LMFDB - Embedded newform 1216.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1216 = 2^{6} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1216.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:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 76)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1216.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+2.00000 q^{3} +1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} +4.00000 q^{13} +2.00000 q^{15} -3.00000 q^{17} -1.00000 q^{19} +6.00000 q^{21} -8.00000 q^{23} -4.00000 q^{25} -4.00000 q^{27} +2.00000 q^{29} -4.00000 q^{31} +10.0000 q^{33} +3.00000 q^{35} -10.0000 q^{37} +8.00000 q^{39} +10.0000 q^{41} +1.00000 q^{43} +1.00000 q^{45} +1.00000 q^{47} +2.00000 q^{49} -6.00000 q^{51} +4.00000 q^{53} +5.00000 q^{55} -2.00000 q^{57} +6.00000 q^{59} +13.0000 q^{61} +3.00000 q^{63} +4.00000 q^{65} -12.0000 q^{67} -16.0000 q^{69} -2.00000 q^{71} +9.00000 q^{73} -8.00000 q^{75} +15.0000 q^{77} -8.00000 q^{79} -11.0000 q^{81} -12.0000 q^{83} -3.00000 q^{85} +4.00000 q^{87} +12.0000 q^{89} +12.0000 q^{91} -8.00000 q^{93} -1.00000 q^{95} -8.00000 q^{97} +5.00000 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
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0	0
$5$	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
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	6.00000	1.30931
$22$	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
$27$	−4.00000	−0.769800
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	10.0000	1.74078
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	1.00000	0.145865	0.0729325	−	0.997337i	$-0.476764\pi$
$47$	1.00000	0.145865	0.0729325	+	0.997337i	$0.476764\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	5.00000	0.674200
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	13.0000	1.66448	0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000	1.66448	0.832240	+	0.554416i	$0.187058\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	−16.0000	−1.92617
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	0	0
$75$	−8.00000	−0.923760
$76$	0	0
$77$	15.0000	1.70941
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−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
$85$	−3.00000	−0.325396
$86$	0	0
$87$	4.00000	0.428845
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	5.00000	0.502519
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	6.00000	0.585540
$106$	0	0
$107$	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
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	−20.0000	−1.89832
$112$	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
$115$	−8.00000	−0.746004
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	−9.00000	−0.825029
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	20.0000	1.80334
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	−11.0000	−0.939793	−0.469897	−	0.882721i	$-0.655709\pi$
$137$	−11.0000	−0.939793	−0.469897	+	0.882721i	$0.655709\pi$
$138$	0	0
$139$	−3.00000	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$139$	−3.00000	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$140$	0	0
$141$	2.00000	0.168430
$142$	0	0
$143$	20.0000	1.67248
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	4.00000	0.329914
$148$	0	0
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	10.0000	0.778499
$166$	0	0
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	26.0000	1.92198
$184$	0	0
$185$	−10.0000	−0.735215
$186$	0	0
$187$	−15.0000	−1.09691
$188$	0	0
$189$	−12.0000	−0.872872
$190$	0	0
$191$	−25.0000	−1.80894	−0.904468	−	0.426541i	$-0.859732\pi$
$191$	−25.0000	−1.80894	−0.904468	+	0.426541i	$0.859732\pi$
$192$	0	0
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	0	0
$195$	8.00000	0.572892
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	−24.0000	−1.69283
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	10.0000	0.698430
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	−5.00000	−0.345857
$210$	0	0
$211$	18.0000	1.23917	0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000	1.23917	0.619586	+	0.784929i	$0.287301\pi$
$212$	0	0
$213$	−4.00000	−0.274075
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	18.0000	1.21633
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	0	0
$229$	−17.0000	−1.12339	−0.561696	−	0.827344i	$-0.689851\pi$
$229$	−17.0000	−1.12339	−0.561696	+	0.827344i	$0.689851\pi$
$230$	0	0
$231$	30.0000	1.97386
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	0	0
$235$	1.00000	0.0652328
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	−21.0000	−1.35838	−0.679189	−	0.733964i	$-0.737668\pi$
$239$	−21.0000	−1.35838	−0.679189	+	0.733964i	$0.737668\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	−10.0000	−0.641500
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−24.0000	−1.52094
$250$	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
$253$	−40.0000	−2.51478
$254$	0	0
$255$	−6.00000	−0.375735
$256$	0	0
$257$	32.0000	1.99611	0.998053	−	0.0623783i	$-0.0198685\pi$
$257$	32.0000	1.99611	0.998053	+	0.0623783i	$0.0198685\pi$
$258$	0	0
$259$	−30.0000	−1.86411
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	24.0000	1.46878
$268$	0	0
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	24.0000	1.45255
$274$	0	0
$275$	−20.0000	−1.20605
$276$	0	0
$277$	−1.00000	−0.0600842	−0.0300421	−	0.999549i	$-0.509564\pi$
$277$	−1.00000	−0.0600842	−0.0300421	+	0.999549i	$0.509564\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	−3.00000	−0.178331	−0.0891657	−	0.996017i	$-0.528420\pi$
$283$	−3.00000	−0.178331	−0.0891657	+	0.996017i	$0.528420\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	30.0000	1.77084
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−16.0000	−0.937937
$292$	0	0
$293$	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
$295$	6.00000	0.349334
$296$	0	0
$297$	−20.0000	−1.16052
$298$	0	0
$299$	−32.0000	−1.85061
$300$	0	0
$301$	3.00000	0.172917
$302$	0	0
$303$	20.0000	1.14897
$304$	0	0
$305$	13.0000	0.744378
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	−7.00000	−0.396934	−0.198467	−	0.980108i	$-0.563596\pi$
$311$	−7.00000	−0.396934	−0.198467	+	0.980108i	$0.563596\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	0	0
$319$	10.0000	0.559893
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	3.00000	0.166924
$324$	0	0
$325$	−16.0000	−0.887520
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	−10.0000	−0.547997
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	−20.0000	−1.08625
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	−16.0000	−0.861411
$346$	0	0
$347$	19.0000	1.01997	0.509987	−	0.860182i	$-0.329650\pi$
$347$	19.0000	1.01997	0.509987	+	0.860182i	$0.329650\pi$
$348$	0	0
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−2.00000	−0.106149
$356$	0	0
$357$	−18.0000	−0.952661
$358$	0	0
$359$	−21.0000	−1.10834	−0.554169	−	0.832404i	$-0.686964\pi$
$359$	−21.0000	−1.10834	−0.554169	+	0.832404i	$0.686964\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	28.0000	1.46962
$364$	0	0
$365$	9.00000	0.471082
$366$	0	0
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	−18.0000	−0.929516
$376$	0	0
$377$	8.00000	0.412021
$378$	0	0
$379$	−30.0000	−1.54100	−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000	−1.54100	−0.770498	+	0.637442i	$0.779993\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	0	0
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	0	0
$385$	15.0000	0.764471
$386$	0	0
$387$	1.00000	0.0508329
$388$	0	0
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	0	0
$391$	24.0000	1.21373
$392$	0	0
$393$	−18.0000	−0.907980
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−5.00000	−0.250943	−0.125471	−	0.992097i	$-0.540044\pi$
$397$	−5.00000	−0.250943	−0.125471	+	0.992097i	$0.540044\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	28.0000	1.39825	0.699127	−	0.714998i	$-0.253572\pi$
$401$	28.0000	1.39825	0.699127	+	0.714998i	$0.253572\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	0	0
$405$	−11.0000	−0.546594
$406$	0	0
$407$	−50.0000	−2.47841
$408$	0	0
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	0	0
$411$	−22.0000	−1.08518
$412$	0	0
$413$	18.0000	0.885722
$414$	0	0
$415$	−12.0000	−0.589057
$416$	0	0
$417$	−6.00000	−0.293821
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	40.0000	1.94948	0.974740	−	0.223341i	$-0.0716964\pi$
$421$	40.0000	1.94948	0.974740	+	0.223341i	$0.0716964\pi$
$422$	0	0
$423$	1.00000	0.0486217
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	39.0000	1.88734
$428$	0	0
$429$	40.0000	1.93122
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	0	0
$437$	8.00000	0.382692
$438$	0	0
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−5.00000	−0.237557	−0.118779	−	0.992921i	$-0.537898\pi$
$443$	−5.00000	−0.237557	−0.118779	+	0.992921i	$0.537898\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	0	0
$447$	30.0000	1.41895
$448$	0	0
$449$	16.0000	0.755087	0.377543	−	0.925992i	$-0.376769\pi$
$449$	16.0000	0.755087	0.377543	+	0.925992i	$0.376769\pi$
$450$	0	0
$451$	50.0000	2.35441
$452$	0	0
$453$	−4.00000	−0.187936
$454$	0	0
$455$	12.0000	0.562569
$456$	0	0
$457$	−13.0000	−0.608114	−0.304057	−	0.952654i	$-0.598341\pi$
$457$	−13.0000	−0.608114	−0.304057	+	0.952654i	$0.598341\pi$
$458$	0	0
$459$	12.0000	0.560112
$460$	0	0
$461$	19.0000	0.884918	0.442459	−	0.896789i	$-0.354106\pi$
$461$	19.0000	0.884918	0.442459	+	0.896789i	$0.354106\pi$
$462$	0	0
$463$	−19.0000	−0.883005	−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000	−0.883005	−0.441502	+	0.897260i	$0.645554\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	−5.00000	−0.231372	−0.115686	−	0.993286i	$-0.536907\pi$
$467$	−5.00000	−0.231372	−0.115686	+	0.993286i	$0.536907\pi$
$468$	0	0
$469$	−36.0000	−1.66233
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	5.00000	0.229900
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	4.00000	0.183147
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	0	0
$483$	−48.0000	−2.18408
$484$	0	0
$485$	−8.00000	−0.363261
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	0	0
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	−6.00000	−0.270226
$494$	0	0
$495$	5.00000	0.224733
$496$	0	0
$497$	−6.00000	−0.269137
$498$	0	0
$499$	19.0000	0.850557	0.425278	−	0.905063i	$-0.360176\pi$
$499$	19.0000	0.850557	0.425278	+	0.905063i	$0.360176\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	6.00000	0.266469
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	27.0000	1.19441
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	6.00000	0.264392
$516$	0	0
$517$	5.00000	0.219900
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	32.0000	1.40195	0.700973	−	0.713188i	$-0.252749\pi$
$521$	32.0000	1.40195	0.700973	+	0.713188i	$0.252749\pi$
$522$	0	0
$523$	26.0000	1.13690	0.568450	−	0.822718i	$-0.307543\pi$
$523$	26.0000	1.13690	0.568450	+	0.822718i	$0.307543\pi$
$524$	0	0
$525$	−24.0000	−1.04745
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	40.0000	1.73259
$534$	0	0
$535$	2.00000	0.0864675
$536$	0	0
$537$	36.0000	1.55351
$538$	0	0
$539$	10.0000	0.430730
$540$	0	0
$541$	−11.0000	−0.472927	−0.236463	−	0.971640i	$-0.575988\pi$
$541$	−11.0000	−0.472927	−0.236463	+	0.971640i	$0.575988\pi$
$542$	0	0
$543$	−20.0000	−0.858282
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	13.0000	0.554826
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	−24.0000	−1.02058
$554$	0	0
$555$	−20.0000	−0.848953
$556$	0	0
$557$	−1.00000	−0.0423714	−0.0211857	−	0.999776i	$-0.506744\pi$
$557$	−1.00000	−0.0423714	−0.0211857	+	0.999776i	$0.506744\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	−30.0000	−1.26660
$562$	0	0
$563$	42.0000	1.77009	0.885044	−	0.465506i	$-0.154128\pi$
$563$	42.0000	1.77009	0.885044	+	0.465506i	$0.154128\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	0	0
$567$	−33.0000	−1.38587
$568$	0	0
$569$	−8.00000	−0.335377	−0.167689	−	0.985840i	$-0.553630\pi$
$569$	−8.00000	−0.335377	−0.167689	+	0.985840i	$0.553630\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	−50.0000	−2.08878
$574$	0	0
$575$	32.0000	1.33449
$576$	0	0
$577$	−13.0000	−0.541197	−0.270599	−	0.962692i	$-0.587222\pi$
$577$	−13.0000	−0.541197	−0.270599	+	0.962692i	$0.587222\pi$
$578$	0	0
$579$	24.0000	0.997406
$580$	0	0
$581$	−36.0000	−1.49353
$582$	0	0
$583$	20.0000	0.828315
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−4.00000	−0.164538
$592$	0	0
$593$	22.0000	0.903432	0.451716	−	0.892162i	$-0.350812\pi$
$593$	22.0000	0.903432	0.451716	+	0.892162i	$0.350812\pi$
$594$	0	0
$595$	−9.00000	−0.368964
$596$	0	0
$597$	14.0000	0.572982
$598$	0	0
$599$	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
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	14.0000	0.569181
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	4.00000	0.161823
$612$	0	0
$613$	47.0000	1.89831	0.949156	−	0.314806i	$-0.101939\pi$
$613$	47.0000	1.89831	0.949156	+	0.314806i	$0.101939\pi$
$614$	0	0
$615$	20.0000	0.806478
$616$	0	0
$617$	9.00000	0.362326	0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000	0.362326	0.181163	+	0.983453i	$0.442014\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	32.0000	1.28412
$622$	0	0
$623$	36.0000	1.44231
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−10.0000	−0.399362
$628$	0	0
$629$	30.0000	1.19618
$630$	0	0
$631$	7.00000	0.278666	0.139333	−	0.990246i	$-0.455504\pi$
$631$	7.00000	0.278666	0.139333	+	0.990246i	$0.455504\pi$
$632$	0	0
$633$	36.0000	1.43087
$634$	0	0
$635$	−6.00000	−0.238103
$636$	0	0
$637$	8.00000	0.316972
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−11.0000	−0.433798	−0.216899	−	0.976194i	$-0.569594\pi$
$643$	−11.0000	−0.433798	−0.216899	+	0.976194i	$0.569594\pi$
$644$	0	0
$645$	2.00000	0.0787499
$646$	0	0
$647$	7.00000	0.275198	0.137599	−	0.990488i	$-0.456061\pi$
$647$	7.00000	0.275198	0.137599	+	0.990488i	$0.456061\pi$
$648$	0	0
$649$	30.0000	1.17760
$650$	0	0
$651$	−24.0000	−0.940634
$652$	0	0
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	0	0
$655$	−9.00000	−0.351659
$656$	0	0
$657$	9.00000	0.351123
$658$	0	0
$659$	−34.0000	−1.32445	−0.662226	−	0.749304i	$-0.730388\pi$
$659$	−34.0000	−1.32445	−0.662226	+	0.749304i	$0.730388\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	−16.0000	−0.619522
$668$	0	0
$669$	−4.00000	−0.154649
$670$	0	0
$671$	65.0000	2.50930
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	16.0000	0.615840
$676$	0	0
$677$	2.00000	0.0768662	0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000	0.0768662	0.0384331	+	0.999261i	$0.487763\pi$
$678$	0	0
$679$	−24.0000	−0.921035
$680$	0	0
$681$	8.00000	0.306561
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	0	0
$685$	−11.0000	−0.420288
$686$	0	0
$687$	−34.0000	−1.29718
$688$	0	0
$689$	16.0000	0.609551
$690$	0	0
$691$	−33.0000	−1.25538	−0.627690	−	0.778464i	$-0.715999\pi$
$691$	−33.0000	−1.25538	−0.627690	+	0.778464i	$0.715999\pi$
$692$	0	0
$693$	15.0000	0.569803
$694$	0	0
$695$	−3.00000	−0.113796
$696$	0	0
$697$	−30.0000	−1.13633
$698$	0	0
$699$	6.00000	0.226941
$700$	0	0
$701$	10.0000	0.377695	0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000	0.377695	0.188847	+	0.982006i	$0.439525\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	30.0000	1.12827
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	32.0000	1.19841
$714$	0	0
$715$	20.0000	0.747958
$716$	0	0
$717$	−42.0000	−1.56852
$718$	0	0
$719$	−13.0000	−0.484818	−0.242409	−	0.970174i	$-0.577938\pi$
$719$	−13.0000	−0.484818	−0.242409	+	0.970174i	$0.577938\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	0	0
$723$	−52.0000	−1.93390
$724$	0	0
$725$	−8.00000	−0.297113
$726$	0	0
$727$	−7.00000	−0.259616	−0.129808	−	0.991539i	$-0.541436\pi$
$727$	−7.00000	−0.259616	−0.129808	+	0.991539i	$0.541436\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−3.00000	−0.110959
$732$	0	0
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	0	0
$735$	4.00000	0.147542
$736$	0	0
$737$	−60.0000	−2.21013
$738$	0	0
$739$	−43.0000	−1.58178	−0.790890	−	0.611958i	$-0.790382\pi$
$739$	−43.0000	−1.58178	−0.790890	+	0.611958i	$0.790382\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	15.0000	0.549557
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	6.00000	0.219235
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	0	0
$753$	22.0000	0.801725
$754$	0	0
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	5.00000	0.181728	0.0908640	−	0.995863i	$-0.471037\pi$
$757$	5.00000	0.181728	0.0908640	+	0.995863i	$0.471037\pi$
$758$	0	0
$759$	−80.0000	−2.90382
$760$	0	0
$761$	9.00000	0.326250	0.163125	−	0.986605i	$-0.447843\pi$
$761$	9.00000	0.326250	0.163125	+	0.986605i	$0.447843\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−3.00000	−0.108465
$766$	0	0
$767$	24.0000	0.866590
$768$	0	0
$769$	31.0000	1.11789	0.558944	−	0.829205i	$-0.311207\pi$
$769$	31.0000	1.11789	0.558944	+	0.829205i	$0.311207\pi$
$770$	0	0
$771$	64.0000	2.30490
$772$	0	0
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	16.0000	0.574737
$776$	0	0
$777$	−60.0000	−2.15249
$778$	0	0
$779$	−10.0000	−0.358287
$780$	0	0
$781$	−10.0000	−0.357828
$782$	0	0
$783$	−8.00000	−0.285897
$784$	0	0
$785$	2.00000	0.0713831
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	0	0
$789$	42.0000	1.49524
$790$	0	0
$791$	−30.0000	−1.06668
$792$	0	0
$793$	52.0000	1.84657
$794$	0	0
$795$	8.00000	0.283731
$796$	0	0
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	−3.00000	−0.106132
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	45.0000	1.58802
$804$	0	0
$805$	−24.0000	−0.845889
$806$	0	0
$807$	48.0000	1.68968
$808$	0	0
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	0	0
$819$	12.0000	0.419314
$820$	0	0
$821$	−45.0000	−1.57051	−0.785255	−	0.619172i	$-0.787468\pi$
$821$	−45.0000	−1.57051	−0.785255	+	0.619172i	$0.787468\pi$
$822$	0	0
$823$	−53.0000	−1.84746	−0.923732	−	0.383040i	$-0.874877\pi$
$823$	−53.0000	−1.84746	−0.923732	+	0.383040i	$0.874877\pi$
$824$	0	0
$825$	−40.0000	−1.39262
$826$	0	0
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	0	0
$829$	48.0000	1.66711	0.833554	−	0.552437i	$-0.186302\pi$
$829$	48.0000	1.66711	0.833554	+	0.552437i	$0.186302\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	6.00000	0.207639
$836$	0	0
$837$	16.0000	0.553041
$838$	0	0
$839$	10.0000	0.345238	0.172619	−	0.984989i	$-0.444777\pi$
$839$	10.0000	0.345238	0.172619	+	0.984989i	$0.444777\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	44.0000	1.51544
$844$	0	0
$845$	3.00000	0.103203
$846$	0	0
$847$	42.0000	1.44314
$848$	0	0
$849$	−6.00000	−0.205919
$850$	0	0
$851$	80.0000	2.74236
$852$	0	0
$853$	−42.0000	−1.43805	−0.719026	−	0.694983i	$-0.755412\pi$
$853$	−42.0000	−1.43805	−0.719026	+	0.694983i	$0.755412\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	1.00000	0.0341196	0.0170598	−	0.999854i	$-0.494569\pi$
$859$	1.00000	0.0341196	0.0170598	+	0.999854i	$0.494569\pi$
$860$	0	0
$861$	60.0000	2.04479
$862$	0	0
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	−16.0000	−0.543388
$868$	0	0
$869$	−40.0000	−1.35691
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	−8.00000	−0.270759
$874$	0	0
$875$	−27.0000	−0.912767
$876$	0	0
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	0	0
$879$	24.0000	0.809500
$880$	0	0
$881$	−51.0000	−1.71823	−0.859117	−	0.511780i	$-0.828986\pi$
$881$	−51.0000	−1.71823	−0.859117	+	0.511780i	$0.828986\pi$
$882$	0	0
$883$	1.00000	0.0336527	0.0168263	−	0.999858i	$-0.494644\pi$
$883$	1.00000	0.0336527	0.0168263	+	0.999858i	$0.494644\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	0	0
$887$	−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
$889$	−18.0000	−0.603701
$890$	0	0
$891$	−55.0000	−1.84257
$892$	0	0
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	18.0000	0.601674
$896$	0	0
$897$	−64.0000	−2.13690
$898$	0	0
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	6.00000	0.199667
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	−60.0000	−1.98571
$914$	0	0
$915$	26.0000	0.859533
$916$	0	0
$917$	−27.0000	−0.891619
$918$	0	0
$919$	28.0000	0.923635	0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000	0.923635	0.461817	+	0.886975i	$0.347198\pi$
$920$	0	0
$921$	24.0000	0.790827
$922$	0	0
$923$	−8.00000	−0.263323
$924$	0	0
$925$	40.0000	1.31519
$926$	0	0
$927$	6.00000	0.197066
$928$	0	0
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	0	0
$933$	−14.0000	−0.458339
$934$	0	0
$935$	−15.0000	−0.490552
$936$	0	0
$937$	−31.0000	−1.01273	−0.506363	−	0.862320i	$-0.669010\pi$
$937$	−31.0000	−1.01273	−0.506363	+	0.862320i	$0.669010\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	0	0
$941$	26.0000	0.847576	0.423788	−	0.905761i	$-0.360700\pi$
$941$	26.0000	0.847576	0.423788	+	0.905761i	$0.360700\pi$
$942$	0	0
$943$	−80.0000	−2.60516
$944$	0	0
$945$	−12.0000	−0.390360
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	36.0000	1.16861
$950$	0	0
$951$	−60.0000	−1.94563
$952$	0	0
$953$	−16.0000	−0.518291	−0.259145	−	0.965838i	$-0.583441\pi$
$953$	−16.0000	−0.518291	−0.259145	+	0.965838i	$0.583441\pi$
$954$	0	0
$955$	−25.0000	−0.808981
$956$	0	0
$957$	20.0000	0.646508
$958$	0	0
$959$	−33.0000	−1.06563
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	2.00000	0.0644491
$964$	0	0
$965$	12.0000	0.386294
$966$	0	0
$967$	−48.0000	−1.54358	−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000	−1.54358	−0.771788	+	0.635880i	$0.780637\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	−9.00000	−0.288527
$974$	0	0
$975$	−32.0000	−1.02482
$976$	0	0
$977$	56.0000	1.79160	0.895799	−	0.444459i	$-0.146604\pi$
$977$	56.0000	1.79160	0.895799	+	0.444459i	$0.146604\pi$
$978$	0	0
$979$	60.0000	1.91761
$980$	0	0
$981$	0	0
$982$	0	0
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	6.00000	0.190982
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	0	0
$993$	−8.00000	−0.253872
$994$	0	0
$995$	7.00000	0.221915
$996$	0	0
$997$	−37.0000	−1.17180	−0.585901	−	0.810383i	$-0.699259\pi$
$997$	−37.0000	−1.17180	−0.585901	+	0.810383i	$0.699259\pi$
$998$	0	0
$999$	40.0000	1.26554

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	1216.2.a.q.1.1		1
4.3	odd	2		1216.2.a.c.1.1		1
8.3	odd	2		76.2.a.a.1.1	✓	1
8.5	even	2		304.2.a.a.1.1		1
24.5	odd	2		2736.2.a.q.1.1		1
24.11	even	2		684.2.a.b.1.1		1
40.3	even	4		1900.2.c.b.1749.2		2
40.19	odd	2		1900.2.a.b.1.1		1
40.27	even	4		1900.2.c.b.1749.1		2
40.29	even	2		7600.2.a.p.1.1		1
56.27	even	2		3724.2.a.a.1.1		1
88.43	even	2		9196.2.a.f.1.1		1
152.11	odd	6		1444.2.e.a.653.1		2
152.27	even	6		1444.2.e.c.653.1		2
152.37	odd	2		5776.2.a.p.1.1		1
152.75	even	2		1444.2.a.a.1.1		1
152.83	odd	6		1444.2.e.a.429.1		2
152.107	even	6		1444.2.e.c.429.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.2.a.a.1.1	✓	1	8.3	odd	2
304.2.a.a.1.1		1	8.5	even	2
684.2.a.b.1.1		1	24.11	even	2
1216.2.a.c.1.1		1	4.3	odd	2
1216.2.a.q.1.1		1	1.1	even	1	trivial
1444.2.a.a.1.1		1	152.75	even	2
1444.2.e.a.429.1		2	152.83	odd	6
1444.2.e.a.653.1		2	152.11	odd	6
1444.2.e.c.429.1		2	152.107	even	6
1444.2.e.c.653.1		2	152.27	even	6
1900.2.a.b.1.1		1	40.19	odd	2
1900.2.c.b.1749.1		2	40.27	even	4
1900.2.c.b.1749.2		2	40.3	even	4
2736.2.a.q.1.1		1	24.5	odd	2
3724.2.a.a.1.1		1	56.27	even	2
5776.2.a.p.1.1		1	152.37	odd	2
7600.2.a.p.1.1		1	40.29	even	2
9196.2.a.f.1.1		1	88.43	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

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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	1216.2.a.q.1.1

Level	$1216$
Weight	$2$
Character	1216.1
Self dual	yes
Analytic conductor	$9.710$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$