LMFDB - Embedded newform 95.9.d.d.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:	yes
Analytic conductor:	$38.7009679558$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{95}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 95 $ x^2 - 95
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			94.2
Root			$9.74679$ of defining polynomial
Character	$\chi$	$=$	95.94

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+29.2404 q^{2} -77.9744 q^{3} +599.000 q^{4} +625.000 q^{5} -2280.00 q^{6} +10029.5 q^{8} -481.000 q^{9} +O(q^{10})$	$q+29.2404 q^{2} -77.9744 q^{3} +599.000 q^{4} +625.000 q^{5} -2280.00 q^{6} +10029.5 q^{8} -481.000 q^{9} +18275.2 q^{10} +25438.0 q^{11} -46706.6 q^{12} -22222.7 q^{13} -48734.0 q^{15} +139921. q^{16} -14064.6 q^{18} +130321. q^{19} +374375. q^{20} +743817. q^{22} -782040. q^{24} +390625. q^{25} -649800. q^{26} +549095. q^{27} -1.42500e6 q^{30} +1.52380e6 q^{32} -1.98351e6 q^{33} -288119. q^{36} +1.88940e6 q^{37} +3.81064e6 q^{38} +1.73280e6 q^{39} +6.26841e6 q^{40} +1.52374e7 q^{44} -300625. q^{45} -1.09102e7 q^{48} +5.76480e6 q^{49} +1.14220e7 q^{50} -1.33114e7 q^{52} +4.40267e6 q^{53} +1.60558e7 q^{54} +1.58988e7 q^{55} -1.01617e7 q^{57} -2.91916e7 q^{60} -2.32594e7 q^{61} +8.73684e6 q^{64} -1.38892e7 q^{65} -5.79986e7 q^{66} -4.01615e7 q^{67} -4.82417e6 q^{72} +5.52467e7 q^{74} -3.04587e7 q^{75} +7.80623e7 q^{76} +5.06677e7 q^{78} +8.74506e7 q^{80} -3.96595e7 q^{81} +2.55129e8 q^{88} -8.79039e6 q^{90} +8.14506e7 q^{95} -1.18818e8 q^{96} -1.66373e8 q^{97} +1.68565e8 q^{98} -1.22357e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 1198 q^{4} + 1250 q^{5} - 4560 q^{6} - 962 q^{9}+O(q^{10}) $ 2 * q + 1198 * q^4 + 1250 * q^5 - 4560 * q^6 - 962 * q^9	$ 2 q + 1198 q^{4} + 1250 q^{5} - 4560 q^{6} - 962 q^{9} + 50876 q^{11} + 279842 q^{16} + 260642 q^{19} + 748750 q^{20} - 1564080 q^{24} + 781250 q^{25} - 1299600 q^{26} - 2850000 q^{30} - 576238 q^{36} + 3465600 q^{39} + 30474724 q^{44} - 601250 q^{45} + 11529602 q^{49} + 32111520 q^{54} + 31797500 q^{55} - 46518724 q^{61} + 17473678 q^{64} - 115997280 q^{66} + 110493360 q^{74} + 156124558 q^{76} + 174901250 q^{80} - 79319038 q^{81} + 162901250 q^{95} - 237635280 q^{96} - 24471356 q^{99}+O(q^{100}) $ 2 * q + 1198 * q^4 + 1250 * q^5 - 4560 * q^6 - 962 * q^9 + 50876 * q^11 + 279842 * q^16 + 260642 * q^19 + 748750 * q^20 - 1564080 * q^24 + 781250 * q^25 - 1299600 * q^26 - 2850000 * q^30 - 576238 * q^36 + 3465600 * q^39 + 30474724 * q^44 - 601250 * q^45 + 11529602 * q^49 + 32111520 * q^54 + 31797500 * q^55 - 46518724 * q^61 + 17473678 * q^64 - 115997280 * q^66 + 110493360 * q^74 + 156124558 * q^76 + 174901250 * q^80 - 79319038 * q^81 + 162901250 * q^95 - 237635280 * q^96 - 24471356 * 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 $

$2$	29.2404	1.82752	0.913762	−	0.406250i	$-0.133164\pi$
$2$	29.2404	1.82752	0.913762	+	0.406250i	$0.133164\pi$
$3$	−77.9744	−0.962646	−0.481323	−	0.876543i	$-0.659844\pi$
$3$	−77.9744	−0.962646	−0.481323	+	0.876543i	$0.659844\pi$
$4$	599.000	2.33984
$5$	625.000	1.00000
$5$	625.000	1.00000
$6$	−2280.00	−1.75926
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	10029.5	2.44860
$9$	−481.000	−0.0733120
$10$	18275.2	1.82752
$11$	25438.0	1.73745	0.868725	−	0.495295i	$-0.164940\pi$
$11$	25438.0	1.73745	0.868725	+	0.495295i	$0.164940\pi$
$12$	−46706.6	−2.25244
$13$	−22222.7	−0.778078	−0.389039	−	0.921221i	$-0.627193\pi$
$13$	−22222.7	−0.778078	−0.389039	+	0.921221i	$0.627193\pi$
$14$	0	0
$15$	−48734.0	−0.962646
$16$	139921.	2.13503
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−14064.6	−0.133979
$19$	130321.	1.00000
$19$	130321.	1.00000
$20$	374375.	2.33984
$21$	0	0
$22$	743817.	3.17523
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−782040.	−2.35713
$25$	390625.	1.00000
$26$	−649800.	−1.42196
$27$	549095.	1.03322
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	−1.42500e6	−1.75926
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	1.52380e6	1.45321
$33$	−1.98351e6	−1.67255
$34$	0	0
$35$	0	0
$36$	−288119.	−0.171539
$37$	1.88940e6	1.00813	0.504065	−	0.863666i	$-0.331837\pi$
$37$	1.88940e6	1.00813	0.504065	+	0.863666i	$0.331837\pi$
$38$	3.81064e6	1.82752
$39$	1.73280e6	0.749014
$40$	6.26841e6	2.44860
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	1.52374e7	4.06536
$45$	−300625.	−0.0733120
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−1.09102e7	−2.05527
$49$	5.76480e6	1.00000
$50$	1.14220e7	1.82752
$51$	0	0
$52$	−1.33114e7	−1.82058
$53$	4.40267e6	0.557972	0.278986	−	0.960295i	$-0.410002\pi$
$53$	4.40267e6	0.557972	0.278986	+	0.960295i	$0.410002\pi$
$54$	1.60558e7	1.88823
$55$	1.58988e7	1.73745
$56$	0	0
$57$	−1.01617e7	−0.962646
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−2.91916e7	−2.25244
$61$	−2.32594e7	−1.67988	−0.839940	−	0.542679i	$-0.817410\pi$
$61$	−2.32594e7	−1.67988	−0.839940	+	0.542679i	$0.817410\pi$
$62$	0	0
$63$	0	0
$64$	8.73684e6	0.520756
$65$	−1.38892e7	−0.778078
$66$	−5.79986e7	−3.05662
$67$	−4.01615e7	−1.99302	−0.996509	−	0.0834847i	$-0.973395\pi$
$67$	−4.01615e7	−1.99302	−0.996509	+	0.0834847i	$0.973395\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−4.82417e6	−0.179511
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	5.52467e7	1.84238
$75$	−3.04587e7	−0.962646
$76$	7.80623e7	2.33984
$77$	0	0
$78$	5.06677e7	1.36884
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	8.74506e7	2.13503
$81$	−3.96595e7	−0.921313
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	2.55129e8	4.25431
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−8.79039e6	−0.133979
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.14506e7	1.00000
$96$	−1.18818e8	−1.39893
$97$	−1.66373e8	−1.87930	−0.939651	−	0.342133i	$-0.888851\pi$
$97$	−1.66373e8	−1.87930	−0.939651	+	0.342133i	$0.888851\pi$
$98$	1.68565e8	1.82752
$99$	−1.22357e7	−0.127376
$100$	2.33984e8	2.33984
$101$	−1.95809e8	−1.88168	−0.940842	−	0.338845i	$-0.889964\pi$
$101$	−1.95809e8	−1.88168	−0.940842	+	0.338845i	$0.889964\pi$
$102$	0	0
$103$	−2.24840e8	−1.99767	−0.998837	−	0.0482057i	$-0.984650\pi$
$103$	−2.24840e8	−1.99767	−0.998837	+	0.0482057i	$0.984650\pi$
$104$	−2.22881e8	−1.90520
$105$	0	0
$106$	1.28736e8	1.01971
$107$	2.59413e8	1.97905	0.989526	−	0.144353i	$-0.0461100\pi$
$107$	2.59413e8	1.97905	0.989526	+	0.144353i	$0.0461100\pi$
$108$	3.28908e8	2.41757
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	4.64886e8	3.17523
$111$	−1.47324e8	−0.970472
$112$	0	0
$113$	−2.16814e8	−1.32976	−0.664880	−	0.746951i	$-0.731517\pi$
$113$	−2.16814e8	−1.32976	−0.664880	+	0.746951i	$0.731517\pi$
$114$	−2.97132e8	−1.75926
$115$	0	0
$116$	0	0
$117$	1.06891e7	0.0570425
$118$	0	0
$119$	0	0
$120$	−4.88775e8	−2.35713
$121$	4.32733e8	2.01873
$122$	−6.80113e8	−3.07002
$123$	0	0
$124$	0	0
$125$	2.44141e8	1.00000
$126$	0	0
$127$	−4.11470e8	−1.58170	−0.790849	−	0.612011i	$-0.790361\pi$
$127$	−4.11470e8	−1.58170	−0.790849	+	0.612011i	$0.790361\pi$
$128$	−1.34625e8	−0.501518
$129$	0	0
$130$	−4.06125e8	−1.42196
$131$	−1.72921e8	−0.587170	−0.293585	−	0.955933i	$-0.594848\pi$
$131$	−1.72921e8	−0.587170	−0.293585	+	0.955933i	$0.594848\pi$
$132$	−1.18812e9	−3.91350
$133$	0	0
$134$	−1.17434e9	−3.64229
$135$	3.43185e8	1.03322
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	7.43150e8	1.99075	0.995376	−	0.0960536i	$-0.0306220\pi$
$139$	7.43150e8	1.99075	0.995376	+	0.0960536i	$0.0306220\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.65301e8	−1.35187
$144$	−6.73020e7	−0.156523
$145$	0	0
$146$	0	0
$147$	−4.49507e8	−0.962646
$148$	1.13175e9	2.35887
$149$	9.27735e8	1.88226	0.941128	−	0.338050i	$-0.109767\pi$
$149$	9.27735e8	1.88226	0.941128	+	0.338050i	$0.109767\pi$
$150$	−8.90625e8	−1.75926
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	1.30705e9	2.44860
$153$	0	0
$154$	0	0
$155$	0	0
$156$	1.03795e9	1.75258
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	−3.43295e8	−0.537130
$160$	9.52378e8	1.45321
$161$	0	0
$162$	−1.15966e9	−1.68372
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−1.23969e9	−1.67255
$166$	0	0
$167$	−6.99256e8	−0.899022	−0.449511	−	0.893275i	$-0.648402\pi$
$167$	−6.99256e8	−0.899022	−0.449511	+	0.893275i	$0.648402\pi$
$168$	0	0
$169$	−3.21883e8	−0.394594
$170$	0	0
$171$	−6.26844e7	−0.0733120
$172$	0	0
$173$	1.54318e9	1.72279	0.861394	−	0.507937i	$-0.169592\pi$
$173$	1.54318e9	1.72279	0.861394	+	0.507937i	$0.169592\pi$
$174$	0	0
$175$	0	0
$176$	3.55931e9	3.70950
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−1.80074e8	−0.171539
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	1.81363e9	1.61713
$184$	0	0
$185$	1.18087e9	1.00813
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	2.38165e9	1.82752
$191$	−2.32896e9	−1.74996	−0.874979	−	0.484160i	$-0.839125\pi$
$191$	−2.32896e9	−1.74996	−0.874979	+	0.484160i	$0.839125\pi$
$192$	−6.81249e8	−0.501304
$193$	1.78121e9	1.28377	0.641884	−	0.766802i	$-0.278153\pi$
$193$	1.78121e9	1.28377	0.641884	+	0.766802i	$0.278153\pi$
$194$	−4.86482e9	−3.43447
$195$	1.08300e9	0.749014
$196$	3.45312e9	2.33984
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−3.57776e8	−0.232783
$199$	−2.53711e9	−1.61781	−0.808904	−	0.587941i	$-0.799939\pi$
$199$	−2.53711e9	−1.61781	−0.808904	+	0.587941i	$0.799939\pi$
$200$	3.91775e9	2.44860
$201$	3.13157e9	1.91857
$202$	−5.72552e9	−3.43882
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−6.57441e9	−3.65080
$207$	0	0
$208$	−3.10942e9	−1.66122
$209$	3.31511e9	1.73745
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	2.63720e9	1.30557
$213$	0	0
$214$	7.58535e9	3.61677
$215$	0	0
$216$	5.50713e9	2.52994
$217$	0	0
$218$	0	0
$219$	0	0
$220$	9.52335e9	4.06536
$221$	0	0
$222$	−4.30782e9	−1.77356
$223$	−4.28518e9	−1.73280	−0.866402	−	0.499347i	$-0.833573\pi$
$223$	−4.28518e9	−1.73280	−0.866402	+	0.499347i	$0.833573\pi$
$224$	0	0
$225$	−1.87891e8	−0.0733120
$226$	−6.33972e9	−2.43017
$227$	4.22128e9	1.58979	0.794897	−	0.606745i	$-0.207525\pi$
$227$	4.22128e9	1.58979	0.794897	+	0.606745i	$0.207525\pi$
$228$	−6.08686e9	−2.25244
$229$	8.62779e8	0.313731	0.156866	−	0.987620i	$-0.449861\pi$
$229$	8.62779e8	0.313731	0.156866	+	0.987620i	$0.449861\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	3.12554e8	0.104246
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.42349e9	1.35573	0.677866	−	0.735186i	$-0.262905\pi$
$239$	4.42349e9	1.35573	0.677866	+	0.735186i	$0.262905\pi$
$240$	−6.81891e9	−2.05527
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.26533e10	3.68928
$243$	−5.10190e8	−0.146321
$244$	−1.39324e10	−3.93066
$245$	3.60300e9	1.00000
$246$	0	0
$247$	−2.89608e9	−0.778078
$248$	0	0
$249$	0	0
$250$	7.13877e9	1.82752
$251$	6.88071e8	0.173356	0.0866779	−	0.996236i	$-0.472375\pi$
$251$	6.88071e8	0.173356	0.0866779	+	0.996236i	$0.472375\pi$
$252$	0	0
$253$	0	0
$254$	−1.20316e10	−2.89059
$255$	0	0
$256$	−6.17313e9	−1.43729
$257$	6.10159e9	1.39865	0.699327	−	0.714802i	$-0.253483\pi$
$257$	6.10159e9	1.39865	0.699327	+	0.714802i	$0.253483\pi$
$258$	0	0
$259$	0	0
$260$	−8.31962e9	−1.82058
$261$	0	0
$262$	−5.05629e9	−1.07307
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	−1.98935e10	−4.09540
$265$	2.75167e9	0.557972
$266$	0	0
$267$	0	0
$268$	−2.40568e10	−4.66335
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	1.00348e10	1.88823
$271$	−1.03139e10	−1.91225	−0.956126	−	0.292955i	$-0.905361\pi$
$271$	−1.03139e10	−1.91225	−0.956126	+	0.292955i	$0.905361\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.93672e9	1.73745
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	2.17300e10	3.63815
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	−6.35106e9	−0.962646
$286$	−1.65296e10	−2.47058
$287$	0	0
$288$	−7.32950e8	−0.106538
$289$	6.97576e9	1.00000
$290$	0	0
$291$	1.29729e10	1.80910
$292$	0	0
$293$	5.95616e8	0.0808157	0.0404079	−	0.999183i	$-0.487134\pi$
$293$	5.95616e8	0.0808157	0.0404079	+	0.999183i	$0.487134\pi$
$294$	−1.31437e10	−1.75926
$295$	0	0
$296$	1.89496e10	2.46850
$297$	1.39679e10	1.79517
$298$	2.71273e10	3.43987
$299$	0	0
$300$	−1.82448e10	−2.25244
$301$	0	0
$302$	0	0
$303$	1.52681e10	1.81140
$304$	1.82346e10	2.13503
$305$	−1.45371e10	−1.67988
$306$	0	0
$307$	−5.39419e9	−0.607257	−0.303628	−	0.952791i	$-0.598198\pi$
$307$	−5.39419e9	−0.607257	−0.303628	+	0.952791i	$0.598198\pi$
$308$	0	0
$309$	1.75318e10	1.92305
$310$	0	0
$311$	1.48383e10	1.58615	0.793073	−	0.609126i	$-0.208480\pi$
$311$	1.48383e10	1.58615	0.793073	+	0.609126i	$0.208480\pi$
$312$	1.73790e10	1.83403
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.24159e10	1.22954	0.614769	−	0.788707i	$-0.289249\pi$
$317$	1.24159e10	1.22954	0.614769	+	0.788707i	$0.289249\pi$
$318$	−1.00381e10	−0.981617
$319$	0	0
$320$	5.46052e9	0.520756
$321$	−2.02276e10	−1.90513
$322$	0	0
$323$	0	0
$324$	−2.37561e10	−2.15573
$325$	−8.68074e9	−0.778078
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	−3.62492e10	−3.05662
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	−9.08800e8	−0.0739080
$334$	−2.04465e10	−1.64298
$335$	−2.51010e10	−1.99302
$336$	0	0
$337$	−9.04476e9	−0.701257	−0.350629	−	0.936515i	$-0.614032\pi$
$337$	−9.04476e9	−0.701257	−0.350629	+	0.936515i	$0.614032\pi$
$338$	−9.41197e9	−0.721131
$339$	1.69059e10	1.28009
$340$	0	0
$341$	0	0
$342$	−1.83292e9	−0.133979
$343$	0	0
$344$	0	0
$345$	0	0
$346$	4.51232e10	3.14844
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	−2.90598e10	−1.95880	−0.979402	−	0.201922i	$-0.935281\pi$
$349$	−2.90598e10	−1.95880	−0.979402	+	0.201922i	$0.935281\pi$
$350$	0	0
$351$	−1.22024e10	−0.803926
$352$	3.87625e10	2.52488
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.09210e10	−1.86156	−0.930778	−	0.365586i	$-0.880869\pi$
$359$	−3.09210e10	−1.86156	−0.930778	+	0.365586i	$0.880869\pi$
$360$	−3.01510e9	−0.179511
$361$	1.69836e10	1.00000
$362$	0	0
$363$	−3.37421e10	−1.94332
$364$	0	0
$365$	0	0
$366$	5.30313e10	2.95535
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	3.45292e10	1.84238
$371$	0	0
$372$	0	0
$373$	3.70714e10	1.91516	0.957578	−	0.288175i	$-0.0930484\pi$
$373$	3.70714e10	1.91516	0.957578	+	0.288175i	$0.0930484\pi$
$374$	0	0
$375$	−1.90367e10	−0.962646
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	4.87889e10	2.33984
$381$	3.20841e10	1.52262
$382$	−6.80996e10	−3.19809
$383$	−2.70081e10	−1.25516	−0.627579	−	0.778553i	$-0.715954\pi$
$383$	−2.70081e10	−1.25516	−0.627579	+	0.778553i	$0.715954\pi$
$384$	1.04973e10	0.482785
$385$	0	0
$386$	5.20833e10	2.34612
$387$	0	0
$388$	−9.96576e10	−4.39728
$389$	1.56692e10	0.684304	0.342152	−	0.939645i	$-0.388844\pi$
$389$	1.56692e10	0.684304	0.342152	+	0.939645i	$0.388844\pi$
$390$	3.16673e10	1.36884
$391$	0	0
$392$	5.78178e10	2.44860
$393$	1.34834e10	0.565237
$394$	0	0
$395$	0	0
$396$	−7.32917e9	−0.298040
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−7.41861e10	−2.95658
$399$	0	0
$400$	5.46566e10	2.13503
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	9.15683e10	3.50624
$403$	0	0
$404$	−1.17289e11	−4.40285
$405$	−2.47872e10	−0.921313
$406$	0	0
$407$	4.80625e10	1.75157
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	−1.34679e11	−4.67425
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−3.38630e10	−1.13071
$417$	−5.79466e10	−1.91639
$418$	9.69350e10	3.17523
$419$	−4.16612e10	−1.35169	−0.675844	−	0.737045i	$-0.736220\pi$
$419$	−4.16612e10	−1.35169	−0.675844	+	0.737045i	$0.736220\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	4.41563e10	1.36625
$425$	0	0
$426$	0	0
$427$	0	0
$428$	1.55389e11	4.63067
$429$	4.40790e10	1.30137
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	7.68300e10	2.20595
$433$	3.16978e10	0.901733	0.450866	−	0.892591i	$-0.351115\pi$
$433$	3.16978e10	0.901733	0.450866	+	0.892591i	$0.351115\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	1.59456e11	4.25431
$441$	−2.77287e9	−0.0733120
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−8.82474e10	−2.27075
$445$	0	0
$446$	−1.25300e11	−3.16674
$447$	−7.23395e10	−1.81195
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−5.49399e9	−0.133979
$451$	0	0
$452$	−1.29871e11	−3.11143
$453$	0	0
$454$	1.23432e11	2.90538
$455$	0	0
$456$	−1.01916e11	−2.35713
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	2.52280e10	0.573351
$459$	0	0
$460$	0	0
$461$	−8.57825e10	−1.89931	−0.949653	−	0.313304i	$-0.898564\pi$
$461$	−8.57825e10	−1.89931	−0.949653	+	0.313304i	$0.898564\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	6.40278e9	0.133470
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	5.09066e10	1.00000
$476$	0	0
$477$	−2.11768e9	−0.0409060
$478$	1.29345e11	2.47763
$479$	−1.66110e10	−0.315539	−0.157770	−	0.987476i	$-0.550430\pi$
$479$	−1.66110e10	−0.315539	−0.157770	+	0.987476i	$0.550430\pi$
$480$	−7.42610e10	−1.39893
$481$	−4.19875e10	−0.784403
$482$	0	0
$483$	0	0
$484$	2.59207e11	4.72352
$485$	−1.03983e11	−1.87930
$486$	−1.49181e10	−0.267405
$487$	2.98415e10	0.530524	0.265262	−	0.964176i	$-0.414542\pi$
$487$	2.98415e10	0.530524	0.265262	+	0.964176i	$0.414542\pi$
$488$	−2.33279e11	−4.11335
$489$	0	0
$490$	1.05353e11	1.82752
$491$	3.84904e10	0.662257	0.331129	−	0.943586i	$-0.392571\pi$
$491$	3.84904e10	0.662257	0.331129	+	0.943586i	$0.392571\pi$
$492$	0	0
$493$	0	0
$494$	−8.46826e10	−1.42196
$495$	−7.64730e9	−0.127376
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−8.87190e10	−1.43092	−0.715459	−	0.698655i	$-0.753782\pi$
$499$	−8.87190e10	−1.43092	−0.715459	+	0.698655i	$0.753782\pi$
$500$	1.46240e11	2.33984
$501$	5.45240e10	0.865440
$502$	2.01195e10	0.316812
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−1.22381e11	−1.88168
$506$	0	0
$507$	2.50986e10	0.379855
$508$	−2.46471e11	−3.70093
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	−1.46041e11	−2.12517
$513$	7.15587e10	1.03322
$514$	1.78413e11	2.55607
$515$	−1.40525e11	−1.99767
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−1.20328e11	−1.65844
$520$	−1.39301e11	−1.90520
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	1.18966e11	1.59007	0.795034	−	0.606565i	$-0.207453\pi$
$523$	1.18966e11	1.59007	0.795034	+	0.606565i	$0.207453\pi$
$524$	−1.03580e11	−1.37389
$525$	0	0
$526$	0	0
$527$	0	0
$528$	−2.77535e11	−3.57094
$529$	7.83110e10	1.00000
$530$	8.04598e10	1.01971
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	1.62133e11	1.97905
$536$	−4.02798e11	−4.88010
$537$	0	0
$538$	0	0
$539$	1.46645e11	1.73745
$540$	2.05568e11	2.41757
$541$	−5.22741e10	−0.610236	−0.305118	−	0.952315i	$-0.598696\pi$
$541$	−5.22741e10	−0.610236	−0.305118	+	0.952315i	$0.598696\pi$
$542$	−3.01582e11	−3.49469
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.61783e10	−0.850907	−0.425453	−	0.904980i	$-0.639885\pi$
$547$	−7.61783e10	−0.850907	−0.425453	+	0.904980i	$0.639885\pi$
$548$	0	0
$549$	1.11878e10	0.123155
$550$	2.90553e11	3.17523
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−9.20778e10	−0.970472
$556$	4.45147e11	4.65805
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.76163e11	1.75340	0.876702	−	0.481034i	$-0.159739\pi$
$563$	1.76163e11	1.75340	0.876702	+	0.481034i	$0.159739\pi$
$564$	0	0
$565$	−1.35509e11	−1.32976
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	−1.85707e11	−1.75926
$571$	−1.69333e10	−0.159293	−0.0796464	−	0.996823i	$-0.525379\pi$
$571$	−1.69333e10	−0.159293	−0.0796464	+	0.996823i	$0.525379\pi$
$572$	−3.38615e11	−3.16317
$573$	1.81599e11	1.68459
$574$	0	0
$575$	0	0
$576$	−4.20242e9	−0.0381777
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	2.03974e11	1.82752
$579$	−1.38889e11	−1.23581
$580$	0	0
$581$	0	0
$582$	3.79331e11	3.30618
$583$	1.11995e11	0.969448
$584$	0	0
$585$	6.68070e9	0.0570425
$586$	1.74160e10	0.147693
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	−2.69254e11	−2.25244
$589$	0	0
$590$	0	0
$591$	0	0
$592$	2.64366e11	2.15238
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	4.08426e11	3.28071
$595$	0	0
$596$	5.55713e11	4.40419
$597$	1.97830e11	1.55738
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−3.05484e11	−2.35713
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	1.93177e10	0.146112
$604$	0	0
$605$	2.70458e11	2.01873
$606$	4.46444e11	3.31037
$607$	2.06910e11	1.52414	0.762072	−	0.647492i	$-0.224182\pi$
$607$	2.06910e11	1.52414	0.762072	+	0.647492i	$0.224182\pi$
$608$	1.98584e11	1.45321
$609$	0	0
$610$	−4.25070e11	−3.07002
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−1.57728e11	−1.10978
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	5.12635e11	3.51443
$619$	−2.93349e11	−1.99812	−0.999061	−	0.0433359i	$-0.986201\pi$
$619$	−2.93349e11	−1.99812	−0.999061	+	0.0433359i	$0.986201\pi$
$620$	0	0
$621$	0	0
$622$	4.33878e11	2.89872
$623$	0	0
$624$	2.42455e11	1.59916
$625$	1.52588e11	1.00000
$626$	0	0
$627$	−2.58493e11	−1.67255
$628$	0	0
$629$	0	0
$630$	0	0
$631$	6.28689e10	0.396569	0.198284	−	0.980145i	$-0.436463\pi$
$631$	6.28689e10	0.396569	0.198284	+	0.980145i	$0.436463\pi$
$632$	0	0
$633$	0	0
$634$	3.63046e11	2.24701
$635$	−2.57169e11	−1.58170
$636$	−2.05634e11	−1.25680
$637$	−1.28109e11	−0.778078
$638$	0	0
$639$	0	0
$640$	−8.41408e10	−0.501518
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	−5.91463e11	−3.48167
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−3.97763e11	−2.25592
$649$	0	0
$650$	−2.53828e11	−1.42196
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−1.08076e11	−0.587170
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	−7.42577e11	−3.91350
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−2.65737e10	−0.135069
$667$	0	0
$668$	−4.18854e11	−2.10357
$669$	3.34134e11	1.66808
$670$	−7.33962e11	−3.64229
$671$	−5.91672e11	−2.91871
$672$	0	0
$673$	−3.92999e11	−1.91572	−0.957859	−	0.287240i	$-0.907262\pi$
$673$	−3.92999e11	−1.91572	−0.957859	+	0.287240i	$0.907262\pi$
$674$	−2.64472e11	−1.28156
$675$	2.14490e11	1.03322
$676$	−1.92808e11	−0.923289
$677$	3.97601e11	1.89275	0.946373	−	0.323075i	$-0.104717\pi$
$677$	3.97601e11	1.89275	0.946373	+	0.323075i	$0.104717\pi$
$678$	4.94335e11	2.33939
$679$	0	0
$680$	0	0
$681$	−3.29151e11	−1.53041
$682$	0	0
$683$	4.34597e11	1.99712	0.998560	−	0.0536512i	$-0.0170859\pi$
$683$	4.34597e11	1.99712	0.998560	+	0.0536512i	$0.0170859\pi$
$684$	−3.75480e10	−0.171539
$685$	0	0
$686$	0	0
$687$	−6.72747e10	−0.302012
$688$	0	0
$689$	−9.78391e10	−0.434146
$690$	0	0
$691$	−4.55396e11	−1.99745	−0.998727	−	0.0504447i	$-0.983936\pi$
$691$	−4.55396e11	−1.99745	−0.998727	+	0.0504447i	$0.983936\pi$
$692$	9.24364e11	4.03106
$693$	0	0
$694$	0	0
$695$	4.64469e11	1.99075
$696$	0	0
$697$	0	0
$698$	−8.49719e11	−3.57976
$699$	0	0
$700$	0	0
$701$	4.77261e11	1.97644	0.988221	−	0.153031i	$-0.0489034\pi$
$701$	4.77261e11	1.97644	0.988221	+	0.153031i	$0.0489034\pi$
$702$	−3.56802e11	−1.46919
$703$	2.46228e11	1.00813
$704$	2.22248e11	0.904788
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	4.24996e11	1.68190	0.840949	−	0.541114i	$-0.181997\pi$
$709$	4.24996e11	1.68190	0.840949	+	0.541114i	$0.181997\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−3.53313e11	−1.35187
$716$	0	0
$717$	−3.44919e11	−1.30509
$718$	−9.04142e11	−3.40204
$719$	2.61075e11	0.976899	0.488450	−	0.872592i	$-0.337563\pi$
$719$	2.61075e11	0.976899	0.488450	+	0.872592i	$0.337563\pi$
$720$	−4.20638e10	−0.156523
$721$	0	0
$722$	4.96606e11	1.82752
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−9.86631e11	−3.55147
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	2.99988e11	1.06217
$730$	0	0
$731$	0	0
$732$	1.08637e12	3.78383
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	−2.80942e11	−0.962646
$736$	0	0
$737$	−1.02163e12	−3.46277
$738$	0	0
$739$	1.66265e11	0.557472	0.278736	−	0.960368i	$-0.410085\pi$
$739$	1.66265e11	0.557472	0.278736	+	0.960368i	$0.410085\pi$
$740$	7.07343e11	2.35887
$741$	2.25820e11	0.749014
$742$	0	0
$743$	1.44136e11	0.472952	0.236476	−	0.971637i	$-0.424008\pi$
$743$	1.44136e11	0.472952	0.236476	+	0.971637i	$0.424008\pi$
$744$	0	0
$745$	5.79834e11	1.88226
$746$	1.08398e12	3.49999
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−5.56641e11	−1.75926
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	−5.36519e10	−0.166880
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	8.16905e11	2.44860
$761$	−1.43885e11	−0.429019	−0.214509	−	0.976722i	$-0.568815\pi$
$761$	−1.43885e11	−0.429019	−0.214509	+	0.976722i	$0.568815\pi$
$762$	9.38153e11	2.78262
$763$	0	0
$764$	−1.39504e12	−4.09463
$765$	0	0
$766$	−7.89726e11	−2.29383
$767$	0	0
$768$	4.81346e11	1.38360
$769$	1.88845e11	0.540009	0.270005	−	0.962859i	$-0.412975\pi$
$769$	1.88845e11	0.540009	0.270005	+	0.962859i	$0.412975\pi$
$770$	0	0
$771$	−4.75767e11	−1.34641
$772$	1.06695e12	3.00382
$773$	−6.76787e11	−1.89554	−0.947772	−	0.318948i	$-0.896671\pi$
$773$	−6.76787e11	−1.89554	−0.947772	+	0.318948i	$0.896671\pi$
$774$	0	0
$775$	0	0
$776$	−1.66863e12	−4.60165
$777$	0	0
$778$	4.58174e11	1.25058
$779$	0	0
$780$	6.48717e11	1.75258
$781$	0	0
$782$	0	0
$783$	0	0
$784$	8.06617e11	2.13503
$785$	0	0
$786$	3.94261e11	1.03298
$787$	4.73234e11	1.23361	0.616803	−	0.787117i	$-0.288427\pi$
$787$	4.73234e11	1.23361	0.616803	+	0.787117i	$0.288427\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−1.22717e11	−0.311892
$793$	5.16886e11	1.30708
$794$	0	0
$795$	−2.14559e11	−0.537130
$796$	−1.51973e12	−3.78542
$797$	−7.38977e11	−1.83146	−0.915730	−	0.401794i	$-0.868387\pi$
$797$	−7.38977e11	−1.83146	−0.915730	+	0.401794i	$0.868387\pi$
$798$	0	0
$799$	0	0
$800$	5.95236e11	1.45321
$801$	0	0
$802$	0	0
$803$	0	0
$804$	1.87581e12	4.48916
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−1.96385e12	−4.60749
$809$	−8.53205e11	−1.99186	−0.995931	−	0.0901140i	$-0.971277\pi$
$809$	−8.53205e11	−1.99186	−0.995931	+	0.0901140i	$0.971277\pi$
$810$	−7.24787e11	−1.68372
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	8.04219e11	1.84082
$814$	1.40537e12	3.20104
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.68754e11	1.91216	0.956080	−	0.293105i	$-0.0946885\pi$
$821$	8.68754e11	1.91216	0.956080	+	0.293105i	$0.0946885\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	−2.25502e12	−4.89150
$825$	−7.74809e11	−1.67255
$826$	0	0
$827$	−9.23631e11	−1.97459	−0.987294	−	0.158902i	$-0.949205\pi$
$827$	−9.23631e11	−1.97459	−0.987294	+	0.158902i	$0.949205\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	−1.94156e11	−0.405189
$833$	0	0
$834$	−1.69438e12	−3.50225
$835$	−4.37035e11	−0.899022
$836$	1.98575e12	4.06536
$837$	0	0
$838$	−1.21819e12	−2.47024
$839$	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
$845$	−2.01177e11	−0.394594
$846$	0	0
$847$	0	0
$848$	6.16025e11	1.19128
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	−3.91778e10	−0.0733120
$856$	2.60177e12	4.84590
$857$	1.06418e12	1.97285	0.986424	−	0.164221i	$-0.0525110\pi$
$857$	1.06418e12	1.97285	0.986424	+	0.164221i	$0.0525110\pi$
$858$	1.28889e12	2.37829
$859$	9.30424e11	1.70887	0.854433	−	0.519561i	$-0.173905\pi$
$859$	9.30424e11	1.70887	0.854433	+	0.519561i	$0.173905\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.37567e11	0.969146	0.484573	−	0.874751i	$-0.338975\pi$
$863$	5.37567e11	0.969146	0.484573	+	0.874751i	$0.338975\pi$
$864$	8.36714e11	1.50149
$865$	9.64487e11	1.72279
$866$	9.26856e11	1.64794
$867$	−5.43930e11	−0.962646
$868$	0	0
$869$	0	0
$870$	0	0
$871$	8.92498e11	1.55072
$872$	0	0
$873$	8.00256e10	0.137775
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.20027e11	−0.202900	−0.101450	−	0.994841i	$-0.532348\pi$
$877$	−1.20027e11	−0.202900	−0.101450	+	0.994841i	$0.532348\pi$
$878$	0	0
$879$	−4.64428e10	−0.0777970
$880$	2.22457e12	3.70950
$881$	−3.83197e11	−0.636090	−0.318045	−	0.948076i	$-0.603026\pi$
$881$	−3.83197e11	−0.636090	−0.318045	+	0.948076i	$0.603026\pi$
$882$	−8.10798e10	−0.133979
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.27492e11	−1.49836	−0.749179	−	0.662368i	$-0.769552\pi$
$887$	−9.27492e11	−1.49836	−0.749179	+	0.662368i	$0.769552\pi$
$888$	−1.47758e12	−2.37629
$889$	0	0
$890$	0	0
$891$	−1.00886e12	−1.60074
$892$	−2.56682e12	−4.05449
$893$	0	0
$894$	−2.11524e12	−3.31138
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−1.12546e11	−0.171539
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−2.17452e12	−3.25604
$905$	0	0
$906$	0	0
$907$	1.00891e12	1.49081	0.745406	−	0.666611i	$-0.232256\pi$
$907$	1.00891e12	1.49081	0.745406	+	0.666611i	$0.232256\pi$
$908$	2.52855e12	3.71987
$909$	9.41840e10	0.137950
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	−1.42183e12	−2.05527
$913$	0	0
$914$	0	0
$915$	1.13352e12	1.61713
$916$	5.16805e11	0.734082
$917$	0	0
$918$	0	0
$919$	−7.64618e11	−1.07197	−0.535985	−	0.844227i	$-0.680060\pi$
$919$	−7.64618e11	−1.07197	−0.535985	+	0.844227i	$0.680060\pi$
$920$	0	0
$921$	4.20608e11	0.584574
$922$	−2.50831e12	−3.47103
$923$	0	0
$924$	0	0
$925$	7.38046e11	1.00813
$926$	0	0
$927$	1.08148e11	0.146454
$928$	0	0
$929$	−5.77276e11	−0.775034	−0.387517	−	0.921863i	$-0.626667\pi$
$929$	−5.77276e11	−0.775034	−0.387517	+	0.921863i	$0.626667\pi$
$930$	0	0
$931$	7.51275e11	1.00000
$932$	0	0
$933$	−1.15701e12	−1.52690
$934$	0	0
$935$	0	0
$936$	1.07206e11	0.139674
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	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
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	1.48853e12	1.82752
$951$	−9.68124e11	−1.18361
$952$	0	0
$953$	1.12800e12	1.36754	0.683768	−	0.729699i	$-0.260340\pi$
$953$	1.12800e12	1.36754	0.683768	+	0.729699i	$0.260340\pi$
$954$	−6.19218e10	−0.0747567
$955$	−1.45560e12	−1.74996
$956$	2.64967e12	3.17220
$957$	0	0
$958$	−4.85712e11	−0.576656
$959$	0	0
$960$	−4.25781e11	−0.501304
$961$	8.52891e11	1.00000
$962$	−1.22773e12	−1.43352
$963$	−1.24778e11	−0.145088
$964$	0	0
$965$	1.11326e12	1.28377
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	4.34007e12	4.94306
$969$	0	0
$970$	−3.04051e12	−3.43447
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−3.05604e11	−0.342368
$973$	0	0
$974$	8.72577e11	0.969545
$975$	6.76875e11	0.749014
$976$	−3.25447e12	−3.58659
$977$	1.52462e12	1.67334	0.836669	−	0.547709i	$-0.184500\pi$
$977$	1.52462e12	1.67334	0.836669	+	0.547709i	$0.184500\pi$
$978$	0	0
$979$	0	0
$980$	2.15820e12	2.33984
$981$	0	0
$982$	1.12547e12	1.21029
$983$	−4.58563e10	−0.0491117	−0.0245559	−	0.999698i	$-0.507817\pi$
$983$	−4.58563e10	−0.0491117	−0.0245559	+	0.999698i	$0.507817\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−1.73475e12	−1.82058
$989$	0	0
$990$	−2.23610e11	−0.232783
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.58569e12	−1.61781
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	−2.59418e12	−2.61504
$999$	1.03746e12	1.04162

Twists

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

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

\(f(q)\)	\(=\)	\(q+29.2404 q^{2} -77.9744 q^{3} +599.000 q^{4} +625.000 q^{5} -2280.00 q^{6} +10029.5 q^{8} -481.000 q^{9} +O(q^{10})\)	\(q+29.2404 q^{2} -77.9744 q^{3} +599.000 q^{4} +625.000 q^{5} -2280.00 q^{6} +10029.5 q^{8} -481.000 q^{9} +18275.2 q^{10} +25438.0 q^{11} -46706.6 q^{12} -22222.7 q^{13} -48734.0 q^{15} +139921. q^{16} -14064.6 q^{18} +130321. q^{19} +374375. q^{20} +743817. q^{22} -782040. q^{24} +390625. q^{25} -649800. q^{26} +549095. q^{27} -1.42500e6 q^{30} +1.52380e6 q^{32} -1.98351e6 q^{33} -288119. q^{36} +1.88940e6 q^{37} +3.81064e6 q^{38} +1.73280e6 q^{39} +6.26841e6 q^{40} +1.52374e7 q^{44} -300625. q^{45} -1.09102e7 q^{48} +5.76480e6 q^{49} +1.14220e7 q^{50} -1.33114e7 q^{52} +4.40267e6 q^{53} +1.60558e7 q^{54} +1.58988e7 q^{55} -1.01617e7 q^{57} -2.91916e7 q^{60} -2.32594e7 q^{61} +8.73684e6 q^{64} -1.38892e7 q^{65} -5.79986e7 q^{66} -4.01615e7 q^{67} -4.82417e6 q^{72} +5.52467e7 q^{74} -3.04587e7 q^{75} +7.80623e7 q^{76} +5.06677e7 q^{78} +8.74506e7 q^{80} -3.96595e7 q^{81} +2.55129e8 q^{88} -8.79039e6 q^{90} +8.14506e7 q^{95} -1.18818e8 q^{96} -1.66373e8 q^{97} +1.68565e8 q^{98} -1.22357e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1198 q^{4} + 1250 q^{5} - 4560 q^{6} - 962 q^{9}+O(q^{10}) \) 2 * q + 1198 * q^4 + 1250 * q^5 - 4560 * q^6 - 962 * q^9	\( 2 q + 1198 q^{4} + 1250 q^{5} - 4560 q^{6} - 962 q^{9} + 50876 q^{11} + 279842 q^{16} + 260642 q^{19} + 748750 q^{20} - 1564080 q^{24} + 781250 q^{25} - 1299600 q^{26} - 2850000 q^{30} - 576238 q^{36} + 3465600 q^{39} + 30474724 q^{44} - 601250 q^{45} + 11529602 q^{49} + 32111520 q^{54} + 31797500 q^{55} - 46518724 q^{61} + 17473678 q^{64} - 115997280 q^{66} + 110493360 q^{74} + 156124558 q^{76} + 174901250 q^{80} - 79319038 q^{81} + 162901250 q^{95} - 237635280 q^{96} - 24471356 q^{99}+O(q^{100}) \) 2 * q + 1198 * q^4 + 1250 * q^5 - 4560 * q^6 - 962 * q^9 + 50876 * q^11 + 279842 * q^16 + 260642 * q^19 + 748750 * q^20 - 1564080 * q^24 + 781250 * q^25 - 1299600 * q^26 - 2850000 * q^30 - 576238 * q^36 + 3465600 * q^39 + 30474724 * q^44 - 601250 * q^45 + 11529602 * q^49 + 32111520 * q^54 + 31797500 * q^55 - 46518724 * q^61 + 17473678 * q^64 - 115997280 * q^66 + 110493360 * q^74 + 156124558 * q^76 + 174901250 * q^80 - 79319038 * q^81 + 162901250 * q^95 - 237635280 * q^96 - 24471356 * q^99

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