Embedded newform 33.5.g.a.13.1

• Label: 33.5.g.a.13.1
• 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			13.1
Character	\(\chi\)	\(=\)	33.13
Dual form			33.5.g.a.28.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.29601 - 2.37062i) q^{2} +(4.20378 - 3.05422i) q^{3} +(34.6676 + 25.1875i) q^{4} +(2.18122 + 6.71310i) q^{5} +(-37.9112 + 12.3181i) q^{6} +(28.6722 - 39.4639i) q^{7} +(-121.078 - 166.650i) q^{8} +(8.34346 - 25.6785i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 76 q^{4} + 36 q^{5} + 150 q^{7} + 480 q^{8} - 216 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.29601	−	2.37062i	−1.82400	−	0.592654i	−0.999647	−	0.0265805i	\(-0.991538\pi\)
							−0.824355	−	0.566074i	\(-0.808462\pi\)
\(3\)	4.20378	−	3.05422i	0.467086	−	0.339358i
							\(5\)	2.18122	+	6.71310i	0.0872488	+	0.268524i	0.985156	−	0.171660i	\(-0.0549131\pi\)
−0.897907	+	0.440184i	\(0.854913\pi\)
\(7\)	28.6722	−	39.4639i	0.585147	−	0.805385i	−0.409101	−	0.912489i	\(-0.634158\pi\)
							0.994248	+	0.107104i	\(0.0341577\pi\)
\(11\)	−31.5805	−	116.806i	−0.260996	−	0.965340i
							\(13\)	5.63636	+	1.83136i	0.0333512	+	0.0108365i	0.325645	−	0.945492i	\(-0.394419\pi\)
−0.292294	+	0.956329i	\(0.594419\pi\)
\(17\)	404.280	−	131.359i	1.39889	−	0.454528i	0.490061	−	0.871688i	\(-0.336975\pi\)
							0.908833	+	0.417160i	\(0.136975\pi\)
\(19\)	−347.268	−	477.974i	−0.961963	−	1.32403i	−0.946004	−	0.324155i	\(-0.894920\pi\)
							−0.0159586	−	0.999873i	\(-0.505080\pi\)
\(23\)	361.694			0.683731			0.341866	−	0.939749i	\(-0.388941\pi\)
							0.341866	+	0.939749i	\(0.388941\pi\)
\(29\)	−755.357	+	1039.66i	−0.898165	+	1.23622i	0.0728848	+	0.997340i	\(0.476779\pi\)
							−0.971050	+	0.238878i	\(0.923221\pi\)
\(31\)	−36.4276	+	112.113i	−0.0379059	+	0.116662i	−0.968219	−	0.250104i	\(-0.919535\pi\)
							0.930313	+	0.366766i	\(0.119535\pi\)
\(37\)	−141.071	−	102.494i	−0.103047	−	0.0748677i	0.535069	−	0.844809i	\(-0.320286\pi\)
							−0.638115	+	0.769941i	\(0.720286\pi\)
\(41\)	1117.70	+	1538.38i	0.664902	+	0.915159i	0.999631	−	0.0271553i	\(-0.00864486\pi\)
							−0.334729	+	0.942314i	\(0.608645\pi\)
\(43\)			1419.27i			0.767585i	0.923419	+	0.383793i	\(0.125382\pi\)
							−0.923419	+	0.383793i	\(0.874618\pi\)
\(47\)	−177.347	+	128.850i	−0.0802840	+	0.0583297i	−0.627203	−	0.778856i	\(-0.715800\pi\)
							0.546919	+	0.837185i	\(0.315800\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.g.a.13.1	✓	32
3.2	odd	2		99.5.k.c.46.8		32
11.4	even	5		363.5.c.e.241.1		32
11.6	odd	10	inner	33.5.g.a.28.1	yes	32
11.7	odd	10		363.5.c.e.241.32		32
33.17	even	10		99.5.k.c.28.8		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.g.a.13.1	✓	32	1.1	even	1	trivial
33.5.g.a.28.1	yes	32	11.6	odd	10	inner
99.5.k.c.28.8		32	33.17	even	10
99.5.k.c.46.8		32	3.2	odd	2
363.5.c.e.241.1		32	11.4	even	5
363.5.c.e.241.32		32	11.7	odd	10