Embedded newform 456.2.u.d.11.66

• Label: 456.2.u.d.11.66
• 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			11.66
Character	\(\chi\)	\(=\)	456.11
Dual form			456.2.u.d.83.66

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39862 - 0.209415i) q^{2} +(-1.72005 - 0.203571i) q^{3} +(1.91229 - 0.585784i) q^{4} +(-1.25093 + 2.16668i) q^{5} +(-2.44833 + 0.0754831i) q^{6} -0.788639i q^{7} +(2.55190 - 1.21975i) q^{8} +(2.91712 + 0.700305i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 2 q^{3} - 4 q^{4} - 9 q^{6} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39862	−	0.209415i	0.988976	−	0.148078i
							\(3\)	−1.72005	−	0.203571i	−0.993069	−	0.117532i
\(5\)	−1.25093	+	2.16668i	−0.559435	+	0.968969i								0.438109	+	0.898922i	\(0.355648\pi\)
							−0.997544	+	0.0700476i	\(0.977685\pi\)
\(7\)		−	0.788639i		−	0.298078i	−0.988831	−	0.149039i	\(-0.952382\pi\)
							0.988831	−	0.149039i	\(-0.0476179\pi\)
\(11\)			5.60321i			1.68943i	0.535215	+	0.844716i	\(0.320231\pi\)
							−0.535215	+	0.844716i	\(0.679769\pi\)
\(13\)	5.46355	−	3.15438i	1.51531	−	0.874867i	0.515476	−	0.856904i	\(-0.327615\pi\)
							0.999839	−	0.0179635i	\(-0.00571828\pi\)
\(17\)	2.43592	+	1.40638i	0.590799	+	0.341098i	0.765413	−	0.643539i	\(-0.222535\pi\)
							−0.174615	+	0.984637i	\(0.555868\pi\)
\(19\)	−2.46826	+	3.59273i	−0.566257	+	0.824229i
							\(23\)	2.30066	+	3.98486i	0.479720	+	0.830900i	0.999729	−	0.0232608i	\(-0.00740481\pi\)
−0.520009	+	0.854161i	\(0.674071\pi\)
\(29\)	−1.39821	−	2.42177i	−0.259641	−	0.449711i	0.706505	−	0.707708i	\(-0.250271\pi\)
							−0.966146	+	0.257997i	\(0.916937\pi\)
\(31\)		−	7.92771i		−	1.42386i	−0.702251	−	0.711929i	\(-0.747822\pi\)
							0.702251	−	0.711929i	\(-0.252178\pi\)
\(37\)			5.33647i			0.877310i	0.898655	+	0.438655i	\(0.144545\pi\)
							−0.898655	+	0.438655i	\(0.855455\pi\)
\(41\)	−6.96017	−	4.01845i	−1.08700	−	0.627577i	−0.154221	−	0.988036i	\(-0.549287\pi\)
							−0.932775	+	0.360459i	\(0.882620\pi\)
\(43\)	−1.39455	+	2.41544i	−0.212668	+	0.368351i	−0.952549	−	0.304387i	\(-0.901548\pi\)
							0.739881	+	0.672738i	\(0.234882\pi\)
\(47\)	−2.96473	−	5.13506i	−0.432450	−	0.749025i	0.564634	−	0.825341i	\(-0.309017\pi\)
							−0.997084	+	0.0763167i	\(0.975684\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.u.d.11.66	yes	136
3.2	odd	2	inner	456.2.u.d.11.3	✓	136
8.3	odd	2	inner	456.2.u.d.11.45	yes	136
19.7	even	3	inner	456.2.u.d.83.24	yes	136
24.11	even	2	inner	456.2.u.d.11.24	yes	136
57.26	odd	6	inner	456.2.u.d.83.45	yes	136
152.83	odd	6	inner	456.2.u.d.83.3	yes	136
456.83	even	6	inner	456.2.u.d.83.66	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.u.d.11.3	✓	136	3.2	odd	2	inner
456.2.u.d.11.24	yes	136	24.11	even	2	inner
456.2.u.d.11.45	yes	136	8.3	odd	2	inner
456.2.u.d.11.66	yes	136	1.1	even	1	trivial
456.2.u.d.83.3	yes	136	152.83	odd	6	inner
456.2.u.d.83.24	yes	136	19.7	even	3	inner
456.2.u.d.83.45	yes	136	57.26	odd	6	inner
456.2.u.d.83.66	yes	136	456.83	even	6	inner