Embedded newform 304.2.bg.a.243.24

• Label: 304.2.bg.a.243.24
• Level: $304$
• Weight: $2$
• Character: 304.243
• Analytic conductor: $2.427$
• Analytic rank: $0$
• Dimension: $456$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	304.bg (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.42745222145\)
Analytic rank:	\(0\)
Dimension:	\(456\)
Relative dimension:	\(38\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			243.24
Character	\(\chi\)	\(=\)	304.243
Dual form			304.2.bg.a.299.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.514486 + 1.31731i) q^{2} +(0.617681 - 0.882140i) q^{3} +(-1.47061 + 1.35547i) q^{4} +(-0.208259 + 2.38041i) q^{5} +(1.47984 + 0.359828i) q^{6} +(1.04812 - 1.81540i) q^{7} +(-2.54219 - 1.23987i) q^{8} +(0.629420 + 1.72932i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.514486	+	1.31731i	0.363797	+	0.931478i
							\(3\)	0.617681	−	0.882140i	0.356618	−	0.509304i	−0.600255	−	0.799808i	\(-0.704934\pi\)
0.956874	+	0.290505i	\(0.0938233\pi\)
\(5\)	−0.208259	+	2.38041i	−0.0931362	+	1.06455i	0.795504	+	0.605948i	\(0.207206\pi\)
							−0.888641	+	0.458604i	\(0.848350\pi\)
\(7\)	1.04812	−	1.81540i	0.396152	−	0.686156i	−0.597095	−	0.802170i	\(-0.703679\pi\)
							0.993248	+	0.116015i	\(0.0370119\pi\)
\(11\)	−1.13934	+	4.25206i	−0.343523	+	1.28204i	0.550806	+	0.834634i	\(0.314321\pi\)
							−0.894328	+	0.447411i	\(0.852346\pi\)
\(13\)	1.60622	+	2.29392i	0.445485	+	0.636218i	0.977604	−	0.210451i	\(-0.0674932\pi\)
							−0.532120	+	0.846669i	\(0.678604\pi\)
\(17\)	2.73780	+	0.996479i	0.664015	+	0.241682i	0.651969	−	0.758246i	\(-0.273943\pi\)
							0.0120460	+	0.999927i	\(0.496166\pi\)
\(19\)	−1.05355	−	4.22966i	−0.241700	−	0.970351i
							\(23\)	−3.44474	−	2.89048i	−0.718277	−	0.602706i	0.208631	−	0.977994i	\(-0.433099\pi\)
−0.926908	+	0.375288i	\(0.877544\pi\)
\(29\)	−1.84862	+	3.96438i	−0.343280	+	0.736166i	−0.999836	−	0.0181110i	\(-0.994235\pi\)
							0.656556	+	0.754277i	\(0.272013\pi\)
\(31\)	3.06762	−	5.31327i	0.550961	−	0.954292i	−0.447245	−	0.894412i	\(-0.647595\pi\)
							0.998206	−	0.0598803i	\(-0.0190719\pi\)
\(37\)	−5.70901	−	5.70901i	−0.938556	−	0.938556i	0.0596629	−	0.998219i	\(-0.480997\pi\)
							−0.998219	+	0.0596629i	\(0.980997\pi\)
\(41\)	−1.56074	−	8.85140i	−0.243747	−	1.38236i	−0.823385	−	0.567483i	\(-0.807917\pi\)
							0.579638	−	0.814874i	\(-0.303194\pi\)
\(43\)	6.21072	+	0.543368i	0.947126	+	0.0828628i	0.550241	−	0.835006i	\(-0.314536\pi\)
							0.396885	+	0.917868i	\(0.370091\pi\)
\(47\)	1.46611	+	4.02811i	0.213854	+	0.587560i	0.999516	−	0.0310957i	\(-0.00989967\pi\)
							−0.785662	+	0.618656i	\(0.787677\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.243.24	yes	456
16.11	odd	4	inner	304.2.bg.a.91.6	✓	456
19.14	odd	18	inner	304.2.bg.a.147.6	yes	456
304.299	even	36	inner	304.2.bg.a.299.24	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.91.6	✓	456	16.11	odd	4	inner
304.2.bg.a.147.6	yes	456	19.14	odd	18	inner
304.2.bg.a.243.24	yes	456	1.1	even	1	trivial
304.2.bg.a.299.24	yes	456	304.299	even	36	inner