Embedded newform 33.4.f.a.8.9

• Label: 33.4.f.a.8.9
• Level: $33$
• Weight: $4$
• Character: 33.8
• Analytic conductor: $1.947$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.94706303019\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			8.9
Character	\(\chi\)	\(=\)	33.8
Dual form			33.4.f.a.29.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.30778 - 2.40324i) q^{2} +(0.900354 + 5.11755i) q^{3} +(2.69368 - 8.29030i) q^{4} +(3.98805 - 5.48909i) q^{5} +(15.2769 + 14.7640i) q^{6} +(-17.8876 - 5.81204i) q^{7} +(-0.905818 - 2.78782i) q^{8} +(-25.3787 + 9.21522i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 3 q^{3} - 38 q^{4} + 45 q^{6} - 10 q^{7} - 65 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.30778	−	2.40324i	1.16948	−	0.849673i	0.178529	−	0.983935i	\(-0.442866\pi\)
							0.990946	+	0.134261i	\(0.0428661\pi\)
\(3\)	0.900354	+	5.11755i	0.173273	+	0.984874i
							\(5\)	3.98805	−	5.48909i	0.356702	−	0.490959i	−0.592524	−	0.805553i	\(-0.701868\pi\)
0.949226	+	0.314594i	\(0.101868\pi\)
\(7\)	−17.8876	−	5.81204i	−0.965841	−	0.313821i	−0.216705	−	0.976237i	\(-0.569531\pi\)
							−0.749136	+	0.662416i	\(0.769531\pi\)
\(11\)	−27.8487	−	23.5680i	−0.763336	−	0.646001i
							\(13\)	24.8886	+	34.2562i	0.530989	+	0.730844i	0.987281	−	0.158986i	\(-0.0508226\pi\)
−0.456291	+	0.889830i	\(0.650823\pi\)
\(17\)	16.8800	+	12.2640i	0.240824	+	0.174969i	0.701650	−	0.712522i	\(-0.252447\pi\)
							−0.460826	+	0.887490i	\(0.652447\pi\)
\(19\)	4.70027	−	1.52721i	0.0567535	−	0.0184403i	−0.280503	−	0.959853i	\(-0.590501\pi\)
							0.337256	+	0.941413i	\(0.390501\pi\)
\(23\)		−	209.129i		−	1.89593i	−0.318372	−	0.947966i	\(-0.603136\pi\)
							0.318372	−	0.947966i	\(-0.396864\pi\)
\(29\)	−69.0033	+	212.370i	−0.441848	+	1.35987i	0.444055	+	0.895999i	\(0.353539\pi\)
							−0.885904	+	0.463869i	\(0.846461\pi\)
\(31\)	195.762	−	142.230i	1.13419	−	0.824039i	0.147892	−	0.989003i	\(-0.452751\pi\)
							0.986299	+	0.164965i	\(0.0527511\pi\)
\(37\)	51.4758	−	158.426i	0.228718	−	0.703922i	−0.769175	−	0.639038i	\(-0.779332\pi\)
							0.997893	−	0.0648834i	\(-0.0206675\pi\)
\(41\)	−25.5653	−	78.6818i	−0.0973811	−	0.299708i	0.890486	−	0.455011i	\(-0.150365\pi\)
							−0.987867	+	0.155303i	\(0.950365\pi\)
\(43\)			255.149i			0.904882i	0.891794	+	0.452441i	\(0.149447\pi\)
							−0.891794	+	0.452441i	\(0.850553\pi\)
\(47\)	−136.188	+	44.2500i	−0.422660	+	0.137330i	−0.512621	−	0.858615i	\(-0.671326\pi\)
							0.0899616	+	0.995945i	\(0.471326\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.4.f.a.8.9	yes	40
3.2	odd	2	inner	33.4.f.a.8.2	✓	40
11.2	odd	10		363.4.d.d.362.33		40
11.7	odd	10	inner	33.4.f.a.29.2	yes	40
11.9	even	5		363.4.d.d.362.7		40
33.2	even	10		363.4.d.d.362.8		40
33.20	odd	10		363.4.d.d.362.34		40
33.29	even	10	inner	33.4.f.a.29.9	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.f.a.8.2	✓	40	3.2	odd	2	inner
33.4.f.a.8.9	yes	40	1.1	even	1	trivial
33.4.f.a.29.2	yes	40	11.7	odd	10	inner
33.4.f.a.29.9	yes	40	33.29	even	10	inner
363.4.d.d.362.7		40	11.9	even	5
363.4.d.d.362.8		40	33.2	even	10
363.4.d.d.362.33		40	11.2	odd	10
363.4.d.d.362.34		40	33.20	odd	10