Embedded newform 304.2.v.a.197.37

• Label: 304.2.v.a.197.37
• 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			197.37
Character	\(\chi\)	\(=\)	304.197
Dual form			304.2.v.a.125.37

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38517 - 0.285137i) q^{2} +(-2.03562 - 0.545442i) q^{3} +(1.83739 - 0.789927i) q^{4} +(-1.08631 + 4.05417i) q^{5} +(-2.97520 - 0.175100i) q^{6} +1.60823i q^{7} +(2.31987 - 1.61809i) q^{8} +(1.24815 + 0.720620i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 20 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38517	−	0.285137i	0.979463	−	0.201622i
							\(3\)	−2.03562	−	0.545442i	−1.17526	−	0.314911i	−0.382217	−	0.924072i	\(-0.624839\pi\)
−0.793046	+	0.609162i	\(0.791506\pi\)
\(5\)	−1.08631	+	4.05417i	−0.485813	+	1.81308i	0.0905563	+	0.995891i	\(0.471135\pi\)
							−0.576370	+	0.817189i	\(0.695531\pi\)
\(7\)			1.60823i			0.607855i	0.952695	+	0.303927i	\(0.0982980\pi\)
							−0.952695	+	0.303927i	\(0.901702\pi\)
\(11\)	1.21957	+	1.21957i	0.367715	+	0.367715i	0.866643	−	0.498928i	\(-0.166273\pi\)
							−0.498928	+	0.866643i	\(0.666273\pi\)
\(13\)	1.68532	+	6.28972i	0.467425	+	1.74445i	0.648722	+	0.761026i	\(0.275304\pi\)
							−0.181297	+	0.983428i	\(0.558029\pi\)
\(17\)	−0.878586	−	1.52176i	−0.213088	−	0.369080i	0.739591	−	0.673056i	\(-0.235019\pi\)
							−0.952680	+	0.303977i	\(0.901686\pi\)
\(19\)	1.10745	−	4.21587i	0.254066	−	0.967187i
							\(23\)	−0.604119	−	0.348788i	−0.125968	−	0.0727274i	0.435692	−	0.900096i	\(-0.356504\pi\)
−0.561660	+	0.827368i	\(0.689837\pi\)
\(29\)	1.09439	+	4.08432i	0.203223	+	0.758440i	0.989984	+	0.141181i	\(0.0450901\pi\)
							−0.786760	+	0.617259i	\(0.788243\pi\)
\(31\)	−2.02990			−0.364581			−0.182291	−	0.983245i	\(-0.558351\pi\)
							−0.182291	+	0.983245i	\(0.558351\pi\)
\(37\)	1.71034	+	1.71034i	0.281179	+	0.281179i	0.833579	−	0.552400i	\(-0.186288\pi\)
							−0.552400	+	0.833579i	\(0.686288\pi\)
\(41\)	0.726513	−	0.419453i	0.113462	−	0.0655075i	−0.442195	−	0.896919i	\(-0.645800\pi\)
							0.555657	+	0.831411i	\(0.312467\pi\)
\(43\)	2.90860	−	10.8550i	0.443558	−	1.65538i	−0.276159	−	0.961112i	\(-0.589062\pi\)
							0.719717	−	0.694268i	\(-0.244272\pi\)
\(47\)	−2.23888	+	3.87786i	−0.326575	+	0.565644i	−0.981830	−	0.189764i	\(-0.939228\pi\)
							0.655255	+	0.755408i	\(0.272561\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.v.a.197.37	yes	152
16.13	even	4	inner	304.2.v.a.45.11	✓	152
19.11	even	3	inner	304.2.v.a.277.11	yes	152
304.125	even	12	inner	304.2.v.a.125.37	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.v.a.45.11	✓	152	16.13	even	4	inner
304.2.v.a.125.37	yes	152	304.125	even	12	inner
304.2.v.a.197.37	yes	152	1.1	even	1	trivial
304.2.v.a.277.11	yes	152	19.11	even	3	inner