LMFDB - Embedded newform 95.9.d.c.94.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,9,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, 9, names="a")

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

chi := DirichletCharacter("95.94");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 95 = 5 \cdot 19 $
Weight:	$ k $	$=$	$ 9 $
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:	$38.7009679558$
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:	$ 3 $
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			95.9.d.c.94.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-256.000 q^{4} +(-144.500 + 608.066i) q^{5} +4772.99i q^{7} -6561.00 q^{9} +O(q^{10})$	$q-256.000 q^{4} +(-144.500 + 608.066i) q^{5} +4772.99i q^{7} -6561.00 q^{9} -25007.0 q^{11} +65536.0 q^{16} +161563. i q^{17} +130321. q^{19} +(36992.0 - 155665. i) q^{20} +165289. i q^{23} +(-348864. - 175731. i) q^{25} -1.22189e6i q^{28} +(-2.90230e6 - 689698. i) q^{35} +1.67962e6 q^{36} -3.92033e6i q^{43} +6.40179e6 q^{44} +(948064. - 3.98952e6i) q^{45} +5.12966e6i q^{47} -1.70167e7 q^{49} +(3.61351e6 - 1.52059e7i) q^{55} +1.76618e7 q^{61} -3.13156e7i q^{63} -1.67772e7 q^{64} -4.13600e7i q^{68} +3.60796e7i q^{73} -3.33622e7 q^{76} -1.19358e8i q^{77} +(-9.46995e6 + 3.98502e7i) q^{80} +4.30467e7 q^{81} +7.12795e7i q^{83} +(-9.82408e7 - 2.33458e7i) q^{85} -4.23141e7i q^{92} +(-1.88314e7 + 7.92438e7i) q^{95} +1.64071e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 512 q^{4} - 289 q^{5} - 13122 q^{9}+O(q^{10}) $ 2 * q - 512 * q^4 - 289 * q^5 - 13122 * q^9	$ 2 q - 512 q^{4} - 289 q^{5} - 13122 q^{9} - 50014 q^{11} + 131072 q^{16} + 260642 q^{19} + 73984 q^{20} - 697729 q^{25} - 5804595 q^{35} + 3359232 q^{36} + 12803584 q^{44} + 1896129 q^{45} - 34033348 q^{49} + 7227023 q^{55} + 35323586 q^{61} - 33554432 q^{64} - 66724352 q^{76} - 18939904 q^{80} + 86093442 q^{81} - 196481565 q^{85} - 37662769 q^{95} + 328141854 q^{99}+O(q^{100}) $ 2 * q - 512 * q^4 - 289 * q^5 - 13122 * q^9 - 50014 * q^11 + 131072 * q^16 + 260642 * q^19 + 73984 * q^20 - 697729 * q^25 - 5804595 * q^35 + 3359232 * q^36 + 12803584 * q^44 + 1896129 * q^45 - 34033348 * q^49 + 7227023 * q^55 + 35323586 * q^61 - 33554432 * q^64 - 66724352 * q^76 - 18939904 * q^80 + 86093442 * q^81 - 196481565 * q^85 - 37662769 * q^95 + 328141854 * 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	$0.5\pi$
$4$	−256.000			−1.00000
$5$	−144.500	+	608.066i	−0.231200	+	0.972906i
$5$	−144.500	+	608.066i	−0.231200	+	0.972906i
$6$		0			0
$7$			4772.99i			1.98792i	0.109746	+	0.993960i	$0.464996\pi$
$7$			4772.99i			1.98792i	−0.109746	+	0.993960i	$0.535004\pi$
$8$		0			0
$9$	−6561.00			−1.00000
$10$		0			0
$11$	−25007.0			−1.70801			−0.854006	−	0.520263i	$-0.825834\pi$
$11$	−25007.0			−1.70801			−0.854006	+	0.520263i	$0.825834\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	65536.0			1.00000
$17$			161563.i			1.93439i	0.254026	+	0.967197i	$0.418245\pi$
$17$			161563.i			1.93439i	−0.254026	+	0.967197i	$0.581755\pi$
$18$		0			0
$19$	130321.			1.00000
$19$	130321.			1.00000
$20$	36992.0	−	155665.i	0.231200	−	0.972906i
$21$		0			0
$22$		0			0
$23$			165289.i			0.590655i	0.955396	+	0.295327i	$0.0954287\pi$
$23$			165289.i			0.590655i	−0.955396	+	0.295327i	$0.904571\pi$
$24$		0			0
$25$	−348864.	−	175731.i	−0.893093	−	0.449872i
$26$		0			0
$27$		0			0
$28$		−	1.22189e6i		−	1.98792i
$29$		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
$35$	−2.90230e6	−	689698.i	−1.93406	−	0.459607i
$36$	1.67962e6			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
$43$		−	3.92033e6i		−	1.14670i	−0.819312	−	0.573348i	$-0.805644\pi$
$43$		−	3.92033e6i		−	1.14670i	0.819312	−	0.573348i	$-0.194356\pi$
$44$	6.40179e6			1.70801
$45$	948064.	−	3.98952e6i	0.231200	−	0.972906i
$46$		0			0
$47$			5.12966e6i			1.05123i	0.850723	+	0.525614i	$0.176165\pi$
$47$			5.12966e6i			1.05123i	−0.850723	+	0.525614i	$0.823835\pi$
$48$		0			0
$49$	−1.70167e7			−2.95182
$50$		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
$55$	3.61351e6	−	1.52059e7i	0.394892	−	1.66174i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	1.76618e7			1.27560			0.637801	−	0.770201i	$-0.279844\pi$
$61$	1.76618e7			1.27560			0.637801	+	0.770201i	$0.279844\pi$
$62$		0			0
$63$		−	3.13156e7i		−	1.98792i
$64$	−1.67772e7			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		−	4.13600e7i		−	1.93439i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$			3.60796e7i			1.27049i	0.772312	+	0.635243i	$0.219100\pi$
$73$			3.60796e7i			1.27049i	−0.772312	+	0.635243i	$0.780900\pi$
$74$		0			0
$75$		0			0
$76$	−3.33622e7			−1.00000
$77$		−	1.19358e8i		−	3.39539i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	−9.46995e6	+	3.98502e7i	−0.231200	+	0.972906i
$81$	4.30467e7			1.00000
$82$		0			0
$83$			7.12795e7i			1.50194i	0.660337	+	0.750970i	$0.270414\pi$
$83$			7.12795e7i			1.50194i	−0.660337	+	0.750970i	$0.729586\pi$
$84$		0			0
$85$	−9.82408e7	−	2.33458e7i	−1.88198	−	0.447232i
$86$		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
$92$		−	4.23141e7i		−	0.590655i
$93$		0			0
$94$		0			0
$95$	−1.88314e7	+	7.92438e7i	−0.231200	+	0.972906i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$	1.64071e8			1.70801
$100$	8.93093e7	+	4.49872e7i	0.893093	+	0.449872i
$101$	1.08161e8			1.03940			0.519702	−	0.854348i	$-0.326043\pi$
$101$	1.08161e8			1.03940			0.519702	+	0.854348i	$0.326043\pi$
$102$		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
$112$			3.12803e8i			1.98792i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	−1.00507e8	−	2.38843e7i	−0.574652	−	0.136559i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.71137e8			−3.84542
$120$		0			0
$121$	4.10991e8			1.91730
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.57267e8	−	1.86740e8i	0.644166	−	0.764886i
$126$		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
$131$	−4.66964e8			−1.58562			−0.792808	−	0.609472i	$-0.791382\pi$
$131$	−4.66964e8			−1.58562			−0.792808	+	0.609472i	$0.791382\pi$
$132$		0			0
$133$			6.22021e8i			1.98792i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			7.02704e8i			1.99476i	0.0723645	+	0.997378i	$0.476946\pi$
$137$			7.02704e8i			1.99476i	−0.0723645	+	0.997378i	$0.523054\pi$
$138$		0			0
$139$	7.46574e8			1.99993			0.999963	−	0.00864336i	$-0.00275130\pi$
$139$	7.46574e8			1.99993			0.999963	+	0.00864336i	$0.00275130\pi$
$140$	7.42988e8	+	1.76563e8i	1.93406	+	0.459607i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−4.29982e8			−1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8.14865e8			1.65326			0.826629	−	0.562747i	$-0.190255\pi$
$149$	8.14865e8			1.65326			0.826629	+	0.562747i	$0.190255\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		−	1.06001e9i		−	1.93439i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			1.54403e8i			0.254131i	0.991894	+	0.127066i	$0.0405559\pi$
$157$			1.54403e8i			0.254131i	−0.991894	+	0.127066i	$0.959444\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−7.88926e8			−1.17417
$162$		0			0
$163$			4.89696e8i			0.693707i	0.937919	+	0.346854i	$0.112750\pi$
$163$			4.89696e8i			0.693707i	−0.937919	+	0.346854i	$0.887250\pi$
$164$		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
$169$	−8.15731e8			−1.00000
$170$		0			0
$171$	−8.55036e8			−1.00000
$172$			1.00360e9i			1.14670i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	8.38764e8	−	1.66513e9i	0.894309	−	1.77540i
$176$	−1.63886e9			−1.70801
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	−2.42705e8	+	1.02132e9i	−0.231200	+	0.972906i
$181$		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
$187$		−	4.04020e9i		−	3.30397i
$188$		−	1.31319e9i		−	1.05123i
$189$		0			0
$190$		0			0
$191$	1.47444e9			1.10788			0.553939	−	0.832557i	$-0.313124\pi$
$191$	1.47444e9			1.10788			0.553939	+	0.832557i	$0.313124\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	4.35627e9			2.95182
$197$		−	2.39993e9i		−	1.59343i	−0.604352	−	0.796717i	$-0.706568\pi$
$197$		−	2.39993e9i		−	1.59343i	0.604352	−	0.796717i	$-0.293432\pi$
$198$		0			0
$199$	−2.37068e9			−1.51168			−0.755842	−	0.654754i	$-0.772772\pi$
$199$	−2.37068e9			−1.51168			−0.755842	+	0.654754i	$0.772772\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	1.08446e9i		−	0.590655i
$208$		0			0
$209$	−3.25894e9			−1.70801
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.38382e9	+	5.66487e8i	1.11563	+	0.265116i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−9.25059e8	+	3.89271e9i	−0.394892	+	1.66174i
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	2.28890e9	+	1.15297e9i	0.893093	+	0.449872i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	−5.43958e9			−1.97799			−0.988993	−	0.147960i	$-0.952729\pi$
$229$	−5.43958e9			−1.97799			−0.988993	+	0.147960i	$0.952729\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1.52732e9i			0.518210i	0.965849	+	0.259105i	$0.0834276\pi$
$233$			1.52732e9i			0.518210i	−0.965849	+	0.259105i	$0.916572\pi$
$234$		0			0
$235$	−3.11917e9	−	7.41236e8i	−1.02275	−	0.243044i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.74938e9			0.536157			0.268079	−	0.963397i	$-0.413611\pi$
$239$	1.74938e9			0.536157			0.268079	+	0.963397i	$0.413611\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$	−4.52142e9			−1.27560
$245$	2.45891e9	−	1.03473e10i	0.682462	−	2.87185i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.75507e9			1.95385			0.976924	−	0.213585i	$-0.0685141\pi$
$251$	7.75507e9			1.95385			0.976924	+	0.213585i	$0.0685141\pi$
$252$			8.01680e9i			1.98792i
$253$		−	4.13339e9i		−	1.00885i
$254$		0			0
$255$		0			0
$256$	4.29497e9			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
$263$			3.49169e9i			0.729815i	0.931044	+	0.364908i	$0.118899\pi$
$263$			3.49169e9i			0.729815i	−0.931044	+	0.364908i	$0.881101\pi$
$264$		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
$271$	−5.27029e9			−0.977141			−0.488571	−	0.872524i	$-0.662482\pi$
$271$	−5.27029e9			−0.977141			−0.488571	+	0.872524i	$0.662482\pi$
$272$			1.05882e10i			1.93439i
$273$		0			0
$274$		0			0
$275$	8.72405e9	+	4.39451e9i	1.52541	+	0.768386i
$276$		0			0
$277$			1.02173e10i			1.73546i	0.497032	+	0.867732i	$0.334423\pi$
$277$			1.02173e10i			1.73546i	−0.497032	+	0.867732i	$0.665577\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		−	6.64996e8i		−	0.103675i	−0.998656	−	0.0518374i	$-0.983492\pi$
$283$		−	6.64996e8i		−	0.103675i	0.998656	−	0.0518374i	$-0.0165077\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.91267e10			−2.74188
$290$		0			0
$291$		0			0
$292$		−	9.23637e9i		−	1.27049i
$293$		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
$301$	1.87117e10			2.27954
$302$		0			0
$303$		0			0
$304$	8.54072e9			1.00000
$305$	−2.55213e9	+	1.07395e10i	−0.294919	+	1.24104i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$			3.05557e10i			3.39539i
$309$		0			0
$310$		0			0
$311$	−1.02469e10			−1.09535			−0.547673	−	0.836693i	$-0.684486\pi$
$311$	−1.02469e10			−1.09535			−0.547673	+	0.836693i	$0.684486\pi$
$312$		0			0
$313$		−	1.87835e10i		−	1.95704i	−0.206160	−	0.978518i	$-0.566097\pi$
$313$		−	1.87835e10i		−	1.95704i	0.206160	−	0.978518i	$-0.433903\pi$
$314$		0			0
$315$	1.90420e10	+	4.52511e9i	1.93406	+	0.459607i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$	2.42431e9	−	1.02017e10i	0.231200	−	0.972906i
$321$		0			0
$322$		0			0
$323$			2.10550e10i			1.93439i
$324$	−1.10200e10			−1.00000
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.44838e10			−2.08976
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		−	1.82476e10i		−	1.50194i
$333$		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
$340$	2.51496e10	+	5.97652e9i	1.88198	+	0.447232i
$341$		0			0
$342$		0			0
$343$		−	5.37051e10i		−	3.88007i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	2.33846e10i		−	1.61292i	−0.591289	−	0.806460i	$-0.701381\pi$
$347$		−	2.33846e10i		−	1.61292i	0.591289	−	0.806460i	$-0.298619\pi$
$348$		0			0
$349$	−2.85063e10			−1.92150			−0.960748	−	0.277423i	$-0.910520\pi$
$349$	−2.85063e10			−1.92150			−0.960748	+	0.277423i	$0.910520\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			2.70940e9i			0.174492i	0.996187	+	0.0872459i	$0.0278066\pi$
$353$			2.70940e9i			0.174492i	−0.996187	+	0.0872459i	$0.972193\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.26623e9			−0.377250			−0.188625	−	0.982049i	$-0.560403\pi$
$359$	−6.26623e9			−0.377250			−0.188625	+	0.982049i	$0.560403\pi$
$360$		0			0
$361$	1.69836e10			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.19388e10	−	5.21350e9i	−1.23606	−	0.293737i
$366$		0			0
$367$		−	9.77167e9i		−	0.538647i	−0.963050	−	0.269324i	$-0.913200\pi$
$367$		−	9.77167e9i		−	0.538647i	0.963050	−	0.269324i	$-0.0868002\pi$
$368$			1.08324e10i			0.590655i
$369$		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			$\pi$
$380$	4.82083e9	−	2.02864e10i	0.231200	−	0.972906i
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$	7.25778e10	+	1.72473e10i	3.30340	+	0.785014i
$386$		0			0
$387$			2.57213e10i			1.14670i
$388$		0			0
$389$	3.21750e10			1.40514			0.702570	−	0.711614i	$-0.252036\pi$
$389$	3.21750e10			1.40514			0.702570	+	0.711614i	$0.252036\pi$
$390$		0			0
$391$	−2.67046e10			−1.14256
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−4.20022e10			−1.70801
$397$		−	4.90609e10i		−	1.97503i	−0.157528	−	0.987514i	$-0.550353\pi$
$397$		−	4.90609e10i		−	1.97503i	0.157528	−	0.987514i	$-0.449647\pi$
$398$		0			0
$399$		0			0
$400$	−2.28632e10	−	1.15167e10i	−0.893093	−	0.449872i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	−2.76892e10			−1.03940
$405$	−6.22025e9	+	2.61753e10i	−0.231200	+	0.972906i
$406$		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
$415$	−4.33427e10	−	1.02999e10i	−1.46125	−	0.347248i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.96298e9			−0.290801			−0.145401	−	0.989373i	$-0.546447\pi$
$419$	−8.96298e9			−0.290801			−0.145401	+	0.989373i	$0.546447\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		−	3.36557e10i		−	1.05123i
$424$		0			0
$425$	2.83916e10	−	5.63635e10i	0.870230	−	1.72759i
$426$		0			0
$427$			8.42996e10i			2.53580i
$428$		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
$437$			2.15407e10i			0.590655i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	1.11646e11			2.95182
$442$		0			0
$443$			1.55481e10i			0.403703i	0.979416	+	0.201851i	$0.0646958\pi$
$443$			1.55481e10i			0.403703i	−0.979416	+	0.201851i	$0.935304\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		−	8.00776e10i		−	1.98792i
$449$		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
$457$			1.12846e10i			0.258714i	0.991598	+	0.129357i	$0.0412913\pi$
$457$			1.12846e10i			0.258714i	−0.991598	+	0.129357i	$0.958709\pi$
$458$		0			0
$459$		0			0
$460$	2.57298e10	+	6.11439e9i	0.574652	+	0.136559i
$461$	−3.96004e10			−0.876792			−0.438396	−	0.898782i	$-0.644453\pi$
$461$	−3.96004e10			−0.876792			−0.438396	+	0.898782i	$0.644453\pi$
$462$		0			0
$463$			5.71914e10i			1.24453i	0.782805	+	0.622267i	$0.213788\pi$
$463$			5.71914e10i			1.24453i	−0.782805	+	0.622267i	$0.786212\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			6.87799e10i			1.44609i	0.690803	+	0.723043i	$0.257257\pi$
$467$			6.87799e10i			1.44609i	−0.690803	+	0.723043i	$0.742743\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			9.80357e10i			1.95857i
$474$		0			0
$475$	−4.54644e10	−	2.29015e10i	−0.893093	−	0.449872i
$476$	1.97411e11			3.84542
$477$		0			0
$478$		0			0
$479$	−7.83105e10			−1.48757			−0.743786	−	0.668418i	$-0.766972\pi$
$479$	−7.83105e10			−1.48757			−0.743786	+	0.668418i	$0.766972\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.05214e11			−1.91730
$485$		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
$491$	1.37649e10			0.236835			0.118417	−	0.992964i	$-0.462218\pi$
$491$	1.37649e10			0.236835			0.118417	+	0.992964i	$0.462218\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−2.37082e10	+	9.97660e10i	−0.394892	+	1.66174i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	7.36149e10			1.18731			0.593654	−	0.804720i	$-0.297685\pi$
$499$	7.36149e10			1.18731			0.593654	+	0.804720i	$0.297685\pi$
$500$	−4.02604e10	+	4.78053e10i	−0.644166	+	0.764886i
$501$		0			0
$502$		0			0
$503$			1.28025e11i			1.99996i	0.00600796	+	0.999982i	$0.498088\pi$
$503$			1.28025e11i			1.99996i	−0.00600796	+	0.999982i	$0.501912\pi$
$504$		0			0
$505$	−1.56292e10	+	6.57689e10i	−0.240310	+	1.01124i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$	−1.72208e11			−2.52563
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	1.28277e11i		−	1.79551i
$518$		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	$0.5\pi$
$524$	1.19543e11			1.58562
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	5.09904e10			0.651127
$530$		0			0
$531$		0			0
$532$		−	1.59237e11i		−	1.98792i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.25536e11			5.04175
$540$		0			0
$541$	2.95623e10			0.345103			0.172552	−	0.985000i	$-0.444799\pi$
$541$	2.95623e10			0.345103			0.172552	+	0.985000i	$0.444799\pi$
$542$		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	$0.5\pi$
$548$		−	1.79892e11i		−	1.99476i
$549$	−1.15879e11			−1.27560
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.91123e11			−1.99993
$557$		−	1.29781e11i		−	1.34832i	−0.738587	−	0.674158i	$-0.764507\pi$
$557$		−	1.29781e11i		−	1.34832i	0.738587	−	0.674158i	$-0.235493\pi$
$558$		0			0
$559$		0			0
$560$	−1.90205e11	−	4.52000e10i	−1.93406	−	0.459607i
$561$		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
$567$			2.05462e11i			1.98792i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	1.69603e10			0.159547			0.0797734	−	0.996813i	$-0.474580\pi$
$571$	1.69603e10			0.159547			0.0797734	+	0.996813i	$0.474580\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.90465e10	−	5.76636e10i	0.265719	−	0.527510i
$576$	1.10075e11			1.00000
$577$		−	1.06260e11i		−	0.958661i	−0.877634	−	0.479331i	$-0.840880\pi$
$577$		−	1.06260e11i		−	0.958661i	0.877634	−	0.479331i	$-0.159120\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.40217e11			−2.98573
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			2.17676e11i			1.83340i	0.399576	+	0.916700i	$0.369157\pi$
$587$			2.17676e11i			1.83340i	−0.399576	+	0.916700i	$0.630843\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	2.49833e10i		−	0.202037i	−0.994885	−	0.101019i	$-0.967790\pi$
$593$		−	2.49833e10i		−	0.202037i	0.994885	−	0.101019i	$-0.0322102\pi$
$594$		0			0
$595$	1.11429e11	−	4.68903e11i	0.889061	−	3.74123i
$596$	−2.08606e11			−1.65326
$597$		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
$605$	−5.93882e10	+	2.49910e11i	−0.443281	+	1.86536i
$606$		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
$612$			2.71363e11i			1.93439i
$613$		−	2.33273e11i		−	1.65205i	−0.563634	−	0.826025i	$-0.690597\pi$
$613$		−	2.33273e11i		−	1.65205i	0.563634	−	0.826025i	$-0.309403\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	2.47510e11i		−	1.70786i	−0.520387	−	0.853930i	$-0.674212\pi$
$617$		−	2.47510e11i		−	1.70786i	0.520387	−	0.853930i	$-0.325788\pi$
$618$		0			0
$619$	1.85990e11			1.26685			0.633426	−	0.773803i	$-0.281648\pi$
$619$	1.85990e11			1.26685			0.633426	+	0.773803i	$0.281648\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.08250e10	+	1.22613e11i	0.595231	+	0.803555i
$626$		0			0
$627$		0			0
$628$		−	3.95273e10i		−	0.254131i
$629$		0			0
$630$		0			0
$631$	2.39197e11			1.50882			0.754412	−	0.656401i	$-0.227922\pi$
$631$	2.39197e11			1.50882			0.754412	+	0.656401i	$0.227922\pi$
$632$		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
$643$		−	2.97443e11i		−	1.74004i	−0.493015	−	0.870021i	$-0.664105\pi$
$643$		−	2.97443e11i		−	1.74004i	0.493015	−	0.870021i	$-0.335895\pi$
$644$	2.01965e11			1.17417
$645$		0			0
$646$		0			0
$647$			1.41562e11i			0.807846i	0.914793	+	0.403923i	$0.132354\pi$
$647$			1.41562e11i			0.807846i	−0.914793	+	0.403923i	$0.867646\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		−	1.25362e11i		−	0.693707i
$653$			3.34109e11i			1.83753i	0.394802	+	0.918766i	$0.370813\pi$
$653$			3.34109e11i			1.83753i	−0.394802	+	0.918766i	$0.629187\pi$
$654$		0			0
$655$	6.74762e10	−	2.83945e11i	0.366594	−	1.54266i
$656$		0			0
$657$		−	2.36718e11i		−	1.27049i
$658$		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
$665$	−3.78230e11	−	8.98821e10i	−1.93406	−	0.459607i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.41668e11			−2.17874
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	2.08827e11			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	$0.5\pi$
$684$	2.18889e11			1.00000
$685$	−4.27290e11	−	1.01541e11i	−1.94071	−	0.461188i
$686$		0			0
$687$		0			0
$688$		−	2.56923e11i		−	1.14670i
$689$		0			0
$690$		0			0
$691$	−4.09506e11			−1.79617			−0.898086	−	0.439819i	$-0.855043\pi$
$691$	−4.09506e11			−1.79617			−0.898086	+	0.439819i	$0.855043\pi$
$692$		0			0
$693$			7.83110e11i			3.39539i
$694$		0			0
$695$	−1.07880e11	+	4.53967e11i	−0.462383	+	1.94574i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−2.14724e11	+	4.26273e11i	−0.894309	+	1.77540i
$701$	4.33309e11			1.79443			0.897213	−	0.441597i	$-0.145588\pi$
$701$	4.33309e11			1.79443			0.897213	+	0.441597i	$0.145588\pi$
$702$		0			0
$703$		0			0
$704$	4.19548e11			1.70801
$705$		0			0
$706$		0			0
$707$			5.16251e11i			2.06625i
$708$		0			0
$709$	3.00992e10			0.119116			0.0595581	−	0.998225i	$-0.481031\pi$
$709$	3.00992e10			0.119116			0.0595581	+	0.998225i	$0.481031\pi$
$710$		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
$719$	5.23144e11			1.95752			0.978758	−	0.205017i	$-0.0657249\pi$
$719$	5.23144e11			1.95752			0.978758	+	0.205017i	$0.0657249\pi$
$720$	6.21324e10	−	2.61457e11i	0.231200	−	0.972906i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			1.29527e11i			0.463684i	0.972753	+	0.231842i	$0.0744753\pi$
$727$			1.29527e11i			0.463684i	−0.972753	+	0.231842i	$0.925525\pi$
$728$		0			0
$729$	−2.82430e11			−1.00000
$730$		0			0
$731$	6.33378e11			2.21816
$732$		0			0
$733$		−	5.00647e11i		−	1.73427i	−0.498076	−	0.867134i	$-0.665960\pi$
$733$		−	5.00647e11i		−	1.73427i	0.498076	−	0.867134i	$-0.334040\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	3.24050e10			0.108651			0.0543256	−	0.998523i	$-0.482699\pi$
$739$	3.24050e10			0.108651			0.0543256	+	0.998523i	$0.482699\pi$
$740$		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
$745$	−1.17748e11	+	4.95492e11i	−0.382233	+	1.60847i
$746$		0			0
$747$		−	4.67665e11i		−	1.50194i
$748$			1.03429e12i			3.30397i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			3.36177e11i			1.05123i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	5.38486e11i		−	1.63980i	−0.572506	−	0.819901i	$-0.694029\pi$
$757$		−	5.38486e11i		−	1.63980i	0.572506	−	0.819901i	$-0.305971\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5.40926e11			−1.61287			−0.806434	−	0.591324i	$-0.798606\pi$
$761$	−5.40926e11			−1.61287			−0.806434	+	0.591324i	$0.798606\pi$
$762$		0			0
$763$		0			0
$764$	−3.77455e11			−1.10788
$765$	6.44558e11	+	1.53172e11i	1.88198	+	0.447232i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	6.96633e11			1.99204			0.996021	−	0.0891173i	$-0.0284046\pi$
$769$	6.96633e11			1.99204			0.996021	+	0.0891173i	$0.0284046\pi$
$770$		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
$784$	−1.11520e12			−2.95182
$785$	−9.38875e10	−	2.23113e10i	−0.247246	−	0.0587552i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$			6.14383e11i			1.59343i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	6.06895e11			1.51168
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	−8.28761e11			−2.03349
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	9.02242e11i		−	2.17001i
$804$		0			0
$805$	1.14000e11	−	4.79719e11i	0.271469	−	1.14236i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.52981e11			1.52443			0.762213	−	0.647326i	$-0.224113\pi$
$809$	6.52981e11			1.52443			0.762213	+	0.647326i	$0.224113\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.97768e11	−	7.07611e10i	−0.674912	−	0.160385i
$816$		0			0
$817$		−	5.10901e11i		−	1.14670i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.08472e11			−1.99958			−0.999790	−	0.0204828i	$-0.993480\pi$
$821$	−9.08472e11			−1.99958			−0.999790	+	0.0204828i	$0.993480\pi$
$822$		0			0
$823$			8.73614e11i			1.90423i	0.305736	+	0.952116i	$0.401098\pi$
$823$			8.73614e11i			1.90423i	−0.305736	+	0.952116i	$0.598902\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$			2.77623e11i			0.590655i
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	2.74926e12i		−	5.70999i
$834$		0			0
$835$		0			0
$836$	8.34288e11			1.70801
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	5.00246e11			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.17873e11	−	4.96018e11i	0.231200	−	0.972906i
$846$		0			0
$847$			1.96166e12i			3.81145i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	5.52351e11i		−	1.04332i	−0.853152	−	0.521662i	$-0.825312\pi$
$853$		−	5.52351e11i		−	1.04332i	0.853152	−	0.521662i	$-0.174688\pi$
$854$		0			0
$855$	1.23553e11	−	5.19919e11i	0.231200	−	0.972906i
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−5.21495e11			−0.957807			−0.478903	−	0.877868i	$-0.658966\pi$
$859$	−5.21495e11			−0.957807			−0.478903	+	0.877868i	$0.658966\pi$
$860$	−6.10258e11	−	1.45021e11i	−1.11563	−	0.265116i
$861$		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
$875$	8.91307e11	+	7.50635e11i	1.52053	+	1.28055i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$	2.36815e11	−	9.96535e11i	0.394892	−	1.66174i
$881$	−3.92725e11			−0.651907			−0.325953	−	0.945386i	$-0.605685\pi$
$881$	−3.92725e11			−0.651907			−0.325953	+	0.945386i	$0.605685\pi$
$882$		0			0
$883$			1.12025e12i			1.84278i	0.388642	+	0.921389i	$0.372944\pi$
$883$			1.12025e12i			1.84278i	−0.388642	+	0.921389i	$0.627056\pi$
$884$		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
$891$	−1.07647e12			−1.70801
$892$		0			0
$893$			6.68503e11i			1.05123i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−5.85958e11	−	2.95161e11i	−0.893093	−	0.449872i
$901$		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
$909$	−7.09643e11			−1.03940
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		−	1.78249e12i		−	2.56533i
$914$		0			0
$915$		0			0
$916$	1.39253e12			1.97799
$917$		−	2.22881e12i		−	3.15208i
$918$		0			0
$919$	5.77136e11			0.809126			0.404563	−	0.914510i	$-0.367424\pi$
$919$	5.77136e11			0.809126			0.404563	+	0.914510i	$0.367424\pi$
$920$		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
$929$	2.36701e11			0.317788			0.158894	−	0.987296i	$-0.449207\pi$
$929$	2.36701e11			0.317788			0.158894	+	0.987296i	$0.449207\pi$
$930$		0			0
$931$	−2.21763e12			−2.95182
$932$		−	3.90994e11i		−	0.518210i
$933$		0			0
$934$		0			0
$935$	2.45671e12	+	5.83808e11i	3.21445	+	0.763878i
$936$		0			0
$937$		−	1.01528e12i		−	1.31713i	−0.752523	−	0.658566i	$-0.771163\pi$
$937$		−	1.01528e12i		−	1.31713i	0.752523	−	0.658566i	$-0.228837\pi$
$938$		0			0
$939$		0			0
$940$	7.98509e11	+	1.89756e11i	1.02275	+	0.243044i
$941$		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
$947$			1.38949e12i			1.72765i	0.503794	+	0.863824i	$0.331937\pi$
$947$			1.38949e12i			1.72765i	−0.503794	+	0.863824i	$0.668063\pi$
$948$		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
$955$	−2.13056e11	+	8.96555e11i	−0.256142	+	1.07786i
$956$	−4.47841e11			−0.536157
$957$		0			0
$958$		0			0
$959$	−3.35400e12			−3.96542
$960$		0			0
$961$	8.52891e11			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1.74852e12i		−	1.99969i	−0.0174779	−	0.999847i	$-0.505564\pi$
$967$		−	1.74852e12i		−	1.99969i	0.0174779	−	0.999847i	$-0.494436\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$			3.56339e12i			3.97569i
$974$		0			0
$975$		0			0
$976$	1.15748e12			1.27560
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$	−6.29481e11	+	2.64890e12i	−0.682462	+	2.87185i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	1.45932e12	+	3.46790e11i	1.55026	+	0.368402i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.47989e11			0.677302
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3.42564e11	−	1.44153e12i	0.349502	−	1.47073i
$996$		0			0
$997$			1.03731e12i			1.04986i	0.851147	+	0.524928i	$0.175908\pi$
$997$			1.03731e12i			1.04986i	−0.851147	+	0.524928i	$0.824092\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.9.d.c.94.2	yes	2
5.4	even	2	inner	95.9.d.c.94.1	✓	2
19.18	odd	2	CM	95.9.d.c.94.2	yes	2
95.94	odd	2	inner	95.9.d.c.94.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.9.d.c.94.1	✓	2	5.4	even	2	inner
95.9.d.c.94.1	✓	2	95.94	odd	2	inner
95.9.d.c.94.2	yes	2	1.1	even	1	trivial
95.9.d.c.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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	95.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-256.000 q^{4} +(-144.500 + 608.066i) q^{5} +4772.99i q^{7} -6561.00 q^{9} +O(q^{10})\)	\(q-256.000 q^{4} +(-144.500 + 608.066i) q^{5} +4772.99i q^{7} -6561.00 q^{9} -25007.0 q^{11} +65536.0 q^{16} +161563. i q^{17} +130321. q^{19} +(36992.0 - 155665. i) q^{20} +165289. i q^{23} +(-348864. - 175731. i) q^{25} -1.22189e6i q^{28} +(-2.90230e6 - 689698. i) q^{35} +1.67962e6 q^{36} -3.92033e6i q^{43} +6.40179e6 q^{44} +(948064. - 3.98952e6i) q^{45} +5.12966e6i q^{47} -1.70167e7 q^{49} +(3.61351e6 - 1.52059e7i) q^{55} +1.76618e7 q^{61} -3.13156e7i q^{63} -1.67772e7 q^{64} -4.13600e7i q^{68} +3.60796e7i q^{73} -3.33622e7 q^{76} -1.19358e8i q^{77} +(-9.46995e6 + 3.98502e7i) q^{80} +4.30467e7 q^{81} +7.12795e7i q^{83} +(-9.82408e7 - 2.33458e7i) q^{85} -4.23141e7i q^{92} +(-1.88314e7 + 7.92438e7i) q^{95} +1.64071e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 512 q^{4} - 289 q^{5} - 13122 q^{9}+O(q^{10}) \) 2 * q - 512 * q^4 - 289 * q^5 - 13122 * q^9	\( 2 q - 512 q^{4} - 289 q^{5} - 13122 q^{9} - 50014 q^{11} + 131072 q^{16} + 260642 q^{19} + 73984 q^{20} - 697729 q^{25} - 5804595 q^{35} + 3359232 q^{36} + 12803584 q^{44} + 1896129 q^{45} - 34033348 q^{49} + 7227023 q^{55} + 35323586 q^{61} - 33554432 q^{64} - 66724352 q^{76} - 18939904 q^{80} + 86093442 q^{81} - 196481565 q^{85} - 37662769 q^{95} + 328141854 q^{99}+O(q^{100}) \) 2 * q - 512 * q^4 - 289 * q^5 - 13122 * q^9 - 50014 * q^11 + 131072 * q^16 + 260642 * q^19 + 73984 * q^20 - 697729 * q^25 - 5804595 * q^35 + 3359232 * q^36 + 12803584 * q^44 + 1896129 * q^45 - 34033348 * q^49 + 7227023 * q^55 + 35323586 * q^61 - 33554432 * q^64 - 66724352 * q^76 - 18939904 * q^80 + 86093442 * q^81 - 196481565 * q^85 - 37662769 * q^95 + 328141854 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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