LMFDB - Embedded newform 8016.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -3.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -3.00000 q^{5} +3.00000 q^{7} +1.00000 q^{9} -6.00000 q^{13} +3.00000 q^{15} -2.00000 q^{17} +2.00000 q^{19} -3.00000 q^{21} +6.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} +4.00000 q^{29} -5.00000 q^{31} -9.00000 q^{35} -1.00000 q^{37} +6.00000 q^{39} +2.00000 q^{41} +4.00000 q^{43} -3.00000 q^{45} +9.00000 q^{47} +2.00000 q^{49} +2.00000 q^{51} +3.00000 q^{53} -2.00000 q^{57} -7.00000 q^{59} -8.00000 q^{61} +3.00000 q^{63} +18.0000 q^{65} -3.00000 q^{67} -6.00000 q^{69} -14.0000 q^{71} -8.00000 q^{73} -4.00000 q^{75} +16.0000 q^{79} +1.00000 q^{81} +3.00000 q^{83} +6.00000 q^{85} -4.00000 q^{87} +15.0000 q^{89} -18.0000 q^{91} +5.00000 q^{93} -6.00000 q^{95} -1.00000 q^{97} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−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
$7$	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
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−9.00000	−1.52128
$36$	0	0
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	0	0
$39$	6.00000	0.960769
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	−3.00000	−0.447214
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	−7.00000	−0.911322	−0.455661	−	0.890153i	$-0.650597\pi$
$59$	−7.00000	−0.911322	−0.455661	+	0.890153i	$0.650597\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	18.0000	2.23263
$66$	0	0
$67$	−3.00000	−0.366508	−0.183254	−	0.983066i	$-0.558663\pi$
$67$	−3.00000	−0.366508	−0.183254	+	0.983066i	$0.558663\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	0	0
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	−18.0000	−1.88691
$92$	0	0
$93$	5.00000	0.518476
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	9.00000	0.878310
$106$	0	0
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	4.00000	0.376288	0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000	0.376288	0.188144	+	0.982141i	$0.439753\pi$
$114$	0	0
$115$	−18.0000	−1.67851
$116$	0	0
$117$	−6.00000	−0.554700
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	−17.0000	−1.50851	−0.754253	−	0.656584i	$-0.772001\pi$
$127$	−17.0000	−1.50851	−0.754253	+	0.656584i	$0.772001\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	3.00000	0.258199
$136$	0	0
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	0	0
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	−1.00000	−0.0819232	−0.0409616	−	0.999161i	$-0.513042\pi$
$149$	−1.00000	−0.0819232	−0.0409616	+	0.999161i	$0.513042\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	15.0000	1.20483
$156$	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
$159$	−3.00000	−0.237915
$160$	0	0
$161$	18.0000	1.41860
$162$	0	0
$163$	5.00000	0.391630	0.195815	−	0.980641i	$-0.437265\pi$
$163$	5.00000	0.391630	0.195815	+	0.980641i	$0.437265\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	0	0
$177$	7.00000	0.526152
$178$	0	0
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	3.00000	0.220564
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3.00000	−0.218218
$190$	0	0
$191$	5.00000	0.361787	0.180894	−	0.983503i	$-0.442101\pi$
$191$	5.00000	0.361787	0.180894	+	0.983503i	$0.442101\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	−18.0000	−1.28901
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	3.00000	0.211604
$202$	0	0
$203$	12.0000	0.842235
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	6.00000	0.417029
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	0	0
$213$	14.0000	0.959264
$214$	0	0
$215$	−12.0000	−0.818393
$216$	0	0
$217$	−15.0000	−1.01827
$218$	0	0
$219$	8.00000	0.540590
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	7.00000	0.464606	0.232303	−	0.972643i	$-0.425374\pi$
$227$	7.00000	0.464606	0.232303	+	0.972643i	$0.425374\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.0000	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$234$	0	0
$235$	−27.0000	−1.76129
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	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
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.00000	−0.383326
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	−3.00000	−0.190117
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−6.00000	−0.375735
$256$	0	0
$257$	−16.0000	−0.998053	−0.499026	−	0.866587i	$-0.666309\pi$
$257$	−16.0000	−0.998053	−0.499026	+	0.866587i	$0.666309\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	4.00000	0.247594
$262$	0	0
$263$	17.0000	1.04826	0.524132	−	0.851637i	$-0.324390\pi$
$263$	17.0000	1.04826	0.524132	+	0.851637i	$0.324390\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	0	0
$267$	−15.0000	−0.917985
$268$	0	0
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	28.0000	1.70088	0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000	1.70088	0.850439	+	0.526073i	$0.176336\pi$
$272$	0	0
$273$	18.0000	1.08941
$274$	0	0
$275$	0	0
$276$	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
$279$	−5.00000	−0.299342
$280$	0	0
$281$	17.0000	1.01413	0.507067	−	0.861906i	$-0.330729\pi$
$281$	17.0000	1.01413	0.507067	+	0.861906i	$0.330729\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	6.00000	0.355409
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	1.00000	0.0586210
$292$	0	0
$293$	−24.0000	−1.40209	−0.701047	−	0.713115i	$-0.747284\pi$
$293$	−24.0000	−1.40209	−0.701047	+	0.713115i	$0.747284\pi$
$294$	0	0
$295$	21.0000	1.22267
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−36.0000	−2.08193
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	−3.00000	−0.172345
$304$	0	0
$305$	24.0000	1.37424
$306$	0	0
$307$	21.0000	1.19853	0.599267	−	0.800549i	$-0.295459\pi$
$307$	21.0000	1.19853	0.599267	+	0.800549i	$0.295459\pi$
$308$	0	0
$309$	−10.0000	−0.568880
$310$	0	0
$311$	15.0000	0.850572	0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000	0.850572	0.425286	+	0.905059i	$0.360174\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	0	0
$315$	−9.00000	−0.507093
$316$	0	0
$317$	24.0000	1.34797	0.673987	−	0.738743i	$-0.264580\pi$
$317$	24.0000	1.34797	0.673987	+	0.738743i	$0.264580\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	2.00000	0.111629
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	−24.0000	−1.33128
$326$	0	0
$327$	−10.0000	−0.553001
$328$	0	0
$329$	27.0000	1.48856
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	9.00000	0.491723
$336$	0	0
$337$	−23.0000	−1.25289	−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000	−1.25289	−0.626445	+	0.779466i	$0.715491\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	18.0000	0.969087
$346$	0	0
$347$	1.00000	0.0536828	0.0268414	−	0.999640i	$-0.491455\pi$
$347$	1.00000	0.0536828	0.0268414	+	0.999640i	$0.491455\pi$
$348$	0	0
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	0	0
$355$	42.0000	2.22913
$356$	0	0
$357$	6.00000	0.317554
$358$	0	0
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	24.0000	1.25622
$366$	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
$369$	2.00000	0.104116
$370$	0	0
$371$	9.00000	0.467257
$372$	0	0
$373$	−31.0000	−1.60512	−0.802560	−	0.596572i	$-0.796529\pi$
$373$	−31.0000	−1.60512	−0.802560	+	0.596572i	$0.796529\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	0	0
$377$	−24.0000	−1.23606
$378$	0	0
$379$	−27.0000	−1.38690	−0.693448	−	0.720506i	$-0.743909\pi$
$379$	−27.0000	−1.38690	−0.693448	+	0.720506i	$0.743909\pi$
$380$	0	0
$381$	17.0000	0.870936
$382$	0	0
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	−48.0000	−2.41514
$396$	0	0
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	30.0000	1.49441
$404$	0	0
$405$	−3.00000	−0.149071
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	13.0000	0.641243
$412$	0	0
$413$	−21.0000	−1.03334
$414$	0	0
$415$	−9.00000	−0.441793
$416$	0	0
$417$	11.0000	0.538672
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	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
$423$	9.00000	0.437595
$424$	0	0
$425$	−8.00000	−0.388057
$426$	0	0
$427$	−24.0000	−1.16144
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.00000	0.337178	0.168589	−	0.985686i	$-0.446079\pi$
$431$	7.00000	0.337178	0.168589	+	0.985686i	$0.446079\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	0	0
$435$	12.0000	0.575356
$436$	0	0
$437$	12.0000	0.574038
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	35.0000	1.66290	0.831450	−	0.555599i	$-0.187511\pi$
$443$	35.0000	1.66290	0.831450	+	0.555599i	$0.187511\pi$
$444$	0	0
$445$	−45.0000	−2.13320
$446$	0	0
$447$	1.00000	0.0472984
$448$	0	0
$449$	−1.00000	−0.0471929	−0.0235965	−	0.999722i	$-0.507512\pi$
$449$	−1.00000	−0.0471929	−0.0235965	+	0.999722i	$0.507512\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−16.0000	−0.751746
$454$	0	0
$455$	54.0000	2.53156
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	−36.0000	−1.67669	−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000	−1.67669	−0.838344	+	0.545142i	$0.816476\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	0	0
$465$	−15.0000	−0.695608
$466$	0	0
$467$	−22.0000	−1.01804	−0.509019	−	0.860755i	$-0.669992\pi$
$467$	−22.0000	−1.01804	−0.509019	+	0.860755i	$0.669992\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	0	0
$474$	0	0
$475$	8.00000	0.367065
$476$	0	0
$477$	3.00000	0.137361
$478$	0	0
$479$	−40.0000	−1.82765	−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000	−1.82765	−0.913823	+	0.406112i	$0.866884\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	−18.0000	−0.819028
$484$	0	0
$485$	3.00000	0.136223
$486$	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
$489$	−5.00000	−0.226108
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	−8.00000	−0.360302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−42.0000	−1.88396
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−9.00000	−0.400495
$506$	0	0
$507$	−23.0000	−1.02147
$508$	0	0
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	−30.0000	−1.32196
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	−12.0000	−0.523723
$526$	0	0
$527$	10.0000	0.435607
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−7.00000	−0.303774
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	16.0000	0.690451
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−23.0000	−0.988847	−0.494424	−	0.869221i	$-0.664621\pi$
$541$	−23.0000	−0.988847	−0.494424	+	0.869221i	$0.664621\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	0	0
$545$	−30.0000	−1.28506
$546$	0	0
$547$	−17.0000	−0.726868	−0.363434	−	0.931620i	$-0.618396\pi$
$547$	−17.0000	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$548$	0	0
$549$	−8.00000	−0.341432
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	48.0000	2.04117
$554$	0	0
$555$	−3.00000	−0.127343
$556$	0	0
$557$	−4.00000	−0.169485	−0.0847427	−	0.996403i	$-0.527007\pi$
$557$	−4.00000	−0.169485	−0.0847427	+	0.996403i	$0.527007\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−46.0000	−1.93867	−0.969334	−	0.245745i	$-0.920967\pi$
$563$	−46.0000	−1.93867	−0.969334	+	0.245745i	$0.920967\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	0	0
$567$	3.00000	0.125988
$568$	0	0
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	33.0000	1.38101	0.690504	−	0.723329i	$-0.257389\pi$
$571$	33.0000	1.38101	0.690504	+	0.723329i	$0.257389\pi$
$572$	0	0
$573$	−5.00000	−0.208878
$574$	0	0
$575$	24.0000	1.00087
$576$	0	0
$577$	43.0000	1.79011	0.895057	−	0.445952i	$-0.147135\pi$
$577$	43.0000	1.79011	0.895057	+	0.445952i	$0.147135\pi$
$578$	0	0
$579$	14.0000	0.581820
$580$	0	0
$581$	9.00000	0.373383
$582$	0	0
$583$	0	0
$584$	0	0
$585$	18.0000	0.744208
$586$	0	0
$587$	−3.00000	−0.123823	−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000	−0.123823	−0.0619116	+	0.998082i	$0.519720\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	10.0000	0.411345
$592$	0	0
$593$	−28.0000	−1.14982	−0.574911	−	0.818216i	$-0.694963\pi$
$593$	−28.0000	−1.14982	−0.574911	+	0.818216i	$0.694963\pi$
$594$	0	0
$595$	18.0000	0.737928
$596$	0	0
$597$	−16.0000	−0.654836
$598$	0	0
$599$	25.0000	1.02147	0.510736	−	0.859738i	$-0.329373\pi$
$599$	25.0000	1.02147	0.510736	+	0.859738i	$0.329373\pi$
$600$	0	0
$601$	−37.0000	−1.50926	−0.754631	−	0.656150i	$-0.772184\pi$
$601$	−37.0000	−1.50926	−0.754631	+	0.656150i	$0.772184\pi$
$602$	0	0
$603$	−3.00000	−0.122169
$604$	0	0
$605$	33.0000	1.34164
$606$	0	0
$607$	−30.0000	−1.21766	−0.608831	−	0.793300i	$-0.708361\pi$
$607$	−30.0000	−1.21766	−0.608831	+	0.793300i	$0.708361\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	−54.0000	−2.18461
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	0	0
$615$	6.00000	0.241943
$616$	0	0
$617$	46.0000	1.85189	0.925945	−	0.377658i	$-0.123271\pi$
$617$	46.0000	1.85189	0.925945	+	0.377658i	$0.123271\pi$
$618$	0	0
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	0	0
$623$	45.0000	1.80289
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.00000	0.0797452
$630$	0	0
$631$	−15.0000	−0.597141	−0.298570	−	0.954388i	$-0.596510\pi$
$631$	−15.0000	−0.597141	−0.298570	+	0.954388i	$0.596510\pi$
$632$	0	0
$633$	22.0000	0.874421
$634$	0	0
$635$	51.0000	2.02387
$636$	0	0
$637$	−12.0000	−0.475457
$638$	0	0
$639$	−14.0000	−0.553831
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	0	0
$645$	12.0000	0.472500
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	15.0000	0.587896
$652$	0	0
$653$	−8.00000	−0.313064	−0.156532	−	0.987673i	$-0.550031\pi$
$653$	−8.00000	−0.313064	−0.156532	+	0.987673i	$0.550031\pi$
$654$	0	0
$655$	−36.0000	−1.40664
$656$	0	0
$657$	−8.00000	−0.312110
$658$	0	0
$659$	39.0000	1.51922	0.759612	−	0.650376i	$-0.225389\pi$
$659$	39.0000	1.51922	0.759612	+	0.650376i	$0.225389\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	−12.0000	−0.466041
$664$	0	0
$665$	−18.0000	−0.698010
$666$	0	0
$667$	24.0000	0.929284
$668$	0	0
$669$	19.0000	0.734582
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	−3.00000	−0.115129
$680$	0	0
$681$	−7.00000	−0.268241
$682$	0	0
$683$	−16.0000	−0.612223	−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000	−0.612223	−0.306111	+	0.951996i	$0.599028\pi$
$684$	0	0
$685$	39.0000	1.49011
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−18.0000	−0.685745
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	33.0000	1.25176
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	−21.0000	−0.794293
$700$	0	0
$701$	16.0000	0.604312	0.302156	−	0.953259i	$-0.402294\pi$
$701$	16.0000	0.604312	0.302156	+	0.953259i	$0.402294\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	27.0000	1.01688
$706$	0	0
$707$	9.00000	0.338480
$708$	0	0
$709$	31.0000	1.16423	0.582115	−	0.813107i	$-0.302225\pi$
$709$	31.0000	1.16423	0.582115	+	0.813107i	$0.302225\pi$
$710$	0	0
$711$	16.0000	0.600047
$712$	0	0
$713$	−30.0000	−1.12351
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	30.0000	1.11726
$722$	0	0
$723$	26.0000	0.966950
$724$	0	0
$725$	16.0000	0.594225
$726$	0	0
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−25.0000	−0.919640	−0.459820	−	0.888012i	$-0.652086\pi$
$739$	−25.0000	−0.919640	−0.459820	+	0.888012i	$0.652086\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	0	0
$743$	−9.00000	−0.330178	−0.165089	−	0.986279i	$-0.552791\pi$
$743$	−9.00000	−0.330178	−0.165089	+	0.986279i	$0.552791\pi$
$744$	0	0
$745$	3.00000	0.109911
$746$	0	0
$747$	3.00000	0.109764
$748$	0	0
$749$	−6.00000	−0.219235
$750$	0	0
$751$	46.0000	1.67856	0.839282	−	0.543696i	$-0.182976\pi$
$751$	46.0000	1.67856	0.839282	+	0.543696i	$0.182976\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	−48.0000	−1.74690
$756$	0	0
$757$	24.0000	0.872295	0.436147	−	0.899875i	$-0.356343\pi$
$757$	24.0000	0.872295	0.436147	+	0.899875i	$0.356343\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35.0000	−1.26875	−0.634375	−	0.773026i	$-0.718742\pi$
$761$	−35.0000	−1.26875	−0.634375	+	0.773026i	$0.718742\pi$
$762$	0	0
$763$	30.0000	1.08607
$764$	0	0
$765$	6.00000	0.216930
$766$	0	0
$767$	42.0000	1.51653
$768$	0	0
$769$	−12.0000	−0.432731	−0.216366	−	0.976312i	$-0.569420\pi$
$769$	−12.0000	−0.432731	−0.216366	+	0.976312i	$0.569420\pi$
$770$	0	0
$771$	16.0000	0.576226
$772$	0	0
$773$	−55.0000	−1.97821	−0.989106	−	0.147203i	$-0.952973\pi$
$773$	−55.0000	−1.97821	−0.989106	+	0.147203i	$0.952973\pi$
$774$	0	0
$775$	−20.0000	−0.718421
$776$	0	0
$777$	3.00000	0.107624
$778$	0	0
$779$	4.00000	0.143315
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−4.00000	−0.142948
$784$	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
$789$	−17.0000	−0.605216
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	48.0000	1.70453
$794$	0	0
$795$	9.00000	0.319197
$796$	0	0
$797$	50.0000	1.77109	0.885545	−	0.464553i	$-0.153785\pi$
$797$	50.0000	1.77109	0.885545	+	0.464553i	$0.153785\pi$
$798$	0	0
$799$	−18.0000	−0.636794
$800$	0	0
$801$	15.0000	0.529999
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−54.0000	−1.90325
$806$	0	0
$807$	15.0000	0.528025
$808$	0	0
$809$	−29.0000	−1.01959	−0.509793	−	0.860297i	$-0.670278\pi$
$809$	−29.0000	−1.01959	−0.509793	+	0.860297i	$0.670278\pi$
$810$	0	0
$811$	3.00000	0.105344	0.0526721	−	0.998612i	$-0.483226\pi$
$811$	3.00000	0.105344	0.0526721	+	0.998612i	$0.483226\pi$
$812$	0	0
$813$	−28.0000	−0.982003
$814$	0	0
$815$	−15.0000	−0.525427
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	−18.0000	−0.628971
$820$	0	0
$821$	46.0000	1.60541	0.802706	−	0.596376i	$-0.203393\pi$
$821$	46.0000	1.60541	0.802706	+	0.596376i	$0.203393\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.0000	0.695468	0.347734	−	0.937593i	$-0.386951\pi$
$827$	20.0000	0.695468	0.347734	+	0.937593i	$0.386951\pi$
$828$	0	0
$829$	37.0000	1.28506	0.642532	−	0.766259i	$-0.277884\pi$
$829$	37.0000	1.28506	0.642532	+	0.766259i	$0.277884\pi$
$830$	0	0
$831$	−19.0000	−0.659103
$832$	0	0
$833$	−4.00000	−0.138592
$834$	0	0
$835$	−3.00000	−0.103819
$836$	0	0
$837$	5.00000	0.172825
$838$	0	0
$839$	−35.0000	−1.20833	−0.604167	−	0.796858i	$-0.706494\pi$
$839$	−35.0000	−1.20833	−0.604167	+	0.796858i	$0.706494\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	−17.0000	−0.585511
$844$	0	0
$845$	−69.0000	−2.37367
$846$	0	0
$847$	−33.0000	−1.13389
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	−6.00000	−0.205677
$852$	0	0
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	0	0
$857$	33.0000	1.12726	0.563629	−	0.826028i	$-0.309405\pi$
$857$	33.0000	1.12726	0.563629	+	0.826028i	$0.309405\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	44.0000	1.49778	0.748889	−	0.662696i	$-0.230588\pi$
$863$	44.0000	1.49778	0.748889	+	0.662696i	$0.230588\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	18.0000	0.609907
$872$	0	0
$873$	−1.00000	−0.0338449
$874$	0	0
$875$	9.00000	0.304256
$876$	0	0
$877$	−4.00000	−0.135070	−0.0675352	−	0.997717i	$-0.521513\pi$
$877$	−4.00000	−0.135070	−0.0675352	+	0.997717i	$0.521513\pi$
$878$	0	0
$879$	24.0000	0.809500
$880$	0	0
$881$	16.0000	0.539054	0.269527	−	0.962993i	$-0.413133\pi$
$881$	16.0000	0.539054	0.269527	+	0.962993i	$0.413133\pi$
$882$	0	0
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	0	0
$885$	−21.0000	−0.705907
$886$	0	0
$887$	−14.0000	−0.470074	−0.235037	−	0.971986i	$-0.575521\pi$
$887$	−14.0000	−0.470074	−0.235037	+	0.971986i	$0.575521\pi$
$888$	0	0
$889$	−51.0000	−1.71049
$890$	0	0
$891$	0	0
$892$	0	0
$893$	18.0000	0.602347
$894$	0	0
$895$	48.0000	1.60446
$896$	0	0
$897$	36.0000	1.20201
$898$	0	0
$899$	−20.0000	−0.667037
$900$	0	0
$901$	−6.00000	−0.199889
$902$	0	0
$903$	−12.0000	−0.399335
$904$	0	0
$905$	42.0000	1.39613
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	3.00000	0.0995037
$910$	0	0
$911$	13.0000	0.430709	0.215355	−	0.976536i	$-0.430909\pi$
$911$	13.0000	0.430709	0.215355	+	0.976536i	$0.430909\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−24.0000	−0.793416
$916$	0	0
$917$	36.0000	1.18882
$918$	0	0
$919$	41.0000	1.35247	0.676233	−	0.736688i	$-0.263611\pi$
$919$	41.0000	1.35247	0.676233	+	0.736688i	$0.263611\pi$
$920$	0	0
$921$	−21.0000	−0.691974
$922$	0	0
$923$	84.0000	2.76489
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	10.0000	0.328443
$928$	0	0
$929$	−15.0000	−0.492134	−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000	−0.492134	−0.246067	+	0.969253i	$0.579138\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	0	0
$933$	−15.0000	−0.491078
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	−8.00000	−0.261070
$940$	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
$943$	12.0000	0.390774
$944$	0	0
$945$	9.00000	0.292770
$946$	0	0
$947$	−54.0000	−1.75476	−0.877382	−	0.479792i	$-0.840712\pi$
$947$	−54.0000	−1.75476	−0.877382	+	0.479792i	$0.840712\pi$
$948$	0	0
$949$	48.0000	1.55815
$950$	0	0
$951$	−24.0000	−0.778253
$952$	0	0
$953$	24.0000	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$954$	0	0
$955$	−15.0000	−0.485389
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−39.0000	−1.25938
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	−2.00000	−0.0644491
$964$	0	0
$965$	42.0000	1.35203
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	−1.00000	−0.0320915	−0.0160458	−	0.999871i	$-0.505108\pi$
$971$	−1.00000	−0.0320915	−0.0160458	+	0.999871i	$0.505108\pi$
$972$	0	0
$973$	−33.0000	−1.05793
$974$	0	0
$975$	24.0000	0.768615
$976$	0	0
$977$	22.0000	0.703842	0.351921	−	0.936030i	$-0.385529\pi$
$977$	22.0000	0.703842	0.351921	+	0.936030i	$0.385529\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	−60.0000	−1.91370	−0.956851	−	0.290578i	$-0.906153\pi$
$983$	−60.0000	−1.91370	−0.956851	+	0.290578i	$0.906153\pi$
$984$	0	0
$985$	30.0000	0.955879
$986$	0	0
$987$	−27.0000	−0.859419
$988$	0	0
$989$	24.0000	0.763156
$990$	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
$993$	8.00000	0.253872
$994$	0	0
$995$	−48.0000	−1.52170
$996$	0	0
$997$	54.0000	1.71020	0.855099	−	0.518465i	$-0.173497\pi$
$997$	54.0000	1.71020	0.855099	+	0.518465i	$0.173497\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.a.1.1		1
4.3	odd	2		1002.2.a.e.1.1	✓	1
12.11	even	2		3006.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.e.1.1	✓	1	4.3	odd	2
3006.2.a.b.1.1		1	12.11	even	2
8016.2.a.a.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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