Embedded newform 304.2.v.a.125.26

• Label: 304.2.v.a.125.26
• Level: $304$
• Weight: $2$
• Character: 304.125
• 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			125.26
Character	\(\chi\)	\(=\)	304.125
Dual form			304.2.v.a.197.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.741835 - 1.20403i) q^{2} +(-2.88720 + 0.773623i) q^{3} +(-0.899361 - 1.78638i) q^{4} +(-0.896570 - 3.34604i) q^{5} +(-1.21036 + 4.05017i) q^{6} +4.13915i q^{7} +(-2.81803 - 0.242345i) q^{8} +(5.13936 - 2.96721i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 20 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.741835	−	1.20403i	0.524557	−	0.851375i
							\(3\)	−2.88720	+	0.773623i	−1.66693	+	0.446651i	−0.964279	−	0.264887i	\(-0.914665\pi\)
−0.702647	+	0.711539i	\(0.747999\pi\)
\(5\)	−0.896570	−	3.34604i	−0.400958	−	1.49640i	−0.811390	−	0.584505i	\(-0.801289\pi\)
							0.410432	−	0.911891i	\(-0.365378\pi\)
\(7\)			4.13915i			1.56445i	0.622996	+	0.782225i	\(0.285915\pi\)
							−0.622996	+	0.782225i	\(0.714085\pi\)
\(11\)	−0.755680	+	0.755680i	−0.227846	+	0.227846i	−0.811792	−	0.583946i	\(-0.801508\pi\)
							0.583946	+	0.811792i	\(0.301508\pi\)
\(13\)	−1.15206	+	4.29953i	−0.319523	+	1.19248i	0.600182	+	0.799864i	\(0.295095\pi\)
							−0.919704	+	0.392611i	\(0.871572\pi\)
\(17\)	−1.59702	+	2.76612i	−0.387335	+	0.670883i	−0.992090	−	0.125528i	\(-0.959938\pi\)
							0.604755	+	0.796411i	\(0.293271\pi\)
\(19\)	−4.05128	−	1.60845i	−0.929428	−	0.369004i
							\(23\)	2.86859	−	1.65618i	0.598143	−	0.345338i	−0.170168	−	0.985415i	\(-0.554431\pi\)
0.768311	+	0.640077i	\(0.221098\pi\)
\(29\)	0.0242852	−	0.0906337i	0.00450965	−	0.0168303i	−0.963634	−	0.267224i	\(-0.913894\pi\)
							0.968144	+	0.250394i	\(0.0805602\pi\)
\(31\)	−8.00981			−1.43860			−0.719302	−	0.694697i	\(-0.755538\pi\)
							−0.719302	+	0.694697i	\(0.755538\pi\)
\(37\)	−0.805323	+	0.805323i	−0.132394	+	0.132394i	−0.770199	−	0.637804i	\(-0.779843\pi\)
							0.637804	+	0.770199i	\(0.279843\pi\)
\(41\)	−1.64412	−	0.949235i	−0.256769	−	0.148246i	0.366091	−	0.930579i	\(-0.380696\pi\)
							−0.622860	+	0.782334i	\(0.714029\pi\)
\(43\)	−0.614418	−	2.29304i	−0.0936979	−	0.349685i	0.903121	−	0.429386i	\(-0.141270\pi\)
							−0.996819	+	0.0797010i	\(0.974603\pi\)
\(47\)	−2.63688	−	4.56721i	−0.384629	−	0.666196i	0.607089	−	0.794634i	\(-0.292337\pi\)
							−0.991718	+	0.128438i	\(0.959004\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.v.a.125.26	yes	152
16.5	even	4	inner	304.2.v.a.277.26	yes	152
19.7	even	3	inner	304.2.v.a.45.26	✓	152
304.197	even	12	inner	304.2.v.a.197.26	yes	152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.v.a.45.26	✓	152	19.7	even	3	inner
304.2.v.a.125.26	yes	152	1.1	even	1	trivial
304.2.v.a.197.26	yes	152	304.197	even	12	inner
304.2.v.a.277.26	yes	152	16.5	even	4	inner