Embedded newform 304.2.bi.a.213.25

• Label: 304.2.bi.a.213.25
• Level: $304$
• Weight: $2$
• Character: 304.213
• 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.bi (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			213.25
Character	\(\chi\)	\(=\)	304.213
Dual form			304.2.bi.a.157.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.491099 + 1.32621i) q^{2} +(-0.463391 - 0.324470i) q^{3} +(-1.51764 + 1.30260i) q^{4} +(2.08276 + 0.182218i) q^{5} +(0.202743 - 0.773899i) q^{6} +(2.46100 + 1.42086i) q^{7} +(-2.47282 - 1.37300i) q^{8} +(-0.916610 - 2.51836i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)	\(1\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.491099	+	1.32621i	0.347259	+	0.937769i
							\(3\)	−0.463391	−	0.324470i	−0.267539	−	0.187333i	0.432107	−	0.901822i	\(-0.357770\pi\)
−0.699646	+	0.714489i	\(0.746659\pi\)
\(5\)	2.08276	+	0.182218i	0.931438	+	0.0814902i	0.542764	−	0.839885i	\(-0.317378\pi\)
							0.388674	+	0.921375i	\(0.372933\pi\)
\(7\)	2.46100	+	1.42086i	0.930169	+	0.537034i	0.886865	−	0.462028i	\(-0.152878\pi\)
							0.0433043	+	0.999062i	\(0.486212\pi\)
\(11\)	−0.910071	+	3.39643i	−0.274397	+	1.02406i	0.681848	+	0.731494i	\(0.261177\pi\)
							−0.956244	+	0.292569i	\(0.905490\pi\)
\(13\)	3.98237	+	5.68742i	1.10451	+	1.57741i	0.773519	+	0.633773i	\(0.218495\pi\)
							0.330993	+	0.943633i	\(0.392616\pi\)
\(17\)	−0.157983	−	0.0575010i	−0.0383165	−	0.0139460i	0.322791	−	0.946470i	\(-0.395379\pi\)
							−0.361107	+	0.932524i	\(0.617601\pi\)
\(19\)	−0.574438	−	4.32088i	−0.131785	−	0.991278i
							\(23\)	−3.82660	+	4.56037i	−0.797902	+	0.950902i	−0.999592	−	0.0285468i	\(-0.990912\pi\)
0.201691	+	0.979449i	\(0.435356\pi\)
\(29\)	4.08278	−	8.75555i	0.758153	−	1.62587i	−0.0215910	−	0.999767i	\(-0.506873\pi\)
							0.779744	−	0.626098i	\(-0.215349\pi\)
\(31\)	−1.60746	+	2.78421i	−0.288709	+	0.500058i	−0.973502	−	0.228680i	\(-0.926559\pi\)
							0.684793	+	0.728738i	\(0.259893\pi\)
\(37\)	−0.980030	−	0.980030i	−0.161116	−	0.161116i	0.621945	−	0.783061i	\(-0.286343\pi\)
							−0.783061	+	0.621945i	\(0.786343\pi\)
\(41\)	4.30125	−	0.758426i	0.671742	−	0.118446i	0.172634	−	0.984986i	\(-0.444772\pi\)
							0.499108	+	0.866540i	\(0.333661\pi\)
\(43\)	4.19770	+	0.367252i	0.640144	+	0.0560053i	0.402607	−	0.915373i	\(-0.368104\pi\)
							0.237537	+	0.971378i	\(0.423660\pi\)
\(47\)	−0.436460	+	0.158858i	−0.0636642	+	0.0231719i	−0.373656	−	0.927567i	\(-0.621896\pi\)
							0.309992	+	0.950739i	\(0.399674\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bi.a.213.25	yes	456
16.13	even	4	inner	304.2.bi.a.61.33	yes	456
19.5	even	9	inner	304.2.bi.a.5.33	✓	456
304.157	even	36	inner	304.2.bi.a.157.25	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bi.a.5.33	✓	456	19.5	even	9	inner
304.2.bi.a.61.33	yes	456	16.13	even	4	inner
304.2.bi.a.157.25	yes	456	304.157	even	36	inner
304.2.bi.a.213.25	yes	456	1.1	even	1	trivial