Embedded newform 456.2.u.d.11.49

• Label: 456.2.u.d.11.49
• Level: $456$
• Weight: $2$
• Character: 456.11
• Analytic conductor: $3.641$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	456.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.64117833217\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(68\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.49
Character	\(\chi\)	\(=\)	456.11
Dual form			456.2.u.d.83.49

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.954387 - 1.04362i) q^{2} +(1.61297 + 0.631141i) q^{3} +(-0.178291 - 1.99204i) q^{4} +(1.99273 - 3.45151i) q^{5} +(2.19807 - 1.08097i) q^{6} +0.938475i q^{7} +(-2.24909 - 1.71511i) q^{8} +(2.20332 + 2.03602i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 2 q^{3} - 4 q^{4} - 9 q^{6} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/456\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(229\)	\(305\)	\(343\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(-1\)	\(-1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.954387	−	1.04362i	0.674853	−	0.737952i
							\(3\)	1.61297	+	0.631141i	0.931247	+	0.364390i
\(5\)	1.99273	−	3.45151i	0.891176	−	1.54356i								0.0527086	−	0.998610i	\(-0.483215\pi\)
							0.838467	−	0.544952i	\(-0.183452\pi\)
\(7\)			0.938475i			0.354710i	0.984147	+	0.177355i	\(0.0567541\pi\)
							−0.984147	+	0.177355i	\(0.943246\pi\)
\(11\)			4.53299i			1.36675i	0.730069	+	0.683374i	\(0.239488\pi\)
							−0.730069	+	0.683374i	\(0.760512\pi\)
\(13\)	−1.90481	+	1.09974i	−0.528298	+	0.305013i	−0.740323	−	0.672251i	\(-0.765328\pi\)
							0.212025	+	0.977264i	\(0.431994\pi\)
\(17\)	−3.59371	−	2.07483i	−0.871602	−	0.503219i	−0.00372152	−	0.999993i	\(-0.501185\pi\)
							−0.867880	+	0.496774i	\(0.834518\pi\)
\(19\)	−3.77512	+	2.17911i	−0.866071	+	0.499921i
							\(23\)	2.04022	+	3.53376i	0.425414	+	0.736839i	0.996459	−	0.0840797i	\(-0.0267950\pi\)
−0.571045	+	0.820919i	\(0.693462\pi\)
\(29\)	2.79078	+	4.83378i	0.518235	+	0.897610i	0.999776	+	0.0211857i	\(0.00674412\pi\)
							−0.481540	+	0.876424i	\(0.659923\pi\)
\(31\)		−	3.09951i		−	0.556689i	−0.960481	−	0.278345i	\(-0.910214\pi\)
							0.960481	−	0.278345i	\(-0.0897857\pi\)
\(37\)		−	9.54582i		−	1.56932i	−0.619924	−	0.784662i	\(-0.712837\pi\)
							0.619924	−	0.784662i	\(-0.287163\pi\)
\(41\)	9.86165	+	5.69363i	1.54013	+	0.889195i	0.998830	+	0.0483668i	\(0.0154016\pi\)
							0.541302	+	0.840828i	\(0.317932\pi\)
\(43\)	2.21581	−	3.83790i	0.337908	−	0.585274i	−0.646131	−	0.763227i	\(-0.723614\pi\)
							0.984039	+	0.177952i	\(0.0569473\pi\)
\(47\)	−1.34559	−	2.33064i	−0.196275	−	0.339958i	0.751043	−	0.660254i	\(-0.229551\pi\)
							−0.947318	+	0.320295i	\(0.896218\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.u.d.11.49	yes	136
3.2	odd	2	inner	456.2.u.d.11.20	✓	136
8.3	odd	2	inner	456.2.u.d.11.26	yes	136
19.7	even	3	inner	456.2.u.d.83.43	yes	136
24.11	even	2	inner	456.2.u.d.11.43	yes	136
57.26	odd	6	inner	456.2.u.d.83.26	yes	136
152.83	odd	6	inner	456.2.u.d.83.20	yes	136
456.83	even	6	inner	456.2.u.d.83.49	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.u.d.11.20	✓	136	3.2	odd	2	inner
456.2.u.d.11.26	yes	136	8.3	odd	2	inner
456.2.u.d.11.43	yes	136	24.11	even	2	inner
456.2.u.d.11.49	yes	136	1.1	even	1	trivial
456.2.u.d.83.20	yes	136	152.83	odd	6	inner
456.2.u.d.83.26	yes	136	57.26	odd	6	inner
456.2.u.d.83.43	yes	136	19.7	even	3	inner
456.2.u.d.83.49	yes	136	456.83	even	6	inner