LMFDB - Embedded newform 7938.2.a.l.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
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)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{8} -3.00000 q^{10} -6.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} +7.00000 q^{19} +3.00000 q^{20} +6.00000 q^{22} +3.00000 q^{23} +4.00000 q^{25} +2.00000 q^{26} +6.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} +2.00000 q^{37} -7.00000 q^{38} -3.00000 q^{40} +2.00000 q^{43} -6.00000 q^{44} -3.00000 q^{46} -4.00000 q^{50} -2.00000 q^{52} +6.00000 q^{53} -18.0000 q^{55} -6.00000 q^{58} -5.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} -6.00000 q^{65} +8.00000 q^{67} -6.00000 q^{68} +3.00000 q^{71} -2.00000 q^{73} -2.00000 q^{74} +7.00000 q^{76} +5.00000 q^{79} +3.00000 q^{80} -12.0000 q^{83} -18.0000 q^{85} -2.00000 q^{86} +6.00000 q^{88} +3.00000 q^{92} +21.0000 q^{95} -2.00000 q^{97} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−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
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	6.00000	1.27920
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	2.00000	0.392232
$27$	0	0
$28$	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
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−7.00000	−1.13555
$39$	0	0
$40$	−3.00000	−0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	−3.00000	−0.442326
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	−4.00000	−0.565685
$51$	0	0
$52$	−2.00000	−0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−18.0000	−2.42712
$56$	0	0
$57$	0	0
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.00000	−0.744208
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	7.00000	0.802955
$77$	0	0
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	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
$85$	−18.0000	−1.95237
$86$	−2.00000	−0.215666
$87$	0	0
$88$	6.00000	0.639602
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	3.00000	0.312772
$93$	0	0
$94$	0	0
$95$	21.0000	2.15455
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	4.00000	0.400000
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	18.0000	1.71623
$111$	0	0
$112$	0	0
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	0	0
$115$	9.00000	0.839254
$116$	6.00000	0.557086
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	25.0000	2.27273
$122$	5.00000	0.452679
$123$	0	0
$124$	−2.00000	−0.179605
$125$	−3.00000	−0.268328
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	6.00000	0.526235
$131$	9.00000	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$131$	9.00000	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$132$	0	0
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	6.00000	0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	0	0
$142$	−3.00000	−0.251754
$143$	12.0000	1.00349
$144$	0	0
$145$	18.0000	1.49482
$146$	2.00000	0.165521
$147$	0	0
$148$	2.00000	0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	23.0000	1.87171	0.935857	−	0.352381i	$-0.114628\pi$
$151$	23.0000	1.87171	0.935857	+	0.352381i	$0.114628\pi$
$152$	−7.00000	−0.567775
$153$	0	0
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	−5.00000	−0.397779
$159$	0	0
$160$	−3.00000	−0.237171
$161$	0	0
$162$	0	0
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	0	0
$165$	0	0
$166$	12.0000	0.931381
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	18.0000	1.38054
$171$	0	0
$172$	2.00000	0.152499
$173$	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
$175$	0	0
$176$	−6.00000	−0.452267
$177$	0	0
$178$	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
$181$	25.0000	1.85824	0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000	1.85824	0.929118	+	0.369784i	$0.120568\pi$
$182$	0	0
$183$	0	0
$184$	−3.00000	−0.221163
$185$	6.00000	0.441129
$186$	0	0
$187$	36.0000	2.63258
$188$	0	0
$189$	0	0
$190$	−21.0000	−1.52350
$191$	−9.00000	−0.651217	−0.325609	−	0.945505i	$-0.605569\pi$
$191$	−9.00000	−0.651217	−0.325609	+	0.945505i	$0.605569\pi$
$192$	0	0
$193$	17.0000	1.22369	0.611843	−	0.790979i	$-0.290428\pi$
$193$	17.0000	1.22369	0.611843	+	0.790979i	$0.290428\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	9.00000	0.633238
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−10.0000	−0.696733
$207$	0	0
$208$	−2.00000	−0.138675
$209$	−42.0000	−2.90520
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	12.0000	0.820303
$215$	6.00000	0.409197
$216$	0	0
$217$	0	0
$218$	10.0000	0.677285
$219$	0	0
$220$	−18.0000	−1.21356
$221$	12.0000	0.807207
$222$	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
$227$	15.0000	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$228$	0	0
$229$	1.00000	0.0660819	0.0330409	−	0.999454i	$-0.489481\pi$
$229$	1.00000	0.0660819	0.0330409	+	0.999454i	$0.489481\pi$
$230$	−9.00000	−0.593442
$231$	0	0
$232$	−6.00000	−0.393919
$233$	9.00000	0.589610	0.294805	−	0.955557i	$-0.404745\pi$
$233$	9.00000	0.589610	0.294805	+	0.955557i	$0.404745\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	−25.0000	−1.60706
$243$	0	0
$244$	−5.00000	−0.320092
$245$	0	0
$246$	0	0
$247$	−14.0000	−0.890799
$248$	2.00000	0.127000
$249$	0	0
$250$	3.00000	0.189737
$251$	−3.00000	−0.189358	−0.0946792	−	0.995508i	$-0.530183\pi$
$251$	−3.00000	−0.189358	−0.0946792	+	0.995508i	$0.530183\pi$
$252$	0	0
$253$	−18.0000	−1.13165
$254$	−17.0000	−1.06667
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	0	0
$260$	−6.00000	−0.372104
$261$	0	0
$262$	−9.00000	−0.556022
$263$	−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
$265$	18.0000	1.10573
$266$	0	0
$267$	0	0
$268$	8.00000	0.488678
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	0	0
$271$	28.0000	1.70088	0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000	1.70088	0.850439	+	0.526073i	$0.176336\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−24.0000	−1.44725
$276$	0	0
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	5.00000	0.299880
$279$	0	0
$280$	0	0
$281$	−27.0000	−1.61068	−0.805342	−	0.592810i	$-0.798019\pi$
$281$	−27.0000	−1.61068	−0.805342	+	0.592810i	$0.798019\pi$
$282$	0	0
$283$	19.0000	1.12943	0.564716	−	0.825285i	$-0.308986\pi$
$283$	19.0000	1.12943	0.564716	+	0.825285i	$0.308986\pi$
$284$	3.00000	0.178017
$285$	0	0
$286$	−12.0000	−0.709575
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	−18.0000	−1.05700
$291$	0	0
$292$	−2.00000	−0.117041
$293$	3.00000	0.175262	0.0876309	−	0.996153i	$-0.472070\pi$
$293$	3.00000	0.175262	0.0876309	+	0.996153i	$0.472070\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	6.00000	0.347571
$299$	−6.00000	−0.346989
$300$	0	0
$301$	0	0
$302$	−23.0000	−1.32350
$303$	0	0
$304$	7.00000	0.401478
$305$	−15.0000	−0.858898
$306$	0	0
$307$	25.0000	1.42683	0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000	1.42683	0.713413	+	0.700744i	$0.247149\pi$
$308$	0	0
$309$	0	0
$310$	6.00000	0.340777
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−13.0000	−0.733632
$315$	0	0
$316$	5.00000	0.281272
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	−36.0000	−2.01561
$320$	3.00000	0.167705
$321$	0	0
$322$	0	0
$323$	−42.0000	−2.33694
$324$	0	0
$325$	−8.00000	−0.443760
$326$	−2.00000	−0.110770
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	0	0
$335$	24.0000	1.31126
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	−18.0000	−0.976187
$341$	12.0000	0.649836
$342$	0	0
$343$	0	0
$344$	−2.00000	−0.107833
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	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	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	9.00000	0.477670
$356$	0	0
$357$	0	0
$358$	−18.0000	−0.951330
$359$	−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
$361$	30.0000	1.57895
$362$	−25.0000	−1.31397
$363$	0	0
$364$	0	0
$365$	−6.00000	−0.314054
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	−6.00000	−0.311925
$371$	0	0
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−36.0000	−1.86152
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	21.0000	1.07728
$381$	0	0
$382$	9.00000	0.460480
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	0	0
$386$	−17.0000	−0.865277
$387$	0	0
$388$	−2.00000	−0.101535
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	0	0
$394$	−18.0000	−0.906827
$395$	15.0000	0.754732
$396$	0	0
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	14.0000	0.701757
$399$	0	0
$400$	4.00000	0.200000
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	−9.00000	−0.447767
$405$	0	0
$406$	0	0
$407$	−12.0000	−0.594818
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	0	0
$412$	10.0000	0.492665
$413$	0	0
$414$	0	0
$415$	−36.0000	−1.76717
$416$	2.00000	0.0980581
$417$	0	0
$418$	42.0000	2.05429
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−24.0000	−1.16417
$426$	0	0
$427$	0	0
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−6.00000	−0.289346
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	21.0000	1.00457
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	18.0000	0.858116
$441$	0	0
$442$	−12.0000	−0.570782
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	0	0
$445$	0	0
$446$	−28.0000	−1.32584
$447$	0	0
$448$	0	0
$449$	33.0000	1.55737	0.778683	−	0.627417i	$-0.215888\pi$
$449$	33.0000	1.55737	0.778683	+	0.627417i	$0.215888\pi$
$450$	0	0
$451$	0	0
$452$	15.0000	0.705541
$453$	0	0
$454$	−15.0000	−0.703985
$455$	0	0
$456$	0	0
$457$	29.0000	1.35656	0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000	1.35656	0.678281	+	0.734802i	$0.262725\pi$
$458$	−1.00000	−0.0467269
$459$	0	0
$460$	9.00000	0.419627
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	0	0
$463$	−13.0000	−0.604161	−0.302081	−	0.953282i	$-0.597681\pi$
$463$	−13.0000	−0.604161	−0.302081	+	0.953282i	$0.597681\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−9.00000	−0.416917
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−12.0000	−0.551761
$474$	0	0
$475$	28.0000	1.28473
$476$	0	0
$477$	0	0
$478$	15.0000	0.686084
$479$	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
$481$	−4.00000	−0.182384
$482$	8.00000	0.364390
$483$	0	0
$484$	25.0000	1.13636
$485$	−6.00000	−0.272446
$486$	0	0
$487$	29.0000	1.31412	0.657058	−	0.753840i	$-0.271801\pi$
$487$	29.0000	1.31412	0.657058	+	0.753840i	$0.271801\pi$
$488$	5.00000	0.226339
$489$	0	0
$490$	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
$493$	−36.0000	−1.62136
$494$	14.0000	0.629890
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	3.00000	0.133897
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	−27.0000	−1.20148
$506$	18.0000	0.800198
$507$	0	0
$508$	17.0000	0.754253
$509$	−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
$514$	18.0000	0.793946
$515$	30.0000	1.32196
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	6.00000	0.263117
$521$	−24.0000	−1.05146	−0.525730	−	0.850652i	$-0.676208\pi$
$521$	−24.0000	−1.05146	−0.525730	+	0.850652i	$0.676208\pi$
$522$	0	0
$523$	13.0000	0.568450	0.284225	−	0.958758i	$-0.408264\pi$
$523$	13.0000	0.568450	0.284225	+	0.958758i	$0.408264\pi$
$524$	9.00000	0.393167
$525$	0	0
$526$	21.0000	0.915644
$527$	12.0000	0.522728
$528$	0	0
$529$	−14.0000	−0.608696
$530$	−18.0000	−0.781870
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−36.0000	−1.55642
$536$	−8.00000	−0.345547
$537$	0	0
$538$	−9.00000	−0.388018
$539$	0	0
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	−28.0000	−1.20270
$543$	0	0
$544$	6.00000	0.257248
$545$	−30.0000	−1.28506
$546$	0	0
$547$	32.0000	1.36822	0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000	1.36822	0.684111	+	0.729378i	$0.260191\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	24.0000	1.02336
$551$	42.0000	1.78926
$552$	0	0
$553$	0	0
$554$	16.0000	0.679775
$555$	0	0
$556$	−5.00000	−0.212047
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	27.0000	1.13893
$563$	33.0000	1.39078	0.695392	−	0.718631i	$-0.255231\pi$
$563$	33.0000	1.39078	0.695392	+	0.718631i	$0.255231\pi$
$564$	0	0
$565$	45.0000	1.89316
$566$	−19.0000	−0.798630
$567$	0	0
$568$	−3.00000	−0.125877
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	12.0000	0.501745
$573$	0	0
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	4.00000	0.166522	0.0832611	−	0.996528i	$-0.473466\pi$
$577$	4.00000	0.166522	0.0832611	+	0.996528i	$0.473466\pi$
$578$	−19.0000	−0.790296
$579$	0	0
$580$	18.0000	0.747409
$581$	0	0
$582$	0	0
$583$	−36.0000	−1.49097
$584$	2.00000	0.0827606
$585$	0	0
$586$	−3.00000	−0.123929
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	0	0
$589$	−14.0000	−0.576860
$590$	0	0
$591$	0	0
$592$	2.00000	0.0821995
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	6.00000	0.245358
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	0	0
$603$	0	0
$604$	23.0000	0.935857
$605$	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	$0.352679\pi$
$608$	−7.00000	−0.283887
$609$	0	0
$610$	15.0000	0.607332
$611$	0	0
$612$	0	0
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	−25.0000	−1.00892
$615$	0	0
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	7.00000	0.281354	0.140677	−	0.990056i	$-0.455072\pi$
$619$	7.00000	0.281354	0.140677	+	0.990056i	$0.455072\pi$
$620$	−6.00000	−0.240966
$621$	0	0
$622$	12.0000	0.481156
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	13.0000	0.518756
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−7.00000	−0.278666	−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000	−0.278666	−0.139333	+	0.990246i	$0.544496\pi$
$632$	−5.00000	−0.198889
$633$	0	0
$634$	−18.0000	−0.714871
$635$	51.0000	2.02387
$636$	0	0
$637$	0	0
$638$	36.0000	1.42525
$639$	0	0
$640$	−3.00000	−0.118585
$641$	27.0000	1.06644	0.533218	−	0.845978i	$-0.320983\pi$
$641$	27.0000	1.06644	0.533218	+	0.845978i	$0.320983\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	0	0
$646$	42.0000	1.65247
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	8.00000	0.313786
$651$	0	0
$652$	2.00000	0.0783260
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	0	0
$655$	27.0000	1.05498
$656$	0	0
$657$	0	0
$658$	0	0
$659$	42.0000	1.63609	0.818044	−	0.575156i	$-0.195059\pi$
$659$	42.0000	1.63609	0.818044	+	0.575156i	$0.195059\pi$
$660$	0	0
$661$	−5.00000	−0.194477	−0.0972387	−	0.995261i	$-0.531001\pi$
$661$	−5.00000	−0.194477	−0.0972387	+	0.995261i	$0.531001\pi$
$662$	−26.0000	−1.01052
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	18.0000	0.696963
$668$	0	0
$669$	0	0
$670$	−24.0000	−0.927201
$671$	30.0000	1.15814
$672$	0	0
$673$	−37.0000	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$673$	−37.0000	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−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
$679$	0	0
$680$	18.0000	0.690268
$681$	0	0
$682$	−12.0000	−0.459504
$683$	−6.00000	−0.229584	−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000	−0.229584	−0.114792	+	0.993390i	$0.536620\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	0	0
$688$	2.00000	0.0762493
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−47.0000	−1.78796	−0.893982	−	0.448103i	$-0.852100\pi$
$691$	−47.0000	−1.78796	−0.893982	+	0.448103i	$0.852100\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	24.0000	0.911028
$695$	−15.0000	−0.568982
$696$	0	0
$697$	0	0
$698$	26.0000	0.984115
$699$	0	0
$700$	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
$703$	14.0000	0.528020
$704$	−6.00000	−0.226134
$705$	0	0
$706$	−18.0000	−0.677439
$707$	0	0
$708$	0	0
$709$	−52.0000	−1.95290	−0.976450	−	0.215742i	$-0.930783\pi$
$709$	−52.0000	−1.95290	−0.976450	+	0.215742i	$0.930783\pi$
$710$	−9.00000	−0.337764
$711$	0	0
$712$	0	0
$713$	−6.00000	−0.224702
$714$	0	0
$715$	36.0000	1.34632
$716$	18.0000	0.672692
$717$	0	0
$718$	3.00000	0.111959
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	0	0
$722$	−30.0000	−1.11648
$723$	0	0
$724$	25.0000	0.929118
$725$	24.0000	0.891338
$726$	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
$730$	6.00000	0.222070
$731$	−12.0000	−0.443836
$732$	0	0
$733$	−29.0000	−1.07114	−0.535570	−	0.844491i	$-0.679903\pi$
$733$	−29.0000	−1.07114	−0.535570	+	0.844491i	$0.679903\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	−3.00000	−0.110581
$737$	−48.0000	−1.76810
$738$	0	0
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	6.00000	0.220564
$741$	0	0
$742$	0	0
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	−14.0000	−0.512576
$747$	0	0
$748$	36.0000	1.31629
$749$	0	0
$750$	0	0
$751$	−31.0000	−1.13121	−0.565603	−	0.824678i	$-0.691357\pi$
$751$	−31.0000	−1.13121	−0.565603	+	0.824678i	$0.691357\pi$
$752$	0	0
$753$	0	0
$754$	12.0000	0.437014
$755$	69.0000	2.51117
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	−2.00000	−0.0726433
$759$	0	0
$760$	−21.0000	−0.761750
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	0	0
$764$	−9.00000	−0.325609
$765$	0	0
$766$	−18.0000	−0.650366
$767$	0	0
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	0	0
$772$	17.0000	0.611843
$773$	51.0000	1.83434	0.917171	−	0.398493i	$-0.130467\pi$
$773$	51.0000	1.83434	0.917171	+	0.398493i	$0.130467\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	2.00000	0.0717958
$777$	0	0
$778$	−24.0000	−0.860442
$779$	0	0
$780$	0	0
$781$	−18.0000	−0.644091
$782$	18.0000	0.643679
$783$	0	0
$784$	0	0
$785$	39.0000	1.39197
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	−15.0000	−0.533676
$791$	0	0
$792$	0	0
$793$	10.0000	0.355110
$794$	26.0000	0.922705
$795$	0	0
$796$	−14.0000	−0.496217
$797$	−3.00000	−0.106265	−0.0531327	−	0.998587i	$-0.516921\pi$
$797$	−3.00000	−0.106265	−0.0531327	+	0.998587i	$0.516921\pi$
$798$	0	0
$799$	0	0
$800$	−4.00000	−0.141421
$801$	0	0
$802$	−3.00000	−0.105934
$803$	12.0000	0.423471
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	9.00000	0.316619
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	0	0
$814$	12.0000	0.420600
$815$	6.00000	0.210171
$816$	0	0
$817$	14.0000	0.489798
$818$	32.0000	1.11885
$819$	0	0
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	−10.0000	−0.348367
$825$	0	0
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	36.0000	1.24958
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−42.0000	−1.45260
$837$	0	0
$838$	15.0000	0.518166
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	10.0000	0.344623
$843$	0	0
$844$	8.00000	0.275371
$845$	−27.0000	−0.928828
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	24.0000	0.823193
$851$	6.00000	0.205677
$852$	0	0
$853$	−35.0000	−1.19838	−0.599189	−	0.800608i	$-0.704510\pi$
$853$	−35.0000	−1.19838	−0.599189	+	0.800608i	$0.704510\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	6.00000	0.204598
$861$	0	0
$862$	−12.0000	−0.408722
$863$	9.00000	0.306364	0.153182	−	0.988198i	$-0.451048\pi$
$863$	9.00000	0.306364	0.153182	+	0.988198i	$0.451048\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	14.0000	0.475739
$867$	0	0
$868$	0	0
$869$	−30.0000	−1.01768
$870$	0	0
$871$	−16.0000	−0.542139
$872$	10.0000	0.338643
$873$	0	0
$874$	−21.0000	−0.710336
$875$	0	0
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	8.00000	0.269987
$879$	0	0
$880$	−18.0000	−0.606780
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	−34.0000	−1.14419	−0.572096	−	0.820187i	$-0.693869\pi$
$883$	−34.0000	−1.14419	−0.572096	+	0.820187i	$0.693869\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−18.0000	−0.604722
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	28.0000	0.937509
$893$	0	0
$894$	0	0
$895$	54.0000	1.80502
$896$	0	0
$897$	0	0
$898$	−33.0000	−1.10122
$899$	−12.0000	−0.400222
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	0	0
$904$	−15.0000	−0.498893
$905$	75.0000	2.49308
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	15.0000	0.497792
$909$	0	0
$910$	0	0
$911$	15.0000	0.496972	0.248486	−	0.968635i	$-0.420067\pi$
$911$	15.0000	0.496972	0.248486	+	0.968635i	$0.420067\pi$
$912$	0	0
$913$	72.0000	2.38285
$914$	−29.0000	−0.959235
$915$	0	0
$916$	1.00000	0.0330409
$917$	0	0
$918$	0	0
$919$	11.0000	0.362857	0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000	0.362857	0.181428	+	0.983404i	$0.441928\pi$
$920$	−9.00000	−0.296721
$921$	0	0
$922$	−33.0000	−1.08680
$923$	−6.00000	−0.197492
$924$	0	0
$925$	8.00000	0.263038
$926$	13.0000	0.427207
$927$	0	0
$928$	−6.00000	−0.196960
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	9.00000	0.294805
$933$	0	0
$934$	12.0000	0.392652
$935$	108.000	3.53198
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−21.0000	−0.684580	−0.342290	−	0.939594i	$-0.611203\pi$
$941$	−21.0000	−0.684580	−0.342290	+	0.939594i	$0.611203\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	12.0000	0.390154
$947$	−24.0000	−0.779895	−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000	−0.779895	−0.389948	+	0.920837i	$0.627507\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	−28.0000	−0.908440
$951$	0	0
$952$	0	0
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	0	0
$955$	−27.0000	−0.873699
$956$	−15.0000	−0.485135
$957$	0	0
$958$	−6.00000	−0.193851
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	4.00000	0.128965
$963$	0	0
$964$	−8.00000	−0.257663
$965$	51.0000	1.64175
$966$	0	0
$967$	17.0000	0.546683	0.273342	−	0.961917i	$-0.411871\pi$
$967$	17.0000	0.546683	0.273342	+	0.961917i	$0.411871\pi$
$968$	−25.0000	−0.803530
$969$	0	0
$970$	6.00000	0.192648
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	0	0
$973$	0	0
$974$	−29.0000	−0.929220
$975$	0	0
$976$	−5.00000	−0.160046
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−18.0000	−0.574403
$983$	−18.0000	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$983$	−18.0000	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$984$	0	0
$985$	54.0000	1.72058
$986$	36.0000	1.14647
$987$	0	0
$988$	−14.0000	−0.445399
$989$	6.00000	0.190789
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	2.00000	0.0635001
$993$	0	0
$994$	0	0
$995$	−42.0000	−1.33149
$996$	0	0
$997$	55.0000	1.74187	0.870934	−	0.491400i	$-0.163515\pi$
$997$	55.0000	1.74187	0.870934	+	0.491400i	$0.163515\pi$
$998$	−32.0000	−1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.l.1.1		1
3.2	odd	2		7938.2.a.u.1.1		1
7.6	odd	2		1134.2.a.a.1.1		1
9.2	odd	6		2646.2.f.c.1765.1		2
9.4	even	3		882.2.f.h.295.1		2
9.5	odd	6		2646.2.f.c.883.1		2
9.7	even	3		882.2.f.h.589.1		2
21.20	even	2		1134.2.a.h.1.1		1
28.27	even	2		9072.2.a.c.1.1		1
63.2	odd	6		2646.2.h.a.361.1		2
63.4	even	3		882.2.h.f.79.1		2
63.5	even	6		2646.2.e.f.2125.1		2
63.11	odd	6		2646.2.e.j.1549.1		2
63.13	odd	6		126.2.f.a.43.1	✓	2
63.16	even	3		882.2.h.f.67.1		2
63.20	even	6		378.2.f.a.253.1		2
63.23	odd	6		2646.2.e.j.2125.1		2
63.25	even	3		882.2.e.d.373.1		2
63.31	odd	6		882.2.h.j.79.1		2
63.32	odd	6		2646.2.h.a.667.1		2
63.34	odd	6		126.2.f.a.85.1	yes	2
63.38	even	6		2646.2.e.f.1549.1		2
63.40	odd	6		882.2.e.b.655.1		2
63.41	even	6		378.2.f.a.127.1		2
63.47	even	6		2646.2.h.e.361.1		2
63.52	odd	6		882.2.e.b.373.1		2
63.58	even	3		882.2.e.d.655.1		2
63.59	even	6		2646.2.h.e.667.1		2
63.61	odd	6		882.2.h.j.67.1		2
84.83	odd	2		9072.2.a.w.1.1		1
252.83	odd	6		3024.2.r.a.1009.1		2
252.139	even	6		1008.2.r.d.673.1		2
252.167	odd	6		3024.2.r.a.2017.1		2
252.223	even	6		1008.2.r.d.337.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.f.a.43.1	✓	2	63.13	odd	6
126.2.f.a.85.1	yes	2	63.34	odd	6
378.2.f.a.127.1		2	63.41	even	6
378.2.f.a.253.1		2	63.20	even	6
882.2.e.b.373.1		2	63.52	odd	6
882.2.e.b.655.1		2	63.40	odd	6
882.2.e.d.373.1		2	63.25	even	3
882.2.e.d.655.1		2	63.58	even	3
882.2.f.h.295.1		2	9.4	even	3
882.2.f.h.589.1		2	9.7	even	3
882.2.h.f.67.1		2	63.16	even	3
882.2.h.f.79.1		2	63.4	even	3
882.2.h.j.67.1		2	63.61	odd	6
882.2.h.j.79.1		2	63.31	odd	6
1008.2.r.d.337.1		2	252.223	even	6
1008.2.r.d.673.1		2	252.139	even	6
1134.2.a.a.1.1		1	7.6	odd	2
1134.2.a.h.1.1		1	21.20	even	2
2646.2.e.f.1549.1		2	63.38	even	6
2646.2.e.f.2125.1		2	63.5	even	6
2646.2.e.j.1549.1		2	63.11	odd	6
2646.2.e.j.2125.1		2	63.23	odd	6
2646.2.f.c.883.1		2	9.5	odd	6
2646.2.f.c.1765.1		2	9.2	odd	6
2646.2.h.a.361.1		2	63.2	odd	6
2646.2.h.a.667.1		2	63.32	odd	6
2646.2.h.e.361.1		2	63.47	even	6
2646.2.h.e.667.1		2	63.59	even	6
3024.2.r.a.1009.1		2	252.83	odd	6
3024.2.r.a.2017.1		2	252.167	odd	6
7938.2.a.l.1.1		1	1.1	even	1	trivial
7938.2.a.u.1.1		1	3.2	odd	2
9072.2.a.c.1.1		1	28.27	even	2
9072.2.a.w.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)

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