Embedded newform 304.2.bg.a.243.32

• Label: 304.2.bg.a.243.32
• 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			243.32
Character	\(\chi\)	\(=\)	304.243
Dual form			304.2.bg.a.299.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.18428 + 0.772970i) q^{2} +(-1.24207 + 1.77385i) q^{3} +(0.805035 + 1.83082i) q^{4} +(-0.255654 + 2.92214i) q^{5} +(-2.84209 + 1.14066i) q^{6} +(1.88433 - 3.26375i) q^{7} +(-0.461786 + 2.79048i) q^{8} +(-0.577768 - 1.58740i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.18428	+	0.772970i	0.837412	+	0.546572i
							\(3\)	−1.24207	+	1.77385i	−0.717107	+	1.02413i	0.280819	+	0.959761i	\(0.409394\pi\)
−0.997926	+	0.0643738i	\(0.979495\pi\)
\(5\)	−0.255654	+	2.92214i	−0.114332	+	1.30682i	0.694805	+	0.719198i	\(0.255491\pi\)
							−0.809137	+	0.587620i	\(0.800065\pi\)
\(7\)	1.88433	−	3.26375i	0.712210	−	1.23358i	−0.251816	−	0.967775i	\(-0.581028\pi\)
							0.964026	−	0.265808i	\(-0.0856387\pi\)
\(11\)	0.159229	−	0.594249i	0.0480092	−	0.179173i	−0.937758	−	0.347290i	\(-0.887102\pi\)
							0.985767	+	0.168117i	\(0.0537687\pi\)
\(13\)	−1.02039	−	1.45727i	−0.283006	−	0.404175i	0.652330	−	0.757935i	\(-0.273792\pi\)
							−0.935336	+	0.353761i	\(0.884903\pi\)
\(17\)	−4.42546	−	1.61073i	−1.07333	−	0.390660i	−0.255909	−	0.966701i	\(-0.582375\pi\)
							−0.817421	+	0.576040i	\(0.804597\pi\)
\(19\)	4.35873	+	0.0379962i	0.999962	+	0.00871692i
							\(23\)	3.92486	+	3.29335i	0.818390	+	0.686711i	0.952594	−	0.304243i	\(-0.0984034\pi\)
−0.134204	+	0.990954i	\(0.542848\pi\)
\(29\)	−1.17893	+	2.52822i	−0.218922	+	0.469479i	−0.984841	−	0.173457i	\(-0.944506\pi\)
							0.765920	+	0.642936i	\(0.222284\pi\)
\(31\)	4.02355	−	6.96899i	0.722650	−	1.25167i	−0.237283	−	0.971440i	\(-0.576257\pi\)
							0.959934	−	0.280227i	\(-0.0904097\pi\)
\(37\)	4.07650	+	4.07650i	0.670173	+	0.670173i	0.957756	−	0.287583i	\(-0.0928517\pi\)
							−0.287583	+	0.957756i	\(0.592852\pi\)
\(41\)	0.342202	+	1.94073i	0.0534430	+	0.303091i	0.999799	−	0.0200346i	\(-0.00637764\pi\)
							−0.946356	+	0.323125i	\(0.895267\pi\)
\(43\)	10.7326	+	0.938979i	1.63670	+	0.143193i	0.868051	−	0.496475i	\(-0.165373\pi\)
							0.768651	+	0.639668i	\(0.220928\pi\)
\(47\)	−0.597021	−	1.64030i	−0.0870845	−	0.239263i	0.888504	−	0.458869i	\(-0.151745\pi\)
							−0.975589	+	0.219606i	\(0.929523\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.243.32	yes	456
16.11	odd	4	inner	304.2.bg.a.91.3	✓	456
19.14	odd	18	inner	304.2.bg.a.147.3	yes	456
304.299	even	36	inner	304.2.bg.a.299.32	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.91.3	✓	456	16.11	odd	4	inner
304.2.bg.a.147.3	yes	456	19.14	odd	18	inner
304.2.bg.a.243.32	yes	456	1.1	even	1	trivial
304.2.bg.a.299.32	yes	456	304.299	even	36	inner