LMFDB - Embedded newform 172.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [172,2,Mod(1,172)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(172, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("172.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 172 = 2^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	172.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:	$1.37342691477$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	172.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{3} -4.00000 q^{7} +1.00000 q^{9} +O(q^{10})$

$q-2.00000 q^{3} -4.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -1.00000 q^{13} -3.00000 q^{17} +2.00000 q^{19} +8.00000 q^{21} -3.00000 q^{23} -5.00000 q^{25} +4.00000 q^{27} +6.00000 q^{29} +5.00000 q^{31} +6.00000 q^{33} +8.00000 q^{37} +2.00000 q^{39} -3.00000 q^{41} +1.00000 q^{43} -12.0000 q^{47} +9.00000 q^{49} +6.00000 q^{51} -9.00000 q^{53} -4.00000 q^{57} -12.0000 q^{59} -10.0000 q^{61} -4.00000 q^{63} +11.0000 q^{67} +6.00000 q^{69} +6.00000 q^{71} -10.0000 q^{73} +10.0000 q^{75} +12.0000 q^{77} +8.00000 q^{79} -11.0000 q^{81} -15.0000 q^{83} -12.0000 q^{87} +4.00000 q^{91} -10.0000 q^{93} -1.00000 q^{97} -3.00000 q^{99} +O(q^{100})$

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	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
$21$	8.00000	1.74574
$22$	0	0
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	6.00000	0.840168
$52$	0	0
$53$	−9.00000	−1.23625	−0.618123	−	0.786082i	$-0.712106\pi$
$53$	−9.00000	−1.23625	−0.618123	+	0.786082i	$0.712106\pi$
$54$	0	0
$55$	0	0
$56$	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
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	−4.00000	−0.503953
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.0000	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	0	0
$75$	10.0000	1.15470
$76$	0	0
$77$	12.0000	1.36753
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−15.0000	−1.64646	−0.823232	−	0.567705i	$-0.807831\pi$
$83$	−15.0000	−1.64646	−0.823232	+	0.567705i	$0.807831\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−12.0000	−1.28654
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	−10.0000	−1.03695
$94$	0	0
$95$	0	0
$96$	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
$99$	−3.00000	−0.301511
$100$	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
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	−16.0000	−1.51865
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	6.00000	0.541002
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	24.0000	2.02116
$142$	0	0
$143$	3.00000	0.250873
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−18.0000	−1.48461
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	18.0000	1.42749
$160$	0	0
$161$	12.0000	0.945732
$162$	0	0
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.0000	1.62503	0.812514	−	0.582941i	$-0.198098\pi$
$167$	21.0000	1.62503	0.812514	+	0.582941i	$0.198098\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	20.0000	1.51186
$176$	0	0
$177$	24.0000	1.80395
$178$	0	0
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	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
$183$	20.0000	1.47844
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.00000	0.658145
$188$	0	0
$189$	−16.0000	−1.16383
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	−22.0000	−1.55176
$202$	0	0
$203$	−24.0000	−1.68447
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.00000	−0.208514
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	−12.0000	−0.822226
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−20.0000	−1.35769
$218$	0	0
$219$	20.0000	1.35147
$220$	0	0
$221$	3.00000	0.201802
$222$	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
$225$	−5.00000	−0.333333
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	0	0
$231$	−24.0000	−1.57908
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	0	0
$236$	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
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	30.0000	1.90117
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	9.00000	0.565825
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	−32.0000	−1.98838
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	27.0000	1.64622	0.823110	−	0.567883i	$-0.192237\pi$
$269$	27.0000	1.64622	0.823110	+	0.567883i	$0.192237\pi$
$270$	0	0
$271$	5.00000	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$271$	5.00000	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	15.0000	0.904534
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	5.00000	0.299342
$280$	0	0
$281$	27.0000	1.61068	0.805342	−	0.592810i	$-0.201981\pi$
$281$	27.0000	1.61068	0.805342	+	0.592810i	$0.201981\pi$
$282$	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
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	2.00000	0.117242
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−12.0000	−0.696311
$298$	0	0
$299$	3.00000	0.173494
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−25.0000	−1.42683	−0.713413	−	0.700744i	$-0.752851\pi$
$307$	−25.0000	−1.42683	−0.713413	+	0.700744i	$0.752851\pi$
$308$	0	0
$309$	26.0000	1.47909
$310$	0	0
$311$	−27.0000	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$311$	−27.0000	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.0000	1.17948	0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000	1.17948	0.589739	+	0.807594i	$0.299231\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	−24.0000	−1.33955
$322$	0	0
$323$	−6.00000	−0.333849
$324$	0	0
$325$	5.00000	0.277350
$326$	0	0
$327$	−22.0000	−1.21660
$328$	0	0
$329$	48.0000	2.64633
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	0	0
$333$	8.00000	0.438397
$334$	0	0
$335$	0	0
$336$	0	0
$337$	29.0000	1.57973	0.789865	−	0.613280i	$-0.210150\pi$
$337$	29.0000	1.57973	0.789865	+	0.613280i	$0.210150\pi$
$338$	0	0
$339$	24.0000	1.30350
$340$	0	0
$341$	−15.0000	−0.812296
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	33.0000	1.75641	0.878206	−	0.478282i	$-0.158740\pi$
$353$	33.0000	1.75641	0.878206	+	0.478282i	$0.158740\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−24.0000	−1.27021
$358$	0	0
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	4.00000	0.209946
$364$	0	0
$365$	0	0
$366$	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
$369$	−3.00000	−0.156174
$370$	0	0
$371$	36.0000	1.86903
$372$	0	0
$373$	8.00000	0.414224	0.207112	−	0.978317i	$-0.433593\pi$
$373$	8.00000	0.414224	0.207112	+	0.978317i	$0.433593\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.00000	−0.309016
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	−22.0000	−1.12709
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.00000	0.0508329
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	9.00000	0.455150
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	16.0000	0.801002
$400$	0	0
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	0	0
$403$	−5.00000	−0.249068
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24.0000	−1.18964
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	36.0000	1.77575
$412$	0	0
$413$	48.0000	2.36193
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−10.0000	−0.489702
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	−12.0000	−0.583460
$424$	0	0
$425$	15.0000	0.727607
$426$	0	0
$427$	40.0000	1.93574
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	9.00000	0.433515	0.216757	−	0.976226i	$-0.430452\pi$
$431$	9.00000	0.433515	0.216757	+	0.976226i	$0.430452\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.00000	−0.287019
$438$	0	0
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−24.0000	−1.13516
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	0	0
$453$	32.0000	1.50349
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	0	0
$459$	−12.0000	−0.560112
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	−44.0000	−2.03173
$470$	0	0
$471$	−28.0000	−1.29017
$472$	0	0
$473$	−3.00000	−0.137940
$474$	0	0
$475$	−10.0000	−0.458831
$476$	0	0
$477$	−9.00000	−0.412082
$478$	0	0
$479$	−33.0000	−1.50781	−0.753904	−	0.656984i	$-0.771832\pi$
$479$	−33.0000	−1.50781	−0.753904	+	0.656984i	$0.771832\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	−24.0000	−1.09204
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	0	0
$489$	−28.0000	−1.26620
$490$	0	0
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	0	0
$493$	−18.0000	−0.810679
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.0000	−1.07655
$498$	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
$501$	−42.0000	−1.87642
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	24.0000	1.06588
$508$	0	0
$509$	−27.0000	−1.19675	−0.598377	−	0.801215i	$-0.704187\pi$
$509$	−27.0000	−1.19675	−0.598377	+	0.801215i	$0.704187\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	0	0
$513$	8.00000	0.353209
$514$	0	0
$515$	0	0
$516$	0	0
$517$	36.0000	1.58328
$518$	0	0
$519$	−12.0000	−0.526742
$520$	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
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	0	0
$525$	−40.0000	−1.74574
$526$	0	0
$527$	−15.0000	−0.653410
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	3.00000	0.129944
$534$	0	0
$535$	0	0
$536$	0	0
$537$	48.0000	2.07135
$538$	0	0
$539$	−27.0000	−1.16297
$540$	0	0
$541$	−43.0000	−1.84871	−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000	−1.84871	−0.924357	+	0.381528i	$0.875398\pi$
$542$	0	0
$543$	44.0000	1.88822
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−19.0000	−0.812381	−0.406191	−	0.913788i	$-0.633143\pi$
$547$	−19.0000	−0.812381	−0.406191	+	0.913788i	$0.633143\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	−32.0000	−1.36078
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.0000	1.39825	0.699127	−	0.714997i	$-0.253572\pi$
$557$	33.0000	1.39825	0.699127	+	0.714997i	$0.253572\pi$
$558$	0	0
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	−18.0000	−0.759961
$562$	0	0
$563$	−21.0000	−0.885044	−0.442522	−	0.896758i	$-0.645916\pi$
$563$	−21.0000	−0.885044	−0.442522	+	0.896758i	$0.645916\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	44.0000	1.84783
$568$	0	0
$569$	−9.00000	−0.377300	−0.188650	−	0.982044i	$-0.560411\pi$
$569$	−9.00000	−0.377300	−0.188650	+	0.982044i	$0.560411\pi$
$570$	0	0
$571$	26.0000	1.08807	0.544033	−	0.839064i	$-0.316897\pi$
$571$	26.0000	1.08807	0.544033	+	0.839064i	$0.316897\pi$
$572$	0	0
$573$	−48.0000	−2.00523
$574$	0	0
$575$	15.0000	0.625543
$576$	0	0
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	0	0
$579$	−22.0000	−0.914289
$580$	0	0
$581$	60.0000	2.48922
$582$	0	0
$583$	27.0000	1.11823
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.0000	1.23823	0.619116	−	0.785299i	$-0.287491\pi$
$587$	30.0000	1.23823	0.619116	+	0.785299i	$0.287491\pi$
$588$	0	0
$589$	10.0000	0.412043
$590$	0	0
$591$	12.0000	0.493614
$592$	0	0
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.00000	−0.163709
$598$	0	0
$599$	−3.00000	−0.122577	−0.0612883	−	0.998120i	$-0.519521\pi$
$599$	−3.00000	−0.122577	−0.0612883	+	0.998120i	$0.519521\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	11.0000	0.447955
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	0	0
$609$	48.0000	1.94506
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−21.0000	−0.845428	−0.422714	−	0.906263i	$-0.638923\pi$
$617$	−21.0000	−0.845428	−0.422714	+	0.906263i	$0.638923\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−12.0000	−0.481543
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	12.0000	0.479234
$628$	0	0
$629$	−24.0000	−0.956943
$630$	0	0
$631$	−10.0000	−0.398094	−0.199047	−	0.979990i	$-0.563785\pi$
$631$	−10.0000	−0.398094	−0.199047	+	0.979990i	$0.563785\pi$
$632$	0	0
$633$	20.0000	0.794929
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−9.00000	−0.356593
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	0	0
$645$	0	0
$646$	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
$649$	36.0000	1.41312
$650$	0	0
$651$	40.0000	1.56772
$652$	0	0
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−10.0000	−0.390137
$658$	0	0
$659$	3.00000	0.116863	0.0584317	−	0.998291i	$-0.481390\pi$
$659$	3.00000	0.116863	0.0584317	+	0.998291i	$0.481390\pi$
$660$	0	0
$661$	35.0000	1.36134	0.680671	−	0.732589i	$-0.261688\pi$
$661$	35.0000	1.36134	0.680671	+	0.732589i	$0.261688\pi$
$662$	0	0
$663$	−6.00000	−0.233021
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.0000	−0.696963
$668$	0	0
$669$	−16.0000	−0.618596
$670$	0	0
$671$	30.0000	1.15814
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	0	0
$675$	−20.0000	−0.769800
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	24.0000	0.919682
$682$	0	0
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	9.00000	0.342873
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	12.0000	0.455842
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9.00000	0.340899
$698$	0	0
$699$	36.0000	1.36165
$700$	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
$703$	16.0000	0.603451
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.0000	−0.451306
$708$	0	0
$709$	−37.0000	−1.38956	−0.694782	−	0.719220i	$-0.744499\pi$
$709$	−37.0000	−1.38956	−0.694782	+	0.719220i	$0.744499\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	−15.0000	−0.561754
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	52.0000	1.93658
$722$	0	0
$723$	8.00000	0.297523
$724$	0	0
$725$	−30.0000	−1.11417
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−3.00000	−0.110959
$732$	0	0
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−33.0000	−1.21557
$738$	0	0
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−15.0000	−0.548821
$748$	0	0
$749$	−48.0000	−1.75388
$750$	0	0
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20.0000	0.726912	0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000	0.726912	0.363456	+	0.931611i	$0.381597\pi$
$758$	0	0
$759$	−18.0000	−0.653359
$760$	0	0
$761$	−36.0000	−1.30500	−0.652499	−	0.757789i	$-0.726280\pi$
$761$	−36.0000	−1.30500	−0.652499	+	0.757789i	$0.726280\pi$
$762$	0	0
$763$	−44.0000	−1.59291
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.0000	0.433295
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	0	0
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	0	0
$775$	−25.0000	−0.898027
$776$	0	0
$777$	64.0000	2.29599
$778$	0	0
$779$	−6.00000	−0.214972
$780$	0	0
$781$	−18.0000	−0.644091
$782$	0	0
$783$	24.0000	0.857690
$784$	0	0
$785$	0	0
$786$	0	0
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	0	0
$791$	48.0000	1.70668
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	0	0
$801$	0	0
$802$	0	0
$803$	30.0000	1.05868
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−54.0000	−1.90089
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	−10.0000	−0.350715
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	4.00000	0.139771
$820$	0	0
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	0	0
$823$	−19.0000	−0.662298	−0.331149	−	0.943578i	$-0.607436\pi$
$823$	−19.0000	−0.662298	−0.331149	+	0.943578i	$0.607436\pi$
$824$	0	0
$825$	−30.0000	−1.04447
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	56.0000	1.94262
$832$	0	0
$833$	−27.0000	−0.935495
$834$	0	0
$835$	0	0
$836$	0	0
$837$	20.0000	0.691301
$838$	0	0
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−54.0000	−1.85986
$844$	0	0
$845$	0	0
$846$	0	0
$847$	8.00000	0.274883
$848$	0	0
$849$	26.0000	0.892318
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	−1.00000	−0.0342393	−0.0171197	−	0.999853i	$-0.505450\pi$
$853$	−1.00000	−0.0342393	−0.0171197	+	0.999853i	$0.505450\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	0	0
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.0000	0.543388
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	−11.0000	−0.372721
$872$	0	0
$873$	−1.00000	−0.0338449
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−43.0000	−1.45201	−0.726003	−	0.687691i	$-0.758624\pi$
$877$	−43.0000	−1.45201	−0.726003	+	0.687691i	$0.758624\pi$
$878$	0	0
$879$	−12.0000	−0.404750
$880$	0	0
$881$	−51.0000	−1.71823	−0.859117	−	0.511780i	$-0.828986\pi$
$881$	−51.0000	−1.71823	−0.859117	+	0.511780i	$0.828986\pi$
$882$	0	0
$883$	−31.0000	−1.04323	−0.521617	−	0.853180i	$-0.674671\pi$
$883$	−31.0000	−1.04323	−0.521617	+	0.853180i	$0.674671\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	0	0
$889$	−44.0000	−1.47571
$890$	0	0
$891$	33.0000	1.10554
$892$	0	0
$893$	−24.0000	−0.803129
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.00000	−0.200334
$898$	0	0
$899$	30.0000	1.00056
$900$	0	0
$901$	27.0000	0.899500
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−7.00000	−0.232431	−0.116216	−	0.993224i	$-0.537076\pi$
$907$	−7.00000	−0.232431	−0.116216	+	0.993224i	$0.537076\pi$
$908$	0	0
$909$	3.00000	0.0995037
$910$	0	0
$911$	−54.0000	−1.78910	−0.894550	−	0.446968i	$-0.852504\pi$
$911$	−54.0000	−1.78910	−0.894550	+	0.446968i	$0.852504\pi$
$912$	0	0
$913$	45.0000	1.48928
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	29.0000	0.956622	0.478311	−	0.878191i	$-0.341249\pi$
$919$	29.0000	0.956622	0.478311	+	0.878191i	$0.341249\pi$
$920$	0	0
$921$	50.0000	1.64756
$922$	0	0
$923$	−6.00000	−0.197492
$924$	0	0
$925$	−40.0000	−1.31519
$926$	0	0
$927$	−13.0000	−0.426976
$928$	0	0
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	0	0
$933$	54.0000	1.76788
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	0	0
$939$	−52.0000	−1.69696
$940$	0	0
$941$	21.0000	0.684580	0.342290	−	0.939594i	$-0.388797\pi$
$941$	21.0000	0.684580	0.342290	+	0.939594i	$0.388797\pi$
$942$	0	0
$943$	9.00000	0.293080
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.00000	0.292461	0.146230	−	0.989251i	$-0.453286\pi$
$947$	9.00000	0.292461	0.146230	+	0.989251i	$0.453286\pi$
$948$	0	0
$949$	10.0000	0.324614
$950$	0	0
$951$	−42.0000	−1.36194
$952$	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
$955$	0	0
$956$	0	0
$957$	36.0000	1.16371
$958$	0	0
$959$	72.0000	2.32500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	0	0
$966$	0	0
$967$	23.0000	0.739630	0.369815	−	0.929105i	$-0.379421\pi$
$967$	23.0000	0.739630	0.369815	+	0.929105i	$0.379421\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	21.0000	0.673922	0.336961	−	0.941519i	$-0.390601\pi$
$971$	21.0000	0.673922	0.336961	+	0.941519i	$0.390601\pi$
$972$	0	0
$973$	−20.0000	−0.641171
$974$	0	0
$975$	−10.0000	−0.320256
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	11.0000	0.351203
$982$	0	0
$983$	60.0000	1.91370	0.956851	−	0.290578i	$-0.0938475\pi$
$983$	60.0000	1.91370	0.956851	+	0.290578i	$0.0938475\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−96.0000	−3.05571
$988$	0	0
$989$	−3.00000	−0.0953945
$990$	0	0
$991$	−22.0000	−0.698853	−0.349427	−	0.936964i	$-0.613624\pi$
$991$	−22.0000	−0.698853	−0.349427	+	0.936964i	$0.613624\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	0	0
$999$	32.0000	1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	172.2.a.a.1.1	✓	1
3.2	odd	2		1548.2.a.c.1.1		1
4.3	odd	2		688.2.a.c.1.1		1
5.2	odd	4		4300.2.d.a.1549.2		2
5.3	odd	4		4300.2.d.a.1549.1		2
5.4	even	2		4300.2.a.c.1.1		1
7.6	odd	2		8428.2.a.c.1.1		1
8.3	odd	2		2752.2.a.a.1.1		1
8.5	even	2		2752.2.a.e.1.1		1
12.11	even	2		6192.2.a.m.1.1		1
43.42	odd	2		7396.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
172.2.a.a.1.1	✓	1	1.1	even	1	trivial
688.2.a.c.1.1		1	4.3	odd	2
1548.2.a.c.1.1		1	3.2	odd	2
2752.2.a.a.1.1		1	8.3	odd	2
2752.2.a.e.1.1		1	8.5	even	2
4300.2.a.c.1.1		1	5.4	even	2
4300.2.d.a.1549.1		2	5.3	odd	4
4300.2.d.a.1549.2		2	5.2	odd	4
6192.2.a.m.1.1		1	12.11	even	2
7396.2.a.a.1.1		1	43.42	odd	2
8428.2.a.c.1.1		1	7.6	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:	\( N \)	\(=\)	\( 172 = 2^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	172.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more