Embedded newform 33.7.g.a.7.2

• Label: 33.7.g.a.7.2
• 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			7.2
Character	\(\chi\)	\(=\)	33.7
Dual form			33.7.g.a.19.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.91015 + 9.51101i) q^{2} +(-4.81710 + 14.8255i) q^{3} +(-22.9320 - 70.5775i) q^{4} +(174.946 - 127.106i) q^{5} +(-107.719 - 148.262i) q^{6} +(472.320 - 153.466i) q^{7} +(114.152 + 37.0902i) q^{8} +(-196.591 - 142.832i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 204 q^{4} - 224 q^{5} + 720 q^{7} - 3200 q^{8} - 2916 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.91015	+	9.51101i	−0.863769	+	1.18888i	0.116888	+	0.993145i	\(0.462708\pi\)
							−0.980658	+	0.195731i	\(0.937292\pi\)
\(3\)	−4.81710	+	14.8255i	−0.178411	+	0.549093i
							\(5\)	174.946	−	127.106i	1.39957	−	1.01685i	0.404834	−	0.914390i	\(-0.367329\pi\)
0.994737	−	0.102458i	\(-0.0326706\pi\)
\(7\)	472.320	−	153.466i	1.37703	−	0.447423i	0.475336	−	0.879805i	\(-0.342327\pi\)
							0.901691	+	0.432382i	\(0.142327\pi\)
\(11\)	360.014	−	1281.39i	0.270483	−	0.962725i
							\(13\)	−760.522	+	1046.77i	−0.346164	+	0.476453i	−0.946229	−	0.323498i	\(-0.895141\pi\)
0.600065	+	0.799951i	\(0.295141\pi\)
\(17\)	−1362.12	−	1874.80i	−0.277249	−	0.381600i	0.647572	−	0.762005i	\(-0.275785\pi\)
							−0.924820	+	0.380405i	\(0.875785\pi\)
\(19\)	11103.9	+	3607.87i	1.61888	+	0.526005i	0.971677	−	0.236315i	\(-0.0759396\pi\)
							0.647201	+	0.762320i	\(0.275940\pi\)
\(23\)	−11131.0			−0.914855			−0.457428	−	0.889247i	\(-0.651229\pi\)
							−0.457428	+	0.889247i	\(0.651229\pi\)
\(29\)	3169.03	−	1029.68i	0.129937	−	0.0422190i	−0.243327	−	0.969944i	\(-0.578239\pi\)
							0.373263	+	0.927725i	\(0.378239\pi\)
\(31\)	−14575.0	−	10589.4i	−0.489242	−	0.355455i	0.315651	−	0.948876i	\(-0.397777\pi\)
							−0.804893	+	0.593420i	\(0.797777\pi\)
\(37\)	21736.6	+	66898.5i	0.429128	+	1.32072i	0.898986	+	0.437978i	\(0.144305\pi\)
							−0.469857	+	0.882742i	\(0.655695\pi\)
\(41\)	41490.7	+	13481.1i	0.602004	+	0.195603i	0.594134	−	0.804366i	\(-0.297495\pi\)
							0.00786972	+	0.999969i	\(0.497495\pi\)
\(43\)			79339.9i			0.997898i	0.866631	+	0.498949i	\(0.166280\pi\)
							−0.866631	+	0.498949i	\(0.833720\pi\)
\(47\)	14819.5	−	45609.7i	0.142738	−	0.439302i	−0.853975	−	0.520314i	\(-0.825815\pi\)
							0.996713	+	0.0810114i	\(0.0258150\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.g.a.7.2	✓	48
3.2	odd	2		99.7.k.b.73.11		48
11.8	odd	10	inner	33.7.g.a.19.2	yes	48
33.8	even	10		99.7.k.b.19.11		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.g.a.7.2	✓	48	1.1	even	1	trivial
33.7.g.a.19.2	yes	48	11.8	odd	10	inner
99.7.k.b.19.11		48	33.8	even	10
99.7.k.b.73.11		48	3.2	odd	2