Embedded newform 304.2.x.c.179.30

• Label: 304.2.x.c.179.30
• Level: $304$
• Weight: $2$
• Character: 304.179
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	304.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.42745222145\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			179.30
Character	\(\chi\)	\(=\)	304.179
Dual form			304.2.x.c.107.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.04505 - 0.952820i) q^{2} +(-2.47333 + 0.662726i) q^{3} +(0.184266 - 1.99149i) q^{4} +(0.0905735 + 0.338025i) q^{5} +(-1.95329 + 3.04922i) q^{6} -1.90034 q^{7} +(-1.70497 - 2.25679i) q^{8} +(3.08006 - 1.77827i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 6 q^{2} - 12 q^{3} - 2 q^{4} + 2 q^{5} - 2 q^{6} - 8 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.04505	−	0.952820i	0.738963	−	0.673746i
							\(3\)	−2.47333	+	0.662726i	−1.42798	+	0.382625i	−0.888306	−	0.459252i	\(-0.848117\pi\)
−0.539669	+	0.841877i	\(0.681451\pi\)
\(5\)	0.0905735	+	0.338025i	0.0405057	+	0.151169i	0.983217	−	0.182441i	\(-0.0583998\pi\)
							−0.942711	+	0.333610i	\(0.891733\pi\)
\(7\)	−1.90034			−0.718261			−0.359130	−	0.933287i	\(-0.616927\pi\)
							−0.359130	+	0.933287i	\(0.616927\pi\)
\(11\)	−3.26470	−	3.26470i	−0.984343	−	0.984343i	0.0155363	−	0.999879i	\(-0.495054\pi\)
							−0.999879	+	0.0155363i	\(0.995054\pi\)
\(13\)	−2.55658	−	0.685033i	−0.709067	−	0.189994i	−0.113779	−	0.993506i	\(-0.536296\pi\)
							−0.595288	+	0.803512i	\(0.702962\pi\)
\(17\)	−2.56099	+	4.43576i	−0.621131	+	1.07583i	0.368144	+	0.929769i	\(0.379993\pi\)
							−0.989275	+	0.146062i	\(0.953340\pi\)
\(19\)	−4.32673	+	0.528627i	−0.992619	+	0.121275i
							\(23\)	−0.732669	−	1.26902i	−0.152772	−	0.264609i	0.779474	−	0.626435i	\(-0.215487\pi\)
−0.932246	+	0.361826i	\(0.882153\pi\)
\(29\)	4.01371	+	1.07547i	0.745328	+	0.199710i	0.611445	−	0.791287i	\(-0.290589\pi\)
							0.133883	+	0.990997i	\(0.457255\pi\)
\(31\)	6.04047			1.08490			0.542451	−	0.840088i	\(-0.317497\pi\)
							0.542451	+	0.840088i	\(0.317497\pi\)
\(37\)	−7.05226	−	7.05226i	−1.15938	−	1.15938i	−0.984608	−	0.174776i	\(-0.944080\pi\)
							−0.174776	−	0.984608i	\(-0.555920\pi\)
\(41\)	4.34244	−	7.52133i	0.678176	−	1.17463i	−0.297354	−	0.954767i	\(-0.596104\pi\)
							0.975530	−	0.219867i	\(-0.0705625\pi\)
\(43\)	4.39900	−	1.17871i	0.670841	−	0.179751i	0.0927072	−	0.995693i	\(-0.470448\pi\)
							0.578134	+	0.815942i	\(0.303781\pi\)
\(47\)	−5.48205	+	3.16506i	−0.799639	+	0.461672i	−0.843345	−	0.537372i	\(-0.819417\pi\)
							0.0437056	+	0.999044i	\(0.486084\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.x.c.179.30	yes	144
16.11	odd	4	inner	304.2.x.c.27.18	✓	144
19.12	odd	6	inner	304.2.x.c.259.18	yes	144
304.107	even	12	inner	304.2.x.c.107.30	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.x.c.27.18	✓	144	16.11	odd	4	inner
304.2.x.c.107.30	yes	144	304.107	even	12	inner
304.2.x.c.179.30	yes	144	1.1	even	1	trivial
304.2.x.c.259.18	yes	144	19.12	odd	6	inner