Embedded newform 304.2.k.b.77.3

• Label: 304.2.k.b.77.3
• Level: $304$
• Weight: $2$
• Character: 304.77
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(2.42745222145\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			77.3
Character	\(\chi\)	\(=\)	304.77
Dual form			304.2.k.b.229.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36784 - 0.359174i) q^{2} +(-1.75469 - 1.75469i) q^{3} +(1.74199 + 0.982587i) q^{4} +(0.519383 - 0.519383i) q^{5} +(1.76991 + 3.03039i) q^{6} -1.79541i q^{7} +(-2.02985 - 1.96970i) q^{8} +3.15791i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 4 q^{3} + 4 q^{4} + 8 q^{5} - 8 q^{6}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36784	−	0.359174i	−0.967211	−	0.253974i
							\(3\)	−1.75469	−	1.75469i	−1.01307	−	1.01307i	−0.999913	−	0.0131599i	\(-0.995811\pi\)
−0.0131599	−	0.999913i	\(-0.504189\pi\)
\(5\)	0.519383	−	0.519383i	0.232275	−	0.232275i	−0.581367	−	0.813642i	\(-0.697482\pi\)
							0.813642	+	0.581367i	\(0.197482\pi\)
\(7\)		−	1.79541i		−	0.678599i	−0.940678	−	0.339300i	\(-0.889810\pi\)
							0.940678	−	0.339300i	\(-0.110190\pi\)
\(11\)	3.97560	−	3.97560i	1.19869	−	1.19869i	0.224128	−	0.974560i	\(-0.428047\pi\)
							0.974560	−	0.224128i	\(-0.0719533\pi\)
\(13\)	0.114860	+	0.114860i	0.0318566	+	0.0318566i	0.722856	−	0.690999i	\(-0.242829\pi\)
							−0.690999	+	0.722856i	\(0.742829\pi\)
\(17\)	−5.90956			−1.43328			−0.716639	−	0.697444i	\(-0.754321\pi\)
							−0.716639	+	0.697444i	\(0.754321\pi\)
\(19\)	−0.707107	−	0.707107i	−0.162221	−	0.162221i
							\(23\)		−	2.02675i		−	0.422607i	−0.977421	−	0.211303i	\(-0.932229\pi\)
0.977421	−	0.211303i	\(-0.0677708\pi\)
\(29\)	−2.66440	−	2.66440i	−0.494767	−	0.494767i	0.415037	−	0.909804i	\(-0.363769\pi\)
							−0.909804	+	0.415037i	\(0.863769\pi\)
\(31\)	−7.42624			−1.33379			−0.666896	−	0.745151i	\(-0.732377\pi\)
							−0.666896	+	0.745151i	\(0.732377\pi\)
\(37\)	−3.31665	+	3.31665i	−0.545255	+	0.545255i	−0.925065	−	0.379810i	\(-0.875989\pi\)
							0.379810	+	0.925065i	\(0.375989\pi\)
\(41\)		−	1.52590i		−	0.238306i	−0.992876	−	0.119153i	\(-0.961982\pi\)
							0.992876	−	0.119153i	\(-0.0380179\pi\)
\(43\)	4.08964	−	4.08964i	0.623664	−	0.623664i	−0.322803	−	0.946466i	\(-0.604625\pi\)
							0.946466	+	0.322803i	\(0.104625\pi\)
\(47\)	−7.56238			−1.10309			−0.551543	−	0.834146i	\(-0.685961\pi\)
							−0.551543	+	0.834146i	\(0.685961\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.k.b.77.3	✓	68
4.3	odd	2		1216.2.k.b.913.29		68
16.5	even	4	inner	304.2.k.b.229.3	yes	68
16.11	odd	4		1216.2.k.b.305.29		68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.k.b.77.3	✓	68	1.1	even	1	trivial
304.2.k.b.229.3	yes	68	16.5	even	4	inner
1216.2.k.b.305.29		68	16.11	odd	4
1216.2.k.b.913.29		68	4.3	odd	2