Embedded newform 33.7.g.a.7.6

• Label: 33.7.g.a.7.6
• Level: $33$
• Weight: $7$
• Character: 33.7
• Analytic conductor: $7.592$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	33.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.59178475946\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			7.6
Character	\(\chi\)	\(=\)	33.7
Dual form			33.7.g.a.19.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.61569 - 2.22380i) q^{2} +(4.81710 - 14.8255i) q^{3} +(17.4422 + 53.6817i) q^{4} +(-125.017 + 90.8304i) q^{5} +(-25.1861 - 34.6656i) q^{6} +(-357.347 + 116.109i) q^{7} +(314.870 + 102.307i) q^{8} +(-196.591 - 142.832i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 204 q^{4} - 224 q^{5} + 720 q^{7} - 3200 q^{8} - 2916 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.61569	−	2.22380i	0.201961	−	0.277975i	−0.696008	−	0.718034i	\(-0.745042\pi\)
							0.897969	+	0.440059i	\(0.145042\pi\)
\(3\)	4.81710	−	14.8255i	0.178411	−	0.549093i
							\(5\)	−125.017	+	90.8304i	−1.00014	+	0.726643i	−0.962118	−	0.272634i	\(-0.912105\pi\)
−0.0380203	+	0.999277i	\(0.512105\pi\)
\(7\)	−357.347	+	116.109i	−1.04183	+	0.338511i	−0.779457	−	0.626456i	\(-0.784505\pi\)
							−0.262372	+	0.964967i	\(0.584505\pi\)
\(11\)	736.647	+	1108.56i	0.553454	+	0.832880i
							\(13\)	−946.161	+	1302.28i	−0.430660	+	0.592753i	−0.968105	−	0.250546i	\(-0.919390\pi\)
0.537444	+	0.843299i	\(0.319390\pi\)
\(17\)	−1030.10	−	1417.82i	−0.209669	−	0.288584i	0.691211	−	0.722653i	\(-0.257078\pi\)
							−0.900880	+	0.434069i	\(0.857078\pi\)
\(19\)	1244.48	+	404.355i	0.181437	+	0.0589525i	0.398327	−	0.917244i	\(-0.369591\pi\)
							−0.216889	+	0.976196i	\(0.569591\pi\)
\(23\)	9846.97			0.809318			0.404659	−	0.914468i	\(-0.367390\pi\)
							0.404659	+	0.914468i	\(0.367390\pi\)
\(29\)	12444.8	−	4043.57i	0.510263	−	0.165795i	−0.0425586	−	0.999094i	\(-0.513551\pi\)
							0.552822	+	0.833299i	\(0.313551\pi\)
\(31\)	−37424.4	−	27190.4i	−1.25623	−	0.912705i	−0.257665	−	0.966234i	\(-0.582953\pi\)
							−0.998566	+	0.0535293i	\(0.982953\pi\)
\(37\)	−20507.1	−	63114.2i	−0.404854	−	1.24601i	−0.921017	−	0.389521i	\(-0.872640\pi\)
							0.516164	−	0.856490i	\(-0.327360\pi\)
\(41\)	38488.6	+	12505.7i	0.558445	+	0.181450i	0.574622	−	0.818419i	\(-0.305149\pi\)
							−0.0161762	+	0.999869i	\(0.505149\pi\)
\(43\)			119921.i			1.50830i	0.656700	+	0.754152i	\(0.271952\pi\)
							−0.656700	+	0.754152i	\(0.728048\pi\)
\(47\)	−6393.80	+	19678.1i	−0.0615836	+	0.189535i	−0.977115	−	0.212712i	\(-0.931771\pi\)
							0.915531	+	0.402247i	\(0.131771\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.g.a.7.6	✓	48
3.2	odd	2		99.7.k.b.73.7		48
11.8	odd	10	inner	33.7.g.a.19.6	yes	48
33.8	even	10		99.7.k.b.19.7		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.g.a.7.6	✓	48	1.1	even	1	trivial
33.7.g.a.19.6	yes	48	11.8	odd	10	inner
99.7.k.b.19.7		48	33.8	even	10
99.7.k.b.73.7		48	3.2	odd	2