Embedded newform 304.2.v.a.197.24

• Label: 304.2.v.a.197.24
• Level: $304$
• Weight: $2$
• Character: 304.197
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $152$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.42745222145\)
Analytic rank:	\(0\)
Dimension:	\(152\)
Relative dimension:	\(38\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			197.24
Character	\(\chi\)	\(=\)	304.197
Dual form			304.2.v.a.125.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.521485 - 1.31455i) q^{2} +(-2.39403 - 0.641478i) q^{3} +(-1.45611 - 1.37104i) q^{4} +(-0.255301 + 0.952796i) q^{5} +(-2.09171 + 2.81256i) q^{6} +3.66664i q^{7} +(-2.56165 + 1.19915i) q^{8} +(2.72180 + 1.57143i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 20 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.521485	−	1.31455i	0.368746	−	0.929530i
							\(3\)	−2.39403	−	0.641478i	−1.38219	−	0.370357i	−0.510275	−	0.860011i	\(-0.670457\pi\)
−0.871917	+	0.489654i	\(0.837123\pi\)
\(5\)	−0.255301	+	0.952796i	−0.114174	+	0.426103i	−0.999224	−	0.0393932i	\(-0.987458\pi\)
							0.885050	+	0.465496i	\(0.154124\pi\)
\(7\)			3.66664i			1.38586i	0.721005	+	0.692930i	\(0.243680\pi\)
							−0.721005	+	0.692930i	\(0.756320\pi\)
\(11\)	−0.283971	−	0.283971i	−0.0856206	−	0.0856206i	0.662999	−	0.748620i	\(-0.269283\pi\)
							−0.748620	+	0.662999i	\(0.769283\pi\)
\(13\)	−0.352288	−	1.31476i	−0.0977072	−	0.364648i	0.899709	−	0.436491i	\(-0.143779\pi\)
							−0.997416	+	0.0718424i	\(0.977112\pi\)
\(17\)	2.74713	+	4.75816i	0.666276	+	1.15402i	0.978938	+	0.204159i	\(0.0654460\pi\)
							−0.312662	+	0.949864i	\(0.601221\pi\)
\(19\)	−4.04524	+	1.62359i	−0.928042	+	0.372477i
							\(23\)	−1.82644	−	1.05449i	−0.380839	−	0.219877i	0.297344	−	0.954770i	\(-0.403899\pi\)
−0.678183	+	0.734893i	\(0.737232\pi\)
\(29\)	0.440216	+	1.64291i	0.0817461	+	0.305081i	0.994678	−	0.103031i	\(-0.0328541\pi\)
							−0.912932	+	0.408112i	\(0.866187\pi\)
\(31\)	−9.23863			−1.65931			−0.829653	−	0.558279i	\(-0.811462\pi\)
							−0.829653	+	0.558279i	\(0.811462\pi\)
\(37\)	−3.30967	−	3.30967i	−0.544106	−	0.544106i	0.380624	−	0.924730i	\(-0.375709\pi\)
							−0.924730	+	0.380624i	\(0.875709\pi\)
\(41\)	−5.57470	+	3.21856i	−0.870622	+	0.502654i	−0.867555	−	0.497341i	\(-0.834310\pi\)
							−0.00306717	+	0.999995i	\(0.500976\pi\)
\(43\)	−2.67772	+	9.99339i	−0.408349	+	1.52398i	0.389447	+	0.921049i	\(0.372666\pi\)
							−0.797795	+	0.602928i	\(0.794000\pi\)
\(47\)	−1.78656	+	3.09442i	−0.260597	+	0.451367i	−0.966401	−	0.257040i	\(-0.917253\pi\)
							0.705804	+	0.708407i	\(0.250586\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.v.a.197.24	yes	152
16.13	even	4	inner	304.2.v.a.45.3	✓	152
19.11	even	3	inner	304.2.v.a.277.3	yes	152
304.125	even	12	inner	304.2.v.a.125.24	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.v.a.45.3	✓	152	16.13	even	4	inner
304.2.v.a.125.24	yes	152	304.125	even	12	inner
304.2.v.a.197.24	yes	152	1.1	even	1	trivial
304.2.v.a.277.3	yes	152	19.11	even	3	inner