Embedded newform 33.8.f.a.2.13

• Label: 33.8.f.a.2.13
• 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			2.13
Character	\(\chi\)	\(=\)	33.2
Dual form			33.8.f.a.17.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.119980 + 0.369259i) q^{2} +(44.9099 - 13.0422i) q^{3} +(103.432 + 75.1479i) q^{4} +(-475.452 + 154.484i) q^{5} +(-0.572341 + 18.1482i) q^{6} +(-851.641 + 1172.18i) q^{7} +(-80.3650 + 58.3886i) q^{8} +(1846.80 - 1171.44i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 42 q^{3} - 1542 q^{4} - 1725 q^{6} - 10 q^{7} + 550 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.119980	+	0.369259i	−0.0106048	+	0.0326382i	−0.956219	−	0.292652i	\(-0.905462\pi\)
							0.945614	+	0.325291i	\(0.105462\pi\)
\(3\)	44.9099	−	13.0422i	0.960325	−	0.278885i
							\(5\)	−475.452	+	154.484i	−1.70103	+	0.552698i	−0.988800	−	0.149248i	\(-0.952315\pi\)
−0.712230	+	0.701946i	\(0.752315\pi\)
\(7\)	−851.641	+	1172.18i	−0.938455	+	1.29167i	0.0180131	+	0.999838i	\(0.494266\pi\)
							−0.956469	+	0.291835i	\(0.905734\pi\)
\(11\)	−4107.26	−	1617.91i	−0.930416	−	0.366504i
							\(13\)	1101.40	+	357.866i	0.139041	+	0.0451771i	0.377711	−	0.925924i	\(-0.376711\pi\)
−0.238670	+	0.971101i	\(0.576711\pi\)
\(17\)	7151.05	+	22008.7i	0.353020	+	1.08648i	0.957149	+	0.289596i	\(0.0935210\pi\)
							−0.604129	+	0.796886i	\(0.706479\pi\)
\(19\)	−5679.28	−	7816.85i	−0.189957	−	0.261453i	0.703407	−	0.710788i	\(-0.251661\pi\)
							−0.893364	+	0.449334i	\(0.851661\pi\)
\(23\)			6857.58i			0.117523i	0.998272	+	0.0587616i	\(0.0187152\pi\)
							−0.998272	+	0.0587616i	\(0.981285\pi\)
\(29\)	145261.	+	105538.i	1.10600	+	0.803559i	0.982030	−	0.188727i	\(-0.0604360\pi\)
							0.123974	+	0.992285i	\(0.460436\pi\)
\(31\)	−33006.5	+	101584.i	−0.198991	+	0.612432i	0.800915	+	0.598777i	\(0.204347\pi\)
							−0.999907	+	0.0136548i	\(0.995653\pi\)
\(37\)	−80661.5	−	58604.0i	−0.261794	−	0.190205i	0.449143	−	0.893460i	\(-0.351729\pi\)
							−0.710938	+	0.703255i	\(0.751729\pi\)
\(41\)	381141.	−	276915.i	0.863659	−	0.627485i	−0.0652193	−	0.997871i	\(-0.520775\pi\)
							0.928878	+	0.370386i	\(0.120775\pi\)
\(43\)			354282.i			0.679531i	0.940510	+	0.339765i	\(0.110348\pi\)
							−0.940510	+	0.339765i	\(0.889652\pi\)
\(47\)	15997.4	+	22018.5i	0.0224753	+	0.0309346i	0.820107	−	0.572211i	\(-0.193914\pi\)
							−0.797631	+	0.603145i	\(0.793914\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.8.f.a.2.13	✓	104
3.2	odd	2	inner	33.8.f.a.2.14	yes	104
11.6	odd	10	inner	33.8.f.a.17.14	yes	104
33.17	even	10	inner	33.8.f.a.17.13	yes	104

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.f.a.2.13	✓	104	1.1	even	1	trivial
33.8.f.a.2.14	yes	104	3.2	odd	2	inner
33.8.f.a.17.13	yes	104	33.17	even	10	inner
33.8.f.a.17.14	yes	104	11.6	odd	10	inner