Embedded newform 304.2.k.b.77.34

• Label: 304.2.k.b.77.34
• Level: $304$
• Weight: $2$
• Character: 304.77
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $68$
• 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			77.34
Character	\(\chi\)	\(=\)	304.77
Dual form			304.2.k.b.229.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41380 - 0.0343004i) q^{2} +(-1.89015 - 1.89015i) q^{3} +(1.99765 - 0.0969877i) q^{4} +(-1.06645 + 1.06645i) q^{5} +(-2.73712 - 2.60746i) q^{6} -4.02959i q^{7} +(2.82094 - 0.205641i) q^{8} +4.14533i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 4 q^{3} + 4 q^{4} + 8 q^{5} - 8 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41380	−	0.0343004i	0.999706	−	0.0242540i
							\(3\)	−1.89015	−	1.89015i	−1.09128	−	1.09128i	−0.995392	−	0.0958864i	\(-0.969431\pi\)
−0.0958864	−	0.995392i	\(-0.530569\pi\)
\(5\)	−1.06645	+	1.06645i	−0.476932	+	0.476932i	−0.904149	−	0.427217i	\(-0.859494\pi\)
							0.427217	+	0.904149i	\(0.359494\pi\)
\(7\)		−	4.02959i		−	1.52304i	−0.648141	−	0.761521i	\(-0.724453\pi\)
							0.648141	−	0.761521i	\(-0.275547\pi\)
\(11\)	0.552303	−	0.552303i	0.166526	−	0.166526i	−0.618925	−	0.785450i	\(-0.712431\pi\)
							0.785450	+	0.618925i	\(0.212431\pi\)
\(13\)	−4.37311	−	4.37311i	−1.21288	−	1.21288i	−0.970075	−	0.242807i	\(-0.921932\pi\)
							−0.242807	−	0.970075i	\(-0.578068\pi\)
\(17\)	0.608142			0.147496			0.0737480	−	0.997277i	\(-0.476504\pi\)
							0.0737480	+	0.997277i	\(0.476504\pi\)
\(19\)	−0.707107	−	0.707107i	−0.162221	−	0.162221i
							\(23\)		−	0.900783i		−	0.187826i	−0.995580	−	0.0939132i	\(-0.970062\pi\)
0.995580	−	0.0939132i	\(-0.0299376\pi\)
\(29\)	6.34029	+	6.34029i	1.17736	+	1.17736i	0.980414	+	0.196949i	\(0.0631033\pi\)
							0.196949	+	0.980414i	\(0.436897\pi\)
\(31\)	7.50604			1.34813			0.674063	−	0.738674i	\(-0.264548\pi\)
							0.674063	+	0.738674i	\(0.264548\pi\)
\(37\)	4.12547	−	4.12547i	0.678223	−	0.678223i	−0.281375	−	0.959598i	\(-0.590790\pi\)
							0.959598	+	0.281375i	\(0.0907905\pi\)
\(41\)		−	3.15335i		−	0.492471i	−0.969210	−	0.246236i	\(-0.920806\pi\)
							0.969210	−	0.246236i	\(-0.0791937\pi\)
\(43\)	−4.34630	+	4.34630i	−0.662805	+	0.662805i	−0.956040	−	0.293235i	\(-0.905268\pi\)
							0.293235	+	0.956040i	\(0.405268\pi\)
\(47\)	12.4949			1.82258			0.911288	−	0.411770i	\(-0.135089\pi\)
							0.911288	+	0.411770i	\(0.135089\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.k.b.77.34	✓	68
4.3	odd	2		1216.2.k.b.913.30		68
16.5	even	4	inner	304.2.k.b.229.34	yes	68
16.11	odd	4		1216.2.k.b.305.30		68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.k.b.77.34	✓	68	1.1	even	1	trivial
304.2.k.b.229.34	yes	68	16.5	even	4	inner
1216.2.k.b.305.30		68	16.11	odd	4
1216.2.k.b.913.30		68	4.3	odd	2