Embedded newform 33.7.g.a.13.3

• Label: 33.7.g.a.13.3
• Level: $33$
• Weight: $7$
• Character: 33.13
• 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			13.3
Character	\(\chi\)	\(=\)	33.13
Dual form			33.7.g.a.28.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.54864 - 3.10254i) q^{2} +(-12.6113 + 9.16267i) q^{3} +(29.7736 + 21.6318i) q^{4} +(33.3013 + 102.491i) q^{5} +(148.849 - 48.3638i) q^{6} +(-90.4877 + 124.546i) q^{7} +(160.505 + 220.916i) q^{8} +(75.0911 - 231.107i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 204 q^{4} - 224 q^{5} + 720 q^{7} - 3200 q^{8} - 2916 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.54864	−	3.10254i	−1.19358	−	0.387818i	−0.356184	−	0.934416i	\(-0.615923\pi\)
							−0.837395	+	0.546598i	\(0.815923\pi\)
\(3\)	−12.6113	+	9.16267i	−0.467086	+	0.339358i
							\(5\)	33.3013	+	102.491i	0.266410	+	0.819927i	0.991365	+	0.131130i	\(0.0418606\pi\)
−0.724955	+	0.688796i	\(0.758139\pi\)
\(7\)	−90.4877	+	124.546i	−0.263813	+	0.363107i	−0.920289	−	0.391240i	\(-0.872046\pi\)
							0.656476	+	0.754347i	\(0.272046\pi\)
\(11\)	−37.8004	−	1330.46i	−0.0284000	−	0.999597i
							\(13\)	−3185.84	−	1035.14i	−1.45009	−	0.471161i	−0.525059	−	0.851066i	\(-0.675957\pi\)
−0.925026	+	0.379904i	\(0.875957\pi\)
\(17\)	4525.60	−	1470.46i	0.921148	−	0.299299i	0.190210	−	0.981743i	\(-0.439083\pi\)
							0.730938	+	0.682444i	\(0.239083\pi\)
\(19\)	−3071.31	−	4227.29i	−0.447778	−	0.616314i	0.524140	−	0.851632i	\(-0.324387\pi\)
							−0.971918	+	0.235318i	\(0.924387\pi\)
\(23\)	8173.50			0.671776			0.335888	−	0.941902i	\(-0.390964\pi\)
							0.335888	+	0.941902i	\(0.390964\pi\)
\(29\)	−3257.25	+	4483.23i	−0.133554	+	0.183822i	−0.870556	−	0.492069i	\(-0.836241\pi\)
							0.737002	+	0.675891i	\(0.236241\pi\)
\(31\)	6185.05	−	19035.6i	0.207615	−	0.638973i	−0.791981	−	0.610546i	\(-0.790950\pi\)
							0.999596	−	0.0284269i	\(-0.00904979\pi\)
\(37\)	−12848.7	−	9335.13i	−0.253661	−	0.184296i	0.453687	−	0.891161i	\(-0.350109\pi\)
							−0.707348	+	0.706866i	\(0.750109\pi\)
\(41\)	−78557.8	−	108126.i	−1.13982	−	1.56883i	−0.767901	−	0.640569i	\(-0.778699\pi\)
							−0.371923	−	0.928264i	\(-0.621301\pi\)
\(43\)		−	111600.i		−	1.40364i	−0.712352	−	0.701822i	\(-0.752370\pi\)
							0.712352	−	0.701822i	\(-0.247630\pi\)
\(47\)	75743.3	−	55030.7i	0.729542	−	0.530043i	−0.159876	−	0.987137i	\(-0.551110\pi\)
							0.889419	+	0.457094i	\(0.151110\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.g.a.13.3	✓	48
3.2	odd	2		99.7.k.b.46.10		48
11.6	odd	10	inner	33.7.g.a.28.3	yes	48
33.17	even	10		99.7.k.b.28.10		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.g.a.13.3	✓	48	1.1	even	1	trivial
33.7.g.a.28.3	yes	48	11.6	odd	10	inner
99.7.k.b.28.10		48	33.17	even	10
99.7.k.b.46.10		48	3.2	odd	2