LMFDB - Embedded newform 24.21.h.a.5.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [24,21,Mod(5,24)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("24.5"); S:= CuspForms(chi, 21); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(24, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 21, names="a")

Level:	$ N $	$=$	$ 24 = 2^{3} \cdot 3 $
Weight:	$ k $	$=$	$ 21 $
Character orbit:	$[\chi]$	$=$	24.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,-1024] 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:	$60.8433036246$
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			5.1
Character	$\chi$	$=$	24.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1024.00 q^{2} -59049.0 q^{3} +1.04858e6 q^{4} -8.36827e6 q^{5} +6.04662e7 q^{6} +5.75520e7 q^{7} -1.07374e9 q^{8} +3.48678e9 q^{9} +8.56911e9 q^{10} +3.23147e8 q^{11} -6.19174e10 q^{12} -5.89333e10 q^{14} +4.94138e11 q^{15} +1.09951e12 q^{16} -3.57047e12 q^{18} -8.77477e12 q^{20} -3.39839e12 q^{21} -3.30902e11 q^{22} +6.34034e13 q^{24} -2.53394e13 q^{25} -2.05891e14 q^{27} +6.03476e13 q^{28} -4.76620e14 q^{29} -5.05998e14 q^{30} -1.55905e15 q^{31} -1.12590e15 q^{32} -1.90815e13 q^{33} -4.81611e14 q^{35} +3.65616e15 q^{36} +8.98537e15 q^{40} +3.47995e15 q^{42} +3.38844e14 q^{44} -2.91784e16 q^{45} -6.49251e16 q^{48} -7.64800e16 q^{49} +2.59476e16 q^{50} -3.21317e16 q^{53} +2.10833e17 q^{54} -2.70418e15 q^{55} -6.17960e16 q^{56} +4.88059e17 q^{58} +6.75827e17 q^{59} +5.18141e17 q^{60} +1.59647e18 q^{62} +2.00671e17 q^{63} +1.15292e18 q^{64} +1.95394e16 q^{66} +4.93170e17 q^{70} -3.74391e18 q^{72} +8.06286e18 q^{73} +1.49627e18 q^{75} +1.85977e16 q^{77} +1.54430e19 q^{79} -9.20101e18 q^{80} +1.21577e19 q^{81} -3.10178e19 q^{83} -3.56347e18 q^{84} +2.81440e19 q^{87} -3.46976e17 q^{88} +2.98786e19 q^{90} +9.20604e19 q^{93} +6.64833e19 q^{96} -6.59247e19 q^{97} +7.83156e19 q^{98} +1.12674e18 q^{99} +O(q^{100})\)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$.

$n$	$7$	$13$	$17$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	−1024.00	−1.00000
$2$	−1024.00	−1.00000
$3$	−59049.0	−1.00000
$3$	−59049.0	−1.00000
$4$	1.04858e6	1.00000
$5$	−8.36827e6	−0.856911	−0.428456	−	0.903563i	$-0.640942\pi$
$5$	−8.36827e6	−0.856911	−0.428456	+	0.903563i	$0.640942\pi$
$6$	6.04662e7	1.00000
$7$	5.75520e7	0.203742	0.101871	−	0.994798i	$-0.467517\pi$
$7$	5.75520e7	0.203742	0.101871	+	0.994798i	$0.467517\pi$
$8$	−1.07374e9	−1.00000
$9$	3.48678e9	1.00000
$10$	8.56911e9	0.856911
$11$	3.23147e8	0.0124587	0.00622935	−	0.999981i	$-0.498017\pi$
$11$	3.23147e8	0.0124587	0.00622935	+	0.999981i	$0.498017\pi$
$12$	−6.19174e10	−1.00000
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−5.89333e10	−0.203742
$15$	4.94138e11	0.856911
$16$	1.09951e12	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−3.57047e12	−1.00000
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−8.77477e12	−0.856911
$21$	−3.39839e12	−0.203742
$22$	−3.30902e11	−0.0124587
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	6.34034e13	1.00000
$25$	−2.53394e13	−0.265703
$26$	0	0
$27$	−2.05891e14	−1.00000
$28$	6.03476e13	0.203742
$29$	−4.76620e14	−1.13290	−0.566451	−	0.824095i	$-0.691684\pi$
$29$	−4.76620e14	−1.13290	−0.566451	+	0.824095i	$0.691684\pi$
$30$	−5.05998e14	−0.856911
$31$	−1.55905e15	−1.90214	−0.951071	−	0.308971i	$-0.900015\pi$
$31$	−1.55905e15	−1.90214	−0.951071	+	0.308971i	$0.900015\pi$
$32$	−1.12590e15	−1.00000
$33$	−1.90815e13	−0.0124587
$34$	0	0
$35$	−4.81611e14	−0.174589
$36$	3.65616e15	1.00000
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	8.98537e15	0.856911
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	3.47995e15	0.203742
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	3.38844e14	0.0124587
$45$	−2.91784e16	−0.856911
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−6.49251e16	−1.00000
$49$	−7.64800e16	−0.958489
$50$	2.59476e16	0.265703
$51$	0	0
$52$	0	0
$53$	−3.21317e16	−0.183728	−0.0918640	−	0.995772i	$-0.529283\pi$
$53$	−3.21317e16	−0.183728	−0.0918640	+	0.995772i	$0.529283\pi$
$54$	2.10833e17	1.00000
$55$	−2.70418e15	−0.0106760
$56$	−6.17960e16	−0.203742
$57$	0	0
$58$	4.88059e17	1.13290
$59$	6.75827e17	1.32226	0.661128	−	0.750273i	$-0.270078\pi$
$59$	6.75827e17	1.32226	0.661128	+	0.750273i	$0.270078\pi$
$60$	5.18141e17	0.856911
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	1.59647e18	1.90214
$63$	2.00671e17	0.203742
$64$	1.15292e18	1.00000
$65$	0	0
$66$	1.95394e16	0.0124587
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	4.93170e17	0.174589
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−3.74391e18	−1.00000
$73$	8.06286e18	1.87612	0.938060	−	0.346473i	$-0.112621\pi$
$73$	8.06286e18	1.87612	0.938060	+	0.346473i	$0.112621\pi$
$74$	0	0
$75$	1.49627e18	0.265703
$76$	0	0
$77$	1.85977e16	0.00253836
$78$	0	0
$79$	1.54430e19	1.63103	0.815514	−	0.578737i	$-0.196454\pi$
$79$	1.54430e19	1.63103	0.815514	+	0.578737i	$0.196454\pi$
$80$	−9.20101e18	−0.856911
$81$	1.21577e19	1.00000
$82$	0	0
$83$	−3.10178e19	−1.99908	−0.999539	−	0.0303526i	$-0.990337\pi$
$83$	−3.10178e19	−1.99908	−0.999539	+	0.0303526i	$0.990337\pi$
$84$	−3.56347e18	−0.203742
$85$	0	0
$86$	0	0
$87$	2.81440e19	1.13290
$88$	−3.46976e17	−0.0124587
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	2.98786e19	0.856911
$91$	0	0
$92$	0	0
$93$	9.20604e19	1.90214
$94$	0	0
$95$	0	0
$96$	6.64833e19	1.00000
$97$	−6.59247e19	−0.893987	−0.446993	−	0.894537i	$-0.647505\pi$
$97$	−6.59247e19	−0.893987	−0.446993	+	0.894537i	$0.647505\pi$
$98$	7.83156e19	0.958489
$99$	1.12674e18	0.0124587
$100$	−2.65703e19	−0.265703
$101$	2.09527e20	1.89682	0.948410	−	0.317046i	$-0.102691\pi$
$101$	2.09527e20	1.89682	0.948410	+	0.317046i	$0.102691\pi$
$102$	0	0
$103$	−2.37707e20	−1.76876	−0.884382	−	0.466764i	$-0.845420\pi$
$103$	−2.37707e20	−1.76876	−0.884382	+	0.466764i	$0.845420\pi$
$104$	0	0
$105$	2.84386e19	0.174589
$106$	3.29029e19	0.183728
$107$	−2.14561e20	−1.09072	−0.545359	−	0.838203i	$-0.683607\pi$
$107$	−2.14561e20	−1.09072	−0.545359	+	0.838203i	$0.683607\pi$
$108$	−2.15892e20	−1.00000
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	2.76908e18	0.0106760
$111$	0	0
$112$	6.32791e19	0.203742
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−4.99773e20	−1.13290
$117$	0	0
$118$	−6.92047e20	−1.32226
$119$	0	0
$120$	−5.30577e20	−0.856911
$121$	−6.72646e20	−0.999845
$122$	0	0
$123$	0	0
$124$	−1.63478e21	−1.90214
$125$	1.01011e21	1.08460
$126$	−2.05488e20	−0.203742
$127$	−7.07964e20	−0.648596	−0.324298	−	0.945955i	$-0.605128\pi$
$127$	−7.07964e20	−0.648596	−0.324298	+	0.945955i	$0.605128\pi$
$128$	−1.18059e21	−1.00000
$129$	0	0
$130$	0	0
$131$	2.96088e21	1.98933	0.994667	−	0.103137i	$-0.0328879\pi$
$131$	2.96088e21	1.98933	0.994667	+	0.103137i	$0.0328879\pi$
$132$	−2.00084e19	−0.0124587
$133$	0	0
$134$	0	0
$135$	1.72295e21	0.856911
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−5.05006e20	−0.174589
$141$	0	0
$142$	0	0
$143$	0	0
$144$	3.83376e21	1.00000
$145$	3.98849e21	0.970797
$146$	−8.25637e21	−1.87612
$147$	4.51607e21	0.958489
$148$	0	0
$149$	7.36498e21	1.36555	0.682777	−	0.730627i	$-0.260772\pi$
$149$	7.36498e21	1.36555	0.682777	+	0.730627i	$0.260772\pi$
$150$	−1.53218e21	−0.265703
$151$	−7.21636e20	−0.117098	−0.0585489	−	0.998285i	$-0.518647\pi$
$151$	−7.21636e20	−0.117098	−0.0585489	+	0.998285i	$0.518647\pi$
$152$	0	0
$153$	0	0
$154$	−1.90441e19	−0.00253836
$155$	1.30466e22	1.62997
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	−1.58137e22	−1.63103
$159$	1.89735e21	0.183728
$160$	9.42184e21	0.856911
$161$	0	0
$162$	−1.24494e22	−1.00000
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	1.59679e20	0.0106760
$166$	3.17622e22	1.99908
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	3.64899e21	0.203742
$169$	1.90050e22	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.12935e22	0.470292	0.235146	−	0.971960i	$-0.424443\pi$
$173$	1.12935e22	0.470292	0.235146	+	0.971960i	$0.424443\pi$
$174$	−2.88194e22	−1.13290
$175$	−1.45833e21	−0.0541348
$176$	3.55304e20	0.0124587
$177$	−3.99069e22	−1.32226
$178$	0	0
$179$	5.19248e22	1.53760	0.768802	−	0.639487i	$-0.220853\pi$
$179$	5.19248e22	1.53760	0.768802	+	0.639487i	$0.220853\pi$
$180$	−3.05957e22	−0.856911
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	−9.42698e22	−1.90214
$187$	0	0
$188$	0	0
$189$	−1.18494e22	−0.203742
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−6.80789e22	−1.00000
$193$	8.63651e22	1.20439	0.602193	−	0.798351i	$-0.294294\pi$
$193$	8.63651e22	1.20439	0.602193	+	0.798351i	$0.294294\pi$
$194$	6.75069e22	0.893987
$195$	0	0
$196$	−8.01951e22	−0.958489
$197$	7.43423e22	0.844450	0.422225	−	0.906491i	$-0.361249\pi$
$197$	7.43423e22	0.844450	0.422225	+	0.906491i	$0.361249\pi$
$198$	−1.15378e21	−0.0124587
$199$	−1.11724e23	−1.14714	−0.573570	−	0.819157i	$-0.694442\pi$
$199$	−1.11724e23	−1.14714	−0.573570	+	0.819157i	$0.694442\pi$
$200$	2.72080e22	0.265703
$201$	0	0
$202$	−2.14556e23	−1.89682
$203$	−2.74305e22	−0.230820
$204$	0	0
$205$	0	0
$206$	2.43412e23	1.76876
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−2.91212e22	−0.174589
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	−3.36926e22	−0.183728
$213$	0	0
$214$	2.19710e23	1.09072
$215$	0	0
$216$	2.21074e23	1.00000
$217$	−8.97265e22	−0.387546
$218$	0	0
$219$	−4.76104e23	−1.87612
$220$	−2.83554e21	−0.0106760
$221$	0	0
$222$	0	0
$223$	−3.94533e23	−1.29728	−0.648642	−	0.761094i	$-0.724663\pi$
$223$	−3.94533e23	−1.29728	−0.648642	+	0.761094i	$0.724663\pi$
$224$	−6.47978e22	−0.203742
$225$	−8.83531e22	−0.265703
$226$	0	0
$227$	−5.27133e23	−1.45098	−0.725491	−	0.688232i	$-0.758387\pi$
$227$	−5.27133e23	−1.45098	−0.725491	+	0.688232i	$0.758387\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	−1.09818e21	−0.00253836
$232$	5.11767e23	1.13290
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	7.08656e23	1.32226
$237$	−9.11895e23	−1.63103
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	5.43311e23	0.856911
$241$	1.27213e24	1.92469	0.962345	−	0.271830i	$-0.0876288\pi$
$241$	1.27213e24	1.92469	0.962345	+	0.271830i	$0.0876288\pi$
$242$	6.88789e23	0.999845
$243$	−7.17898e23	−1.00000
$244$	0	0
$245$	6.40006e23	0.821340
$246$	0	0
$247$	0	0
$248$	1.67402e24	1.90214
$249$	1.83157e24	1.99908
$250$	−1.03435e24	−1.08460
$251$	1.78894e24	1.80243	0.901217	−	0.433368i	$-0.142675\pi$
$251$	1.78894e24	1.80243	0.901217	+	0.433368i	$0.142675\pi$
$252$	2.10419e23	0.203742
$253$	0	0
$254$	7.24955e23	0.648596
$255$	0	0
$256$	1.20893e24	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.66187e24	−1.13290
$262$	−3.03194e24	−1.98933
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	2.04886e22	0.0124587
$265$	2.68887e23	0.157439
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.91445e24	−1.97310	−0.986548	−	0.163470i	$-0.947731\pi$
$269$	−3.91445e24	−1.97310	−0.986548	+	0.163470i	$0.947731\pi$
$270$	−1.76430e24	−0.856911
$271$	−2.04341e24	−0.956450	−0.478225	−	0.878237i	$-0.658720\pi$
$271$	−2.04341e24	−0.956450	−0.478225	+	0.878237i	$0.658720\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.18835e21	−0.00331032
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	−5.43607e24	−1.90214
$280$	5.17126e23	0.174589
$281$	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
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−3.92577e24	−1.00000
$289$	4.06423e24	1.00000
$290$	−4.08421e24	−0.970797
$291$	3.89279e24	0.893987
$292$	8.45452e24	1.87612
$293$	5.74047e24	1.23104	0.615519	−	0.788122i	$-0.288946\pi$
$293$	5.74047e24	1.23104	0.615519	+	0.788122i	$0.288946\pi$
$294$	−4.62446e24	−0.958489
$295$	−5.65551e24	−1.13306
$296$	0	0
$297$	−6.65330e22	−0.0124587
$298$	−7.54174e24	−1.36555
$299$	0	0
$300$	1.56895e24	0.265703
$301$	0	0
$302$	7.38955e23	0.117098
$303$	−1.23724e25	−1.89682
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	1.95011e22	0.00253836
$309$	1.40364e25	1.76876
$310$	−1.33597e25	−1.62997
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	1.41286e25	1.56551	0.782754	−	0.622331i	$-0.213814\pi$
$313$	1.41286e25	1.56551	0.782754	+	0.622331i	$0.213814\pi$
$314$	0	0
$315$	−1.67927e24	−0.174589
$316$	1.61932e25	1.63103
$317$	1.52069e25	1.48405	0.742027	−	0.670371i	$-0.233865\pi$
$317$	1.52069e25	1.48405	0.742027	+	0.670371i	$0.233865\pi$
$318$	−1.94288e24	−0.183728
$319$	−1.54018e23	−0.0141145
$320$	−9.64796e24	−0.856911
$321$	1.26696e25	1.09072
$322$	0	0
$323$	0	0
$324$	1.27482e25	1.00000
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	−1.63511e23	−0.0106760
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−3.25245e25	−1.99908
$333$	0	0
$334$	0	0
$335$	0	0
$336$	−3.73657e24	−0.203742
$337$	3.62737e25	1.91996	0.959981	−	0.280067i	$-0.0903565\pi$
$337$	3.62737e25	1.91996	0.959981	+	0.280067i	$0.0903565\pi$
$338$	−1.94611e25	−1.00000
$339$	0	0
$340$	0	0
$341$	−5.03802e23	−0.0236982
$342$	0	0
$343$	−8.99378e24	−0.399026
$344$	0	0
$345$	0	0
$346$	−1.15645e25	−0.470292
$347$	1.89361e24	0.0748164	0.0374082	−	0.999300i	$-0.488090\pi$
$347$	1.89361e24	0.0748164	0.0374082	+	0.999300i	$0.488090\pi$
$348$	2.95111e25	1.13290
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	1.49333e24	0.0541348
$351$	0	0
$352$	−3.63831e23	−0.0124587
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	4.08647e25	1.32226
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−5.31710e25	−1.53760
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	3.13300e25	0.856911
$361$	3.75900e25	1.00000
$362$	0	0
$363$	3.97190e25	0.999845
$364$	0	0
$365$	−6.74722e25	−1.60767
$366$	0	0
$367$	−7.42841e25	−1.67585	−0.837925	−	0.545786i	$-0.816231\pi$
$367$	−7.42841e25	−1.67585	−0.837925	+	0.545786i	$0.816231\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.84925e24	−0.0374331
$372$	9.65323e25	1.90214
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−5.96459e25	−1.08460
$376$	0	0
$377$	0	0
$378$	1.21338e25	0.203742
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	4.18046e25	0.648596
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	6.97128e25	1.00000
$385$	−1.55631e23	−0.00217515
$386$	−8.84379e25	−1.20439
$387$	0	0
$388$	−6.91271e25	−0.893987
$389$	−1.23935e26	−1.56206	−0.781031	−	0.624493i	$-0.785306\pi$
$389$	−1.23935e26	−1.56206	−0.781031	+	0.624493i	$0.785306\pi$
$390$	0	0
$391$	0	0
$392$	8.21198e25	0.958489
$393$	−1.74837e26	−1.98933
$394$	−7.61265e25	−0.844450
$395$	−1.29231e26	−1.39765
$396$	1.18148e24	0.0124587
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	1.14406e26	1.14714
$399$	0	0
$400$	−2.78610e25	−0.265703
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	2.19705e26	1.89682
$405$	−1.01739e26	−0.856911
$406$	2.80888e25	0.230820
$407$	0	0
$408$	0	0
$409$	7.34316e25	0.560596	0.280298	−	0.959913i	$-0.409567\pi$
$409$	7.34316e25	0.560596	0.280298	+	0.959913i	$0.409567\pi$
$410$	0	0
$411$	0	0
$412$	−2.49254e26	−1.76876
$413$	3.88952e25	0.269399
$414$	0	0
$415$	2.59565e26	1.71303
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.36241e26	−0.816899	−0.408449	−	0.912781i	$-0.633930\pi$
$419$	−1.36241e26	−0.816899	−0.408449	+	0.912781i	$0.633930\pi$
$420$	2.98201e25	0.174589
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	3.45012e25	0.183728
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−2.24983e26	−1.09072
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−2.26380e26	−1.00000
$433$	3.87927e26	1.67444	0.837222	−	0.546863i	$-0.184178\pi$
$433$	3.87927e26	1.67444	0.837222	+	0.546863i	$0.184178\pi$
$434$	9.18799e25	0.387546
$435$	−2.35516e26	−0.970797
$436$	0	0
$437$	0	0
$438$	4.87530e26	1.87612
$439$	5.30004e26	1.99358	0.996791	−	0.0800537i	$-0.0255092\pi$
$439$	5.30004e26	1.99358	0.996791	+	0.0800537i	$0.0255092\pi$
$440$	2.90359e24	0.0106760
$441$	−2.66669e26	−0.958489
$442$	0	0
$443$	3.28063e26	1.12699	0.563496	−	0.826119i	$-0.309456\pi$
$443$	3.28063e26	1.12699	0.563496	+	0.826119i	$0.309456\pi$
$444$	0	0
$445$	0	0
$446$	4.04002e26	1.29728
$447$	−4.34895e26	−1.36555
$448$	6.63529e25	0.203742
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	9.04736e25	0.265703
$451$	0	0
$452$	0	0
$453$	4.26119e25	0.117098
$454$	5.39784e26	1.45098
$455$	0	0
$456$	0	0
$457$	−7.90904e26	−1.99050	−0.995250	−	0.0973555i	$-0.968962\pi$
$457$	−7.90904e26	−1.99050	−0.995250	+	0.0973555i	$0.968962\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.56085e26	1.97473	0.987365	−	0.158461i	$-0.0506533\pi$
$461$	8.56085e26	1.97473	0.987365	+	0.158461i	$0.0506533\pi$
$462$	1.12453e24	0.00253836
$463$	8.31121e26	1.83592	0.917961	−	0.396670i	$-0.129834\pi$
$463$	8.31121e26	1.83592	0.917961	+	0.396670i	$0.129834\pi$
$464$	−5.24050e26	−1.13290
$465$	−7.70386e26	−1.62997
$466$	0	0
$467$	3.77604e26	0.765364	0.382682	−	0.923880i	$-0.375000\pi$
$467$	3.77604e26	0.765364	0.382682	+	0.923880i	$0.375000\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	−7.25664e26	−1.32226
$473$	0	0
$474$	9.33781e26	1.63103
$475$	0	0
$476$	0	0
$477$	−1.12036e26	−0.183728
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−5.56350e26	−0.856911
$481$	0	0
$482$	−1.30266e27	−1.92469
$483$	0	0
$484$	−7.05320e26	−0.999845
$485$	5.51676e26	0.766067
$486$	7.35128e26	1.00000
$487$	9.25130e26	1.23286	0.616429	−	0.787411i	$-0.288579\pi$
$487$	9.25130e26	1.23286	0.616429	+	0.787411i	$0.288579\pi$
$488$	0	0
$489$	0	0
$490$	−6.55366e26	−0.821340
$491$	4.47565e26	0.549593	0.274797	−	0.961502i	$-0.411389\pi$
$491$	4.47565e26	0.549593	0.274797	+	0.961502i	$0.411389\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−9.42889e24	−0.0106760
$496$	−1.71419e27	−1.90214
$497$	0	0
$498$	−1.87553e27	−1.99908
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	1.05918e27	1.08460
$501$	0	0
$502$	−1.83188e27	−1.80243
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−2.15469e26	−0.203742
$505$	−1.75338e27	−1.62541
$506$	0	0
$507$	−1.12222e27	−1.00000
$508$	−7.42354e26	−0.648596
$509$	−7.26165e26	−0.622096	−0.311048	−	0.950394i	$-0.600680\pi$
$509$	−7.26165e26	−0.622096	−0.311048	+	0.950394i	$0.600680\pi$
$510$	0	0
$511$	4.64034e26	0.382244
$512$	−1.23794e27	−1.00000
$513$	0	0
$514$	0	0
$515$	1.98920e27	1.51567
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−6.66870e26	−0.470292
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	1.70176e27	1.13290
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	3.10471e27	1.98933
$525$	8.61132e25	0.0541348
$526$	0	0
$527$	0	0
$528$	−2.09803e25	−0.0124587
$529$	1.71616e27	1.00000
$530$	−2.75340e26	−0.157439
$531$	2.35646e27	1.32226
$532$	0	0
$533$	0	0
$534$	0	0
$535$	1.79550e27	0.934648
$536$	0	0
$537$	−3.06611e27	−1.53760
$538$	4.00840e27	1.97310
$539$	−2.47143e25	−0.0119415
$540$	1.80665e27	0.856911
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	2.09245e27	0.956450
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	8.38487e24	0.00331032
$551$	0	0
$552$	0	0
$553$	8.88777e26	0.332309
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.63711e27	−1.96112	−0.980562	−	0.196211i	$-0.937136\pi$
$557$	−5.63711e27	−1.96112	−0.980562	+	0.196211i	$0.937136\pi$
$558$	5.56654e27	1.90214
$559$	0	0
$560$	−5.29537e26	−0.174589
$561$	0	0
$562$	0	0
$563$	−6.26590e27	−1.95839	−0.979196	−	0.202916i	$-0.934958\pi$
$563$	−6.26590e27	−1.95839	−0.979196	+	0.202916i	$0.934958\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6.99698e26	0.203742
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	4.01999e27	1.00000
$577$	6.74548e27	1.64913	0.824565	−	0.565767i	$-0.191420\pi$
$577$	6.74548e27	1.64913	0.824565	+	0.565767i	$0.191420\pi$
$578$	−4.16177e27	−1.00000
$579$	−5.09977e27	−1.20439
$580$	4.18224e27	0.970797
$581$	−1.78514e27	−0.407296
$582$	−3.98622e27	−0.893987
$583$	−1.03833e25	−0.00228901
$584$	−8.65743e27	−1.87612
$585$	0	0
$586$	−5.87824e27	−1.23104
$587$	−5.02215e27	−1.03397	−0.516986	−	0.855994i	$-0.672946\pi$
$587$	−5.02215e27	−1.03397	−0.516986	+	0.855994i	$0.672946\pi$
$588$	4.73544e27	0.958489
$589$	0	0
$590$	5.79124e27	1.13306
$591$	−4.38984e27	−0.844450
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	6.81298e25	0.0124587
$595$	0	0
$596$	7.72274e27	1.36555
$597$	6.59720e27	1.14714
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−1.60661e27	−0.265703
$601$	8.42431e27	1.37022	0.685109	−	0.728441i	$-0.259755\pi$
$601$	8.42431e27	1.37022	0.685109	+	0.728441i	$0.259755\pi$
$602$	0	0
$603$	0	0
$604$	−7.56690e26	−0.117098
$605$	5.62888e27	0.856778
$606$	1.26693e28	1.89682
$607$	−1.33886e28	−1.97174	−0.985869	−	0.167517i	$-0.946425\pi$
$607$	−1.33886e28	−1.97174	−0.985869	+	0.167517i	$0.946425\pi$
$608$	0	0
$609$	1.61974e27	0.230820
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	−1.99692e25	−0.00253836
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	−1.43732e28	−1.76876
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	1.36803e28	1.62997
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−6.03631e27	−0.663699
$626$	−1.44677e28	−1.56551
$627$	0	0
$628$	0	0
$629$	0	0
$630$	1.71958e27	0.174589
$631$	−7.47099e27	−0.746594	−0.373297	−	0.927712i	$-0.621773\pi$
$631$	−7.47099e27	−0.746594	−0.373297	+	0.927712i	$0.621773\pi$
$632$	−1.65818e28	−1.63103
$633$	0	0
$634$	−1.55719e28	−1.48405
$635$	5.92444e27	0.555789
$636$	1.98951e27	0.183728
$637$	0	0
$638$	1.57715e26	0.0141145
$639$	0	0
$640$	9.87951e27	0.856911
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	−1.29737e28	−1.09072
$643$	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			$\pi$
$648$	−1.30542e28	−1.00000
$649$	2.18391e26	0.0164736
$650$	0	0
$651$	5.29826e27	0.387546
$652$	0	0
$653$	−2.27518e28	−1.61393	−0.806963	−	0.590602i	$-0.798890\pi$
$653$	−2.27518e28	−1.61393	−0.806963	+	0.590602i	$0.798890\pi$
$654$	0	0
$655$	−2.47775e28	−1.70468
$656$	0	0
$657$	2.81135e28	1.87612
$658$	0	0
$659$	2.82954e28	1.83173	0.915865	−	0.401487i	$-0.131506\pi$
$659$	2.82954e28	1.83173	0.915865	+	0.401487i	$0.131506\pi$
$660$	1.67436e26	0.0106760
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	3.33051e28	1.99908
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	2.32968e28	1.29728
$670$	0	0
$671$	0	0
$672$	3.82624e27	0.203742
$673$	−5.86561e27	−0.307725	−0.153862	−	0.988092i	$-0.549171\pi$
$673$	−5.86561e27	−0.307725	−0.153862	+	0.988092i	$0.549171\pi$
$674$	−3.71442e28	−1.91996
$675$	5.21716e27	0.265703
$676$	1.99281e28	1.00000
$677$	−1.87939e28	−0.929244	−0.464622	−	0.885509i	$-0.653810\pi$
$677$	−1.87939e28	−0.929244	−0.464622	+	0.885509i	$0.653810\pi$
$678$	0	0
$679$	−3.79410e27	−0.182142
$680$	0	0
$681$	3.11267e28	1.45098
$682$	5.15893e26	0.0236982
$683$	2.48731e28	1.12596	0.562980	−	0.826471i	$-0.309655\pi$
$683$	2.48731e28	1.12596	0.562980	+	0.826471i	$0.309655\pi$
$684$	0	0
$685$	0	0
$686$	9.20963e27	0.399026
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	1.18421e28	0.470292
$693$	6.48463e25	0.00253836
$694$	−1.93906e27	−0.0748164
$695$	0	0
$696$	−3.02193e28	−1.13290
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−1.52917e27	−0.0541348
$701$	5.36999e28	1.87410	0.937051	−	0.349192i	$-0.113544\pi$
$701$	5.36999e28	1.87410	0.937051	+	0.349192i	$0.113544\pi$
$702$	0	0
$703$	0	0
$704$	3.72563e26	0.0124587
$705$	0	0
$706$	0	0
$707$	1.20587e28	0.386461
$708$	−4.18454e28	−1.32226
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	5.38465e28	1.63103
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	5.44471e28	1.53760
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−3.20820e28	−0.856911
$721$	−1.36805e28	−0.360371
$722$	−3.84921e28	−1.00000
$723$	−7.51180e28	−1.92469
$724$	0	0
$725$	1.20773e28	0.301016
$726$	−4.06723e28	−0.999845
$727$	6.08050e28	1.47433	0.737166	−	0.675711i	$-0.236163\pi$
$727$	6.08050e28	1.47433	0.737166	+	0.675711i	$0.236163\pi$
$728$	0	0
$729$	4.23912e28	1.00000
$730$	6.90916e28	1.60767
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	7.60669e28	1.67585
$735$	−3.77917e28	−0.821340
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	1.89363e27	0.0374331
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−9.88491e28	−1.90214
$745$	−6.16322e28	−1.17016
$746$	0	0
$747$	−1.08152e29	−1.99908
$748$	0	0
$749$	−1.23484e28	−0.222225
$750$	6.10774e28	1.08460
$751$	−9.85933e28	−1.72762	−0.863809	−	0.503819i	$-0.831928\pi$
$751$	−9.85933e28	−1.72762	−0.863809	+	0.503819i	$0.831928\pi$
$752$	0	0
$753$	−1.05635e29	−1.80243
$754$	0	0
$755$	6.03885e27	0.100342
$756$	−1.24250e28	−0.203742
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	−4.28079e28	−0.648596
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−7.13859e28	−1.00000
$769$	−1.36664e29	−1.88969	−0.944845	−	0.327518i	$-0.893788\pi$
$769$	−1.36664e29	−1.88969	−0.944845	+	0.327518i	$0.893788\pi$
$770$	1.59366e26	0.00217515
$771$	0	0
$772$	9.05604e28	1.20439
$773$	−8.25961e28	−1.08434	−0.542169	−	0.840269i	$-0.682397\pi$
$773$	−8.25961e28	−1.08434	−0.542169	+	0.840269i	$0.682397\pi$
$774$	0	0
$775$	3.95054e28	0.505405
$776$	7.07862e28	0.893987
$777$	0	0
$778$	1.26909e29	1.56206
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	9.81319e28	1.13290
$784$	−8.40907e28	−0.958489
$785$	0	0
$786$	1.79033e29	1.98933
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	7.79536e28	0.844450
$789$	0	0
$790$	1.32333e29	1.39765
$791$	0	0
$792$	−1.20983e27	−0.0124587
$793$	0	0
$794$	0	0
$795$	−1.58775e28	−0.157439
$796$	−1.17151e29	−1.14714
$797$	−2.06598e29	−1.99775	−0.998877	−	0.0473862i	$-0.984911\pi$
$797$	−2.06598e29	−1.99775	−0.998877	+	0.0473862i	$0.984911\pi$
$798$	0	0
$799$	0	0
$800$	2.85297e28	0.265703
$801$	0	0
$802$	0	0
$803$	2.60549e27	0.0233740
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.31145e29	1.97310
$808$	−2.24978e29	−1.89682
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	1.04180e29	0.856911
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	−2.87629e28	−0.230820
$813$	1.20661e29	0.956450
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−7.51940e28	−0.560596
$819$	0	0
$820$	0	0
$821$	−1.35683e29	−0.975200	−0.487600	−	0.873067i	$-0.662127\pi$
$821$	−1.35683e29	−0.975200	−0.487600	+	0.873067i	$0.662127\pi$
$822$	0	0
$823$	2.76545e29	1.93984	0.969922	−	0.243417i	$-0.0782682\pi$
$823$	2.76545e29	1.93984	0.969922	+	0.243417i	$0.0782682\pi$
$824$	2.55236e29	1.76876
$825$	4.83514e26	0.00331032
$826$	−3.98287e28	−0.269399
$827$	2.63769e29	1.76266	0.881331	−	0.472499i	$-0.156648\pi$
$827$	2.63769e29	1.76266	0.881331	+	0.472499i	$0.156648\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	−2.65795e29	−1.71303
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.20995e29	1.90214
$838$	1.39511e29	0.816899
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	−3.05358e28	−0.174589
$841$	5.01724e28	0.283469
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.59039e29	−0.856911
$846$	0	0
$847$	−3.87121e28	−0.203710
$848$	−3.53292e28	−0.183728
$849$	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
$855$	0	0
$856$	2.30383e29	1.09072
$857$	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			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	2.31813e29	1.00000
$865$	−9.45071e28	−0.402998
$866$	−3.97237e29	−1.67444
$867$	−2.39989e29	−1.00000
$868$	−9.40850e28	−0.387546
$869$	4.99036e27	0.0203205
$870$	2.41169e29	0.970797
$871$	0	0
$872$	0	0
$873$	−2.29865e29	−0.893987
$874$	0	0
$875$	5.81337e28	0.220977
$876$	−4.99231e29	−1.87612
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−5.42724e29	−1.99358
$879$	−3.38969e29	−1.23104
$880$	−2.97328e27	−0.0106760
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	2.73069e29	0.958489
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	3.33952e29	1.13306
$886$	−3.35937e29	−1.12699
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−4.07447e28	−0.132146
$890$	0	0
$891$	3.92871e27	0.0124587
$892$	−4.13698e29	−1.29728
$893$	0	0
$894$	4.45332e29	1.36555
$895$	−4.34521e29	−1.31759
$896$	−6.79454e28	−0.203742
$897$	0	0
$898$	0	0
$899$	7.43075e29	2.15494
$900$	−9.26449e28	−0.265703
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−4.36346e28	−0.117098
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−5.52739e29	−1.45098
$909$	7.30575e29	1.89682
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−1.00233e28	−0.0249059
$914$	8.09886e29	1.99050
$915$	0	0
$916$	0	0
$917$	1.70405e29	0.405310
$918$	0	0
$919$	7.35790e29	1.71237	0.856187	−	0.516666i	$-0.172827\pi$
$919$	7.35790e29	1.71237	0.856187	+	0.516666i	$0.172827\pi$
$920$	0	0
$921$	0	0
$922$	−8.76631e29	−1.97473
$923$	0	0
$924$	−1.15152e27	−0.00253836
$925$	0	0
$926$	−8.51068e29	−1.83592
$927$	−8.28833e29	−1.76876
$928$	5.36627e29	1.13290
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	7.88876e29	1.62997
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−3.86667e29	−0.765364
$935$	0	0
$936$	0	0
$937$	1.01510e30	1.94586	0.972930	−	0.231098i	$-0.0742320\pi$
$937$	1.01510e30	1.94586	0.972930	+	0.231098i	$0.0742320\pi$
$938$	0	0
$939$	−8.34279e29	−1.56551
$940$	0	0
$941$	−7.29588e29	−1.34024	−0.670118	−	0.742254i	$-0.733757\pi$
$941$	−7.29588e29	−1.34024	−0.670118	+	0.742254i	$0.733757\pi$
$942$	0	0
$943$	0	0
$944$	7.43080e29	1.32226
$945$	9.91594e28	0.174589
$946$	0	0
$947$	2.81820e29	0.485816	0.242908	−	0.970049i	$-0.421899\pi$
$947$	2.81820e29	0.485816	0.242908	+	0.970049i	$0.421899\pi$
$948$	−9.56192e29	−1.63103
$949$	0	0
$950$	0	0
$951$	−8.97955e29	−1.48405
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	1.14725e29	0.183728
$955$	0	0
$956$	0	0
$957$	9.09463e27	0.0141145
$958$	0	0
$959$	0	0
$960$	5.69703e29	0.856911
$961$	1.75885e30	2.61815
$962$	0	0
$963$	−7.48127e29	−1.09072
$964$	1.33392e30	1.92469
$965$	−7.22727e29	−1.03205
$966$	0	0
$967$	−8.92659e27	−0.0124859	−0.00624296	−	0.999981i	$-0.501987\pi$
$967$	−8.92659e27	−0.0124859	−0.00624296	+	0.999981i	$0.501987\pi$
$968$	7.22248e29	0.999845
$969$	0	0
$970$	−5.64917e29	−0.766067
$971$	−6.59587e29	−0.885279	−0.442639	−	0.896700i	$-0.645958\pi$
$971$	−6.59587e29	−0.885279	−0.442639	+	0.896700i	$0.645958\pi$
$972$	−7.52771e29	−1.00000
$973$	0	0
$974$	−9.47333e29	−1.23286
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	6.71095e29	0.821340
$981$	0	0
$982$	−4.58307e29	−0.549593
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−6.22117e29	−0.723619
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	9.65519e27	0.0106760
$991$	−2.51499e29	−0.275296	−0.137648	−	0.990481i	$-0.543954\pi$
$991$	−2.51499e29	−0.275296	−0.137648	+	0.990481i	$0.543954\pi$
$992$	1.75533e30	1.90214
$993$	0	0
$994$	0	0
$995$	9.34939e29	0.982997
$996$	1.92054e30	1.99908
$997$	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	24.21.h.a.5.1	✓	1
3.2	odd	2		24.21.h.b.5.1	yes	1
8.5	even	2		24.21.h.b.5.1	yes	1
24.5	odd	2	CM	24.21.h.a.5.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.21.h.a.5.1	✓	1	1.1	even	1	trivial
24.21.h.a.5.1	✓	1	24.5	odd	2	CM
24.21.h.b.5.1	yes	1	3.2	odd	2
24.21.h.b.5.1	yes	1	8.5	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:	\( N \)	\(=\)	\( 24 = 2^{3} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 21 \)
Character orbit:	\([\chi]\)	\(=\)	24.h (of order \(2\), degree \(1\), minimal)

\(n\)	\(7\)	\(13\)	\(17\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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