Embedded newform 304.2.x.c.179.27

• Label: 304.2.x.c.179.27
• Level: $304$
• Weight: $2$
• Character: 304.179
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	304.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.42745222145\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			179.27
Character	\(\chi\)	\(=\)	304.179
Dual form			304.2.x.c.107.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.924019 + 1.07060i) q^{2} +(2.13514 - 0.572110i) q^{3} +(-0.292376 + 1.97851i) q^{4} +(0.404588 + 1.50994i) q^{5} +(2.58542 + 1.75725i) q^{6} -1.53949 q^{7} +(-2.38836 + 1.51517i) q^{8} +(1.63345 - 0.943074i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 6 q^{2} - 12 q^{3} - 2 q^{4} + 2 q^{5} - 2 q^{6} - 8 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.924019	+	1.07060i	0.653380	+	0.757030i
							\(3\)	2.13514	−	0.572110i	1.23273	−	0.330308i	0.417085	−	0.908867i	\(-0.363052\pi\)
0.815640	+	0.578559i	\(0.196385\pi\)
\(5\)	0.404588	+	1.50994i	0.180937	+	0.675267i	0.995464	+	0.0951410i	\(0.0303302\pi\)
							−0.814526	+	0.580126i	\(0.803003\pi\)
\(7\)	−1.53949			−0.581871			−0.290936	−	0.956743i	\(-0.593967\pi\)
							−0.290936	+	0.956743i	\(0.593967\pi\)
\(11\)	−0.933623	−	0.933623i	−0.281498	−	0.281498i	0.552208	−	0.833706i	\(-0.313785\pi\)
							−0.833706	+	0.552208i	\(0.813785\pi\)
\(13\)	1.45016	+	0.388568i	0.402201	+	0.107769i	0.454248	−	0.890875i	\(-0.349908\pi\)
							−0.0520471	+	0.998645i	\(0.516575\pi\)
\(17\)	3.04826	−	5.27974i	0.739312	−	1.28053i	−0.213494	−	0.976944i	\(-0.568484\pi\)
							0.952806	−	0.303581i	\(-0.0981824\pi\)
\(19\)	0.0424658	−	4.35869i	0.00974231	−	0.999953i
							\(23\)	−0.626603	−	1.08531i	−0.130656	−	0.226303i	0.793274	−	0.608865i	\(-0.208375\pi\)
−0.923930	+	0.382563i	\(0.875042\pi\)
\(29\)	2.45618	+	0.658130i	0.456100	+	0.122212i	0.479552	−	0.877513i	\(-0.340799\pi\)
							−0.0234521	+	0.999725i	\(0.507466\pi\)
\(31\)	3.48231			0.625442			0.312721	−	0.949845i	\(-0.398759\pi\)
							0.312721	+	0.949845i	\(0.398759\pi\)
\(37\)	−0.806686	−	0.806686i	−0.132618	−	0.132618i	0.637682	−	0.770300i	\(-0.279894\pi\)
							−0.770300	+	0.637682i	\(0.779894\pi\)
\(41\)	−2.68451	+	4.64972i	−0.419251	+	0.726164i	−0.995864	−	0.0908535i	\(-0.971041\pi\)
							0.576614	+	0.817017i	\(0.304374\pi\)
\(43\)	−4.03300	+	1.08064i	−0.615027	+	0.164796i	−0.552866	−	0.833270i	\(-0.686466\pi\)
							−0.0621611	+	0.998066i	\(0.519799\pi\)
\(47\)	−4.49319	+	2.59415i	−0.655400	+	0.378395i	−0.790522	−	0.612434i	\(-0.790191\pi\)
							0.135122	+	0.990829i	\(0.456857\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.x.c.179.27	yes	144
16.11	odd	4	inner	304.2.x.c.27.35	✓	144
19.12	odd	6	inner	304.2.x.c.259.35	yes	144
304.107	even	12	inner	304.2.x.c.107.27	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.x.c.27.35	✓	144	16.11	odd	4	inner
304.2.x.c.107.27	yes	144	304.107	even	12	inner
304.2.x.c.179.27	yes	144	1.1	even	1	trivial
304.2.x.c.259.35	yes	144	19.12	odd	6	inner