LMFDB - Embedded newform 1216.2.a.p.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 152)
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)

$f(q)$

$=$

$q+2.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} +4.00000 q^{13} +2.00000 q^{15} +5.00000 q^{17} +1.00000 q^{19} -6.00000 q^{21} -4.00000 q^{25} -4.00000 q^{27} -2.00000 q^{29} +8.00000 q^{31} +6.00000 q^{33} -3.00000 q^{35} +10.0000 q^{37} +8.00000 q^{39} +6.00000 q^{41} +7.00000 q^{43} +1.00000 q^{45} -9.00000 q^{47} +2.00000 q^{49} +10.0000 q^{51} +8.00000 q^{53} +3.00000 q^{55} +2.00000 q^{57} -14.0000 q^{59} +5.00000 q^{61} -3.00000 q^{63} +4.00000 q^{65} -6.00000 q^{71} -15.0000 q^{73} -8.00000 q^{75} -9.00000 q^{77} -4.00000 q^{79} -11.0000 q^{81} -4.00000 q^{83} +5.00000 q^{85} -4.00000 q^{87} -12.0000 q^{91} +16.0000 q^{93} +1.00000 q^{95} +16.0000 q^{97} +3.00000 q^{99} +O(q^{100})$

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
$3$	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	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	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	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−6.00000	−1.30931
$22$	0	0
$23$	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
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	−3.00000	−0.507093
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	7.00000	1.06749	0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000	1.06749	0.533745	+	0.845645i	$0.320784\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	10.0000	1.40028
$52$	0	0
$53$	8.00000	1.09888	0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000	1.09888	0.549442	+	0.835532i	$0.314840\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	−14.0000	−1.82264	−0.911322	−	0.411693i	$-0.864937\pi$
$59$	−14.0000	−1.82264	−0.911322	+	0.411693i	$0.864937\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−15.0000	−1.75562	−0.877809	−	0.479012i	$-0.840995\pi$
$73$	−15.0000	−1.75562	−0.877809	+	0.479012i	$0.840995\pi$
$74$	0	0
$75$	−8.00000	−0.923760
$76$	0	0
$77$	−9.00000	−1.02565
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	0	0
$93$	16.0000	1.65912
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	0	0
$109$	−12.0000	−1.14939	−0.574696	−	0.818367i	$-0.694880\pi$
$109$	−12.0000	−1.14939	−0.574696	+	0.818367i	$0.694880\pi$
$110$	0	0
$111$	20.0000	1.89832
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	−15.0000	−1.37505
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	12.0000	1.08200
$124$	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
$129$	14.0000	1.23263
$130$	0	0
$131$	9.00000	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$131$	9.00000	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	21.0000	1.79415	0.897076	−	0.441877i	$-0.145687\pi$
$137$	21.0000	1.79415	0.897076	+	0.441877i	$0.145687\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	−18.0000	−1.51587
$142$	0	0
$143$	12.0000	1.00349
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	4.00000	0.329914
$148$	0	0
$149$	−17.0000	−1.39269	−0.696347	−	0.717705i	$-0.745193\pi$
$149$	−17.0000	−1.39269	−0.696347	+	0.717705i	$0.745193\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	5.00000	0.404226
$154$	0	0
$155$	8.00000	0.642575
$156$	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
$159$	16.0000	1.26888
$160$	0	0
$161$	0	0
$162$	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
$165$	6.00000	0.467099
$166$	0	0
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	0	0
$177$	−28.0000	−2.10461
$178$	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
$183$	10.0000	0.739221
$184$	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
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	0	0
$193$	−24.0000	−1.72756	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−24.0000	−1.72756	−0.863779	+	0.503871i	$0.831909\pi$
$194$	0	0
$195$	8.00000	0.572892
$196$	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
$199$	−15.0000	−1.06332	−0.531661	−	0.846957i	$-0.678432\pi$
$199$	−15.0000	−1.06332	−0.531661	+	0.846957i	$0.678432\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.00000	0.207514
$210$	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
$213$	−12.0000	−0.822226
$214$	0	0
$215$	7.00000	0.477396
$216$	0	0
$217$	−24.0000	−1.62923
$218$	0	0
$219$	−30.0000	−2.02721
$220$	0	0
$221$	20.0000	1.34535
$222$	0	0
$223$	22.0000	1.47323	0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000	1.47323	0.736614	+	0.676313i	$0.236423\pi$
$224$	0	0
$225$	−4.00000	−0.266667
$226$	0	0
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	−1.00000	−0.0660819	−0.0330409	−	0.999454i	$-0.510519\pi$
$229$	−1.00000	−0.0660819	−0.0330409	+	0.999454i	$0.510519\pi$
$230$	0	0
$231$	−18.0000	−1.18431
$232$	0	0
$233$	−13.0000	−0.851658	−0.425829	−	0.904804i	$-0.640018\pi$
$233$	−13.0000	−0.851658	−0.425829	+	0.904804i	$0.640018\pi$
$234$	0	0
$235$	−9.00000	−0.587095
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−3.00000	−0.194054	−0.0970269	−	0.995282i	$-0.530933\pi$
$239$	−3.00000	−0.194054	−0.0970269	+	0.995282i	$0.530933\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	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
$249$	−8.00000	−0.506979
$250$	0	0
$251$	13.0000	0.820553	0.410276	−	0.911961i	$-0.365432\pi$
$251$	13.0000	0.820553	0.410276	+	0.911961i	$0.365432\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	10.0000	0.626224
$256$	0	0
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	0	0
$259$	−30.0000	−1.86411
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−5.00000	−0.308313	−0.154157	−	0.988046i	$-0.549266\pi$
$263$	−5.00000	−0.308313	−0.154157	+	0.988046i	$0.549266\pi$
$264$	0	0
$265$	8.00000	0.491436
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	−24.0000	−1.45255
$274$	0	0
$275$	−12.0000	−0.723627
$276$	0	0
$277$	−9.00000	−0.540758	−0.270379	−	0.962754i	$-0.587149\pi$
$277$	−9.00000	−0.540758	−0.270379	+	0.962754i	$0.587149\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	0	0
$287$	−18.0000	−1.06251
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	32.0000	1.87587
$292$	0	0
$293$	4.00000	0.233682	0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000	0.233682	0.116841	+	0.993151i	$0.462723\pi$
$294$	0	0
$295$	−14.0000	−0.815112
$296$	0	0
$297$	−12.0000	−0.696311
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−21.0000	−1.21042
$302$	0	0
$303$	36.0000	2.06815
$304$	0	0
$305$	5.00000	0.286299
$306$	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
$309$	−28.0000	−1.59286
$310$	0	0
$311$	31.0000	1.75785	0.878924	−	0.476961i	$-0.158262\pi$
$311$	31.0000	1.75785	0.878924	+	0.476961i	$0.158262\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	−20.0000	−1.11629
$322$	0	0
$323$	5.00000	0.278207
$324$	0	0
$325$	−16.0000	−0.887520
$326$	0	0
$327$	−24.0000	−1.32720
$328$	0	0
$329$	27.0000	1.48856
$330$	0	0
$331$	8.00000	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	4.00000	0.217250
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−27.0000	−1.44944	−0.724718	−	0.689046i	$-0.758030\pi$
$347$	−27.0000	−1.44944	−0.724718	+	0.689046i	$0.758030\pi$
$348$	0	0
$349$	19.0000	1.01705	0.508523	−	0.861048i	$-0.330192\pi$
$349$	19.0000	1.01705	0.508523	+	0.861048i	$0.330192\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	0	0
$357$	−30.0000	−1.58777
$358$	0	0
$359$	−11.0000	−0.580558	−0.290279	−	0.956942i	$-0.593748\pi$
$359$	−11.0000	−0.580558	−0.290279	+	0.956942i	$0.593748\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−4.00000	−0.209946
$364$	0	0
$365$	−15.0000	−0.785136
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	−24.0000	−1.24602
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	0	0
$375$	−18.0000	−0.929516
$376$	0	0
$377$	−8.00000	−0.412021
$378$	0	0
$379$	−18.0000	−0.924598	−0.462299	−	0.886724i	$-0.652975\pi$
$379$	−18.0000	−0.924598	−0.462299	+	0.886724i	$0.652975\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	−9.00000	−0.458682
$386$	0	0
$387$	7.00000	0.355830
$388$	0	0
$389$	29.0000	1.47036	0.735179	−	0.677873i	$-0.237098\pi$
$389$	29.0000	1.47036	0.735179	+	0.677873i	$0.237098\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	18.0000	0.907980
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−13.0000	−0.652451	−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000	−0.652451	−0.326226	+	0.945292i	$0.605777\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	32.0000	1.59403
$404$	0	0
$405$	−11.0000	−0.546594
$406$	0	0
$407$	30.0000	1.48704
$408$	0	0
$409$	−8.00000	−0.395575	−0.197787	−	0.980245i	$-0.563376\pi$
$409$	−8.00000	−0.395575	−0.197787	+	0.980245i	$0.563376\pi$
$410$	0	0
$411$	42.0000	2.07171
$412$	0	0
$413$	42.0000	2.06668
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	−10.0000	−0.489702
$418$	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
$421$	−20.0000	−0.974740	−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000	−0.974740	−0.487370	+	0.873195i	$0.662044\pi$
$422$	0	0
$423$	−9.00000	−0.437595
$424$	0	0
$425$	−20.0000	−0.970143
$426$	0	0
$427$	−15.0000	−0.725901
$428$	0	0
$429$	24.0000	1.15873
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	−19.0000	−0.902717	−0.451359	−	0.892343i	$-0.649060\pi$
$443$	−19.0000	−0.902717	−0.451359	+	0.892343i	$0.649060\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−34.0000	−1.60814
$448$	0	0
$449$	24.0000	1.13263	0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000	1.13263	0.566315	+	0.824189i	$0.308369\pi$
$450$	0	0
$451$	18.0000	0.847587
$452$	0	0
$453$	4.00000	0.187936
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	−29.0000	−1.35656	−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000	−1.35656	−0.678281	+	0.734802i	$0.737275\pi$
$458$	0	0
$459$	−20.0000	−0.933520
$460$	0	0
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	0	0
$463$	−37.0000	−1.71954	−0.859768	−	0.510685i	$-0.829392\pi$
$463$	−37.0000	−1.71954	−0.859768	+	0.510685i	$0.829392\pi$
$464$	0	0
$465$	16.0000	0.741982
$466$	0	0
$467$	13.0000	0.601568	0.300784	−	0.953692i	$-0.402752\pi$
$467$	13.0000	0.601568	0.300784	+	0.953692i	$0.402752\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−28.0000	−1.29017
$472$	0	0
$473$	21.0000	0.965581
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	8.00000	0.366295
$478$	0	0
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	0	0
$481$	40.0000	1.82384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.0000	0.726523
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	−8.00000	−0.361773
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	−10.0000	−0.450377
$494$	0	0
$495$	3.00000	0.134840
$496$	0	0
$497$	18.0000	0.807410
$498$	0	0
$499$	29.0000	1.29822	0.649109	−	0.760695i	$-0.275142\pi$
$499$	29.0000	1.29822	0.649109	+	0.760695i	$0.275142\pi$
$500$	0	0
$501$	−4.00000	−0.178707
$502$	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
$509$	8.00000	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$509$	8.00000	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$510$	0	0
$511$	45.0000	1.99068
$512$	0	0
$513$	−4.00000	−0.176604
$514$	0	0
$515$	−14.0000	−0.616914
$516$	0	0
$517$	−27.0000	−1.18746
$518$	0	0
$519$	4.00000	0.175581
$520$	0	0
$521$	4.00000	0.175243	0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000	0.175243	0.0876216	+	0.996154i	$0.472073\pi$
$522$	0	0
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	0	0
$525$	24.0000	1.04745
$526$	0	0
$527$	40.0000	1.74243
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−14.0000	−0.607548
$532$	0	0
$533$	24.0000	1.03956
$534$	0	0
$535$	−10.0000	−0.432338
$536$	0	0
$537$	36.0000	1.55351
$538$	0	0
$539$	6.00000	0.258438
$540$	0	0
$541$	−19.0000	−0.816874	−0.408437	−	0.912787i	$-0.633926\pi$
$541$	−19.0000	−0.816874	−0.408437	+	0.912787i	$0.633926\pi$
$542$	0	0
$543$	−4.00000	−0.171656
$544$	0	0
$545$	−12.0000	−0.514024
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	0	0
$549$	5.00000	0.213395
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	20.0000	0.848953
$556$	0	0
$557$	7.00000	0.296600	0.148300	−	0.988942i	$-0.452620\pi$
$557$	7.00000	0.296600	0.148300	+	0.988942i	$0.452620\pi$
$558$	0	0
$559$	28.0000	1.18427
$560$	0	0
$561$	30.0000	1.26660
$562$	0	0
$563$	−14.0000	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$563$	−14.0000	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	33.0000	1.38587
$568$	0	0
$569$	8.00000	0.335377	0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000	0.335377	0.167689	+	0.985840i	$0.446370\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	−30.0000	−1.25327
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−37.0000	−1.54033	−0.770165	−	0.637845i	$-0.779826\pi$
$577$	−37.0000	−1.54033	−0.770165	+	0.637845i	$0.779826\pi$
$578$	0	0
$579$	−48.0000	−1.99481
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	24.0000	0.993978
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	29.0000	1.19696	0.598479	−	0.801138i	$-0.295772\pi$
$587$	29.0000	1.19696	0.598479	+	0.801138i	$0.295772\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	−36.0000	−1.48084
$592$	0	0
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	−15.0000	−0.614940
$596$	0	0
$597$	−30.0000	−1.22782
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	−36.0000	−1.45640
$612$	0	0
$613$	31.0000	1.25208	0.626039	−	0.779792i	$-0.284675\pi$
$613$	31.0000	1.25208	0.626039	+	0.779792i	$0.284675\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	33.0000	1.32853	0.664265	−	0.747497i	$-0.268745\pi$
$617$	33.0000	1.32853	0.664265	+	0.747497i	$0.268745\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	6.00000	0.239617
$628$	0	0
$629$	50.0000	1.99363
$630$	0	0
$631$	25.0000	0.995234	0.497617	−	0.867397i	$-0.334208\pi$
$631$	25.0000	0.995234	0.497617	+	0.867397i	$0.334208\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	0	0
$635$	−6.00000	−0.238103
$636$	0	0
$637$	8.00000	0.316972
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	43.0000	1.69575	0.847877	−	0.530193i	$-0.177880\pi$
$643$	43.0000	1.69575	0.847877	+	0.530193i	$0.177880\pi$
$644$	0	0
$645$	14.0000	0.551249
$646$	0	0
$647$	49.0000	1.92639	0.963194	−	0.268806i	$-0.0866290\pi$
$647$	49.0000	1.92639	0.963194	+	0.268806i	$0.0866290\pi$
$648$	0	0
$649$	−42.0000	−1.64864
$650$	0	0
$651$	−48.0000	−1.88127
$652$	0	0
$653$	−21.0000	−0.821794	−0.410897	−	0.911682i	$-0.634784\pi$
$653$	−21.0000	−0.821794	−0.410897	+	0.911682i	$0.634784\pi$
$654$	0	0
$655$	9.00000	0.351659
$656$	0	0
$657$	−15.0000	−0.585206
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	16.0000	0.622328	0.311164	−	0.950356i	$-0.399281\pi$
$661$	16.0000	0.622328	0.311164	+	0.950356i	$0.399281\pi$
$662$	0	0
$663$	40.0000	1.55347
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	0	0
$668$	0	0
$669$	44.0000	1.70114
$670$	0	0
$671$	15.0000	0.579069
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	0	0
$675$	16.0000	0.615840
$676$	0	0
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	0	0
$679$	−48.0000	−1.84207
$680$	0	0
$681$	40.0000	1.53280
$682$	0	0
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	0	0
$685$	21.0000	0.802369
$686$	0	0
$687$	−2.00000	−0.0763048
$688$	0	0
$689$	32.0000	1.21910
$690$	0	0
$691$	−15.0000	−0.570627	−0.285313	−	0.958434i	$-0.592098\pi$
$691$	−15.0000	−0.570627	−0.285313	+	0.958434i	$0.592098\pi$
$692$	0	0
$693$	−9.00000	−0.341882
$694$	0	0
$695$	−5.00000	−0.189661
$696$	0	0
$697$	30.0000	1.13633
$698$	0	0
$699$	−26.0000	−0.983410
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	0	0
$705$	−18.0000	−0.677919
$706$	0	0
$707$	−54.0000	−2.03088
$708$	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
$711$	−4.00000	−0.150012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	12.0000	0.448775
$716$	0	0
$717$	−6.00000	−0.224074
$718$	0	0
$719$	29.0000	1.08152	0.540759	−	0.841178i	$-0.318137\pi$
$719$	29.0000	1.08152	0.540759	+	0.841178i	$0.318137\pi$
$720$	0	0
$721$	42.0000	1.56416
$722$	0	0
$723$	36.0000	1.33885
$724$	0	0
$725$	8.00000	0.297113
$726$	0	0
$727$	7.00000	0.259616	0.129808	−	0.991539i	$-0.458564\pi$
$727$	7.00000	0.259616	0.129808	+	0.991539i	$0.458564\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	35.0000	1.29452
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	4.00000	0.147542
$736$	0	0
$737$	0	0
$738$	0	0
$739$	35.0000	1.28750	0.643748	−	0.765238i	$-0.277379\pi$
$739$	35.0000	1.28750	0.643748	+	0.765238i	$0.277379\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−17.0000	−0.622832
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	30.0000	1.09618
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	26.0000	0.947493
$754$	0	0
$755$	2.00000	0.0727875
$756$	0	0
$757$	37.0000	1.34479	0.672394	−	0.740193i	$-0.265266\pi$
$757$	37.0000	1.34479	0.672394	+	0.740193i	$0.265266\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.00000	−0.253750	−0.126875	−	0.991919i	$-0.540495\pi$
$761$	−7.00000	−0.253750	−0.126875	+	0.991919i	$0.540495\pi$
$762$	0	0
$763$	36.0000	1.30329
$764$	0	0
$765$	5.00000	0.180775
$766$	0	0
$767$	−56.0000	−2.02204
$768$	0	0
$769$	−1.00000	−0.0360609	−0.0180305	−	0.999837i	$-0.505740\pi$
$769$	−1.00000	−0.0360609	−0.0180305	+	0.999837i	$0.505740\pi$
$770$	0	0
$771$	−48.0000	−1.72868
$772$	0	0
$773$	−34.0000	−1.22290	−0.611448	−	0.791285i	$-0.709412\pi$
$773$	−34.0000	−1.22290	−0.611448	+	0.791285i	$0.709412\pi$
$774$	0	0
$775$	−32.0000	−1.14947
$776$	0	0
$777$	−60.0000	−2.15249
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	−18.0000	−0.644091
$782$	0	0
$783$	8.00000	0.285897
$784$	0	0
$785$	−14.0000	−0.499681
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	0	0
$789$	−10.0000	−0.356009
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	20.0000	0.710221
$794$	0	0
$795$	16.0000	0.567462
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	−45.0000	−1.59199
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−45.0000	−1.58802
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.00000	−0.281613
$808$	0	0
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	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
$817$	7.00000	0.244899
$818$	0	0
$819$	−12.0000	−0.419314
$820$	0	0
$821$	11.0000	0.383903	0.191951	−	0.981404i	$-0.438518\pi$
$821$	11.0000	0.383903	0.191951	+	0.981404i	$0.438518\pi$
$822$	0	0
$823$	37.0000	1.28974	0.644869	−	0.764293i	$-0.276912\pi$
$823$	37.0000	1.28974	0.644869	+	0.764293i	$0.276912\pi$
$824$	0	0
$825$	−24.0000	−0.835573
$826$	0	0
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	0	0
$829$	−24.0000	−0.833554	−0.416777	−	0.909009i	$-0.636840\pi$
$829$	−24.0000	−0.833554	−0.416777	+	0.909009i	$0.636840\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	0	0
$833$	10.0000	0.346479
$834$	0	0
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	−32.0000	−1.10608
$838$	0	0
$839$	30.0000	1.03572	0.517858	−	0.855467i	$-0.326730\pi$
$839$	30.0000	1.03572	0.517858	+	0.855467i	$0.326730\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−20.0000	−0.688837
$844$	0	0
$845$	3.00000	0.103203
$846$	0	0
$847$	6.00000	0.206162
$848$	0	0
$849$	−26.0000	−0.892318
$850$	0	0
$851$	0	0
$852$	0	0
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	0	0
$859$	−17.0000	−0.580033	−0.290016	−	0.957022i	$-0.593661\pi$
$859$	−17.0000	−0.580033	−0.290016	+	0.957022i	$0.593661\pi$
$860$	0	0
$861$	−36.0000	−1.22688
$862$	0	0
$863$	−30.0000	−1.02121	−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000	−1.02121	−0.510606	+	0.859815i	$0.670579\pi$
$864$	0	0
$865$	2.00000	0.0680020
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	0	0
$872$	0	0
$873$	16.0000	0.541518
$874$	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
$879$	8.00000	0.269833
$880$	0	0
$881$	37.0000	1.24656	0.623281	−	0.781998i	$-0.285799\pi$
$881$	37.0000	1.24656	0.623281	+	0.781998i	$0.285799\pi$
$882$	0	0
$883$	−41.0000	−1.37976	−0.689880	−	0.723924i	$-0.742337\pi$
$883$	−41.0000	−1.37976	−0.689880	+	0.723924i	$0.742337\pi$
$884$	0	0
$885$	−28.0000	−0.941210
$886$	0	0
$887$	38.0000	1.27592	0.637958	−	0.770072i	$-0.279780\pi$
$887$	38.0000	1.27592	0.637958	+	0.770072i	$0.279780\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	−33.0000	−1.10554
$892$	0	0
$893$	−9.00000	−0.301174
$894$	0	0
$895$	18.0000	0.601674
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−16.0000	−0.533630
$900$	0	0
$901$	40.0000	1.33259
$902$	0	0
$903$	−42.0000	−1.39767
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	0	0
$909$	18.0000	0.597022
$910$	0	0
$911$	−2.00000	−0.0662630	−0.0331315	−	0.999451i	$-0.510548\pi$
$911$	−2.00000	−0.0662630	−0.0331315	+	0.999451i	$0.510548\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	10.0000	0.330590
$916$	0	0
$917$	−27.0000	−0.891619
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	−32.0000	−1.05444
$922$	0	0
$923$	−24.0000	−0.789970
$924$	0	0
$925$	−40.0000	−1.31519
$926$	0	0
$927$	−14.0000	−0.459820
$928$	0	0
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	62.0000	2.02979
$934$	0	0
$935$	15.0000	0.490552
$936$	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
$939$	28.0000	0.913745
$940$	0	0
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	12.0000	0.390360
$946$	0	0
$947$	−20.0000	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	0	0
$951$	36.0000	1.16738
$952$	0	0
$953$	4.00000	0.129573	0.0647864	−	0.997899i	$-0.479363\pi$
$953$	4.00000	0.129573	0.0647864	+	0.997899i	$0.479363\pi$
$954$	0	0
$955$	−15.0000	−0.485389
$956$	0	0
$957$	−12.0000	−0.387905
$958$	0	0
$959$	−63.0000	−2.03438
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−10.0000	−0.322245
$964$	0	0
$965$	−24.0000	−0.772587
$966$	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
$969$	10.0000	0.321246
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	15.0000	0.480878
$974$	0	0
$975$	−32.0000	−1.02482
$976$	0	0
$977$	52.0000	1.66363	0.831814	−	0.555055i	$-0.187303\pi$
$977$	52.0000	1.66363	0.831814	+	0.555055i	$0.187303\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−12.0000	−0.383131
$982$	0	0
$983$	20.0000	0.637901	0.318950	−	0.947771i	$-0.396670\pi$
$983$	20.0000	0.637901	0.318950	+	0.947771i	$0.396670\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	54.0000	1.71884
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	0	0
$993$	16.0000	0.507745
$994$	0	0
$995$	−15.0000	−0.475532
$996$	0	0
$997$	−13.0000	−0.411714	−0.205857	−	0.978582i	$-0.565998\pi$
$997$	−13.0000	−0.411714	−0.205857	+	0.978582i	$0.565998\pi$
$998$	0	0
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.a.p.1.1		1
4.3	odd	2		1216.2.a.d.1.1		1
8.3	odd	2		304.2.a.e.1.1		1
8.5	even	2		152.2.a.a.1.1	✓	1
24.5	odd	2		1368.2.a.h.1.1		1
24.11	even	2		2736.2.a.p.1.1		1
40.13	odd	4		3800.2.d.d.3649.1		2
40.19	odd	2		7600.2.a.b.1.1		1
40.29	even	2		3800.2.a.i.1.1		1
40.37	odd	4		3800.2.d.d.3649.2		2
56.13	odd	2		7448.2.a.s.1.1		1
152.37	odd	2		2888.2.a.f.1.1		1
152.75	even	2		5776.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.a.a.1.1	✓	1	8.5	even	2
304.2.a.e.1.1		1	8.3	odd	2
1216.2.a.d.1.1		1	4.3	odd	2
1216.2.a.p.1.1		1	1.1	even	1	trivial
1368.2.a.h.1.1		1	24.5	odd	2
2736.2.a.p.1.1		1	24.11	even	2
2888.2.a.f.1.1		1	152.37	odd	2
3800.2.a.i.1.1		1	40.29	even	2
3800.2.d.d.3649.1		2	40.13	odd	4
3800.2.d.d.3649.2		2	40.37	odd	4
5776.2.a.b.1.1		1	152.75	even	2
7448.2.a.s.1.1		1	56.13	odd	2
7600.2.a.b.1.1		1	40.19	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

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L-functions

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