Embedded newform 304.2.x.c.27.8

• Label: 304.2.x.c.27.8
• Level: $304$
• Weight: $2$
• Character: 304.27
• 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			27.8
Character	\(\chi\)	\(=\)	304.27
Dual form			304.2.x.c.259.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18640 + 0.769715i) q^{2} +(-0.552843 - 2.06324i) q^{3} +(0.815079 - 1.82638i) q^{4} +(-0.617154 + 0.165366i) q^{5} +(2.24399 + 2.02229i) q^{6} +4.97623 q^{7} +(0.438780 + 2.79419i) q^{8} +(-1.35323 + 0.781290i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 6 q^{2} - 12 q^{3} - 2 q^{4} + 2 q^{5} - 2 q^{6} - 8 q^{7}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18640	+	0.769715i	−0.838910	+	0.544270i
							\(3\)	−0.552843	−	2.06324i	−0.319184	−	1.19121i	−0.920031	−	0.391846i	\(-0.871837\pi\)
0.600847	−	0.799364i	\(-0.294830\pi\)
\(5\)	−0.617154	+	0.165366i	−0.276000	+	0.0739539i	−0.394164	−	0.919040i	\(-0.628966\pi\)
							0.118164	+	0.992994i	\(0.462299\pi\)
\(7\)	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	−0.0136985	+	0.999906i	\(0.504360\pi\)
							−0.999906	+	0.0136985i	\(0.995640\pi\)
\(13\)	1.25848	−	4.69673i	0.349041	−	1.30264i	−0.538779	−	0.842447i	\(-0.681114\pi\)
							0.887820	−	0.460191i	\(-0.152219\pi\)
\(17\)	2.36087	−	4.08914i	0.572594	−	0.991762i	−0.423704	−	0.905801i	\(-0.639270\pi\)
							0.996298	−	0.0859617i	\(-0.0273963\pi\)
\(19\)	3.65834	−	2.36993i	0.839281	−	0.543698i
							\(23\)	0.383364	+	0.664006i	0.0799370	+	0.138455i	0.903223	−	0.429173i	\(-0.141195\pi\)
−0.823286	+	0.567627i	\(0.807861\pi\)
\(29\)	1.27238	−	4.74860i	0.236276	−	0.881793i	−0.741294	−	0.671181i	\(-0.765787\pi\)
							0.977570	−	0.210613i	\(-0.0675459\pi\)
\(31\)	−3.10309			−0.557332			−0.278666	−	0.960388i	\(-0.589892\pi\)
							−0.278666	+	0.960388i	\(0.589892\pi\)
\(37\)	−4.24500	+	4.24500i	−0.697873	+	0.697873i	−0.963951	−	0.266078i	\(-0.914272\pi\)
							0.266078	+	0.963951i	\(0.414272\pi\)
\(41\)	5.47688	−	9.48624i	0.855345	−	1.48150i	−0.0209787	−	0.999780i	\(-0.506678\pi\)
							0.876324	−	0.481722i	\(-0.159988\pi\)
\(43\)	1.82206	+	6.80000i	0.277861	+	1.03699i	0.953900	+	0.300125i	\(0.0970284\pi\)
							−0.676039	+	0.736866i	\(0.736305\pi\)
\(47\)	−0.925559	+	0.534372i	−0.135007	+	0.0779461i	−0.565982	−	0.824418i	\(-0.691503\pi\)
							0.430975	+	0.902364i	\(0.358170\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.x.c.27.8	✓	144	1.1	even	1	trivial
304.2.x.c.107.21	yes	144	19.12	odd	6	inner
304.2.x.c.179.21	yes	144	16.3	odd	4	inner
304.2.x.c.259.8	yes	144	304.259	even	12	inner