Embedded newform 33.5.g.a.7.6

• Label: 33.5.g.a.7.6
• Level: $33$
• Weight: $5$
• Character: 33.7
• 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			7.6
Character	\(\chi\)	\(=\)	33.7
Dual form			33.5.g.a.19.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.01888 - 1.40237i) q^{2} +(-1.60570 + 4.94183i) q^{3} +(4.01574 + 12.3592i) q^{4} +(-32.0081 + 23.2552i) q^{5} +(5.29428 + 7.28695i) q^{6} +(27.3116 - 8.87408i) q^{7} +(47.8012 + 15.5316i) q^{8} +(-21.8435 - 15.8702i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 76 q^{4} + 36 q^{5} + 150 q^{7} + 480 q^{8} - 216 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{7}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.01888	−	1.40237i	0.254721	−	0.350594i	−0.662437	−	0.749118i	\(-0.730478\pi\)
							0.917158	+	0.398524i	\(0.130478\pi\)
\(3\)	−1.60570	+	4.94183i	−0.178411	+	0.549093i
							\(5\)	−32.0081	+	23.2552i	−1.28032	+	0.930209i	−0.999563	−	0.0295695i	\(-0.990586\pi\)
−0.280760	+	0.959778i	\(0.590586\pi\)
\(7\)	27.3116	−	8.87408i	0.557380	−	0.181104i	−0.0167621	−	0.999860i	\(-0.505336\pi\)
							0.574142	+	0.818756i	\(0.305336\pi\)
\(11\)	30.6027	+	117.066i	0.252915	+	0.967489i
							\(13\)	94.6157	−	130.227i	0.559856	−	0.770576i	−0.431452	−	0.902136i	\(-0.641999\pi\)
0.991308	+	0.131560i	\(0.0419986\pi\)
\(17\)	−141.875	−	195.274i	−0.490917	−	0.675689i	0.489640	−	0.871925i	\(-0.337128\pi\)
							−0.980557	+	0.196236i	\(0.937128\pi\)
\(19\)	439.357	+	142.756i	1.21705	+	0.395445i	0.846008	−	0.533170i	\(-0.178999\pi\)
							0.371046	+	0.928615i	\(0.378999\pi\)
\(23\)	−241.540			−0.456598			−0.228299	−	0.973591i	\(-0.573316\pi\)
							−0.228299	+	0.973591i	\(0.573316\pi\)
\(29\)	553.272	−	179.769i	0.657874	−	0.213756i	0.0389915	−	0.999240i	\(-0.487585\pi\)
							0.618883	+	0.785483i	\(0.287585\pi\)
\(31\)	580.162	+	421.513i	0.603707	+	0.438619i	0.847193	−	0.531286i	\(-0.178291\pi\)
							−0.243486	+	0.969904i	\(0.578291\pi\)
\(37\)	757.807	+	2332.29i	0.553548	+	1.70365i	0.699748	+	0.714390i	\(0.253296\pi\)
							−0.146200	+	0.989255i	\(0.546704\pi\)
\(41\)	529.997	+	172.207i	0.315287	+	0.102443i	0.462386	−	0.886679i	\(-0.346994\pi\)
							−0.147099	+	0.989122i	\(0.546994\pi\)
\(43\)		−	3168.88i		−	1.71383i	−0.515456	−	0.856916i	\(-0.672377\pi\)
							0.515456	−	0.856916i	\(-0.327623\pi\)
\(47\)	777.518	−	2392.95i	0.351977	−	1.08327i	−0.605764	−	0.795644i	\(-0.707133\pi\)
							0.957741	−	0.287630i	\(-0.0928675\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.g.a.7.6	✓	32
3.2	odd	2		99.5.k.c.73.3		32
11.5	even	5		363.5.c.e.241.21		32
11.6	odd	10		363.5.c.e.241.12		32
11.8	odd	10	inner	33.5.g.a.19.6	yes	32
33.8	even	10		99.5.k.c.19.3		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.g.a.7.6	✓	32	1.1	even	1	trivial
33.5.g.a.19.6	yes	32	11.8	odd	10	inner
99.5.k.c.19.3		32	33.8	even	10
99.5.k.c.73.3		32	3.2	odd	2
363.5.c.e.241.12		32	11.6	odd	10
363.5.c.e.241.21		32	11.5	even	5