Embedded newform 304.2.x.c.259.19

• Label: 304.2.x.c.259.19
• Level: $304$
• Weight: $2$
• Character: 304.259
• 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			259.19
Character	\(\chi\)	\(=\)	304.259
Dual form			304.2.x.c.27.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.103117 - 1.41045i) q^{2} +(-0.0465071 + 0.173567i) q^{3} +(-1.97873 - 0.290882i) q^{4} +(2.65024 + 0.710129i) q^{5} +(0.240011 + 0.0834935i) q^{6} +1.52112 q^{7} +(-0.614315 + 2.76091i) q^{8} +(2.57011 + 1.48386i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 6 q^{2} - 12 q^{3} - 2 q^{4} + 2 q^{5} - 2 q^{6} - 8 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.103117	−	1.41045i	0.0729145	−	0.997338i
							\(3\)	−0.0465071	+	0.173567i	−0.0268509	+	0.100209i	−0.978051	−	0.208366i	\(-0.933185\pi\)
0.951200	+	0.308575i	\(0.0998521\pi\)
\(5\)	2.65024	+	0.710129i	1.18522	+	0.317579i	0.796996	−	0.603985i	\(-0.206421\pi\)
							0.388226	+	0.921564i	\(0.373088\pi\)
\(7\)	1.52112			0.574931			0.287466	−	0.957791i	\(-0.407187\pi\)
							0.287466	+	0.957791i	\(0.407187\pi\)
\(11\)	−0.988586	−	0.988586i	−0.298070	−	0.298070i	0.542188	−	0.840258i	\(-0.317596\pi\)
							−0.840258	+	0.542188i	\(0.817596\pi\)
\(13\)	0.507893	+	1.89548i	0.140864	+	0.525712i	0.999905	+	0.0138046i	\(0.00439428\pi\)
							−0.859041	+	0.511908i	\(0.828939\pi\)
\(17\)	−0.914341	−	1.58369i	−0.221760	−	0.384100i	0.733582	−	0.679601i	\(-0.237847\pi\)
							−0.955343	+	0.295501i	\(0.904514\pi\)
\(19\)	−0.426907	−	4.33794i	−0.0979392	−	0.995192i
							\(23\)	1.63532	−	2.83245i	0.340987	−	0.590607i	−0.643629	−	0.765338i	\(-0.722572\pi\)
0.984616	+	0.174730i	\(0.0559053\pi\)
\(29\)	−0.143901	−	0.537046i	−0.0267217	−	0.0997269i	0.951277	−	0.308337i	\(-0.0997726\pi\)
							−0.977999	+	0.208611i	\(0.933106\pi\)
\(31\)	−4.51679			−0.811240			−0.405620	−	0.914042i	\(-0.632944\pi\)
							−0.405620	+	0.914042i	\(0.632944\pi\)
\(37\)	−3.73898	−	3.73898i	−0.614684	−	0.614684i	0.329479	−	0.944163i	\(-0.393127\pi\)
							−0.944163	+	0.329479i	\(0.893127\pi\)
\(41\)	6.10731	+	10.5782i	0.953802	+	1.65203i	0.737087	+	0.675798i	\(0.236201\pi\)
							0.216715	+	0.976235i	\(0.430466\pi\)
\(43\)	−1.63952	+	6.11876i	−0.250024	+	0.933102i	0.720767	+	0.693177i	\(0.243790\pi\)
							−0.970791	+	0.239925i	\(0.922877\pi\)
\(47\)	2.37955	+	1.37384i	0.347093	+	0.200394i	0.663404	−	0.748261i	\(-0.269111\pi\)
							−0.316311	+	0.948656i	\(0.602444\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.x.c.259.19	yes	144
16.11	odd	4	inner	304.2.x.c.107.32	yes	144
19.8	odd	6	inner	304.2.x.c.179.32	yes	144
304.27	even	12	inner	304.2.x.c.27.19	✓	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.x.c.27.19	✓	144	304.27	even	12	inner
304.2.x.c.107.32	yes	144	16.11	odd	4	inner
304.2.x.c.179.32	yes	144	19.8	odd	6	inner
304.2.x.c.259.19	yes	144	1.1	even	1	trivial