Embedded newform 304.2.k.b.77.7

• Label: 304.2.k.b.77.7
• 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			77.7
Character	\(\chi\)	\(=\)	304.77
Dual form			304.2.k.b.229.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19470 - 0.756759i) q^{2} +(-0.0648127 - 0.0648127i) q^{3} +(0.854632 + 1.80820i) q^{4} +(2.89124 - 2.89124i) q^{5} +(0.0283843 + 0.126480i) q^{6} +2.82413i q^{7} +(0.347344 - 2.80702i) q^{8} -2.99160i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 4 q^{3} + 4 q^{4} + 8 q^{5} - 8 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19470	−	0.756759i	−0.844783	−	0.535109i
							\(3\)	−0.0648127	−	0.0648127i	−0.0374196	−	0.0374196i	0.688149	−	0.725569i	\(-0.258423\pi\)
−0.725569	+	0.688149i	\(0.758423\pi\)
\(5\)	2.89124	−	2.89124i	1.29300	−	1.29300i	0.360084	−	0.932920i	\(-0.382748\pi\)
							0.932920	−	0.360084i	\(-0.117252\pi\)
\(7\)			2.82413i			1.06742i	0.845667	+	0.533710i	\(0.179203\pi\)
							−0.845667	+	0.533710i	\(0.820797\pi\)
\(11\)	−0.671611	+	0.671611i	−0.202498	+	0.202498i	−0.801070	−	0.598571i	\(-0.795735\pi\)
							0.598571	+	0.801070i	\(0.295735\pi\)
\(13\)	3.42708	+	3.42708i	0.950501	+	0.950501i	0.998831	−	0.0483307i	\(-0.0153901\pi\)
							−0.0483307	+	0.998831i	\(0.515390\pi\)
\(17\)	1.21626			0.294987			0.147493	−	0.989063i	\(-0.452879\pi\)
							0.147493	+	0.989063i	\(0.452879\pi\)
\(19\)	0.707107	+	0.707107i	0.162221	+	0.162221i
							\(23\)		−	7.28974i		−	1.52002i	−0.649913	−	0.760008i	\(-0.725195\pi\)
0.649913	−	0.760008i	\(-0.274805\pi\)
\(29\)	−3.97741	−	3.97741i	−0.738587	−	0.738587i	0.233717	−	0.972305i	\(-0.424911\pi\)
							−0.972305	+	0.233717i	\(0.924911\pi\)
\(31\)	2.01280			0.361510			0.180755	−	0.983528i	\(-0.442146\pi\)
							0.180755	+	0.983528i	\(0.442146\pi\)
\(37\)	−2.14508	+	2.14508i	−0.352650	+	0.352650i	−0.861095	−	0.508445i	\(-0.830221\pi\)
							0.508445	+	0.861095i	\(0.330221\pi\)
\(41\)			6.19492i			0.967484i	0.875211	+	0.483742i	\(0.160723\pi\)
							−0.875211	+	0.483742i	\(0.839277\pi\)
\(43\)	4.41155	−	4.41155i	0.672755	−	0.672755i	−0.285596	−	0.958350i	\(-0.592191\pi\)
							0.958350	+	0.285596i	\(0.0921914\pi\)
\(47\)	2.66677			0.388988			0.194494	−	0.980904i	\(-0.437694\pi\)
							0.194494	+	0.980904i	\(0.437694\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.k.b.77.7	✓	68
4.3	odd	2		1216.2.k.b.913.19		68
16.5	even	4	inner	304.2.k.b.229.7	yes	68
16.11	odd	4		1216.2.k.b.305.19		68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.k.b.77.7	✓	68	1.1	even	1	trivial
304.2.k.b.229.7	yes	68	16.5	even	4	inner
1216.2.k.b.305.19		68	16.11	odd	4
1216.2.k.b.913.19		68	4.3	odd	2