Embedded newform 33.4.f.a.8.2

• Label: 33.4.f.a.8.2
• 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			8.2
Character	\(\chi\)	\(=\)	33.8
Dual form			33.4.f.a.29.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.30778 + 2.40324i) q^{2} +(2.27962 + 4.66940i) q^{3} +(2.69368 - 8.29030i) q^{4} +(-3.98805 + 5.48909i) q^{5} +(-18.7622 - 9.96686i) q^{6} +(-17.8876 - 5.81204i) q^{7} +(0.905818 + 2.78782i) q^{8} +(-16.6067 + 21.2889i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 3 q^{3} - 38 q^{4} + 45 q^{6} - 10 q^{7} - 65 q^{9}+O(q^{10}) \)

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	−0.990946	−	0.134261i	\(-0.957134\pi\)
							−0.178529	+	0.983935i	\(0.557134\pi\)
\(3\)	2.27962	+	4.66940i	0.438713	+	0.898627i
							\(5\)	−3.98805	+	5.48909i	−0.356702	+	0.490959i	−0.949226	−	0.314594i	\(-0.898132\pi\)
0.592524	+	0.805553i	\(0.298132\pi\)
\(7\)	−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	0.460826	−	0.887490i	\(-0.347553\pi\)
							−0.701650	+	0.712522i	\(0.747553\pi\)
\(19\)	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	\(0.396864\pi\)
							−0.318372	+	0.947966i	\(0.603136\pi\)
\(29\)	69.0033	−	212.370i	0.441848	−	1.35987i	−0.444055	−	0.895999i	\(-0.646461\pi\)
							0.885904	−	0.463869i	\(-0.153539\pi\)
\(31\)	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	0.987867	−	0.155303i	\(-0.0496353\pi\)
							−0.890486	+	0.455011i	\(0.849635\pi\)
\(43\)			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	−0.0899616	−	0.995945i	\(-0.528674\pi\)
							0.512621	+	0.858615i	\(0.328674\pi\)

(See \(a_n\) instead)

Twists

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

    

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