LMFDB - Embedded newform 3712.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3712,2,Mod(1,3712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3712, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3712 = 2^{7} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3712.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:	$29.6404692303$
Analytic rank:	$2$
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	$\chi$	$=$	3712.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} -1.00000 q^{5} -4.00000 q^{7} -2.00000 q^{9} +O(q^{10})$

$q-1.00000 q^{3} -1.00000 q^{5} -4.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} -3.00000 q^{13} +1.00000 q^{15} -4.00000 q^{19} +4.00000 q^{21} -8.00000 q^{23} -4.00000 q^{25} +5.00000 q^{27} -1.00000 q^{29} -5.00000 q^{31} +3.00000 q^{33} +4.00000 q^{35} +4.00000 q^{37} +3.00000 q^{39} -2.00000 q^{41} +1.00000 q^{43} +2.00000 q^{45} -5.00000 q^{47} +9.00000 q^{49} -1.00000 q^{53} +3.00000 q^{55} +4.00000 q^{57} +4.00000 q^{59} -8.00000 q^{61} +8.00000 q^{63} +3.00000 q^{65} -2.00000 q^{67} +8.00000 q^{69} -12.0000 q^{71} +6.00000 q^{73} +4.00000 q^{75} +12.0000 q^{77} -7.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} +1.00000 q^{87} +8.00000 q^{89} +12.0000 q^{91} +5.00000 q^{93} +4.00000 q^{95} -12.0000 q^{97} +6.00000 q^{99} +O(q^{100})$

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	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	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
$9$	−2.00000	−0.666667
$10$	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
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	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
$33$	3.00000	0.522233
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−5.00000	−0.729325	−0.364662	−	0.931140i	$-0.618816\pi$
$47$	−5.00000	−0.729325	−0.364662	+	0.931140i	$0.618816\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.00000	−0.137361	−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000	−0.137361	−0.0686803	+	0.997639i	$0.521879\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	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
$63$	8.00000	1.00791
$64$	0	0
$65$	3.00000	0.372104
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	4.00000	0.461880
$76$	0	0
$77$	12.0000	1.36753
$78$	0	0
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	0	0
$93$	5.00000	0.518476
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	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	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	19.0000	1.81987	0.909935	−	0.414751i	$-0.136131\pi$
$109$	19.0000	1.81987	0.909935	+	0.414751i	$0.136131\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	0	0
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	0	0
$117$	6.00000	0.554700
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	−1.00000	−0.0880451
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	0	0
$141$	5.00000	0.421076
$142$	0	0
$143$	9.00000	0.752618
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	−9.00000	−0.742307
$148$	0	0
$149$	−11.0000	−0.901155	−0.450578	−	0.892737i	$-0.648782\pi$
$149$	−11.0000	−0.901155	−0.450578	+	0.892737i	$0.648782\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.00000	0.401610
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	0	0
$159$	1.00000	0.0793052
$160$	0	0
$161$	32.0000	2.52195
$162$	0	0
$163$	17.0000	1.33154	0.665771	−	0.746156i	$-0.268103\pi$
$163$	17.0000	1.33154	0.665771	+	0.746156i	$0.268103\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	8.00000	0.611775
$172$	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
$175$	16.0000	1.20949
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	1.00000	0.0743294	0.0371647	−	0.999309i	$-0.488167\pi$
$181$	1.00000	0.0743294	0.0371647	+	0.999309i	$0.488167\pi$
$182$	0	0
$183$	8.00000	0.591377
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−20.0000	−1.45479
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	−3.00000	−0.214834
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	4.00000	0.280745
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	16.0000	1.11208
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	20.0000	1.35769
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	23.0000	1.50678	0.753390	−	0.657574i	$-0.228417\pi$
$233$	23.0000	1.50678	0.753390	+	0.657574i	$0.228417\pi$
$234$	0	0
$235$	5.00000	0.326164
$236$	0	0
$237$	7.00000	0.454699
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	21.0000	1.35273	0.676364	−	0.736567i	$-0.263554\pi$
$241$	21.0000	1.35273	0.676364	+	0.736567i	$0.263554\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	−9.00000	−0.574989
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	−13.0000	−0.820553	−0.410276	−	0.911961i	$-0.634568\pi$
$251$	−13.0000	−0.820553	−0.410276	+	0.911961i	$0.634568\pi$
$252$	0	0
$253$	24.0000	1.50887
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	0	0
$265$	1.00000	0.0614295
$266$	0	0
$267$	−8.00000	−0.489592
$268$	0	0
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	0	0
$271$	25.0000	1.51864	0.759321	−	0.650716i	$-0.225531\pi$
$271$	25.0000	1.51864	0.759321	+	0.650716i	$0.225531\pi$
$272$	0	0
$273$	−12.0000	−0.726273
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	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
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	12.0000	0.703452
$292$	0	0
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	−15.0000	−0.870388
$298$	0	0
$299$	24.0000	1.38796
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	2.00000	0.114897
$304$	0	0
$305$	8.00000	0.458079
$306$	0	0
$307$	5.00000	0.285365	0.142683	−	0.989769i	$-0.454427\pi$
$307$	5.00000	0.285365	0.142683	+	0.989769i	$0.454427\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	0	0
$313$	29.0000	1.63918	0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000	1.63918	0.819588	+	0.572953i	$0.194202\pi$
$314$	0	0
$315$	−8.00000	−0.450749
$316$	0	0
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	3.00000	0.167968
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	0	0
$324$	0	0
$325$	12.0000	0.665640
$326$	0	0
$327$	−19.0000	−1.05070
$328$	0	0
$329$	20.0000	1.10264
$330$	0	0
$331$	33.0000	1.81384	0.906922	−	0.421299i	$-0.138426\pi$
$331$	33.0000	1.81384	0.906922	+	0.421299i	$0.138426\pi$
$332$	0	0
$333$	−8.00000	−0.438397
$334$	0	0
$335$	2.00000	0.109272
$336$	0	0
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	0	0
$339$	−8.00000	−0.434500
$340$	0	0
$341$	15.0000	0.812296
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	−8.00000	−0.430706
$346$	0	0
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	0	0
$349$	−37.0000	−1.98056	−0.990282	−	0.139072i	$-0.955588\pi$
$349$	−37.0000	−1.98056	−0.990282	+	0.139072i	$0.955588\pi$
$350$	0	0
$351$	−15.0000	−0.800641
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.00000	−0.158334	−0.0791670	−	0.996861i	$-0.525226\pi$
$359$	−3.00000	−0.158334	−0.0791670	+	0.996861i	$0.525226\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	4.00000	0.208232
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	−7.00000	−0.362446	−0.181223	−	0.983442i	$-0.558006\pi$
$373$	−7.00000	−0.362446	−0.181223	+	0.983442i	$0.558006\pi$
$374$	0	0
$375$	−9.00000	−0.464758
$376$	0	0
$377$	3.00000	0.154508
$378$	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
$381$	8.00000	0.409852
$382$	0	0
$383$	26.0000	1.32854	0.664269	−	0.747494i	$-0.268743\pi$
$383$	26.0000	1.32854	0.664269	+	0.747494i	$0.268743\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	7.00000	0.352208
$396$	0	0
$397$	5.00000	0.250943	0.125471	−	0.992097i	$-0.459956\pi$
$397$	5.00000	0.250943	0.125471	+	0.992097i	$0.459956\pi$
$398$	0	0
$399$	−16.0000	−0.801002
$400$	0	0
$401$	19.0000	0.948815	0.474407	−	0.880305i	$-0.342662\pi$
$401$	19.0000	0.948815	0.474407	+	0.880305i	$0.342662\pi$
$402$	0	0
$403$	15.0000	0.747203
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	0	0
$411$	14.0000	0.690569
$412$	0	0
$413$	−16.0000	−0.787309
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	−6.00000	−0.293821
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	10.0000	0.486217
$424$	0	0
$425$	0	0
$426$	0	0
$427$	32.0000	1.54859
$428$	0	0
$429$	−9.00000	−0.434524
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	−1.00000	−0.0479463
$436$	0	0
$437$	32.0000	1.53077
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	0	0
$447$	11.0000	0.520282
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	0	0
$453$	−2.00000	−0.0939682
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	0	0
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	0	0
$465$	−5.00000	−0.231869
$466$	0	0
$467$	−27.0000	−1.24941	−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000	−1.24941	−0.624705	+	0.780860i	$0.714781\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	−12.0000	−0.552931
$472$	0	0
$473$	−3.00000	−0.137940
$474$	0	0
$475$	16.0000	0.734130
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	33.0000	1.50781	0.753904	−	0.656984i	$-0.228168\pi$
$479$	33.0000	1.50781	0.753904	+	0.656984i	$0.228168\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	0	0
$483$	−32.0000	−1.45605
$484$	0	0
$485$	12.0000	0.544892
$486$	0	0
$487$	28.0000	1.26880	0.634401	−	0.773004i	$-0.281247\pi$
$487$	28.0000	1.26880	0.634401	+	0.773004i	$0.281247\pi$
$488$	0	0
$489$	−17.0000	−0.768767
$490$	0	0
$491$	−17.0000	−0.767199	−0.383600	−	0.923499i	$-0.625316\pi$
$491$	−17.0000	−0.767199	−0.383600	+	0.923499i	$0.625316\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	48.0000	2.15309
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	0	0
$503$	5.00000	0.222939	0.111469	−	0.993768i	$-0.464444\pi$
$503$	5.00000	0.222939	0.111469	+	0.993768i	$0.464444\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	4.00000	0.177646
$508$	0	0
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	0	0
$513$	−20.0000	−0.883022
$514$	0	0
$515$	0	0
$516$	0	0
$517$	15.0000	0.659699
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	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
$525$	−16.0000	−0.698297
$526$	0	0
$527$	0	0
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	6.00000	0.259889
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	−18.0000	−0.776757
$538$	0	0
$539$	−27.0000	−1.16297
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	0	0
$543$	−1.00000	−0.0429141
$544$	0	0
$545$	−19.0000	−0.813871
$546$	0	0
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	0	0
$549$	16.0000	0.682863
$550$	0	0
$551$	4.00000	0.170406
$552$	0	0
$553$	28.0000	1.19068
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	0	0
$559$	−3.00000	−0.126886
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17.0000	0.716465	0.358232	−	0.933632i	$-0.383380\pi$
$563$	17.0000	0.716465	0.358232	+	0.933632i	$0.383380\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	0	0
$567$	−4.00000	−0.167984
$568$	0	0
$569$	−20.0000	−0.838444	−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000	−0.838444	−0.419222	+	0.907884i	$0.637697\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	0	0
$575$	32.0000	1.33449
$576$	0	0
$577$	−26.0000	−1.08239	−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000	−1.08239	−0.541197	+	0.840896i	$0.682029\pi$
$578$	0	0
$579$	16.0000	0.664937
$580$	0	0
$581$	48.0000	1.99138
$582$	0	0
$583$	3.00000	0.124247
$584$	0	0
$585$	−6.00000	−0.248069
$586$	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
$589$	20.0000	0.824086
$590$	0	0
$591$	18.0000	0.740421
$592$	0	0
$593$	23.0000	0.944497	0.472248	−	0.881466i	$-0.343443\pi$
$593$	23.0000	0.944497	0.472248	+	0.881466i	$0.343443\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	18.0000	0.736691
$598$	0	0
$599$	1.00000	0.0408589	0.0204294	−	0.999791i	$-0.493497\pi$
$599$	1.00000	0.0408589	0.0204294	+	0.999791i	$0.493497\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	2.00000	0.0813116
$606$	0	0
$607$	−19.0000	−0.771186	−0.385593	−	0.922669i	$-0.626003\pi$
$607$	−19.0000	−0.771186	−0.385593	+	0.922669i	$0.626003\pi$
$608$	0	0
$609$	−4.00000	−0.162088
$610$	0	0
$611$	15.0000	0.606835
$612$	0	0
$613$	−45.0000	−1.81753	−0.908766	−	0.417305i	$-0.862975\pi$
$613$	−45.0000	−1.81753	−0.908766	+	0.417305i	$0.862975\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	−7.00000	−0.281354	−0.140677	−	0.990056i	$-0.544928\pi$
$619$	−7.00000	−0.281354	−0.140677	+	0.990056i	$0.544928\pi$
$620$	0	0
$621$	−40.0000	−1.60514
$622$	0	0
$623$	−32.0000	−1.28205
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−2.00000	−0.0796187	−0.0398094	−	0.999207i	$-0.512675\pi$
$631$	−2.00000	−0.0796187	−0.0398094	+	0.999207i	$0.512675\pi$
$632$	0	0
$633$	19.0000	0.755182
$634$	0	0
$635$	8.00000	0.317470
$636$	0	0
$637$	−27.0000	−1.06978
$638$	0	0
$639$	24.0000	0.949425
$640$	0	0
$641$	−42.0000	−1.65890	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000	−1.65890	−0.829450	+	0.558581i	$0.811346\pi$
$642$	0	0
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	0	0
$645$	1.00000	0.0393750
$646$	0	0
$647$	42.0000	1.65119	0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000	1.65119	0.825595	+	0.564263i	$0.190840\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−20.0000	−0.783862
$652$	0	0
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	0	0
$657$	−12.0000	−0.468165
$658$	0	0
$659$	−25.0000	−0.973862	−0.486931	−	0.873441i	$-0.661884\pi$
$659$	−25.0000	−0.973862	−0.486931	+	0.873441i	$0.661884\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−16.0000	−0.620453
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	14.0000	0.541271
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	−31.0000	−1.19496	−0.597481	−	0.801883i	$-0.703832\pi$
$673$	−31.0000	−1.19496	−0.597481	+	0.801883i	$0.703832\pi$
$674$	0	0
$675$	−20.0000	−0.769800
$676$	0	0
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	48.0000	1.84207
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	46.0000	1.76014	0.880071	−	0.474843i	$-0.157495\pi$
$683$	46.0000	1.76014	0.880071	+	0.474843i	$0.157495\pi$
$684$	0	0
$685$	14.0000	0.534913
$686$	0	0
$687$	20.0000	0.763048
$688$	0	0
$689$	3.00000	0.114291
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	−24.0000	−0.911685
$694$	0	0
$695$	−6.00000	−0.227593
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−23.0000	−0.869940
$700$	0	0
$701$	5.00000	0.188847	0.0944237	−	0.995532i	$-0.469899\pi$
$701$	5.00000	0.188847	0.0944237	+	0.995532i	$0.469899\pi$
$702$	0	0
$703$	−16.0000	−0.603451
$704$	0	0
$705$	−5.00000	−0.188311
$706$	0	0
$707$	8.00000	0.300871
$708$	0	0
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	0	0
$711$	14.0000	0.525041
$712$	0	0
$713$	40.0000	1.49801
$714$	0	0
$715$	−9.00000	−0.336581
$716$	0	0
$717$	6.00000	0.224074
$718$	0	0
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−21.0000	−0.780998
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−42.0000	−1.55131	−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000	−1.55131	−0.775653	+	0.631160i	$0.782579\pi$
$734$	0	0
$735$	9.00000	0.331970
$736$	0	0
$737$	6.00000	0.221013
$738$	0	0
$739$	−29.0000	−1.06678	−0.533391	−	0.845869i	$-0.679083\pi$
$739$	−29.0000	−1.06678	−0.533391	+	0.845869i	$0.679083\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	0	0
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	11.0000	0.403009
$746$	0	0
$747$	24.0000	0.878114
$748$	0	0
$749$	−48.0000	−1.75388
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	0	0
$753$	13.0000	0.473746
$754$	0	0
$755$	−2.00000	−0.0727875
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	0	0
$759$	−24.0000	−0.871145
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	0	0
$763$	−76.0000	−2.75138
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12.0000	−0.433295
$768$	0	0
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	7.00000	0.252099
$772$	0	0
$773$	28.0000	1.00709	0.503545	−	0.863969i	$-0.332029\pi$
$773$	28.0000	1.00709	0.503545	+	0.863969i	$0.332029\pi$
$774$	0	0
$775$	20.0000	0.718421
$776$	0	0
$777$	16.0000	0.573997
$778$	0	0
$779$	8.00000	0.286630
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	−5.00000	−0.178685
$784$	0	0
$785$	−12.0000	−0.428298
$786$	0	0
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	0	0
$789$	21.0000	0.747620
$790$	0	0
$791$	−32.0000	−1.13779
$792$	0	0
$793$	24.0000	0.852265
$794$	0	0
$795$	−1.00000	−0.0354663
$796$	0	0
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−16.0000	−0.565332
$802$	0	0
$803$	−18.0000	−0.635206
$804$	0	0
$805$	−32.0000	−1.12785
$806$	0	0
$807$	20.0000	0.704033
$808$	0	0
$809$	−44.0000	−1.54696	−0.773479	−	0.633822i	$-0.781485\pi$
$809$	−44.0000	−1.54696	−0.773479	+	0.633822i	$0.781485\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	−25.0000	−0.876788
$814$	0	0
$815$	−17.0000	−0.595484
$816$	0	0
$817$	−4.00000	−0.139942
$818$	0	0
$819$	−24.0000	−0.838628
$820$	0	0
$821$	1.00000	0.0349002	0.0174501	−	0.999848i	$-0.494445\pi$
$821$	1.00000	0.0349002	0.0174501	+	0.999848i	$0.494445\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	37.0000	1.28662	0.643308	−	0.765607i	$-0.277561\pi$
$827$	37.0000	1.28662	0.643308	+	0.765607i	$0.277561\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	14.0000	0.485655
$832$	0	0
$833$	0	0
$834$	0	0
$835$	18.0000	0.622916
$836$	0	0
$837$	−25.0000	−0.864126
$838$	0	0
$839$	−9.00000	−0.310715	−0.155357	−	0.987858i	$-0.549653\pi$
$839$	−9.00000	−0.310715	−0.155357	+	0.987858i	$0.549653\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−27.0000	−0.929929
$844$	0	0
$845$	4.00000	0.137604
$846$	0	0
$847$	8.00000	0.274883
$848$	0	0
$849$	6.00000	0.205919
$850$	0	0
$851$	−32.0000	−1.09695
$852$	0	0
$853$	−54.0000	−1.84892	−0.924462	−	0.381273i	$-0.875486\pi$
$853$	−54.0000	−1.84892	−0.924462	+	0.381273i	$0.875486\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	5.00000	0.170797	0.0853984	−	0.996347i	$-0.472784\pi$
$857$	5.00000	0.170797	0.0853984	+	0.996347i	$0.472784\pi$
$858$	0	0
$859$	−45.0000	−1.53538	−0.767690	−	0.640821i	$-0.778594\pi$
$859$	−45.0000	−1.53538	−0.767690	+	0.640821i	$0.778594\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	−30.0000	−1.02121	−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000	−1.02121	−0.510606	+	0.859815i	$0.670579\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	21.0000	0.712376
$870$	0	0
$871$	6.00000	0.203302
$872$	0	0
$873$	24.0000	0.812277
$874$	0	0
$875$	−36.0000	−1.21702
$876$	0	0
$877$	−5.00000	−0.168838	−0.0844190	−	0.996430i	$-0.526903\pi$
$877$	−5.00000	−0.168838	−0.0844190	+	0.996430i	$0.526903\pi$
$878$	0	0
$879$	−16.0000	−0.539667
$880$	0	0
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	0	0
$883$	48.0000	1.61533	0.807664	−	0.589643i	$-0.200731\pi$
$883$	48.0000	1.61533	0.807664	+	0.589643i	$0.200731\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	39.0000	1.30949	0.654746	−	0.755849i	$-0.272776\pi$
$887$	39.0000	1.30949	0.654746	+	0.755849i	$0.272776\pi$
$888$	0	0
$889$	32.0000	1.07325
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	20.0000	0.669274
$894$	0	0
$895$	−18.0000	−0.601674
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	5.00000	0.166759
$900$	0	0
$901$	0	0
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	−1.00000	−0.0332411
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	4.00000	0.132672
$910$	0	0
$911$	9.00000	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	9.00000	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	0	0
$913$	36.0000	1.19143
$914$	0	0
$915$	−8.00000	−0.264472
$916$	0	0
$917$	16.0000	0.528367
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−5.00000	−0.164756
$922$	0	0
$923$	36.0000	1.18495
$924$	0	0
$925$	−16.0000	−0.526077
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	0	0
$933$	16.0000	0.523816
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	−29.0000	−0.946379
$940$	0	0
$941$	51.0000	1.66255	0.831276	−	0.555860i	$-0.187611\pi$
$941$	51.0000	1.66255	0.831276	+	0.555860i	$0.187611\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	0	0
$945$	20.0000	0.650600
$946$	0	0
$947$	45.0000	1.46230	0.731152	−	0.682215i	$-0.238983\pi$
$947$	45.0000	1.46230	0.731152	+	0.682215i	$0.238983\pi$
$948$	0	0
$949$	−18.0000	−0.584305
$950$	0	0
$951$	−22.0000	−0.713399
$952$	0	0
$953$	−33.0000	−1.06897	−0.534487	−	0.845176i	$-0.679495\pi$
$953$	−33.0000	−1.06897	−0.534487	+	0.845176i	$0.679495\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	0	0
$957$	−3.00000	−0.0969762
$958$	0	0
$959$	56.0000	1.80833
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	−24.0000	−0.773389
$964$	0	0
$965$	16.0000	0.515058
$966$	0	0
$967$	31.0000	0.996893	0.498446	−	0.866921i	$-0.333904\pi$
$967$	31.0000	0.996893	0.498446	+	0.866921i	$0.333904\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	0	0
$975$	−12.0000	−0.384308
$976$	0	0
$977$	−47.0000	−1.50366	−0.751832	−	0.659355i	$-0.770829\pi$
$977$	−47.0000	−1.50366	−0.751832	+	0.659355i	$0.770829\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	−38.0000	−1.21325
$982$	0	0
$983$	−33.0000	−1.05254	−0.526268	−	0.850319i	$-0.676409\pi$
$983$	−33.0000	−1.05254	−0.526268	+	0.850319i	$0.676409\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	−20.0000	−0.636607
$988$	0	0
$989$	−8.00000	−0.254385
$990$	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
$993$	−33.0000	−1.04722
$994$	0	0
$995$	18.0000	0.570638
$996$	0	0
$997$	34.0000	1.07679	0.538395	−	0.842692i	$-0.319031\pi$
$997$	34.0000	1.07679	0.538395	+	0.842692i	$0.319031\pi$
$998$	0	0
$999$	20.0000	0.632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3712.2.a.e.1.1	✓	1
4.3	odd	2		3712.2.a.j.1.1	yes	1
8.3	odd	2		3712.2.a.h.1.1	yes	1
8.5	even	2		3712.2.a.k.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3712.2.a.e.1.1	✓	1	1.1	even	1	trivial
3712.2.a.h.1.1	yes	1	8.3	odd	2
3712.2.a.j.1.1	yes	1	4.3	odd	2
3712.2.a.k.1.1	yes	1	8.5	even	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:	\( N \)	\(=\)	\( 3712 = 2^{7} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3712.a (trivial)

Introduction

L-functions

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