Embedded newform 304.2.k.b.77.24

• Label: 304.2.k.b.77.24
• 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			77.24
Character	\(\chi\)	\(=\)	304.77
Dual form			304.2.k.b.229.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.880445 - 1.10671i) q^{2} +(-1.52443 - 1.52443i) q^{3} +(-0.449632 - 1.94880i) q^{4} +(-2.81667 + 2.81667i) q^{5} +(-3.02928 + 0.344930i) q^{6} +4.66200i q^{7} +(-2.55264 - 1.21820i) q^{8} +1.64775i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 4 q^{3} + 4 q^{4} + 8 q^{5} - 8 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.880445	−	1.10671i	0.622569	−	0.782565i
							\(3\)	−1.52443	−	1.52443i	−0.880128	−	0.880128i	0.113419	−	0.993547i	\(-0.463820\pi\)
−0.993547	+	0.113419i	\(0.963820\pi\)
\(5\)	−2.81667	+	2.81667i	−1.25965	+	1.25965i	−0.308397	+	0.951258i	\(0.599792\pi\)
							−0.951258	+	0.308397i	\(0.900208\pi\)
\(7\)			4.66200i			1.76207i	0.473050	+	0.881036i	\(0.343153\pi\)
							−0.473050	+	0.881036i	\(0.656847\pi\)
\(11\)	0.655371	−	0.655371i	0.197602	−	0.197602i	−0.601369	−	0.798971i	\(-0.705378\pi\)
							0.798971	+	0.601369i	\(0.205378\pi\)
\(13\)	−3.00741	−	3.00741i	−0.834105	−	0.834105i	0.153971	−	0.988075i	\(-0.450794\pi\)
							−0.988075	+	0.153971i	\(0.950794\pi\)
\(17\)	−4.83672			−1.17308			−0.586538	−	0.809922i	\(-0.699510\pi\)
							−0.586538	+	0.809922i	\(0.699510\pi\)
\(19\)	−0.707107	−	0.707107i	−0.162221	−	0.162221i
							\(23\)			5.34579i			1.11467i	0.830287	+	0.557337i	\(0.188177\pi\)
−0.830287	+	0.557337i	\(0.811823\pi\)
\(29\)	−1.34953	−	1.34953i	−0.250602	−	0.250602i	0.570615	−	0.821217i	\(-0.306705\pi\)
							−0.821217	+	0.570615i	\(0.806705\pi\)
\(31\)	5.09551			0.915181			0.457591	−	0.889163i	\(-0.348713\pi\)
							0.457591	+	0.889163i	\(0.348713\pi\)
\(37\)	−2.94155	+	2.94155i	−0.483587	+	0.483587i	−0.906275	−	0.422688i	\(-0.861087\pi\)
							0.422688	+	0.906275i	\(0.361087\pi\)
\(41\)			2.93776i			0.458801i	0.973332	+	0.229400i	\(0.0736765\pi\)
							−0.973332	+	0.229400i	\(0.926323\pi\)
\(43\)	4.63582	−	4.63582i	0.706957	−	0.706957i	−0.258937	−	0.965894i	\(-0.583372\pi\)
							0.965894	+	0.258937i	\(0.0833724\pi\)
\(47\)	−10.0513			−1.46614			−0.733068	−	0.680156i	\(-0.761912\pi\)
							−0.733068	+	0.680156i	\(0.761912\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.k.b.77.24	✓	68
4.3	odd	2		1216.2.k.b.913.27		68
16.5	even	4	inner	304.2.k.b.229.24	yes	68
16.11	odd	4		1216.2.k.b.305.27		68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.k.b.77.24	✓	68	1.1	even	1	trivial
304.2.k.b.229.24	yes	68	16.5	even	4	inner
1216.2.k.b.305.27		68	16.11	odd	4
1216.2.k.b.913.27		68	4.3	odd	2