LMFDB - Embedded newform 7938.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7938,2,Mod(1,7938)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7938, 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("7938.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7938 = 2 \cdot 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7938.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:	$63.3852491245$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1134)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7938.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{8} +3.00000 q^{10} -6.00000 q^{11} +1.00000 q^{13} +1.00000 q^{16} -3.00000 q^{17} -2.00000 q^{19} +3.00000 q^{20} -6.00000 q^{22} -6.00000 q^{23} +4.00000 q^{25} +1.00000 q^{26} -9.00000 q^{29} +10.0000 q^{31} +1.00000 q^{32} -3.00000 q^{34} -7.00000 q^{37} -2.00000 q^{38} +3.00000 q^{40} -6.00000 q^{41} -4.00000 q^{43} -6.00000 q^{44} -6.00000 q^{46} -6.00000 q^{47} +4.00000 q^{50} +1.00000 q^{52} -6.00000 q^{53} -18.0000 q^{55} -9.00000 q^{58} +6.00000 q^{59} -11.0000 q^{61} +10.0000 q^{62} +1.00000 q^{64} +3.00000 q^{65} +2.00000 q^{67} -3.00000 q^{68} +12.0000 q^{71} +7.00000 q^{73} -7.00000 q^{74} -2.00000 q^{76} +2.00000 q^{79} +3.00000 q^{80} -6.00000 q^{82} -6.00000 q^{83} -9.00000 q^{85} -4.00000 q^{86} -6.00000 q^{88} -3.00000 q^{89} -6.00000 q^{92} -6.00000 q^{94} -6.00000 q^{95} -14.0000 q^{97} +O(q^{100})$

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	3.00000	0.948683
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−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	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−6.00000	−1.27920
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	1.00000	0.196116
$27$	0	0
$28$	0	0
$29$	−9.00000	−1.67126	−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000	−1.67126	−0.835629	+	0.549294i	$0.814897\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	0	0
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	3.00000	0.474342
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	0	0
$50$	4.00000	0.565685
$51$	0	0
$52$	1.00000	0.138675
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−18.0000	−2.42712
$56$	0	0
$57$	0	0
$58$	−9.00000	−1.18176
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	10.0000	1.27000
$63$	0	0
$64$	1.00000	0.125000
$65$	3.00000	0.372104
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	−2.00000	−0.229416
$77$	0	0
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	−9.00000	−0.976187
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−6.00000	−0.639602
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	0	0
$94$	−6.00000	−0.618853
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	0	0
$100$	4.00000	0.400000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−6.00000	−0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	5.00000	0.478913	0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000	0.478913	0.239457	+	0.970907i	$0.423031\pi$
$110$	−18.0000	−1.71623
$111$	0	0
$112$	0	0
$113$	−15.0000	−1.41108	−0.705541	−	0.708669i	$-0.749296\pi$
$113$	−15.0000	−1.41108	−0.705541	+	0.708669i	$0.749296\pi$
$114$	0	0
$115$	−18.0000	−1.67851
$116$	−9.00000	−0.835629
$117$	0	0
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	−11.0000	−0.995893
$123$	0	0
$124$	10.0000	0.898027
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.00000	0.263117
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	2.00000	0.172774
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−3.00000	−0.256307	−0.128154	−	0.991754i	$-0.540905\pi$
$137$	−3.00000	−0.256307	−0.128154	+	0.991754i	$0.540905\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	0	0
$142$	12.0000	1.00702
$143$	−6.00000	−0.501745
$144$	0	0
$145$	−27.0000	−2.24223
$146$	7.00000	0.579324
$147$	0	0
$148$	−7.00000	−0.575396
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	0	0
$155$	30.0000	2.40966
$156$	0	0
$157$	−23.0000	−1.83560	−0.917800	−	0.397043i	$-0.870036\pi$
$157$	−23.0000	−1.83560	−0.917800	+	0.397043i	$0.870036\pi$
$158$	2.00000	0.159111
$159$	0	0
$160$	3.00000	0.237171
$161$	0	0
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−9.00000	−0.690268
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	0	0
$175$	0	0
$176$	−6.00000	−0.452267
$177$	0	0
$178$	−3.00000	−0.224860
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	0	0
$184$	−6.00000	−0.442326
$185$	−21.0000	−1.54395
$186$	0	0
$187$	18.0000	1.31629
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−6.00000	−0.435286
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	0	0
$197$	−9.00000	−0.641223	−0.320612	−	0.947211i	$-0.603888\pi$
$197$	−9.00000	−0.641223	−0.320612	+	0.947211i	$0.603888\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	18.0000	1.26648
$203$	0	0
$204$	0	0
$205$	−18.0000	−1.25717
$206$	16.0000	1.11477
$207$	0	0
$208$	1.00000	0.0693375
$209$	12.0000	0.830057
$210$	0	0
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	0	0
$215$	−12.0000	−0.818393
$216$	0	0
$217$	0	0
$218$	5.00000	0.338643
$219$	0	0
$220$	−18.0000	−1.21356
$221$	−3.00000	−0.201802
$222$	0	0
$223$	28.0000	1.87502	0.937509	−	0.347960i	$-0.113126\pi$
$223$	28.0000	1.87502	0.937509	+	0.347960i	$0.113126\pi$
$224$	0	0
$225$	0	0
$226$	−15.0000	−0.997785
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	25.0000	1.65205	0.826023	−	0.563636i	$-0.190598\pi$
$229$	25.0000	1.65205	0.826023	+	0.563636i	$0.190598\pi$
$230$	−18.0000	−1.18688
$231$	0	0
$232$	−9.00000	−0.590879
$233$	21.0000	1.37576	0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000	1.37576	0.687878	+	0.725826i	$0.258542\pi$
$234$	0	0
$235$	−18.0000	−1.17419
$236$	6.00000	0.390567
$237$	0	0
$238$	0	0
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	0	0
$241$	19.0000	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$241$	19.0000	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$242$	25.0000	1.60706
$243$	0	0
$244$	−11.0000	−0.704203
$245$	0	0
$246$	0	0
$247$	−2.00000	−0.127257
$248$	10.0000	0.635001
$249$	0	0
$250$	−3.00000	−0.189737
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	36.0000	2.26330
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	0	0
$259$	0	0
$260$	3.00000	0.186052
$261$	0	0
$262$	0	0
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	−18.0000	−1.10573
$266$	0	0
$267$	0	0
$268$	2.00000	0.122169
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−3.00000	−0.181237
$275$	−24.0000	−1.44725
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	−14.0000	−0.839664
$279$	0	0
$280$	0	0
$281$	21.0000	1.25275	0.626377	−	0.779520i	$-0.284537\pi$
$281$	21.0000	1.25275	0.626377	+	0.779520i	$0.284537\pi$
$282$	0	0
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	−6.00000	−0.354787
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−27.0000	−1.58549
$291$	0	0
$292$	7.00000	0.409644
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	0	0
$295$	18.0000	1.04800
$296$	−7.00000	−0.406867
$297$	0	0
$298$	3.00000	0.173785
$299$	−6.00000	−0.346989
$300$	0	0
$301$	0	0
$302$	2.00000	0.115087
$303$	0	0
$304$	−2.00000	−0.114708
$305$	−33.0000	−1.88957
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	0	0
$310$	30.0000	1.70389
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	−23.0000	−1.29797
$315$	0	0
$316$	2.00000	0.112509
$317$	−21.0000	−1.17948	−0.589739	−	0.807594i	$-0.700769\pi$
$317$	−21.0000	−1.17948	−0.589739	+	0.807594i	$0.700769\pi$
$318$	0	0
$319$	54.0000	3.02342
$320$	3.00000	0.167705
$321$	0	0
$322$	0	0
$323$	6.00000	0.333849
$324$	0	0
$325$	4.00000	0.221880
$326$	−16.0000	−0.886158
$327$	0	0
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	−6.00000	−0.328305
$335$	6.00000	0.327815
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	−9.00000	−0.488094
$341$	−60.0000	−3.24918
$342$	0	0
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−9.00000	−0.483843
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	0	0
$352$	−6.00000	−0.319801
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	36.0000	1.91068
$356$	−3.00000	−0.159000
$357$	0	0
$358$	6.00000	0.317110
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−2.00000	−0.105118
$363$	0	0
$364$	0	0
$365$	21.0000	1.09919
$366$	0	0
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	−21.0000	−1.09174
$371$	0	0
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	18.0000	0.930758
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−9.00000	−0.463524
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	−6.00000	−0.307794
$381$	0	0
$382$	6.00000	0.306987
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	11.0000	0.559885
$387$	0	0
$388$	−14.0000	−0.710742
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	0	0
$394$	−9.00000	−0.453413
$395$	6.00000	0.301893
$396$	0	0
$397$	13.0000	0.652451	0.326226	−	0.945292i	$-0.394223\pi$
$397$	13.0000	0.652451	0.326226	+	0.945292i	$0.394223\pi$
$398$	10.0000	0.501255
$399$	0	0
$400$	4.00000	0.200000
$401$	21.0000	1.04869	0.524345	−	0.851506i	$-0.324310\pi$
$401$	21.0000	1.04869	0.524345	+	0.851506i	$0.324310\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	18.0000	0.895533
$405$	0	0
$406$	0	0
$407$	42.0000	2.08186
$408$	0	0
$409$	−5.00000	−0.247234	−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000	−0.247234	−0.123617	+	0.992330i	$0.539449\pi$
$410$	−18.0000	−0.888957
$411$	0	0
$412$	16.0000	0.788263
$413$	0	0
$414$	0	0
$415$	−18.0000	−0.883585
$416$	1.00000	0.0490290
$417$	0	0
$418$	12.0000	0.586939
$419$	−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
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	2.00000	0.0973585
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−12.0000	−0.582086
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	−12.0000	−0.578691
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	−17.0000	−0.816968	−0.408484	−	0.912766i	$-0.633942\pi$
$433$	−17.0000	−0.816968	−0.408484	+	0.912766i	$0.633942\pi$
$434$	0	0
$435$	0	0
$436$	5.00000	0.239457
$437$	12.0000	0.574038
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	−18.0000	−0.858116
$441$	0	0
$442$	−3.00000	−0.142695
$443$	−30.0000	−1.42534	−0.712672	−	0.701498i	$-0.752515\pi$
$443$	−30.0000	−1.42534	−0.712672	+	0.701498i	$0.752515\pi$
$444$	0	0
$445$	−9.00000	−0.426641
$446$	28.0000	1.32584
$447$	0	0
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	−15.0000	−0.705541
$453$	0	0
$454$	24.0000	1.12638
$455$	0	0
$456$	0	0
$457$	23.0000	1.07589	0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000	1.07589	0.537947	+	0.842978i	$0.319200\pi$
$458$	25.0000	1.16817
$459$	0	0
$460$	−18.0000	−0.839254
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−22.0000	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	21.0000	0.972806
$467$	30.0000	1.38823	0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000	1.38823	0.694117	+	0.719862i	$0.255795\pi$
$468$	0	0
$469$	0	0
$470$	−18.0000	−0.830278
$471$	0	0
$472$	6.00000	0.276172
$473$	24.0000	1.10352
$474$	0	0
$475$	−8.00000	−0.367065
$476$	0	0
$477$	0	0
$478$	18.0000	0.823301
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−7.00000	−0.319173
$482$	19.0000	0.865426
$483$	0	0
$484$	25.0000	1.13636
$485$	−42.0000	−1.90712
$486$	0	0
$487$	−34.0000	−1.54069	−0.770344	−	0.637629i	$-0.779915\pi$
$487$	−34.0000	−1.54069	−0.770344	+	0.637629i	$0.779915\pi$
$488$	−11.0000	−0.497947
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	27.0000	1.21602
$494$	−2.00000	−0.0899843
$495$	0	0
$496$	10.0000	0.449013
$497$	0	0
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	6.00000	0.267793
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	54.0000	2.40297
$506$	36.0000	1.60040
$507$	0	0
$508$	−4.00000	−0.177471
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−27.0000	−1.19092
$515$	48.0000	2.11513
$516$	0	0
$517$	36.0000	1.58328
$518$	0	0
$519$	0	0
$520$	3.00000	0.131559
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	−44.0000	−1.92399	−0.961993	−	0.273075i	$-0.911959\pi$
$523$	−44.0000	−1.92399	−0.961993	+	0.273075i	$0.911959\pi$
$524$	0	0
$525$	0	0
$526$	6.00000	0.261612
$527$	−30.0000	−1.30682
$528$	0	0
$529$	13.0000	0.565217
$530$	−18.0000	−0.781870
$531$	0	0
$532$	0	0
$533$	−6.00000	−0.259889
$534$	0	0
$535$	0	0
$536$	2.00000	0.0863868
$537$	0	0
$538$	3.00000	0.129339
$539$	0	0
$540$	0	0
$541$	−43.0000	−1.84871	−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000	−1.84871	−0.924357	+	0.381528i	$0.875398\pi$
$542$	−14.0000	−0.601351
$543$	0	0
$544$	−3.00000	−0.128624
$545$	15.0000	0.642529
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	−3.00000	−0.128154
$549$	0	0
$550$	−24.0000	−1.02336
$551$	18.0000	0.766826
$552$	0	0
$553$	0	0
$554$	14.0000	0.594803
$555$	0	0
$556$	−14.0000	−0.593732
$557$	−9.00000	−0.381342	−0.190671	−	0.981654i	$-0.561066\pi$
$557$	−9.00000	−0.381342	−0.190671	+	0.981654i	$0.561066\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	21.0000	0.885832
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	−45.0000	−1.89316
$566$	16.0000	0.672530
$567$	0	0
$568$	12.0000	0.503509
$569$	−3.00000	−0.125767	−0.0628833	−	0.998021i	$-0.520030\pi$
$569$	−3.00000	−0.125767	−0.0628833	+	0.998021i	$0.520030\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	−6.00000	−0.250873
$573$	0	0
$574$	0	0
$575$	−24.0000	−1.00087
$576$	0	0
$577$	−17.0000	−0.707719	−0.353860	−	0.935299i	$-0.615131\pi$
$577$	−17.0000	−0.707719	−0.353860	+	0.935299i	$0.615131\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	−27.0000	−1.12111
$581$	0	0
$582$	0	0
$583$	36.0000	1.49097
$584$	7.00000	0.289662
$585$	0	0
$586$	−21.0000	−0.867502
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	−20.0000	−0.824086
$590$	18.0000	0.741048
$591$	0	0
$592$	−7.00000	−0.287698
$593$	−3.00000	−0.123195	−0.0615976	−	0.998101i	$-0.519620\pi$
$593$	−3.00000	−0.123195	−0.0615976	+	0.998101i	$0.519620\pi$
$594$	0	0
$595$	0	0
$596$	3.00000	0.122885
$597$	0	0
$598$	−6.00000	−0.245358
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	7.00000	0.285536	0.142768	−	0.989756i	$-0.454400\pi$
$601$	7.00000	0.285536	0.142768	+	0.989756i	$0.454400\pi$
$602$	0	0
$603$	0	0
$604$	2.00000	0.0813788
$605$	75.0000	3.04918
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	−33.0000	−1.33613
$611$	−6.00000	−0.242734
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	0	0
$617$	−15.0000	−0.603877	−0.301939	−	0.953327i	$-0.597634\pi$
$617$	−15.0000	−0.603877	−0.301939	+	0.953327i	$0.597634\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	30.0000	1.20483
$621$	0	0
$622$	30.0000	1.20289
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	19.0000	0.759393
$627$	0	0
$628$	−23.0000	−0.917800
$629$	21.0000	0.837325
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	2.00000	0.0795557
$633$	0	0
$634$	−21.0000	−0.834017
$635$	−12.0000	−0.476205
$636$	0	0
$637$	0	0
$638$	54.0000	2.13788
$639$	0	0
$640$	3.00000	0.118585
$641$	9.00000	0.355479	0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000	0.355479	0.177739	+	0.984078i	$0.443122\pi$
$642$	0	0
$643$	22.0000	0.867595	0.433798	−	0.901010i	$-0.357173\pi$
$643$	22.0000	0.867595	0.433798	+	0.901010i	$0.357173\pi$
$644$	0	0
$645$	0	0
$646$	6.00000	0.236067
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	4.00000	0.156893
$651$	0	0
$652$	−16.0000	−0.626608
$653$	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
$656$	−6.00000	−0.234261
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	13.0000	0.505641	0.252821	−	0.967513i	$-0.418642\pi$
$661$	13.0000	0.505641	0.252821	+	0.967513i	$0.418642\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	−6.00000	−0.232845
$665$	0	0
$666$	0	0
$667$	54.0000	2.09089
$668$	−6.00000	−0.232147
$669$	0	0
$670$	6.00000	0.231800
$671$	66.0000	2.54790
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−12.0000	−0.461538
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	0	0
$680$	−9.00000	−0.345134
$681$	0	0
$682$	−60.0000	−2.29752
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	0	0
$685$	−9.00000	−0.343872
$686$	0	0
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−6.00000	−0.228582
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	−9.00000	−0.342129
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−42.0000	−1.59315
$696$	0	0
$697$	18.0000	0.681799
$698$	−26.0000	−0.984115
$699$	0	0
$700$	0	0
$701$	27.0000	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$701$	27.0000	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$702$	0	0
$703$	14.0000	0.528020
$704$	−6.00000	−0.226134
$705$	0	0
$706$	−18.0000	−0.677439
$707$	0	0
$708$	0	0
$709$	5.00000	0.187779	0.0938895	−	0.995583i	$-0.470070\pi$
$709$	5.00000	0.187779	0.0938895	+	0.995583i	$0.470070\pi$
$710$	36.0000	1.35106
$711$	0	0
$712$	−3.00000	−0.112430
$713$	−60.0000	−2.24702
$714$	0	0
$715$	−18.0000	−0.673162
$716$	6.00000	0.224231
$717$	0	0
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	0	0
$722$	−15.0000	−0.558242
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	−36.0000	−1.33701
$726$	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
$729$	0	0
$730$	21.0000	0.777245
$731$	12.0000	0.443836
$732$	0	0
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	−6.00000	−0.221163
$737$	−12.0000	−0.442026
$738$	0	0
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	−21.0000	−0.771975
$741$	0	0
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	9.00000	0.329734
$746$	−22.0000	−0.805477
$747$	0	0
$748$	18.0000	0.658145
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	−9.00000	−0.327761
$755$	6.00000	0.218362
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	2.00000	0.0726433
$759$	0	0
$760$	−6.00000	−0.217643
$761$	33.0000	1.19625	0.598125	−	0.801403i	$-0.295913\pi$
$761$	33.0000	1.19625	0.598125	+	0.801403i	$0.295913\pi$
$762$	0	0
$763$	0	0
$764$	6.00000	0.217072
$765$	0	0
$766$	0	0
$767$	6.00000	0.216647
$768$	0	0
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	0	0
$772$	11.0000	0.395899
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	0	0
$775$	40.0000	1.43684
$776$	−14.0000	−0.502571
$777$	0	0
$778$	−30.0000	−1.07555
$779$	12.0000	0.429945
$780$	0	0
$781$	−72.0000	−2.57636
$782$	18.0000	0.643679
$783$	0	0
$784$	0	0
$785$	−69.0000	−2.46272
$786$	0	0
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	−9.00000	−0.320612
$789$	0	0
$790$	6.00000	0.213470
$791$	0	0
$792$	0	0
$793$	−11.0000	−0.390621
$794$	13.0000	0.461353
$795$	0	0
$796$	10.0000	0.354441
$797$	−9.00000	−0.318796	−0.159398	−	0.987214i	$-0.550955\pi$
$797$	−9.00000	−0.318796	−0.159398	+	0.987214i	$0.550955\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	4.00000	0.141421
$801$	0	0
$802$	21.0000	0.741536
$803$	−42.0000	−1.48215
$804$	0	0
$805$	0	0
$806$	10.0000	0.352235
$807$	0	0
$808$	18.0000	0.633238
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	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
$813$	0	0
$814$	42.0000	1.47210
$815$	−48.0000	−1.68137
$816$	0	0
$817$	8.00000	0.279885
$818$	−5.00000	−0.174821
$819$	0	0
$820$	−18.0000	−0.628587
$821$	−45.0000	−1.57051	−0.785255	−	0.619172i	$-0.787468\pi$
$821$	−45.0000	−1.57051	−0.785255	+	0.619172i	$0.787468\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	−18.0000	−0.624789
$831$	0	0
$832$	1.00000	0.0346688
$833$	0	0
$834$	0	0
$835$	−18.0000	−0.622916
$836$	12.0000	0.415029
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	17.0000	0.585859
$843$	0	0
$844$	2.00000	0.0688428
$845$	−36.0000	−1.23844
$846$	0	0
$847$	0	0
$848$	−6.00000	−0.206041
$849$	0	0
$850$	−12.0000	−0.411597
$851$	42.0000	1.43974
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.00000	0.307434	0.153717	−	0.988115i	$-0.450876\pi$
$857$	9.00000	0.307434	0.153717	+	0.988115i	$0.450876\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	−12.0000	−0.409197
$861$	0	0
$862$	−36.0000	−1.22616
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	0	0
$865$	−27.0000	−0.918028
$866$	−17.0000	−0.577684
$867$	0	0
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	2.00000	0.0677674
$872$	5.00000	0.169321
$873$	0	0
$874$	12.0000	0.405906
$875$	0	0
$876$	0	0
$877$	41.0000	1.38447	0.692236	−	0.721671i	$-0.256626\pi$
$877$	41.0000	1.38447	0.692236	+	0.721671i	$0.256626\pi$
$878$	28.0000	0.944954
$879$	0	0
$880$	−18.0000	−0.606780
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	−30.0000	−1.00787
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	0	0
$889$	0	0
$890$	−9.00000	−0.301681
$891$	0	0
$892$	28.0000	0.937509
$893$	12.0000	0.401565
$894$	0	0
$895$	18.0000	0.601674
$896$	0	0
$897$	0	0
$898$	−30.0000	−1.00111
$899$	−90.0000	−3.00167
$900$	0	0
$901$	18.0000	0.599667
$902$	36.0000	1.19867
$903$	0	0
$904$	−15.0000	−0.498893
$905$	−6.00000	−0.199447
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	24.0000	0.796468
$909$	0	0
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	36.0000	1.19143
$914$	23.0000	0.760772
$915$	0	0
$916$	25.0000	0.826023
$917$	0	0
$918$	0	0
$919$	2.00000	0.0659739	0.0329870	−	0.999456i	$-0.489498\pi$
$919$	2.00000	0.0659739	0.0329870	+	0.999456i	$0.489498\pi$
$920$	−18.0000	−0.593442
$921$	0	0
$922$	−6.00000	−0.197599
$923$	12.0000	0.394985
$924$	0	0
$925$	−28.0000	−0.920634
$926$	−22.0000	−0.722965
$927$	0	0
$928$	−9.00000	−0.295439
$929$	21.0000	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$929$	21.0000	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$930$	0	0
$931$	0	0
$932$	21.0000	0.687878
$933$	0	0
$934$	30.0000	0.981630
$935$	54.0000	1.76599
$936$	0	0
$937$	7.00000	0.228680	0.114340	−	0.993442i	$-0.463525\pi$
$937$	7.00000	0.228680	0.114340	+	0.993442i	$0.463525\pi$
$938$	0	0
$939$	0	0
$940$	−18.0000	−0.587095
$941$	3.00000	0.0977972	0.0488986	−	0.998804i	$-0.484429\pi$
$941$	3.00000	0.0977972	0.0488986	+	0.998804i	$0.484429\pi$
$942$	0	0
$943$	36.0000	1.17232
$944$	6.00000	0.195283
$945$	0	0
$946$	24.0000	0.780307
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	0	0
$949$	7.00000	0.227230
$950$	−8.00000	−0.259554
$951$	0	0
$952$	0	0
$953$	−3.00000	−0.0971795	−0.0485898	−	0.998819i	$-0.515473\pi$
$953$	−3.00000	−0.0971795	−0.0485898	+	0.998819i	$0.515473\pi$
$954$	0	0
$955$	18.0000	0.582466
$956$	18.0000	0.582162
$957$	0	0
$958$	6.00000	0.193851
$959$	0	0
$960$	0	0
$961$	69.0000	2.22581
$962$	−7.00000	−0.225689
$963$	0	0
$964$	19.0000	0.611949
$965$	33.0000	1.06231
$966$	0	0
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	−42.0000	−1.34854
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	0	0
$973$	0	0
$974$	−34.0000	−1.08943
$975$	0	0
$976$	−11.0000	−0.352101
$977$	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
$979$	18.0000	0.575282
$980$	0	0
$981$	0	0
$982$	−12.0000	−0.382935
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	−27.0000	−0.860292
$986$	27.0000	0.859855
$987$	0	0
$988$	−2.00000	−0.0636285
$989$	24.0000	0.763156
$990$	0	0
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	10.0000	0.317500
$993$	0	0
$994$	0	0
$995$	30.0000	0.951064
$996$	0	0
$997$	37.0000	1.17180	0.585901	−	0.810383i	$-0.300741\pi$
$997$	37.0000	1.17180	0.585901	+	0.810383i	$0.300741\pi$
$998$	14.0000	0.443162
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.bc.1.1		1
3.2	odd	2		7938.2.a.d.1.1		1
7.6	odd	2		1134.2.a.e.1.1	yes	1
21.20	even	2		1134.2.a.d.1.1	✓	1
28.27	even	2		9072.2.a.b.1.1		1
63.13	odd	6		1134.2.f.h.379.1		2
63.20	even	6		1134.2.f.i.757.1		2
63.34	odd	6		1134.2.f.h.757.1		2
63.41	even	6		1134.2.f.i.379.1		2
84.83	odd	2		9072.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1134.2.a.d.1.1	✓	1	21.20	even	2
1134.2.a.e.1.1	yes	1	7.6	odd	2
1134.2.f.h.379.1		2	63.13	odd	6
1134.2.f.h.757.1		2	63.34	odd	6
1134.2.f.i.379.1		2	63.41	even	6
1134.2.f.i.757.1		2	63.20	even	6
7938.2.a.d.1.1		1	3.2	odd	2
7938.2.a.bc.1.1		1	1.1	even	1	trivial
9072.2.a.b.1.1		1	28.27	even	2
9072.2.a.v.1.1		1	84.83	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:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more