LMFDB - Embedded newform 1216.2.a.o.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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 19)
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} -3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +4.00000 q^{13} -6.00000 q^{15} -3.00000 q^{17} -1.00000 q^{19} -2.00000 q^{21} +4.00000 q^{25} -4.00000 q^{27} -6.00000 q^{29} -4.00000 q^{31} -6.00000 q^{33} +3.00000 q^{35} -2.00000 q^{37} +8.00000 q^{39} -6.00000 q^{41} +1.00000 q^{43} -3.00000 q^{45} -3.00000 q^{47} -6.00000 q^{49} -6.00000 q^{51} -12.0000 q^{53} +9.00000 q^{55} -2.00000 q^{57} +6.00000 q^{59} +1.00000 q^{61} -1.00000 q^{63} -12.0000 q^{65} +4.00000 q^{67} +6.00000 q^{71} -7.00000 q^{73} +8.00000 q^{75} +3.00000 q^{77} +8.00000 q^{79} -11.0000 q^{81} -12.0000 q^{83} +9.00000 q^{85} -12.0000 q^{87} +12.0000 q^{89} -4.00000 q^{91} -8.00000 q^{93} +3.00000 q^{95} +8.00000 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
$3$	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
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	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
$15$	−6.00000	−1.54919
$16$	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
$21$	−2.00000	−0.436436
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	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
$33$	−6.00000	−1.04447
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	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
$45$	−3.00000	−0.447214
$46$	0	0
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	9.00000	1.21356
$56$	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
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−12.0000	−1.48842
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	8.00000	0.923760
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	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
$85$	9.00000	0.976187
$86$	0	0
$87$	−12.0000	−1.28654
$88$	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
$91$	−4.00000	−0.419314
$92$	0	0
$93$	−8.00000	−0.829561
$94$	0	0
$95$	3.00000	0.307794
$96$	0	0
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	6.00000	0.585540
$106$	0	0
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−12.0000	−1.08200
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	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
$133$	1.00000	0.0867110
$134$	0	0
$135$	12.0000	1.03280
$136$	0	0
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	18.0000	1.49482
$146$	0	0
$147$	−12.0000	−0.989743
$148$	0	0
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	12.0000	0.963863
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	−24.0000	−1.90332
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	18.0000	1.40130
$166$	0	0
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	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
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	9.00000	0.658145
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	−24.0000	−1.71868
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	18.0000	1.25717
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	−3.00000	−0.204598
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	−21.0000	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	0	0
$237$	16.0000	1.03931
$238$	0	0
$239$	15.0000	0.970269	0.485135	−	0.874439i	$-0.338771\pi$
$239$	15.0000	0.970269	0.485135	+	0.874439i	$0.338771\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	−10.0000	−0.641500
$244$	0	0
$245$	18.0000	1.14998
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	−24.0000	−1.52094
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	18.0000	1.12720
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	9.00000	0.554964	0.277482	−	0.960731i	$-0.410500\pi$
$263$	9.00000	0.554964	0.277482	+	0.960731i	$0.410500\pi$
$264$	0	0
$265$	36.0000	2.21146
$266$	0	0
$267$	24.0000	1.46878
$268$	0	0
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	−12.0000	−0.723627
$276$	0	0
$277$	19.0000	1.14160	0.570800	−	0.821089i	$-0.306633\pi$
$277$	19.0000	1.14160	0.570800	+	0.821089i	$0.306633\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	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
$283$	13.0000	0.772770	0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000	0.772770	0.386385	+	0.922338i	$0.373724\pi$
$284$	0	0
$285$	6.00000	0.355409
$286$	0	0
$287$	6.00000	0.354169
$288$	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
$295$	−18.0000	−1.04800
$296$	0	0
$297$	12.0000	0.696311
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−1.00000	−0.0576390
$302$	0	0
$303$	−12.0000	−0.689382
$304$	0	0
$305$	−3.00000	−0.171780
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	28.0000	1.59286
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	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
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	0	0
$321$	36.0000	2.00932
$322$	0	0
$323$	3.00000	0.166924
$324$	0	0
$325$	16.0000	0.887520
$326$	0	0
$327$	32.0000	1.76960
$328$	0	0
$329$	3.00000	0.165395
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	32.0000	1.74315	0.871576	−	0.490261i	$-0.163099\pi$
$337$	32.0000	1.74315	0.871576	+	0.490261i	$0.163099\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	12.0000	0.649836
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.0000	−1.12734	−0.563670	−	0.826000i	$-0.690611\pi$
$347$	−21.0000	−1.12734	−0.563670	+	0.826000i	$0.690611\pi$
$348$	0	0
$349$	−17.0000	−0.909989	−0.454995	−	0.890494i	$-0.650359\pi$
$349$	−17.0000	−0.909989	−0.454995	+	0.890494i	$0.650359\pi$
$350$	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
$355$	−18.0000	−0.955341
$356$	0	0
$357$	6.00000	0.317554
$358$	0	0
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−4.00000	−0.209946
$364$	0	0
$365$	21.0000	1.09919
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	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
$375$	6.00000	0.309839
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	−9.00000	−0.458682
$386$	0	0
$387$	1.00000	0.0508329
$388$	0	0
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	30.0000	1.51330
$394$	0	0
$395$	−24.0000	−1.20757
$396$	0	0
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	0	0
$405$	33.0000	1.63978
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	−6.00000	−0.295241
$414$	0	0
$415$	36.0000	1.76717
$416$	0	0
$417$	26.0000	1.27323
$418$	0	0
$419$	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
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	0	0
$423$	−3.00000	−0.145865
$424$	0	0
$425$	−12.0000	−0.582086
$426$	0	0
$427$	−1.00000	−0.0483934
$428$	0	0
$429$	−24.0000	−1.15873
$430$	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
$435$	36.0000	1.72607
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	3.00000	0.142534	0.0712672	−	0.997457i	$-0.477296\pi$
$443$	3.00000	0.142534	0.0712672	+	0.997457i	$0.477296\pi$
$444$	0	0
$445$	−36.0000	−1.70656
$446$	0	0
$447$	−42.0000	−1.98653
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	18.0000	0.847587
$452$	0	0
$453$	−20.0000	−0.939682
$454$	0	0
$455$	12.0000	0.562569
$456$	0	0
$457$	−37.0000	−1.73079	−0.865393	−	0.501093i	$-0.832931\pi$
$457$	−37.0000	−1.73079	−0.865393	+	0.501093i	$0.832931\pi$
$458$	0	0
$459$	12.0000	0.560112
$460$	0	0
$461$	−9.00000	−0.419172	−0.209586	−	0.977790i	$-0.567212\pi$
$461$	−9.00000	−0.419172	−0.209586	+	0.977790i	$0.567212\pi$
$462$	0	0
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	0	0
$465$	24.0000	1.11297
$466$	0	0
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−28.0000	−1.29017
$472$	0	0
$473$	−3.00000	−0.137940
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−24.0000	−1.08978
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	−40.0000	−1.80886
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	18.0000	0.810679
$494$	0	0
$495$	9.00000	0.404520
$496$	0	0
$497$	−6.00000	−0.269137
$498$	0	0
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	0	0
$501$	−36.0000	−1.60836
$502$	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
$505$	18.0000	0.800989
$506$	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
$511$	7.00000	0.309662
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	−42.0000	−1.85074
$516$	0	0
$517$	9.00000	0.395820
$518$	0	0
$519$	36.0000	1.58022
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−38.0000	−1.66162	−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−38.0000	−1.66162	−0.830812	+	0.556553i	$0.812124\pi$
$524$	0	0
$525$	−8.00000	−0.349149
$526$	0	0
$527$	12.0000	0.522728
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	−24.0000	−1.03956
$534$	0	0
$535$	−54.0000	−2.33462
$536$	0	0
$537$	36.0000	1.55351
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	25.0000	1.07483	0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000	1.07483	0.537417	+	0.843317i	$0.319400\pi$
$542$	0	0
$543$	−4.00000	−0.171656
$544$	0	0
$545$	−48.0000	−2.05609
$546$	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
$549$	1.00000	0.0426790
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	12.0000	0.509372
$556$	0	0
$557$	−21.0000	−0.889799	−0.444899	−	0.895581i	$-0.646761\pi$
$557$	−21.0000	−0.889799	−0.444899	+	0.895581i	$0.646761\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	18.0000	0.759961
$562$	0	0
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	0	0
$565$	−18.0000	−0.757266
$566$	0	0
$567$	11.0000	0.461957
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	6.00000	0.250654
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.0000	0.457936	0.228968	−	0.973434i	$-0.426465\pi$
$577$	11.0000	0.457936	0.228968	+	0.973434i	$0.426465\pi$
$578$	0	0
$579$	−8.00000	−0.332469
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	36.0000	1.49097
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	−45.0000	−1.85735	−0.928674	−	0.370896i	$-0.879051\pi$
$587$	−45.0000	−1.85735	−0.928674	+	0.370896i	$0.879051\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−36.0000	−1.48084
$592$	0	0
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	−9.00000	−0.368964
$596$	0	0
$597$	22.0000	0.900400
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	6.00000	0.243935
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	−29.0000	−1.17130	−0.585649	−	0.810564i	$-0.699160\pi$
$613$	−29.0000	−1.17130	−0.585649	+	0.810564i	$0.699160\pi$
$614$	0	0
$615$	36.0000	1.45166
$616$	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
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	6.00000	0.239617
$628$	0	0
$629$	6.00000	0.239236
$630$	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
$633$	−28.0000	−1.11290
$634$	0	0
$635$	−6.00000	−0.238103
$636$	0	0
$637$	−24.0000	−0.950915
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	13.0000	0.512670	0.256335	−	0.966588i	$-0.417485\pi$
$643$	13.0000	0.512670	0.256335	+	0.966588i	$0.417485\pi$
$644$	0	0
$645$	−6.00000	−0.236250
$646$	0	0
$647$	27.0000	1.06148	0.530740	−	0.847535i	$-0.321914\pi$
$647$	27.0000	1.06148	0.530740	+	0.847535i	$0.321914\pi$
$648$	0	0
$649$	−18.0000	−0.706562
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	39.0000	1.52619	0.763094	−	0.646288i	$-0.223679\pi$
$653$	39.0000	1.52619	0.763094	+	0.646288i	$0.223679\pi$
$654$	0	0
$655$	−45.0000	−1.75830
$656$	0	0
$657$	−7.00000	−0.273096
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−20.0000	−0.773245
$670$	0	0
$671$	−3.00000	−0.115814
$672$	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
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	9.00000	0.343872
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	−48.0000	−1.82865
$690$	0	0
$691$	−17.0000	−0.646710	−0.323355	−	0.946278i	$-0.604811\pi$
$691$	−17.0000	−0.646710	−0.323355	+	0.946278i	$0.604811\pi$
$692$	0	0
$693$	3.00000	0.113961
$694$	0	0
$695$	−39.0000	−1.47935
$696$	0	0
$697$	18.0000	0.681799
$698$	0	0
$699$	−42.0000	−1.58859
$700$	0	0
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	18.0000	0.677919
$706$	0	0
$707$	6.00000	0.225653
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	36.0000	1.34632
$716$	0	0
$717$	30.0000	1.12037
$718$	0	0
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	−14.0000	−0.521387
$722$	0	0
$723$	−20.0000	−0.743808
$724$	0	0
$725$	−24.0000	−0.891338
$726$	0	0
$727$	−19.0000	−0.704671	−0.352335	−	0.935874i	$-0.614612\pi$
$727$	−19.0000	−0.704671	−0.352335	+	0.935874i	$0.614612\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−3.00000	−0.110959
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	0	0
$735$	36.0000	1.32788
$736$	0	0
$737$	−12.0000	−0.442026
$738$	0	0
$739$	−11.0000	−0.404642	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000	−0.404642	−0.202321	+	0.979319i	$0.564848\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	63.0000	2.30814
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	−18.0000	−0.657706
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	−42.0000	−1.53057
$754$	0	0
$755$	30.0000	1.09181
$756$	0	0
$757$	25.0000	0.908640	0.454320	−	0.890838i	$-0.349882\pi$
$757$	25.0000	0.908640	0.454320	+	0.890838i	$0.349882\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.0000	1.19625	0.598125	−	0.801403i	$-0.295913\pi$
$761$	33.0000	1.19625	0.598125	+	0.801403i	$0.295913\pi$
$762$	0	0
$763$	−16.0000	−0.579239
$764$	0	0
$765$	9.00000	0.325396
$766$	0	0
$767$	24.0000	0.866590
$768$	0	0
$769$	23.0000	0.829401	0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000	0.829401	0.414701	+	0.909958i	$0.363886\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	−18.0000	−0.644091
$782$	0	0
$783$	24.0000	0.857690
$784$	0	0
$785$	42.0000	1.49904
$786$	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
$789$	18.0000	0.640817
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	72.0000	2.55358
$796$	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
$799$	9.00000	0.318397
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	21.0000	0.741074
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−48.0000	−1.68968
$808$	0	0
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	−32.0000	−1.12229
$814$	0	0
$815$	60.0000	2.10171
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−33.0000	−1.15171	−0.575854	−	0.817553i	$-0.695330\pi$
$821$	−33.0000	−1.15171	−0.575854	+	0.817553i	$0.695330\pi$
$822$	0	0
$823$	−49.0000	−1.70803	−0.854016	−	0.520246i	$-0.825840\pi$
$823$	−49.0000	−1.70803	−0.854016	+	0.520246i	$0.825840\pi$
$824$	0	0
$825$	−24.0000	−0.835573
$826$	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
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	0	0
$831$	38.0000	1.31821
$832$	0	0
$833$	18.0000	0.623663
$834$	0	0
$835$	54.0000	1.86875
$836$	0	0
$837$	16.0000	0.553041
$838$	0	0
$839$	18.0000	0.621429	0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000	0.621429	0.310715	+	0.950503i	$0.399432\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	12.0000	0.413302
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	26.0000	0.892318
$850$	0	0
$851$	0	0
$852$	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
$855$	3.00000	0.102598
$856$	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
$859$	49.0000	1.67186	0.835929	−	0.548837i	$-0.184929\pi$
$859$	49.0000	1.67186	0.835929	+	0.548837i	$0.184929\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	0	0
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	−54.0000	−1.83606
$866$	0	0
$867$	−16.0000	−0.543388
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	16.0000	0.542139
$872$	0	0
$873$	8.00000	0.270759
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	0	0
$879$	24.0000	0.809500
$880$	0	0
$881$	−27.0000	−0.909653	−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000	−0.909653	−0.454827	+	0.890580i	$0.650299\pi$
$882$	0	0
$883$	−47.0000	−1.58168	−0.790838	−	0.612026i	$-0.790355\pi$
$883$	−47.0000	−1.58168	−0.790838	+	0.612026i	$0.790355\pi$
$884$	0	0
$885$	−36.0000	−1.21013
$886$	0	0
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	33.0000	1.10554
$892$	0	0
$893$	3.00000	0.100391
$894$	0	0
$895$	−54.0000	−1.80502
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	−2.00000	−0.0665558
$904$	0	0
$905$	6.00000	0.199447
$906$	0	0
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	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
$913$	36.0000	1.19143
$914$	0	0
$915$	−6.00000	−0.198354
$916$	0	0
$917$	−15.0000	−0.495344
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	−40.0000	−1.31804
$922$	0	0
$923$	24.0000	0.789970
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	14.0000	0.459820
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	0	0
$933$	−6.00000	−0.196431
$934$	0	0
$935$	−27.0000	−0.882994
$936$	0	0
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−12.0000	−0.390360
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	−28.0000	−0.908918
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	−48.0000	−1.55487	−0.777436	−	0.628962i	$-0.783480\pi$
$953$	−48.0000	−1.55487	−0.777436	+	0.628962i	$0.783480\pi$
$954$	0	0
$955$	−9.00000	−0.291233
$956$	0	0
$957$	36.0000	1.16371
$958$	0	0
$959$	3.00000	0.0968751
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	18.0000	0.580042
$964$	0	0
$965$	12.0000	0.386294
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	0	0
$969$	6.00000	0.192748
$970$	0	0
$971$	−60.0000	−1.92549	−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000	−1.92549	−0.962746	+	0.270408i	$0.912841\pi$
$972$	0	0
$973$	−13.0000	−0.416761
$974$	0	0
$975$	32.0000	1.02482
$976$	0	0
$977$	24.0000	0.767828	0.383914	−	0.923369i	$-0.374576\pi$
$977$	24.0000	0.767828	0.383914	+	0.923369i	$0.374576\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	0	0
$981$	16.0000	0.510841
$982$	0	0
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	54.0000	1.72058
$986$	0	0
$987$	6.00000	0.190982
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	0	0
$993$	56.0000	1.77711
$994$	0	0
$995$	−33.0000	−1.04617
$996$	0	0
$997$	−17.0000	−0.538395	−0.269198	−	0.963085i	$-0.586759\pi$
$997$	−17.0000	−0.538395	−0.269198	+	0.963085i	$0.586759\pi$
$998$	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	1216.2.a.o.1.1		1
4.3	odd	2		1216.2.a.b.1.1		1
8.3	odd	2		304.2.a.f.1.1		1
8.5	even	2		19.2.a.a.1.1	✓	1
24.5	odd	2		171.2.a.b.1.1		1
24.11	even	2		2736.2.a.c.1.1		1
40.13	odd	4		475.2.b.a.324.1		2
40.19	odd	2		7600.2.a.c.1.1		1
40.29	even	2		475.2.a.b.1.1		1
40.37	odd	4		475.2.b.a.324.2		2
56.5	odd	6		931.2.f.b.704.1		2
56.13	odd	2		931.2.a.a.1.1		1
56.37	even	6		931.2.f.c.704.1		2
56.45	odd	6		931.2.f.b.324.1		2
56.53	even	6		931.2.f.c.324.1		2
88.21	odd	2		2299.2.a.b.1.1		1
104.77	even	2		3211.2.a.a.1.1		1
120.29	odd	2		4275.2.a.i.1.1		1
136.101	even	2		5491.2.a.b.1.1		1
152.5	even	18		361.2.e.d.234.1		6
152.13	odd	18		361.2.e.e.245.1		6
152.21	odd	18		361.2.e.e.99.1		6
152.29	odd	18		361.2.e.e.62.1		6
152.37	odd	2		361.2.a.b.1.1		1
152.45	even	6		361.2.c.c.68.1		2
152.53	odd	18		361.2.e.e.54.1		6
152.61	even	18		361.2.e.d.54.1		6
152.69	odd	6		361.2.c.a.68.1		2
152.75	even	2		5776.2.a.c.1.1		1
152.85	even	18		361.2.e.d.62.1		6
152.93	even	18		361.2.e.d.99.1		6
152.101	even	18		361.2.e.d.245.1		6
152.109	odd	18		361.2.e.e.234.1		6
152.117	odd	18		361.2.e.e.28.1		6
152.125	even	6		361.2.c.c.292.1		2
152.141	odd	6		361.2.c.a.292.1		2
152.149	even	18		361.2.e.d.28.1		6
168.125	even	2		8379.2.a.j.1.1		1
456.341	even	2		3249.2.a.d.1.1		1
760.189	odd	2		9025.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.2.a.a.1.1	✓	1	8.5	even	2
171.2.a.b.1.1		1	24.5	odd	2
304.2.a.f.1.1		1	8.3	odd	2
361.2.a.b.1.1		1	152.37	odd	2
361.2.c.a.68.1		2	152.69	odd	6
361.2.c.a.292.1		2	152.141	odd	6
361.2.c.c.68.1		2	152.45	even	6
361.2.c.c.292.1		2	152.125	even	6
361.2.e.d.28.1		6	152.149	even	18
361.2.e.d.54.1		6	152.61	even	18
361.2.e.d.62.1		6	152.85	even	18
361.2.e.d.99.1		6	152.93	even	18
361.2.e.d.234.1		6	152.5	even	18
361.2.e.d.245.1		6	152.101	even	18
361.2.e.e.28.1		6	152.117	odd	18
361.2.e.e.54.1		6	152.53	odd	18
361.2.e.e.62.1		6	152.29	odd	18
361.2.e.e.99.1		6	152.21	odd	18
361.2.e.e.234.1		6	152.109	odd	18
361.2.e.e.245.1		6	152.13	odd	18
475.2.a.b.1.1		1	40.29	even	2
475.2.b.a.324.1		2	40.13	odd	4
475.2.b.a.324.2		2	40.37	odd	4
931.2.a.a.1.1		1	56.13	odd	2
931.2.f.b.324.1		2	56.45	odd	6
931.2.f.b.704.1		2	56.5	odd	6
931.2.f.c.324.1		2	56.53	even	6
931.2.f.c.704.1		2	56.37	even	6
1216.2.a.b.1.1		1	4.3	odd	2
1216.2.a.o.1.1		1	1.1	even	1	trivial
2299.2.a.b.1.1		1	88.21	odd	2
2736.2.a.c.1.1		1	24.11	even	2
3211.2.a.a.1.1		1	104.77	even	2
3249.2.a.d.1.1		1	456.341	even	2
4275.2.a.i.1.1		1	120.29	odd	2
5491.2.a.b.1.1		1	136.101	even	2
5776.2.a.c.1.1		1	152.75	even	2
7600.2.a.c.1.1		1	40.19	odd	2
8379.2.a.j.1.1		1	168.125	even	2
9025.2.a.d.1.1		1	760.189	odd	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

L-functions

Modular forms

Varieties

Fields

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Database

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	1216.2.a.o.1.1

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