Embedded newform 304.2.bg.a.3.1

• Label: 304.2.bg.a.3.1
• Level: $304$
• Weight: $2$
• Character: 304.3
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $456$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	304.bg (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.42745222145\)
Analytic rank:	\(0\)
Dimension:	\(456\)
Relative dimension:	\(38\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			3.1
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41273 + 0.0647022i) q^{2} +(0.459790 + 0.214404i) q^{3} +(1.99163 - 0.182814i) q^{4} +(-1.87793 - 2.68196i) q^{5} +(-0.663433 - 0.273146i) q^{6} +(0.233325 + 0.404130i) q^{7} +(-2.80181 + 0.387130i) q^{8} +(-1.76292 - 2.10097i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41273	+	0.0647022i	−0.998953	+	0.0457514i
							\(3\)	0.459790	+	0.214404i	0.265460	+	0.123786i	0.550792	−	0.834643i	\(-0.314326\pi\)
−0.285331	+	0.958429i	\(0.592104\pi\)
\(5\)	−1.87793	−	2.68196i	−0.839834	−	1.19941i	−0.977918	−	0.208987i	\(-0.932983\pi\)
							0.138084	−	0.990420i	\(-0.455905\pi\)
\(7\)	0.233325	+	0.404130i	0.0881885	+	0.152747i	0.906745	−	0.421678i	\(-0.138559\pi\)
							−0.818557	+	0.574425i	\(0.805226\pi\)
\(11\)	−1.01331	+	0.271517i	−0.305526	+	0.0818654i	−0.408325	−	0.912837i	\(-0.633887\pi\)
							0.102799	+	0.994702i	\(0.467220\pi\)
\(13\)	−5.64008	+	2.63001i	−1.56428	+	0.729434i	−0.995651	−	0.0931627i	\(-0.970302\pi\)
							−0.568625	+	0.822597i	\(0.692525\pi\)
\(17\)	0.0648918	+	0.0544507i	0.0157386	+	0.0132062i	0.650623	−	0.759401i	\(-0.274508\pi\)
							−0.634884	+	0.772607i	\(0.718952\pi\)
\(19\)	−1.32430	+	4.15286i	−0.303816	+	0.952731i
							\(23\)	−1.21059	−	6.86560i	−0.252426	−	1.43158i	−0.802596	−	0.596524i	\(-0.796548\pi\)
0.550170	−	0.835053i	\(-0.314563\pi\)
\(29\)	0.0201044	−	0.229794i	0.00373330	−	0.0426718i	−0.994100	−	0.108472i	\(-0.965404\pi\)
							0.997833	+	0.0657999i	\(0.0209599\pi\)
\(31\)	−2.25707	−	3.90937i	−0.405383	−	0.702143i	0.588983	−	0.808145i	\(-0.299528\pi\)
							−0.994366	+	0.106002i	\(0.966195\pi\)
\(37\)	−1.70665	−	1.70665i	−0.280571	−	0.280571i	0.552766	−	0.833337i	\(-0.313572\pi\)
							−0.833337	+	0.552766i	\(0.813572\pi\)
\(41\)	1.18135	−	0.429976i	0.184496	−	0.0671509i	−0.248120	−	0.968729i	\(-0.579813\pi\)
							0.432616	+	0.901578i	\(0.357591\pi\)
\(43\)	1.69059	−	1.18377i	0.257813	−	0.180523i	−0.437523	−	0.899207i	\(-0.644144\pi\)
							0.695336	+	0.718684i	\(0.255255\pi\)
\(47\)	−6.35508	−	7.57369i	−0.926984	−	1.10474i	−0.994259	−	0.107002i	\(-0.965875\pi\)
							0.0672748	−	0.997734i	\(-0.478570\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.1	✓	456
16.11	odd	4	inner	304.2.bg.a.155.21	yes	456
19.13	odd	18	inner	304.2.bg.a.51.21	yes	456
304.203	even	36	inner	304.2.bg.a.203.1	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.1	✓	456	1.1	even	1	trivial
304.2.bg.a.51.21	yes	456	19.13	odd	18	inner
304.2.bg.a.155.21	yes	456	16.11	odd	4	inner
304.2.bg.a.203.1	yes	456	304.203	even	36	inner