LMFDB - Embedded newform 4080.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4080,2,Mod(1,4080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4080.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:	$32.5789640247$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{3} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{3} -1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} -2.00000 q^{13} -1.00000 q^{15} +1.00000 q^{17} +4.00000 q^{19} +4.00000 q^{21} +4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} -4.00000 q^{31} +4.00000 q^{33} -4.00000 q^{35} -6.00000 q^{37} -2.00000 q^{39} +2.00000 q^{41} +12.0000 q^{43} -1.00000 q^{45} -8.00000 q^{47} +9.00000 q^{49} +1.00000 q^{51} -2.00000 q^{53} -4.00000 q^{55} +4.00000 q^{57} -12.0000 q^{59} +2.00000 q^{61} +4.00000 q^{63} +2.00000 q^{65} -4.00000 q^{67} +4.00000 q^{69} +4.00000 q^{71} -14.0000 q^{73} +1.00000 q^{75} +16.0000 q^{77} +12.0000 q^{79} +1.00000 q^{81} +4.00000 q^{83} -1.00000 q^{85} +2.00000 q^{87} +10.0000 q^{89} -8.00000 q^{91} -4.00000 q^{93} -4.00000 q^{95} +18.0000 q^{97} +4.00000 q^{99} +O(q^{100})$

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	4.00000	0.696311
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	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
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	16.0000	1.82337
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	18.0000	1.82762	0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000	1.82762	0.913812	+	0.406138i	$0.133125\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	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
$111$	−6.00000	−0.569495
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	9.00000	0.742307
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	−4.00000	−0.311400
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	4.00000	0.290957
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	4.00000	0.274075
$214$	0	0
$215$	−12.0000	−0.818393
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	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
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	12.0000	0.779484
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−9.00000	−0.574989
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	28.0000	1.76734	0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000	1.76734	0.883672	+	0.468106i	$0.155064\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	0	0
$255$	−1.00000	−0.0626224
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	18.0000	1.05518
$292$	0	0
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	4.00000	0.232104
$298$	0	0
$299$	−8.00000	−0.462652
$300$	0	0
$301$	48.0000	2.76667
$302$	0	0
$303$	−10.0000	−0.574485
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	−4.00000	−0.225374
$316$	0	0
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	10.0000	0.553001
$328$	0	0
$329$	−32.0000	−1.76422
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	−4.00000	−0.215353
$346$	0	0
$347$	−20.0000	−1.07366	−0.536828	−	0.843692i	$-0.680378\pi$
$347$	−20.0000	−1.07366	−0.536828	+	0.843692i	$0.680378\pi$
$348$	0	0
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	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
$355$	−4.00000	−0.212298
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	−8.00000	−0.415339
$372$	0	0
$373$	38.0000	1.96757	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	38.0000	1.96757	0.983783	+	0.179364i	$0.0574041\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	0	0
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	12.0000	0.609994
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	−20.0000	−1.00887
$394$	0	0
$395$	−12.0000	−0.603786
$396$	0	0
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	0	0
$399$	16.0000	0.801002
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	−48.0000	−2.36193
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	8.00000	0.387147
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	4.00000	0.192673	0.0963366	−	0.995349i	$-0.469287\pi$
$431$	4.00000	0.192673	0.0963366	+	0.995349i	$0.469287\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	0	0
$437$	16.0000	0.765384
$438$	0	0
$439$	12.0000	0.572729	0.286364	−	0.958121i	$-0.407553\pi$
$439$	12.0000	0.572729	0.286364	+	0.958121i	$0.407553\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	−10.0000	−0.474045
$446$	0	0
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	0	0
$453$	8.00000	0.375873
$454$	0	0
$455$	8.00000	0.375046
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	48.0000	2.20704
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	16.0000	0.728025
$484$	0	0
$485$	−18.0000	−0.817338
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	2.00000	0.0900755
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	16.0000	0.717698
$498$	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
$501$	−12.0000	−0.536120
$502$	0	0
$503$	−44.0000	−1.96186	−0.980932	−	0.194354i	$-0.937739\pi$
$503$	−44.0000	−1.96186	−0.980932	+	0.194354i	$0.937739\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	−9.00000	−0.399704
$508$	0	0
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	0	0
$511$	−56.0000	−2.47729
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−32.0000	−1.40736
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	0	0
$525$	4.00000	0.174574
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	−4.00000	−0.173259
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	36.0000	1.55063
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	−22.0000	−0.944110
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	48.0000	2.04117
$554$	0	0
$555$	6.00000	0.254686
$556$	0	0
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	4.00000	0.168880
$562$	0	0
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	4.00000	0.167984
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	18.0000	0.748054
$580$	0	0
$581$	16.0000	0.663792
$582$	0	0
$583$	−8.00000	−0.331326
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	−44.0000	−1.81607	−0.908037	−	0.418890i	$-0.862419\pi$
$587$	−44.0000	−1.81607	−0.908037	+	0.418890i	$0.862419\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	−14.0000	−0.575883
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	−4.00000	−0.163709
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	−5.00000	−0.203279
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	40.0000	1.60257
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	16.0000	0.638978
$628$	0	0
$629$	−6.00000	−0.239236
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	−16.0000	−0.634941
$636$	0	0
$637$	−18.0000	−0.713186
$638$	0	0
$639$	4.00000	0.158238
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	−28.0000	−1.10421	−0.552106	−	0.833774i	$-0.686176\pi$
$643$	−28.0000	−1.10421	−0.552106	+	0.833774i	$0.686176\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	16.0000	0.629025	0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000	0.629025	0.314512	+	0.949253i	$0.398159\pi$
$648$	0	0
$649$	−48.0000	−1.88416
$650$	0	0
$651$	−16.0000	−0.627089
$652$	0	0
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	20.0000	0.781465
$656$	0	0
$657$	−14.0000	−0.546192
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	−16.0000	−0.620453
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	72.0000	2.76311
$680$	0	0
$681$	20.0000	0.766402
$682$	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
$685$	6.00000	0.229248
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	4.00000	0.152388
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	0	0
$693$	16.0000	0.607790
$694$	0	0
$695$	−20.0000	−0.758643
$696$	0	0
$697$	2.00000	0.0757554
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	−40.0000	−1.50435
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	8.00000	0.299183
$716$	0	0
$717$	24.0000	0.896296
$718$	0	0
$719$	44.0000	1.64092	0.820462	−	0.571702i	$-0.193717\pi$
$719$	44.0000	1.64092	0.820462	+	0.571702i	$0.193717\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	10.0000	0.371904
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	0	0
$735$	−9.00000	−0.331970
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−28.0000	−1.02722	−0.513610	−	0.858024i	$-0.671692\pi$
$743$	−28.0000	−1.02722	−0.513610	+	0.858024i	$0.671692\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	−48.0000	−1.75388
$750$	0	0
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	0	0
$753$	28.0000	1.02038
$754$	0	0
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	0	0
$759$	16.0000	0.580763
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	40.0000	1.44810
$764$	0	0
$765$	−1.00000	−0.0361551
$766$	0	0
$767$	24.0000	0.866590
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	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
$775$	−4.00000	−0.143684
$776$	0	0
$777$	−24.0000	−0.860995
$778$	0	0
$779$	8.00000	0.286630
$780$	0	0
$781$	16.0000	0.572525
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	−36.0000	−1.28326	−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000	−1.28326	−0.641631	+	0.767014i	$0.721742\pi$
$788$	0	0
$789$	16.0000	0.569615
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	2.00000	0.0709327
$796$	0	0
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	−56.0000	−1.97620
$804$	0	0
$805$	−16.0000	−0.563926
$806$	0	0
$807$	−30.0000	−1.05605
$808$	0	0
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−20.0000	−0.700569
$816$	0	0
$817$	48.0000	1.67931
$818$	0	0
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	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
$825$	4.00000	0.139262
$826$	0	0
$827$	−44.0000	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	9.00000	0.311832
$834$	0	0
$835$	12.0000	0.415277
$836$	0	0
$837$	−4.00000	−0.138260
$838$	0	0
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	10.0000	0.344418
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	20.0000	0.687208
$848$	0	0
$849$	20.0000	0.686398
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	8.00000	0.272639
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	48.0000	1.62829
$870$	0	0
$871$	8.00000	0.271070
$872$	0	0
$873$	18.0000	0.609208
$874$	0	0
$875$	−4.00000	−0.135225
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	0	0
$879$	−26.0000	−0.876958
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	0	0
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	64.0000	2.14649
$890$	0	0
$891$	4.00000	0.134005
$892$	0	0
$893$	−32.0000	−1.07084
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	−8.00000	−0.267112
$898$	0	0
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−2.00000	−0.0666297
$902$	0	0
$903$	48.0000	1.59734
$904$	0	0
$905$	22.0000	0.731305
$906$	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
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−20.0000	−0.662630	−0.331315	−	0.943520i	$-0.607492\pi$
$911$	−20.0000	−0.662630	−0.331315	+	0.943520i	$0.607492\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	0	0
$915$	−2.00000	−0.0661180
$916$	0	0
$917$	−80.0000	−2.64183
$918$	0	0
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	0	0
$933$	−20.0000	−0.654771
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	0	0
$951$	10.0000	0.324272
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	0	0
$957$	8.00000	0.258603
$958$	0	0
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	−18.0000	−0.579441
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	0	0
$973$	80.0000	2.56468
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	40.0000	1.27841
$980$	0	0
$981$	10.0000	0.319275
$982$	0	0
$983$	28.0000	0.893061	0.446531	−	0.894768i	$-0.352659\pi$
$983$	28.0000	0.893061	0.446531	+	0.894768i	$0.352659\pi$
$984$	0	0
$985$	14.0000	0.446077
$986$	0	0
$987$	−32.0000	−1.01857
$988$	0	0
$989$	48.0000	1.52631
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	0	0
$993$	−28.0000	−0.888553
$994$	0	0
$995$	4.00000	0.126809
$996$	0	0
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	0	0
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4080.2.a.x.1.1		1
4.3	odd	2		510.2.a.c.1.1	✓	1
12.11	even	2		1530.2.a.d.1.1		1
20.3	even	4		2550.2.d.b.2449.1		2
20.7	even	4		2550.2.d.b.2449.2		2
20.19	odd	2		2550.2.a.n.1.1		1
60.59	even	2		7650.2.a.cn.1.1		1
68.67	odd	2		8670.2.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.c.1.1	✓	1	4.3	odd	2
1530.2.a.d.1.1		1	12.11	even	2
2550.2.a.n.1.1		1	20.19	odd	2
2550.2.d.b.2449.1		2	20.3	even	4
2550.2.d.b.2449.2		2	20.7	even	4
4080.2.a.x.1.1		1	1.1	even	1	trivial
7650.2.a.cn.1.1		1	60.59	even	2
8670.2.a.bb.1.1		1	68.67	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:	\( N \)	\(=\)	\( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4080.a (trivial)

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