Embedded newform 304.2.k.b.77.29

• Label: 304.2.k.b.77.29
• 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			77.29
Character	\(\chi\)	\(=\)	304.77
Dual form			304.2.k.b.229.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.22770 - 0.701956i) q^{2} +(2.26925 + 2.26925i) q^{3} +(1.01451 - 1.72359i) q^{4} +(-1.08984 + 1.08984i) q^{5} +(4.37888 + 1.19305i) q^{6} -4.17557i q^{7} +(0.0356401 - 2.82820i) q^{8} +7.29898i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 4 q^{3} + 4 q^{4} + 8 q^{5} - 8 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.22770	−	0.701956i	0.868118	−	0.496358i
							\(3\)	2.26925	+	2.26925i	1.31015	+	1.31015i	0.921301	+	0.388851i	\(0.127128\pi\)
0.388851	+	0.921301i	\(0.372872\pi\)
\(5\)	−1.08984	+	1.08984i	−0.487392	+	0.487392i	−0.907482	−	0.420090i	\(-0.861998\pi\)
							0.420090	+	0.907482i	\(0.361998\pi\)
\(7\)		−	4.17557i		−	1.57822i	−0.614254	−	0.789108i	\(-0.710543\pi\)
							0.614254	−	0.789108i	\(-0.289457\pi\)
\(11\)	−0.984617	+	0.984617i	−0.296873	+	0.296873i	−0.839788	−	0.542915i	\(-0.817321\pi\)
							0.542915	+	0.839788i	\(0.317321\pi\)
\(13\)	−0.198232	−	0.198232i	−0.0549796	−	0.0549796i	0.679082	−	0.734062i	\(-0.262378\pi\)
							−0.734062	+	0.679082i	\(0.762378\pi\)
\(17\)	−4.10877			−0.996524			−0.498262	−	0.867026i	\(-0.666028\pi\)
							−0.498262	+	0.867026i	\(0.666028\pi\)
\(19\)	−0.707107	−	0.707107i	−0.162221	−	0.162221i
							\(23\)			7.37494i			1.53778i	0.639381	+	0.768890i	\(0.279191\pi\)
−0.639381	+	0.768890i	\(0.720809\pi\)
\(29\)	−4.06522	−	4.06522i	−0.754892	−	0.754892i	0.220496	−	0.975388i	\(-0.429232\pi\)
							−0.975388	+	0.220496i	\(0.929232\pi\)
\(31\)	0.696897			0.125166			0.0625832	−	0.998040i	\(-0.480066\pi\)
							0.0625832	+	0.998040i	\(0.480066\pi\)
\(37\)	6.01110	−	6.01110i	0.988218	−	0.988218i	−0.0117131	−	0.999931i	\(-0.503728\pi\)
							0.999931	+	0.0117131i	\(0.00372848\pi\)
\(41\)		−	3.09499i		−	0.483356i	−0.970357	−	0.241678i	\(-0.922302\pi\)
							0.970357	−	0.241678i	\(-0.0776977\pi\)
\(43\)	6.73708	−	6.73708i	1.02739	−	1.02739i	0.0277804	−	0.999614i	\(-0.491156\pi\)
							0.999614	−	0.0277804i	\(-0.00884390\pi\)
\(47\)	−6.28321			−0.916500			−0.458250	−	0.888823i	\(-0.651524\pi\)
							−0.458250	+	0.888823i	\(0.651524\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.k.b.77.29	✓	68
4.3	odd	2		1216.2.k.b.913.1		68
16.5	even	4	inner	304.2.k.b.229.29	yes	68
16.11	odd	4		1216.2.k.b.305.1		68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.k.b.77.29	✓	68	1.1	even	1	trivial
304.2.k.b.229.29	yes	68	16.5	even	4	inner
1216.2.k.b.305.1		68	16.11	odd	4
1216.2.k.b.913.1		68	4.3	odd	2