LMFDB - Embedded newform 9408.2.a.cq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9408,2,Mod(1,9408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9408, 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("9408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$9408 = 2^{6} \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9408.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:	$75.1232582216$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9408.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{9} +3.00000 q^{11} -4.00000 q^{13} +1.00000 q^{15} -4.00000 q^{19} -8.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} +3.00000 q^{29} +5.00000 q^{31} +3.00000 q^{33} -8.00000 q^{37} -4.00000 q^{39} +8.00000 q^{41} +6.00000 q^{43} +1.00000 q^{45} -10.0000 q^{47} -9.00000 q^{53} +3.00000 q^{55} -4.00000 q^{57} -5.00000 q^{59} +10.0000 q^{61} -4.00000 q^{65} +6.00000 q^{67} -8.00000 q^{69} -10.0000 q^{71} +2.00000 q^{73} -4.00000 q^{75} -11.0000 q^{79} +1.00000 q^{81} +7.00000 q^{83} +3.00000 q^{87} -18.0000 q^{89} +5.00000 q^{93} -4.00000 q^{95} -17.0000 q^{97} +3.00000 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	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
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−10.0000	−1.45865	−0.729325	−	0.684167i	$-0.760166\pi$
$47$	−10.0000	−1.45865	−0.729325	+	0.684167i	$0.760166\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−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
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.00000	0.768350	0.384175	−	0.923260i	$-0.374486\pi$
$83$	7.00000	0.768350	0.384175	+	0.923260i	$0.374486\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.00000	0.518476
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.0000	−1.06341	−0.531705	−	0.846930i	$-0.678449\pi$
$107$	−11.0000	−1.06341	−0.531705	+	0.846930i	$0.678449\pi$
$108$	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
$111$	−8.00000	−0.759326
$112$	0	0
$113$	−8.00000	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$113$	−8.00000	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	8.00000	0.721336
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	6.00000	0.528271
$130$	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
$135$	1.00000	0.0860663
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	−22.0000	−1.86602	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−22.0000	−1.86602	−0.933008	+	0.359856i	$0.882826\pi$
$140$	0	0
$141$	−10.0000	−0.842152
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	3.00000	0.249136
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.00000	0.401610
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	0	0
$159$	−9.00000	−0.713746
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	0	0
$165$	3.00000	0.233550
$166$	0	0
$167$	2.00000	0.154765	0.0773823	−	0.997001i	$-0.475344\pi$
$167$	2.00000	0.154765	0.0773823	+	0.997001i	$0.475344\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.00000	−0.375823
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	0	0
$193$	−27.0000	−1.94350	−0.971751	−	0.236007i	$-0.924161\pi$
$193$	−27.0000	−1.94350	−0.971751	+	0.236007i	$0.924161\pi$
$194$	0	0
$195$	−4.00000	−0.286446
$196$	0	0
$197$	26.0000	1.85242	0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000	1.85242	0.926212	+	0.377004i	$0.123046\pi$
$198$	0	0
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	0	0
$201$	6.00000	0.423207
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.00000	0.558744
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	6.00000	0.413057	0.206529	−	0.978441i	$-0.433783\pi$
$211$	6.00000	0.413057	0.206529	+	0.978441i	$0.433783\pi$
$212$	0	0
$213$	−10.0000	−0.685189
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	21.0000	1.39382	0.696909	−	0.717159i	$-0.254558\pi$
$227$	21.0000	1.39382	0.696909	+	0.717159i	$0.254558\pi$
$228$	0	0
$229$	12.0000	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$229$	12.0000	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	−10.0000	−0.652328
$236$	0	0
$237$	−11.0000	−0.714527
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	23.0000	1.48156	0.740780	−	0.671748i	$-0.234456\pi$
$241$	23.0000	1.48156	0.740780	+	0.671748i	$0.234456\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	7.00000	0.443607
$250$	0	0
$251$	11.0000	0.694314	0.347157	−	0.937807i	$-0.387147\pi$
$251$	11.0000	0.694314	0.347157	+	0.937807i	$0.387147\pi$
$252$	0	0
$253$	−24.0000	−1.50887
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3.00000	0.185695
$262$	0	0
$263$	14.0000	0.863277	0.431638	−	0.902047i	$-0.357936\pi$
$263$	14.0000	0.863277	0.431638	+	0.902047i	$0.357936\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	0	0
$267$	−18.0000	−1.10158
$268$	0	0
$269$	−1.00000	−0.0609711	−0.0304855	−	0.999535i	$-0.509705\pi$
$269$	−1.00000	−0.0609711	−0.0304855	+	0.999535i	$0.509705\pi$
$270$	0	0
$271$	17.0000	1.03268	0.516338	−	0.856385i	$-0.327295\pi$
$271$	17.0000	1.03268	0.516338	+	0.856385i	$0.327295\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−12.0000	−0.723627
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	0	0
$279$	5.00000	0.299342
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−18.0000	−1.06999	−0.534994	−	0.844856i	$-0.679686\pi$
$283$	−18.0000	−1.06999	−0.534994	+	0.844856i	$0.679686\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−17.0000	−0.996558
$292$	0	0
$293$	19.0000	1.10999	0.554996	−	0.831853i	$-0.312720\pi$
$293$	19.0000	1.10999	0.554996	+	0.831853i	$0.312720\pi$
$294$	0	0
$295$	−5.00000	−0.291111
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	32.0000	1.85061
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.00000	0.114897
$304$	0	0
$305$	10.0000	0.572598
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	9.00000	0.508710	0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000	0.508710	0.254355	+	0.967111i	$0.418137\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.0000	−0.730153	−0.365076	−	0.930978i	$-0.618957\pi$
$317$	−13.0000	−0.730153	−0.365076	+	0.930978i	$0.618957\pi$
$318$	0	0
$319$	9.00000	0.503903
$320$	0	0
$321$	−11.0000	−0.613960
$322$	0	0
$323$	0	0
$324$	0	0
$325$	16.0000	0.887520
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	−8.00000	−0.438397
$334$	0	0
$335$	6.00000	0.327815
$336$	0	0
$337$	−15.0000	−0.817102	−0.408551	−	0.912735i	$-0.633966\pi$
$337$	−15.0000	−0.817102	−0.408551	+	0.912735i	$0.633966\pi$
$338$	0	0
$339$	−8.00000	−0.434500
$340$	0	0
$341$	15.0000	0.812296
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−8.00000	−0.430706
$346$	0	0
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	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
$351$	−4.00000	−0.213504
$352$	0	0
$353$	8.00000	0.425797	0.212899	−	0.977074i	$-0.431710\pi$
$353$	8.00000	0.425797	0.212899	+	0.977074i	$0.431710\pi$
$354$	0	0
$355$	−10.0000	−0.530745
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	7.00000	0.365397	0.182699	−	0.983169i	$-0.441517\pi$
$367$	7.00000	0.365397	0.182699	+	0.983169i	$0.441517\pi$
$368$	0	0
$369$	8.00000	0.416463
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	7.00000	0.358621
$382$	0	0
$383$	−2.00000	−0.102195	−0.0510976	−	0.998694i	$-0.516272\pi$
$383$	−2.00000	−0.102195	−0.0510976	+	0.998694i	$0.516272\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.00000	0.304997
$388$	0	0
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	15.0000	0.756650
$394$	0	0
$395$	−11.0000	−0.553470
$396$	0	0
$397$	−28.0000	−1.40528	−0.702640	−	0.711546i	$-0.747995\pi$
$397$	−28.0000	−1.40528	−0.702640	+	0.711546i	$0.747995\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	0	0
$403$	−20.0000	−0.996271
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	0	0
$413$	0	0
$414$	0	0
$415$	7.00000	0.343616
$416$	0	0
$417$	−22.0000	−1.07734
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	−10.0000	−0.486217
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−12.0000	−0.579365
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	3.00000	0.143839
$436$	0	0
$437$	32.0000	1.53077
$438$	0	0
$439$	9.00000	0.429547	0.214773	−	0.976664i	$-0.431099\pi$
$439$	9.00000	0.429547	0.214773	+	0.976664i	$0.431099\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.00000	−0.0475114	−0.0237557	−	0.999718i	$-0.507562\pi$
$443$	−1.00000	−0.0475114	−0.0237557	+	0.999718i	$0.507562\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	0	0
$447$	−18.0000	−0.851371
$448$	0	0
$449$	4.00000	0.188772	0.0943858	−	0.995536i	$-0.469911\pi$
$449$	4.00000	0.188772	0.0943858	+	0.995536i	$0.469911\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	0	0
$453$	5.00000	0.234920
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	5.00000	0.231869
$466$	0	0
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	18.0000	0.827641
$474$	0	0
$475$	16.0000	0.734130
$476$	0	0
$477$	−9.00000	−0.412082
$478$	0	0
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−17.0000	−0.771930
$486$	0	0
$487$	−37.0000	−1.67663	−0.838315	−	0.545186i	$-0.816459\pi$
$487$	−37.0000	−1.67663	−0.838315	+	0.545186i	$0.816459\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	3.00000	0.134840
$496$	0	0
$497$	0	0
$498$	0	0
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	0	0
$501$	2.00000	0.0893534
$502$	0	0
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	39.0000	1.72864	0.864322	−	0.502938i	$-0.167748\pi$
$509$	39.0000	1.72864	0.864322	+	0.502938i	$0.167748\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−30.0000	−1.31940
$518$	0	0
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	−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
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−5.00000	−0.216982
$532$	0	0
$533$	−32.0000	−1.38607
$534$	0	0
$535$	−11.0000	−0.475571
$536$	0	0
$537$	12.0000	0.517838
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−8.00000	−0.339581
$556$	0	0
$557$	−15.0000	−0.635570	−0.317785	−	0.948163i	$-0.602939\pi$
$557$	−15.0000	−0.635570	−0.317785	+	0.948163i	$0.602939\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	0	0
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	0	0
$571$	22.0000	0.920671	0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000	0.920671	0.460336	+	0.887745i	$0.347729\pi$
$572$	0	0
$573$	16.0000	0.668410
$574$	0	0
$575$	32.0000	1.33449
$576$	0	0
$577$	−33.0000	−1.37381	−0.686904	−	0.726748i	$-0.741031\pi$
$577$	−33.0000	−1.37381	−0.686904	+	0.726748i	$0.741031\pi$
$578$	0	0
$579$	−27.0000	−1.12208
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−27.0000	−1.11823
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	45.0000	1.85735	0.928674	−	0.370896i	$-0.120949\pi$
$587$	45.0000	1.85735	0.928674	+	0.370896i	$0.120949\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	0	0
$591$	26.0000	1.06950
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12.0000	−0.491127
$598$	0	0
$599$	42.0000	1.71607	0.858037	−	0.513588i	$-0.171684\pi$
$599$	42.0000	1.71607	0.858037	+	0.513588i	$0.171684\pi$
$600$	0	0
$601$	−13.0000	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$602$	0	0
$603$	6.00000	0.244339
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	−13.0000	−0.527654	−0.263827	−	0.964570i	$-0.584985\pi$
$607$	−13.0000	−0.527654	−0.263827	+	0.964570i	$0.584985\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	40.0000	1.61823
$612$	0	0
$613$	−8.00000	−0.323117	−0.161558	−	0.986863i	$-0.551652\pi$
$613$	−8.00000	−0.323117	−0.161558	+	0.986863i	$0.551652\pi$
$614$	0	0
$615$	8.00000	0.322591
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	6.00000	0.241160	0.120580	−	0.992704i	$-0.461525\pi$
$619$	6.00000	0.241160	0.120580	+	0.992704i	$0.461525\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	0	0
$630$	0	0
$631$	11.0000	0.437903	0.218952	−	0.975736i	$-0.429736\pi$
$631$	11.0000	0.437903	0.218952	+	0.975736i	$0.429736\pi$
$632$	0	0
$633$	6.00000	0.238479
$634$	0	0
$635$	7.00000	0.277787
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.0000	−0.395594
$640$	0	0
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	0	0
$643$	46.0000	1.81406	0.907031	−	0.421063i	$-0.138343\pi$
$643$	46.0000	1.81406	0.907031	+	0.421063i	$0.138343\pi$
$644$	0	0
$645$	6.00000	0.236250
$646$	0	0
$647$	50.0000	1.96570	0.982851	−	0.184399i	$-0.0590339\pi$
$647$	50.0000	1.96570	0.982851	+	0.184399i	$0.0590339\pi$
$648$	0	0
$649$	−15.0000	−0.588802
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15.0000	−0.586995	−0.293498	−	0.955960i	$-0.594819\pi$
$653$	−15.0000	−0.586995	−0.293498	+	0.955960i	$0.594819\pi$
$654$	0	0
$655$	15.0000	0.586098
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−24.0000	−0.929284
$668$	0	0
$669$	−1.00000	−0.0386622
$670$	0	0
$671$	30.0000	1.15814
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−19.0000	−0.730229	−0.365115	−	0.930963i	$-0.618970\pi$
$677$	−19.0000	−0.730229	−0.365115	+	0.930963i	$0.618970\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	21.0000	0.804722
$682$	0	0
$683$	17.0000	0.650487	0.325243	−	0.945630i	$-0.394554\pi$
$683$	17.0000	0.650487	0.325243	+	0.945630i	$0.394554\pi$
$684$	0	0
$685$	−14.0000	−0.534913
$686$	0	0
$687$	12.0000	0.457829
$688$	0	0
$689$	36.0000	1.37149
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−22.0000	−0.834508
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−24.0000	−0.907763
$700$	0	0
$701$	−5.00000	−0.188847	−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000	−0.188847	−0.0944237	+	0.995532i	$0.530101\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	0	0
$705$	−10.0000	−0.376622
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	−11.0000	−0.412532
$712$	0	0
$713$	−40.0000	−1.49801
$714$	0	0
$715$	−12.0000	−0.448775
$716$	0	0
$717$	0	0
$718$	0	0
$719$	34.0000	1.26799	0.633993	−	0.773339i	$-0.281415\pi$
$719$	34.0000	1.26799	0.633993	+	0.773339i	$0.281415\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	23.0000	0.855379
$724$	0	0
$725$	−12.0000	−0.445669
$726$	0	0
$727$	−47.0000	−1.74313	−0.871567	−	0.490277i	$-0.836896\pi$
$727$	−47.0000	−1.74313	−0.871567	+	0.490277i	$0.836896\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.0000	0.663039
$738$	0	0
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	0	0
$743$	−34.0000	−1.24734	−0.623670	−	0.781688i	$-0.714359\pi$
$743$	−34.0000	−1.24734	−0.623670	+	0.781688i	$0.714359\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	7.00000	0.256117
$748$	0	0
$749$	0	0
$750$	0	0
$751$	27.0000	0.985244	0.492622	−	0.870243i	$-0.336039\pi$
$751$	27.0000	0.985244	0.492622	+	0.870243i	$0.336039\pi$
$752$	0	0
$753$	11.0000	0.400862
$754$	0	0
$755$	5.00000	0.181969
$756$	0	0
$757$	10.0000	0.363456	0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000	0.363456	0.181728	+	0.983349i	$0.441831\pi$
$758$	0	0
$759$	−24.0000	−0.871145
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.0000	0.722158
$768$	0	0
$769$	37.0000	1.33425	0.667127	−	0.744944i	$-0.267524\pi$
$769$	37.0000	1.33425	0.667127	+	0.744944i	$0.267524\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	0	0
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−32.0000	−1.14652
$780$	0	0
$781$	−30.0000	−1.07348
$782$	0	0
$783$	3.00000	0.107211
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	0	0
$789$	14.0000	0.498413
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−40.0000	−1.42044
$794$	0	0
$795$	−9.00000	−0.319197
$796$	0	0
$797$	−3.00000	−0.106265	−0.0531327	−	0.998587i	$-0.516921\pi$
$797$	−3.00000	−0.106265	−0.0531327	+	0.998587i	$0.516921\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−18.0000	−0.635999
$802$	0	0
$803$	6.00000	0.211735
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.00000	−0.0352017
$808$	0	0
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	17.0000	0.596216
$814$	0	0
$815$	8.00000	0.280228
$816$	0	0
$817$	−24.0000	−0.839654
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.0000	1.36111	0.680555	−	0.732697i	$-0.261739\pi$
$821$	39.0000	1.36111	0.680555	+	0.732697i	$0.261739\pi$
$822$	0	0
$823$	−32.0000	−1.11545	−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000	−1.11545	−0.557725	+	0.830026i	$0.688326\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−41.0000	−1.42571	−0.712855	−	0.701312i	$-0.752598\pi$
$827$	−41.0000	−1.42571	−0.712855	+	0.701312i	$0.752598\pi$
$828$	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
$831$	−8.00000	−0.277517
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.00000	0.0692129
$836$	0	0
$837$	5.00000	0.172825
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	6.00000	0.206651
$844$	0	0
$845$	3.00000	0.103203
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−18.0000	−0.617758
$850$	0	0
$851$	64.0000	2.19389
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22.0000	−0.748889	−0.374444	−	0.927249i	$-0.622167\pi$
$863$	−22.0000	−0.748889	−0.374444	+	0.927249i	$0.622167\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	0	0
$867$	−17.0000	−0.577350
$868$	0	0
$869$	−33.0000	−1.11945
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	−17.0000	−0.575363
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	0	0
$879$	19.0000	0.640854
$880$	0	0
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	0	0
$883$	40.0000	1.34611	0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000	1.34611	0.673054	+	0.739594i	$0.264982\pi$
$884$	0	0
$885$	−5.00000	−0.168073
$886$	0	0
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	40.0000	1.33855
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	32.0000	1.06845
$898$	0	0
$899$	15.0000	0.500278
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−16.0000	−0.531271	−0.265636	−	0.964073i	$-0.585582\pi$
$907$	−16.0000	−0.531271	−0.265636	+	0.964073i	$0.585582\pi$
$908$	0	0
$909$	2.00000	0.0663358
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	21.0000	0.694999
$914$	0	0
$915$	10.0000	0.330590
$916$	0	0
$917$	0	0
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	0	0
$923$	40.0000	1.31662
$924$	0	0
$925$	32.0000	1.05215
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.0000	0.620703	0.310351	−	0.950622i	$-0.399553\pi$
$937$	19.0000	0.620703	0.310351	+	0.950622i	$0.399553\pi$
$938$	0	0
$939$	9.00000	0.293704
$940$	0	0
$941$	13.0000	0.423788	0.211894	−	0.977293i	$-0.432037\pi$
$941$	13.0000	0.423788	0.211894	+	0.977293i	$0.432037\pi$
$942$	0	0
$943$	−64.0000	−2.08413
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	0	0
$951$	−13.0000	−0.421554
$952$	0	0
$953$	−38.0000	−1.23094	−0.615470	−	0.788160i	$-0.711034\pi$
$953$	−38.0000	−1.23094	−0.615470	+	0.788160i	$0.711034\pi$
$954$	0	0
$955$	16.0000	0.517748
$956$	0	0
$957$	9.00000	0.290929
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	−11.0000	−0.354470
$964$	0	0
$965$	−27.0000	−0.869161
$966$	0	0
$967$	−3.00000	−0.0964735	−0.0482367	−	0.998836i	$-0.515360\pi$
$967$	−3.00000	−0.0964735	−0.0482367	+	0.998836i	$0.515360\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	25.0000	0.802288	0.401144	−	0.916015i	$-0.368613\pi$
$971$	25.0000	0.802288	0.401144	+	0.916015i	$0.368613\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	16.0000	0.512410
$976$	0	0
$977$	−22.0000	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$978$	0	0
$979$	−54.0000	−1.72585
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	−44.0000	−1.40338	−0.701691	−	0.712481i	$-0.747571\pi$
$983$	−44.0000	−1.40338	−0.701691	+	0.712481i	$0.747571\pi$
$984$	0	0
$985$	26.0000	0.828429
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−48.0000	−1.52631
$990$	0	0
$991$	−15.0000	−0.476491	−0.238245	−	0.971205i	$-0.576572\pi$
$991$	−15.0000	−0.476491	−0.238245	+	0.971205i	$0.576572\pi$
$992$	0	0
$993$	4.00000	0.126936
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	0	0
$999$	−8.00000	−0.253109

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	9408.2.a.cq.1.1		1
4.3	odd	2		9408.2.a.ba.1.1		1
7.2	even	3		1344.2.q.d.193.1		2
7.4	even	3		1344.2.q.d.961.1		2
7.6	odd	2		9408.2.a.p.1.1		1
8.3	odd	2		1176.2.a.g.1.1		1
8.5	even	2		2352.2.a.g.1.1		1
24.5	odd	2		7056.2.a.bk.1.1		1
24.11	even	2		3528.2.a.q.1.1		1
28.11	odd	6		1344.2.q.o.961.1		2
28.23	odd	6		1344.2.q.o.193.1		2
28.27	even	2		9408.2.a.cf.1.1		1
56.3	even	6		1176.2.q.g.961.1		2
56.5	odd	6		2352.2.q.f.1537.1		2
56.11	odd	6		168.2.q.a.121.1	yes	2
56.13	odd	2		2352.2.a.u.1.1		1
56.19	even	6		1176.2.q.g.361.1		2
56.27	even	2		1176.2.a.c.1.1		1
56.37	even	6		336.2.q.e.193.1		2
56.45	odd	6		2352.2.q.f.961.1		2
56.51	odd	6		168.2.q.a.25.1	✓	2
56.53	even	6		336.2.q.e.289.1		2
168.11	even	6		504.2.s.d.289.1		2
168.53	odd	6		1008.2.s.f.289.1		2
168.59	odd	6		3528.2.s.p.3313.1		2
168.83	odd	2		3528.2.a.i.1.1		1
168.107	even	6		504.2.s.d.361.1		2
168.125	even	2		7056.2.a.t.1.1		1
168.131	odd	6		3528.2.s.p.361.1		2
168.149	odd	6		1008.2.s.f.865.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.q.a.25.1	✓	2	56.51	odd	6
168.2.q.a.121.1	yes	2	56.11	odd	6
336.2.q.e.193.1		2	56.37	even	6
336.2.q.e.289.1		2	56.53	even	6
504.2.s.d.289.1		2	168.11	even	6
504.2.s.d.361.1		2	168.107	even	6
1008.2.s.f.289.1		2	168.53	odd	6
1008.2.s.f.865.1		2	168.149	odd	6
1176.2.a.c.1.1		1	56.27	even	2
1176.2.a.g.1.1		1	8.3	odd	2
1176.2.q.g.361.1		2	56.19	even	6
1176.2.q.g.961.1		2	56.3	even	6
1344.2.q.d.193.1		2	7.2	even	3
1344.2.q.d.961.1		2	7.4	even	3
1344.2.q.o.193.1		2	28.23	odd	6
1344.2.q.o.961.1		2	28.11	odd	6
2352.2.a.g.1.1		1	8.5	even	2
2352.2.a.u.1.1		1	56.13	odd	2
2352.2.q.f.961.1		2	56.45	odd	6
2352.2.q.f.1537.1		2	56.5	odd	6
3528.2.a.i.1.1		1	168.83	odd	2
3528.2.a.q.1.1		1	24.11	even	2
3528.2.s.p.361.1		2	168.131	odd	6
3528.2.s.p.3313.1		2	168.59	odd	6
7056.2.a.t.1.1		1	168.125	even	2
7056.2.a.bk.1.1		1	24.5	odd	2
9408.2.a.p.1.1		1	7.6	odd	2
9408.2.a.ba.1.1		1	4.3	odd	2
9408.2.a.cf.1.1		1	28.27	even	2
9408.2.a.cq.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

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Properties

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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	9408.2.a.cq.1.1

Level	$9408$
Weight	$2$
Character	9408.1
Self dual	yes
Analytic conductor	$75.123$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$