LMFDB - Embedded newform 20.27.d.a.19.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [20,27,Mod(19,20)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(20, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 27, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("20.19"); S:= CuspForms(chi, 27); N := Newforms(S);

Level:	$ N $	$=$	$ 20 = 2^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 27 $
Character orbit:	$[\chi]$	$=$	20.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,-8192] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$85.6585841459$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			19.1
Character	$\chi$	$=$	20.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-8192.00 q^{2} -196556. q^{3} +6.71089e7 q^{4} -1.22070e9 q^{5} +1.61019e9 q^{6} +1.13407e11 q^{7} -5.49756e11 q^{8} -2.50323e12 q^{9} +1.00000e13 q^{10} -1.31906e13 q^{12} -9.29032e14 q^{14} +2.39937e14 q^{15} +4.50360e15 q^{16} +2.05065e16 q^{18} -8.19200e16 q^{20} -2.22909e16 q^{21} -7.67155e17 q^{23} +1.08058e17 q^{24} +1.49012e18 q^{25} +9.91644e17 q^{27} +7.61063e18 q^{28} +1.93102e19 q^{29} -1.96556e18 q^{30} -3.68935e19 q^{32} -1.38437e20 q^{35} -1.67989e20 q^{36} +6.71089e20 q^{40} -1.82951e21 q^{41} +1.82607e20 q^{42} -3.43251e21 q^{43} +3.05570e21 q^{45} +6.28453e21 q^{46} -5.74003e21 q^{47} -8.85210e20 q^{48} +3.47371e21 q^{49} -1.22070e22 q^{50} -8.12355e21 q^{54} -6.23463e22 q^{56} -1.58189e23 q^{58} +1.61019e22 q^{60} -5.07615e22 q^{61} -2.83884e23 q^{63} +3.02231e23 q^{64} +9.44504e23 q^{67} +1.50789e23 q^{69} +1.13407e24 q^{70} +1.37617e24 q^{72} -2.92891e23 q^{75} -5.49756e24 q^{80} +6.16797e24 q^{81} +1.49874e25 q^{82} +1.78871e24 q^{83} -1.49591e24 q^{84} +2.81191e25 q^{86} -3.79553e24 q^{87} +2.33570e25 q^{89} -2.50323e25 q^{90} -5.14829e25 q^{92} +4.70223e25 q^{94} +7.25164e24 q^{96} -2.84566e25 q^{98} +O(q^{100})\)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$.

$n$	$11$	$17$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−8192.00	−1.00000
$2$	−8192.00	−1.00000
$3$	−196556.	−0.123285	−0.0616425	−	0.998098i	$-0.519634\pi$
$3$	−196556.	−0.123285	−0.0616425	+	0.998098i	$0.519634\pi$
$4$	6.71089e7	1.00000
$5$	−1.22070e9	−1.00000
$5$	−1.22070e9	−1.00000
$6$	1.61019e9	0.123285
$7$	1.13407e11	1.17049	0.585243	−	0.810858i	$-0.300999\pi$
$7$	1.13407e11	1.17049	0.585243	+	0.810858i	$0.300999\pi$
$8$	−5.49756e11	−1.00000
$9$	−2.50323e12	−0.984801
$10$	1.00000e13	1.00000
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−1.31906e13	−0.123285
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−9.29032e14	−1.17049
$15$	2.39937e14	0.123285
$16$	4.50360e15	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	2.05065e16	0.984801
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−8.19200e16	−1.00000
$21$	−2.22909e16	−0.144303
$22$	0	0
$23$	−7.67155e17	−1.52202	−0.761012	−	0.648738i	$-0.775297\pi$
$23$	−7.67155e17	−1.52202	−0.761012	+	0.648738i	$0.775297\pi$
$24$	1.08058e17	0.123285
$25$	1.49012e18	1.00000
$26$	0	0
$27$	9.91644e17	0.244696
$28$	7.61063e18	1.17049
$29$	1.93102e19	1.88197	0.940984	−	0.338452i	$-0.109903\pi$
$29$	1.93102e19	1.88197	0.940984	+	0.338452i	$0.109903\pi$
$30$	−1.96556e18	−0.123285
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	−3.68935e19	−1.00000
$33$	0	0
$34$	0	0
$35$	−1.38437e20	−1.17049
$36$	−1.67989e20	−0.984801
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	6.71089e20	1.00000
$41$	−1.82951e21	−1.97763	−0.988814	−	0.149151i	$-0.952346\pi$
$41$	−1.82951e21	−1.97763	−0.988814	+	0.149151i	$0.952346\pi$
$42$	1.82607e20	0.144303
$43$	−3.43251e21	−1.99766	−0.998831	−	0.0483468i	$-0.984605\pi$
$43$	−3.43251e21	−1.99766	−0.998831	+	0.0483468i	$0.984605\pi$
$44$	0	0
$45$	3.05570e21	0.984801
$46$	6.28453e21	1.52202
$47$	−5.74003e21	−1.05110	−0.525548	−	0.850764i	$-0.676140\pi$
$47$	−5.74003e21	−1.05110	−0.525548	+	0.850764i	$0.676140\pi$
$48$	−8.85210e20	−0.123285
$49$	3.47371e21	0.370037
$50$	−1.22070e22	−1.00000
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−8.12355e21	−0.244696
$55$	0	0
$56$	−6.23463e22	−1.17049
$57$	0	0
$58$	−1.58189e23	−1.88197
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	1.61019e22	0.123285
$61$	−5.07615e22	−0.313506	−0.156753	−	0.987638i	$-0.550103\pi$
$61$	−5.07615e22	−0.313506	−0.156753	+	0.987638i	$0.550103\pi$
$62$	0	0
$63$	−2.83884e23	−1.15270
$64$	3.02231e23	1.00000
$65$	0	0
$66$	0	0
$67$	9.44504e23	1.72279	0.861393	−	0.507939i	$-0.169593\pi$
$67$	9.44504e23	1.72279	0.861393	+	0.507939i	$0.169593\pi$
$68$	0	0
$69$	1.50789e23	0.187643
$70$	1.13407e24	1.17049
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1.37617e24	0.984801
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−2.92891e23	−0.123285
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−5.49756e24	−1.00000
$81$	6.16797e24	0.954633
$82$	1.49874e25	1.97763
$83$	1.78871e24	0.201615	0.100808	−	0.994906i	$-0.467857\pi$
$83$	1.78871e24	0.201615	0.100808	+	0.994906i	$0.467857\pi$
$84$	−1.49591e24	−0.144303
$85$	0	0
$86$	2.81191e25	1.99766
$87$	−3.79553e24	−0.232018
$88$	0	0
$89$	2.33570e25	1.06254	0.531272	−	0.847201i	$-0.321714\pi$
$89$	2.33570e25	1.06254	0.531272	+	0.847201i	$0.321714\pi$
$90$	−2.50323e25	−0.984801
$91$	0	0
$92$	−5.14829e25	−1.52202
$93$	0	0
$94$	4.70223e25	1.05110
$95$	0	0
$96$	7.25164e24	0.123285
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−2.84566e25	−0.370037
$99$	0	0
$100$	1.00000e26	1.00000
$101$	1.90891e26	1.67729	0.838643	−	0.544682i	$-0.183349\pi$
$101$	1.90891e26	1.67729	0.838643	+	0.544682i	$0.183349\pi$
$102$	0	0
$103$	9.88703e25	0.673259	0.336629	−	0.941637i	$-0.390713\pi$
$103$	9.88703e25	0.673259	0.336629	+	0.941637i	$0.390713\pi$
$104$	0	0
$105$	2.72105e25	0.144303
$106$	0	0
$107$	−4.74135e26	−1.96749	−0.983746	−	0.179566i	$-0.942531\pi$
$107$	−4.74135e26	−1.96749	−0.983746	+	0.179566i	$0.942531\pi$
$108$	6.65481e25	0.244696
$109$	4.66237e26	1.52077	0.760383	−	0.649475i	$-0.225011\pi$
$109$	4.66237e26	1.52077	0.760383	+	0.649475i	$0.225011\pi$
$110$	0	0
$111$	0	0
$112$	5.10741e26	1.17049
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	9.36469e26	1.52202
$116$	1.29588e27	1.88197
$117$	0	0
$118$	0	0
$119$	0	0
$120$	−1.31906e26	−0.123285
$121$	1.19182e27	1.00000
$122$	4.15838e26	0.313506
$123$	3.59601e26	0.243812
$124$	0	0
$125$	−1.81899e27	−1.00000
$126$	2.32558e27	1.15270
$127$	9.37166e26	0.419149	0.209574	−	0.977793i	$-0.432792\pi$
$127$	9.37166e26	0.419149	0.209574	+	0.977793i	$0.432792\pi$
$128$	−2.47588e27	−1.00000
$129$	6.74680e26	0.246282
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	−7.73737e27	−1.72279
$135$	−1.21050e27	−0.244696
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	−1.23526e27	−0.187643
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−9.29032e27	−1.17049
$141$	1.12824e27	0.129584
$142$	0	0
$143$	0	0
$144$	−1.12736e28	−0.984801
$145$	−2.35720e28	−1.88197
$146$	0	0
$147$	−6.82779e26	−0.0456199
$148$	0	0
$149$	2.76227e27	0.154826	0.0774132	−	0.996999i	$-0.475334\pi$
$149$	2.76227e27	0.154826	0.0774132	+	0.996999i	$0.475334\pi$
$150$	2.39937e27	0.123285
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	4.50360e28	1.00000
$161$	−8.70009e28	−1.78151
$162$	−5.05280e28	−0.954633
$163$	6.18526e28	1.07875	0.539373	−	0.842067i	$-0.318661\pi$
$163$	6.18526e28	1.07875	0.539373	+	0.842067i	$0.318661\pi$
$164$	−1.22776e29	−1.97763
$165$	0	0
$166$	−1.46531e28	−0.201615
$167$	1.47786e29	1.88070	0.940352	−	0.340204i	$-0.110496\pi$
$167$	1.47786e29	1.88070	0.940352	+	0.340204i	$0.110496\pi$
$168$	1.22545e28	0.144303
$169$	9.17333e28	1.00000
$170$	0	0
$171$	0	0
$172$	−2.30352e29	−1.99766
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	3.10930e28	0.232018
$175$	1.68990e29	1.17049
$176$	0	0
$177$	0	0
$178$	−1.91340e29	−1.06254
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	2.05065e29	0.984801
$181$	1.59151e29	0.711193	0.355597	−	0.934639i	$-0.384278\pi$
$181$	1.59151e29	0.711193	0.355597	+	0.934639i	$0.384278\pi$
$182$	0	0
$183$	9.97747e27	0.0386506
$184$	4.21748e29	1.52202
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−3.85207e29	−1.05110
$189$	1.12460e29	0.286413
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−5.94054e28	−0.123285
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	2.33117e29	0.370037
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	−8.19200e29	−1.00000
$201$	−1.85648e29	−0.212393
$202$	−1.56378e30	−1.67729
$203$	2.18991e30	2.20282
$204$	0	0
$205$	2.23329e30	1.97763
$206$	−8.09946e29	−0.673259
$207$	1.92037e30	1.49889
$208$	0	0
$209$	0	0
$210$	−2.22909e29	−0.144303
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	3.88411e30	1.96749
$215$	4.19008e30	1.99766
$216$	−5.45162e29	−0.244696
$217$	0	0
$218$	−3.81941e30	−1.52077
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	6.25516e30	1.85471	0.927353	−	0.374187i	$-0.122078\pi$
$223$	6.25516e30	1.85471	0.927353	+	0.374187i	$0.122078\pi$
$224$	−4.18399e30	−1.17049
$225$	−3.73011e30	−0.984801
$226$	0	0
$227$	−6.38136e30	−1.50168	−0.750839	−	0.660485i	$-0.770351\pi$
$227$	−6.38136e30	−1.50168	−0.750839	+	0.660485i	$0.770351\pi$
$228$	0	0
$229$	−9.50662e30	−1.99602	−0.998010	−	0.0630538i	$-0.979916\pi$
$229$	−9.50662e30	−1.99602	−0.998010	+	0.0630538i	$0.979916\pi$
$230$	−7.67155e30	−1.52202
$231$	0	0
$232$	−1.06159e31	−1.88197
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	7.00687e30	1.05110
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	1.08058e30	0.123285
$241$	1.07832e31	1.16554	0.582771	−	0.812636i	$-0.301968\pi$
$241$	1.07832e31	1.16554	0.582771	+	0.812636i	$0.301968\pi$
$242$	−9.76337e30	−1.00000
$243$	−3.73298e30	−0.362388
$244$	−3.40655e30	−0.313506
$245$	−4.24037e30	−0.370037
$246$	−2.94585e30	−0.243812
$247$	0	0
$248$	0	0
$249$	−3.51581e29	−0.0248561
$250$	1.49012e31	1.00000
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−1.90512e31	−1.15270
$253$	0	0
$254$	−7.67726e30	−0.419149
$255$	0	0
$256$	2.02824e31	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−5.52698e30	−0.246282
$259$	0	0
$260$	0	0
$261$	−4.83378e31	−1.85336
$262$	0	0
$263$	5.21305e31	1.80995	0.904976	−	0.425463i	$-0.139889\pi$
$263$	5.21305e31	1.80995	0.904976	+	0.425463i	$0.139889\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.59096e30	−0.130996
$268$	6.33846e31	1.72279
$269$	6.13999e31	1.58997	0.794984	−	0.606631i	$-0.207479\pi$
$269$	6.13999e31	1.58997	0.794984	+	0.606631i	$0.207479\pi$
$270$	9.91644e30	0.244696
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	1.01193e31	0.187643
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	7.61063e31	1.17049
$281$	1.35628e32	1.99145	0.995723	−	0.0923919i	$-0.0294512\pi$
$281$	1.35628e32	1.99145	0.995723	+	0.0923919i	$0.0294512\pi$
$282$	−9.24252e30	−0.129584
$283$	−1.47609e32	−1.97647	−0.988233	−	0.152959i	$-0.951120\pi$
$283$	−1.47609e32	−1.97647	−0.988233	+	0.152959i	$0.951120\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.07480e32	−2.31479
$288$	9.23529e31	0.984801
$289$	9.81007e31	1.00000
$290$	1.93102e32	1.88197
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	5.59332e30	0.0456199
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−2.26285e31	−0.154826
$299$	0	0
$300$	−1.96556e31	−0.123285
$301$	−3.89271e32	−2.33823
$302$	0	0
$303$	−3.75207e31	−0.206784
$304$	0	0
$305$	6.19647e31	0.313506
$306$	0	0
$307$	−4.30315e32	−1.99980	−0.999902	−	0.0139720i	$-0.995552\pi$
$307$	−4.30315e32	−1.99980	−0.999902	+	0.0139720i	$0.995552\pi$
$308$	0	0
$309$	−1.94336e31	−0.0830027
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	3.46539e32	1.15270
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	−3.68935e32	−1.00000
$321$	9.31941e31	0.242562
$322$	7.12711e32	1.78151
$323$	0	0
$324$	4.13925e32	0.954633
$325$	0	0
$326$	−5.06697e32	−1.07875
$327$	−9.16417e31	−0.187487
$328$	1.00578e33	1.97763
$329$	−6.50961e32	−1.23029
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	1.20038e32	0.201615
$333$	0	0
$334$	−1.21067e33	−1.88070
$335$	−1.15296e33	−1.72279
$336$	−1.00389e32	−0.144303
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−7.51479e32	−1.00000
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−6.70664e32	−0.737363
$344$	1.88704e33	1.99766
$345$	−1.84069e32	−0.187643
$346$	0	0
$347$	1.73070e33	1.63659	0.818293	−	0.574801i	$-0.194921\pi$
$347$	1.73070e33	1.63659	0.818293	+	0.574801i	$0.194921\pi$
$348$	−2.54714e32	−0.232018
$349$	−9.08067e32	−0.796870	−0.398435	−	0.917197i	$-0.630447\pi$
$349$	−9.08067e32	−0.796870	−0.398435	+	0.917197i	$0.630447\pi$
$350$	−1.38437e33	−1.17049
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	1.56746e33	1.06254
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−1.67989e33	−0.984801
$361$	1.76845e33	1.00000
$362$	−1.30376e33	−0.711193
$363$	−2.34259e32	−0.123285
$364$	0	0
$365$	0	0
$366$	−8.17355e31	−0.0386506
$367$	2.39255e33	1.09195	0.545974	−	0.837802i	$-0.316160\pi$
$367$	2.39255e33	1.09195	0.545974	+	0.837802i	$0.316160\pi$
$368$	−3.45496e33	−1.52202
$369$	4.57969e33	1.94757
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	3.57533e32	0.123285
$376$	3.15561e33	1.05110
$377$	0	0
$378$	−9.21269e32	−0.286413
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−1.84206e32	−0.0516747
$382$	0	0
$383$	5.18603e33	1.35910	0.679549	−	0.733630i	$-0.262176\pi$
$383$	5.18603e33	1.35910	0.679549	+	0.733630i	$0.262176\pi$
$384$	4.86649e32	0.123285
$385$	0	0
$386$	0	0
$387$	8.59237e33	1.96730
$388$	0	0
$389$	2.51709e33	0.538956	0.269478	−	0.963007i	$-0.413149\pi$
$389$	2.51709e33	0.538956	0.269478	+	0.963007i	$0.413149\pi$
$390$	0	0
$391$	0	0
$392$	−1.90969e33	−0.370037
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	6.71089e33	1.00000
$401$	−1.25035e34	−1.80366	−0.901828	−	0.432096i	$-0.857774\pi$
$401$	−1.25035e34	−1.80366	−0.901828	+	0.432096i	$0.857774\pi$
$402$	1.52083e33	0.212393
$403$	0	0
$404$	1.28105e34	1.67729
$405$	−7.52925e33	−0.954633
$406$	−1.79398e34	−2.20282
$407$	0	0
$408$	0	0
$409$	1.51273e34	1.68795	0.843973	−	0.536385i	$-0.180211\pi$
$409$	1.51273e34	1.68795	0.843973	+	0.536385i	$0.180211\pi$
$410$	−1.82951e34	−1.97763
$411$	0	0
$412$	6.63508e33	0.673259
$413$	0	0
$414$	−1.57316e34	−1.49889
$415$	−2.18348e33	−0.201615
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	1.82607e33	0.144303
$421$	−9.89833e33	−0.758394	−0.379197	−	0.925316i	$-0.623800\pi$
$421$	−9.89833e33	−0.758394	−0.379197	+	0.925316i	$0.623800\pi$
$422$	0	0
$423$	1.43686e34	1.03512
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5.75672e33	−0.366955
$428$	−3.18187e34	−1.96749
$429$	0	0
$430$	−3.43251e34	−1.99766
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	4.46597e33	0.244696
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	4.63321e33	0.232018
$436$	3.12886e34	1.52077
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−8.69550e33	−0.364412
$442$	0	0
$443$	−4.71366e34	−1.86256	−0.931280	−	0.364304i	$-0.881307\pi$
$443$	−4.71366e34	−1.86256	−0.931280	+	0.364304i	$0.881307\pi$
$444$	0	0
$445$	−2.85119e34	−1.06254
$446$	−5.12423e34	−1.85471
$447$	−5.42941e32	−0.0190878
$448$	3.42752e34	1.17049
$449$	−1.86171e34	−0.617604	−0.308802	−	0.951126i	$-0.599928\pi$
$449$	−1.86171e34	−0.617604	−0.308802	+	0.951126i	$0.599928\pi$
$450$	3.05570e34	0.984801
$451$	0	0
$452$	0	0
$453$	0	0
$454$	5.22761e34	1.50168
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	7.78782e34	1.99602
$459$	0	0
$460$	6.28453e34	1.52202
$461$	5.74687e34	1.35307	0.676533	−	0.736412i	$-0.263481\pi$
$461$	5.74687e34	1.35307	0.676533	+	0.736412i	$0.263481\pi$
$462$	0	0
$463$	4.26692e33	0.0949646	0.0474823	−	0.998872i	$-0.484880\pi$
$463$	4.26692e33	0.0949646	0.0474823	+	0.998872i	$0.484880\pi$
$464$	8.69653e34	1.88197
$465$	0	0
$466$	0	0
$467$	−9.92089e34	−1.97438	−0.987191	−	0.159540i	$-0.948999\pi$
$467$	−9.92089e34	−1.97438	−0.987191	+	0.159540i	$0.948999\pi$
$468$	0	0
$469$	1.07114e35	2.01650
$470$	−5.74003e34	−1.05110
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−8.85210e33	−0.123285
$481$	0	0
$482$	−8.83363e34	−1.16554
$483$	1.71005e34	0.219633
$484$	7.99815e34	1.00000
$485$	0	0
$486$	3.05805e34	0.362388
$487$	−7.96478e34	−0.918962	−0.459481	−	0.888188i	$-0.651965\pi$
$487$	−7.96478e34	−0.918962	−0.459481	+	0.888188i	$0.651965\pi$
$488$	2.79064e34	0.313506
$489$	−1.21575e34	−0.132993
$490$	3.47371e34	0.370037
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	2.41324e34	0.243812
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	2.88015e33	0.0248561
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−1.22070e35	−1.00000
$501$	−2.90483e34	−0.231862
$502$	0	0
$503$	−1.94446e35	−1.47372	−0.736861	−	0.676044i	$-0.763693\pi$
$503$	−1.94446e35	−1.47372	−0.736861	+	0.676044i	$0.763693\pi$
$504$	1.56067e35	1.15270
$505$	−2.33021e35	−1.67729
$506$	0	0
$507$	−1.80307e34	−0.123285
$508$	6.28921e34	0.419149
$509$	2.91654e35	1.89468	0.947342	−	0.320225i	$-0.103759\pi$
$509$	2.91654e35	1.89468	0.947342	+	0.320225i	$0.103759\pi$
$510$	0	0
$511$	0	0
$512$	−1.66153e35	−1.00000
$513$	0	0
$514$	0	0
$515$	−1.20691e35	−0.673259
$516$	4.52770e34	0.246282
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.09527e35	−1.00543	−0.502713	−	0.864453i	$-0.667665\pi$
$521$	−2.09527e35	−1.00543	−0.502713	+	0.864453i	$0.667665\pi$
$522$	3.95983e35	1.85336
$523$	3.89402e35	1.77777	0.888886	−	0.458129i	$-0.151480\pi$
$523$	3.89402e35	1.77777	0.888886	+	0.458129i	$0.151480\pi$
$524$	0	0
$525$	−3.32160e34	−0.144303
$526$	−4.27053e35	−1.80995
$527$	0	0
$528$	0	0
$529$	3.34474e35	1.31655
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	3.76091e34	0.130996
$535$	5.78778e35	1.96749
$536$	−5.19246e35	−1.72279
$537$	0	0
$538$	−5.02988e35	−1.58997
$539$	0	0
$540$	−8.12355e34	−0.244696
$541$	−6.54112e35	−1.92348	−0.961740	−	0.273965i	$-0.911665\pi$
$541$	−6.54112e35	−1.92348	−0.961740	+	0.273965i	$0.911665\pi$
$542$	0	0
$543$	−3.12821e34	−0.0876794
$544$	0	0
$545$	−5.69137e35	−1.52077
$546$	0	0
$547$	1.46915e35	0.374310	0.187155	−	0.982330i	$-0.440073\pi$
$547$	1.46915e35	0.374310	0.187155	+	0.982330i	$0.440073\pi$
$548$	0	0
$549$	1.27068e35	0.308741
$550$	0	0
$551$	0	0
$552$	−8.28971e34	−0.187643
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	−6.23463e35	−1.17049
$561$	0	0
$562$	−1.11107e36	−1.99145
$563$	8.15866e35	1.42893	0.714463	−	0.699674i	$-0.246671\pi$
$563$	8.15866e35	1.42893	0.714463	+	0.699674i	$0.246671\pi$
$564$	7.57147e34	0.129584
$565$	0	0
$566$	1.20921e36	1.97647
$567$	6.99492e35	1.11738
$568$	0	0
$569$	1.27310e36	1.94269	0.971344	−	0.237678i	$-0.0763862\pi$
$569$	1.27310e36	1.94269	0.971344	+	0.237678i	$0.0763862\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	1.69967e36	2.31479
$575$	−1.14315e36	−1.52202
$576$	−7.56555e35	−0.984801
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−8.03641e35	−1.00000
$579$	0	0
$580$	−1.58189e36	−1.88197
$581$	2.02852e35	0.235988
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.16869e36	1.18960	0.594802	−	0.803872i	$-0.297230\pi$
$587$	1.16869e36	1.18960	0.594802	+	0.803872i	$0.297230\pi$
$588$	−4.58205e34	−0.0456199
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	1.85373e35	0.154826
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	1.61019e35	0.123285
$601$	2.65724e36	1.99096	0.995479	−	0.0949804i	$-0.0302788\pi$
$601$	2.65724e36	1.99096	0.995479	+	0.0949804i	$0.0302788\pi$
$602$	3.18891e36	2.33823
$603$	−2.36431e36	−1.69660
$604$	0	0
$605$	−1.45486e36	−1.00000
$606$	3.07370e35	0.206784
$607$	1.81189e36	1.19311	0.596553	−	0.802573i	$-0.296536\pi$
$607$	1.81189e36	1.19311	0.596553	+	0.802573i	$0.296536\pi$
$608$	0	0
$609$	−4.30440e35	−0.271574
$610$	−5.07615e35	−0.313506
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	3.52514e36	1.99980
$615$	−4.38966e35	−0.243812
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	1.59200e35	0.0830027
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	−7.60745e35	−0.372433
$622$	0	0
$623$	2.64885e36	1.24369
$624$	0	0
$625$	2.22045e36	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−2.83884e36	−1.15270
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.14400e36	−0.419149
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	3.02231e36	1.00000
$641$	−6.15657e36	−1.99611	−0.998055	−	0.0623319i	$-0.980146\pi$
$641$	−6.15657e36	−1.99611	−0.998055	+	0.0623319i	$0.980146\pi$
$642$	−7.63446e35	−0.242562
$643$	−5.44896e36	−1.69657	−0.848284	−	0.529542i	$-0.822364\pi$
$643$	−5.44896e36	−1.69657	−0.848284	+	0.529542i	$0.822364\pi$
$644$	−5.83853e36	−1.78151
$645$	−8.23584e35	−0.246282
$646$	0	0
$647$	−6.83892e36	−1.96441	−0.982204	−	0.187816i	$-0.939859\pi$
$647$	−6.83892e36	−1.96441	−0.982204	+	0.187816i	$0.939859\pi$
$648$	−3.39087e36	−0.954633
$649$	0	0
$650$	0	0
$651$	0	0
$652$	4.15086e36	1.07875
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	7.50729e35	0.187487
$655$	0	0
$656$	−8.23938e36	−1.97763
$657$	0	0
$658$	5.33267e36	1.23029
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	−1.69884e36	−0.369430	−0.184715	−	0.982792i	$-0.559136\pi$
$661$	−1.69884e36	−0.369430	−0.184715	+	0.982792i	$0.559136\pi$
$662$	0	0
$663$	0	0
$664$	−9.83351e35	−0.201615
$665$	0	0
$666$	0	0
$667$	−1.48139e37	−2.86440
$668$	9.91778e36	1.88070
$669$	−1.22949e36	−0.228657
$670$	9.44504e36	1.72279
$671$	0	0
$672$	8.22388e35	0.144303
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	1.47766e36	0.244696
$676$	6.15612e36	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1.25429e36	0.185134
$682$	0	0
$683$	1.04219e37	1.48073	0.740366	−	0.672204i	$-0.234652\pi$
$683$	1.04219e37	1.48073	0.740366	+	0.672204i	$0.234652\pi$
$684$	0	0
$685$	0	0
$686$	5.49408e36	0.737363
$687$	1.86858e36	0.246079
$688$	−1.54586e37	−1.99766
$689$	0	0
$690$	1.50789e36	0.187643
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	−1.41779e37	−1.63659
$695$	0	0
$696$	2.08661e36	0.232018
$697$	0	0
$698$	7.43889e36	0.796870
$699$	0	0
$700$	1.13407e37	1.17049
$701$	6.56909e36	0.665535	0.332767	−	0.943009i	$-0.392018\pi$
$701$	6.56909e36	0.665535	0.332767	+	0.943009i	$0.392018\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−1.37724e36	−0.129584
$706$	0	0
$707$	2.16484e37	1.96324
$708$	0	0
$709$	9.06611e36	0.792538	0.396269	−	0.918134i	$-0.370305\pi$
$709$	9.06611e36	0.792538	0.396269	+	0.918134i	$0.370305\pi$
$710$	0	0
$711$	0	0
$712$	−1.28406e37	−1.06254
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	1.37617e37	0.984801
$721$	1.12126e37	0.788040
$722$	−1.44872e37	−1.00000
$723$	−2.11951e36	−0.143694
$724$	1.06804e37	0.711193
$725$	2.87744e37	1.88197
$726$	1.91905e36	0.123285
$727$	−1.45479e37	−0.918020	−0.459010	−	0.888431i	$-0.651796\pi$
$727$	−1.45479e37	−0.918020	−0.459010	+	0.888431i	$0.651796\pi$
$728$	0	0
$729$	−1.49444e37	−0.909957
$730$	0	0
$731$	0	0
$732$	6.69577e35	0.0386506
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−1.95998e37	−1.09195
$735$	8.33470e35	0.0456199
$736$	2.83030e37	1.52202
$737$	0	0
$738$	−3.75168e37	−1.94757
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.82605e37	1.34377	0.671886	−	0.740654i	$-0.265484\pi$
$743$	2.82605e37	1.34377	0.671886	+	0.740654i	$0.265484\pi$
$744$	0	0
$745$	−3.37191e36	−0.154826
$746$	0	0
$747$	−4.47754e36	−0.198551
$748$	0	0
$749$	−5.37703e37	−2.30292
$750$	−2.92891e36	−0.123285
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	−2.58508e37	−1.05110
$753$	0	0
$754$	0	0
$755$	0	0
$756$	7.54703e36	0.286413
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.02479e37	1.75032	0.875160	−	0.483834i	$-0.160756\pi$
$761$	5.02479e37	1.75032	0.875160	+	0.483834i	$0.160756\pi$
$762$	1.50901e36	0.0516747
$763$	5.28746e37	1.78003
$764$	0	0
$765$	0	0
$766$	−4.24839e37	−1.35910
$767$	0	0
$768$	−3.98663e36	−0.123285
$769$	−6.17509e37	−1.87759	−0.938796	−	0.344475i	$-0.888057\pi$
$769$	−6.17509e37	−1.87759	−0.938796	+	0.344475i	$0.888057\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−7.03887e37	−1.96730
$775$	0	0
$776$	0	0
$777$	0	0
$778$	−2.06200e37	−0.538956
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1.91488e37	0.460510
$784$	1.56442e37	0.370037
$785$	0	0
$786$	0	0
$787$	7.27384e37	1.63716	0.818582	−	0.574390i	$-0.194761\pi$
$787$	7.27384e37	1.63716	0.818582	+	0.574390i	$0.194761\pi$
$788$	0	0
$789$	−1.02466e37	−0.223140
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−5.49756e37	−1.00000
$801$	−5.84680e37	−1.04639
$802$	1.02428e38	1.80366
$803$	0	0
$804$	−1.24586e37	−0.212393
$805$	1.06202e38	1.78151
$806$	0	0
$807$	−1.20685e37	−0.196019
$808$	−1.04943e38	−1.67729
$809$	−5.68187e37	−0.893636	−0.446818	−	0.894625i	$-0.647443\pi$
$809$	−5.68187e37	−0.893636	−0.446818	+	0.894625i	$0.647443\pi$
$810$	6.16797e37	0.954633
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	1.46962e38	2.20282
$813$	0	0
$814$	0	0
$815$	−7.55037e37	−1.07875
$816$	0	0
$817$	0	0
$818$	−1.23923e38	−1.68795
$819$	0	0
$820$	1.49874e38	1.97763
$821$	−1.19517e38	−1.55227	−0.776135	−	0.630567i	$-0.782822\pi$
$821$	−1.19517e38	−1.55227	−0.776135	+	0.630567i	$0.782822\pi$
$822$	0	0
$823$	−8.90985e37	−1.12117	−0.560586	−	0.828096i	$-0.689424\pi$
$823$	−8.90985e37	−1.12117	−0.560586	+	0.828096i	$0.689424\pi$
$824$	−5.43545e37	−0.673259
$825$	0	0
$826$	0	0
$827$	−1.68740e38	−1.99363	−0.996817	−	0.0797270i	$-0.974595\pi$
$827$	−1.68740e38	−1.99363	−0.996817	+	0.0797270i	$0.974595\pi$
$828$	1.28874e38	1.49889
$829$	1.72179e38	1.97138	0.985691	−	0.168565i	$-0.0539133\pi$
$829$	1.72179e38	1.97138	0.985691	+	0.168565i	$0.0539133\pi$
$830$	1.78871e37	0.201615
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.80403e38	−1.88070
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	−1.49591e37	−0.144303
$841$	2.67602e38	2.54180
$842$	8.10871e37	0.758394
$843$	−2.66586e37	−0.245515
$844$	0	0
$845$	−1.11979e38	−1.00000
$846$	−1.17708e38	−1.03512
$847$	1.35161e38	1.17049
$848$	0	0
$849$	2.90134e37	0.243668
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	4.71590e37	0.366955
$855$	0	0
$856$	2.60659e38	1.96749
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	2.81191e38	1.99766
$861$	4.07814e37	0.285378
$862$	0	0
$863$	1.40123e38	0.951413	0.475707	−	0.879604i	$-0.342192\pi$
$863$	1.40123e38	0.951413	0.475707	+	0.879604i	$0.342192\pi$
$864$	−3.65852e37	−0.244696
$865$	0	0
$866$	0	0
$867$	−1.92823e37	−0.123285
$868$	0	0
$869$	0	0
$870$	−3.79553e37	−0.232018
$871$	0	0
$872$	−2.56316e38	−1.52077
$873$	0	0
$874$	0	0
$875$	−2.06286e38	−1.17049
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.11066e38	−1.09580	−0.547901	−	0.836543i	$-0.684573\pi$
$881$	−2.11066e38	−1.09580	−0.547901	+	0.836543i	$0.684573\pi$
$882$	7.12336e37	0.364412
$883$	3.48461e38	1.75657	0.878285	−	0.478137i	$-0.158688\pi$
$883$	3.48461e38	1.75657	0.878285	+	0.478137i	$0.158688\pi$
$884$	0	0
$885$	0	0
$886$	3.86143e38	1.86256
$887$	−3.89860e38	−1.85311	−0.926557	−	0.376155i	$-0.877246\pi$
$887$	−3.89860e38	−1.85311	−0.926557	+	0.376155i	$0.877246\pi$
$888$	0	0
$889$	1.06281e38	0.490607
$890$	2.33570e38	1.06254
$891$	0	0
$892$	4.19777e38	1.85471
$893$	0	0
$894$	4.44777e36	0.0190878
$895$	0	0
$896$	−2.80783e38	−1.17049
$897$	0	0
$898$	1.52512e38	0.617604
$899$	0	0
$900$	−2.50323e38	−0.984801
$901$	0	0
$902$	0	0
$903$	7.65136e37	0.288269
$904$	0	0
$905$	−1.94276e38	−0.711193
$906$	0	0
$907$	−1.45678e38	−0.518201	−0.259101	−	0.965850i	$-0.583426\pi$
$907$	−1.45678e38	−0.518201	−0.259101	+	0.965850i	$0.583426\pi$
$908$	−4.28246e38	−1.50168
$909$	−4.77844e38	−1.65179
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−1.21795e37	−0.0386506
$916$	−6.37979e38	−1.99602
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	−5.14829e38	−1.52202
$921$	8.45810e37	0.246546
$922$	−4.70783e38	−1.35307
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−3.49546e37	−0.0949646
$927$	−2.47495e38	−0.663026
$928$	−7.12419e38	−1.88197
$929$	5.78416e38	1.50673	0.753365	−	0.657602i	$-0.228429\pi$
$929$	5.78416e38	1.50673	0.753365	+	0.657602i	$0.228429\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	8.12720e38	1.97438
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	−8.77474e38	−2.01650
$939$	0	0
$940$	4.70223e38	1.05110
$941$	−6.60570e38	−1.45631	−0.728154	−	0.685413i	$-0.759622\pi$
$941$	−6.60570e38	−1.45631	−0.728154	+	0.685413i	$0.759622\pi$
$942$	0	0
$943$	1.40352e39	3.01000
$944$	0	0
$945$	−1.37280e38	−0.286413
$946$	0	0
$947$	6.26207e38	1.27107	0.635533	−	0.772074i	$-0.280780\pi$
$947$	6.26207e38	1.27107	0.635533	+	0.772074i	$0.280780\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	7.25164e37	0.123285
$961$	5.96217e38	1.00000
$962$	0	0
$963$	1.18687e39	1.93759
$964$	7.23651e38	1.16554
$965$	0	0
$966$	−1.40088e38	−0.219633
$967$	−1.28942e39	−1.99457	−0.997285	−	0.0736435i	$-0.976537\pi$
$967$	−1.28942e39	−1.99457	−0.997285	+	0.0736435i	$0.976537\pi$
$968$	−6.55209e38	−1.00000
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−2.50516e38	−0.362388
$973$	0	0
$974$	6.52475e38	0.918962
$975$	0	0
$976$	−2.28609e38	−0.313506
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	9.95943e37	0.132993
$979$	0	0
$980$	−2.84566e38	−0.370037
$981$	−1.16710e39	−1.49765
$982$	0	0
$983$	1.52228e39	1.90238	0.951192	−	0.308601i	$-0.0998607\pi$
$983$	1.52228e39	1.90238	0.951192	+	0.308601i	$0.0998607\pi$
$984$	−1.97693e38	−0.243812
$985$	0	0
$986$	0	0
$987$	1.27950e38	0.151676
$988$	0	0
$989$	2.63327e39	3.04049
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	−2.35942e37	−0.0248561
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	20.27.d.a.19.1	✓	1
4.3	odd	2		20.27.d.b.19.1	yes	1
5.4	even	2		20.27.d.b.19.1	yes	1
20.19	odd	2	CM	20.27.d.a.19.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.27.d.a.19.1	✓	1	1.1	even	1	trivial
20.27.d.a.19.1	✓	1	20.19	odd	2	CM
20.27.d.b.19.1	yes	1	4.3	odd	2
20.27.d.b.19.1	yes	1	5.4	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 20 = 2^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 27 \)
Character orbit:	\([\chi]\)	\(=\)	20.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(11\)	\(17\)
\(\chi(n)\)	\(-1\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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