LMFDB - Embedded newform 7938.2.a.cp.1.3

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:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 882)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.517638$ of defining polynomial
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.03528 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.03528 q^{5} +1.00000 q^{8} +1.03528 q^{10} -0.267949 q^{11} -1.79315 q^{13} +1.00000 q^{16} +6.83083 q^{17} -4.38134 q^{19} +1.03528 q^{20} -0.267949 q^{22} +5.46410 q^{23} -3.92820 q^{25} -1.79315 q^{26} -4.00000 q^{29} -6.69213 q^{31} +1.00000 q^{32} +6.83083 q^{34} +7.46410 q^{37} -4.38134 q^{38} +1.03528 q^{40} +8.62398 q^{41} +0.267949 q^{43} -0.267949 q^{44} +5.46410 q^{46} -0.757875 q^{47} -3.92820 q^{50} -1.79315 q^{52} +10.9282 q^{53} -0.277401 q^{55} -4.00000 q^{58} +1.27551 q^{59} +12.6264 q^{61} -6.69213 q^{62} +1.00000 q^{64} -1.85641 q^{65} +12.4641 q^{67} +6.83083 q^{68} +9.46410 q^{71} -5.41662 q^{73} +7.46410 q^{74} -4.38134 q^{76} +8.92820 q^{79} +1.03528 q^{80} +8.62398 q^{82} -6.59059 q^{83} +7.07180 q^{85} +0.267949 q^{86} -0.267949 q^{88} -7.07107 q^{89} +5.46410 q^{92} -0.757875 q^{94} -4.53590 q^{95} +18.1445 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 8 q^{11} + 4 q^{16} - 8 q^{22} + 8 q^{23} + 12 q^{25} - 16 q^{29} + 4 q^{32} + 16 q^{37} + 8 q^{43} - 8 q^{44} + 8 q^{46} + 12 q^{50} + 16 q^{53} - 16 q^{58} + 4 q^{64} + 48 q^{65} + 36 q^{67} + 24 q^{71} + 16 q^{74} + 8 q^{79} + 56 q^{85} + 8 q^{86} - 8 q^{88} + 8 q^{92} - 32 q^{95}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 8 * q^11 + 4 * q^16 - 8 * q^22 + 8 * q^23 + 12 * q^25 - 16 * q^29 + 4 * q^32 + 16 * q^37 + 8 * q^43 - 8 * q^44 + 8 * q^46 + 12 * q^50 + 16 * q^53 - 16 * q^58 + 4 * q^64 + 48 * q^65 + 36 * q^67 + 24 * q^71 + 16 * q^74 + 8 * q^79 + 56 * q^85 + 8 * q^86 - 8 * q^88 + 8 * q^92 - 32 * q^95

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
$5$	1.03528	0.462990	0.231495	−	0.972836i	$-0.425638\pi$
$5$	1.03528	0.462990	0.231495	+	0.972836i	$0.425638\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	1.03528	0.327383
$11$	−0.267949	−0.0807897	−0.0403949	−	0.999184i	$-0.512862\pi$
$11$	−0.267949	−0.0807897	−0.0403949	+	0.999184i	$0.512862\pi$
$12$	0	0
$13$	−1.79315	−0.497331	−0.248665	−	0.968589i	$-0.579992\pi$
$13$	−1.79315	−0.497331	−0.248665	+	0.968589i	$0.579992\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.83083	1.65672	0.828360	−	0.560196i	$-0.189274\pi$
$17$	6.83083	1.65672	0.828360	+	0.560196i	$0.189274\pi$
$18$	0	0
$19$	−4.38134	−1.00515	−0.502574	−	0.864534i	$-0.667614\pi$
$19$	−4.38134	−1.00515	−0.502574	+	0.864534i	$0.667614\pi$
$20$	1.03528	0.231495
$21$	0	0
$22$	−0.267949	−0.0571270
$23$	5.46410	1.13934	0.569672	−	0.821872i	$-0.307070\pi$
$23$	5.46410	1.13934	0.569672	+	0.821872i	$0.307070\pi$
$24$	0	0
$25$	−3.92820	−0.785641
$26$	−1.79315	−0.351666
$27$	0	0
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−6.69213	−1.20194	−0.600971	−	0.799271i	$-0.705219\pi$
$31$	−6.69213	−1.20194	−0.600971	+	0.799271i	$0.705219\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.83083	1.17148
$35$	0	0
$36$	0	0
$37$	7.46410	1.22709	0.613545	−	0.789659i	$-0.289743\pi$
$37$	7.46410	1.22709	0.613545	+	0.789659i	$0.289743\pi$
$38$	−4.38134	−0.710747
$39$	0	0
$40$	1.03528	0.163692
$41$	8.62398	1.34684	0.673420	−	0.739260i	$-0.264825\pi$
$41$	8.62398	1.34684	0.673420	+	0.739260i	$0.264825\pi$
$42$	0	0
$43$	0.267949	0.0408619	0.0204309	−	0.999791i	$-0.493496\pi$
$43$	0.267949	0.0408619	0.0204309	+	0.999791i	$0.493496\pi$
$44$	−0.267949	−0.0403949
$45$	0	0
$46$	5.46410	0.805638
$47$	−0.757875	−0.110547	−0.0552737	−	0.998471i	$-0.517603\pi$
$47$	−0.757875	−0.110547	−0.0552737	+	0.998471i	$0.517603\pi$
$48$	0	0
$49$	0	0
$50$	−3.92820	−0.555532
$51$	0	0
$52$	−1.79315	−0.248665
$53$	10.9282	1.50110	0.750552	−	0.660811i	$-0.229788\pi$
$53$	10.9282	1.50110	0.750552	+	0.660811i	$0.229788\pi$
$54$	0	0
$55$	−0.277401	−0.0374048
$56$	0	0
$57$	0	0
$58$	−4.00000	−0.525226
$59$	1.27551	0.166058	0.0830288	−	0.996547i	$-0.473541\pi$
$59$	1.27551	0.166058	0.0830288	+	0.996547i	$0.473541\pi$
$60$	0	0
$61$	12.6264	1.61664	0.808322	−	0.588741i	$-0.200376\pi$
$61$	12.6264	1.61664	0.808322	+	0.588741i	$0.200376\pi$
$62$	−6.69213	−0.849901
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.85641	−0.230259
$66$	0	0
$67$	12.4641	1.52273	0.761366	−	0.648322i	$-0.224529\pi$
$67$	12.4641	1.52273	0.761366	+	0.648322i	$0.224529\pi$
$68$	6.83083	0.828360
$69$	0	0
$70$	0	0
$71$	9.46410	1.12318	0.561591	−	0.827415i	$-0.310189\pi$
$71$	9.46410	1.12318	0.561591	+	0.827415i	$0.310189\pi$
$72$	0	0
$73$	−5.41662	−0.633967	−0.316984	−	0.948431i	$-0.602670\pi$
$73$	−5.41662	−0.633967	−0.316984	+	0.948431i	$0.602670\pi$
$74$	7.46410	0.867684
$75$	0	0
$76$	−4.38134	−0.502574
$77$	0	0
$78$	0	0
$79$	8.92820	1.00450	0.502251	−	0.864722i	$-0.332505\pi$
$79$	8.92820	1.00450	0.502251	+	0.864722i	$0.332505\pi$
$80$	1.03528	0.115747
$81$	0	0
$82$	8.62398	0.952360
$83$	−6.59059	−0.723412	−0.361706	−	0.932292i	$-0.617806\pi$
$83$	−6.59059	−0.723412	−0.361706	+	0.932292i	$0.617806\pi$
$84$	0	0
$85$	7.07180	0.767044
$86$	0.267949	0.0288937
$87$	0	0
$88$	−0.267949	−0.0285635
$89$	−7.07107	−0.749532	−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107	−0.749532	−0.374766	+	0.927119i	$0.622277\pi$
$90$	0	0
$91$	0	0
$92$	5.46410	0.569672
$93$	0	0
$94$	−0.757875	−0.0781688
$95$	−4.53590	−0.465373
$96$	0	0
$97$	18.1445	1.84230	0.921149	−	0.389209i	$-0.127252\pi$
$97$	18.1445	1.84230	0.921149	+	0.389209i	$0.127252\pi$
$98$	0	0
$99$	0	0
$100$	−3.92820	−0.392820
$101$	4.89898	0.487467	0.243733	−	0.969842i	$-0.421628\pi$
$101$	4.89898	0.487467	0.243733	+	0.969842i	$0.421628\pi$
$102$	0	0
$103$	−12.3490	−1.21678	−0.608391	−	0.793638i	$-0.708185\pi$
$103$	−12.3490	−1.21678	−0.608391	+	0.793638i	$0.708185\pi$
$104$	−1.79315	−0.175833
$105$	0	0
$106$	10.9282	1.06144
$107$	−17.3923	−1.68138	−0.840689	−	0.541519i	$-0.817850\pi$
$107$	−17.3923	−1.68138	−0.840689	+	0.541519i	$0.817850\pi$
$108$	0	0
$109$	4.92820	0.472036	0.236018	−	0.971749i	$-0.424158\pi$
$109$	4.92820	0.472036	0.236018	+	0.971749i	$0.424158\pi$
$110$	−0.277401	−0.0264492
$111$	0	0
$112$	0	0
$113$	−6.92820	−0.651751	−0.325875	−	0.945413i	$-0.605659\pi$
$113$	−6.92820	−0.651751	−0.325875	+	0.945413i	$0.605659\pi$
$114$	0	0
$115$	5.65685	0.527504
$116$	−4.00000	−0.371391
$117$	0	0
$118$	1.27551	0.117420
$119$	0	0
$120$	0	0
$121$	−10.9282	−0.993473
$122$	12.6264	1.14314
$123$	0	0
$124$	−6.69213	−0.600971
$125$	−9.24316	−0.826733
$126$	0	0
$127$	13.4641	1.19475	0.597373	−	0.801964i	$-0.296211\pi$
$127$	13.4641	1.19475	0.597373	+	0.801964i	$0.296211\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.85641	−0.162818
$131$	−10.9348	−0.955375	−0.477688	−	0.878530i	$-0.658525\pi$
$131$	−10.9348	−0.955375	−0.477688	+	0.878530i	$0.658525\pi$
$132$	0	0
$133$	0	0
$134$	12.4641	1.07673
$135$	0	0
$136$	6.83083	0.585739
$137$	8.66025	0.739895	0.369948	−	0.929053i	$-0.379376\pi$
$137$	8.66025	0.739895	0.369948	+	0.929053i	$0.379376\pi$
$138$	0	0
$139$	−0.795040	−0.0674344	−0.0337172	−	0.999431i	$-0.510735\pi$
$139$	−0.795040	−0.0674344	−0.0337172	+	0.999431i	$0.510735\pi$
$140$	0	0
$141$	0	0
$142$	9.46410	0.794210
$143$	0.480473	0.0401792
$144$	0	0
$145$	−4.14110	−0.343900
$146$	−5.41662	−0.448282
$147$	0	0
$148$	7.46410	0.613545
$149$	22.9282	1.87835	0.939176	−	0.343437i	$-0.111591\pi$
$149$	22.9282	1.87835	0.939176	+	0.343437i	$0.111591\pi$
$150$	0	0
$151$	18.3923	1.49674	0.748372	−	0.663279i	$-0.230836\pi$
$151$	18.3923	1.49674	0.748372	+	0.663279i	$0.230836\pi$
$152$	−4.38134	−0.355374
$153$	0	0
$154$	0	0
$155$	−6.92820	−0.556487
$156$	0	0
$157$	−9.52056	−0.759823	−0.379912	−	0.925023i	$-0.624046\pi$
$157$	−9.52056	−0.759823	−0.379912	+	0.925023i	$0.624046\pi$
$158$	8.92820	0.710290
$159$	0	0
$160$	1.03528	0.0818458
$161$	0	0
$162$	0	0
$163$	−13.3205	−1.04334	−0.521671	−	0.853147i	$-0.674691\pi$
$163$	−13.3205	−1.04334	−0.521671	+	0.853147i	$0.674691\pi$
$164$	8.62398	0.673420
$165$	0	0
$166$	−6.59059	−0.511529
$167$	−1.51575	−0.117292	−0.0586461	−	0.998279i	$-0.518678\pi$
$167$	−1.51575	−0.117292	−0.0586461	+	0.998279i	$0.518678\pi$
$168$	0	0
$169$	−9.78461	−0.752662
$170$	7.07180	0.542382
$171$	0	0
$172$	0.267949	0.0204309
$173$	−6.69213	−0.508793	−0.254397	−	0.967100i	$-0.581877\pi$
$173$	−6.69213	−0.508793	−0.254397	+	0.967100i	$0.581877\pi$
$174$	0	0
$175$	0	0
$176$	−0.267949	−0.0201974
$177$	0	0
$178$	−7.07107	−0.529999
$179$	5.07180	0.379084	0.189542	−	0.981873i	$-0.439300\pi$
$179$	5.07180	0.379084	0.189542	+	0.981873i	$0.439300\pi$
$180$	0	0
$181$	16.9706	1.26141	0.630706	−	0.776022i	$-0.282765\pi$
$181$	16.9706	1.26141	0.630706	+	0.776022i	$0.282765\pi$
$182$	0	0
$183$	0	0
$184$	5.46410	0.402819
$185$	7.72741	0.568130
$186$	0	0
$187$	−1.83032	−0.133846
$188$	−0.757875	−0.0552737
$189$	0	0
$190$	−4.53590	−0.329069
$191$	14.9282	1.08017	0.540083	−	0.841611i	$-0.318393\pi$
$191$	14.9282	1.08017	0.540083	+	0.841611i	$0.318393\pi$
$192$	0	0
$193$	15.0526	1.08351	0.541753	−	0.840537i	$-0.317761\pi$
$193$	15.0526	1.08351	0.541753	+	0.840537i	$0.317761\pi$
$194$	18.1445	1.30270
$195$	0	0
$196$	0	0
$197$	−16.9282	−1.20608	−0.603042	−	0.797709i	$-0.706045\pi$
$197$	−16.9282	−1.20608	−0.603042	+	0.797709i	$0.706045\pi$
$198$	0	0
$199$	−26.2880	−1.86351	−0.931755	−	0.363087i	$-0.881723\pi$
$199$	−26.2880	−1.86351	−0.931755	+	0.363087i	$0.881723\pi$
$200$	−3.92820	−0.277766
$201$	0	0
$202$	4.89898	0.344691
$203$	0	0
$204$	0	0
$205$	8.92820	0.623573
$206$	−12.3490	−0.860395
$207$	0	0
$208$	−1.79315	−0.124333
$209$	1.17398	0.0812057
$210$	0	0
$211$	18.9282	1.30307	0.651536	−	0.758618i	$-0.274125\pi$
$211$	18.9282	1.30307	0.651536	+	0.758618i	$0.274125\pi$
$212$	10.9282	0.750552
$213$	0	0
$214$	−17.3923	−1.18891
$215$	0.277401	0.0189186
$216$	0	0
$217$	0	0
$218$	4.92820	0.333780
$219$	0	0
$220$	−0.277401	−0.0187024
$221$	−12.2487	−0.823937
$222$	0	0
$223$	7.17260	0.480313	0.240157	−	0.970734i	$-0.422801\pi$
$223$	7.17260	0.480313	0.240157	+	0.970734i	$0.422801\pi$
$224$	0	0
$225$	0	0
$226$	−6.92820	−0.460857
$227$	27.6651	1.83620	0.918098	−	0.396352i	$-0.129724\pi$
$227$	27.6651	1.83620	0.918098	+	0.396352i	$0.129724\pi$
$228$	0	0
$229$	−0.480473	−0.0317506	−0.0158753	−	0.999874i	$-0.505053\pi$
$229$	−0.480473	−0.0317506	−0.0158753	+	0.999874i	$0.505053\pi$
$230$	5.65685	0.373002
$231$	0	0
$232$	−4.00000	−0.262613
$233$	0.124356	0.00814681	0.00407340	−	0.999992i	$-0.498703\pi$
$233$	0.124356	0.00814681	0.00407340	+	0.999992i	$0.498703\pi$
$234$	0	0
$235$	−0.784610	−0.0511823
$236$	1.27551	0.0830288
$237$	0	0
$238$	0	0
$239$	0.928203	0.0600405	0.0300202	−	0.999549i	$-0.490443\pi$
$239$	0.928203	0.0600405	0.0300202	+	0.999549i	$0.490443\pi$
$240$	0	0
$241$	−6.27603	−0.404275	−0.202137	−	0.979357i	$-0.564789\pi$
$241$	−6.27603	−0.404275	−0.202137	+	0.979357i	$0.564789\pi$
$242$	−10.9282	−0.702492
$243$	0	0
$244$	12.6264	0.808322
$245$	0	0
$246$	0	0
$247$	7.85641	0.499891
$248$	−6.69213	−0.424951
$249$	0	0
$250$	−9.24316	−0.584589
$251$	−0.795040	−0.0501824	−0.0250912	−	0.999685i	$-0.507988\pi$
$251$	−0.795040	−0.0501824	−0.0250912	+	0.999685i	$0.507988\pi$
$252$	0	0
$253$	−1.46410	−0.0920473
$254$	13.4641	0.844813
$255$	0	0
$256$	1.00000	0.0625000
$257$	−10.0382	−0.626165	−0.313083	−	0.949726i	$-0.601362\pi$
$257$	−10.0382	−0.626165	−0.313083	+	0.949726i	$0.601362\pi$
$258$	0	0
$259$	0	0
$260$	−1.85641	−0.115129
$261$	0	0
$262$	−10.9348	−0.675552
$263$	−8.53590	−0.526346	−0.263173	−	0.964749i	$-0.584769\pi$
$263$	−8.53590	−0.526346	−0.263173	+	0.964749i	$0.584769\pi$
$264$	0	0
$265$	11.3137	0.694996
$266$	0	0
$267$	0	0
$268$	12.4641	0.761366
$269$	5.65685	0.344904	0.172452	−	0.985018i	$-0.444831\pi$
$269$	5.65685	0.344904	0.172452	+	0.985018i	$0.444831\pi$
$270$	0	0
$271$	−13.1069	−0.796185	−0.398093	−	0.917345i	$-0.630328\pi$
$271$	−13.1069	−0.796185	−0.398093	+	0.917345i	$0.630328\pi$
$272$	6.83083	0.414180
$273$	0	0
$274$	8.66025	0.523185
$275$	1.05256	0.0634717
$276$	0	0
$277$	31.4641	1.89049	0.945247	−	0.326355i	$-0.105820\pi$
$277$	31.4641	1.89049	0.945247	+	0.326355i	$0.105820\pi$
$278$	−0.795040	−0.0476833
$279$	0	0
$280$	0	0
$281$	17.8564	1.06522	0.532612	−	0.846360i	$-0.321211\pi$
$281$	17.8564	1.06522	0.532612	+	0.846360i	$0.321211\pi$
$282$	0	0
$283$	15.0759	0.896168	0.448084	−	0.893992i	$-0.352107\pi$
$283$	15.0759	0.896168	0.448084	+	0.893992i	$0.352107\pi$
$284$	9.46410	0.561591
$285$	0	0
$286$	0.480473	0.0284110
$287$	0	0
$288$	0	0
$289$	29.6603	1.74472
$290$	−4.14110	−0.243174
$291$	0	0
$292$	−5.41662	−0.316984
$293$	9.24316	0.539991	0.269995	−	0.962862i	$-0.412978\pi$
$293$	9.24316	0.539991	0.269995	+	0.962862i	$0.412978\pi$
$294$	0	0
$295$	1.32051	0.0768830
$296$	7.46410	0.433842
$297$	0	0
$298$	22.9282	1.32820
$299$	−9.79796	−0.566631
$300$	0	0
$301$	0	0
$302$	18.3923	1.05836
$303$	0	0
$304$	−4.38134	−0.251287
$305$	13.0718	0.748489
$306$	0	0
$307$	−1.17398	−0.0670024	−0.0335012	−	0.999439i	$-0.510666\pi$
$307$	−1.17398	−0.0670024	−0.0335012	+	0.999439i	$0.510666\pi$
$308$	0	0
$309$	0	0
$310$	−6.92820	−0.393496
$311$	7.45001	0.422451	0.211226	−	0.977437i	$-0.432255\pi$
$311$	7.45001	0.422451	0.211226	+	0.977437i	$0.432255\pi$
$312$	0	0
$313$	−6.27603	−0.354742	−0.177371	−	0.984144i	$-0.556759\pi$
$313$	−6.27603	−0.354742	−0.177371	+	0.984144i	$0.556759\pi$
$314$	−9.52056	−0.537276
$315$	0	0
$316$	8.92820	0.502251
$317$	−26.0000	−1.46031	−0.730153	−	0.683284i	$-0.760551\pi$
$317$	−26.0000	−1.46031	−0.730153	+	0.683284i	$0.760551\pi$
$318$	0	0
$319$	1.07180	0.0600091
$320$	1.03528	0.0578737
$321$	0	0
$322$	0	0
$323$	−29.9282	−1.66525
$324$	0	0
$325$	7.04386	0.390723
$326$	−13.3205	−0.737755
$327$	0	0
$328$	8.62398	0.476180
$329$	0	0
$330$	0	0
$331$	−11.4641	−0.630124	−0.315062	−	0.949071i	$-0.602025\pi$
$331$	−11.4641	−0.630124	−0.315062	+	0.949071i	$0.602025\pi$
$332$	−6.59059	−0.361706
$333$	0	0
$334$	−1.51575	−0.0829381
$335$	12.9038	0.705009
$336$	0	0
$337$	−7.00000	−0.381314	−0.190657	−	0.981657i	$-0.561062\pi$
$337$	−7.00000	−0.381314	−0.190657	+	0.981657i	$0.561062\pi$
$338$	−9.78461	−0.532213
$339$	0	0
$340$	7.07180	0.383522
$341$	1.79315	0.0971046
$342$	0	0
$343$	0	0
$344$	0.267949	0.0144469
$345$	0	0
$346$	−6.69213	−0.359771
$347$	−9.58846	−0.514735	−0.257368	−	0.966314i	$-0.582855\pi$
$347$	−9.58846	−0.514735	−0.257368	+	0.966314i	$0.582855\pi$
$348$	0	0
$349$	8.00481	0.428488	0.214244	−	0.976780i	$-0.431271\pi$
$349$	8.00481	0.428488	0.214244	+	0.976780i	$0.431271\pi$
$350$	0	0
$351$	0	0
$352$	−0.267949	−0.0142817
$353$	−25.0125	−1.33128	−0.665641	−	0.746272i	$-0.731842\pi$
$353$	−25.0125	−1.33128	−0.665641	+	0.746272i	$0.731842\pi$
$354$	0	0
$355$	9.79796	0.520022
$356$	−7.07107	−0.374766
$357$	0	0
$358$	5.07180	0.268053
$359$	7.46410	0.393940	0.196970	−	0.980409i	$-0.436890\pi$
$359$	7.46410	0.393940	0.196970	+	0.980409i	$0.436890\pi$
$360$	0	0
$361$	0.196152	0.0103238
$362$	16.9706	0.891953
$363$	0	0
$364$	0	0
$365$	−5.60770	−0.293520
$366$	0	0
$367$	−18.5606	−0.968858	−0.484429	−	0.874831i	$-0.660973\pi$
$367$	−18.5606	−0.968858	−0.484429	+	0.874831i	$0.660973\pi$
$368$	5.46410	0.284836
$369$	0	0
$370$	7.72741	0.401729
$371$	0	0
$372$	0	0
$373$	−10.7846	−0.558406	−0.279203	−	0.960232i	$-0.590070\pi$
$373$	−10.7846	−0.558406	−0.279203	+	0.960232i	$0.590070\pi$
$374$	−1.83032	−0.0946434
$375$	0	0
$376$	−0.757875	−0.0390844
$377$	7.17260	0.369408
$378$	0	0
$379$	13.5885	0.697992	0.348996	−	0.937124i	$-0.386523\pi$
$379$	13.5885	0.697992	0.348996	+	0.937124i	$0.386523\pi$
$380$	−4.53590	−0.232687
$381$	0	0
$382$	14.9282	0.763793
$383$	−27.3233	−1.39616	−0.698078	−	0.716021i	$-0.745961\pi$
$383$	−27.3233	−1.39616	−0.698078	+	0.716021i	$0.745961\pi$
$384$	0	0
$385$	0	0
$386$	15.0526	0.766155
$387$	0	0
$388$	18.1445	0.921149
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	37.3244	1.88757
$392$	0	0
$393$	0	0
$394$	−16.9282	−0.852831
$395$	9.24316	0.465074
$396$	0	0
$397$	−13.1069	−0.657814	−0.328907	−	0.944362i	$-0.606680\pi$
$397$	−13.1069	−0.657814	−0.328907	+	0.944362i	$0.606680\pi$
$398$	−26.2880	−1.31770
$399$	0	0
$400$	−3.92820	−0.196410
$401$	−23.7846	−1.18775	−0.593873	−	0.804559i	$-0.702402\pi$
$401$	−23.7846	−1.18775	−0.593873	+	0.804559i	$0.702402\pi$
$402$	0	0
$403$	12.0000	0.597763
$404$	4.89898	0.243733
$405$	0	0
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−4.48288	−0.221664	−0.110832	−	0.993839i	$-0.535352\pi$
$409$	−4.48288	−0.221664	−0.110832	+	0.993839i	$0.535352\pi$
$410$	8.92820	0.440933
$411$	0	0
$412$	−12.3490	−0.608391
$413$	0	0
$414$	0	0
$415$	−6.82309	−0.334932
$416$	−1.79315	−0.0879165
$417$	0	0
$418$	1.17398	0.0574211
$419$	−36.1875	−1.76788	−0.883939	−	0.467603i	$-0.845118\pi$
$419$	−36.1875	−1.76788	−0.883939	+	0.467603i	$0.845118\pi$
$420$	0	0
$421$	7.60770	0.370776	0.185388	−	0.982665i	$-0.440646\pi$
$421$	7.60770	0.370776	0.185388	+	0.982665i	$0.440646\pi$
$422$	18.9282	0.921411
$423$	0	0
$424$	10.9282	0.530720
$425$	−26.8329	−1.30159
$426$	0	0
$427$	0	0
$428$	−17.3923	−0.840689
$429$	0	0
$430$	0.277401	0.0133775
$431$	−10.1436	−0.488600	−0.244300	−	0.969700i	$-0.578558\pi$
$431$	−10.1436	−0.488600	−0.244300	+	0.969700i	$0.578558\pi$
$432$	0	0
$433$	19.8362	0.953265	0.476632	−	0.879103i	$-0.341857\pi$
$433$	19.8362	0.953265	0.476632	+	0.879103i	$0.341857\pi$
$434$	0	0
$435$	0	0
$436$	4.92820	0.236018
$437$	−23.9401	−1.14521
$438$	0	0
$439$	−19.5959	−0.935262	−0.467631	−	0.883924i	$-0.654892\pi$
$439$	−19.5959	−0.935262	−0.467631	+	0.883924i	$0.654892\pi$
$440$	−0.277401	−0.0132246
$441$	0	0
$442$	−12.2487	−0.582612
$443$	16.3205	0.775411	0.387705	−	0.921783i	$-0.373268\pi$
$443$	16.3205	0.775411	0.387705	+	0.921783i	$0.373268\pi$
$444$	0	0
$445$	−7.32051	−0.347025
$446$	7.17260	0.339633
$447$	0	0
$448$	0	0
$449$	23.7846	1.12247	0.561233	−	0.827658i	$-0.310327\pi$
$449$	23.7846	1.12247	0.561233	+	0.827658i	$0.310327\pi$
$450$	0	0
$451$	−2.31079	−0.108811
$452$	−6.92820	−0.325875
$453$	0	0
$454$	27.6651	1.29839
$455$	0	0
$456$	0	0
$457$	−31.0526	−1.45258	−0.726289	−	0.687390i	$-0.758756\pi$
$457$	−31.0526	−1.45258	−0.726289	+	0.687390i	$0.758756\pi$
$458$	−0.480473	−0.0224510
$459$	0	0
$460$	5.65685	0.263752
$461$	11.0363	0.514012	0.257006	−	0.966410i	$-0.417264\pi$
$461$	11.0363	0.514012	0.257006	+	0.966410i	$0.417264\pi$
$462$	0	0
$463$	−30.6410	−1.42401	−0.712004	−	0.702175i	$-0.752212\pi$
$463$	−30.6410	−1.42401	−0.712004	+	0.702175i	$0.752212\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	0.124356	0.00576066
$467$	−9.17878	−0.424744	−0.212372	−	0.977189i	$-0.568119\pi$
$467$	−9.17878	−0.424744	−0.212372	+	0.977189i	$0.568119\pi$
$468$	0	0
$469$	0	0
$470$	−0.784610	−0.0361913
$471$	0	0
$472$	1.27551	0.0587102
$473$	−0.0717968	−0.00330122
$474$	0	0
$475$	17.2108	0.789686
$476$	0	0
$477$	0	0
$478$	0.928203	0.0424550
$479$	4.14110	0.189212	0.0946060	−	0.995515i	$-0.469841\pi$
$479$	4.14110	0.189212	0.0946060	+	0.995515i	$0.469841\pi$
$480$	0	0
$481$	−13.3843	−0.610270
$482$	−6.27603	−0.285865
$483$	0	0
$484$	−10.9282	−0.496737
$485$	18.7846	0.852965
$486$	0	0
$487$	−2.78461	−0.126183	−0.0630914	−	0.998008i	$-0.520096\pi$
$487$	−2.78461	−0.126183	−0.0630914	+	0.998008i	$0.520096\pi$
$488$	12.6264	0.571570
$489$	0	0
$490$	0	0
$491$	19.3923	0.875162	0.437581	−	0.899179i	$-0.355835\pi$
$491$	19.3923	0.875162	0.437581	+	0.899179i	$0.355835\pi$
$492$	0	0
$493$	−27.3233	−1.23058
$494$	7.85641	0.353476
$495$	0	0
$496$	−6.69213	−0.300486
$497$	0	0
$498$	0	0
$499$	33.3923	1.49484	0.747422	−	0.664349i	$-0.231291\pi$
$499$	33.3923	1.49484	0.747422	+	0.664349i	$0.231291\pi$
$500$	−9.24316	−0.413367
$501$	0	0
$502$	−0.795040	−0.0354843
$503$	12.3490	0.550614	0.275307	−	0.961356i	$-0.411220\pi$
$503$	12.3490	0.550614	0.275307	+	0.961356i	$0.411220\pi$
$504$	0	0
$505$	5.07180	0.225692
$506$	−1.46410	−0.0650873
$507$	0	0
$508$	13.4641	0.597373
$509$	−21.5921	−0.957055	−0.478527	−	0.878073i	$-0.658829\pi$
$509$	−21.5921	−0.957055	−0.478527	+	0.878073i	$0.658829\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−10.0382	−0.442766
$515$	−12.7846	−0.563357
$516$	0	0
$517$	0.203072	0.00893109
$518$	0	0
$519$	0	0
$520$	−1.85641	−0.0814088
$521$	33.7752	1.47972	0.739860	−	0.672761i	$-0.234892\pi$
$521$	33.7752	1.47972	0.739860	+	0.672761i	$0.234892\pi$
$522$	0	0
$523$	−20.7327	−0.906579	−0.453289	−	0.891363i	$-0.649750\pi$
$523$	−20.7327	−0.906579	−0.453289	+	0.891363i	$0.649750\pi$
$524$	−10.9348	−0.477688
$525$	0	0
$526$	−8.53590	−0.372183
$527$	−45.7128	−1.99128
$528$	0	0
$529$	6.85641	0.298105
$530$	11.3137	0.491436
$531$	0	0
$532$	0	0
$533$	−15.4641	−0.669825
$534$	0	0
$535$	−18.0058	−0.778460
$536$	12.4641	0.538367
$537$	0	0
$538$	5.65685	0.243884
$539$	0	0
$540$	0	0
$541$	15.3205	0.658680	0.329340	−	0.944211i	$-0.393174\pi$
$541$	15.3205	0.658680	0.329340	+	0.944211i	$0.393174\pi$
$542$	−13.1069	−0.562988
$543$	0	0
$544$	6.83083	0.292869
$545$	5.10205	0.218548
$546$	0	0
$547$	34.3731	1.46969	0.734843	−	0.678237i	$-0.237256\pi$
$547$	34.3731	1.46969	0.734843	+	0.678237i	$0.237256\pi$
$548$	8.66025	0.369948
$549$	0	0
$550$	1.05256	0.0448813
$551$	17.5254	0.746606
$552$	0	0
$553$	0	0
$554$	31.4641	1.33678
$555$	0	0
$556$	−0.795040	−0.0337172
$557$	−6.92820	−0.293557	−0.146779	−	0.989169i	$-0.546891\pi$
$557$	−6.92820	−0.293557	−0.146779	+	0.989169i	$0.546891\pi$
$558$	0	0
$559$	−0.480473	−0.0203219
$560$	0	0
$561$	0	0
$562$	17.8564	0.753227
$563$	18.2461	0.768980	0.384490	−	0.923129i	$-0.374377\pi$
$563$	18.2461	0.768980	0.384490	+	0.923129i	$0.374377\pi$
$564$	0	0
$565$	−7.17260	−0.301754
$566$	15.0759	0.633686
$567$	0	0
$568$	9.46410	0.397105
$569$	−25.7846	−1.08095	−0.540474	−	0.841361i	$-0.681755\pi$
$569$	−25.7846	−1.08095	−0.540474	+	0.841361i	$0.681755\pi$
$570$	0	0
$571$	−33.0526	−1.38321	−0.691603	−	0.722278i	$-0.743095\pi$
$571$	−33.0526	−1.38321	−0.691603	+	0.722278i	$0.743095\pi$
$572$	0.480473	0.0200896
$573$	0	0
$574$	0	0
$575$	−21.4641	−0.895115
$576$	0	0
$577$	27.4892	1.14439	0.572196	−	0.820117i	$-0.306092\pi$
$577$	27.4892	1.14439	0.572196	+	0.820117i	$0.306092\pi$
$578$	29.6603	1.23370
$579$	0	0
$580$	−4.14110	−0.171950
$581$	0	0
$582$	0	0
$583$	−2.92820	−0.121274
$584$	−5.41662	−0.224141
$585$	0	0
$586$	9.24316	0.381831
$587$	41.8816	1.72864	0.864319	−	0.502945i	$-0.167750\pi$
$587$	41.8816	1.72864	0.864319	+	0.502945i	$0.167750\pi$
$588$	0	0
$589$	29.3205	1.20813
$590$	1.32051	0.0543645
$591$	0	0
$592$	7.46410	0.306773
$593$	−16.8690	−0.692728	−0.346364	−	0.938100i	$-0.612584\pi$
$593$	−16.8690	−0.692728	−0.346364	+	0.938100i	$0.612584\pi$
$594$	0	0
$595$	0	0
$596$	22.9282	0.939176
$597$	0	0
$598$	−9.79796	−0.400668
$599$	4.78461	0.195494	0.0977469	−	0.995211i	$-0.468836\pi$
$599$	4.78461	0.195494	0.0977469	+	0.995211i	$0.468836\pi$
$600$	0	0
$601$	3.34607	0.136489	0.0682444	−	0.997669i	$-0.478260\pi$
$601$	3.34607	0.136489	0.0682444	+	0.997669i	$0.478260\pi$
$602$	0	0
$603$	0	0
$604$	18.3923	0.748372
$605$	−11.3137	−0.459968
$606$	0	0
$607$	2.27362	0.0922836	0.0461418	−	0.998935i	$-0.485307\pi$
$607$	2.27362	0.0922836	0.0461418	+	0.998935i	$0.485307\pi$
$608$	−4.38134	−0.177687
$609$	0	0
$610$	13.0718	0.529262
$611$	1.35898	0.0549786
$612$	0	0
$613$	24.9282	1.00684	0.503420	−	0.864042i	$-0.332075\pi$
$613$	24.9282	1.00684	0.503420	+	0.864042i	$0.332075\pi$
$614$	−1.17398	−0.0473779
$615$	0	0
$616$	0	0
$617$	−3.14359	−0.126556	−0.0632782	−	0.997996i	$-0.520156\pi$
$617$	−3.14359	−0.126556	−0.0632782	+	0.997996i	$0.520156\pi$
$618$	0	0
$619$	31.7047	1.27432	0.637159	−	0.770732i	$-0.280109\pi$
$619$	31.7047	1.27432	0.637159	+	0.770732i	$0.280109\pi$
$620$	−6.92820	−0.278243
$621$	0	0
$622$	7.45001	0.298718
$623$	0	0
$624$	0	0
$625$	10.0718	0.402872
$626$	−6.27603	−0.250841
$627$	0	0
$628$	−9.52056	−0.379912
$629$	50.9860	2.03295
$630$	0	0
$631$	−19.7128	−0.784755	−0.392377	−	0.919804i	$-0.628347\pi$
$631$	−19.7128	−0.784755	−0.392377	+	0.919804i	$0.628347\pi$
$632$	8.92820	0.355145
$633$	0	0
$634$	−26.0000	−1.03259
$635$	13.9391	0.553155
$636$	0	0
$637$	0	0
$638$	1.07180	0.0424328
$639$	0	0
$640$	1.03528	0.0409229
$641$	4.07180	0.160826	0.0804132	−	0.996762i	$-0.474376\pi$
$641$	4.07180	0.160826	0.0804132	+	0.996762i	$0.474376\pi$
$642$	0	0
$643$	44.9131	1.77120	0.885599	−	0.464450i	$-0.153748\pi$
$643$	44.9131	1.77120	0.885599	+	0.464450i	$0.153748\pi$
$644$	0	0
$645$	0	0
$646$	−29.9282	−1.17751
$647$	−21.8695	−0.859780	−0.429890	−	0.902881i	$-0.641448\pi$
$647$	−21.8695	−0.859780	−0.429890	+	0.902881i	$0.641448\pi$
$648$	0	0
$649$	−0.341773	−0.0134157
$650$	7.04386	0.276283
$651$	0	0
$652$	−13.3205	−0.521671
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	−11.3205	−0.442329
$656$	8.62398	0.336710
$657$	0	0
$658$	0	0
$659$	−48.2487	−1.87950	−0.939751	−	0.341858i	$-0.888944\pi$
$659$	−48.2487	−1.87950	−0.939751	+	0.341858i	$0.888944\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	−11.4641	−0.445565
$663$	0	0
$664$	−6.59059	−0.255765
$665$	0	0
$666$	0	0
$667$	−21.8564	−0.846283
$668$	−1.51575	−0.0586461
$669$	0	0
$670$	12.9038	0.498517
$671$	−3.38323	−0.130608
$672$	0	0
$673$	41.5692	1.60238	0.801188	−	0.598413i	$-0.204202\pi$
$673$	41.5692	1.60238	0.801188	+	0.598413i	$0.204202\pi$
$674$	−7.00000	−0.269630
$675$	0	0
$676$	−9.78461	−0.376331
$677$	5.37945	0.206749	0.103375	−	0.994642i	$-0.467036\pi$
$677$	5.37945	0.206749	0.103375	+	0.994642i	$0.467036\pi$
$678$	0	0
$679$	0	0
$680$	7.07180	0.271191
$681$	0	0
$682$	1.79315	0.0686633
$683$	−38.3205	−1.46629	−0.733147	−	0.680070i	$-0.761949\pi$
$683$	−38.3205	−1.46629	−0.733147	+	0.680070i	$0.761949\pi$
$684$	0	0
$685$	8.96575	0.342564
$686$	0	0
$687$	0	0
$688$	0.267949	0.0102155
$689$	−19.5959	−0.746545
$690$	0	0
$691$	−24.3190	−0.925140	−0.462570	−	0.886583i	$-0.653073\pi$
$691$	−24.3190	−0.925140	−0.462570	+	0.886583i	$0.653073\pi$
$692$	−6.69213	−0.254397
$693$	0	0
$694$	−9.58846	−0.363973
$695$	−0.823085	−0.0312214
$696$	0	0
$697$	58.9090	2.23134
$698$	8.00481	0.302986
$699$	0	0
$700$	0	0
$701$	−20.7846	−0.785024	−0.392512	−	0.919747i	$-0.628394\pi$
$701$	−20.7846	−0.785024	−0.392512	+	0.919747i	$0.628394\pi$
$702$	0	0
$703$	−32.7028	−1.23341
$704$	−0.267949	−0.0100987
$705$	0	0
$706$	−25.0125	−0.941359
$707$	0	0
$708$	0	0
$709$	−12.3923	−0.465403	−0.232701	−	0.972548i	$-0.574756\pi$
$709$	−12.3923	−0.465403	−0.232701	+	0.972548i	$0.574756\pi$
$710$	9.79796	0.367711
$711$	0	0
$712$	−7.07107	−0.264999
$713$	−36.5665	−1.36943
$714$	0	0
$715$	0.497423	0.0186026
$716$	5.07180	0.189542
$717$	0	0
$718$	7.46410	0.278558
$719$	49.6733	1.85250	0.926251	−	0.376906i	$-0.123012\pi$
$719$	49.6733	1.85250	0.926251	+	0.376906i	$0.123012\pi$
$720$	0	0
$721$	0	0
$722$	0.196152	0.00730004
$723$	0	0
$724$	16.9706	0.630706
$725$	15.7128	0.583559
$726$	0	0
$727$	−32.7028	−1.21288	−0.606439	−	0.795130i	$-0.707403\pi$
$727$	−32.7028	−1.21288	−0.606439	+	0.795130i	$0.707403\pi$
$728$	0	0
$729$	0	0
$730$	−5.60770	−0.207550
$731$	1.83032	0.0676967
$732$	0	0
$733$	−8.00481	−0.295664	−0.147832	−	0.989012i	$-0.547230\pi$
$733$	−8.00481	−0.295664	−0.147832	+	0.989012i	$0.547230\pi$
$734$	−18.5606	−0.685086
$735$	0	0
$736$	5.46410	0.201409
$737$	−3.33975	−0.123021
$738$	0	0
$739$	6.12436	0.225288	0.112644	−	0.993635i	$-0.464068\pi$
$739$	6.12436	0.225288	0.112644	+	0.993635i	$0.464068\pi$
$740$	7.72741	0.284065
$741$	0	0
$742$	0	0
$743$	−31.5692	−1.15816	−0.579081	−	0.815270i	$-0.696589\pi$
$743$	−31.5692	−1.15816	−0.579081	+	0.815270i	$0.696589\pi$
$744$	0	0
$745$	23.7370	0.869657
$746$	−10.7846	−0.394853
$747$	0	0
$748$	−1.83032	−0.0669230
$749$	0	0
$750$	0	0
$751$	34.7846	1.26931	0.634654	−	0.772796i	$-0.281143\pi$
$751$	34.7846	1.26931	0.634654	+	0.772796i	$0.281143\pi$
$752$	−0.757875	−0.0276368
$753$	0	0
$754$	7.17260	0.261211
$755$	19.0411	0.692977
$756$	0	0
$757$	19.3205	0.702216	0.351108	−	0.936335i	$-0.385805\pi$
$757$	19.3205	0.702216	0.351108	+	0.936335i	$0.385805\pi$
$758$	13.5885	0.493555
$759$	0	0
$760$	−4.53590	−0.164534
$761$	40.2543	1.45922	0.729609	−	0.683865i	$-0.239702\pi$
$761$	40.2543	1.45922	0.729609	+	0.683865i	$0.239702\pi$
$762$	0	0
$763$	0	0
$764$	14.9282	0.540083
$765$	0	0
$766$	−27.3233	−0.987232
$767$	−2.28719	−0.0825855
$768$	0	0
$769$	−38.1838	−1.37694	−0.688471	−	0.725264i	$-0.741718\pi$
$769$	−38.1838	−1.37694	−0.688471	+	0.725264i	$0.741718\pi$
$770$	0	0
$771$	0	0
$772$	15.0526	0.541753
$773$	−39.3949	−1.41694	−0.708468	−	0.705743i	$-0.750613\pi$
$773$	−39.3949	−1.41694	−0.708468	+	0.705743i	$0.750613\pi$
$774$	0	0
$775$	26.2880	0.944295
$776$	18.1445	0.651351
$777$	0	0
$778$	8.00000	0.286814
$779$	−37.7846	−1.35377
$780$	0	0
$781$	−2.53590	−0.0907416
$782$	37.3244	1.33472
$783$	0	0
$784$	0	0
$785$	−9.85641	−0.351790
$786$	0	0
$787$	−7.14540	−0.254706	−0.127353	−	0.991857i	$-0.540648\pi$
$787$	−7.14540	−0.254706	−0.127353	+	0.991857i	$0.540648\pi$
$788$	−16.9282	−0.603042
$789$	0	0
$790$	9.24316	0.328857
$791$	0	0
$792$	0	0
$793$	−22.6410	−0.804006
$794$	−13.1069	−0.465145
$795$	0	0
$796$	−26.2880	−0.931755
$797$	37.8792	1.34175	0.670874	−	0.741571i	$-0.265919\pi$
$797$	37.8792	1.34175	0.670874	+	0.741571i	$0.265919\pi$
$798$	0	0
$799$	−5.17691	−0.183146
$800$	−3.92820	−0.138883
$801$	0	0
$802$	−23.7846	−0.839864
$803$	1.45138	0.0512180
$804$	0	0
$805$	0	0
$806$	12.0000	0.422682
$807$	0	0
$808$	4.89898	0.172345
$809$	−29.7321	−1.04532	−0.522662	−	0.852540i	$-0.675061\pi$
$809$	−29.7321	−1.04532	−0.522662	+	0.852540i	$0.675061\pi$
$810$	0	0
$811$	24.9110	0.874744	0.437372	−	0.899281i	$-0.355909\pi$
$811$	24.9110	0.874744	0.437372	+	0.899281i	$0.355909\pi$
$812$	0	0
$813$	0	0
$814$	−2.00000	−0.0701000
$815$	−13.7904	−0.483057
$816$	0	0
$817$	−1.17398	−0.0410723
$818$	−4.48288	−0.156740
$819$	0	0
$820$	8.92820	0.311786
$821$	10.3923	0.362694	0.181347	−	0.983419i	$-0.441954\pi$
$821$	10.3923	0.362694	0.181347	+	0.983419i	$0.441954\pi$
$822$	0	0
$823$	−10.7846	−0.375928	−0.187964	−	0.982176i	$-0.560189\pi$
$823$	−10.7846	−0.375928	−0.187964	+	0.982176i	$0.560189\pi$
$824$	−12.3490	−0.430197
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−7.93048	−0.275437	−0.137718	−	0.990471i	$-0.543977\pi$
$829$	−7.93048	−0.275437	−0.137718	+	0.990471i	$0.543977\pi$
$830$	−6.82309	−0.236833
$831$	0	0
$832$	−1.79315	−0.0621663
$833$	0	0
$834$	0	0
$835$	−1.56922	−0.0543051
$836$	1.17398	0.0406028
$837$	0	0
$838$	−36.1875	−1.25008
$839$	−22.6274	−0.781185	−0.390593	−	0.920564i	$-0.627730\pi$
$839$	−22.6274	−0.781185	−0.390593	+	0.920564i	$0.627730\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	7.60770	0.262178
$843$	0	0
$844$	18.9282	0.651536
$845$	−10.1298	−0.348475
$846$	0	0
$847$	0	0
$848$	10.9282	0.375276
$849$	0	0
$850$	−26.8329	−0.920361
$851$	40.7846	1.39808
$852$	0	0
$853$	−32.9802	−1.12922	−0.564610	−	0.825358i	$-0.690973\pi$
$853$	−32.9802	−1.12922	−0.564610	+	0.825358i	$0.690973\pi$
$854$	0	0
$855$	0	0
$856$	−17.3923	−0.594457
$857$	11.2122	0.383001	0.191500	−	0.981493i	$-0.438665\pi$
$857$	11.2122	0.383001	0.191500	+	0.981493i	$0.438665\pi$
$858$	0	0
$859$	41.6042	1.41952	0.709758	−	0.704446i	$-0.248804\pi$
$859$	41.6042	1.41952	0.709758	+	0.704446i	$0.248804\pi$
$860$	0.277401	0.00945931
$861$	0	0
$862$	−10.1436	−0.345492
$863$	−54.1051	−1.84176	−0.920880	−	0.389847i	$-0.872528\pi$
$863$	−54.1051	−1.84176	−0.920880	+	0.389847i	$0.872528\pi$
$864$	0	0
$865$	−6.92820	−0.235566
$866$	19.8362	0.674060
$867$	0	0
$868$	0	0
$869$	−2.39230	−0.0811534
$870$	0	0
$871$	−22.3500	−0.757301
$872$	4.92820	0.166890
$873$	0	0
$874$	−23.9401	−0.809786
$875$	0	0
$876$	0	0
$877$	−29.1769	−0.985234	−0.492617	−	0.870246i	$-0.663960\pi$
$877$	−29.1769	−0.985234	−0.492617	+	0.870246i	$0.663960\pi$
$878$	−19.5959	−0.661330
$879$	0	0
$880$	−0.277401	−0.00935120
$881$	12.7279	0.428815	0.214407	−	0.976744i	$-0.431218\pi$
$881$	12.7279	0.428815	0.214407	+	0.976744i	$0.431218\pi$
$882$	0	0
$883$	14.4641	0.486756	0.243378	−	0.969932i	$-0.421744\pi$
$883$	14.4641	0.486756	0.243378	+	0.969932i	$0.421744\pi$
$884$	−12.2487	−0.411969
$885$	0	0
$886$	16.3205	0.548298
$887$	−53.2596	−1.78828	−0.894142	−	0.447784i	$-0.852213\pi$
$887$	−53.2596	−1.78828	−0.894142	+	0.447784i	$0.852213\pi$
$888$	0	0
$889$	0	0
$890$	−7.32051	−0.245384
$891$	0	0
$892$	7.17260	0.240157
$893$	3.32051	0.111117
$894$	0	0
$895$	5.25071	0.175512
$896$	0	0
$897$	0	0
$898$	23.7846	0.793703
$899$	26.7685	0.892780
$900$	0	0
$901$	74.6487	2.48691
$902$	−2.31079	−0.0769409
$903$	0	0
$904$	−6.92820	−0.230429
$905$	17.5692	0.584021
$906$	0	0
$907$	−5.24871	−0.174281	−0.0871403	−	0.996196i	$-0.527773\pi$
$907$	−5.24871	−0.174281	−0.0871403	+	0.996196i	$0.527773\pi$
$908$	27.6651	0.918098
$909$	0	0
$910$	0	0
$911$	−33.0718	−1.09572	−0.547859	−	0.836571i	$-0.684557\pi$
$911$	−33.0718	−1.09572	−0.547859	+	0.836571i	$0.684557\pi$
$912$	0	0
$913$	1.76594	0.0584442
$914$	−31.0526	−1.02713
$915$	0	0
$916$	−0.480473	−0.0158753
$917$	0	0
$918$	0	0
$919$	9.07180	0.299251	0.149625	−	0.988743i	$-0.452193\pi$
$919$	9.07180	0.299251	0.149625	+	0.988743i	$0.452193\pi$
$920$	5.65685	0.186501
$921$	0	0
$922$	11.0363	0.363461
$923$	−16.9706	−0.558593
$924$	0	0
$925$	−29.3205	−0.964052
$926$	−30.6410	−1.00693
$927$	0	0
$928$	−4.00000	−0.131306
$929$	−30.8081	−1.01078	−0.505390	−	0.862891i	$-0.668651\pi$
$929$	−30.8081	−1.01078	−0.505390	+	0.862891i	$0.668651\pi$
$930$	0	0
$931$	0	0
$932$	0.124356	0.00407340
$933$	0	0
$934$	−9.17878	−0.300339
$935$	−1.89488	−0.0619693
$936$	0	0
$937$	9.89949	0.323402	0.161701	−	0.986840i	$-0.448302\pi$
$937$	9.89949	0.323402	0.161701	+	0.986840i	$0.448302\pi$
$938$	0	0
$939$	0	0
$940$	−0.784610	−0.0255911
$941$	47.8802	1.56085	0.780425	−	0.625250i	$-0.215003\pi$
$941$	47.8802	1.56085	0.780425	+	0.625250i	$0.215003\pi$
$942$	0	0
$943$	47.1223	1.53451
$944$	1.27551	0.0415144
$945$	0	0
$946$	−0.0717968	−0.00233431
$947$	−18.1244	−0.588962	−0.294481	−	0.955657i	$-0.595147\pi$
$947$	−18.1244	−0.588962	−0.294481	+	0.955657i	$0.595147\pi$
$948$	0	0
$949$	9.71281	0.315291
$950$	17.2108	0.558392
$951$	0	0
$952$	0	0
$953$	19.0000	0.615470	0.307735	−	0.951472i	$-0.400429\pi$
$953$	19.0000	0.615470	0.307735	+	0.951472i	$0.400429\pi$
$954$	0	0
$955$	15.4548	0.500106
$956$	0.928203	0.0300202
$957$	0	0
$958$	4.14110	0.133793
$959$	0	0
$960$	0	0
$961$	13.7846	0.444665
$962$	−13.3843	−0.431526
$963$	0	0
$964$	−6.27603	−0.202137
$965$	15.5836	0.501652
$966$	0	0
$967$	−47.5692	−1.52972	−0.764861	−	0.644195i	$-0.777193\pi$
$967$	−47.5692	−1.52972	−0.764861	+	0.644195i	$0.777193\pi$
$968$	−10.9282	−0.351246
$969$	0	0
$970$	18.7846	0.603137
$971$	−31.2886	−1.00410	−0.502049	−	0.864839i	$-0.667420\pi$
$971$	−31.2886	−1.00410	−0.502049	+	0.864839i	$0.667420\pi$
$972$	0	0
$973$	0	0
$974$	−2.78461	−0.0892246
$975$	0	0
$976$	12.6264	0.404161
$977$	1.98076	0.0633702	0.0316851	−	0.999498i	$-0.489913\pi$
$977$	1.98076	0.0633702	0.0316851	+	0.999498i	$0.489913\pi$
$978$	0	0
$979$	1.89469	0.0605545
$980$	0	0
$981$	0	0
$982$	19.3923	0.618833
$983$	−12.6264	−0.402719	−0.201360	−	0.979517i	$-0.564536\pi$
$983$	−12.6264	−0.402719	−0.201360	+	0.979517i	$0.564536\pi$
$984$	0	0
$985$	−17.5254	−0.558405
$986$	−27.3233	−0.870152
$987$	0	0
$988$	7.85641	0.249946
$989$	1.46410	0.0465557
$990$	0	0
$991$	50.2487	1.59620	0.798101	−	0.602523i	$-0.205838\pi$
$991$	50.2487	1.59620	0.798101	+	0.602523i	$0.205838\pi$
$992$	−6.69213	−0.212475
$993$	0	0
$994$	0	0
$995$	−27.2154	−0.862786
$996$	0	0
$997$	37.6018	1.19086	0.595430	−	0.803407i	$-0.296982\pi$
$997$	37.6018	1.19086	0.595430	+	0.803407i	$0.296982\pi$
$998$	33.3923	1.05701
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.cp.1.3		4
3.2	odd	2		7938.2.a.ci.1.2		4
7.6	odd	2	inner	7938.2.a.cp.1.2		4
9.2	odd	6		2646.2.f.r.1765.3		8
9.4	even	3		882.2.f.q.295.4	yes	8
9.5	odd	6		2646.2.f.r.883.3		8
9.7	even	3		882.2.f.q.589.4	yes	8
21.20	even	2		7938.2.a.ci.1.3		4
63.2	odd	6		2646.2.h.t.361.2		8
63.4	even	3		882.2.h.q.79.1		8
63.5	even	6		2646.2.e.q.2125.2		8
63.11	odd	6		2646.2.e.q.1549.3		8
63.13	odd	6		882.2.f.q.295.1	✓	8
63.16	even	3		882.2.h.q.67.2		8
63.20	even	6		2646.2.f.r.1765.2		8
63.23	odd	6		2646.2.e.q.2125.3		8
63.25	even	3		882.2.e.s.373.2		8
63.31	odd	6		882.2.h.q.79.4		8
63.32	odd	6		2646.2.h.t.667.2		8
63.34	odd	6		882.2.f.q.589.1	yes	8
63.38	even	6		2646.2.e.q.1549.2		8
63.40	odd	6		882.2.e.s.655.3		8
63.41	even	6		2646.2.f.r.883.2		8
63.47	even	6		2646.2.h.t.361.3		8
63.52	odd	6		882.2.e.s.373.3		8
63.58	even	3		882.2.e.s.655.2		8
63.59	even	6		2646.2.h.t.667.3		8
63.61	odd	6		882.2.h.q.67.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
882.2.e.s.373.2		8	63.25	even	3
882.2.e.s.373.3		8	63.52	odd	6
882.2.e.s.655.2		8	63.58	even	3
882.2.e.s.655.3		8	63.40	odd	6
882.2.f.q.295.1	✓	8	63.13	odd	6
882.2.f.q.295.4	yes	8	9.4	even	3
882.2.f.q.589.1	yes	8	63.34	odd	6
882.2.f.q.589.4	yes	8	9.7	even	3
882.2.h.q.67.2		8	63.16	even	3
882.2.h.q.67.3		8	63.61	odd	6
882.2.h.q.79.1		8	63.4	even	3
882.2.h.q.79.4		8	63.31	odd	6
2646.2.e.q.1549.2		8	63.38	even	6
2646.2.e.q.1549.3		8	63.11	odd	6
2646.2.e.q.2125.2		8	63.5	even	6
2646.2.e.q.2125.3		8	63.23	odd	6
2646.2.f.r.883.2		8	63.41	even	6
2646.2.f.r.883.3		8	9.5	odd	6
2646.2.f.r.1765.2		8	63.20	even	6
2646.2.f.r.1765.3		8	9.2	odd	6
2646.2.h.t.361.2		8	63.2	odd	6
2646.2.h.t.361.3		8	63.47	even	6
2646.2.h.t.667.2		8	63.32	odd	6
2646.2.h.t.667.3		8	63.59	even	6
7938.2.a.ci.1.2		4	3.2	odd	2
7938.2.a.ci.1.3		4	21.20	even	2
7938.2.a.cp.1.2		4	7.6	odd	2	inner
7938.2.a.cp.1.3		4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.03528 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.03528 q^{5} +1.00000 q^{8} +1.03528 q^{10} -0.267949 q^{11} -1.79315 q^{13} +1.00000 q^{16} +6.83083 q^{17} -4.38134 q^{19} +1.03528 q^{20} -0.267949 q^{22} +5.46410 q^{23} -3.92820 q^{25} -1.79315 q^{26} -4.00000 q^{29} -6.69213 q^{31} +1.00000 q^{32} +6.83083 q^{34} +7.46410 q^{37} -4.38134 q^{38} +1.03528 q^{40} +8.62398 q^{41} +0.267949 q^{43} -0.267949 q^{44} +5.46410 q^{46} -0.757875 q^{47} -3.92820 q^{50} -1.79315 q^{52} +10.9282 q^{53} -0.277401 q^{55} -4.00000 q^{58} +1.27551 q^{59} +12.6264 q^{61} -6.69213 q^{62} +1.00000 q^{64} -1.85641 q^{65} +12.4641 q^{67} +6.83083 q^{68} +9.46410 q^{71} -5.41662 q^{73} +7.46410 q^{74} -4.38134 q^{76} +8.92820 q^{79} +1.03528 q^{80} +8.62398 q^{82} -6.59059 q^{83} +7.07180 q^{85} +0.267949 q^{86} -0.267949 q^{88} -7.07107 q^{89} +5.46410 q^{92} -0.757875 q^{94} -4.53590 q^{95} +18.1445 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 8 q^{11} + 4 q^{16} - 8 q^{22} + 8 q^{23} + 12 q^{25} - 16 q^{29} + 4 q^{32} + 16 q^{37} + 8 q^{43} - 8 q^{44} + 8 q^{46} + 12 q^{50} + 16 q^{53} - 16 q^{58} + 4 q^{64} + 48 q^{65} + 36 q^{67} + 24 q^{71} + 16 q^{74} + 8 q^{79} + 56 q^{85} + 8 q^{86} - 8 q^{88} + 8 q^{92} - 32 q^{95}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 8 * q^11 + 4 * q^16 - 8 * q^22 + 8 * q^23 + 12 * q^25 - 16 * q^29 + 4 * q^32 + 16 * q^37 + 8 * q^43 - 8 * q^44 + 8 * q^46 + 12 * q^50 + 16 * q^53 - 16 * q^58 + 4 * q^64 + 48 * q^65 + 36 * q^67 + 24 * q^71 + 16 * q^74 + 8 * q^79 + 56 * q^85 + 8 * q^86 - 8 * q^88 + 8 * q^92 - 32 * q^95

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