Embedded newform 6498.2.a.bp.1.2

• Label: 6498.2.a.bp.1.2
• Level: $6498$
• Weight: $2$
• Character: 6498.1
• Self dual: yes
• Analytic conductor: $51.887$
• Analytic rank: $0$
• Dimension: $3$
• CM: no
• Inner twists: $1$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	\(51.8867912334\)
Analytic rank:	\(0\)
Dimension:	\(3\)
Coefficient field:	\(\Q(\zeta_{18})^+\)
Defining polynomial:	\( x^{3} - 3x - 1 \)
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

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

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−0.707107
			\(3\)	0	0
\(5\)	2.34730	1.04974				0.524871	−	0.851182i	\(-0.324113\pi\)
			0.524871	+	0.851182i	\(0.324113\pi\)
\(7\)	3.57398	1.35084	0.675418	−	0.737435i	\(-0.263963\pi\)
			0.675418	+	0.737435i	\(0.263963\pi\)
\(11\)	−2.71688	−0.819171	−0.409585	−	0.912272i	\(-0.634327\pi\)
			−0.409585	+	0.912272i	\(0.634327\pi\)
\(13\)	5.41147	1.50087	0.750436	−	0.660943i	\(-0.229843\pi\)
			0.750436	+	0.660943i	\(0.229843\pi\)
\(17\)	3.87939	0.940889	0.470445	−	0.882430i	\(-0.344094\pi\)
			0.470445	+	0.882430i	\(0.344094\pi\)
\(19\)	0	0
			\(23\)	8.23442	1.71700	0.858498	−	0.512817i	\(-0.171398\pi\)
0.858498	+	0.512817i				\(0.171398\pi\)
\(29\)	−3.53209	−0.655892	−0.327946	−	0.944696i	\(-0.606356\pi\)
			−0.327946	+	0.944696i	\(0.606356\pi\)
\(31\)	6.53209	1.17320	0.586599	−	0.809878i	\(-0.300467\pi\)
			0.586599	+	0.809878i	\(0.300467\pi\)
\(37\)	0.389185	0.0639817	0.0319908	−	0.999488i	\(-0.489815\pi\)
			0.0319908	+	0.999488i	\(0.489815\pi\)
\(41\)	1.94356	0.303534	0.151767	−	0.988416i	\(-0.451504\pi\)
			0.151767	+	0.988416i	\(0.451504\pi\)
\(43\)	5.02229	0.765892	0.382946	−	0.923771i	\(-0.374910\pi\)
			0.382946	+	0.923771i	\(0.374910\pi\)
\(47\)	−2.98545	−0.435473	−0.217736	−	0.976008i	\(-0.569867\pi\)
			−0.217736	+	0.976008i	\(0.569867\pi\)
\(53\)	−8.30541	−1.14084	−0.570418	−	0.821355i	\(-0.693219\pi\)
			−0.570418	+	0.821355i	\(0.693219\pi\)
\(59\)	−2.73143	−0.355602	−0.177801	−	0.984066i	\(-0.556898\pi\)
			−0.177801	+	0.984066i	\(0.556898\pi\)
\(61\)	6.29086	0.805462	0.402731	−	0.915318i	\(-0.368061\pi\)
			0.402731	+	0.915318i	\(0.368061\pi\)
\(67\)	−14.9368	−1.82482	−0.912408	−	0.409283i	\(-0.865779\pi\)
			−0.912408	+	0.409283i	\(0.865779\pi\)
\(71\)	−9.02229	−1.07075	−0.535374	−	0.844615i	\(-0.679829\pi\)
			−0.535374	+	0.844615i	\(0.679829\pi\)
\(73\)	10.2909	1.20445	0.602227	−	0.798325i	\(-0.294280\pi\)
			0.602227	+	0.798325i	\(0.294280\pi\)
\(79\)	13.0077	1.46349	0.731743	−	0.681581i	\(-0.238707\pi\)
			0.731743	+	0.681581i	\(0.238707\pi\)
\(83\)	8.17024	0.896801	0.448400	−	0.893833i	\(-0.351994\pi\)
			0.448400	+	0.893833i	\(0.351994\pi\)
\(89\)	−11.7246	−1.24281	−0.621404	−	0.783491i	\(-0.713437\pi\)
			−0.621404	+	0.783491i	\(0.713437\pi\)
\(97\)	−8.60401	−0.873605	−0.436802	−	0.899558i	\(-0.643889\pi\)
			−0.436802	+	0.899558i	\(0.643889\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bp.1.2		3
3.2	odd	2		2166.2.a.r.1.2		3
19.6	even	9		342.2.u.b.55.1		6
19.16	even	9		342.2.u.b.199.1		6
19.18	odd	2		6498.2.a.bu.1.2		3
57.35	odd	18		114.2.i.c.85.1	yes	6
57.44	odd	18		114.2.i.c.55.1	✓	6
57.56	even	2		2166.2.a.p.1.2		3
228.35	even	18		912.2.bo.d.769.1		6
228.215	even	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	57.44	odd	18
114.2.i.c.85.1	yes	6	57.35	odd	18
342.2.u.b.55.1		6	19.6	even	9
342.2.u.b.199.1		6	19.16	even	9
912.2.bo.d.625.1		6	228.215	even	18
912.2.bo.d.769.1		6	228.35	even	18
2166.2.a.p.1.2		3	57.56	even	2
2166.2.a.r.1.2		3	3.2	odd	2
6498.2.a.bp.1.2		3	1.1	even	1	trivial
6498.2.a.bu.1.2		3	19.18	odd	2