LMFDB - Embedded newform 95.11.d.a.94.2

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1024.00 q^{4} +(1975.50 + 2421.37i) q^{5} +8486.78i q^{7} -59049.0 q^{9} +O(q^{10})$	$q-1024.00 q^{4} +(1975.50 + 2421.37i) q^{5} +8486.78i q^{7} -59049.0 q^{9} -203523. q^{11} +1.04858e6 q^{16} -1.85908e6i q^{17} -2.47610e6 q^{19} +(-2.02291e6 - 2.47948e6i) q^{20} +1.18025e7i q^{23} +(-1.96042e6 + 9.56683e6i) q^{25} -8.69046e6i q^{28} +(-2.05496e7 + 1.67656e7i) q^{35} +6.04662e7 q^{36} -2.03243e8i q^{43} +2.08408e8 q^{44} +(-1.16651e8 - 1.42979e8i) q^{45} +4.28715e7i q^{47} +2.10450e8 q^{49} +(-4.02060e8 - 4.92804e8i) q^{55} +1.60684e9 q^{61} -5.01136e8i q^{63} -1.07374e9 q^{64} +1.90370e9i q^{68} -2.70383e9i q^{73} +2.53553e9 q^{76} -1.72725e9i q^{77} +(2.07146e9 + 2.53899e9i) q^{80} +3.48678e9 q^{81} -7.57882e9i q^{83} +(4.50153e9 - 3.67262e9i) q^{85} -1.20857e10i q^{92} +(-4.89153e9 - 5.99555e9i) q^{95} +1.20178e10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.0812453\pi$
$7$			8486.78i			0.504955i	−0.967603	+	0.252477i	$0.918755\pi$
$8$		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	$-0.772806\pi$
$17$		−	1.85908e6i		−	1.30935i	0.755912	−	0.654673i	$-0.227194\pi$
$18$		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	$0.369286\pi$
$23$			1.18025e7i			1.83372i	−0.399206	+	0.916861i	$0.630714\pi$
$24$		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	$-0.757055\pi$
$43$		−	2.03243e8i		−	1.38252i	0.722605	−	0.691262i	$-0.242945\pi$
$44$	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	$0.0297943\pi$
$47$			4.28715e7i			0.186930i	−0.995623	+	0.0934651i	$0.970206\pi$
$48$		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	$-0.773875\pi$
$73$		−	2.70383e9i		−	1.30426i	0.758106	−	0.652131i	$-0.226125\pi$
$74$		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	$-0.588017\pi$
$83$		−	7.57882e9i		−	1.92403i	0.273003	−	0.962013i	$-0.411983\pi$
$84$		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	$-0.596182\pi$
$137$		−	9.21504e10i		−	1.90939i	0.297587	−	0.954695i	$-0.403818\pi$
$138$		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	$0.449305\pi$
$157$			1.88364e11i			1.97469i	−0.158590	+	0.987344i	$0.550695\pi$
$158$		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	$0.109063\pi$
$163$			7.73148e10i			0.671931i	−0.941874	+	0.335966i	$0.890937\pi$
$164$		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	$-0.866790\pi$
$197$		−	2.41155e11i		−	0.812765i	0.913703	−	0.406382i	$-0.133210\pi$
$198$		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	$0.104285\pi$
$233$			4.41959e11i			0.643580i	−0.946811	+	0.321790i	$0.895715\pi$
$234$		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	$-0.898624\pi$
$263$		−	7.87999e11i		−	0.626249i	0.949712	−	0.313124i	$-0.101376\pi$
$264$		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	$-0.581971\pi$
$277$		−	3.15403e12i		−	1.93405i	0.254683	−	0.967024i	$-0.418029\pi$
$278$		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	$0.270635\pi$
$283$			2.72803e12i			1.50286i	−0.659815	+	0.751428i	$0.729365\pi$
$284$		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	$-0.542379\pi$
$313$		−	5.95513e12i		−	1.98230i	0.132744	−	0.991150i	$-0.457621\pi$
$314$		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	$0.373274\pi$
$347$			9.27491e12i			1.84358i	−0.387687	+	0.921791i	$0.626726\pi$
$348$		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	$-0.784758\pi$
$353$		−	6.86062e12i		−	1.25167i	0.779955	−	0.625835i	$-0.215242\pi$
$354$		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	$-0.608500\pi$
$367$		−	1.25495e13i		−	1.88493i	0.334301	−	0.942466i	$-0.391500\pi$
$368$			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	$-0.562059\pi$
$397$		−	1.93498e13i		−	1.96211i	0.193732	−	0.981054i	$-0.437941\pi$
$398$		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	$-0.580870\pi$
$443$		−	3.30278e13i		−	1.93580i	0.251335	−	0.967900i	$-0.419130\pi$
$444$		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	$0.301614\pi$
$457$			3.23713e13i			1.62397i	−0.583675	+	0.811987i	$0.698386\pi$
$458$		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	$-0.982765\pi$
$463$		−	2.30301e12i		−	0.108241i	0.998534	−	0.0541204i	$-0.0172355\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	3.76252e13i		−	1.69393i	−0.531652	−	0.846963i	$-0.678429\pi$
$467$		−	3.76252e13i		−	1.69393i	0.531652	−	0.846963i	$-0.321571\pi$
$468$		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	$-0.622609\pi$
$503$		−	5.96791e13i		−	1.85345i	0.375734	−	0.926727i	$-0.377391\pi$
$504$		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	$0.294367\pi$
$557$			8.56199e13i			1.59698i	−0.602010	+	0.798488i	$0.705633\pi$
$558$		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	$-0.801099\pi$
$577$		−	7.48265e13i		−	1.16998i	0.811042	−	0.584988i	$-0.198901\pi$
$578$		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	$0.461446\pi$
$587$			1.38365e14i			1.98535i	−0.120823	+	0.992674i	$0.538554\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			1.45486e14i			1.98402i	0.126152	+	0.992011i	$0.459737\pi$
$593$			1.45486e14i			1.98402i	−0.126152	+	0.992011i	$0.540263\pi$
$594$		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	$-0.886754\pi$
$613$		−	6.02984e13i		−	0.696632i	0.937377	−	0.348316i	$-0.113246\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	8.47903e13i		−	0.948244i	−0.880459	−	0.474122i	$-0.842766\pi$
$617$		−	8.47903e13i		−	0.948244i	0.880459	−	0.474122i	$-0.157234\pi$
$618$		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	$0.169869\pi$
$643$			1.11824e14i			1.01737i	−0.860952	+	0.508686i	$0.830131\pi$
$644$	1.02569e14			0.925947
$645$		0			0
$646$		0			0
$647$		−	5.95300e13i		−	0.525066i	−0.964923	−	0.262533i	$-0.915442\pi$
$647$		−	5.95300e13i		−	0.525066i	0.964923	−	0.262533i	$-0.0845579\pi$
$648$		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	$0.0364837\pi$
$653$			2.71577e13i			0.228732i	−0.993439	+	0.114366i	$0.963516\pi$
$654$		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	$-0.593094\pi$
$727$		−	3.88917e14i		−	1.91507i	0.288312	−	0.957536i	$-0.406906\pi$
$728$		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	$-0.832450\pi$
$733$		−	2.12619e14i		−	1.00480i	0.864634	−	0.502402i	$-0.167550\pi$
$734$		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	$-0.632464\pi$
$757$		−	4.54743e14i		−	1.82931i	0.404241	−	0.914653i	$-0.367536\pi$
$758$		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
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−6.82082e13			−0.213438
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	7.97017e13			0.244756
$800$		0			0
$801$		0			0
$802$		0			0
$803$			5.50291e14i			1.64822i
$804$		0			0
$805$	−1.97876e14	−	2.42536e14i	−0.585347	−	0.717459i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−4.41538e14			−1.27417			−0.637083	−	0.770796i	$-0.719859\pi$
$809$	−4.41538e14			−1.27417			−0.637083	+	0.770796i	$0.719859\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.87208e14	+	1.52735e14i	−0.520638	+	0.424768i
$816$		0			0
$817$			5.03249e14i			1.38252i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5.40843e14			−1.44996			−0.724979	−	0.688771i	$-0.758150\pi$
$821$	−5.40843e14			−1.44996			−0.724979	+	0.688771i	$0.758150\pi$
$822$		0			0
$823$			7.55136e14i			1.99998i	0.00430994	+	0.999991i	$0.498628\pi$
$823$			7.55136e14i			1.99998i	−0.00430994	+	0.999991i	$0.501372\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$			7.13650e14i			1.83372i
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	3.91244e14i		−	0.975489i
$834$		0			0
$835$		0			0
$836$	−5.16038e14			−1.26372
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	4.20707e14			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.72339e14	−	3.33806e14i	−0.632160	−	0.774838i
$846$		0			0
$847$			1.31411e14i			0.301449i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	8.98723e14i		−	1.99013i	−0.0992374	−	0.995064i	$-0.531640\pi$
$853$		−	8.98723e14i		−	1.99013i	0.0992374	−	0.995064i	$-0.468360\pi$
$854$		0			0
$855$	2.88840e14	+	3.54031e14i	0.632160	+	0.774838i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	7.95565e14			1.70102			0.850511	−	0.525958i	$-0.176293\pi$
$859$	7.95565e14			1.70102			0.850511	+	0.525958i	$0.176293\pi$
$860$	−5.03936e14	+	4.11142e14i	−1.07123	+	0.873976i
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.20108e14	−	2.29462e14i	−0.234170	−	0.447374i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$	−4.21590e14	−	5.16743e14i	−0.798872	−	0.979176i
$881$	1.06121e15			1.99950			0.999751	−	0.0223239i	$-0.00710652\pi$
$881$	1.06121e15			1.99950			0.999751	+	0.0223239i	$0.00710652\pi$
$882$		0			0
$883$		−	8.35357e14i		−	1.55621i	−0.628133	−	0.778106i	$-0.716181\pi$
$883$		−	8.35357e14i		−	1.55621i	0.628133	−	0.778106i	$-0.283819\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−7.09641e14			−1.26372
$892$		0			0
$893$		−	1.06154e14i		−	0.186930i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−1.18539e14	+	5.78469e14i	−0.200747	+	0.979643i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$	−5.89586e14			−0.950010
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$			1.54246e15i			2.43143i
$914$		0			0
$915$		0			0
$916$	−2.38047e14			−0.369136
$917$			6.54456e14i			1.00933i
$918$		0			0
$919$	1.67257e14			0.255157			0.127578	−	0.991828i	$-0.459280\pi$
$919$	1.67257e14			0.255157			0.127578	+	0.991828i	$0.459280\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.35815e15			1.96278			0.981388	−	0.192036i	$-0.0615090\pi$
$929$	1.35815e15			1.96278			0.981388	+	0.192036i	$0.0615090\pi$
$930$		0			0
$931$	−5.21095e14			−0.745021
$932$		−	4.52566e14i		−	0.643580i
$933$		0			0
$934$		0			0
$935$	−9.16164e14	+	7.47463e14i	−1.28208	+	1.04600i
$936$		0			0
$937$		−	8.99475e14i		−	1.24535i	−0.782481	−	0.622675i	$-0.786046\pi$
$937$		−	8.99475e14i		−	1.24535i	0.782481	−	0.622675i	$-0.213954\pi$
$938$		0			0
$939$		0			0
$940$	1.06299e14	−	8.67253e13i	0.144841	−	0.118170i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1.46919e15i			1.92898i	0.264112	+	0.964492i	$0.414921\pi$
$947$			1.46919e15i			1.92898i	−0.264112	+	0.964492i	$0.585079\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	−9.46442e14	−	1.16005e15i	−1.19145	−	1.46036i
$956$	−8.53244e13			−0.106852
$957$		0			0
$958$		0			0
$959$	7.82060e14			0.964155
$960$		0			0
$961$	8.19628e14			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1.57611e15i		−	1.86404i	−0.362408	−	0.932019i	$-0.618045\pi$
$967$		−	1.57611e15i		−	1.86404i	0.362408	−	0.932019i	$-0.381955\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		−	6.29468e14i		−	0.721788i
$974$		0			0
$975$		0			0
$976$	1.68489e15			1.90249
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$	−4.25722e14	−	5.21807e14i	−0.470972	−	0.577270i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	5.83925e14	−	4.76401e14i	0.629761	−	0.513797i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.39876e15			2.53516
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.31587e14	+	1.61286e14i	0.134927	+	0.165380i
$996$		0			0
$997$		−	1.25529e15i		−	1.27429i	−0.770744	−	0.637145i	$-0.780115\pi$
$997$		−	1.25529e15i		−	1.27429i	0.770744	−	0.637145i	$-0.219885\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.11.d.a.94.2	yes	2
5.4	even	2	inner	95.11.d.a.94.1	✓	2
19.18	odd	2	CM	95.11.d.a.94.2	yes	2
95.94	odd	2	inner	95.11.d.a.94.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.11.d.a.94.1	✓	2	5.4	even	2	inner
95.11.d.a.94.1	✓	2	95.94	odd	2	inner
95.11.d.a.94.2	yes	2	1.1	even	1	trivial
95.11.d.a.94.2	yes	2	19.18	odd	2	CM

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

Introduction

L-functions

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