Embedded newform 33.8.f.a.2.20

• Label: 33.8.f.a.2.20
• Level: $33$
• Weight: $8$
• Character: 33.2
• Analytic conductor: $10.309$
• Analytic rank: $0$
• Dimension: $104$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	33.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3087058410\)
Analytic rank:	\(0\)
Dimension:	\(104\)
Relative dimension:	\(26\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			2.20
Character	\(\chi\)	\(=\)	33.2
Dual form			33.8.f.a.17.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.82493 - 11.7719i) q^{2} +(44.8842 + 13.1306i) q^{3} +(-20.3938 - 14.8170i) q^{4} +(22.0489 - 7.16411i) q^{5} +(326.251 - 478.149i) q^{6} +(49.3778 - 67.9627i) q^{7} +(1029.34 - 747.857i) q^{8} +(1842.17 + 1178.71i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 42 q^{3} - 1542 q^{4} - 1725 q^{6} - 10 q^{7} + 550 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.82493	−	11.7719i	0.338079	−	1.04050i	−0.627106	−	0.778934i	\(-0.715761\pi\)
							0.965185	−	0.261567i	\(-0.0842392\pi\)
\(3\)	44.8842	+	13.1306i	0.959773	+	0.280776i
							\(5\)	22.0489	−	7.16411i	0.0788844	−	0.0256311i	−0.269309	−	0.963054i	\(-0.586795\pi\)
0.348193	+	0.937423i	\(0.386795\pi\)
\(7\)	49.3778	−	67.9627i	0.0544112	−	0.0748907i	−0.780944	−	0.624601i	\(-0.785262\pi\)
							0.835356	+	0.549710i	\(0.185262\pi\)
\(11\)	4347.42	−	766.243i	0.984820	−	0.173577i
							\(13\)	−2166.72	−	704.010i	−0.273527	−	0.0888744i	0.169042	−	0.985609i	\(-0.445933\pi\)
−0.442569	+	0.896734i	\(0.645933\pi\)
\(17\)	2457.83	+	7564.41i	0.121333	+	0.373425i	0.993215	−	0.116291i	\(-0.0371004\pi\)
							−0.871882	+	0.489716i	\(0.837100\pi\)
\(19\)	−26304.9	−	36205.6i	−0.879830	−	1.21098i	−0.976468	−	0.215663i	\(-0.930809\pi\)
							0.0966373	−	0.995320i	\(-0.469191\pi\)
\(23\)		−	4847.59i		−	0.0830765i	−0.999137	−	0.0415382i	\(-0.986774\pi\)
							0.999137	−	0.0415382i	\(-0.0132258\pi\)
\(29\)	54996.6	+	39957.4i	0.418739	+	0.304231i	0.777130	−	0.629340i	\(-0.216675\pi\)
							−0.358392	+	0.933571i	\(0.616675\pi\)
\(31\)	−66917.5	+	205951.i	−0.403435	+	1.24165i	0.518759	+	0.854920i	\(0.326394\pi\)
							−0.922195	+	0.386726i	\(0.873606\pi\)
\(37\)	−27808.6	−	20204.1i	−0.0902553	−	0.0655743i	0.541742	−	0.840545i	\(-0.317765\pi\)
							−0.631998	+	0.774970i	\(0.717765\pi\)
\(41\)	−429481.	+	312036.i	−0.973196	+	0.707068i	−0.956178	−	0.292787i	\(-0.905417\pi\)
							−0.0170180	+	0.999855i	\(0.505417\pi\)
\(43\)		−	156862.i		−	0.300870i	−0.988620	−	0.150435i	\(-0.951933\pi\)
							0.988620	−	0.150435i	\(-0.0480674\pi\)
\(47\)	60816.6	+	83706.9i	0.0854436	+	0.117603i	0.849599	−	0.527429i	\(-0.176844\pi\)
							−0.764155	+	0.645032i	\(0.776844\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.f.a.2.20	yes	104
3.2	odd	2	inner	33.8.f.a.2.7	✓	104
11.6	odd	10	inner	33.8.f.a.17.7	yes	104
33.17	even	10	inner	33.8.f.a.17.20	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.f.a.2.7	✓	104	3.2	odd	2	inner
33.8.f.a.2.20	yes	104	1.1	even	1	trivial
33.8.f.a.17.7	yes	104	11.6	odd	10	inner
33.8.f.a.17.20	yes	104	33.17	even	10	inner