Embedded newform 368.3.v.a.5.1

• Label: 368.3.v.a.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	368.5
Dual form			368.3.v.a.221.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99866 + 0.0732790i) q^{2} +(0.278870 + 3.89911i) q^{3} +(3.98926 - 0.292919i) q^{4} +(-3.40401 - 4.54722i) q^{5} +(-0.843088 - 7.77254i) q^{6} +(-4.72083 - 10.3372i) q^{7} +(-7.95170 + 0.877774i) q^{8} +(-6.21688 + 0.893852i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1880 q - 18 q^{2} - 18 q^{3} - 6 q^{4} - 22 q^{5} - 24 q^{6} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99866	+	0.0732790i	−0.999329	+	0.0366395i
							\(3\)	0.278870	+	3.89911i	0.0929566	+	1.29970i	0.803184	+	0.595731i	\(0.203137\pi\)
−0.710228	+	0.703972i	\(0.751408\pi\)
\(5\)	−3.40401	−	4.54722i	−0.680802	−	0.909444i	0.318457	−	0.947937i	\(-0.396835\pi\)
							−0.999259	+	0.0384928i	\(0.987744\pi\)
\(7\)	−4.72083	−	10.3372i	−0.674404	−	1.47674i	−0.868466	−	0.495749i	\(-0.834894\pi\)
							0.194062	−	0.980989i	\(-0.437834\pi\)
\(11\)	−4.13822	+	7.57858i	−0.376202	+	0.688962i	−0.995351	−	0.0963112i	\(-0.969296\pi\)
							0.619150	+	0.785273i	\(0.287477\pi\)
\(13\)	−1.05944	−	2.84046i	−0.0814951	−	0.218497i	0.889811	−	0.456329i	\(-0.150836\pi\)
							−0.971306	+	0.237832i	\(0.923563\pi\)
\(17\)	16.1259	+	25.0924i	0.948582	+	1.47602i	0.878082	+	0.478511i	\(0.158823\pi\)
							0.0705006	+	0.997512i	\(0.477540\pi\)
\(19\)	2.07806	+	9.55269i	0.109372	+	0.502773i	0.998943	+	0.0459761i	\(0.0146398\pi\)
							−0.889571	+	0.456797i	\(0.848997\pi\)
\(23\)	15.3069	+	17.1668i	0.665518	+	0.746382i
							\(29\)	0.183049	−	0.841462i	0.00631203	−	0.0290159i	−0.973883	−	0.227050i	\(-0.927092\pi\)
0.980195	+	0.198034i	\(0.0634556\pi\)
\(31\)	36.7861	+	42.4534i	1.18665	+	1.36947i	0.913165	+	0.407589i	\(0.133630\pi\)
							0.273483	+	0.961877i	\(0.411824\pi\)
\(37\)	−49.3606	−	36.9509i	−1.33407	−	0.998672i	−0.998392	−	0.0566952i	\(-0.981944\pi\)
							−0.335678	−	0.941977i	\(-0.608965\pi\)
\(41\)	46.6525	+	6.70762i	1.13787	+	0.163600i	0.685390	−	0.728176i	\(-0.259632\pi\)
							0.452476	+	0.891777i	\(0.350541\pi\)
\(43\)	−0.346654	−	4.84686i	−0.00806173	−	0.112718i	0.991828	−	0.127584i	\(-0.0407223\pi\)
							−0.999889	+	0.0148663i	\(0.995268\pi\)
\(47\)	61.7766			1.31440			0.657198	−	0.753718i	\(-0.271741\pi\)
							0.657198	+	0.753718i	\(0.271741\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	368.3.v.a.5.1	✓	1880
16.13	even	4	inner	368.3.v.a.189.18	yes	1880
23.14	odd	22	inner	368.3.v.a.37.18	yes	1880
368.221	odd	44	inner	368.3.v.a.221.1	yes	1880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
368.3.v.a.5.1	✓	1880	1.1	even	1	trivial
368.3.v.a.37.18	yes	1880	23.14	odd	22	inner
368.3.v.a.189.18	yes	1880	16.13	even	4	inner
368.3.v.a.221.1	yes	1880	368.221	odd	44	inner