Embedded newform 304.2.k.b.77.6

• Label: 304.2.k.b.77.6
• 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			77.6
Character	\(\chi\)	\(=\)	304.77
Dual form			304.2.k.b.229.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26466 - 0.632956i) q^{2} +(1.26193 + 1.26193i) q^{3} +(1.19873 + 1.60095i) q^{4} +(0.588594 - 0.588594i) q^{5} +(-0.797168 - 2.39466i) q^{6} -3.53958i q^{7} +(-0.502658 - 2.78340i) q^{8} +0.184946i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 4 q^{3} + 4 q^{4} + 8 q^{5} - 8 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26466	−	0.632956i	−0.894250	−	0.447568i
							\(3\)	1.26193	+	1.26193i	0.728577	+	0.728577i	0.970336	−	0.241759i	\(-0.0777244\pi\)
−0.241759	+	0.970336i	\(0.577724\pi\)
\(5\)	0.588594	−	0.588594i	0.263227	−	0.263227i	−0.563136	−	0.826364i	\(-0.690405\pi\)
							0.826364	+	0.563136i	\(0.190405\pi\)
\(7\)		−	3.53958i		−	1.33783i	−0.743337	−	0.668917i	\(-0.766758\pi\)
							0.743337	−	0.668917i	\(-0.233242\pi\)
\(11\)	1.70629	−	1.70629i	0.514466	−	0.514466i	−0.401425	−	0.915892i	\(-0.631485\pi\)
							0.915892	+	0.401425i	\(0.131485\pi\)
\(13\)	−2.66076	−	2.66076i	−0.737961	−	0.737961i	0.234222	−	0.972183i	\(-0.424746\pi\)
							−0.972183	+	0.234222i	\(0.924746\pi\)
\(17\)	7.43076			1.80223			0.901113	−	0.433585i	\(-0.142752\pi\)
							0.901113	+	0.433585i	\(0.142752\pi\)
\(19\)	0.707107	+	0.707107i	0.162221	+	0.162221i
							\(23\)			9.22644i			1.92385i	0.273320	+	0.961923i	\(0.411878\pi\)
−0.273320	+	0.961923i	\(0.588122\pi\)
\(29\)	−0.767522	−	0.767522i	−0.142525	−	0.142525i	0.632244	−	0.774769i	\(-0.282134\pi\)
							−0.774769	+	0.632244i	\(0.782134\pi\)
\(31\)	−10.2761			−1.84563			−0.922817	−	0.385239i	\(-0.874119\pi\)
							−0.922817	+	0.385239i	\(0.874119\pi\)
\(37\)	4.19450	−	4.19450i	0.689572	−	0.689572i	−0.272565	−	0.962137i	\(-0.587872\pi\)
							0.962137	+	0.272565i	\(0.0878721\pi\)
\(41\)		−	2.96619i		−	0.463242i	−0.972806	−	0.231621i	\(-0.925597\pi\)
							0.972806	−	0.231621i	\(-0.0744029\pi\)
\(43\)	−6.52927	+	6.52927i	−0.995704	+	0.995704i	−0.999991	−	0.00428636i	\(-0.998636\pi\)
							0.00428636	+	0.999991i	\(0.498636\pi\)
\(47\)	7.69612			1.12260			0.561298	−	0.827614i	\(-0.310302\pi\)
							0.561298	+	0.827614i	\(0.310302\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.k.b.77.6	✓	68
4.3	odd	2		1216.2.k.b.913.9		68
16.5	even	4	inner	304.2.k.b.229.6	yes	68
16.11	odd	4		1216.2.k.b.305.9		68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.k.b.77.6	✓	68	1.1	even	1	trivial
304.2.k.b.229.6	yes	68	16.5	even	4	inner
1216.2.k.b.305.9		68	16.11	odd	4
1216.2.k.b.913.9		68	4.3	odd	2