Embedded newform 304.2.bg.a.203.35

• Label: 304.2.bg.a.203.35
• 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			203.35
Character	\(\chi\)	\(=\)	304.203
Dual form			304.2.bg.a.3.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28817 + 0.583615i) q^{2} +(1.41948 - 0.661916i) q^{3} +(1.31879 + 1.50359i) q^{4} +(0.799685 - 1.14207i) q^{5} +(2.21485 - 0.0242323i) q^{6} +(-0.0847274 + 0.146752i) q^{7} +(0.821310 + 2.70656i) q^{8} +(-0.351561 + 0.418974i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 456 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.28817	+	0.583615i	0.910877	+	0.412678i
							\(3\)	1.41948	−	0.661916i	0.819539	−	0.382158i	0.0328147	−	0.999461i	\(-0.489553\pi\)
0.786725	+	0.617304i	\(0.211775\pi\)
\(5\)	0.799685	−	1.14207i	0.357630	−	0.510748i	−0.599513	−	0.800365i	\(-0.704639\pi\)
							0.957143	+	0.289617i	\(0.0935279\pi\)
\(7\)	−0.0847274	+	0.146752i	−0.0320239	+	0.0554671i	−0.881593	−	0.472010i	\(-0.843529\pi\)
							0.849569	+	0.527477i	\(0.176862\pi\)
\(11\)	−2.03950	−	0.546481i	−0.614931	−	0.164770i	−0.0621086	−	0.998069i	\(-0.519783\pi\)
							−0.552822	+	0.833299i	\(0.686449\pi\)
\(13\)	−4.86801	−	2.26999i	−1.35014	−	0.629582i	−0.393092	−	0.919499i	\(-0.628594\pi\)
							−0.957052	+	0.289917i	\(0.906372\pi\)
\(17\)	0.223989	−	0.187949i	0.0543254	−	0.0455844i	−0.615221	−	0.788355i	\(-0.710933\pi\)
							0.669546	+	0.742771i	\(0.266489\pi\)
\(19\)	2.41985	−	3.62551i	0.555151	−	0.831750i
							\(23\)	−0.157625	+	0.893937i	−0.0328671	+	0.186399i	−0.996821	−	0.0796694i	\(-0.974614\pi\)
0.963954	+	0.266068i	\(0.0857246\pi\)
\(29\)	−0.126498	−	1.44587i	−0.0234900	−	0.268492i	−0.998853	−	0.0478853i	\(-0.984752\pi\)
							0.975363	−	0.220607i	\(-0.0708038\pi\)
\(31\)	1.76886	−	3.06376i	0.317697	−	0.550268i	−0.662310	−	0.749230i	\(-0.730424\pi\)
							0.980007	+	0.198962i	\(0.0637571\pi\)
\(37\)	−5.24468	+	5.24468i	−0.862220	+	0.862220i	−0.991596	−	0.129376i	\(-0.958703\pi\)
							0.129376	+	0.991596i	\(0.458703\pi\)
\(41\)	−0.800764	−	0.291454i	−0.125058	−	0.0455175i	0.278733	−	0.960369i	\(-0.410086\pi\)
							−0.403791	+	0.914851i	\(0.632308\pi\)
\(43\)	−1.02038	−	0.714480i	−0.155607	−	0.108957i	0.493180	−	0.869927i	\(-0.335834\pi\)
							−0.648787	+	0.760970i	\(0.724723\pi\)
\(47\)	6.20173	−	7.39094i	0.904616	−	1.07808i	−0.0919902	−	0.995760i	\(-0.529323\pi\)
							0.996606	−	0.0823192i	\(-0.0262327\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.2.bg.a.203.35	yes	456
16.3	odd	4	inner	304.2.bg.a.51.22	yes	456
19.3	odd	18	inner	304.2.bg.a.155.22	yes	456
304.3	even	36	inner	304.2.bg.a.3.35	✓	456

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
304.2.bg.a.3.35	✓	456	304.3	even	36	inner
304.2.bg.a.51.22	yes	456	16.3	odd	4	inner
304.2.bg.a.155.22	yes	456	19.3	odd	18	inner
304.2.bg.a.203.35	yes	456	1.1	even	1	trivial