Embedded newform 304.2.x.c.179.21

• Label: 304.2.x.c.179.21
• 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			179.21
Character	\(\chi\)	\(=\)	304.179
Dual form			304.2.x.c.107.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0733936 - 1.41231i) q^{2} +(-2.06324 + 0.552843i) q^{3} +(-1.98923 - 0.207309i) q^{4} +(0.165366 + 0.617154i) q^{5} +(0.629356 + 2.95450i) q^{6} +4.97623 q^{7} +(-0.438780 + 2.79419i) q^{8} +(1.35323 - 0.781290i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 6 q^{2} - 12 q^{3} - 2 q^{4} + 2 q^{5} - 2 q^{6} - 8 q^{7}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0733936	−	1.41231i	0.0518971	−	0.998652i
							\(3\)	−2.06324	+	0.552843i	−1.19121	+	0.319184i	−0.799364	−	0.600847i	\(-0.794830\pi\)
−0.391846	+	0.920031i	\(0.628163\pi\)
\(5\)	0.165366	+	0.617154i	0.0739539	+	0.276000i	0.992994	−	0.118164i	\(-0.0377009\pi\)
							−0.919040	+	0.394164i	\(0.871034\pi\)
\(7\)	4.97623			1.88084			0.940419	−	0.340019i	\(-0.110433\pi\)
							0.940419	+	0.340019i	\(0.110433\pi\)
\(11\)	−3.36175	−	3.36175i	−1.01360	−	1.01360i	−0.999906	−	0.0136985i	\(-0.995640\pi\)
							−0.0136985	−	0.999906i	\(-0.504360\pi\)
\(13\)	4.69673	+	1.25848i	1.30264	+	0.349041i	0.842447	−	0.538779i	\(-0.181114\pi\)
							0.460191	+	0.887820i	\(0.347781\pi\)
\(17\)	2.36087	−	4.08914i	0.572594	−	0.991762i	−0.423704	−	0.905801i	\(-0.639270\pi\)
							0.996298	−	0.0859617i	\(-0.0273963\pi\)
\(19\)	−2.36993	−	3.65834i	−0.543698	−	0.839281i
							\(23\)	0.383364	+	0.664006i	0.0799370	+	0.138455i	0.903223	−	0.429173i	\(-0.141195\pi\)
−0.823286	+	0.567627i	\(0.807861\pi\)
\(29\)	4.74860	+	1.27238i	0.881793	+	0.236276i	0.671181	−	0.741294i	\(-0.265787\pi\)
							0.210613	+	0.977570i	\(0.432454\pi\)
\(31\)	3.10309			0.557332			0.278666	−	0.960388i	\(-0.410108\pi\)
							0.278666	+	0.960388i	\(0.410108\pi\)
\(37\)	4.24500	+	4.24500i	0.697873	+	0.697873i	0.963951	−	0.266078i	\(-0.0857279\pi\)
							−0.266078	+	0.963951i	\(0.585728\pi\)
\(41\)	−5.47688	+	9.48624i	−0.855345	+	1.48150i	0.0209787	+	0.999780i	\(0.493322\pi\)
							−0.876324	+	0.481722i	\(0.840012\pi\)
\(43\)	−6.80000	+	1.82206i	−1.03699	+	0.277861i	−0.736866	−	0.676039i	\(-0.763695\pi\)
							−0.300125	+	0.953900i	\(0.597028\pi\)
\(47\)	0.925559	−	0.534372i	0.135007	−	0.0779461i	−0.430975	−	0.902364i	\(-0.641830\pi\)
							0.565982	+	0.824418i	\(0.308497\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.x.c.179.21	yes	144
16.11	odd	4	inner	304.2.x.c.27.8	✓	144
19.12	odd	6	inner	304.2.x.c.259.8	yes	144
304.107	even	12	inner	304.2.x.c.107.21	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.x.c.27.8	✓	144	16.11	odd	4	inner
304.2.x.c.107.21	yes	144	304.107	even	12	inner
304.2.x.c.179.21	yes	144	1.1	even	1	trivial
304.2.x.c.259.8	yes	144	19.12	odd	6	inner