Embedded newform 33.7.g.a.7.4

• Label: 33.7.g.a.7.4
• Level: $33$
• Weight: $7$
• Character: 33.7
• Analytic conductor: $7.592$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.59178475946\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			7.4
Character	\(\chi\)	\(=\)	33.7
Dual form			33.7.g.a.19.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.06354 + 6.96937i) q^{2} +(4.81710 - 14.8255i) q^{3} +(-3.15555 - 9.71180i) q^{4} +(-8.79332 + 6.38872i) q^{5} +(78.9328 + 108.642i) q^{6} +(-242.092 + 78.6605i) q^{7} +(-440.688 - 143.188i) q^{8} +(-196.591 - 142.832i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 204 q^{4} - 224 q^{5} + 720 q^{7} - 3200 q^{8} - 2916 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.06354	+	6.96937i	−0.632943	+	0.871171i	−0.998215	−	0.0597298i	\(-0.980976\pi\)
							0.365272	+	0.930901i	\(0.380976\pi\)
\(3\)	4.81710	−	14.8255i	0.178411	−	0.549093i
							\(5\)	−8.79332	+	6.38872i	−0.0703466	+	0.0511098i	−0.622403	−	0.782697i	\(-0.713843\pi\)
0.552056	+	0.833807i	\(0.313843\pi\)
\(7\)	−242.092	+	78.6605i	−0.705808	+	0.229331i	−0.639859	−	0.768492i	\(-0.721007\pi\)
							−0.0659488	+	0.997823i	\(0.521007\pi\)
\(11\)	−153.630	−	1322.10i	−0.115425	−	0.993316i
							\(13\)	617.742	−	850.249i	0.281175	−	0.387005i	−0.644947	−	0.764227i	\(-0.723121\pi\)
0.926123	+	0.377222i	\(0.123121\pi\)
\(17\)	−5016.34	−	6904.40i	−1.02103	−	1.40533i	−0.911476	−	0.411353i	\(-0.865056\pi\)
							−0.109558	−	0.993980i	\(-0.534944\pi\)
\(19\)	−1828.21	−	594.021i	−0.266542	−	0.0866047i	0.172697	−	0.984975i	\(-0.444752\pi\)
							−0.439239	+	0.898370i	\(0.644752\pi\)
\(23\)	−10413.4			−0.855874			−0.427937	−	0.903809i	\(-0.640760\pi\)
							−0.427937	+	0.903809i	\(0.640760\pi\)
\(29\)	−15159.4	+	4925.58i	−0.621566	+	0.201959i	−0.602836	−	0.797865i	\(-0.705963\pi\)
							−0.0187307	+	0.999825i	\(0.505963\pi\)
\(31\)	23632.4	+	17169.9i	0.793271	+	0.576345i	0.908933	−	0.416943i	\(-0.136899\pi\)
							−0.115661	+	0.993289i	\(0.536899\pi\)
\(37\)	−3320.13	−	10218.3i	−0.0655466	−	0.201732i	0.912919	−	0.408140i	\(-0.133822\pi\)
							−0.978466	+	0.206408i	\(0.933822\pi\)
\(41\)	3334.93	+	1083.58i	0.0483877	+	0.0157221i	0.333111	−	0.942888i	\(-0.391902\pi\)
							−0.284723	+	0.958610i	\(0.591902\pi\)
\(43\)		−	48322.2i		−	0.607773i	−0.952708	−	0.303887i	\(-0.901716\pi\)
							0.952708	−	0.303887i	\(-0.0982844\pi\)
\(47\)	−43811.1	+	134837.i	−0.421979	+	1.29872i	0.483878	+	0.875135i	\(0.339228\pi\)
							−0.905857	+	0.423583i	\(0.860772\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.g.a.7.4	✓	48
3.2	odd	2		99.7.k.b.73.9		48
11.8	odd	10	inner	33.7.g.a.19.4	yes	48
33.8	even	10		99.7.k.b.19.9		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.g.a.7.4	✓	48	1.1	even	1	trivial
33.7.g.a.19.4	yes	48	11.8	odd	10	inner
99.7.k.b.19.9		48	33.8	even	10
99.7.k.b.73.9		48	3.2	odd	2