LMFDB - Embedded newform 1323.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, 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("1323.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1323 = 3^{3} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1323.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:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1323.1

$q$ -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} +4.00000 q^{5} -3.00000 q^{8} +4.00000 q^{10} +2.00000 q^{11} -1.00000 q^{13} -1.00000 q^{16} +6.00000 q^{17} -4.00000 q^{19} -4.00000 q^{20} +2.00000 q^{22} +6.00000 q^{23} +11.0000 q^{25} -1.00000 q^{26} +2.00000 q^{29} -3.00000 q^{31} +5.00000 q^{32} +6.00000 q^{34} +3.00000 q^{37} -4.00000 q^{38} -12.0000 q^{40} +2.00000 q^{41} -1.00000 q^{43} -2.00000 q^{44} +6.00000 q^{46} -6.00000 q^{47} +11.0000 q^{50} +1.00000 q^{52} -6.00000 q^{53} +8.00000 q^{55} +2.00000 q^{58} -6.00000 q^{59} +5.00000 q^{61} -3.00000 q^{62} +7.00000 q^{64} -4.00000 q^{65} +7.00000 q^{67} -6.00000 q^{68} +6.00000 q^{73} +3.00000 q^{74} +4.00000 q^{76} +11.0000 q^{79} -4.00000 q^{80} +2.00000 q^{82} -6.00000 q^{83} +24.0000 q^{85} -1.00000 q^{86} -6.00000 q^{88} +4.00000 q^{89} -6.00000 q^{92} -6.00000 q^{94} -16.0000 q^{95} -9.00000 q^{97} +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

)

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

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−3.00000	−1.06066
$9$	0	0
$10$	4.00000	1.26491
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−4.00000	−0.894427
$21$	0	0
$22$	2.00000	0.426401
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	11.0000	2.20000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	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
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	−4.00000	−0.648886
$39$	0	0
$40$	−12.0000	−1.89737
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	6.00000	0.884652
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	0	0
$50$	11.0000	1.55563
$51$	0	0
$52$	1.00000	0.138675
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	0	0
$58$	2.00000	0.262613
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	−3.00000	−0.381000
$63$	0	0
$64$	7.00000	0.875000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	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	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	2.00000	0.220863
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	24.0000	2.60317
$86$	−1.00000	−0.107833
$87$	0	0
$88$	−6.00000	−0.639602
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	0	0
$94$	−6.00000	−0.618853
$95$	−16.0000	−1.64157
$96$	0	0
$97$	−9.00000	−0.913812	−0.456906	−	0.889515i	$-0.651042\pi$
$97$	−9.00000	−0.913812	−0.456906	+	0.889515i	$0.651042\pi$
$98$	0	0
$99$	0	0
$100$	−11.0000	−1.10000
$101$	8.00000	0.796030	0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000	0.796030	0.398015	+	0.917379i	$0.369699\pi$
$102$	0	0
$103$	−17.0000	−1.67506	−0.837530	−	0.546392i	$-0.816001\pi$
$103$	−17.0000	−1.67506	−0.837530	+	0.546392i	$0.816001\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−6.00000	−0.582772
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	0	0
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	8.00000	0.762770
$111$	0	0
$112$	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
$115$	24.0000	2.23801
$116$	−2.00000	−0.185695
$117$	0	0
$118$	−6.00000	−0.552345
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	5.00000	0.452679
$123$	0	0
$124$	3.00000	0.269408
$125$	24.0000	2.14663
$126$	0	0
$127$	−9.00000	−0.798621	−0.399310	−	0.916816i	$-0.630750\pi$
$127$	−9.00000	−0.798621	−0.399310	+	0.916816i	$0.630750\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	−4.00000	−0.350823
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0	0
$134$	7.00000	0.604708
$135$	0	0
$136$	−18.0000	−1.54349
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−9.00000	−0.763370	−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000	−0.763370	−0.381685	+	0.924292i	$0.624656\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	8.00000	0.664364
$146$	6.00000	0.496564
$147$	0	0
$148$	−3.00000	−0.246598
$149$	−24.0000	−1.96616	−0.983078	−	0.183186i	$-0.941359\pi$
$149$	−24.0000	−1.96616	−0.983078	+	0.183186i	$0.941359\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	12.0000	0.973329
$153$	0	0
$154$	0	0
$155$	−12.0000	−0.963863
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	11.0000	0.875113
$159$	0	0
$160$	20.0000	1.58114
$161$	0	0
$162$	0	0
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−14.0000	−1.08335	−0.541676	−	0.840587i	$-0.682210\pi$
$167$	−14.0000	−1.08335	−0.541676	+	0.840587i	$0.682210\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	24.0000	1.84072
$171$	0	0
$172$	1.00000	0.0762493
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	4.00000	0.299813
$179$	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
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	0	0
$184$	−18.0000	−1.32698
$185$	12.0000	0.882258
$186$	0	0
$187$	12.0000	0.877527
$188$	6.00000	0.437595
$189$	0	0
$190$	−16.0000	−1.16076
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	−25.0000	−1.79954	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−25.0000	−1.79954	−0.899770	+	0.436365i	$0.856266\pi$
$194$	−9.00000	−0.646162
$195$	0	0
$196$	0	0
$197$	20.0000	1.42494	0.712470	−	0.701702i	$-0.247576\pi$
$197$	20.0000	1.42494	0.712470	+	0.701702i	$0.247576\pi$
$198$	0	0
$199$	9.00000	0.637993	0.318997	−	0.947756i	$-0.396654\pi$
$199$	9.00000	0.637993	0.318997	+	0.947756i	$0.396654\pi$
$200$	−33.0000	−2.33345
$201$	0	0
$202$	8.00000	0.562878
$203$	0	0
$204$	0	0
$205$	8.00000	0.558744
$206$	−17.0000	−1.18445
$207$	0	0
$208$	1.00000	0.0693375
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	2.00000	0.136717
$215$	−4.00000	−0.272798
$216$	0	0
$217$	0	0
$218$	9.00000	0.609557
$219$	0	0
$220$	−8.00000	−0.539360
$221$	−6.00000	−0.403604
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	0	0
$226$	−16.0000	−1.06430
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	24.0000	1.58251
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	−24.0000	−1.56559
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−17.0000	−1.09507	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000	−1.09507	−0.547533	+	0.836784i	$0.684433\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	−5.00000	−0.320092
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	9.00000	0.571501
$249$	0	0
$250$	24.0000	1.51789
$251$	−26.0000	−1.64111	−0.820553	−	0.571571i	$-0.806334\pi$
$251$	−26.0000	−1.64111	−0.820553	+	0.571571i	$0.806334\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	−9.00000	−0.564710
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−10.0000	−0.623783	−0.311891	−	0.950118i	$-0.600963\pi$
$257$	−10.0000	−0.623783	−0.311891	+	0.950118i	$0.600963\pi$
$258$	0	0
$259$	0	0
$260$	4.00000	0.248069
$261$	0	0
$262$	4.00000	0.247121
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	−24.0000	−1.47431
$266$	0	0
$267$	0	0
$268$	−7.00000	−0.427593
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	18.0000	1.08742
$275$	22.0000	1.32665
$276$	0	0
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	−9.00000	−0.539784
$279$	0	0
$280$	0	0
$281$	16.0000	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$281$	16.0000	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$282$	0	0
$283$	5.00000	0.297219	0.148610	−	0.988896i	$-0.452520\pi$
$283$	5.00000	0.297219	0.148610	+	0.988896i	$0.452520\pi$
$284$	0	0
$285$	0	0
$286$	−2.00000	−0.118262
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	8.00000	0.469776
$291$	0	0
$292$	−6.00000	−0.351123
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	−9.00000	−0.523114
$297$	0	0
$298$	−24.0000	−1.39028
$299$	−6.00000	−0.346989
$300$	0	0
$301$	0	0
$302$	5.00000	0.287718
$303$	0	0
$304$	4.00000	0.229416
$305$	20.0000	1.14520
$306$	0	0
$307$	−25.0000	−1.42683	−0.713413	−	0.700744i	$-0.752851\pi$
$307$	−25.0000	−1.42683	−0.713413	+	0.700744i	$0.752851\pi$
$308$	0	0
$309$	0	0
$310$	−12.0000	−0.681554
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−11.0000	−0.618798
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	28.0000	1.56525
$321$	0	0
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	−11.0000	−0.610170
$326$	19.0000	1.05231
$327$	0	0
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	−14.0000	−0.766046
$335$	28.0000	1.52980
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	−24.0000	−1.30158
$341$	−6.00000	−0.324918
$342$	0	0
$343$	0	0
$344$	3.00000	0.161749
$345$	0	0
$346$	−16.0000	−0.860165
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	5.00000	0.267644	0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000	0.267644	0.133822	+	0.991005i	$0.457275\pi$
$350$	0	0
$351$	0	0
$352$	10.0000	0.533002
$353$	28.0000	1.49029	0.745145	−	0.666903i	$-0.232380\pi$
$353$	28.0000	1.49029	0.745145	+	0.666903i	$0.232380\pi$
$354$	0	0
$355$	0	0
$356$	−4.00000	−0.212000
$357$	0	0
$358$	4.00000	0.211407
$359$	22.0000	1.16112	0.580558	−	0.814219i	$-0.302835\pi$
$359$	22.0000	1.16112	0.580558	+	0.814219i	$0.302835\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	2.00000	0.105118
$363$	0	0
$364$	0	0
$365$	24.0000	1.25622
$366$	0	0
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	12.0000	0.623850
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	18.0000	0.928279
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−9.00000	−0.462299	−0.231149	−	0.972918i	$-0.574249\pi$
$379$	−9.00000	−0.462299	−0.231149	+	0.972918i	$0.574249\pi$
$380$	16.0000	0.820783
$381$	0	0
$382$	−4.00000	−0.204658
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	0	0
$386$	−25.0000	−1.27247
$387$	0	0
$388$	9.00000	0.456906
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	36.0000	1.82060
$392$	0	0
$393$	0	0
$394$	20.0000	1.00759
$395$	44.0000	2.21388
$396$	0	0
$397$	33.0000	1.65622	0.828111	−	0.560564i	$-0.189416\pi$
$397$	33.0000	1.65622	0.828111	+	0.560564i	$0.189416\pi$
$398$	9.00000	0.451129
$399$	0	0
$400$	−11.0000	−0.550000
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	3.00000	0.149441
$404$	−8.00000	−0.398015
$405$	0	0
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	8.00000	0.395092
$411$	0	0
$412$	17.0000	0.837530
$413$	0	0
$414$	0	0
$415$	−24.0000	−1.17811
$416$	−5.00000	−0.245145
$417$	0	0
$418$	−8.00000	−0.391293
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−5.00000	−0.243396
$423$	0	0
$424$	18.0000	0.874157
$425$	66.0000	3.20147
$426$	0	0
$427$	0	0
$428$	−2.00000	−0.0966736
$429$	0	0
$430$	−4.00000	−0.192897
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	17.0000	0.816968	0.408484	−	0.912766i	$-0.366058\pi$
$433$	17.0000	0.816968	0.408484	+	0.912766i	$0.366058\pi$
$434$	0	0
$435$	0	0
$436$	−9.00000	−0.431022
$437$	−24.0000	−1.14808
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	−24.0000	−1.14416
$441$	0	0
$442$	−6.00000	−0.285391
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	16.0000	0.758473
$446$	−16.0000	−0.757622
$447$	0	0
$448$	0	0
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	16.0000	0.752577
$453$	0	0
$454$	−18.0000	−0.844782
$455$	0	0
$456$	0	0
$457$	13.0000	0.608114	0.304057	−	0.952654i	$-0.401659\pi$
$457$	13.0000	0.608114	0.304057	+	0.952654i	$0.401659\pi$
$458$	7.00000	0.327089
$459$	0	0
$460$	−24.0000	−1.11901
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−24.0000	−1.11178
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	−24.0000	−1.10704
$471$	0	0
$472$	18.0000	0.828517
$473$	−2.00000	−0.0919601
$474$	0	0
$475$	−44.0000	−2.01886
$476$	0	0
$477$	0	0
$478$	−12.0000	−0.548867
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	−17.0000	−0.774329
$483$	0	0
$484$	7.00000	0.318182
$485$	−36.0000	−1.63468
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	−15.0000	−0.679018
$489$	0	0
$490$	0	0
$491$	34.0000	1.53440	0.767199	−	0.641409i	$-0.221650\pi$
$491$	34.0000	1.53440	0.767199	+	0.641409i	$0.221650\pi$
$492$	0	0
$493$	12.0000	0.540453
$494$	4.00000	0.179969
$495$	0	0
$496$	3.00000	0.134704
$497$	0	0
$498$	0	0
$499$	−17.0000	−0.761025	−0.380512	−	0.924776i	$-0.624252\pi$
$499$	−17.0000	−0.761025	−0.380512	+	0.924776i	$0.624252\pi$
$500$	−24.0000	−1.07331
$501$	0	0
$502$	−26.0000	−1.16044
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	32.0000	1.42398
$506$	12.0000	0.533465
$507$	0	0
$508$	9.00000	0.399310
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	0	0
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−10.0000	−0.441081
$515$	−68.0000	−2.99644
$516$	0	0
$517$	−12.0000	−0.527759
$518$	0	0
$519$	0	0
$520$	12.0000	0.526235
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	17.0000	0.743358	0.371679	−	0.928361i	$-0.378782\pi$
$523$	17.0000	0.743358	0.371679	+	0.928361i	$0.378782\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	2.00000	0.0872041
$527$	−18.0000	−0.784092
$528$	0	0
$529$	13.0000	0.565217
$530$	−24.0000	−1.04249
$531$	0	0
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	8.00000	0.345870
$536$	−21.0000	−0.907062
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	11.0000	0.472490
$543$	0	0
$544$	30.0000	1.28624
$545$	36.0000	1.54207
$546$	0	0
$547$	43.0000	1.83855	0.919274	−	0.393619i	$-0.128777\pi$
$547$	43.0000	1.83855	0.919274	+	0.393619i	$0.128777\pi$
$548$	−18.0000	−0.768922
$549$	0	0
$550$	22.0000	0.938083
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	13.0000	0.552317
$555$	0	0
$556$	9.00000	0.381685
$557$	−28.0000	−1.18640	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000	−1.18640	−0.593199	+	0.805056i	$0.702135\pi$
$558$	0	0
$559$	1.00000	0.0422955
$560$	0	0
$561$	0	0
$562$	16.0000	0.674919
$563$	10.0000	0.421450	0.210725	−	0.977545i	$-0.432418\pi$
$563$	10.0000	0.421450	0.210725	+	0.977545i	$0.432418\pi$
$564$	0	0
$565$	−64.0000	−2.69250
$566$	5.00000	0.210166
$567$	0	0
$568$	0	0
$569$	32.0000	1.34151	0.670755	−	0.741679i	$-0.265970\pi$
$569$	32.0000	1.34151	0.670755	+	0.741679i	$0.265970\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	0	0
$575$	66.0000	2.75239
$576$	0	0
$577$	17.0000	0.707719	0.353860	−	0.935299i	$-0.384869\pi$
$577$	17.0000	0.707719	0.353860	+	0.935299i	$0.384869\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	−8.00000	−0.332182
$581$	0	0
$582$	0	0
$583$	−12.0000	−0.496989
$584$	−18.0000	−0.744845
$585$	0	0
$586$	−22.0000	−0.908812
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	−24.0000	−0.988064
$591$	0	0
$592$	−3.00000	−0.123299
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	24.0000	0.983078
$597$	0	0
$598$	−6.00000	−0.245358
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	33.0000	1.34610	0.673049	−	0.739598i	$-0.264984\pi$
$601$	33.0000	1.34610	0.673049	+	0.739598i	$0.264984\pi$
$602$	0	0
$603$	0	0
$604$	−5.00000	−0.203447
$605$	−28.0000	−1.13836
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	−20.0000	−0.811107
$609$	0	0
$610$	20.0000	0.809776
$611$	6.00000	0.242734
$612$	0	0
$613$	7.00000	0.282727	0.141364	−	0.989958i	$-0.454851\pi$
$613$	7.00000	0.282727	0.141364	+	0.989958i	$0.454851\pi$
$614$	−25.0000	−1.00892
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	41.0000	1.64793	0.823965	−	0.566641i	$-0.191757\pi$
$619$	41.0000	1.64793	0.823965	+	0.566641i	$0.191757\pi$
$620$	12.0000	0.481932
$621$	0	0
$622$	−24.0000	−0.962312
$623$	0	0
$624$	0	0
$625$	41.0000	1.64000
$626$	22.0000	0.879297
$627$	0	0
$628$	10.0000	0.399043
$629$	18.0000	0.717707
$630$	0	0
$631$	5.00000	0.199047	0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000	0.199047	0.0995234	+	0.995035i	$0.468268\pi$
$632$	−33.0000	−1.31267
$633$	0	0
$634$	30.0000	1.19145
$635$	−36.0000	−1.42862
$636$	0	0
$637$	0	0
$638$	4.00000	0.158362
$639$	0	0
$640$	−12.0000	−0.474342
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	25.0000	0.985904	0.492952	−	0.870057i	$-0.335918\pi$
$643$	25.0000	0.985904	0.492952	+	0.870057i	$0.335918\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	−11.0000	−0.431455
$651$	0	0
$652$	−19.0000	−0.744097
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	18.0000	0.698535
$665$	0	0
$666$	0	0
$667$	12.0000	0.464642
$668$	14.0000	0.541676
$669$	0	0
$670$	28.0000	1.08173
$671$	10.0000	0.386046
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	12.0000	0.461538
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	0	0
$679$	0	0
$680$	−72.0000	−2.76107
$681$	0	0
$682$	−6.00000	−0.229752
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	72.0000	2.75098
$686$	0	0
$687$	0	0
$688$	1.00000	0.0381246
$689$	6.00000	0.228582
$690$	0	0
$691$	−41.0000	−1.55971	−0.779857	−	0.625958i	$-0.784708\pi$
$691$	−41.0000	−1.55971	−0.779857	+	0.625958i	$0.784708\pi$
$692$	16.0000	0.608229
$693$	0	0
$694$	16.0000	0.607352
$695$	−36.0000	−1.36556
$696$	0	0
$697$	12.0000	0.454532
$698$	5.00000	0.189253
$699$	0	0
$700$	0	0
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	14.0000	0.527645
$705$	0	0
$706$	28.0000	1.05379
$707$	0	0
$708$	0	0
$709$	21.0000	0.788672	0.394336	−	0.918966i	$-0.370975\pi$
$709$	21.0000	0.788672	0.394336	+	0.918966i	$0.370975\pi$
$710$	0	0
$711$	0	0
$712$	−12.0000	−0.449719
$713$	−18.0000	−0.674105
$714$	0	0
$715$	−8.00000	−0.299183
$716$	−4.00000	−0.149487
$717$	0	0
$718$	22.0000	0.821033
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	0	0
$722$	−3.00000	−0.111648
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	22.0000	0.817059
$726$	0	0
$727$	−41.0000	−1.52061	−0.760303	−	0.649569i	$-0.774949\pi$
$727$	−41.0000	−1.52061	−0.760303	+	0.649569i	$0.774949\pi$
$728$	0	0
$729$	0	0
$730$	24.0000	0.888280
$731$	−6.00000	−0.221918
$732$	0	0
$733$	−21.0000	−0.775653	−0.387826	−	0.921732i	$-0.626774\pi$
$733$	−21.0000	−0.775653	−0.387826	+	0.921732i	$0.626774\pi$
$734$	−12.0000	−0.442928
$735$	0	0
$736$	30.0000	1.10581
$737$	14.0000	0.515697
$738$	0	0
$739$	−9.00000	−0.331070	−0.165535	−	0.986204i	$-0.552935\pi$
$739$	−9.00000	−0.331070	−0.165535	+	0.986204i	$0.552935\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	0	0
$743$	−42.0000	−1.54083	−0.770415	−	0.637542i	$-0.779951\pi$
$743$	−42.0000	−1.54083	−0.770415	+	0.637542i	$0.779951\pi$
$744$	0	0
$745$	−96.0000	−3.51717
$746$	26.0000	0.951928
$747$	0	0
$748$	−12.0000	−0.438763
$749$	0	0
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	−2.00000	−0.0728357
$755$	20.0000	0.727875
$756$	0	0
$757$	17.0000	0.617876	0.308938	−	0.951082i	$-0.400027\pi$
$757$	17.0000	0.617876	0.308938	+	0.951082i	$0.400027\pi$
$758$	−9.00000	−0.326895
$759$	0	0
$760$	48.0000	1.74114
$761$	48.0000	1.74000	0.869999	−	0.493053i	$-0.164119\pi$
$761$	48.0000	1.74000	0.869999	+	0.493053i	$0.164119\pi$
$762$	0	0
$763$	0	0
$764$	4.00000	0.144715
$765$	0	0
$766$	18.0000	0.650366
$767$	6.00000	0.216647
$768$	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
$771$	0	0
$772$	25.0000	0.899770
$773$	−16.0000	−0.575480	−0.287740	−	0.957709i	$-0.592904\pi$
$773$	−16.0000	−0.575480	−0.287740	+	0.957709i	$0.592904\pi$
$774$	0	0
$775$	−33.0000	−1.18539
$776$	27.0000	0.969244
$777$	0	0
$778$	−12.0000	−0.430221
$779$	−8.00000	−0.286630
$780$	0	0
$781$	0	0
$782$	36.0000	1.28736
$783$	0	0
$784$	0	0
$785$	−40.0000	−1.42766
$786$	0	0
$787$	47.0000	1.67537	0.837685	−	0.546154i	$-0.183909\pi$
$787$	47.0000	1.67537	0.837685	+	0.546154i	$0.183909\pi$
$788$	−20.0000	−0.712470
$789$	0	0
$790$	44.0000	1.56545
$791$	0	0
$792$	0	0
$793$	−5.00000	−0.177555
$794$	33.0000	1.17113
$795$	0	0
$796$	−9.00000	−0.318997
$797$	−14.0000	−0.495905	−0.247953	−	0.968772i	$-0.579758\pi$
$797$	−14.0000	−0.495905	−0.247953	+	0.968772i	$0.579758\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	55.0000	1.94454
$801$	0	0
$802$	30.0000	1.05934
$803$	12.0000	0.423471
$804$	0	0
$805$	0	0
$806$	3.00000	0.105670
$807$	0	0
$808$	−24.0000	−0.844317
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−32.0000	−1.12367	−0.561836	−	0.827249i	$-0.689905\pi$
$811$	−32.0000	−1.12367	−0.561836	+	0.827249i	$0.689905\pi$
$812$	0	0
$813$	0	0
$814$	6.00000	0.210300
$815$	76.0000	2.66216
$816$	0	0
$817$	4.00000	0.139942
$818$	−5.00000	−0.174821
$819$	0	0
$820$	−8.00000	−0.279372
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	0	0
$823$	45.0000	1.56860	0.784301	−	0.620381i	$-0.213022\pi$
$823$	45.0000	1.56860	0.784301	+	0.620381i	$0.213022\pi$
$824$	51.0000	1.77667
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	−24.0000	−0.833052
$831$	0	0
$832$	−7.00000	−0.242681
$833$	0	0
$834$	0	0
$835$	−56.0000	−1.93796
$836$	8.00000	0.276686
$837$	0	0
$838$	12.0000	0.414533
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−10.0000	−0.344623
$843$	0	0
$844$	5.00000	0.172107
$845$	−48.0000	−1.65125
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	66.0000	2.26378
$851$	18.0000	0.617032
$852$	0	0
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	0	0
$856$	−6.00000	−0.205076
$857$	16.0000	0.546550	0.273275	−	0.961936i	$-0.411893\pi$
$857$	16.0000	0.546550	0.273275	+	0.961936i	$0.411893\pi$
$858$	0	0
$859$	25.0000	0.852989	0.426494	−	0.904490i	$-0.359748\pi$
$859$	25.0000	0.852989	0.426494	+	0.904490i	$0.359748\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	18.0000	0.613082
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	0	0
$865$	−64.0000	−2.17607
$866$	17.0000	0.577684
$867$	0	0
$868$	0	0
$869$	22.0000	0.746299
$870$	0	0
$871$	−7.00000	−0.237186
$872$	−27.0000	−0.914335
$873$	0	0
$874$	−24.0000	−0.811812
$875$	0	0
$876$	0	0
$877$	7.00000	0.236373	0.118187	−	0.992991i	$-0.462292\pi$
$877$	7.00000	0.236373	0.118187	+	0.992991i	$0.462292\pi$
$878$	−24.0000	−0.809961
$879$	0	0
$880$	−8.00000	−0.269680
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	−24.0000	−0.806296
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	0	0
$889$	0	0
$890$	16.0000	0.536321
$891$	0	0
$892$	16.0000	0.535720
$893$	24.0000	0.803129
$894$	0	0
$895$	16.0000	0.534821
$896$	0	0
$897$	0	0
$898$	−36.0000	−1.20134
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−36.0000	−1.19933
$902$	4.00000	0.133185
$903$	0	0
$904$	48.0000	1.59646
$905$	8.00000	0.265929
$906$	0	0
$907$	−13.0000	−0.431658	−0.215829	−	0.976431i	$-0.569245\pi$
$907$	−13.0000	−0.431658	−0.215829	+	0.976431i	$0.569245\pi$
$908$	18.0000	0.597351
$909$	0	0
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	13.0000	0.430002
$915$	0	0
$916$	−7.00000	−0.231287
$917$	0	0
$918$	0	0
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	−72.0000	−2.37377
$921$	0	0
$922$	2.00000	0.0658665
$923$	0	0
$924$	0	0
$925$	33.0000	1.08503
$926$	−32.0000	−1.05159
$927$	0	0
$928$	10.0000	0.328266
$929$	−4.00000	−0.131236	−0.0656179	−	0.997845i	$-0.520902\pi$
$929$	−4.00000	−0.131236	−0.0656179	+	0.997845i	$0.520902\pi$
$930$	0	0
$931$	0	0
$932$	24.0000	0.786146
$933$	0	0
$934$	0	0
$935$	48.0000	1.56977
$936$	0	0
$937$	21.0000	0.686040	0.343020	−	0.939328i	$-0.388550\pi$
$937$	21.0000	0.686040	0.343020	+	0.939328i	$0.388550\pi$
$938$	0	0
$939$	0	0
$940$	24.0000	0.782794
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	0	0
$943$	12.0000	0.390774
$944$	6.00000	0.195283
$945$	0	0
$946$	−2.00000	−0.0650256
$947$	−26.0000	−0.844886	−0.422443	−	0.906389i	$-0.638827\pi$
$947$	−26.0000	−0.844886	−0.422443	+	0.906389i	$0.638827\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	−44.0000	−1.42755
$951$	0	0
$952$	0	0
$953$	−14.0000	−0.453504	−0.226752	−	0.973952i	$-0.572811\pi$
$953$	−14.0000	−0.453504	−0.226752	+	0.973952i	$0.572811\pi$
$954$	0	0
$955$	−16.0000	−0.517748
$956$	12.0000	0.388108
$957$	0	0
$958$	−16.0000	−0.516937
$959$	0	0
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−3.00000	−0.0967239
$963$	0	0
$964$	17.0000	0.547533
$965$	−100.000	−3.21911
$966$	0	0
$967$	−17.0000	−0.546683	−0.273342	−	0.961917i	$-0.588129\pi$
$967$	−17.0000	−0.546683	−0.273342	+	0.961917i	$0.588129\pi$
$968$	21.0000	0.674966
$969$	0	0
$970$	−36.0000	−1.15589
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	0	0
$974$	16.0000	0.512673
$975$	0	0
$976$	−5.00000	−0.160046
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	0	0
$982$	34.0000	1.08498
$983$	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
$985$	80.0000	2.54901
$986$	12.0000	0.382158
$987$	0	0
$988$	−4.00000	−0.127257
$989$	−6.00000	−0.190789
$990$	0	0
$991$	−55.0000	−1.74713	−0.873566	−	0.486705i	$-0.838199\pi$
$991$	−55.0000	−1.74713	−0.873566	+	0.486705i	$0.838199\pi$
$992$	−15.0000	−0.476250
$993$	0	0
$994$	0	0
$995$	36.0000	1.14128
$996$	0	0
$997$	29.0000	0.918439	0.459220	−	0.888323i	$-0.348129\pi$
$997$	29.0000	0.918439	0.459220	+	0.888323i	$0.348129\pi$
$998$	−17.0000	−0.538126
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.a.p.1.1		1
3.2	odd	2		1323.2.a.d.1.1		1
7.3	odd	6		189.2.e.a.163.1	yes	2
7.5	odd	6		189.2.e.a.109.1	✓	2
7.6	odd	2		1323.2.a.m.1.1		1
21.5	even	6		189.2.e.c.109.1	yes	2
21.17	even	6		189.2.e.c.163.1	yes	2
21.20	even	2		1323.2.a.g.1.1		1
63.5	even	6		567.2.h.b.298.1		2
63.31	odd	6		567.2.g.b.541.1		2
63.38	even	6		567.2.h.b.352.1		2
63.40	odd	6		567.2.h.e.298.1		2
63.47	even	6		567.2.g.e.109.1		2
63.52	odd	6		567.2.h.e.352.1		2
63.59	even	6		567.2.g.e.541.1		2
63.61	odd	6		567.2.g.b.109.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.e.a.109.1	✓	2	7.5	odd	6
189.2.e.a.163.1	yes	2	7.3	odd	6
189.2.e.c.109.1	yes	2	21.5	even	6
189.2.e.c.163.1	yes	2	21.17	even	6
567.2.g.b.109.1		2	63.61	odd	6
567.2.g.b.541.1		2	63.31	odd	6
567.2.g.e.109.1		2	63.47	even	6
567.2.g.e.541.1		2	63.59	even	6
567.2.h.b.298.1		2	63.5	even	6
567.2.h.b.352.1		2	63.38	even	6
567.2.h.e.298.1		2	63.40	odd	6
567.2.h.e.352.1		2	63.52	odd	6
1323.2.a.d.1.1		1	3.2	odd	2
1323.2.a.g.1.1		1	21.20	even	2
1323.2.a.m.1.1		1	7.6	odd	2
1323.2.a.p.1.1		1	1.1	even	1	trivial

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1323.2.a.p.1.1

Level	$1323$
Weight	$2$
Character	1323.1
Self dual	yes
Analytic conductor	$10.564$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$