Embedded newform 33.8.f.a.2.9

• Label: 33.8.f.a.2.9
• 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			2.9
Character	\(\chi\)	\(=\)	33.2
Dual form			33.8.f.a.17.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.85991 + 8.80190i) q^{2} +(44.0288 - 15.7627i) q^{3} +(34.2599 + 24.8913i) q^{4} +(118.636 - 38.5472i) q^{5} +(12.8232 + 432.617i) q^{6} +(768.087 - 1057.18i) q^{7} +(-1275.45 + 926.669i) q^{8} +(1690.07 - 1388.03i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 42 q^{3} - 1542 q^{4} - 1725 q^{6} - 10 q^{7} + 550 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.85991	+	8.80190i	−0.252783	+	0.777985i	0.741476	+	0.670980i	\(0.234126\pi\)
							−0.994258	+	0.107005i	\(0.965874\pi\)
\(3\)	44.0288	−	15.7627i	0.941483	−	0.337059i
							\(5\)	118.636	−	38.5472i	0.424446	−	0.137911i	−0.0890023	−	0.996031i	\(-0.528368\pi\)
0.513448	+	0.858121i	\(0.328368\pi\)
\(7\)	768.087	−	1057.18i	0.846384	−	1.16495i	−0.138264	−	0.990395i	\(-0.544152\pi\)
							0.984648	−	0.174552i	\(-0.0558477\pi\)
\(11\)	1939.11	−	3965.73i	0.439267	−	0.898357i
							\(13\)	−3694.44	−	1200.40i	−0.466387	−	0.151538i	0.0663902	−	0.997794i	\(-0.478852\pi\)
−0.532778	+	0.846255i	\(0.678852\pi\)
\(17\)	10747.4	+	33077.0i	0.530556	+	1.63288i	0.753060	+	0.657951i	\(0.228577\pi\)
							−0.222505	+	0.974932i	\(0.571423\pi\)
\(19\)	23672.9	+	32583.0i	0.791797	+	1.08982i	0.993882	+	0.110447i	\(0.0352283\pi\)
							−0.202085	+	0.979368i	\(0.564772\pi\)
\(23\)		−	7071.57i		−	0.121190i	−0.998162	−	0.0605952i	\(-0.980700\pi\)
							0.998162	−	0.0605952i	\(-0.0192999\pi\)
\(29\)	−151594.	−	110140.i	−1.15422	−	0.838591i	−0.165186	−	0.986263i	\(-0.552822\pi\)
							−0.989037	+	0.147671i	\(0.952822\pi\)
\(31\)	38067.2	−	117159.i	0.229501	−	0.706333i	−0.768302	−	0.640088i	\(-0.778898\pi\)
							0.997803	−	0.0662452i	\(-0.0211020\pi\)
\(37\)	280907.	+	204091.i	0.911710	+	0.662396i	0.941447	−	0.337162i	\(-0.109467\pi\)
							−0.0297367	+	0.999558i	\(0.509467\pi\)
\(41\)	−386476.	+	280791.i	−0.875748	+	0.636268i	−0.932123	−	0.362141i	\(-0.882046\pi\)
							0.0563750	+	0.998410i	\(0.482046\pi\)
\(43\)			320485.i			0.614707i	0.951595	+	0.307354i	\(0.0994435\pi\)
							−0.951595	+	0.307354i	\(0.900557\pi\)
\(47\)	−582947.	−	802358.i	−0.819005	−	1.12726i	−0.989871	−	0.141970i	\(-0.954656\pi\)
							0.170866	−	0.985294i	\(-0.445344\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.f.a.2.9	✓	104
3.2	odd	2	inner	33.8.f.a.2.18	yes	104
11.6	odd	10	inner	33.8.f.a.17.18	yes	104
33.17	even	10	inner	33.8.f.a.17.9	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.f.a.2.9	✓	104	1.1	even	1	trivial
33.8.f.a.2.18	yes	104	3.2	odd	2	inner
33.8.f.a.17.9	yes	104	33.17	even	10	inner
33.8.f.a.17.18	yes	104	11.6	odd	10	inner