Embedded newform 33.5.g.a.13.5

• Label: 33.5.g.a.13.5
• 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			13.5
Character	\(\chi\)	\(=\)	33.13
Dual form			33.5.g.a.28.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.743344 + 0.241527i) q^{2} +(-4.20378 + 3.05422i) q^{3} +(-12.4500 - 9.04549i) q^{4} +(-9.71709 - 29.9061i) q^{5} +(-3.86253 + 1.25501i) q^{6} +(16.8776 - 23.2301i) q^{7} +(-14.4205 - 19.8482i) q^{8} +(8.34346 - 25.6785i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 76 q^{4} + 36 q^{5} + 150 q^{7} + 480 q^{8} - 216 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.743344	+	0.241527i	0.185836	+	0.0603818i	0.400457	−	0.916316i	\(-0.368851\pi\)
							−0.214621	+	0.976697i	\(0.568851\pi\)
\(3\)	−4.20378	+	3.05422i	−0.467086	+	0.339358i
							\(5\)	−9.71709	−	29.9061i	−0.388684	−	1.19625i	−0.933773	−	0.357867i	\(-0.883504\pi\)
0.545089	−	0.838378i	\(-0.316496\pi\)
\(7\)	16.8776	−	23.2301i	0.344441	−	0.474083i	−0.601291	−	0.799030i	\(-0.705347\pi\)
							0.945732	+	0.324947i	\(0.105347\pi\)
\(11\)	−24.4924	+	118.495i	−0.202416	+	0.979300i
							\(13\)	−294.943	−	95.8329i	−1.74523	−	0.567058i	−0.749720	−	0.661755i	\(-0.769812\pi\)
−0.995506	+	0.0946966i	\(0.969812\pi\)
\(17\)	397.531	−	129.166i	1.37554	−	0.446940i	0.474340	−	0.880342i	\(-0.342687\pi\)
							0.901201	+	0.433402i	\(0.142687\pi\)
\(19\)	−144.131	−	198.379i	−0.399255	−	0.549527i	0.561302	−	0.827611i	\(-0.310301\pi\)
							−0.960557	+	0.278084i	\(0.910301\pi\)
\(23\)	529.873			1.00165			0.500825	−	0.865548i	\(-0.333030\pi\)
							0.500825	+	0.865548i	\(0.333030\pi\)
\(29\)	164.437	−	226.329i	0.195526	−	0.269119i	−0.699985	−	0.714157i	\(-0.746810\pi\)
							0.895511	+	0.445039i	\(0.146810\pi\)
\(31\)	443.155	−	1363.89i	0.461140	−	1.41924i	−0.402634	−	0.915361i	\(-0.631905\pi\)
							0.863773	−	0.503881i	\(-0.168095\pi\)
\(37\)	−1466.49	−	1065.47i	−1.07122	−	0.778284i	−0.0950852	−	0.995469i	\(-0.530312\pi\)
							−0.976130	+	0.217185i	\(0.930312\pi\)
\(41\)	−289.849	−	398.943i	−0.172427	−	0.237325i	0.714054	−	0.700091i	\(-0.246857\pi\)
							−0.886481	+	0.462766i	\(0.846857\pi\)
\(43\)			287.561i			0.155522i	0.996972	+	0.0777612i	\(0.0247772\pi\)
							−0.996972	+	0.0777612i	\(0.975223\pi\)
\(47\)	−369.914	+	268.759i	−0.167458	+	0.121665i	−0.668358	−	0.743840i	\(-0.733002\pi\)
							0.500900	+	0.865505i	\(0.333002\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.g.a.13.5	✓	32
3.2	odd	2		99.5.k.c.46.4		32
11.4	even	5		363.5.c.e.241.20		32
11.6	odd	10	inner	33.5.g.a.28.5	yes	32
11.7	odd	10		363.5.c.e.241.13		32
33.17	even	10		99.5.k.c.28.4		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.g.a.13.5	✓	32	1.1	even	1	trivial
33.5.g.a.28.5	yes	32	11.6	odd	10	inner
99.5.k.c.28.4		32	33.17	even	10
99.5.k.c.46.4		32	3.2	odd	2
363.5.c.e.241.13		32	11.7	odd	10
363.5.c.e.241.20		32	11.4	even	5