Embedded newform 33.7.g.a.7.5

• Label: 33.7.g.a.7.5
• 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			7.5
Character	\(\chi\)	\(=\)	33.7
Dual form			33.7.g.a.19.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16939 + 1.60953i) q^{2} +(-4.81710 + 14.8255i) q^{3} +(18.5540 + 57.1033i) q^{4} +(-1.49745 + 1.08796i) q^{5} +(-18.2291 - 25.0902i) q^{6} +(-25.7895 + 8.37953i) q^{7} +(-234.702 - 76.2594i) q^{8} +(-196.591 - 142.832i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 204 q^{4} - 224 q^{5} + 720 q^{7} - 3200 q^{8} - 2916 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16939	+	1.60953i	−0.146174	+	0.201192i	−0.875826	−	0.482628i	\(-0.839682\pi\)
							0.729651	+	0.683819i	\(0.239682\pi\)
\(3\)	−4.81710	+	14.8255i	−0.178411	+	0.549093i
							\(5\)	−1.49745	+	1.08796i	−0.0119796	+	0.00870367i	−0.593759	−	0.804643i	\(-0.702357\pi\)
0.581779	+	0.813347i	\(0.302357\pi\)
\(7\)	−25.7895	+	8.37953i	−0.0751882	+	0.0244301i	−0.346370	−	0.938098i	\(-0.612586\pi\)
							0.271181	+	0.962528i	\(0.412586\pi\)
\(11\)	−1312.59	+	220.601i	−0.986169	+	0.165740i
							\(13\)	−1022.63	+	1407.53i	−0.465466	+	0.640659i	−0.975631	−	0.219417i	\(-0.929584\pi\)
0.510165	+	0.860077i	\(0.329584\pi\)
\(17\)	−1486.92	−	2046.57i	−0.302651	−	0.416563i	0.630421	−	0.776253i	\(-0.282882\pi\)
							−0.933072	+	0.359691i	\(0.882882\pi\)
\(19\)	−1806.32	−	586.908i	−0.263350	−	0.0855676i	0.174366	−	0.984681i	\(-0.444212\pi\)
							−0.437716	+	0.899113i	\(0.644212\pi\)
\(23\)	9114.71			0.749133			0.374567	−	0.927200i	\(-0.377791\pi\)
							0.374567	+	0.927200i	\(0.377791\pi\)
\(29\)	24962.7	−	8110.87i	1.02352	−	0.332563i	0.251296	−	0.967910i	\(-0.419143\pi\)
							0.772226	+	0.635347i	\(0.219143\pi\)
\(31\)	28999.9	+	21069.6i	0.973444	+	0.707249i	0.956234	−	0.292603i	\(-0.0945215\pi\)
							0.0172105	+	0.999852i	\(0.494521\pi\)
\(37\)	15279.9	+	47026.6i	0.301658	+	0.928407i	0.980903	+	0.194496i	\(0.0623071\pi\)
							−0.679246	+	0.733911i	\(0.737693\pi\)
\(41\)	−35995.1	−	11695.5i	−0.522267	−	0.169695i	0.0360072	−	0.999352i	\(-0.488536\pi\)
							−0.558274	+	0.829657i	\(0.688536\pi\)
\(43\)			76694.5i			0.964626i	0.875999	+	0.482313i	\(0.160203\pi\)
							−0.875999	+	0.482313i	\(0.839797\pi\)
\(47\)	−18622.0	+	57312.7i	−0.179363	+	0.552023i	−0.999806	−	0.0197056i	\(-0.993727\pi\)
							0.820443	+	0.571729i	\(0.193727\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.g.a.7.5	✓	48
3.2	odd	2		99.7.k.b.73.8		48
11.8	odd	10	inner	33.7.g.a.19.5	yes	48
33.8	even	10		99.7.k.b.19.8		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.g.a.7.5	✓	48	1.1	even	1	trivial
33.7.g.a.19.5	yes	48	11.8	odd	10	inner
99.7.k.b.19.8		48	33.8	even	10
99.7.k.b.73.8		48	3.2	odd	2