LMFDB - Embedded newform 24.17.h.a.5.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [24,17,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, 17); 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, 17, names="a")

Level:	$ N $	$=$	$ 24 = 2^{3} \cdot 3 $
Weight:	$ k $	$=$	$ 17 $
Character orbit:	$[\chi]$	$=$	24.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,-256] 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:	$38.9578905256$
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)

$f(q)$

$=$

\(q-256.000 q^{2} -6561.00 q^{3} +65536.0 q^{4} +31294.0 q^{5} +1.67962e6 q^{6} +1.14912e7 q^{7} -1.67772e7 q^{8} +4.30467e7 q^{9} -8.01126e6 q^{10} +3.45580e8 q^{11} -4.29982e8 q^{12} -2.94175e9 q^{14} -2.05320e8 q^{15} +4.29497e9 q^{16} -1.10200e10 q^{18} +2.05088e9 q^{20} -7.53938e10 q^{21} -8.84684e10 q^{22} +1.10075e11 q^{24} -1.51609e11 q^{25} -2.82430e11 q^{27} +7.53087e11 q^{28} +4.44815e11 q^{29} +5.25619e10 q^{30} +9.13948e11 q^{31} -1.09951e12 q^{32} -2.26735e12 q^{33} +3.59606e11 q^{35} +2.82111e12 q^{36} -5.25026e11 q^{40} +1.93008e13 q^{42} +2.26479e13 q^{44} +1.34710e12 q^{45} -2.81793e13 q^{48} +9.88148e13 q^{49} +3.88118e13 q^{50} +9.50840e13 q^{53} +7.23020e13 q^{54} +1.08146e13 q^{55} -1.92790e14 q^{56} -1.13873e14 q^{58} -2.28568e14 q^{59} -1.34558e13 q^{60} -2.33971e14 q^{62} +4.94659e14 q^{63} +2.81475e14 q^{64} +5.80441e14 q^{66} -9.20591e13 q^{70} -7.22204e14 q^{72} -1.51773e15 q^{73} +9.94704e14 q^{75} +3.97113e15 q^{77} -3.00678e15 q^{79} +1.34407e14 q^{80} +1.85302e15 q^{81} -1.49562e15 q^{83} -4.94101e15 q^{84} -2.91843e15 q^{87} -5.79787e15 q^{88} -3.44859e14 q^{90} -5.99641e15 q^{93} +7.21390e15 q^{96} -8.86128e14 q^{97} -2.52966e16 q^{98} +1.48761e16 q^{99} +O(q^{100})\)

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 $

$2$	−256.000	−1.00000
$2$	−256.000	−1.00000
$3$	−6561.00	−1.00000
$3$	−6561.00	−1.00000
$4$	65536.0	1.00000
$5$	31294.0	0.0801126	0.0400563	−	0.999197i	$-0.487246\pi$
$5$	31294.0	0.0801126	0.0400563	+	0.999197i	$0.487246\pi$
$6$	1.67962e6	1.00000
$7$	1.14912e7	1.99334	0.996669	−	0.0815477i	$-0.0259863\pi$
$7$	1.14912e7	1.99334	0.996669	+	0.0815477i	$0.0259863\pi$
$8$	−1.67772e7	−1.00000
$9$	4.30467e7	1.00000
$10$	−8.01126e6	−0.0801126
$11$	3.45580e8	1.61216	0.806078	−	0.591810i	$-0.201586\pi$
$11$	3.45580e8	1.61216	0.806078	+	0.591810i	$0.201586\pi$
$12$	−4.29982e8	−1.00000
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−2.94175e9	−1.99334
$15$	−2.05320e8	−0.0801126
$16$	4.29497e9	1.00000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	−1.10200e10	−1.00000
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	2.05088e9	0.0801126
$21$	−7.53938e10	−1.99334
$22$	−8.84684e10	−1.61216
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.10075e11	1.00000
$25$	−1.51609e11	−0.993582
$26$	0	0
$27$	−2.82430e11	−1.00000
$28$	7.53087e11	1.99334
$29$	4.44815e11	0.889192	0.444596	−	0.895731i	$-0.353347\pi$
$29$	4.44815e11	0.889192	0.444596	+	0.895731i	$0.353347\pi$
$30$	5.25619e10	0.0801126
$31$	9.13948e11	1.07159	0.535794	−	0.844349i	$-0.320012\pi$
$31$	9.13948e11	1.07159	0.535794	+	0.844349i	$0.320012\pi$
$32$	−1.09951e12	−1.00000
$33$	−2.26735e12	−1.61216
$34$	0	0
$35$	3.59606e11	0.159692
$36$	2.82111e12	1.00000
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	−5.25026e11	−0.0801126
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	1.93008e13	1.99334
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	2.26479e13	1.61216
$45$	1.34710e12	0.0801126
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−2.81793e13	−1.00000
$49$	9.88148e13	2.97340
$50$	3.88118e13	0.993582
$51$	0	0
$52$	0	0
$53$	9.50840e13	1.52722	0.763608	−	0.645680i	$-0.223426\pi$
$53$	9.50840e13	1.52722	0.763608	+	0.645680i	$0.223426\pi$
$54$	7.23020e13	1.00000
$55$	1.08146e13	0.129154
$56$	−1.92790e14	−1.99334
$57$	0	0
$58$	−1.13873e14	−0.889192
$59$	−2.28568e14	−1.55668	−0.778339	−	0.627844i	$-0.783938\pi$
$59$	−2.28568e14	−1.55668	−0.778339	+	0.627844i	$0.783938\pi$
$60$	−1.34558e13	−0.0801126
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−2.33971e14	−1.07159
$63$	4.94659e14	1.99334
$64$	2.81475e14	1.00000
$65$	0	0
$66$	5.80441e14	1.61216
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	−9.20591e13	−0.159692
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−7.22204e14	−1.00000
$73$	−1.51773e15	−1.88197	−0.940983	−	0.338453i	$-0.890097\pi$
$73$	−1.51773e15	−1.88197	−0.940983	+	0.338453i	$0.890097\pi$
$74$	0	0
$75$	9.94704e14	0.993582
$76$	0	0
$77$	3.97113e15	3.21357
$78$	0	0
$79$	−3.00678e15	−1.98192	−0.990958	−	0.134169i	$-0.957163\pi$
$79$	−3.00678e15	−1.98192	−0.990958	+	0.134169i	$0.957163\pi$
$80$	1.34407e14	0.0801126
$81$	1.85302e15	1.00000
$82$	0	0
$83$	−1.49562e15	−0.664042	−0.332021	−	0.943272i	$-0.607730\pi$
$83$	−1.49562e15	−0.664042	−0.332021	+	0.943272i	$0.607730\pi$
$84$	−4.94101e15	−1.99334
$85$	0	0
$86$	0	0
$87$	−2.91843e15	−0.889192
$88$	−5.79787e15	−1.61216
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−3.44859e14	−0.0801126
$91$	0	0
$92$	0	0
$93$	−5.99641e15	−1.07159
$94$	0	0
$95$	0	0
$96$	7.21390e15	1.00000
$97$	−8.86128e14	−0.113063	−0.0565317	−	0.998401i	$-0.518004\pi$
$97$	−8.86128e14	−0.113063	−0.0565317	+	0.998401i	$0.518004\pi$
$98$	−2.52966e16	−2.97340
$99$	1.48761e16	1.61216
$100$	−9.93582e15	−0.993582
$101$	2.01898e16	1.86449	0.932245	−	0.361828i	$-0.117847\pi$
$101$	2.01898e16	1.86449	0.932245	+	0.361828i	$0.117847\pi$
$102$	0	0
$103$	2.34473e16	1.85095	0.925477	−	0.378803i	$-0.123664\pi$
$103$	2.34473e16	1.85095	0.925477	+	0.378803i	$0.123664\pi$
$104$	0	0
$105$	−2.35937e15	−0.159692
$106$	−2.43415e16	−1.52722
$107$	3.38864e16	1.97222	0.986110	−	0.166091i	$-0.0531147\pi$
$107$	3.38864e16	1.97222	0.986110	+	0.166091i	$0.0531147\pi$
$108$	−1.85093e16	−1.00000
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−2.76853e15	−0.129154
$111$	0	0
$112$	4.93543e16	1.99334
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	2.91514e16	0.889192
$117$	0	0
$118$	5.85133e16	1.55668
$119$	0	0
$120$	3.44470e15	0.0801126
$121$	7.34757e16	1.59905
$122$	0	0
$123$	0	0
$124$	5.98965e16	1.07159
$125$	−9.51952e15	−0.159711
$126$	−1.26633e17	−1.99334
$127$	−1.26477e17	−1.86889	−0.934444	−	0.356111i	$-0.884102\pi$
$127$	−1.26477e17	−1.86889	−0.934444	+	0.356111i	$0.884102\pi$
$128$	−7.20576e16	−1.00000
$129$	0	0
$130$	0	0
$131$	1.31435e17	1.51545	0.757726	−	0.652573i	$-0.226310\pi$
$131$	1.31435e17	1.51545	0.757726	+	0.652573i	$0.226310\pi$
$132$	−1.48593e17	−1.61216
$133$	0	0
$134$	0	0
$135$	−8.83835e15	−0.0801126
$136$	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			$\pi$
$140$	2.35671e16	0.159692
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.84884e17	1.00000
$145$	1.39200e16	0.0712355
$146$	3.88539e17	1.88197
$147$	−6.48324e17	−2.97340
$148$	0	0
$149$	1.37578e17	0.566318	0.283159	−	0.959073i	$-0.408618\pi$
$149$	1.37578e17	0.566318	0.283159	+	0.959073i	$0.408618\pi$
$150$	−2.54644e17	−0.993582
$151$	1.42774e17	0.528244	0.264122	−	0.964489i	$-0.414918\pi$
$151$	1.42774e17	0.528244	0.264122	+	0.964489i	$0.414918\pi$
$152$	0	0
$153$	0	0
$154$	−1.01661e18	−3.21357
$155$	2.86011e16	0.0858478
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	7.69737e17	1.98192
$159$	−6.23846e17	−1.52722
$160$	−3.44081e16	−0.0801126
$161$	0	0
$162$	−4.74373e17	−1.00000
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−7.09544e16	−0.129154
$166$	3.82878e17	0.664042
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	1.26490e18	1.99334
$169$	6.65417e17	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.75046e17	−0.965959	−0.482979	−	0.875632i	$-0.660445\pi$
$173$	−7.75046e17	−0.965959	−0.482979	+	0.875632i	$0.660445\pi$
$174$	7.47118e17	0.889192
$175$	−1.74216e18	−1.98055
$176$	1.48425e18	1.61216
$177$	1.49963e18	1.55668
$178$	0	0
$179$	−1.61009e18	−1.52766	−0.763828	−	0.645419i	$-0.776683\pi$
$179$	−1.61009e18	−1.52766	−0.763828	+	0.645419i	$0.776683\pi$
$180$	8.82838e16	0.0801126
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	1.53508e18	1.07159
$187$	0	0
$188$	0	0
$189$	−3.24545e18	−1.99334
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−1.84676e18	−1.00000
$193$	3.34705e18	1.73862	0.869308	−	0.494271i	$-0.164565\pi$
$193$	3.34705e18	1.73862	0.869308	+	0.494271i	$0.164565\pi$
$194$	2.26849e17	0.113063
$195$	0	0
$196$	6.47593e18	2.97340
$197$	2.53831e18	1.11896	0.559481	−	0.828843i	$-0.311001\pi$
$197$	2.53831e18	1.11896	0.559481	+	0.828843i	$0.311001\pi$
$198$	−3.80828e18	−1.61216
$199$	−2.15627e18	−0.876754	−0.438377	−	0.898791i	$-0.644446\pi$
$199$	−2.15627e18	−0.876754	−0.438377	+	0.898791i	$0.644446\pi$
$200$	2.54357e18	0.993582
$201$	0	0
$202$	−5.16858e18	−1.86449
$203$	5.11146e18	1.77246
$204$	0	0
$205$	0	0
$206$	−6.00252e18	−1.85095
$207$	0	0
$208$	0	0
$209$	0	0
$210$	6.03999e17	0.159692
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	6.23143e18	1.52722
$213$	0	0
$214$	−8.67492e18	−1.97222
$215$	0	0
$216$	4.73838e18	1.00000
$217$	1.05024e19	2.13604
$218$	0	0
$219$	9.95783e18	1.88197
$220$	7.08744e17	0.129154
$221$	0	0
$222$	0	0
$223$	−4.51225e18	−0.737826	−0.368913	−	0.929464i	$-0.620270\pi$
$223$	−4.51225e18	−0.737826	−0.368913	+	0.929464i	$0.620270\pi$
$224$	−1.26347e19	−1.99334
$225$	−6.52625e18	−0.993582
$226$	0	0
$227$	−1.12304e19	−1.59290	−0.796448	−	0.604707i	$-0.793290\pi$
$227$	−1.12304e19	−1.59290	−0.796448	+	0.604707i	$0.793290\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	−2.60546e19	−3.21357
$232$	−7.46276e18	−0.889192
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	−1.49794e19	−1.55668
$237$	1.97275e19	1.98192
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−8.81842e17	−0.0801126
$241$	1.15920e19	1.01864	0.509320	−	0.860577i	$-0.329897\pi$
$241$	1.15920e19	1.01864	0.509320	+	0.860577i	$0.329897\pi$
$242$	−1.88098e19	−1.59905
$243$	−1.21577e19	−1.00000
$244$	0	0
$245$	3.09231e18	0.238207
$246$	0	0
$247$	0	0
$248$	−1.53335e19	−1.07159
$249$	9.81273e18	0.664042
$250$	2.43700e18	0.159711
$251$	3.03686e19	1.92768	0.963841	−	0.266478i	$-0.0858600\pi$
$251$	3.03686e19	1.92768	0.963841	+	0.266478i	$0.0858600\pi$
$252$	3.24179e19	1.99334
$253$	0	0
$254$	3.23782e19	1.86889
$255$	0	0
$256$	1.84467e19	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.91478e19	0.889192
$262$	−3.36475e19	−1.51545
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	3.80398e19	1.61216
$265$	2.97556e18	0.122349
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.36296e19	−0.861863	−0.430932	−	0.902385i	$-0.641815\pi$
$269$	−2.36296e19	−0.861863	−0.430932	+	0.902385i	$0.641815\pi$
$270$	2.26262e18	0.0801126
$271$	3.80597e19	1.30831	0.654155	−	0.756360i	$-0.273024\pi$
$271$	3.80597e19	1.30831	0.654155	+	0.756360i	$0.273024\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.23929e19	−1.60181
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	3.93425e19	1.07159
$280$	−6.03318e18	−0.159692
$281$	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
$288$	−4.73304e19	−1.00000
$289$	4.86612e19	1.00000
$290$	−3.56353e18	−0.0712355
$291$	5.81388e18	0.113063
$292$	−9.94660e19	−1.88197
$293$	−9.37074e19	−1.72517	−0.862587	−	0.505909i	$-0.831157\pi$
$293$	−9.37074e19	−1.72517	−0.862587	+	0.505909i	$0.831157\pi$
$294$	1.65971e20	2.97340
$295$	−7.15280e18	−0.124710
$296$	0	0
$297$	−9.76020e19	−1.61216
$298$	−3.52201e19	−0.566318
$299$	0	0
$300$	6.51889e19	0.993582
$301$	0	0
$302$	−3.65502e19	−0.528244
$303$	−1.32465e20	−1.86449
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	2.60252e20	3.21357
$309$	−1.53838e20	−1.85095
$310$	−7.32188e18	−0.0858478
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−4.07844e19	−0.442731	−0.221365	−	0.975191i	$-0.571051\pi$
$313$	−4.07844e19	−0.442731	−0.221365	+	0.975191i	$0.571051\pi$
$314$	0	0
$315$	1.54798e19	0.159692
$316$	−1.97053e20	−1.98192
$317$	5.51061e19	0.540412	0.270206	−	0.962802i	$-0.412908\pi$
$317$	5.51061e19	0.540412	0.270206	+	0.962802i	$0.412908\pi$
$318$	1.59705e20	1.52722
$319$	1.53719e20	1.43352
$320$	8.80848e18	0.0801126
$321$	−2.22329e20	−1.97222
$322$	0	0
$323$	0	0
$324$	1.21440e20	1.00000
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	1.81643e19	0.129154
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	−9.80167e19	−0.664042
$333$	0	0
$334$	0	0
$335$	0	0
$336$	−3.23814e20	−1.99334
$337$	−2.18229e20	−1.31182	−0.655910	−	0.754839i	$-0.727715\pi$
$337$	−2.18229e20	−1.31182	−0.655910	+	0.754839i	$0.727715\pi$
$338$	−1.70347e20	−1.00000
$339$	0	0
$340$	0	0
$341$	3.15842e20	1.72757
$342$	0	0
$343$	7.53614e20	3.93365
$344$	0	0
$345$	0	0
$346$	1.98412e20	0.965959
$347$	−1.41820e20	−0.674686	−0.337343	−	0.941382i	$-0.609528\pi$
$347$	−1.41820e20	−0.674686	−0.337343	+	0.941382i	$0.609528\pi$
$348$	−1.91262e20	−0.889192
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	4.45994e20	1.98055
$351$	0	0
$352$	−3.79969e20	−1.61216
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	−3.83906e20	−1.55668
$355$	0	0
$356$	0	0
$357$	0	0
$358$	4.12183e20	1.52766
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−2.26007e19	−0.0801126
$361$	2.88441e20	1.00000
$362$	0	0
$363$	−4.82074e20	−1.59905
$364$	0	0
$365$	−4.74959e19	−0.150769
$366$	0	0
$367$	−3.04303e20	−0.924652	−0.462326	−	0.886710i	$-0.652985\pi$
$367$	−3.04303e20	−0.924652	−0.462326	+	0.886710i	$0.652985\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.09263e21	3.04426
$372$	−3.92981e20	−1.07159
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	6.24576e19	0.159711
$376$	0	0
$377$	0	0
$378$	8.30836e20	1.99334
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	8.29818e20	1.86889
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	4.72770e20	1.00000
$385$	1.24272e20	0.257448
$386$	−8.56845e20	−1.73862
$387$	0	0
$388$	−5.80733e19	−0.113063
$389$	4.11152e20	0.784161	0.392081	−	0.919931i	$-0.371755\pi$
$389$	4.11152e20	0.784161	0.392081	+	0.919931i	$0.371755\pi$
$390$	0	0
$391$	0	0
$392$	−1.65784e21	−2.97340
$393$	−8.62348e20	−1.51545
$394$	−6.49808e20	−1.11896
$395$	−9.40943e19	−0.158777
$396$	9.74919e20	1.61216
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	5.52004e20	0.876754
$399$	0	0
$400$	−6.51154e20	−0.993582
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	1.32316e21	1.86449
$405$	5.79884e19	0.0801126
$406$	−1.30853e21	−1.77246
$407$	0	0
$408$	0	0
$409$	−1.02698e21	−1.31153	−0.655764	−	0.754966i	$-0.727653\pi$
$409$	−1.02698e21	−1.31153	−0.655764	+	0.754966i	$0.727653\pi$
$410$	0	0
$411$	0	0
$412$	1.53665e21	1.85095
$413$	−2.62652e21	−3.10299
$414$	0	0
$415$	−4.68038e19	−0.0531981
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.08198e21	1.13896	0.569481	−	0.822005i	$-0.307144\pi$
$419$	1.08198e21	1.13896	0.569481	+	0.822005i	$0.307144\pi$
$420$	−1.54624e20	−0.159692
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	−1.59524e21	−1.52722
$425$	0	0
$426$	0	0
$427$	0	0
$428$	2.22078e21	1.97222
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−1.21303e21	−1.00000
$433$	1.73285e21	1.40236	0.701179	−	0.712985i	$-0.252657\pi$
$433$	1.73285e21	1.40236	0.701179	+	0.712985i	$0.252657\pi$
$434$	−2.68860e21	−2.13604
$435$	−9.13294e19	−0.0712355
$436$	0	0
$437$	0	0
$438$	−2.54921e21	−1.88197
$439$	−2.33136e21	−1.69003	−0.845013	−	0.534746i	$-0.820407\pi$
$439$	−2.33136e21	−1.69003	−0.845013	+	0.534746i	$0.820407\pi$
$440$	−1.81438e20	−0.129154
$441$	4.25365e21	2.97340
$442$	0	0
$443$	−2.11369e21	−1.42499	−0.712495	−	0.701677i	$-0.752435\pi$
$443$	−2.11369e21	−1.42499	−0.712495	+	0.701677i	$0.752435\pi$
$444$	0	0
$445$	0	0
$446$	1.15514e21	0.737826
$447$	−9.02652e20	−0.566318
$448$	3.23449e21	1.99334
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	1.67072e21	0.993582
$451$	0	0
$452$	0	0
$453$	−9.36742e20	−0.528244
$454$	2.87498e21	1.59290
$455$	0	0
$456$	0	0
$457$	8.90237e20	0.467925	0.233962	−	0.972246i	$-0.424831\pi$
$457$	8.90237e20	0.467925	0.233962	+	0.972246i	$0.424831\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.57894e20	−0.371536	−0.185768	−	0.982594i	$-0.559477\pi$
$461$	−7.57894e20	−0.371536	−0.185768	+	0.982594i	$0.559477\pi$
$462$	6.66997e21	3.21357
$463$	−2.44084e21	−1.15583	−0.577913	−	0.816099i	$-0.696133\pi$
$463$	−2.44084e21	−1.15583	−0.577913	+	0.816099i	$0.696133\pi$
$464$	1.91047e21	0.889192
$465$	−1.87652e20	−0.0858478
$466$	0	0
$467$	−4.30300e21	−1.90211	−0.951057	−	0.309016i	$-0.900000\pi$
$467$	−4.30300e21	−1.90211	−0.951057	+	0.309016i	$0.900000\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	3.83473e21	1.55668
$473$	0	0
$474$	−5.05024e21	−1.98192
$475$	0	0
$476$	0	0
$477$	4.09305e21	1.52722
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	2.25752e20	0.0801126
$481$	0	0
$482$	−2.96754e21	−1.01864
$483$	0	0
$484$	4.81530e21	1.59905
$485$	−2.77305e19	−0.00905781
$486$	3.11236e21	1.00000
$487$	4.73529e21	1.49663	0.748315	−	0.663343i	$-0.230863\pi$
$487$	4.73529e21	1.49663	0.748315	+	0.663343i	$0.230863\pi$
$488$	0	0
$489$	0	0
$490$	−7.91631e20	−0.238207
$491$	4.45358e21	1.31843	0.659216	−	0.751954i	$-0.270888\pi$
$491$	4.45358e21	1.31843	0.659216	+	0.751954i	$0.270888\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4.65532e20	0.129154
$496$	3.92538e21	1.07159
$497$	0	0
$498$	−2.51206e21	−0.664042
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−6.23872e20	−0.159711
$501$	0	0
$502$	−7.77437e21	−1.92768
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	−8.29899e21	−1.99334
$505$	6.31818e20	0.149369
$506$	0	0
$507$	−4.36580e21	−1.00000
$508$	−8.28882e21	−1.86889
$509$	−4.84132e21	−1.07454	−0.537269	−	0.843411i	$-0.680544\pi$
$509$	−4.84132e21	−1.07454	−0.537269	+	0.843411i	$0.680544\pi$
$510$	0	0
$511$	−1.74406e22	−3.75140
$512$	−4.72237e21	−1.00000
$513$	0	0
$514$	0	0
$515$	7.33761e20	0.148285
$516$	0	0
$517$	0	0
$518$	0	0
$519$	5.08508e21	0.965959
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	−4.90184e21	−0.889192
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	8.61376e21	1.51545
$525$	1.14303e22	1.98055
$526$	0	0
$527$	0	0
$528$	−9.73819e21	−1.61216
$529$	6.13261e21	1.00000
$530$	−7.61743e20	−0.122349
$531$	−9.83909e21	−1.55668
$532$	0	0
$533$	0	0
$534$	0	0
$535$	1.06044e21	0.158000
$536$	0	0
$537$	1.05638e22	1.52766
$538$	6.04918e21	0.861863
$539$	3.41484e22	4.79358
$540$	−5.79230e20	−0.0801126
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	−9.74328e21	−1.30831
$543$	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
$550$	1.34126e22	1.60181
$551$	0	0
$552$	0	0
$553$	−3.45516e22	−3.95063
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.88197e21	−0.311063	−0.155531	−	0.987831i	$-0.549709\pi$
$557$	−2.88197e21	−0.311063	−0.155531	+	0.987831i	$0.549709\pi$
$558$	−1.00717e22	−1.07159
$559$	0	0
$560$	1.54449e21	0.159692
$561$	0	0
$562$	0	0
$563$	1.41837e22	1.40515	0.702575	−	0.711610i	$-0.252033\pi$
$563$	1.41837e22	1.40515	0.702575	+	0.711610i	$0.252033\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.12934e22	1.99334
$568$	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
$576$	1.21166e22	1.00000
$577$	−1.09391e22	−0.890381	−0.445191	−	0.895436i	$-0.646864\pi$
$577$	−1.09391e22	−0.890381	−0.445191	+	0.895436i	$0.646864\pi$
$578$	−1.24573e22	−1.00000
$579$	−2.19600e22	−1.73862
$580$	9.12264e20	0.0712355
$581$	−1.71864e22	−1.32366
$582$	−1.48835e21	−0.113063
$583$	3.28591e22	2.46211
$584$	2.54633e22	1.88197
$585$	0	0
$586$	2.39891e22	1.72517
$587$	−2.55710e22	−1.81403	−0.907013	−	0.421103i	$-0.861643\pi$
$587$	−2.55710e22	−1.81403	−0.907013	+	0.421103i	$0.861643\pi$
$588$	−4.24886e22	−2.97340
$589$	0	0
$590$	1.83112e21	0.124710
$591$	−1.66539e22	−1.11896
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	2.49861e22	1.61216
$595$	0	0
$596$	9.01634e21	0.566318
$597$	1.41473e22	0.876754
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−1.66884e22	−0.993582
$601$	2.70424e22	1.58873	0.794365	−	0.607441i	$-0.207804\pi$
$601$	2.70424e22	1.58873	0.794365	+	0.607441i	$0.207804\pi$
$602$	0	0
$603$	0	0
$604$	9.35685e21	0.528244
$605$	2.29935e21	0.128104
$606$	3.39110e22	1.86449
$607$	1.59927e22	0.867783	0.433892	−	0.900965i	$-0.357140\pi$
$607$	1.59927e22	0.867783	0.433892	+	0.900965i	$0.357140\pi$
$608$	0	0
$609$	−3.35363e22	−1.77246
$610$	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
$616$	−6.66245e22	−3.21357
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	3.93825e22	1.85095
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	1.87440e21	0.0858478
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.28357e22	0.980787
$626$	1.04408e22	0.442731
$627$	0	0
$628$	0	0
$629$	0	0
$630$	−3.96284e21	−0.159692
$631$	4.07681e20	0.0162213	0.00811066	−	0.999967i	$-0.497418\pi$
$631$	4.07681e20	0.0162213	0.00811066	+	0.999967i	$0.497418\pi$
$632$	5.04455e22	1.98192
$633$	0	0
$634$	−1.41071e22	−0.540412
$635$	−3.95798e21	−0.149721
$636$	−4.08844e22	−1.52722
$637$	0	0
$638$	−3.93521e22	−1.43352
$639$	0	0
$640$	−2.25497e21	−0.0801126
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	5.69162e22	1.97222
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−3.10885e22	−1.00000
$649$	−7.89884e22	−2.50961
$650$	0	0
$651$	−6.89060e22	−2.13604
$652$	0	0
$653$	−4.83257e22	−1.46175	−0.730875	−	0.682511i	$-0.760888\pi$
$653$	−4.83257e22	−1.46175	−0.730875	+	0.682511i	$0.760888\pi$
$654$	0	0
$655$	4.11314e21	0.121407
$656$	0	0
$657$	−6.53334e22	−1.88197
$658$	0	0
$659$	1.16326e21	0.0327035	0.0163518	−	0.999866i	$-0.494795\pi$
$659$	1.16326e21	0.0327035	0.0163518	+	0.999866i	$0.494795\pi$
$660$	−4.65007e21	−0.129154
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	2.50923e22	0.664042
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	2.96049e22	0.737826
$670$	0	0
$671$	0	0
$672$	8.28963e22	1.99334
$673$	3.56786e22	0.847787	0.423894	−	0.905712i	$-0.360663\pi$
$673$	3.56786e22	0.847787	0.423894	+	0.905712i	$0.360663\pi$
$674$	5.58667e22	1.31182
$675$	4.28187e22	0.993582
$676$	4.36087e22	1.00000
$677$	−5.69044e22	−1.28954	−0.644772	−	0.764375i	$-0.723048\pi$
$677$	−5.69044e22	−1.28954	−0.644772	+	0.764375i	$0.723048\pi$
$678$	0	0
$679$	−1.01827e22	−0.225374
$680$	0	0
$681$	7.36825e22	1.59290
$682$	−8.08555e22	−1.72757
$683$	1.54844e22	0.326986	0.163493	−	0.986545i	$-0.447724\pi$
$683$	1.54844e22	0.326986	0.163493	+	0.986545i	$0.447724\pi$
$684$	0	0
$685$	0	0
$686$	−1.92925e23	−3.93365
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	−5.07934e22	−0.965959
$693$	1.70944e23	3.21357
$694$	3.63059e22	0.674686
$695$	0	0
$696$	4.89632e22	0.889192
$697$	0	0
$698$	0	0
$699$	0	0
$700$	−1.14175e23	−1.98055
$701$	−1.11904e23	−1.91912	−0.959558	−	0.281510i	$-0.909165\pi$
$701$	−1.11904e23	−1.91912	−0.959558	+	0.281510i	$0.909165\pi$
$702$	0	0
$703$	0	0
$704$	9.72721e22	1.61216
$705$	0	0
$706$	0	0
$707$	2.32005e23	3.71656
$708$	9.82799e22	1.55668
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	−1.29432e23	−1.98192
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−1.05519e23	−1.52766
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	5.78577e21	0.0801126
$721$	2.69438e23	3.68958
$722$	−7.38410e22	−1.00000
$723$	−7.60549e22	−1.01864
$724$	0	0
$725$	−6.74378e22	−0.883485
$726$	1.23411e23	1.59905
$727$	1.29372e23	1.65793	0.828964	−	0.559302i	$-0.188931\pi$
$727$	1.29372e23	1.65793	0.828964	+	0.559302i	$0.188931\pi$
$728$	0	0
$729$	7.97664e22	1.00000
$730$	1.21589e22	0.150769
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	7.79017e22	0.924652
$735$	−2.02886e22	−0.238207
$736$	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
$742$	−2.79713e23	−3.04426
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	1.00603e23	1.07159
$745$	4.30538e21	0.0453692
$746$	0	0
$747$	−6.43813e22	−0.664042
$748$	0	0
$749$	3.89396e23	3.93130
$750$	−1.59891e22	−0.159711
$751$	−1.98097e23	−1.95775	−0.978876	−	0.204455i	$-0.934458\pi$
$751$	−1.98097e23	−1.95775	−0.978876	+	0.204455i	$0.934458\pi$
$752$	0	0
$753$	−1.99249e23	−1.92768
$754$	0	0
$755$	4.46798e21	0.0423190
$756$	−2.12694e23	−1.99334
$757$	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			$\pi$
$762$	−2.12433e23	−1.86889
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−1.21029e23	−1.00000
$769$	1.15450e22	0.0944023	0.0472011	−	0.998885i	$-0.484970\pi$
$769$	1.15450e22	0.0944023	0.0472011	+	0.998885i	$0.484970\pi$
$770$	−3.18138e22	−0.257448
$771$	0	0
$772$	2.19352e23	1.73862
$773$	−2.28634e23	−1.79351	−0.896756	−	0.442526i	$-0.854083\pi$
$773$	−2.28634e23	−1.79351	−0.896756	+	0.442526i	$0.854083\pi$
$774$	0	0
$775$	−1.38562e23	−1.06471
$776$	1.48668e22	0.113063
$777$	0	0
$778$	−1.05255e23	−0.784161
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−1.25629e23	−0.889192
$784$	4.24406e23	2.97340
$785$	0	0
$786$	2.20761e23	1.51545
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	1.66351e23	1.11896
$789$	0	0
$790$	2.40881e22	0.158777
$791$	0	0
$792$	−2.49579e23	−1.61216
$793$	0	0
$794$	0	0
$795$	−1.95226e22	−0.122349
$796$	−1.41313e23	−0.876754
$797$	−8.88055e22	−0.545473	−0.272736	−	0.962089i	$-0.587929\pi$
$797$	−8.88055e22	−0.545473	−0.272736	+	0.962089i	$0.587929\pi$
$798$	0	0
$799$	0	0
$800$	1.66695e23	0.993582
$801$	0	0
$802$	0	0
$803$	−5.24497e23	−3.03402
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.55034e23	0.861863
$808$	−3.38728e23	−1.86449
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−1.48450e22	−0.0801126
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	3.34985e23	1.77246
$813$	−2.49710e23	−1.30831
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	2.62908e23	1.31153
$819$	0	0
$820$	0	0
$821$	4.02807e23	1.95142	0.975712	−	0.219057i	$-0.0702980\pi$
$821$	4.02807e23	1.95142	0.975712	+	0.219057i	$0.0702980\pi$
$822$	0	0
$823$	−2.85629e23	−1.35707	−0.678536	−	0.734567i	$-0.737385\pi$
$823$	−2.85629e23	−1.35707	−0.678536	+	0.734567i	$0.737385\pi$
$824$	−3.93381e23	−1.85095
$825$	3.43750e23	1.60181
$826$	6.72389e23	3.10299
$827$	4.25608e23	1.94521	0.972603	−	0.232473i	$-0.0746816\pi$
$827$	4.25608e23	1.94521	0.972603	+	0.232473i	$0.0746816\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	1.19818e22	0.0531981
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.58126e23	−1.07159
$838$	−2.76988e23	−1.13896
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	3.95837e22	0.159692
$841$	−5.23861e22	−0.209338
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.08235e22	0.0801126
$846$	0	0
$847$	8.44324e23	3.18744
$848$	4.08383e23	1.52722
$849$	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
$856$	−5.68520e23	−1.97222
$857$	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			$\pi$
$864$	3.10535e23	1.00000
$865$	−2.42543e22	−0.0773855
$866$	−4.43611e23	−1.40236
$867$	−3.19266e23	−1.00000
$868$	6.88283e23	2.13604
$869$	−1.03908e24	−3.19516
$870$	2.33803e22	0.0712355
$871$	0	0
$872$	0	0
$873$	−3.81449e22	−0.113063
$874$	0	0
$875$	−1.09391e23	−0.318358
$876$	6.52597e23	1.88197
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	5.96828e23	1.69003
$879$	6.14814e23	1.72517
$880$	4.64482e22	0.129154
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−1.08894e24	−2.97340
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	4.69295e22	0.124710
$886$	5.41105e23	1.42499
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−1.45338e24	−3.72533
$890$	0	0
$891$	6.40366e23	1.61216
$892$	−2.95715e23	−0.737826
$893$	0	0
$894$	2.31079e23	0.566318
$895$	−5.03861e22	−0.122385
$896$	−8.28028e23	−1.99334
$897$	0	0
$898$	0	0
$899$	4.06538e23	0.952847
$900$	−4.27704e23	−0.993582
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	2.39806e23	0.528244
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−7.35994e23	−1.59290
$909$	8.69103e23	1.86449
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−5.16855e23	−1.07054
$914$	−2.27901e23	−0.467925
$915$	0	0
$916$	0	0
$917$	1.51035e24	3.02081
$918$	0	0
$919$	−9.98466e23	−1.96250	−0.981249	−	0.192743i	$-0.938262\pi$
$919$	−9.98466e23	−1.96250	−0.981249	+	0.192743i	$0.938262\pi$
$920$	0	0
$921$	0	0
$922$	1.94021e23	0.371536
$923$	0	0
$924$	−1.70751e24	−3.21357
$925$	0	0
$926$	6.24856e23	1.15583
$927$	1.00933e24	1.85095
$928$	−4.89079e23	−0.889192
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	4.80388e22	0.0858478
$931$	0	0
$932$	0	0
$933$	0	0
$934$	1.10157e24	1.90211
$935$	0	0
$936$	0	0
$937$	5.70484e23	0.960122	0.480061	−	0.877235i	$-0.340614\pi$
$937$	5.70484e23	0.960122	0.480061	+	0.877235i	$0.340614\pi$
$938$	0	0
$939$	2.67587e23	0.442731
$940$	0	0
$941$	3.31911e23	0.539889	0.269944	−	0.962876i	$-0.412995\pi$
$941$	3.31911e23	0.539889	0.269944	+	0.962876i	$0.412995\pi$
$942$	0	0
$943$	0	0
$944$	−9.81691e23	−1.55668
$945$	−1.01563e23	−0.159692
$946$	0	0
$947$	−1.26885e24	−1.96159	−0.980797	−	0.195032i	$-0.937519\pi$
$947$	−1.26885e24	−1.96159	−0.980797	+	0.195032i	$0.937519\pi$
$948$	1.29286e24	1.98192
$949$	0	0
$950$	0	0
$951$	−3.61551e23	−0.540412
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	−1.04782e24	−1.52722
$955$	0	0
$956$	0	0
$957$	−1.00855e24	−1.43352
$958$	0	0
$959$	0	0
$960$	−5.77924e22	−0.0801126
$961$	1.07878e23	0.148301
$962$	0	0
$963$	1.45870e24	1.97222
$964$	7.59691e23	1.01864
$965$	1.04743e23	0.139285
$966$	0	0
$967$	4.65255e23	0.608526	0.304263	−	0.952588i	$-0.401590\pi$
$967$	4.65255e23	0.608526	0.304263	+	0.952588i	$0.401590\pi$
$968$	−1.23272e24	−1.59905
$969$	0	0
$970$	7.09900e21	0.00905781
$971$	−9.95001e23	−1.25913	−0.629563	−	0.776949i	$-0.716766\pi$
$971$	−9.95001e23	−1.25913	−0.629563	+	0.776949i	$0.716766\pi$
$972$	−7.96765e23	−1.00000
$973$	0	0
$974$	−1.21223e24	−1.49663
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	2.02658e23	0.238207
$981$	0	0
$982$	−1.14012e24	−1.31843
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	7.94339e22	0.0896430
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	−1.19176e23	−0.129154
$991$	1.84911e24	1.98781	0.993904	−	0.110245i	$-0.0351635\pi$
$991$	1.84911e24	1.98781	0.993904	+	0.110245i	$0.0351635\pi$
$992$	−1.00490e24	−1.07159
$993$	0	0
$994$	0	0
$995$	−6.74782e22	−0.0702390
$996$	6.43087e23	0.664042
$997$	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	24.17.h.a.5.1	✓	1
3.2	odd	2		24.17.h.b.5.1	yes	1
4.3	odd	2		96.17.h.b.17.1		1
8.3	odd	2		96.17.h.a.17.1		1
8.5	even	2		24.17.h.b.5.1	yes	1
12.11	even	2		96.17.h.a.17.1		1
24.5	odd	2	CM	24.17.h.a.5.1	✓	1
24.11	even	2		96.17.h.b.17.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.17.h.a.5.1	✓	1	1.1	even	1	trivial
24.17.h.a.5.1	✓	1	24.5	odd	2	CM
24.17.h.b.5.1	yes	1	3.2	odd	2
24.17.h.b.5.1	yes	1	8.5	even	2
96.17.h.a.17.1		1	8.3	odd	2
96.17.h.a.17.1		1	12.11	even	2
96.17.h.b.17.1		1	4.3	odd	2
96.17.h.b.17.1		1	24.11	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:	\( k \)	\(=\)	\( 17 \)
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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