Embedded newform 304.2.bg.a.203.31

• Label: 304.2.bg.a.203.31
• Level: $304$
• Weight: $2$
• Character: 304.203
• 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			203.31
Character	\(\chi\)	\(=\)	304.203
Dual form			304.2.bg.a.3.31

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.20620 + 0.738303i) q^{2} +(0.681394 - 0.317739i) q^{3} +(0.909818 + 1.78108i) q^{4} +(-2.29150 + 3.27260i) q^{5} +(1.05648 + 0.119819i) q^{6} +(2.44491 - 4.23471i) q^{7} +(-0.217555 + 2.82005i) q^{8} +(-1.56502 + 1.86512i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.20620	+	0.738303i	0.852909	+	0.522059i
							\(3\)	0.681394	−	0.317739i	0.393403	−	0.183447i	−0.215843	−	0.976428i	\(-0.569250\pi\)
0.609246	+	0.792981i	\(0.291472\pi\)
\(5\)	−2.29150	+	3.27260i	−1.02479	+	1.46355i	−0.144274	+	0.989538i	\(0.546085\pi\)
							−0.880517	+	0.474015i	\(0.842804\pi\)
\(7\)	2.44491	−	4.23471i	0.924090	−	1.60057i	0.131071	−	0.991373i	\(-0.458158\pi\)
							0.793019	−	0.609197i	\(-0.208508\pi\)
\(11\)	1.70548	+	0.456982i	0.514221	+	0.137785i	0.506592	−	0.862186i	\(-0.330905\pi\)
							0.00762901	+	0.999971i	\(0.497572\pi\)
\(13\)	3.65501	+	1.70436i	1.01372	+	0.472704i	0.857207	−	0.514973i	\(-0.172198\pi\)
							0.156510	+	0.987676i	\(0.449976\pi\)
\(17\)	−1.14503	+	0.960795i	−0.277711	+	0.233027i	−0.770995	−	0.636841i	\(-0.780241\pi\)
							0.493284	+	0.869868i	\(0.335796\pi\)
\(19\)	−3.55082	−	2.52817i	−0.814614	−	0.580003i
							\(23\)	0.963427	−	5.46387i	0.200889	−	1.13930i	−0.702891	−	0.711297i	\(-0.748108\pi\)
0.903780	−	0.427998i	\(-0.140781\pi\)
\(29\)	−0.181435	−	2.07382i	−0.0336917	−	0.385098i	−0.994255	−	0.107037i	\(-0.965864\pi\)
							0.960563	−	0.278061i	\(-0.0896918\pi\)
\(31\)	2.78124	−	4.81725i	0.499525	−	0.865203i	−0.500475	−	0.865751i	\(-0.666841\pi\)
							1.00000		0.000548098i	\(0.000174465\pi\)
\(37\)	5.74063	−	5.74063i	0.943754	−	0.943754i	−0.0547461	−	0.998500i	\(-0.517435\pi\)
							0.998500	+	0.0547461i	\(0.0174349\pi\)
\(41\)	3.28077	+	1.19410i	0.512370	+	0.186487i	0.585249	−	0.810853i	\(-0.300997\pi\)
							−0.0728794	+	0.997341i	\(0.523219\pi\)
\(43\)	−2.02572	−	1.41843i	−0.308920	−	0.216308i	0.408827	−	0.912612i	\(-0.365938\pi\)
							−0.717747	+	0.696304i	\(0.754827\pi\)
\(47\)	0.704603	−	0.839713i	0.102777	−	0.122485i	−0.712204	−	0.701973i	\(-0.752303\pi\)
							0.814981	+	0.579488i	\(0.196747\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.203.31	yes	456
16.3	odd	4	inner	304.2.bg.a.51.24	yes	456
19.3	odd	18	inner	304.2.bg.a.155.24	yes	456
304.3	even	36	inner	304.2.bg.a.3.31	✓	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.31	✓	456	304.3	even	36	inner
304.2.bg.a.51.24	yes	456	16.3	odd	4	inner
304.2.bg.a.155.24	yes	456	19.3	odd	18	inner
304.2.bg.a.203.31	yes	456	1.1	even	1	trivial