Embedded newform 304.2.bg.a.3.5

• Label: 304.2.bg.a.3.5
• Level: $304$
• Weight: $2$
• Character: 304.3
• 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			3.5
Character	\(\chi\)	\(=\)	304.3
Dual form			304.2.bg.a.203.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.25803 + 0.646026i) q^{2} +(-0.566911 - 0.264355i) q^{3} +(1.16530 - 1.62545i) q^{4} +(0.256415 + 0.366198i) q^{5} +(0.883974 - 0.0336715i) q^{6} +(1.25795 + 2.17884i) q^{7} +(-0.415910 + 2.79768i) q^{8} +(-1.67686 - 1.99840i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.25803	+	0.646026i	−0.889565	+	0.456809i
							\(3\)	−0.566911	−	0.264355i	−0.327306	−	0.152625i	0.252018	−	0.967723i	\(-0.418906\pi\)
−0.579324	+	0.815097i	\(0.696684\pi\)
\(5\)	0.256415	+	0.366198i	0.114672	+	0.163769i	0.872385	−	0.488819i	\(-0.162572\pi\)
							−0.757713	+	0.652588i	\(0.773683\pi\)
\(7\)	1.25795	+	2.17884i	0.475462	+	0.823524i	0.999605	−	0.0281063i	\(-0.00894770\pi\)
							−0.524143	+	0.851630i	\(0.675614\pi\)
\(11\)	0.832774	−	0.223141i	0.251091	−	0.0672796i	−0.131078	−	0.991372i	\(-0.541844\pi\)
							0.382169	+	0.924092i	\(0.375177\pi\)
\(13\)	1.36533	−	0.636665i	0.378675	−	0.176579i	−0.223955	−	0.974599i	\(-0.571897\pi\)
							0.602631	+	0.798020i	\(0.294119\pi\)
\(17\)	4.44273	+	3.72789i	1.07752	+	0.904146i	0.995712	−	0.0925023i	\(-0.0294865\pi\)
							0.0818067	+	0.996648i	\(0.473931\pi\)
\(19\)	4.31158	+	0.640500i	0.989145	+	0.146941i
							\(23\)	0.791682	+	4.48985i	0.165077	+	0.936199i	0.948985	+	0.315322i	\(0.102112\pi\)
−0.783908	+	0.620877i	\(0.786776\pi\)
\(29\)	−0.0961428	+	1.09892i	−0.0178533	+	0.204064i	0.982029	+	0.188729i	\(0.0604369\pi\)
							−0.999882	+	0.0153346i	\(0.995119\pi\)
\(31\)	4.69403	+	8.13030i	0.843073	+	1.46024i	0.887285	+	0.461222i	\(0.152589\pi\)
							−0.0442121	+	0.999022i	\(0.514078\pi\)
\(37\)	−2.26352	−	2.26352i	−0.372120	−	0.372120i	0.496129	−	0.868249i	\(-0.334754\pi\)
							−0.868249	+	0.496129i	\(0.834754\pi\)
\(41\)	−8.36068	+	3.04304i	−1.30572	+	0.475243i	−0.898855	−	0.438246i	\(-0.855600\pi\)
							−0.406864	+	0.913489i	\(0.633378\pi\)
\(43\)	6.02331	−	4.21757i	0.918546	−	0.643173i	−0.0156715	−	0.999877i	\(-0.504989\pi\)
							0.934217	+	0.356705i	\(0.116100\pi\)
\(47\)	4.50379	+	5.36741i	0.656946	+	0.782918i	0.986944	−	0.161064i	\(-0.0514926\pi\)
							−0.329998	+	0.943982i	\(0.607048\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.3.5	✓	456
16.11	odd	4	inner	304.2.bg.a.155.18	yes	456
19.13	odd	18	inner	304.2.bg.a.51.18	yes	456
304.203	even	36	inner	304.2.bg.a.203.5	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.5	✓	456	1.1	even	1	trivial
304.2.bg.a.51.18	yes	456	19.13	odd	18	inner
304.2.bg.a.155.18	yes	456	16.11	odd	4	inner
304.2.bg.a.203.5	yes	456	304.203	even	36	inner