LMFDB - Embedded newform 95.11.d.a.94.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,11,Mod(94,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("95.94");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 95 = 5 \cdot 19 $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	95.d (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$60.3589390040$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 5 $ x^2 - x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 11 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			94.1
Root			$0.500000 - 2.17945i$ of defining polynomial
Character	$\chi$	$=$	95.94
Dual form			95.11.d.a.94.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1024.00 q^{4} +(1975.50 - 2421.37i) q^{5} -8486.78i q^{7} -59049.0 q^{9} +O(q^{10})$	$q-1024.00 q^{4} +(1975.50 - 2421.37i) q^{5} -8486.78i q^{7} -59049.0 q^{9} -203523. q^{11} +1.04858e6 q^{16} +1.85908e6i q^{17} -2.47610e6 q^{19} +(-2.02291e6 + 2.47948e6i) q^{20} -1.18025e7i q^{23} +(-1.96042e6 - 9.56683e6i) q^{25} +8.69046e6i q^{28} +(-2.05496e7 - 1.67656e7i) q^{35} +6.04662e7 q^{36} +2.03243e8i q^{43} +2.08408e8 q^{44} +(-1.16651e8 + 1.42979e8i) q^{45} -4.28715e7i q^{47} +2.10450e8 q^{49} +(-4.02060e8 + 4.92804e8i) q^{55} +1.60684e9 q^{61} +5.01136e8i q^{63} -1.07374e9 q^{64} -1.90370e9i q^{68} +2.70383e9i q^{73} +2.53553e9 q^{76} +1.72725e9i q^{77} +(2.07146e9 - 2.53899e9i) q^{80} +3.48678e9 q^{81} +7.57882e9i q^{83} +(4.50153e9 + 3.67262e9i) q^{85} +1.20857e10i q^{92} +(-4.89153e9 + 5.99555e9i) q^{95} +1.20178e10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2048 q^{4} + 3951 q^{5} - 118098 q^{9}+O(q^{10}) $ 2 * q - 2048 * q^4 + 3951 * q^5 - 118098 * q^9	$ 2 q - 2048 q^{4} + 3951 q^{5} - 118098 q^{9} - 407046 q^{11} + 2097152 q^{16} - 4952198 q^{19} - 4045824 q^{20} - 3920849 q^{25} - 41099223 q^{35} + 120932352 q^{36} + 416815104 q^{44} - 233302599 q^{45} + 420899756 q^{49} - 804119373 q^{55} + 3213673954 q^{61} - 2147483648 q^{64} + 5071050752 q^{76} + 4142923776 q^{80} + 6973568802 q^{81} + 9003051827 q^{85} - 9783067149 q^{95} + 24035659254 q^{99}+O(q^{100}) $ 2 * q - 2048 * q^4 + 3951 * q^5 - 118098 * q^9 - 407046 * q^11 + 2097152 * q^16 - 4952198 * q^19 - 4045824 * q^20 - 3920849 * q^25 - 41099223 * q^35 + 120932352 * q^36 + 416815104 * q^44 - 233302599 * q^45 + 420899756 * q^49 - 804119373 * q^55 + 3213673954 * q^61 - 2147483648 * q^64 + 5071050752 * q^76 + 4142923776 * q^80 + 6973568802 * q^81 + 9003051827 * q^85 - 9783067149 * q^95 + 24035659254 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$.

$n$	$21$	$77$
$\chi(n)$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$		0			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$	−1024.00			−1.00000
$5$	1975.50	−	2421.37i	0.632160	−	0.774838i
$5$	1975.50	−	2421.37i	0.632160	−	0.774838i
$6$		0			0
$7$		−	8486.78i		−	0.504955i	−0.967603	−	0.252477i	$-0.918755\pi$
$7$		−	8486.78i		−	0.504955i	0.967603	−	0.252477i	$-0.0812453\pi$
$8$		0			0
$9$	−59049.0			−1.00000
$10$		0			0
$11$	−203523.			−1.26372			−0.631859	−	0.775083i	$-0.717708\pi$
$11$	−203523.			−1.26372			−0.631859	+	0.775083i	$0.717708\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	1.04858e6			1.00000
$17$			1.85908e6i			1.30935i	0.755912	+	0.654673i	$0.227194\pi$
$17$			1.85908e6i			1.30935i	−0.755912	+	0.654673i	$0.772806\pi$
$18$		0			0
$19$	−2.47610e6			−1.00000
$19$	−2.47610e6			−1.00000
$20$	−2.02291e6	+	2.47948e6i	−0.632160	+	0.774838i
$21$		0			0
$22$		0			0
$23$		−	1.18025e7i		−	1.83372i	−0.399206	−	0.916861i	$-0.630714\pi$
$23$		−	1.18025e7i		−	1.83372i	0.399206	−	0.916861i	$-0.369286\pi$
$24$		0			0
$25$	−1.96042e6	−	9.56683e6i	−0.200747	−	0.979643i
$26$		0			0
$27$		0			0
$28$			8.69046e6i			0.504955i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.05496e7	−	1.67656e7i	−0.391258	−	0.319212i
$36$	6.04662e7			1.00000
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$			2.03243e8i			1.38252i	0.722605	+	0.691262i	$0.242945\pi$
$43$			2.03243e8i			1.38252i	−0.722605	+	0.691262i	$0.757055\pi$
$44$	2.08408e8			1.26372
$45$	−1.16651e8	+	1.42979e8i	−0.632160	+	0.774838i
$46$		0			0
$47$		−	4.28715e7i		−	0.186930i	−0.995623	−	0.0934651i	$-0.970206\pi$
$47$		−	4.28715e7i		−	0.186930i	0.995623	−	0.0934651i	$-0.0297943\pi$
$48$		0			0
$49$	2.10450e8			0.745021
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−4.02060e8	+	4.92804e8i	−0.798872	+	0.979176i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	1.60684e9			1.90249			0.951246	−	0.308435i	$-0.0998051\pi$
$61$	1.60684e9			1.90249			0.951246	+	0.308435i	$0.0998051\pi$
$62$		0			0
$63$			5.01136e8i			0.504955i
$64$	−1.07374e9			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		−	1.90370e9i		−	1.30935i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			2.70383e9i			1.30426i	0.758106	+	0.652131i	$0.226125\pi$
$73$			2.70383e9i			1.30426i	−0.758106	+	0.652131i	$0.773875\pi$
$74$		0			0
$75$		0			0
$76$	2.53553e9			1.00000
$77$			1.72725e9i			0.638120i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	2.07146e9	−	2.53899e9i	0.632160	−	0.774838i
$81$	3.48678e9			1.00000
$82$		0			0
$83$			7.57882e9i			1.92403i	0.273003	+	0.962013i	$0.411983\pi$
$83$			7.57882e9i			1.92403i	−0.273003	+	0.962013i	$0.588017\pi$
$84$		0			0
$85$	4.50153e9	+	3.67262e9i	1.01453	+	0.827716i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$			1.20857e10i			1.83372i
$93$		0			0
$94$		0			0
$95$	−4.89153e9	+	5.99555e9i	−0.632160	+	0.774838i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$	1.20178e10			1.26372
$100$	2.00747e9	+	9.79643e9i	0.200747	+	0.979643i
$101$	9.98470e9			0.950010			0.475005	−	0.879983i	$-0.342446\pi$
$101$	9.98470e9			0.950010			0.475005	+	0.879983i	$0.342446\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		−	8.89903e9i		−	0.504955i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	−2.85781e10	−	2.33158e10i	−1.42084	−	1.15921i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.57776e10			0.661160
$120$		0			0
$121$	1.54842e10			0.596982
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.70376e10	−	1.41524e10i	−0.885969	−	0.463744i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.71148e10			1.99885			0.999427	−	0.0338381i	$-0.0107731\pi$
$131$	7.71148e10			1.99885			0.999427	+	0.0338381i	$0.0107731\pi$
$132$		0			0
$133$			2.10141e10i			0.504955i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			9.21504e10i			1.90939i	0.297587	+	0.954695i	$0.403818\pi$
$137$			9.21504e10i			1.90939i	−0.297587	+	0.954695i	$0.596182\pi$
$138$		0			0
$139$	−7.41705e10			−1.42941			−0.714705	−	0.699426i	$-0.753439\pi$
$139$	−7.41705e10			−1.42941			−0.714705	+	0.699426i	$0.753439\pi$
$140$	2.10428e10	+	1.71680e10i	0.391258	+	0.319212i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−6.19174e10			−1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.61929e9			−0.0765156			−0.0382578	−	0.999268i	$-0.512181\pi$
$149$	−5.61929e9			−0.0765156			−0.0382578	+	0.999268i	$0.512181\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		−	1.09777e11i		−	1.30935i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	1.88364e11i		−	1.97469i	−0.158590	−	0.987344i	$-0.550695\pi$
$157$		−	1.88364e11i		−	1.97469i	0.158590	−	0.987344i	$-0.449305\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.00165e11			−0.925947
$162$		0			0
$163$		−	7.73148e10i		−	0.671931i	−0.941874	−	0.335966i	$-0.890937\pi$
$163$		−	7.73148e10i		−	0.671931i	0.941874	−	0.335966i	$-0.109063\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−1.37858e11			−1.00000
$170$		0			0
$171$	1.46211e11			1.00000
$172$		−	2.08120e11i		−	1.38252i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−8.11915e10	+	1.66377e10i	−0.494676	+	0.101368i
$176$	−2.13409e11			−1.26372
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	1.19451e11	−	1.46411e11i	0.632160	−	0.774838i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	3.78366e11i		−	1.65464i
$188$			4.39004e10i			0.186930i
$189$		0			0
$190$		0			0
$191$	−4.79090e11			−1.88473			−0.942367	−	0.334582i	$-0.891405\pi$
$191$	−4.79090e11			−1.88473			−0.942367	+	0.334582i	$0.891405\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	−2.15501e11			−0.745021
$197$			2.41155e11i			0.812765i	0.913703	+	0.406382i	$0.133210\pi$
$197$			2.41155e11i			0.812765i	−0.913703	+	0.406382i	$0.866790\pi$
$198$		0			0
$199$	6.66096e10			0.213438			0.106719	−	0.994289i	$-0.465966\pi$
$199$	6.66096e10			0.213438			0.106719	+	0.994289i	$0.465966\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$			6.96924e11i			1.83372i
$208$		0			0
$209$	5.03943e11			1.26372
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.92125e11	+	4.01506e11i	1.07123	+	0.873976i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	4.11709e11	−	5.04631e11i	0.798872	−	0.979176i
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	1.15761e11	+	5.64912e11i	0.200747	+	0.979643i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	2.32468e11			0.369136			0.184568	−	0.982820i	$-0.440911\pi$
$229$	2.32468e11			0.369136			0.184568	+	0.982820i	$0.440911\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4.41959e11i		−	0.643580i	−0.946811	−	0.321790i	$-0.895715\pi$
$233$		−	4.41959e11i		−	0.643580i	0.946811	−	0.321790i	$-0.104285\pi$
$234$		0			0
$235$	−1.03808e11	−	8.46927e10i	−0.144841	−	0.118170i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	8.33246e10			0.106852			0.0534261	−	0.998572i	$-0.482986\pi$
$239$	8.33246e10			0.106852			0.0534261	+	0.998572i	$0.482986\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$	−1.64540e12			−1.90249
$245$	4.15744e11	−	5.09577e11i	0.470972	−	0.577270i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5.29646e11			0.531639			0.265820	−	0.964023i	$-0.414357\pi$
$251$	5.29646e11			0.531639			0.265820	+	0.964023i	$0.414357\pi$
$252$		−	5.13163e11i		−	0.504955i
$253$			2.40207e12i			2.31731i
$254$		0			0
$255$		0			0
$256$	1.09951e12			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$			7.87999e11i			0.626249i	0.949712	+	0.313124i	$0.101376\pi$
$263$			7.87999e11i			0.626249i	−0.949712	+	0.313124i	$0.898624\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	2.83576e12			1.94010			0.970049	−	0.242910i	$-0.0781019\pi$
$271$	2.83576e12			1.94010			0.970049	+	0.242910i	$0.0781019\pi$
$272$			1.94939e12i			1.30935i
$273$		0			0
$274$		0			0
$275$	3.98991e11	+	1.94707e12i	0.253688	+	1.23799i
$276$		0			0
$277$			3.15403e12i			1.93405i	0.254683	+	0.967024i	$0.418029\pi$
$277$			3.15403e12i			1.93405i	−0.254683	+	0.967024i	$0.581971\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		−	2.72803e12i		−	1.50286i	−0.659815	−	0.751428i	$-0.729365\pi$
$283$		−	2.72803e12i		−	1.50286i	0.659815	−	0.751428i	$-0.270635\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.44020e12			−0.714386
$290$		0			0
$291$		0			0
$292$		−	2.76872e12i		−	1.30426i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	1.72487e12			0.698112
$302$		0			0
$303$		0			0
$304$	−2.59638e12			−1.00000
$305$	3.17431e12	−	3.89074e12i	1.20268	−	1.47412i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		−	1.76871e12i		−	0.638120i
$309$		0			0
$310$		0			0
$311$	−1.89531e12			−0.651446			−0.325723	−	0.945465i	$-0.605608\pi$
$311$	−1.89531e12			−0.651446			−0.325723	+	0.945465i	$0.605608\pi$
$312$		0			0
$313$			5.95513e12i			1.98230i	0.132744	+	0.991150i	$0.457621\pi$
$313$			5.95513e12i			1.98230i	−0.132744	+	0.991150i	$0.542379\pi$
$314$		0			0
$315$	1.21343e12	+	9.89993e11i	0.391258	+	0.319212i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$	−2.12118e12	+	2.59992e12i	−0.632160	+	0.774838i
$321$		0			0
$322$		0			0
$323$		−	4.60327e12i		−	1.30935i
$324$	−3.57047e12			−1.00000
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.63841e11			−0.0943913
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		−	7.76071e12i		−	1.92403i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$		0			0
$340$	−4.60956e12	−	3.76076e12i	−1.01453	−	0.827716i
$341$		0			0
$342$		0			0
$343$		−	4.18335e12i		−	0.881157i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	9.27491e12i		−	1.84358i	−0.387687	−	0.921791i	$-0.626726\pi$
$347$		−	9.27491e12i		−	1.84358i	0.387687	−	0.921791i	$-0.373274\pi$
$348$		0			0
$349$	−9.39511e12			−1.81457			−0.907287	−	0.420512i	$-0.861851\pi$
$349$	−9.39511e12			−1.81457			−0.907287	+	0.420512i	$0.861851\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			6.86062e12i			1.25167i	0.779955	+	0.625835i	$0.215242\pi$
$353$			6.86062e12i			1.25167i	−0.779955	+	0.625835i	$0.784758\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.17823e13			−1.97587			−0.987935	−	0.154871i	$-0.950504\pi$
$359$	−1.17823e13			−1.97587			−0.987935	+	0.154871i	$0.950504\pi$
$360$		0			0
$361$	6.13107e12			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.54696e12	+	5.34141e12i	1.01059	+	0.824502i
$366$		0			0
$367$			1.25495e13i			1.88493i	0.334301	+	0.942466i	$0.391500\pi$
$367$			1.25495e13i			1.88493i	−0.334301	+	0.942466i	$0.608500\pi$
$368$		−	1.23758e13i		−	1.83372i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	5.00893e12	−	6.13944e12i	0.632160	−	0.774838i
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$	4.18232e12	+	3.41219e12i	0.494440	+	0.403394i
$386$		0			0
$387$		−	1.20013e13i		−	1.38252i
$388$		0			0
$389$	−1.48910e13			−1.67177			−0.835884	−	0.548906i	$-0.815044\pi$
$389$	−1.48910e13			−1.67177			−0.835884	+	0.548906i	$0.815044\pi$
$390$		0			0
$391$	2.19418e13			2.40098
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−1.23063e13			−1.26372
$397$			1.93498e13i			1.96211i	0.193732	+	0.981054i	$0.437941\pi$
$397$			1.93498e13i			1.96211i	−0.193732	+	0.981054i	$0.562059\pi$
$398$		0			0
$399$		0			0
$400$	−2.05565e12	−	1.00315e13i	−0.200747	−	0.979643i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	−1.02243e13			−0.950010
$405$	6.88814e12	−	8.44279e12i	0.632160	−	0.774838i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1.83511e13	+	1.49720e13i	1.49081	+	1.21629i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.40486e13			−1.08783			−0.543916	−	0.839140i	$-0.683059\pi$
$419$	−1.40486e13			−1.08783			−0.543916	+	0.839140i	$0.683059\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$			2.53152e12i			0.186930i
$424$		0			0
$425$	1.77855e13	−	3.64459e12i	1.28269	−	0.262848i
$426$		0			0
$427$		−	1.36369e13i		−	0.960672i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			2.92241e13i			1.83372i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−1.24269e13			−0.745021
$442$		0			0
$443$			3.30278e13i			1.93580i	0.251335	+	0.967900i	$0.419130\pi$
$443$			3.30278e13i			1.93580i	−0.251335	+	0.967900i	$0.580870\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$			9.11261e12i			0.504955i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	3.23713e13i		−	1.62397i	−0.583675	−	0.811987i	$-0.698386\pi$
$457$		−	3.23713e13i		−	1.62397i	0.583675	−	0.811987i	$-0.301614\pi$
$458$		0			0
$459$		0			0
$460$	2.92640e13	+	2.38754e13i	1.42084	+	1.15921i
$461$	2.38839e13			1.14710			0.573548	−	0.819172i	$-0.305566\pi$
$461$	2.38839e13			1.14710			0.573548	+	0.819172i	$0.305566\pi$
$462$		0			0
$463$			2.30301e12i			0.108241i	0.998534	+	0.0541204i	$0.0172355\pi$
$463$			2.30301e12i			0.108241i	−0.998534	+	0.0541204i	$0.982765\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			3.76252e13i			1.69393i	0.531652	+	0.846963i	$0.321571\pi$
$467$			3.76252e13i			1.69393i	−0.531652	+	0.846963i	$0.678429\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	4.13645e13i		−	1.74712i
$474$		0			0
$475$	4.85421e12	+	2.36884e13i	0.200747	+	0.979643i
$476$	−1.61563e13			−0.661160
$477$		0			0
$478$		0			0
$479$	3.07491e13			1.21942			0.609711	−	0.792624i	$-0.291285\pi$
$479$	3.07491e13			1.21942			0.609711	+	0.792624i	$0.291285\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.58558e13			−0.596982
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.38065e13			0.483811			0.241905	−	0.970300i	$-0.422228\pi$
$491$	1.38065e13			0.483811			0.241905	+	0.970300i	$0.422228\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	2.37412e13	−	2.90996e13i	0.798872	−	0.979176i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.31588e13			−0.748536			−0.374268	−	0.927321i	$-0.622106\pi$
$499$	−2.31588e13			−0.748536			−0.374268	+	0.927321i	$0.622106\pi$
$500$	2.76865e13	+	1.44920e13i	0.885969	+	0.463744i
$501$		0			0
$502$		0			0
$503$			5.96791e13i			1.85345i	0.375734	+	0.926727i	$0.377391\pi$
$503$			5.96791e13i			1.85345i	−0.375734	+	0.926727i	$0.622609\pi$
$504$		0			0
$505$	1.97248e13	−	2.41766e13i	0.600558	−	0.736104i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$	2.29468e13			0.658593
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			8.72534e12i			0.236227i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	−7.89655e13			−1.99885
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−9.78718e13			−2.36254
$530$		0			0
$531$		0			0
$532$		−	2.15184e13i		−	0.504955i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.28314e13			−0.941496
$540$		0			0
$541$	7.59982e13			1.63990			0.819949	−	0.572436i	$-0.194001\pi$
$541$	7.59982e13			1.63990			0.819949	+	0.572436i	$0.194001\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		−	9.43620e13i		−	1.90939i
$549$	−9.48821e13			−1.90249
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	7.59505e13			1.42941
$557$		−	8.56199e13i		−	1.59698i	−0.602010	−	0.798488i	$-0.705633\pi$
$557$		−	8.56199e13i		−	1.59698i	0.602010	−	0.798488i	$-0.294367\pi$
$558$		0			0
$559$		0			0
$560$	−2.15478e13	−	1.75800e13i	−0.391258	−	0.319212i
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	2.95916e13i		−	0.504955i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−1.06969e14			−1.76229			−0.881143	−	0.472850i	$-0.843225\pi$
$571$	−1.06969e14			−1.76229			−0.881143	+	0.472850i	$0.843225\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.12912e14	+	2.31378e13i	−1.79639	+	0.368115i
$576$	6.34034e13			1.00000
$577$			7.48265e13i			1.16998i	0.811042	+	0.584988i	$0.198901\pi$
$577$			7.48265e13i			1.16998i	−0.811042	+	0.584988i	$0.801099\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	6.43197e13			0.971546
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.38365e14i		−	1.98535i	−0.120823	−	0.992674i	$-0.538554\pi$
$587$		−	1.38365e14i		−	1.98535i	0.120823	−	0.992674i	$-0.461446\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	1.45486e14i		−	1.98402i	−0.126152	−	0.992011i	$-0.540263\pi$
$593$		−	1.45486e14i		−	1.98402i	0.126152	−	0.992011i	$-0.459737\pi$
$594$		0			0
$595$	3.11687e13	−	3.82034e13i	0.417959	−	0.512292i
$596$	5.75415e12			0.0765156
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.05890e13	−	3.74929e13i	0.377388	−	0.462565i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$			1.12412e14i			1.30935i
$613$			6.02984e13i			0.696632i	0.937377	+	0.348316i	$0.113246\pi$
$613$			6.02984e13i			0.696632i	−0.937377	+	0.348316i	$0.886754\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			8.47903e13i			0.948244i	0.880459	+	0.474122i	$0.157234\pi$
$617$			8.47903e13i			0.948244i	−0.880459	+	0.474122i	$0.842766\pi$
$618$		0			0
$619$	5.72825e13			0.630332			0.315166	−	0.949037i	$-0.397940\pi$
$619$	5.72825e13			0.630332			0.315166	+	0.949037i	$0.397940\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−8.76809e13	+	3.75101e13i	−0.919401	+	0.393322i
$626$		0			0
$627$		0			0
$628$			1.92884e14i			1.97469i
$629$		0			0
$630$		0			0
$631$	−1.98866e14			−1.98799			−0.993994	−	0.109430i	$-0.965097\pi$
$631$	−1.98866e14			−1.98799			−0.993994	+	0.109430i	$0.965097\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		−	1.11824e14i		−	1.01737i	−0.860952	−	0.508686i	$-0.830131\pi$
$643$		−	1.11824e14i		−	1.01737i	0.860952	−	0.508686i	$-0.169869\pi$
$644$	1.02569e14			0.925947
$645$		0			0
$646$		0			0
$647$			5.95300e13i			0.525066i	0.964923	+	0.262533i	$0.0845579\pi$
$647$			5.95300e13i			0.525066i	−0.964923	+	0.262533i	$0.915442\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$			7.91704e13i			0.671931i
$653$		−	2.71577e13i		−	0.228732i	−0.993439	−	0.114366i	$-0.963516\pi$
$653$		−	2.71577e13i		−	0.228732i	0.993439	−	0.114366i	$-0.0364837\pi$
$654$		0			0
$655$	1.52340e14	−	1.86723e14i	1.26360	−	1.54879i
$656$		0			0
$657$		−	1.59658e14i		−	1.30426i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.08829e13	+	4.15134e13i	0.391258	+	0.319212i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.27028e14			−2.40421
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	1.41167e14			1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−1.49720e14			−1.00000
$685$	2.23130e14	+	1.82043e14i	1.47947	+	1.20704i
$686$		0			0
$687$		0			0
$688$			2.13115e14i			1.38252i
$689$		0			0
$690$		0			0
$691$	1.69827e14			1.07799			0.538997	−	0.842307i	$-0.318803\pi$
$691$	1.69827e14			1.07799			0.538997	+	0.842307i	$0.318803\pi$
$692$		0			0
$693$		−	1.01993e14i		−	0.638120i
$694$		0			0
$695$	−1.46524e14	+	1.79594e14i	−0.903616	+	1.10756i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	8.31401e13	−	1.70370e13i	0.494676	−	0.101368i
$701$	3.30849e14			1.95452			0.977258	−	0.212054i	$-0.0680151\pi$
$701$	3.30849e14			1.95452			0.977258	+	0.212054i	$0.0680151\pi$
$702$		0			0
$703$		0			0
$704$	2.18531e14			1.26372
$705$		0			0
$706$		0			0
$707$		−	8.47379e13i		−	0.479712i
$708$		0			0
$709$	1.12103e14			0.625729			0.312864	−	0.949798i	$-0.398711\pi$
$709$	1.12103e14			0.625729			0.312864	+	0.949798i	$0.398711\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.93381e14			1.00640			0.503199	−	0.864170i	$-0.332156\pi$
$719$	1.93381e14			1.00640			0.503199	+	0.864170i	$0.332156\pi$
$720$	−1.22318e14	+	1.49925e14i	−0.632160	+	0.774838i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			3.88917e14i			1.91507i	0.288312	+	0.957536i	$0.406906\pi$
$727$			3.88917e14i			1.91507i	−0.288312	+	0.957536i	$0.593094\pi$
$728$		0			0
$729$	−2.05891e14			−1.00000
$730$		0			0
$731$	−3.77845e14			−1.81020
$732$		0			0
$733$			2.12619e14i			1.00480i	0.864634	+	0.502402i	$0.167550\pi$
$733$			2.12619e14i			1.00480i	−0.864634	+	0.502402i	$0.832450\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	4.17769e14			1.89546			0.947728	−	0.319080i	$-0.103374\pi$
$739$	4.17769e14			1.89546			0.947728	+	0.319080i	$0.103374\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	−1.11009e13	+	1.36064e13i	−0.0483701	+	0.0592872i
$746$		0			0
$747$		−	4.47522e14i		−	1.92403i
$748$			3.87447e14i			1.65464i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		−	4.49540e13i		−	0.186930i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			4.54743e14i			1.82931i	0.404241	+	0.914653i	$0.367536\pi$
$757$			4.54743e14i			1.82931i	−0.404241	+	0.914653i	$0.632464\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.58961e14			−1.40645			−0.703224	−	0.710968i	$-0.748257\pi$
$761$	−3.58961e14			−1.40645			−0.703224	+	0.710968i	$0.748257\pi$
$762$		0			0
$763$		0			0
$764$	4.90588e14			1.88473
$765$	−2.65811e14	−	2.16865e14i	−1.01453	−	0.827716i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	3.35620e14			1.24800			0.624002	−	0.781423i	$-0.285506\pi$
$769$	3.35620e14			1.24800			0.624002	+	0.781423i	$0.285506\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	2.20673e14			0.745021
$785$	−4.56098e14	−	3.72112e14i	−1.53006	−	1.24832i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		−	2.46942e14i		−	0.812765i
$789$		0			0
$790$		0			0

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:	\( N \)	\(=\)	\( 95 = 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	95.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1024.00 q^{4} +(1975.50 - 2421.37i) q^{5} -8486.78i q^{7} -59049.0 q^{9} +O(q^{10})\)	\(q-1024.00 q^{4} +(1975.50 - 2421.37i) q^{5} -8486.78i q^{7} -59049.0 q^{9} -203523. q^{11} +1.04858e6 q^{16} +1.85908e6i q^{17} -2.47610e6 q^{19} +(-2.02291e6 + 2.47948e6i) q^{20} -1.18025e7i q^{23} +(-1.96042e6 - 9.56683e6i) q^{25} +8.69046e6i q^{28} +(-2.05496e7 - 1.67656e7i) q^{35} +6.04662e7 q^{36} +2.03243e8i q^{43} +2.08408e8 q^{44} +(-1.16651e8 + 1.42979e8i) q^{45} -4.28715e7i q^{47} +2.10450e8 q^{49} +(-4.02060e8 + 4.92804e8i) q^{55} +1.60684e9 q^{61} +5.01136e8i q^{63} -1.07374e9 q^{64} -1.90370e9i q^{68} +2.70383e9i q^{73} +2.53553e9 q^{76} +1.72725e9i q^{77} +(2.07146e9 - 2.53899e9i) q^{80} +3.48678e9 q^{81} +7.57882e9i q^{83} +(4.50153e9 + 3.67262e9i) q^{85} +1.20857e10i q^{92} +(-4.89153e9 + 5.99555e9i) q^{95} +1.20178e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2048 q^{4} + 3951 q^{5} - 118098 q^{9}+O(q^{10}) \) 2 * q - 2048 * q^4 + 3951 * q^5 - 118098 * q^9	\( 2 q - 2048 q^{4} + 3951 q^{5} - 118098 q^{9} - 407046 q^{11} + 2097152 q^{16} - 4952198 q^{19} - 4045824 q^{20} - 3920849 q^{25} - 41099223 q^{35} + 120932352 q^{36} + 416815104 q^{44} - 233302599 q^{45} + 420899756 q^{49} - 804119373 q^{55} + 3213673954 q^{61} - 2147483648 q^{64} + 5071050752 q^{76} + 4142923776 q^{80} + 6973568802 q^{81} + 9003051827 q^{85} - 9783067149 q^{95} + 24035659254 q^{99}+O(q^{100}) \) 2 * q - 2048 * q^4 + 3951 * q^5 - 118098 * q^9 - 407046 * q^11 + 2097152 * q^16 - 4952198 * q^19 - 4045824 * q^20 - 3920849 * q^25 - 41099223 * q^35 + 120932352 * q^36 + 416815104 * q^44 - 233302599 * q^45 + 420899756 * q^49 - 804119373 * q^55 + 3213673954 * q^61 - 2147483648 * q^64 + 5071050752 * q^76 + 4142923776 * q^80 + 6973568802 * q^81 + 9003051827 * q^85 - 9783067149 * q^95 + 24035659254 * q^99

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