Embedded newform 304.2.bi.a.237.34

• Label: 304.2.bi.a.237.34
• Level: $304$
• Weight: $2$
• Character: 304.237
• 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			237.34
Character	\(\chi\)	\(=\)	304.237
Dual form			304.2.bi.a.245.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.29221 + 0.574630i) q^{2} +(1.64395 + 0.143827i) q^{3} +(1.33960 + 1.48508i) q^{4} +(-3.55647 + 1.65841i) q^{5} +(2.04167 + 1.13051i) q^{6} +(1.19273 + 0.688626i) q^{7} +(0.877668 + 2.68881i) q^{8} +(-0.272552 - 0.0480582i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(97\)	\(191\)	\(229\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.29221	+	0.574630i	0.913729	+	0.406325i
							\(3\)	1.64395	+	0.143827i	0.949133	+	0.0830383i	0.551197	−	0.834375i	\(-0.314171\pi\)
0.397935	+	0.917413i	\(0.369727\pi\)
\(5\)	−3.55647	+	1.65841i	−1.59050	+	0.741664i	−0.997958	−	0.0638735i	\(-0.979655\pi\)
							−0.592545	+	0.805537i	\(0.701877\pi\)
\(7\)	1.19273	+	0.688626i	0.450811	+	0.260276i	0.708173	−	0.706039i	\(-0.249520\pi\)
							−0.257361	+	0.966315i	\(0.582853\pi\)
\(11\)	3.66652	+	0.982442i	1.10550	+	0.296217i	0.765002	−	0.644028i	\(-0.222738\pi\)
							0.340497	+	0.940246i	\(0.389405\pi\)
\(13\)	−0.569888	−	6.51385i	−0.158058	−	1.80662i	−0.499074	−	0.866559i	\(-0.666327\pi\)
							0.341016	−	0.940058i	\(-0.389229\pi\)
\(17\)	0.0136591	+	0.0774649i	0.00331283	+	0.0187880i	0.986419	−	0.164246i	\(-0.0525192\pi\)
							−0.983107	+	0.183034i	\(0.941408\pi\)
\(19\)	3.84180	+	2.05926i	0.881370	+	0.472427i
							\(23\)	−2.02671	−	5.56833i	−0.422598	−	1.16108i	−0.950215	−	0.311595i	\(-0.899137\pi\)
0.527617	−	0.849482i	\(-0.323086\pi\)
\(29\)	−0.238823	+	0.167226i	−0.0443483	+	0.0310530i	−0.595540	−	0.803325i	\(-0.703062\pi\)
							0.551192	+	0.834378i	\(0.314173\pi\)
\(31\)	4.34923	−	7.53309i	0.781145	−	1.35298i	−0.150130	−	0.988666i	\(-0.547969\pi\)
							0.931275	−	0.364317i	\(-0.118697\pi\)
\(37\)	−4.72159	+	4.72159i	−0.776224	+	0.776224i	−0.979187	−	0.202962i	\(-0.934943\pi\)
							0.202962	+	0.979187i	\(0.434943\pi\)
\(41\)	1.85495	+	2.21064i	0.289694	+	0.345244i	0.891189	−	0.453633i	\(-0.149872\pi\)
							−0.601495	+	0.798877i	\(0.705428\pi\)
\(43\)	−2.78055	+	1.29659i	−0.424030	+	0.197728i	−0.622907	−	0.782296i	\(-0.714048\pi\)
							0.198877	+	0.980025i	\(0.436271\pi\)
\(47\)	−0.0378307	+	0.214549i	−0.00551817	+	0.0312951i	−0.987443	−	0.157979i	\(-0.949502\pi\)
							0.981924	+	0.189274i	\(0.0606134\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bi.a.237.34	yes	456
16.5	even	4	inner	304.2.bi.a.85.9	✓	456
19.17	even	9	inner	304.2.bi.a.93.9	yes	456
304.245	even	36	inner	304.2.bi.a.245.34	yes	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bi.a.85.9	✓	456	16.5	even	4	inner
304.2.bi.a.93.9	yes	456	19.17	even	9	inner
304.2.bi.a.237.34	yes	456	1.1	even	1	trivial
304.2.bi.a.245.34	yes	456	304.245	even	36	inner