Embedded newform 368.3.v.a.5.2

• Label: 368.3.v.a.5.2
• 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			5.2
Character	\(\chi\)	\(=\)	368.5
Dual form			368.3.v.a.221.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99574 + 0.130451i) q^{2} +(-0.357948 - 5.00476i) q^{3} +(3.96597 - 0.520691i) q^{4} +(3.32974 + 4.44801i) q^{5} +(1.36725 + 9.94152i) q^{6} +(1.42980 + 3.13082i) q^{7} +(-7.84712 + 1.55653i) q^{8} +(-16.0112 + 2.30206i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1880 q - 18 q^{2} - 18 q^{3} - 6 q^{4} - 22 q^{5} - 24 q^{6} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99574	+	0.130451i	−0.997871	+	0.0652253i
							\(3\)	−0.357948	−	5.00476i	−0.119316	−	1.66825i	−0.606597	−	0.795010i	\(-0.707466\pi\)
0.487281	−	0.873245i	\(-0.337989\pi\)
\(5\)	3.32974	+	4.44801i	0.665948	+	0.889603i	0.998570	−	0.0534574i	\(-0.0170241\pi\)
							−0.332622	+	0.943060i	\(0.607933\pi\)
\(7\)	1.42980	+	3.13082i	0.204257	+	0.447260i	0.983843	−	0.179036i	\(-0.0572977\pi\)
							−0.779586	+	0.626295i	\(0.784570\pi\)
\(11\)	−0.103244	+	0.189077i	−0.00938579	+	0.0171888i	−0.882330	−	0.470632i	\(-0.844026\pi\)
							0.872944	+	0.487820i	\(0.162208\pi\)
\(13\)	−2.24731	−	6.02528i	−0.172870	−	0.463483i	0.821403	−	0.570348i	\(-0.193192\pi\)
							−0.994274	+	0.106865i	\(0.965919\pi\)
\(17\)	−17.5086	−	27.2440i	−1.02992	−	1.60259i	−0.771157	−	0.636645i	\(-0.780322\pi\)
							−0.258763	−	0.965941i	\(-0.583315\pi\)
\(19\)	1.87156	+	8.60342i	0.0985032	+	0.452812i	0.999807	+	0.0196251i	\(0.00624726\pi\)
							−0.901304	+	0.433187i	\(0.857389\pi\)
\(23\)	1.81466	−	22.9283i	0.0788982	−	0.996883i
							\(29\)	11.3935	−	52.3749i	0.392878	−	1.80603i	−0.176895	−	0.984230i	\(-0.556605\pi\)
0.569773	−	0.821802i	\(-0.307031\pi\)
\(31\)	1.74756	+	2.01679i	0.0563730	+	0.0650579i	0.783236	−	0.621725i	\(-0.213568\pi\)
							−0.726863	+	0.686783i	\(0.759022\pi\)
\(37\)	−43.9859	−	32.9274i	−1.18881	−	0.889930i	−0.192982	−	0.981202i	\(-0.561816\pi\)
							−0.995826	+	0.0912719i	\(0.970907\pi\)
\(41\)	−19.7118	−	2.83413i	−0.480776	−	0.0691252i	−0.102334	−	0.994750i	\(-0.532631\pi\)
							−0.378442	+	0.925625i	\(0.623540\pi\)
\(43\)	0.380197	+	5.31584i	0.00884178	+	0.123624i	0.999956	−	0.00938994i	\(-0.00298896\pi\)
							−0.991114	+	0.133014i	\(0.957534\pi\)
\(47\)	25.1459			0.535019			0.267509	−	0.963555i	\(-0.413799\pi\)
							0.267509	+	0.963555i	\(0.413799\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	368.3.v.a.5.2	✓	1880
16.13	even	4	inner	368.3.v.a.189.19	yes	1880
23.14	odd	22	inner	368.3.v.a.37.19	yes	1880
368.221	odd	44	inner	368.3.v.a.221.2	yes	1880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
368.3.v.a.5.2	✓	1880	1.1	even	1	trivial
368.3.v.a.37.19	yes	1880	23.14	odd	22	inner
368.3.v.a.189.19	yes	1880	16.13	even	4	inner
368.3.v.a.221.2	yes	1880	368.221	odd	44	inner