Embedded newform 304.2.bg.a.219.8

• Label: 304.2.bg.a.219.8
• Level: $304$
• Weight: $2$
• Character: 304.219
• 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			219.8
Character	\(\chi\)	\(=\)	304.219
Dual form			304.2.bg.a.211.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18222 + 0.776119i) q^{2} +(0.0312064 - 0.356691i) q^{3} +(0.795277 - 1.83508i) q^{4} +(-0.653263 - 1.40093i) q^{5} +(0.239942 + 0.445906i) q^{6} +(-0.194818 + 0.337434i) q^{7} +(0.484054 + 2.78670i) q^{8} +(2.82817 + 0.498682i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18222	+	0.776119i	−0.835954	+	0.548799i
							\(3\)	0.0312064	−	0.356691i	0.0180170	−	0.205935i	−0.981850	−	0.189657i	\(-0.939263\pi\)
0.999867	−	0.0162788i	\(-0.00518193\pi\)
\(5\)	−0.653263	−	1.40093i	−0.292148	−	0.626514i	0.704375	−	0.709828i	\(-0.251227\pi\)
							−0.996523	+	0.0833140i	\(0.973450\pi\)
\(7\)	−0.194818	+	0.337434i	−0.0736342	+	0.127538i	−0.900492	−	0.434874i	\(-0.856793\pi\)
							0.826857	+	0.562412i	\(0.190126\pi\)
\(11\)	−3.89654	−	1.04407i	−1.17485	−	0.314800i	−0.381969	−	0.924175i	\(-0.624754\pi\)
							−0.792882	+	0.609375i	\(0.791421\pi\)
\(13\)	−0.256979	−	2.93729i	−0.0712733	−	0.814657i	−0.944750	−	0.327791i	\(-0.893696\pi\)
							0.873477	−	0.486866i	\(-0.161860\pi\)
\(17\)	−0.936888	−	5.31336i	−0.227229	−	1.28868i	−0.858378	−	0.513017i	\(-0.828528\pi\)
							0.631150	−	0.775661i	\(-0.282583\pi\)
\(19\)	1.31261	−	4.15657i	0.301133	−	0.953582i
							\(23\)	4.60497	−	1.67607i	0.960203	−	0.349485i	0.186090	−	0.982533i	\(-0.440418\pi\)
0.774113	+	0.633047i	\(0.218196\pi\)
\(29\)	−8.65034	+	6.05704i	−1.60633	+	1.12476i	−0.690396	+	0.723432i	\(0.742564\pi\)
							−0.915933	+	0.401332i	\(0.868547\pi\)
\(31\)	−2.73136	+	4.73086i	−0.490567	+	0.849688i	−0.999941	−	0.0108577i	\(-0.996544\pi\)
							0.509374	+	0.860545i	\(0.329877\pi\)
\(37\)	7.64023	−	7.64023i	1.25605	−	1.25605i	0.303082	−	0.952964i	\(-0.401984\pi\)
							0.952964	−	0.303082i	\(-0.0980158\pi\)
\(41\)	8.86814	−	7.44125i	1.38497	−	1.16213i	0.417639	−	0.908613i	\(-0.362858\pi\)
							0.967331	−	0.253516i	\(-0.0815869\pi\)
\(43\)	0.633249	−	0.295289i	0.0965695	−	0.0450311i	−0.373733	−	0.927536i	\(-0.621922\pi\)
							0.470303	+	0.882505i	\(0.344145\pi\)
\(47\)	−10.9878	−	1.93744i	−1.60273	−	0.282605i	−0.700433	−	0.713718i	\(-0.747010\pi\)
							−0.902298	+	0.431113i	\(0.858121\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.219.8	yes	456
16.3	odd	4	inner	304.2.bg.a.67.3	yes	456
19.2	odd	18	inner	304.2.bg.a.59.3	✓	456
304.211	even	36	inner	304.2.bg.a.211.8	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.59.3	✓	456	19.2	odd	18	inner
304.2.bg.a.67.3	yes	456	16.3	odd	4	inner
304.2.bg.a.211.8	yes	456	304.211	even	36	inner
304.2.bg.a.219.8	yes	456	1.1	even	1	trivial