LMFDB - Embedded newform 6498.2.a.bu.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6498,2,Mod(1,6498)] 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("6498.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6498, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6498 = 2 \cdot 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6498.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,0,3,6,0,3,3,0,6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$51.8867912334$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	6498.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.34730 q^{5} +3.57398 q^{7} +1.00000 q^{8} +2.34730 q^{10} -2.71688 q^{11} -5.41147 q^{13} +3.57398 q^{14} +1.00000 q^{16} +3.87939 q^{17} +2.34730 q^{20} -2.71688 q^{22} +8.23442 q^{23} +0.509800 q^{25} -5.41147 q^{26} +3.57398 q^{28} +3.53209 q^{29} -6.53209 q^{31} +1.00000 q^{32} +3.87939 q^{34} +8.38919 q^{35} -0.389185 q^{37} +2.34730 q^{40} -1.94356 q^{41} +5.02229 q^{43} -2.71688 q^{44} +8.23442 q^{46} -2.98545 q^{47} +5.77332 q^{49} +0.509800 q^{50} -5.41147 q^{52} +8.30541 q^{53} -6.37733 q^{55} +3.57398 q^{56} +3.53209 q^{58} +2.73143 q^{59} +6.29086 q^{61} -6.53209 q^{62} +1.00000 q^{64} -12.7023 q^{65} +14.9368 q^{67} +3.87939 q^{68} +8.38919 q^{70} +9.02229 q^{71} +10.2909 q^{73} -0.389185 q^{74} -9.71007 q^{77} -13.0077 q^{79} +2.34730 q^{80} -1.94356 q^{82} +8.17024 q^{83} +9.10607 q^{85} +5.02229 q^{86} -2.71688 q^{88} +11.7246 q^{89} -19.3405 q^{91} +8.23442 q^{92} -2.98545 q^{94} +8.60401 q^{97} +5.77332 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8} + 6 q^{10} - 6 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} + 6 q^{20} - 6 q^{23} + 3 q^{25} - 6 q^{26} + 3 q^{28} + 6 q^{29} - 15 q^{31} + 3 q^{32} + 6 q^{34}+ \cdots + 24 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8 + 6 * q^10 - 6 * q^13 + 3 * q^14 + 3 * q^16 + 6 * q^17 + 6 * q^20 - 6 * q^23 + 3 * q^25 - 6 * q^26 + 3 * q^28 + 6 * q^29 - 15 * q^31 + 3 * q^32 + 6 * q^34 + 21 * q^35 + 3 * q^37 + 6 * q^40 + 9 * q^41 + 9 * q^43 - 6 * q^46 + 9 * q^47 + 24 * q^49 + 3 * q^50 - 6 * q^52 + 27 * q^53 + 12 * q^55 + 3 * q^56 + 6 * q^58 + 18 * q^59 + 3 * q^61 - 15 * q^62 + 3 * q^64 - 12 * q^65 - 12 * q^67 + 6 * q^68 + 21 * q^70 + 21 * q^71 + 15 * q^73 + 3 * q^74 + 21 * q^77 - 15 * q^79 + 6 * q^80 + 9 * q^82 + 3 * q^83 + 15 * q^85 + 9 * q^86 + 3 * q^89 - 15 * q^91 - 6 * q^92 + 9 * q^94 - 12 * q^97 + 24 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.34730	1.04974	0.524871	−	0.851182i	$-0.324113\pi$
$5$	2.34730	1.04974	0.524871	+	0.851182i	$0.324113\pi$
$6$	0	0
$7$	3.57398	1.35084	0.675418	−	0.737435i	$-0.263963\pi$
$7$	3.57398	1.35084	0.675418	+	0.737435i	$0.263963\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.34730	0.742280
$11$	−2.71688	−0.819171	−0.409585	−	0.912272i	$-0.634327\pi$
$11$	−2.71688	−0.819171	−0.409585	+	0.912272i	$0.634327\pi$
$12$	0	0
$13$	−5.41147	−1.50087	−0.750436	−	0.660943i	$-0.770157\pi$
$13$	−5.41147	−1.50087	−0.750436	+	0.660943i	$0.770157\pi$
$14$	3.57398	0.955186
$15$	0	0
$16$	1.00000	0.250000
$17$	3.87939	0.940889	0.470445	−	0.882430i	$-0.344094\pi$
$17$	3.87939	0.940889	0.470445	+	0.882430i	$0.344094\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	2.34730	0.524871
$21$	0	0
$22$	−2.71688	−0.579241
$23$	8.23442	1.71700	0.858498	−	0.512817i	$-0.171398\pi$
$23$	8.23442	1.71700	0.858498	+	0.512817i	$0.171398\pi$
$24$	0	0
$25$	0.509800	0.101960
$26$	−5.41147	−1.06128
$27$	0	0
$28$	3.57398	0.675418
$29$	3.53209	0.655892	0.327946	−	0.944696i	$-0.393644\pi$
$29$	3.53209	0.655892	0.327946	+	0.944696i	$0.393644\pi$
$30$	0	0
$31$	−6.53209	−1.17320	−0.586599	−	0.809878i	$-0.699533\pi$
$31$	−6.53209	−1.17320	−0.586599	+	0.809878i	$0.699533\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.87939	0.665309
$35$	8.38919	1.41803
$36$	0	0
$37$	−0.389185	−0.0639817	−0.0319908	−	0.999488i	$-0.510185\pi$
$37$	−0.389185	−0.0639817	−0.0319908	+	0.999488i	$0.510185\pi$
$38$	0	0
$39$	0	0
$40$	2.34730	0.371140
$41$	−1.94356	−0.303534	−0.151767	−	0.988416i	$-0.548496\pi$
$41$	−1.94356	−0.303534	−0.151767	+	0.988416i	$0.548496\pi$
$42$	0	0
$43$	5.02229	0.765892	0.382946	−	0.923771i	$-0.374910\pi$
$43$	5.02229	0.765892	0.382946	+	0.923771i	$0.374910\pi$
$44$	−2.71688	−0.409585
$45$	0	0
$46$	8.23442	1.21410
$47$	−2.98545	−0.435473	−0.217736	−	0.976008i	$-0.569867\pi$
$47$	−2.98545	−0.435473	−0.217736	+	0.976008i	$0.569867\pi$
$48$	0	0
$49$	5.77332	0.824760
$50$	0.509800	0.0720966
$51$	0	0
$52$	−5.41147	−0.750436
$53$	8.30541	1.14084	0.570418	−	0.821355i	$-0.306781\pi$
$53$	8.30541	1.14084	0.570418	+	0.821355i	$0.306781\pi$
$54$	0	0
$55$	−6.37733	−0.859918
$56$	3.57398	0.477593
$57$	0	0
$58$	3.53209	0.463786
$59$	2.73143	0.355602	0.177801	−	0.984066i	$-0.443102\pi$
$59$	2.73143	0.355602	0.177801	+	0.984066i	$0.443102\pi$
$60$	0	0
$61$	6.29086	0.805462	0.402731	−	0.915318i	$-0.368061\pi$
$61$	6.29086	0.805462	0.402731	+	0.915318i	$0.368061\pi$
$62$	−6.53209	−0.829576
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.7023	−1.57553
$66$	0	0
$67$	14.9368	1.82482	0.912408	−	0.409283i	$-0.134221\pi$
$67$	14.9368	1.82482	0.912408	+	0.409283i	$0.134221\pi$
$68$	3.87939	0.470445
$69$	0	0
$70$	8.38919	1.00270
$71$	9.02229	1.07075	0.535374	−	0.844615i	$-0.320171\pi$
$71$	9.02229	1.07075	0.535374	+	0.844615i	$0.320171\pi$
$72$	0	0
$73$	10.2909	1.20445	0.602227	−	0.798325i	$-0.294280\pi$
$73$	10.2909	1.20445	0.602227	+	0.798325i	$0.294280\pi$
$74$	−0.389185	−0.0452419
$75$	0	0
$76$	0	0
$77$	−9.71007	−1.10657
$78$	0	0
$79$	−13.0077	−1.46349	−0.731743	−	0.681581i	$-0.761293\pi$
$79$	−13.0077	−1.46349	−0.731743	+	0.681581i	$0.761293\pi$
$80$	2.34730	0.262436
$81$	0	0
$82$	−1.94356	−0.214631
$83$	8.17024	0.896801	0.448400	−	0.893833i	$-0.351994\pi$
$83$	8.17024	0.896801	0.448400	+	0.893833i	$0.351994\pi$
$84$	0	0
$85$	9.10607	0.987692
$86$	5.02229	0.541567
$87$	0	0
$88$	−2.71688	−0.289621
$89$	11.7246	1.24281	0.621404	−	0.783491i	$-0.286563\pi$
$89$	11.7246	1.24281	0.621404	+	0.783491i	$0.286563\pi$
$90$	0	0
$91$	−19.3405	−2.02743
$92$	8.23442	0.858498
$93$	0	0
$94$	−2.98545	−0.307926
$95$	0	0
$96$	0	0
$97$	8.60401	0.873605	0.436802	−	0.899558i	$-0.356111\pi$
$97$	8.60401	0.873605	0.436802	+	0.899558i	$0.356111\pi$
$98$	5.77332	0.583193
$99$	0	0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bu.1.2		3
3.2	odd	2		2166.2.a.p.1.2		3
19.3	odd	18		342.2.u.b.199.1		6
19.13	odd	18		342.2.u.b.55.1		6
19.18	odd	2		6498.2.a.bp.1.2		3
57.32	even	18		114.2.i.c.55.1	✓	6
57.41	even	18		114.2.i.c.85.1	yes	6
57.56	even	2		2166.2.a.r.1.2		3
228.155	odd	18		912.2.bo.d.769.1		6
228.203	odd	18		912.2.bo.d.625.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.c.55.1	✓	6	57.32	even	18
114.2.i.c.85.1	yes	6	57.41	even	18
342.2.u.b.55.1		6	19.13	odd	18
342.2.u.b.199.1		6	19.3	odd	18
912.2.bo.d.625.1		6	228.203	odd	18
912.2.bo.d.769.1		6	228.155	odd	18
2166.2.a.p.1.2		3	3.2	odd	2
2166.2.a.r.1.2		3	57.56	even	2
6498.2.a.bp.1.2		3	19.18	odd	2
6498.2.a.bu.1.2		3	1.1	even	1	trivial

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Level:	\( N \)	\(=\)	\( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6498.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.34730 q^{5} +3.57398 q^{7} +1.00000 q^{8} +2.34730 q^{10} -2.71688 q^{11} -5.41147 q^{13} +3.57398 q^{14} +1.00000 q^{16} +3.87939 q^{17} +2.34730 q^{20} -2.71688 q^{22} +8.23442 q^{23} +0.509800 q^{25} -5.41147 q^{26} +3.57398 q^{28} +3.53209 q^{29} -6.53209 q^{31} +1.00000 q^{32} +3.87939 q^{34} +8.38919 q^{35} -0.389185 q^{37} +2.34730 q^{40} -1.94356 q^{41} +5.02229 q^{43} -2.71688 q^{44} +8.23442 q^{46} -2.98545 q^{47} +5.77332 q^{49} +0.509800 q^{50} -5.41147 q^{52} +8.30541 q^{53} -6.37733 q^{55} +3.57398 q^{56} +3.53209 q^{58} +2.73143 q^{59} +6.29086 q^{61} -6.53209 q^{62} +1.00000 q^{64} -12.7023 q^{65} +14.9368 q^{67} +3.87939 q^{68} +8.38919 q^{70} +9.02229 q^{71} +10.2909 q^{73} -0.389185 q^{74} -9.71007 q^{77} -13.0077 q^{79} +2.34730 q^{80} -1.94356 q^{82} +8.17024 q^{83} +9.10607 q^{85} +5.02229 q^{86} -2.71688 q^{88} +11.7246 q^{89} -19.3405 q^{91} +8.23442 q^{92} -2.98545 q^{94} +8.60401 q^{97} +5.77332 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8} + 6 q^{10} - 6 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} + 6 q^{20} - 6 q^{23} + 3 q^{25} - 6 q^{26} + 3 q^{28} + 6 q^{29} - 15 q^{31} + 3 q^{32} + 6 q^{34}+ \cdots + 24 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8 + 6 * q^10 - 6 * q^13 + 3 * q^14 + 3 * q^16 + 6 * q^17 + 6 * q^20 - 6 * q^23 + 3 * q^25 - 6 * q^26 + 3 * q^28 + 6 * q^29 - 15 * q^31 + 3 * q^32 + 6 * q^34 + 21 * q^35 + 3 * q^37 + 6 * q^40 + 9 * q^41 + 9 * q^43 - 6 * q^46 + 9 * q^47 + 24 * q^49 + 3 * q^50 - 6 * q^52 + 27 * q^53 + 12 * q^55 + 3 * q^56 + 6 * q^58 + 18 * q^59 + 3 * q^61 - 15 * q^62 + 3 * q^64 - 12 * q^65 - 12 * q^67 + 6 * q^68 + 21 * q^70 + 21 * q^71 + 15 * q^73 + 3 * q^74 + 21 * q^77 - 15 * q^79 + 6 * q^80 + 9 * q^82 + 3 * q^83 + 15 * q^85 + 9 * q^86 + 3 * q^89 - 15 * q^91 - 6 * q^92 + 9 * q^94 - 12 * q^97 + 24 * q^98

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