Embedded newform 33.8.f.a.2.1

• Label: 33.8.f.a.2.1
• 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			2.1
Character	\(\chi\)	\(=\)	33.2
Dual form			33.8.f.a.17.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.60852 + 20.3389i) q^{2} +(-0.462116 + 46.7631i) q^{3} +(-266.445 - 193.584i) q^{4} +(133.323 - 43.3192i) q^{5} +(-948.057 - 318.434i) q^{6} +(864.199 - 1189.47i) q^{7} +(3483.52 - 2530.93i) q^{8} +(-2186.57 - 43.2200i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 42 q^{3} - 1542 q^{4} - 1725 q^{6} - 10 q^{7} + 550 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.60852	+	20.3389i	−0.584116	+	1.79772i	0.0186767	+	0.999826i	\(0.494055\pi\)
							−0.602793	+	0.797898i	\(0.705945\pi\)
\(3\)	−0.462116	+	46.7631i	−0.00988159	+	0.999951i
							\(5\)	133.323	−	43.3192i	0.476990	−	0.154983i	−0.0606482	−	0.998159i	\(-0.519317\pi\)
0.537638	+	0.843176i	\(0.319317\pi\)
\(7\)	864.199	−	1189.47i	0.952293	−	1.31072i	0.00179099	−	0.999998i	\(-0.499430\pi\)
							0.950502	−	0.310720i	\(-0.100570\pi\)
\(11\)	−3621.82	−	2523.80i	−0.820451	−	0.571717i
							\(13\)	−11236.2	−	3650.87i	−1.41846	−	0.460887i	−0.503350	−	0.864083i	\(-0.667899\pi\)
−0.915114	+	0.403196i	\(0.867899\pi\)
\(17\)	−23.7809	−	73.1901i	−0.00117397	−	0.00361311i	0.950468	−	0.310823i	\(-0.100605\pi\)
							−0.951642	+	0.307210i	\(0.900605\pi\)
\(19\)	−8070.50	−	11108.1i	−0.269937	−	0.371537i	0.652431	−	0.757848i	\(-0.273749\pi\)
							−0.922369	+	0.386311i	\(0.873749\pi\)
\(23\)		−	27539.8i		−	0.471969i	−0.971757	−	0.235985i	\(-0.924168\pi\)
							0.971757	−	0.235985i	\(-0.0758315\pi\)
\(29\)	10328.8	+	7504.29i	0.0786421	+	0.0571369i	0.626412	−	0.779492i	\(-0.284523\pi\)
							−0.547770	+	0.836629i	\(0.684523\pi\)
\(31\)	−80.5914	+	248.035i	−0.000485873	+	0.00149536i	−0.951299	−	0.308269i	\(-0.900250\pi\)
							0.950813	+	0.309765i	\(0.100250\pi\)
\(37\)	−215419.	−	156511.i	−0.699161	−	0.507970i	0.180498	−	0.983575i	\(-0.442229\pi\)
							−0.879659	+	0.475605i	\(0.842229\pi\)
\(41\)	−263411.	+	191379.i	−0.596885	+	0.433662i	−0.844772	−	0.535127i	\(-0.820264\pi\)
							0.247887	+	0.968789i	\(0.420264\pi\)
\(43\)		−	66393.8i		−	0.127347i	−0.997971	−	0.0636734i	\(-0.979718\pi\)
							0.997971	−	0.0636734i	\(-0.0202816\pi\)
\(47\)	231025.	+	317978.i	0.324575	+	0.446740i	0.939857	−	0.341567i	\(-0.110958\pi\)
							−0.615282	+	0.788307i	\(0.710958\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.f.a.2.1	✓	104
3.2	odd	2	inner	33.8.f.a.2.26	yes	104
11.6	odd	10	inner	33.8.f.a.17.26	yes	104
33.17	even	10	inner	33.8.f.a.17.1	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.f.a.2.1	✓	104	1.1	even	1	trivial
33.8.f.a.2.26	yes	104	3.2	odd	2	inner
33.8.f.a.17.1	yes	104	33.17	even	10	inner
33.8.f.a.17.26	yes	104	11.6	odd	10	inner