Embedded newform 368.3.v.a.5.4

• Label: 368.3.v.a.5.4
• Level: $368$
• Weight: $3$
• Character: 368.5
• Analytic conductor: $10.027$
• Analytic rank: $0$
• Dimension: $1880$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	368.v (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.0272737285\)
Analytic rank:	\(0\)
Dimension:	\(1880\)
Relative dimension:	\(94\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label			5.4
Character	\(\chi\)	\(=\)	368.5
Dual form			368.3.v.a.221.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.98354 + 0.256098i) q^{2} +(0.0694366 + 0.970850i) q^{3} +(3.86883 - 1.01596i) q^{4} +(0.750260 + 1.00223i) q^{5} +(-0.386362 - 1.90793i) q^{6} +(5.35949 + 11.7356i) q^{7} +(-7.41377 + 3.00599i) q^{8} +(7.97066 - 1.14601i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1880 q - 18 q^{2} - 18 q^{3} - 6 q^{4} - 22 q^{5} - 24 q^{6} - 18 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/368\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(97\)	\(277\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{22}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.98354	+	0.256098i	−0.991768	+	0.128049i
							\(3\)	0.0694366	+	0.970850i	0.0231455	+	0.323617i	0.995813	+	0.0914141i	\(0.0291387\pi\)
−0.972667	+	0.232203i	\(0.925407\pi\)
\(5\)	0.750260	+	1.00223i	0.150052	+	0.200446i	0.869269	−	0.494340i	\(-0.164590\pi\)
							−0.719217	+	0.694786i	\(0.755499\pi\)
\(7\)	5.35949	+	11.7356i	0.765641	+	1.67652i	0.736020	+	0.676959i	\(0.236703\pi\)
							0.0296210	+	0.999561i	\(0.490570\pi\)
\(11\)	−8.60990	+	15.7679i	−0.782718	+	1.43344i	0.115230	+	0.993339i	\(0.463240\pi\)
							−0.897948	+	0.440102i	\(0.854942\pi\)
\(13\)	−4.52628	−	12.1354i	−0.348175	−	0.933494i	−0.986505	−	0.163731i	\(-0.947647\pi\)
							0.638330	−	0.769763i	\(-0.279626\pi\)
\(17\)	−2.22171	−	3.45705i	−0.130689	−	0.203356i	0.769742	−	0.638355i	\(-0.220385\pi\)
							−0.900431	+	0.435000i	\(0.856749\pi\)
\(19\)	0.602041	+	2.76754i	0.0316863	+	0.145660i	0.990272	−	0.139144i	\(-0.0444351\pi\)
							−0.958586	+	0.284804i	\(0.908072\pi\)
\(23\)	20.8495	+	9.71071i	0.906501	+	0.422205i
							\(29\)	−10.1400	+	46.6126i	−0.349654	+	1.60733i	0.380994	+	0.924578i	\(0.375582\pi\)
−0.730647	+	0.682755i	\(0.760782\pi\)
\(31\)	−31.0698	−	35.8565i	−1.00225	−	1.15666i	−0.987635	−	0.156768i	\(-0.949893\pi\)
							−0.0146172	−	0.999893i	\(-0.504653\pi\)
\(37\)	−12.7878	−	9.57286i	−0.345617	−	0.258726i	0.412340	−	0.911030i	\(-0.364712\pi\)
							−0.757957	+	0.652304i	\(0.773803\pi\)
\(41\)	22.2962	+	3.20572i	0.543811	+	0.0781882i	0.408745	−	0.912648i	\(-0.365966\pi\)
							0.135065	+	0.990837i	\(0.456876\pi\)
\(43\)	0.605363	+	8.46408i	0.0140782	+	0.196839i	0.999624	+	0.0274285i	\(0.00873187\pi\)
							−0.985546	+	0.169411i	\(0.945814\pi\)
\(47\)	−24.8496			−0.528715			−0.264358	−	0.964425i	\(-0.585160\pi\)
							−0.264358	+	0.964425i	\(0.585160\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	368.3.v.a.5.4	✓	1880
16.13	even	4	inner	368.3.v.a.189.21	yes	1880
23.14	odd	22	inner	368.3.v.a.37.21	yes	1880
368.221	odd	44	inner	368.3.v.a.221.4	yes	1880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
368.3.v.a.5.4	✓	1880	1.1	even	1	trivial
368.3.v.a.37.21	yes	1880	23.14	odd	22	inner
368.3.v.a.189.21	yes	1880	16.13	even	4	inner
368.3.v.a.221.4	yes	1880	368.221	odd	44	inner