Embedded newform 33.5.g.a.13.4

• Label: 33.5.g.a.13.4
• 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			13.4
Character	\(\chi\)	\(=\)	33.13
Dual form			33.5.g.a.28.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.619915 + 0.201423i) q^{2} +(-4.20378 + 3.05422i) q^{3} +(-12.6005 - 9.15483i) q^{4} +(12.5302 + 38.5641i) q^{5} +(-3.22117 + 1.04662i) q^{6} +(-44.8315 + 61.7053i) q^{7} +(-12.0973 - 16.6506i) q^{8} +(8.34346 - 25.6785i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 76 q^{4} + 36 q^{5} + 150 q^{7} + 480 q^{8} - 216 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.619915	+	0.201423i	0.154979	+	0.0503557i	0.385479	−	0.922717i	\(-0.374036\pi\)
							−0.230500	+	0.973072i	\(0.574036\pi\)
\(3\)	−4.20378	+	3.05422i	−0.467086	+	0.339358i
							\(5\)	12.5302	+	38.5641i	0.501209	+	1.54256i	0.807052	+	0.590481i	\(0.201062\pi\)
−0.305842	+	0.952082i	\(0.598938\pi\)
\(7\)	−44.8315	+	61.7053i	−0.914928	+	1.25929i	0.0505270	+	0.998723i	\(0.483910\pi\)
							−0.965455	+	0.260568i	\(0.916090\pi\)
\(11\)	−96.5563	−	72.9239i	−0.797986	−	0.602677i
							\(13\)	181.545	+	58.9875i	1.07423	+	0.349039i	0.792134	−	0.610347i	\(-0.208970\pi\)
0.282097	+	0.959386i	\(0.408970\pi\)
\(17\)	116.485	−	37.8483i	0.403063	−	0.130963i	−0.100468	−	0.994940i	\(-0.532034\pi\)
							0.503531	+	0.863977i	\(0.332034\pi\)
\(19\)	−45.9553	−	63.2520i	−0.127300	−	0.175213i	0.740610	−	0.671935i	\(-0.234537\pi\)
							−0.867910	+	0.496722i	\(0.834537\pi\)
\(23\)	262.718			0.496632			0.248316	−	0.968679i	\(-0.420123\pi\)
							0.248316	+	0.968679i	\(0.420123\pi\)
\(29\)	−13.2120	+	18.1848i	−0.0157099	+	0.0216229i	−0.816800	−	0.576921i	\(-0.804254\pi\)
							0.801090	+	0.598544i	\(0.204254\pi\)
\(31\)	−385.287	+	1185.79i	−0.400923	+	1.23391i	0.523328	+	0.852131i	\(0.324690\pi\)
							−0.924252	+	0.381784i	\(0.875310\pi\)
\(37\)	−361.544	−	262.677i	−0.264094	−	0.191875i	0.447856	−	0.894106i	\(-0.352188\pi\)
							−0.711950	+	0.702230i	\(0.752188\pi\)
\(41\)	1079.64	+	1486.00i	0.642261	+	0.883996i	0.998734	−	0.0503097i	\(-0.0160208\pi\)
							−0.356473	+	0.934306i	\(0.616021\pi\)
\(43\)			3041.96i			1.64519i	0.568628	+	0.822595i	\(0.307475\pi\)
							−0.568628	+	0.822595i	\(0.692525\pi\)
\(47\)	3212.96	−	2334.35i	1.45449	−	1.05675i	0.469729	−	0.882811i	\(-0.344352\pi\)
							0.984757	−	0.173935i	\(-0.0556482\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.g.a.13.4	✓	32
3.2	odd	2		99.5.k.c.46.5		32
11.4	even	5		363.5.c.e.241.19		32
11.6	odd	10	inner	33.5.g.a.28.4	yes	32
11.7	odd	10		363.5.c.e.241.14		32
33.17	even	10		99.5.k.c.28.5		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.g.a.13.4	✓	32	1.1	even	1	trivial
33.5.g.a.28.4	yes	32	11.6	odd	10	inner
99.5.k.c.28.5		32	33.17	even	10
99.5.k.c.46.5		32	3.2	odd	2
363.5.c.e.241.14		32	11.7	odd	10
363.5.c.e.241.19		32	11.4	even	5