Embedded newform 33.5.g.a.13.3

• Label: 33.5.g.a.13.3
• 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			13.3
Character	\(\chi\)	\(=\)	33.13
Dual form			33.5.g.a.28.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.173259 - 0.0562952i) q^{2} +(4.20378 - 3.05422i) q^{3} +(-12.9174 - 9.38506i) q^{4} +(-6.40673 - 19.7179i) q^{5} +(-0.900279 + 0.292519i) q^{6} +(12.6762 - 17.4473i) q^{7} +(3.42300 + 4.71136i) q^{8} +(8.34346 - 25.6785i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 76 q^{4} + 36 q^{5} + 150 q^{7} + 480 q^{8} - 216 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.173259	−	0.0562952i	−0.0433147	−	0.0140738i	0.287280	−	0.957847i	\(-0.407249\pi\)
							−0.330594	+	0.943773i	\(0.607249\pi\)
\(3\)	4.20378	−	3.05422i	0.467086	−	0.339358i
							\(5\)	−6.40673	−	19.7179i	−0.256269	−	0.788715i	−0.993577	−	0.113158i	\(-0.963903\pi\)
0.737308	−	0.675557i	\(-0.236097\pi\)
\(7\)	12.6762	−	17.4473i	0.258698	−	0.356067i	−0.659836	−	0.751410i	\(-0.729374\pi\)
							0.918534	+	0.395343i	\(0.129374\pi\)
\(11\)	−24.1350	−	118.569i	−0.199463	−	0.979905i
							\(13\)	121.444	+	39.4595i	0.718602	+	0.233488i	0.645417	−	0.763830i	\(-0.276684\pi\)
0.0731850	+	0.997318i	\(0.476684\pi\)
\(17\)	−88.1403	+	28.6385i	−0.304984	+	0.0990952i	−0.457511	−	0.889204i	\(-0.651259\pi\)
							0.152527	+	0.988299i	\(0.451259\pi\)
\(19\)	348.902	+	480.222i	0.966487	+	1.33026i	0.943802	+	0.330513i	\(0.107222\pi\)
							0.0226856	+	0.999743i	\(0.492778\pi\)
\(23\)	−279.707			−0.528747			−0.264374	−	0.964420i	\(-0.585165\pi\)
							−0.264374	+	0.964420i	\(0.585165\pi\)
\(29\)	682.232	−	939.012i	0.811216	−	1.11654i	−0.179919	−	0.983681i	\(-0.557584\pi\)
							0.991135	−	0.132861i	\(-0.0424164\pi\)
\(31\)	11.7103	−	36.0407i	0.0121856	−	0.0375033i	−0.944779	−	0.327709i	\(-0.893724\pi\)
							0.956964	+	0.290205i	\(0.0937236\pi\)
\(37\)	375.020	+	272.468i	0.273937	+	0.199027i	0.716269	−	0.697825i	\(-0.245848\pi\)
							−0.442331	+	0.896852i	\(0.645848\pi\)
\(41\)	−532.885	−	733.453i	−0.317005	−	0.436319i	0.620545	−	0.784171i	\(-0.286911\pi\)
							−0.937550	+	0.347851i	\(0.886911\pi\)
\(43\)		−	2632.39i		−	1.42369i	−0.702339	−	0.711843i	\(-0.747861\pi\)
							0.702339	−	0.711843i	\(-0.252139\pi\)
\(47\)	1881.08	−	1366.68i	0.851551	−	0.618688i	−0.0740224	−	0.997257i	\(-0.523584\pi\)
							0.925573	+	0.378569i	\(0.123584\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.g.a.13.3	✓	32
3.2	odd	2		99.5.k.c.46.6		32
11.4	even	5		363.5.c.e.241.15		32
11.6	odd	10	inner	33.5.g.a.28.3	yes	32
11.7	odd	10		363.5.c.e.241.18		32
33.17	even	10		99.5.k.c.28.6		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.g.a.13.3	✓	32	1.1	even	1	trivial
33.5.g.a.28.3	yes	32	11.6	odd	10	inner
99.5.k.c.28.6		32	33.17	even	10
99.5.k.c.46.6		32	3.2	odd	2
363.5.c.e.241.15		32	11.4	even	5
363.5.c.e.241.18		32	11.7	odd	10