Embedded newform 33.5.g.a.13.7

• Label: 33.5.g.a.13.7
• Level: $33$
• Weight: $5$
• Character: 33.13
• Analytic conductor: $3.411$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	33.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.41120878177\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			13.7
Character	\(\chi\)	\(=\)	33.13
Dual form			33.5.g.a.28.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.41780 + 2.08527i) q^{2} +(4.20378 - 3.05422i) q^{3} +(23.8955 + 17.3611i) q^{4} +(-9.80980 - 30.1915i) q^{5} +(33.3478 - 10.8354i) q^{6} +(-46.3989 + 63.8626i) q^{7} +(53.6911 + 73.8995i) q^{8} +(8.34346 - 25.6785i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 76 q^{4} + 36 q^{5} + 150 q^{7} + 480 q^{8} - 216 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.41780	+	2.08527i	1.60445	+	0.521317i	0.968202	−	0.250169i	\(-0.0804861\pi\)
							0.636247	+	0.771486i	\(0.280486\pi\)
\(3\)	4.20378	−	3.05422i	0.467086	−	0.339358i
							\(5\)	−9.80980	−	30.1915i	−0.392392	−	1.20766i	−0.930974	−	0.365086i	\(-0.881040\pi\)
0.538582	−	0.842573i	\(-0.318960\pi\)
\(7\)	−46.3989	+	63.8626i	−0.946916	+	1.30332i	0.00596768	+	0.999982i	\(0.498100\pi\)
							−0.952884	+	0.303336i	\(0.901900\pi\)
\(11\)	−96.8877	+	72.4830i	−0.800725	+	0.599033i
							\(13\)	69.5034	+	22.5830i	0.411263	+	0.133627i	0.507339	−	0.861747i	\(-0.330629\pi\)
−0.0960762	+	0.995374i	\(0.530629\pi\)
\(17\)	445.953	−	144.899i	1.54309	−	0.501381i	0.590864	−	0.806771i	\(-0.298787\pi\)
							0.952227	+	0.305391i	\(0.0987870\pi\)
\(19\)	−154.694	−	212.918i	−0.428516	−	0.589801i	0.539096	−	0.842244i	\(-0.318766\pi\)
							−0.967612	+	0.252443i	\(0.918766\pi\)
\(23\)	−96.1864			−0.181827			−0.0909134	−	0.995859i	\(-0.528979\pi\)
							−0.0909134	+	0.995859i	\(0.528979\pi\)
\(29\)	240.426	−	330.919i	0.285882	−	0.393482i	−0.641789	−	0.766881i	\(-0.721808\pi\)
							0.927671	+	0.373399i	\(0.121808\pi\)
\(31\)	−174.101	+	535.829i	−0.181167	+	0.557574i	−0.999861	−	0.0166553i	\(-0.994698\pi\)
							0.818695	+	0.574229i	\(0.194698\pi\)
\(37\)	94.1211	+	68.3830i	0.0687517	+	0.0499511i	0.621630	−	0.783311i	\(-0.286471\pi\)
							−0.552879	+	0.833262i	\(0.686471\pi\)
\(41\)	1897.25	+	2611.34i	1.12864	+	1.55344i	0.790623	+	0.612303i	\(0.209757\pi\)
							0.338018	+	0.941140i	\(0.390243\pi\)
\(43\)			729.533i			0.394555i	0.980348	+	0.197278i	\(0.0632100\pi\)
							−0.980348	+	0.197278i	\(0.936790\pi\)
\(47\)	−57.5441	+	41.8083i	−0.0260499	+	0.0189263i	−0.600734	−	0.799449i	\(-0.705125\pi\)
							0.574684	+	0.818375i	\(0.305125\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.g.a.13.7	✓	32
3.2	odd	2		99.5.k.c.46.2		32
11.4	even	5		363.5.c.e.241.28		32
11.6	odd	10	inner	33.5.g.a.28.7	yes	32
11.7	odd	10		363.5.c.e.241.5		32
33.17	even	10		99.5.k.c.28.2		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.g.a.13.7	✓	32	1.1	even	1	trivial
33.5.g.a.28.7	yes	32	11.6	odd	10	inner
99.5.k.c.28.2		32	33.17	even	10
99.5.k.c.46.2		32	3.2	odd	2
363.5.c.e.241.5		32	11.7	odd	10
363.5.c.e.241.28		32	11.4	even	5