Embedded newform 33.7.h.a.5.2

• Label: 33.7.h.a.5.2
• Level: $33$
• Weight: $7$
• Character: 33.5
• Analytic conductor: $7.592$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	33.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.59178475946\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			5.2
Character	\(\chi\)	\(=\)	33.5
Dual form			33.7.h.a.20.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.6998 + 4.12642i) q^{2} +(-26.9979 + 0.338512i) q^{3} +(92.4810 - 67.1914i) q^{4} +(73.9941 + 24.0421i) q^{5} +(341.471 - 115.704i) q^{6} +(162.344 - 117.950i) q^{7} +(-394.901 + 543.535i) q^{8} +(728.771 - 18.2782i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 33 q^{3} + 634 q^{4} + 109 q^{6} + 402 q^{7} - 2471 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.6998	+	4.12642i	−1.58748	+	0.515803i	−0.963969	−	0.266014i	\(-0.914293\pi\)
							−0.623508	+	0.781817i	\(0.714293\pi\)
\(3\)	−26.9979	+	0.338512i	−0.999921	+	0.0125375i
							\(5\)	73.9941	+	24.0421i	0.591953	+	0.192337i	0.589648	−	0.807660i	\(-0.299266\pi\)
0.00230466	+	0.999997i	\(0.499266\pi\)
\(7\)	162.344	−	117.950i	0.473307	−	0.343878i	−0.325422	−	0.945569i	\(-0.605506\pi\)
							0.798729	+	0.601691i	\(0.205506\pi\)
\(11\)	−1242.90	+	476.193i	−0.933809	+	0.357771i
							\(13\)	−189.703	−	583.845i	−0.0863463	−	0.265747i	0.898556	−	0.438860i	\(-0.144617\pi\)
−0.984902	+	0.173113i	\(0.944617\pi\)
\(17\)	−3714.77	−	1207.00i	−0.756110	−	0.245675i	−0.0945020	−	0.995525i	\(-0.530126\pi\)
							−0.661608	+	0.749849i	\(0.730126\pi\)
\(19\)	8401.01	+	6103.69i	1.22482	+	0.889881i	0.996491	−	0.0837039i	\(-0.0266750\pi\)
							0.228325	+	0.973585i	\(0.426675\pi\)
\(23\)			12840.3i			1.05534i	0.849450	+	0.527669i	\(0.176934\pi\)
							−0.849450	+	0.527669i	\(0.823066\pi\)
\(29\)	−7068.82	−	9729.40i	−0.289837	−	0.398926i	0.639124	−	0.769103i	\(-0.279297\pi\)
							−0.928961	+	0.370177i	\(0.879297\pi\)
\(31\)	12095.1	+	37225.0i	0.406000	+	1.24954i	0.920057	+	0.391784i	\(0.128142\pi\)
							−0.514057	+	0.857756i	\(0.671858\pi\)
\(37\)	−70747.9	+	51401.4i	−1.39672	+	1.01477i	−0.401627	+	0.915803i	\(0.631555\pi\)
							−0.995090	+	0.0989715i	\(0.968445\pi\)
\(41\)	−49768.9	+	68501.1i	−0.722116	+	0.993907i	0.277335	+	0.960773i	\(0.410549\pi\)
							−0.999451	+	0.0331339i	\(0.989451\pi\)
\(43\)	−57595.8			−0.724412			−0.362206	−	0.932098i	\(-0.617976\pi\)
							−0.362206	+	0.932098i	\(0.617976\pi\)
\(47\)	−13786.4	+	18975.3i	−0.132787	+	0.182766i	−0.870233	−	0.492641i	\(-0.836032\pi\)
							0.737446	+	0.675406i	\(0.236032\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.h.a.5.2	✓	88
3.2	odd	2	inner	33.7.h.a.5.21	yes	88
11.9	even	5	inner	33.7.h.a.20.21	yes	88
33.20	odd	10	inner	33.7.h.a.20.2	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.h.a.5.2	✓	88	1.1	even	1	trivial
33.7.h.a.5.21	yes	88	3.2	odd	2	inner
33.7.h.a.20.2	yes	88	33.20	odd	10	inner
33.7.h.a.20.21	yes	88	11.9	even	5	inner