Embedded newform 368.3.v.a.5.3

• Label: 368.3.v.a.5.3
• 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			5.3
Character	\(\chi\)	\(=\)	368.5
Dual form			368.3.v.a.221.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99447 - 0.148577i) q^{2} +(0.388837 + 5.43666i) q^{3} +(3.95585 + 0.592668i) q^{4} +(-2.80954 - 3.75311i) q^{5} +(0.0322389 - 10.9010i) q^{6} +(2.99409 + 6.55615i) q^{7} +(-7.80178 - 1.76981i) q^{8} +(-20.4977 + 2.94712i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1880 q - 18 q^{2} - 18 q^{3} - 6 q^{4} - 22 q^{5} - 24 q^{6} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99447	−	0.148577i	−0.997237	−	0.0742887i
							\(3\)	0.388837	+	5.43666i	0.129612	+	1.81222i	0.481791	+	0.876286i	\(0.339986\pi\)
−0.352178	+	0.935933i	\(0.614559\pi\)
\(5\)	−2.80954	−	3.75311i	−0.561909	−	0.750622i	0.426158	−	0.904649i	\(-0.359867\pi\)
							−0.988067	+	0.154026i	\(0.950776\pi\)
\(7\)	2.99409	+	6.55615i	0.427727	+	0.936592i	0.993690	+	0.112161i	\(0.0357773\pi\)
							−0.565963	+	0.824431i	\(0.691495\pi\)
\(11\)	9.66738	−	17.7045i	0.878853	−	1.60950i	0.0891426	−	0.996019i	\(-0.471587\pi\)
							0.789710	−	0.613480i	\(-0.210231\pi\)
\(13\)	−5.92753	−	15.8923i	−0.455964	−	1.22249i	−0.938092	−	0.346385i	\(-0.887409\pi\)
							0.482129	−	0.876100i	\(-0.339864\pi\)
\(17\)	−17.2218	−	26.7976i	−1.01304	−	1.57633i	−0.800639	−	0.599147i	\(-0.795506\pi\)
							−0.212406	−	0.977181i	\(-0.568130\pi\)
\(19\)	3.02454	+	13.9036i	0.159186	+	0.731768i	0.985653	+	0.168783i	\(0.0539836\pi\)
							−0.826467	+	0.562985i	\(0.809653\pi\)
\(23\)	−20.9958	+	9.39022i	−0.912861	+	0.408271i
							\(29\)	−1.57034	+	7.21876i	−0.0541498	+	0.248923i	−0.996263	−	0.0863726i	\(-0.972472\pi\)
0.942113	+	0.335295i	\(0.108836\pi\)
\(31\)	−10.4732	−	12.0867i	−0.337846	−	0.389895i	0.561250	−	0.827646i	\(-0.310321\pi\)
							−0.899096	+	0.437751i	\(0.855775\pi\)
\(37\)	−19.7302	−	14.7698i	−0.533247	−	0.399184i	0.298440	−	0.954428i	\(-0.403534\pi\)
							−0.831687	+	0.555244i	\(0.812625\pi\)
\(41\)	27.6694	+	3.97826i	0.674863	+	0.0970307i	0.471222	−	0.882015i	\(-0.343813\pi\)
							0.203642	+	0.979046i	\(0.434722\pi\)
\(43\)	0.931213	+	13.0201i	0.0216561	+	0.302792i	0.996724	+	0.0808834i	\(0.0257742\pi\)
							−0.975067	+	0.221909i	\(0.928771\pi\)
\(47\)	18.5231			0.394108			0.197054	−	0.980393i	\(-0.436863\pi\)
							0.197054	+	0.980393i	\(0.436863\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	368.3.v.a.5.3	✓	1880
16.13	even	4	inner	368.3.v.a.189.15	yes	1880
23.14	odd	22	inner	368.3.v.a.37.15	yes	1880
368.221	odd	44	inner	368.3.v.a.221.3	yes	1880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
368.3.v.a.5.3	✓	1880	1.1	even	1	trivial
368.3.v.a.37.15	yes	1880	23.14	odd	22	inner
368.3.v.a.189.15	yes	1880	16.13	even	4	inner
368.3.v.a.221.3	yes	1880	368.221	odd	44	inner