LMFDB - Embedded newform 24.33.h.b.5.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [24,33,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, 33); 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, 33, names="a")

Level:	$ N $	$=$	$ 24 = 2^{3} \cdot 3 $
Weight:	$ k $	$=$	$ 33 $
Character orbit:	$[\chi]$	$=$	24.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,65536] 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:	$155.679972341$
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)

$f(q)$

$=$

\(q+65536.0 q^{2} +4.30467e7 q^{3} +4.29497e9 q^{4} -3.04196e11 q^{5} +2.82111e12 q^{6} +6.55819e13 q^{7} +2.81475e14 q^{8} +1.85302e15 q^{9} -1.99358e16 q^{10} +2.75260e16 q^{11} +1.84884e17 q^{12} +4.29797e18 q^{14} -1.30947e19 q^{15} +1.84467e19 q^{16} +1.21440e20 q^{18} -1.30651e21 q^{20} +2.82308e21 q^{21} +1.80394e21 q^{22} +1.21166e22 q^{24} +6.92524e22 q^{25} +7.97664e22 q^{27} +2.81672e23 q^{28} -3.02633e23 q^{29} -8.58172e23 q^{30} -6.19545e23 q^{31} +1.20893e24 q^{32} +1.18490e24 q^{33} -1.99498e25 q^{35} +7.95866e24 q^{36} -8.56237e25 q^{40} +1.85014e26 q^{42} +1.18223e26 q^{44} -5.63682e26 q^{45} +7.94072e26 q^{48} +3.19655e27 q^{49} +4.53853e27 q^{50} +1.28843e27 q^{53} +5.22757e27 q^{54} -8.37330e27 q^{55} +1.84597e28 q^{56} -1.98333e28 q^{58} +9.12486e27 q^{59} -5.62411e28 q^{60} -4.06025e28 q^{62} +1.21525e29 q^{63} +7.92282e28 q^{64} +7.76538e28 q^{66} -1.30743e30 q^{70} +5.21579e29 q^{72} +1.00275e30 q^{73} +2.98109e30 q^{75} +1.80520e30 q^{77} +4.43751e30 q^{79} -5.61143e30 q^{80} +3.43368e30 q^{81} -7.90877e30 q^{83} +1.21251e31 q^{84} -1.30273e31 q^{87} +7.74787e30 q^{88} -3.69415e31 q^{90} -2.66694e31 q^{93} +5.20403e31 q^{96} -1.22066e32 q^{97} +2.09489e32 q^{98} +5.10062e31 q^{99} +O(q^{100})\)

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 $

$2$	65536.0	1.00000
$2$	65536.0	1.00000
$3$	4.30467e7	1.00000
$3$	4.30467e7	1.00000
$4$	4.29497e9	1.00000
$5$	−3.04196e11	−1.99358	−0.996791	−	0.0800483i	$-0.974493\pi$
$5$	−3.04196e11	−1.99358	−0.996791	+	0.0800483i	$0.974493\pi$
$6$	2.82111e12	1.00000
$7$	6.55819e13	1.97340	0.986700	−	0.162552i	$-0.0519726\pi$
$7$	6.55819e13	1.97340	0.986700	+	0.162552i	$0.0519726\pi$
$8$	2.81475e14	1.00000
$9$	1.85302e15	1.00000
$10$	−1.99358e16	−1.99358
$11$	2.75260e16	0.599045	0.299523	−	0.954089i	$-0.403173\pi$
$11$	2.75260e16	0.599045	0.299523	+	0.954089i	$0.403173\pi$
$12$	1.84884e17	1.00000
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	4.29797e18	1.97340
$15$	−1.30947e19	−1.99358
$16$	1.84467e19	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	1.21440e20	1.00000
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−1.30651e21	−1.99358
$21$	2.82308e21	1.97340
$22$	1.80394e21	0.599045
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.21166e22	1.00000
$25$	6.92524e22	2.97437
$26$	0	0
$27$	7.97664e22	1.00000
$28$	2.81672e23	1.97340
$29$	−3.02633e23	−1.20934	−0.604669	−	0.796477i	$-0.706695\pi$
$29$	−3.02633e23	−1.20934	−0.604669	+	0.796477i	$0.706695\pi$
$30$	−8.58172e23	−1.99358
$31$	−6.19545e23	−0.851699	−0.425849	−	0.904794i	$-0.640025\pi$
$31$	−6.19545e23	−0.851699	−0.425849	+	0.904794i	$0.640025\pi$
$32$	1.20893e24	1.00000
$33$	1.18490e24	0.599045
$34$	0	0
$35$	−1.99498e25	−3.93413
$36$	7.95866e24	1.00000
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	−8.56237e25	−1.99358
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	1.85014e26	1.97340
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	1.18223e26	0.599045
$45$	−5.63682e26	−1.99358
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	7.94072e26	1.00000
$49$	3.19655e27	2.89431
$50$	4.53853e27	2.97437
$51$	0	0
$52$	0	0
$53$	1.28843e27	0.332389	0.166195	−	0.986093i	$-0.446852\pi$
$53$	1.28843e27	0.332389	0.166195	+	0.986093i	$0.446852\pi$
$54$	5.22757e27	1.00000
$55$	−8.37330e27	−1.19425
$56$	1.84597e28	1.97340
$57$	0	0
$58$	−1.98333e28	−1.20934
$59$	9.12486e27	0.423247	0.211624	−	0.977351i	$-0.432125\pi$
$59$	9.12486e27	0.423247	0.211624	+	0.977351i	$0.432125\pi$
$60$	−5.62411e28	−1.99358
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−4.06025e28	−0.851699
$63$	1.21525e29	1.97340
$64$	7.92282e28	1.00000
$65$	0	0
$66$	7.76538e28	0.599045
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	−1.30743e30	−3.93413
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	5.21579e29	1.00000
$73$	1.00275e30	1.54180	0.770900	−	0.636956i	$-0.219807\pi$
$73$	1.00275e30	1.54180	0.770900	+	0.636956i	$0.219807\pi$
$74$	0	0
$75$	2.98109e30	2.97437
$76$	0	0
$77$	1.80520e30	1.18216
$78$	0	0
$79$	4.43751e30	1.92799	0.963997	−	0.265913i	$-0.0856733\pi$
$79$	4.43751e30	1.92799	0.963997	+	0.265913i	$0.0856733\pi$
$80$	−5.61143e30	−1.99358
$81$	3.43368e30	1.00000
$82$	0	0
$83$	−7.90877e30	−1.55905	−0.779524	−	0.626372i	$-0.784539\pi$
$83$	−7.90877e30	−1.55905	−0.779524	+	0.626372i	$0.784539\pi$
$84$	1.21251e31	1.97340
$85$	0	0
$86$	0	0
$87$	−1.30273e31	−1.20934
$88$	7.74787e30	0.599045
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−3.69415e31	−1.99358
$91$	0	0
$92$	0	0
$93$	−2.66694e31	−0.851699
$94$	0	0
$95$	0	0
$96$	5.20403e31	1.00000
$97$	−1.22066e32	−1.98722	−0.993608	−	0.112883i	$-0.963992\pi$
$97$	−1.22066e32	−1.98722	−0.993608	+	0.112883i	$0.963992\pi$
$98$	2.09489e32	2.89431
$99$	5.10062e31	0.599045
$100$	2.97437e32	2.97437
$101$	1.73110e32	1.47632	0.738161	−	0.674624i	$-0.235694\pi$
$101$	1.73110e32	1.47632	0.738161	+	0.674624i	$0.235694\pi$
$102$	0	0
$103$	2.28837e32	1.42603	0.713017	−	0.701147i	$-0.247328\pi$
$103$	2.28837e32	1.42603	0.713017	+	0.701147i	$0.247328\pi$
$104$	0	0
$105$	−8.58772e32	−3.93413
$106$	8.44385e31	0.332389
$107$	5.57857e32	1.88965	0.944827	−	0.327569i	$-0.106229\pi$
$107$	5.57857e32	1.88965	0.944827	+	0.327569i	$0.106229\pi$
$108$	3.42594e32	1.00000
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−5.48753e32	−1.19425
$111$	0	0
$112$	1.20977e33	1.97340
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	−1.29980e33	−1.20934
$117$	0	0
$118$	5.98007e32	0.423247
$119$	0	0
$120$	−3.68582e33	−1.99358
$121$	−1.35370e33	−0.641145
$122$	0	0
$123$	0	0
$124$	−2.66093e33	−0.851699
$125$	−1.39837e34	−3.93607
$126$	7.96423e33	1.97340
$127$	6.83665e33	1.49274	0.746369	−	0.665532i	$-0.231795\pi$
$127$	6.83665e33	1.49274	0.746369	+	0.665532i	$0.231795\pi$
$128$	5.19230e33	1.00000
$129$	0	0
$130$	0	0
$131$	2.23103e33	0.296595	0.148298	−	0.988943i	$-0.452621\pi$
$131$	2.23103e33	0.296595	0.148298	+	0.988943i	$0.452621\pi$
$132$	5.08912e33	0.599045
$133$	0	0
$134$	0	0
$135$	−2.42647e34	−1.99358
$136$	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			$\pi$
$140$	−8.56836e34	−3.93413
$141$	0	0
$142$	0	0
$143$	0	0
$144$	3.41822e34	1.00000
$145$	9.20598e34	2.41091
$146$	6.57164e34	1.54180
$147$	1.37601e35	2.89431
$148$	0	0
$149$	−9.91070e34	−1.67928	−0.839642	−	0.543140i	$-0.817235\pi$
$149$	−9.91070e34	−1.67928	−0.839642	+	0.543140i	$0.817235\pi$
$150$	1.95369e35	2.97437
$151$	−1.25719e35	−1.72096	−0.860479	−	0.509485i	$-0.829836\pi$
$151$	−1.25719e35	−1.72096	−0.860479	+	0.509485i	$0.829836\pi$
$152$	0	0
$153$	0	0
$154$	1.18306e35	1.18216
$155$	1.88464e35	1.69793
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	2.90817e35	1.92799
$159$	5.54627e34	0.332389
$160$	−3.67751e35	−1.99358
$161$	0	0
$162$	2.25030e35	1.00000
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−3.60443e35	−1.19425
$166$	−5.18309e35	−1.55905
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	7.94628e35	1.97340
$169$	4.42779e35	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.86864e35	−1.06692	−0.533462	−	0.845824i	$-0.679109\pi$
$173$	−6.86864e35	−1.06692	−0.533462	+	0.845824i	$0.679109\pi$
$174$	−8.53760e35	−1.20934
$175$	4.54170e36	5.86962
$176$	5.07764e35	0.599045
$177$	3.92795e35	0.423247
$178$	0	0
$179$	3.70724e35	0.333735	0.166868	−	0.985979i	$-0.446635\pi$
$179$	3.70724e35	0.333735	0.166868	+	0.985979i	$0.446635\pi$
$180$	−2.42100e36	−1.99358
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	−1.74781e36	−0.851699
$187$	0	0
$188$	0	0
$189$	5.23123e36	1.97340
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	3.41051e36	1.00000
$193$	3.79055e36	1.02279	0.511393	−	0.859347i	$-0.329130\pi$
$193$	3.79055e36	1.02279	0.511393	+	0.859347i	$0.329130\pi$
$194$	−7.99969e36	−1.98722
$195$	0	0
$196$	1.37291e37	2.89431
$197$	−3.84873e36	−0.747925	−0.373963	−	0.927444i	$-0.622001\pi$
$197$	−3.84873e36	−0.747925	−0.373963	+	0.927444i	$0.622001\pi$
$198$	3.34274e36	0.599045
$199$	−7.44756e36	−1.23130	−0.615652	−	0.788018i	$-0.711107\pi$
$199$	−7.44756e36	−1.23130	−0.615652	+	0.788018i	$0.711107\pi$
$200$	1.94928e37	2.97437
$201$	0	0
$202$	1.13450e37	1.47632
$203$	−1.98472e37	−2.38651
$204$	0	0
$205$	0	0
$206$	1.49970e37	1.42603
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−5.62805e37	−3.93413
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	5.53376e36	0.332389
$213$	0	0
$214$	3.65597e37	1.88965
$215$	0	0
$216$	2.24523e37	1.00000
$217$	−4.06309e37	−1.68074
$218$	0	0
$219$	4.31652e37	1.54180
$220$	−3.59631e37	−1.19425
$221$	0	0
$222$	0	0
$223$	−5.44407e37	−1.45561	−0.727806	−	0.685783i	$-0.759460\pi$
$223$	−5.44407e37	−1.45561	−0.727806	+	0.685783i	$0.759460\pi$
$224$	7.92836e37	1.97340
$225$	1.28326e38	2.97437
$226$	0	0
$227$	2.67083e37	0.537319	0.268660	−	0.963235i	$-0.413419\pi$
$227$	2.67083e37	0.537319	0.268660	+	0.963235i	$0.413419\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	7.77081e37	1.18216
$232$	−8.51835e37	−1.20934
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	3.91910e37	0.423247
$237$	1.91020e38	1.92799
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−2.41554e38	−1.99358
$241$	−1.24628e38	−0.962373	−0.481186	−	0.876618i	$-0.659794\pi$
$241$	−1.24628e38	−0.962373	−0.481186	+	0.876618i	$0.659794\pi$
$242$	−8.87160e37	−0.641145
$243$	1.47809e38	1.00000
$244$	0	0
$245$	−9.72380e38	−5.77004
$246$	0	0
$247$	0	0
$248$	−1.74387e38	−0.851699
$249$	−3.40447e38	−1.55905
$250$	−9.16437e38	−3.93607
$251$	4.25879e38	1.71596	0.857979	−	0.513685i	$-0.171720\pi$
$251$	4.25879e38	1.71596	0.857979	+	0.513685i	$0.171720\pi$
$252$	5.21944e38	1.97340
$253$	0	0
$254$	4.48047e38	1.49274
$255$	0	0
$256$	3.40282e38	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−5.60784e38	−1.20934
$262$	1.46213e38	0.296595
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	3.33520e38	0.599045
$265$	−3.91936e38	−0.662645
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−9.45013e38	−1.25719	−0.628596	−	0.777732i	$-0.716370\pi$
$269$	−9.45013e38	−1.25719	−0.628596	+	0.777732i	$0.716370\pi$
$270$	−1.59021e39	−1.99358
$271$	−2.43999e38	−0.288323	−0.144161	−	0.989554i	$-0.546048\pi$
$271$	−2.43999e38	−0.288323	−0.144161	+	0.989554i	$0.546048\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.90624e39	1.78178
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	−1.14803e39	−0.851699
$280$	−5.61536e39	−3.93413
$281$	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
$288$	2.24016e39	1.00000
$289$	2.36791e39	1.00000
$290$	6.03323e39	2.41091
$291$	−5.25452e39	−1.98722
$292$	4.30679e39	1.54180
$293$	2.88026e39	0.976225	0.488112	−	0.872781i	$-0.337686\pi$
$293$	2.88026e39	0.976225	0.488112	+	0.872781i	$0.337686\pi$
$294$	9.01783e39	2.89431
$295$	−2.77575e39	−0.843778
$296$	0	0
$297$	2.19565e39	0.599045
$298$	−6.49508e39	−1.67928
$299$	0	0
$300$	1.28037e40	2.97437
$301$	0	0
$302$	−8.23913e39	−1.72096
$303$	7.45184e39	1.47632
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	7.75329e39	1.18216
$309$	9.85067e39	1.42603
$310$	1.23511e40	1.69793
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−1.53089e40	−1.80399	−0.901995	−	0.431747i	$-0.857897\pi$
$313$	−1.53089e40	−1.80399	−0.901995	+	0.431747i	$0.857897\pi$
$314$	0	0
$315$	−3.69673e40	−3.93413
$316$	1.90590e40	1.92799
$317$	−1.77592e40	−1.70795	−0.853977	−	0.520310i	$-0.825816\pi$
$317$	−1.77592e40	−1.70795	−0.853977	+	0.520310i	$0.825816\pi$
$318$	3.63480e39	0.332389
$319$	−8.33025e39	−0.724448
$320$	−2.41009e40	−1.99358
$321$	2.40139e40	1.88965
$322$	0	0
$323$	0	0
$324$	1.47476e40	1.00000
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	−2.36220e40	−1.19425
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−3.39679e40	−1.55905
$333$	0	0
$334$	0	0
$335$	0	0
$336$	5.20767e40	1.97340
$337$	−7.72477e39	−0.279130	−0.139565	−	0.990213i	$-0.544570\pi$
$337$	−7.72477e39	−0.279130	−0.139565	+	0.990213i	$0.544570\pi$
$338$	2.90180e40	1.00000
$339$	0	0
$340$	0	0
$341$	−1.70536e40	−0.510206
$342$	0	0
$343$	1.37205e41	3.73823
$344$	0	0
$345$	0	0
$346$	−4.50143e40	−1.06692
$347$	−6.82564e40	−1.54480	−0.772400	−	0.635137i	$-0.780944\pi$
$347$	−6.82564e40	−1.54480	−0.772400	+	0.635137i	$0.780944\pi$
$348$	−5.59520e40	−1.20934
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	2.97645e41	5.86962
$351$	0	0
$352$	3.32768e40	0.599045
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	2.57422e40	0.423247
$355$	0	0
$356$	0	0
$357$	0	0
$358$	2.42958e40	0.333735
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−1.58662e41	−1.99358
$361$	8.31984e40	1.00000
$362$	0	0
$363$	−5.82723e40	−0.641145
$364$	0	0
$365$	−3.05034e41	−3.07370
$366$	0	0
$367$	−1.24014e41	−1.14502	−0.572509	−	0.819898i	$-0.694030\pi$
$367$	−1.24014e41	−1.14502	−0.572509	+	0.819898i	$0.694030\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.44976e40	0.655937
$372$	−1.14544e41	−0.851699
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−6.01953e41	−3.93607
$376$	0	0
$377$	0	0
$378$	3.42834e41	1.97340
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	2.94295e41	1.49274
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	2.23511e41	1.00000
$385$	−5.49137e41	−2.35672
$386$	2.48417e41	1.02279
$387$	0	0
$388$	−5.24267e41	−1.98722
$389$	−3.80778e41	−1.38509	−0.692546	−	0.721374i	$-0.743511\pi$
$389$	−3.80778e41	−1.38509	−0.692546	+	0.721374i	$0.743511\pi$
$390$	0	0
$391$	0	0
$392$	8.99750e41	2.89431
$393$	9.60384e40	0.296595
$394$	−2.52231e41	−0.747925
$395$	−1.34987e42	−3.84361
$396$	2.19070e41	0.599045
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−4.88084e41	−1.23130
$399$	0	0
$400$	1.27748e42	2.97437
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	7.43504e41	1.47632
$405$	−1.04451e42	−1.99358
$406$	−1.30071e42	−2.38651
$407$	0	0
$408$	0	0
$409$	−1.71620e41	−0.279895	−0.139947	−	0.990159i	$-0.544693\pi$
$409$	−1.71620e41	−0.279895	−0.139947	+	0.990159i	$0.544693\pi$
$410$	0	0
$411$	0	0
$412$	9.82846e41	1.42603
$413$	5.98425e41	0.835236
$414$	0	0
$415$	2.40582e42	3.10809
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6.34214e41	−0.702767	−0.351383	−	0.936232i	$-0.614289\pi$
$419$	−6.34214e41	−0.702767	−0.351383	+	0.936232i	$0.614289\pi$
$420$	−3.68840e42	−3.93413
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	3.62661e41	0.332389
$425$	0	0
$426$	0	0
$427$	0	0
$428$	2.39598e42	1.88965
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	1.47143e42	1.00000
$433$	−5.09860e40	−0.0333921	−0.0166961	−	0.999861i	$-0.505315\pi$
$433$	−5.09860e40	−0.0333921	−0.0166961	+	0.999861i	$0.505315\pi$
$434$	−2.66279e42	−1.68074
$435$	3.96287e42	2.41091
$436$	0	0
$437$	0	0
$438$	2.82887e42	1.54180
$439$	1.62930e42	0.856187	0.428094	−	0.903734i	$-0.359185\pi$
$439$	1.62930e42	0.856187	0.428094	+	0.903734i	$0.359185\pi$
$440$	−2.35687e42	−1.19425
$441$	5.92328e42	2.89431
$442$	0	0
$443$	6.73222e40	0.0305984	0.0152992	−	0.999883i	$-0.495130\pi$
$443$	6.73222e40	0.0305984	0.0152992	+	0.999883i	$0.495130\pi$
$444$	0	0
$445$	0	0
$446$	−3.56783e42	−1.45561
$447$	−4.26623e42	−1.67928
$448$	5.19593e42	1.97340
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	8.40998e42	2.97437
$451$	0	0
$452$	0	0
$453$	−5.41180e42	−1.72096
$454$	1.75036e42	0.537319
$455$	0	0
$456$	0	0
$457$	−6.44665e42	−1.78105	−0.890523	−	0.454938i	$-0.849661\pi$
$457$	−6.44665e42	−1.78105	−0.890523	+	0.454938i	$0.849661\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.74792e42	−1.86196	−0.930980	−	0.365069i	$-0.881045\pi$
$461$	−7.74792e42	−1.86196	−0.930980	+	0.365069i	$0.881045\pi$
$462$	5.09268e42	1.18216
$463$	−2.96148e42	−0.664069	−0.332034	−	0.943267i	$-0.607735\pi$
$463$	−2.96148e42	−0.664069	−0.332034	+	0.943267i	$0.607735\pi$
$464$	−5.58259e42	−1.20934
$465$	8.11274e42	1.69793
$466$	0	0
$467$	8.28054e42	1.61804	0.809018	−	0.587783i	$-0.199999\pi$
$467$	8.28054e42	1.61804	0.809018	+	0.587783i	$0.199999\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	2.56842e42	0.423247
$473$	0	0
$474$	1.25187e43	1.92799
$475$	0	0
$476$	0	0
$477$	2.38749e42	0.332389
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	−1.58305e43	−1.99358
$481$	0	0
$482$	−8.16763e42	−0.962373
$483$	0	0
$484$	−5.81409e42	−0.641145
$485$	3.71319e43	3.96168
$486$	9.68680e42	1.00000
$487$	2.40159e42	0.239903	0.119952	−	0.992780i	$-0.461726\pi$
$487$	2.40159e42	0.239903	0.119952	+	0.992780i	$0.461726\pi$
$488$	0	0
$489$	0	0
$490$	−6.37259e43	−5.77004
$491$	−2.98656e42	−0.261739	−0.130869	−	0.991400i	$-0.541777\pi$
$491$	−2.98656e42	−0.261739	−0.130869	+	0.991400i	$0.541777\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−1.55159e43	−1.19425
$496$	−1.14286e43	−0.851699
$497$	0	0
$498$	−2.23115e43	−1.55905
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−6.00596e43	−3.93607
$501$	0	0
$502$	2.79104e43	1.71596
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	3.42061e43	1.97340
$505$	−5.26596e43	−2.94317
$506$	0	0
$507$	1.90602e43	1.00000
$508$	2.93632e43	1.49274
$509$	−1.71605e43	−0.845368	−0.422684	−	0.906277i	$-0.638912\pi$
$509$	−1.71605e43	−0.845368	−0.422684	+	0.906277i	$0.638912\pi$
$510$	0	0
$511$	6.57624e43	3.04259
$512$	2.23007e43	1.00000
$513$	0	0
$514$	0	0
$515$	−6.96113e43	−2.84292
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−2.95673e43	−1.06692
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−3.67516e43	−1.20934
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	9.58219e42	0.296595
$525$	1.95505e44	5.86962
$526$	0	0
$527$	0	0
$528$	2.18576e43	0.599045
$529$	3.76089e43	1.00000
$530$	−2.56859e43	−0.662645
$531$	1.69085e43	0.423247
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−1.69698e44	−3.76718
$536$	0	0
$537$	1.59584e43	0.333735
$538$	−6.19324e43	−1.25719
$539$	8.79882e43	1.73382
$540$	−1.04216e44	−1.99358
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−1.59907e43	−0.288323
$543$	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
$550$	1.24927e44	1.78178
$551$	0	0
$552$	0	0
$553$	2.91020e44	3.80470
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.63372e44	−1.90324	−0.951620	−	0.307277i	$-0.900582\pi$
$557$	−1.63372e44	−1.90324	−0.951620	+	0.307277i	$0.900582\pi$
$558$	−7.52373e43	−0.851699
$559$	0	0
$560$	−3.68008e44	−3.93413
$561$	0	0
$562$	0	0
$563$	−2.60358e42	−0.0255527	−0.0127763	−	0.999918i	$-0.504067\pi$
$563$	−2.60358e42	−0.0255527	−0.0127763	+	0.999918i	$0.504067\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.25187e44	1.97340
$568$	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
$576$	1.46811e44	1.00000
$577$	−1.82222e44	−1.20722	−0.603610	−	0.797279i	$-0.706272\pi$
$577$	−1.82222e44	−1.20722	−0.603610	+	0.797279i	$0.706272\pi$
$578$	1.55183e44	1.00000
$579$	1.63171e44	1.02279
$580$	3.95394e44	2.41091
$581$	−5.18672e44	−3.07663
$582$	−3.44360e44	−1.98722
$583$	3.54653e43	0.199116
$584$	2.82250e44	1.54180
$585$	0	0
$586$	1.88761e44	0.976225
$587$	2.56467e44	1.29069	0.645345	−	0.763891i	$-0.276714\pi$
$587$	2.56467e44	1.29069	0.645345	+	0.763891i	$0.276714\pi$
$588$	5.90992e44	2.89431
$589$	0	0
$590$	−1.81912e44	−0.843778
$591$	−1.65675e44	−0.747925
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	1.43894e44	0.599045
$595$	0	0
$596$	−4.25662e44	−1.67928
$597$	−3.20593e44	−1.23130
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	8.39102e44	2.97437
$601$	1.51836e44	0.524063	0.262032	−	0.965059i	$-0.415608\pi$
$601$	1.51836e44	0.524063	0.262032	+	0.965059i	$0.415608\pi$
$602$	0	0
$603$	0	0
$604$	−5.39960e44	−1.72096
$605$	4.11790e44	1.27817
$606$	4.88364e44	1.47632
$607$	−4.23515e44	−1.24695	−0.623476	−	0.781842i	$-0.714280\pi$
$607$	−4.23515e44	−1.24695	−0.623476	+	0.781842i	$0.714280\pi$
$608$	0	0
$609$	−8.54357e44	−2.38651
$610$	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
$616$	5.08120e44	1.18216
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	6.45573e44	1.42603
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	8.09445e44	1.69793
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.64139e45	4.87250
$626$	−1.00328e45	−1.80399
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−2.42269e45	−3.93413
$631$	−1.26311e45	−1.99974	−0.999868	−	0.0162208i	$-0.994837\pi$
$631$	−1.26311e45	−1.99974	−0.999868	+	0.0162208i	$0.994837\pi$
$632$	1.24905e45	1.92799
$633$	0	0
$634$	−1.16387e45	−1.70795
$635$	−2.07969e45	−2.97590
$636$	2.38210e44	0.332389
$637$	0	0
$638$	−5.45931e44	−0.724448
$639$	0	0
$640$	−1.57948e45	−1.99358
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	1.57378e45	1.88965
$643$	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			$\pi$
$648$	9.66496e44	1.00000
$649$	2.51171e44	0.253544
$650$	0	0
$651$	−1.74903e45	−1.68074
$652$	0	0
$653$	1.49425e44	0.136713	0.0683567	−	0.997661i	$-0.478224\pi$
$653$	1.49425e44	0.136713	0.0683567	+	0.997661i	$0.478224\pi$
$654$	0	0
$655$	−6.78670e44	−0.591287
$656$	0	0
$657$	1.85812e45	1.54180
$658$	0	0
$659$	−2.52909e45	−1.99893	−0.999465	−	0.0326991i	$-0.989590\pi$
$659$	−2.52909e45	−1.99893	−0.999465	+	0.0326991i	$0.989590\pi$
$660$	−1.54809e45	−1.19425
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	−2.22612e45	−1.55905
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−2.34349e45	−1.45561
$670$	0	0
$671$	0	0
$672$	3.41290e45	1.97340
$673$	−2.26922e45	−1.28126	−0.640628	−	0.767851i	$-0.721326\pi$
$673$	−2.26922e45	−1.28126	−0.640628	+	0.767851i	$0.721326\pi$
$674$	−5.06250e44	−0.279130
$675$	5.52402e45	2.97437
$676$	1.90172e45	1.00000
$677$	−6.56365e44	−0.337075	−0.168537	−	0.985695i	$-0.553904\pi$
$677$	−6.56365e44	−0.337075	−0.168537	+	0.985695i	$0.553904\pi$
$678$	0	0
$679$	−8.00528e45	−3.92157
$680$	0	0
$681$	1.14970e45	0.537319
$682$	−1.11762e45	−0.510206
$683$	−4.24522e45	−1.89308	−0.946540	−	0.322586i	$-0.895448\pi$
$683$	−4.24522e45	−1.89308	−0.946540	+	0.322586i	$0.895448\pi$
$684$	0	0
$685$	0	0
$686$	8.99190e45	3.73823
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−2.95006e45	−1.06692
$693$	3.34508e45	1.18216
$694$	−4.47325e45	−1.54480
$695$	0	0
$696$	−3.66687e45	−1.20934
$697$	0	0
$698$	0	0
$699$	0	0
$700$	1.95065e46	5.86962
$701$	5.72235e45	1.68301	0.841504	−	0.540250i	$-0.181670\pi$
$701$	5.72235e45	1.68301	0.841504	+	0.540250i	$0.181670\pi$
$702$	0	0
$703$	0	0
$704$	2.18083e45	0.599045
$705$	0	0
$706$	0	0
$707$	1.13529e46	2.91337
$708$	1.68704e45	0.423247
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	8.22279e45	1.92799
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	1.59225e45	0.333735
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−1.03981e46	−1.99358
$721$	1.50075e46	2.81414
$722$	5.45249e45	1.00000
$723$	−5.36483e45	−0.962373
$724$	0	0
$725$	−2.09580e46	−3.59702
$726$	−3.81893e45	−0.641145
$727$	4.55904e45	0.748726	0.374363	−	0.927282i	$-0.377861\pi$
$727$	4.55904e45	0.748726	0.374363	+	0.927282i	$0.377861\pi$
$728$	0	0
$729$	6.36269e45	1.00000
$730$	−1.99907e46	−3.07370
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−8.12736e45	−1.14502
$735$	−4.18578e46	−5.77004
$736$	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
$742$	5.53764e45	0.655937
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−7.50677e45	−0.851699
$745$	3.01480e46	3.34779
$746$	0	0
$747$	−1.46551e46	−1.55905
$748$	0	0
$749$	3.65853e46	3.72904
$750$	−3.94496e46	−3.93607
$751$	1.87652e46	1.83279	0.916396	−	0.400273i	$-0.131085\pi$
$751$	1.87652e46	1.83279	0.916396	+	0.400273i	$0.131085\pi$
$752$	0	0
$753$	1.83327e46	1.71596
$754$	0	0
$755$	3.82433e46	3.43087
$756$	2.24680e46	1.97340
$757$	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			$\pi$
$762$	1.92869e46	1.49274
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	1.46480e46	1.00000
$769$	−2.97791e46	−1.99109	−0.995544	−	0.0942971i	$-0.969940\pi$
$769$	−2.97791e46	−1.99109	−0.995544	+	0.0942971i	$0.969940\pi$
$770$	−3.59882e46	−2.35672
$771$	0	0
$772$	1.62803e46	1.02279
$773$	1.97720e46	1.21668	0.608342	−	0.793675i	$-0.291835\pi$
$773$	1.97720e46	1.21668	0.608342	+	0.793675i	$0.291835\pi$
$774$	0	0
$775$	−4.29050e46	−2.53327
$776$	−3.43584e46	−1.98722
$777$	0	0
$778$	−2.49547e46	−1.38509
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−2.41399e46	−1.20934
$784$	5.89660e46	2.89431
$785$	0	0
$786$	6.29397e45	0.296595
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	−1.65302e46	−0.747925
$789$	0	0
$790$	−8.84654e46	−3.84361
$791$	0	0
$792$	1.43570e46	0.599045
$793$	0	0
$794$	0	0
$795$	−1.68715e46	−0.662645
$796$	−3.19870e46	−1.23130
$797$	−4.51242e46	−1.70246	−0.851230	−	0.524793i	$-0.824143\pi$
$797$	−4.51242e46	−1.70246	−0.851230	+	0.524793i	$0.824143\pi$
$798$	0	0
$799$	0	0
$800$	8.37210e46	2.97437
$801$	0	0
$802$	0	0
$803$	2.76017e46	0.923608
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.06797e46	−1.25719
$808$	4.87263e46	1.47632
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−6.84533e46	−1.99358
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	−8.52431e46	−2.38651
$813$	−1.05034e46	−0.288323
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	−1.12473e46	−0.279895
$819$	0	0
$820$	0	0
$821$	7.70376e46	1.80806	0.904028	−	0.427473i	$-0.140596\pi$
$821$	7.70376e46	1.80806	0.904028	+	0.427473i	$0.140596\pi$
$822$	0	0
$823$	−7.01496e45	−0.158354	−0.0791768	−	0.996861i	$-0.525229\pi$
$823$	−7.01496e45	−0.158354	−0.0791768	+	0.996861i	$0.525229\pi$
$824$	6.44118e46	1.42603
$825$	8.20574e46	1.78178
$826$	3.92184e46	0.835236
$827$	8.53966e46	1.78383	0.891913	−	0.452207i	$-0.149363\pi$
$827$	8.53966e46	1.78383	0.891913	+	0.452207i	$0.149363\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	1.57668e47	3.10809
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.94189e46	−0.851699
$838$	−4.15638e46	−0.702767
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	−2.41723e47	−3.93413
$841$	2.89632e46	0.462498
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.34692e47	−1.99358
$846$	0	0
$847$	−8.87781e46	−1.26524
$848$	2.37673e46	0.332389
$849$	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
$856$	1.57023e47	1.88965
$857$	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			$\pi$
$864$	9.64317e46	1.00000
$865$	2.08942e47	2.12700
$866$	−3.34142e45	−0.0333921
$867$	1.01931e47	1.00000
$868$	−1.74509e47	−1.68074
$869$	1.22147e47	1.15496
$870$	2.59711e47	2.41091
$871$	0	0
$872$	0	0
$873$	−2.26190e47	−1.98722
$874$	0	0
$875$	−9.17078e47	−7.76743
$876$	1.85393e47	1.54180
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	1.06778e47	0.856187
$879$	1.23986e47	0.976225
$880$	−1.54460e47	−1.19425
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	3.88188e47	2.89431
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	−1.19487e47	−0.843778
$886$	4.41203e45	0.0305984
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	4.48360e47	2.94577
$890$	0	0
$891$	9.45155e46	0.599045
$892$	−2.33821e47	−1.45561
$893$	0	0
$894$	−2.79592e47	−1.67928
$895$	−1.12773e47	−0.665328
$896$	3.40520e47	1.97340
$897$	0	0
$898$	0	0
$899$	1.87495e47	1.02999
$900$	5.51157e47	2.97437
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−3.54668e47	−1.72096
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	1.14711e47	0.537319
$909$	3.20777e47	1.47632
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−2.17697e47	−0.933941
$914$	−4.22488e47	−1.78105
$915$	0	0
$916$	0	0
$917$	1.46315e47	0.585301
$918$	0	0
$919$	4.79235e47	1.85140	0.925700	−	0.378257i	$-0.123477\pi$
$919$	4.79235e47	1.85140	0.925700	+	0.378257i	$0.123477\pi$
$920$	0	0
$921$	0	0
$922$	−5.07768e47	−1.86196
$923$	0	0
$924$	3.33754e47	1.18216
$925$	0	0
$926$	−1.94084e47	−0.664069
$927$	4.24039e47	1.42603
$928$	−3.65860e47	−1.20934
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	5.31676e47	1.69793
$931$	0	0
$932$	0	0
$933$	0	0
$934$	5.42673e47	1.61804
$935$	0	0
$936$	0	0
$937$	−3.80644e47	−1.07816	−0.539082	−	0.842253i	$-0.681229\pi$
$937$	−3.80644e47	−1.07816	−0.539082	+	0.842253i	$0.681229\pi$
$938$	0	0
$939$	−6.58997e47	−1.80399
$940$	0	0
$941$	−6.45735e47	−1.70852	−0.854260	−	0.519846i	$-0.825989\pi$
$941$	−6.45735e47	−1.70852	−0.854260	+	0.519846i	$0.825989\pi$
$942$	0	0
$943$	0	0
$944$	1.68324e47	0.423247
$945$	−1.59132e48	−3.93413
$946$	0	0
$947$	7.73155e47	1.84785	0.923925	−	0.382573i	$-0.124962\pi$
$947$	7.73155e47	1.84785	0.923925	+	0.382573i	$0.124962\pi$
$948$	8.20425e47	1.92799
$949$	0	0
$950$	0	0
$951$	−7.64477e47	−1.70795
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	1.56466e47	0.332389
$955$	0	0
$956$	0	0
$957$	−3.58590e47	−0.724448
$958$	0	0
$959$	0	0
$960$	−1.03747e48	−1.99358
$961$	−1.45308e47	−0.274609
$962$	0	0
$963$	1.03372e48	1.88965
$964$	−5.35274e47	−0.962373
$965$	−1.15307e48	−2.03901
$966$	0	0
$967$	−9.52643e47	−1.62970	−0.814848	−	0.579675i	$-0.803180\pi$
$967$	−9.52643e47	−1.62970	−0.814848	+	0.579675i	$0.803180\pi$
$968$	−3.81032e47	−0.641145
$969$	0	0
$970$	2.43348e48	3.96168
$971$	−2.58902e47	−0.414599	−0.207300	−	0.978277i	$-0.566468\pi$
$971$	−2.58902e47	−0.414599	−0.207300	+	0.978277i	$0.566468\pi$
$972$	6.34834e47	1.00000
$973$	0	0
$974$	1.57391e47	0.239903
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	−4.17634e48	−5.77004
$981$	0	0
$982$	−1.95727e47	−0.261739
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	1.17077e48	1.49105
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−1.01685e48	−1.19425
$991$	1.68858e48	1.95138	0.975692	−	0.219145i	$-0.0703269\pi$
$991$	1.68858e48	1.95138	0.975692	+	0.219145i	$0.0703269\pi$
$992$	−7.48984e47	−0.851699
$993$	0	0
$994$	0	0
$995$	2.26552e48	2.45470
$996$	−1.46221e48	−1.55905
$997$	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	24.33.h.b.5.1	yes	1
3.2	odd	2		24.33.h.a.5.1	✓	1
8.5	even	2		24.33.h.a.5.1	✓	1
24.5	odd	2	CM	24.33.h.b.5.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.33.h.a.5.1	✓	1	3.2	odd	2
24.33.h.a.5.1	✓	1	8.5	even	2
24.33.h.b.5.1	yes	1	1.1	even	1	trivial
24.33.h.b.5.1	yes	1	24.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 24 = 2^{3} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 33 \)
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

Downloads

Learn more