354286
$965$	0	0
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	42.0000	1.34993
$969$	0	0
$970$	0	0
$971$	−45.0000	−1.44412	−0.722059	−	0.691831i	$-0.756804\pi$
$971$	−45.0000	−1.44412	−0.722059	+	0.691831i	$0.756804\pi$
$972$	0	0
$973$	0	0
$974$	−40.0000	−1.28168
$975$	0	0
$976$	11.0000	0.352101
$977$	−40.0000	−1.27971	−0.639857	−	0.768494i	$-0.721006\pi$
$977$	−40.0000	−1.27971	−0.639857	+	0.768494i	$0.721006\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−20.0000	−0.638226
$983$	−27.0000	−0.861166	−0.430583	−	0.902551i	$-0.641692\pi$
$983$	−27.0000	−0.861166	−0.430583	+	0.902551i	$0.641692\pi$
$984$	0	0
$985$	0	0
$986$	−40.0000	−1.27386
$987$	0	0
$988$	10.0000	0.318142
$989$	−30.0000	−0.953945
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−30.0000	−0.952501
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	15.0000	0.475055	0.237527	−	0.971381i	$-0.423663\pi$
$997$	15.0000	0.475055	0.237527	+	0.971381i	$0.423663\pi$
$998$	8.00000	0.253236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.2.a.b.1.1	✓	1
3.2	odd	2		675.2.a.h.1.1	yes	1
5.2	odd	4		675.2.b.c.649.1		2
5.3	odd	4		675.2.b.c.649.2		2
5.4	even	2		675.2.a.g.1.1	yes	1
15.2	even	4		675.2.b.d.649.2		2
15.8	even	4		675.2.b.d.649.1		2
15.14	odd	2		675.2.a.c.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
675.2.a.b.1.1	✓	1	1.1	even	1	trivial
675.2.a.c.1.1	yes	1	15.14	odd	2
675.2.a.g.1.1	yes	1	5.4	even	2
675.2.a.h.1.1	yes	1	3.2	odd	2
675.2.b.c.649.1		2	5.2	odd	4
675.2.b.c.649.2		2	5.3	odd	4
675.2.b.d.649.1		2	15.8	even	4
675.2.b.d.649.2		2	15.2	even	4

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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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