Embedded newform 33.7.g.a.7.10

• Label: 33.7.g.a.7.10
• 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			7.10
Character	\(\chi\)	\(=\)	33.7
Dual form			33.7.g.a.19.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.26570 - 8.62400i) q^{2} +(4.81710 - 14.8255i) q^{3} +(-15.3373 - 47.2032i) q^{4} +(161.919 - 117.641i) q^{5} +(-97.6727 - 134.435i) q^{6} +(-310.129 + 100.767i) q^{7} +(145.660 + 47.3278i) q^{8} +(-196.591 - 142.832i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 204 q^{4} - 224 q^{5} + 720 q^{7} - 3200 q^{8} - 2916 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.26570	−	8.62400i	0.783213	−	1.07800i	−0.211707	−	0.977333i	\(-0.567902\pi\)
							0.994920	−	0.100667i	\(-0.0320977\pi\)
\(3\)	4.81710	−	14.8255i	0.178411	−	0.549093i
							\(5\)	161.919	−	117.641i	1.29536	−	0.941131i	0.295456	−	0.955356i	\(-0.404528\pi\)
0.999899	+	0.0142255i	\(0.00452827\pi\)
\(7\)	−310.129	+	100.767i	−0.904166	+	0.293781i	−0.723955	−	0.689847i	\(-0.757678\pi\)
							−0.180211	+	0.983628i	\(0.557678\pi\)
\(11\)	316.242	−	1292.89i	0.237597	−	0.971364i
							\(13\)	−2520.25	+	3468.83i	−1.14713	+	1.57889i	−0.396695	+	0.917951i	\(0.629843\pi\)
−0.750437	+	0.660942i	\(0.770157\pi\)
\(17\)	729.960	+	1004.70i	0.148577	+	0.204499i	0.876818	−	0.480822i	\(-0.159662\pi\)
							−0.728241	+	0.685321i	\(0.759662\pi\)
\(19\)	135.672	+	44.0825i	0.0197801	+	0.00642696i	0.318890	−	0.947792i	\(-0.396690\pi\)
							−0.299110	+	0.954219i	\(0.596690\pi\)
\(23\)	15403.5			1.26601			0.633004	−	0.774149i	\(-0.281822\pi\)
							0.633004	+	0.774149i	\(0.281822\pi\)
\(29\)	7679.09	−	2495.09i	0.314859	−	0.102304i	−0.147325	−	0.989088i	\(-0.547066\pi\)
							0.462183	+	0.886784i	\(0.347066\pi\)
\(31\)	46122.2	+	33509.8i	1.54819	+	1.12483i	0.944923	+	0.327292i	\(0.106136\pi\)
							0.603271	+	0.797536i	\(0.293864\pi\)
\(37\)	5037.61	+	15504.2i	0.0994533	+	0.306086i	0.988389	−	0.151947i	\(-0.0485542\pi\)
							−0.888935	+	0.458033i	\(0.848554\pi\)
\(41\)	−78923.7	−	25643.8i	−1.14513	−	0.372076i	−0.325825	−	0.945430i	\(-0.605642\pi\)
							−0.819308	+	0.573354i	\(0.805642\pi\)
\(43\)			57907.2i			0.728328i	0.931335	+	0.364164i	\(0.118645\pi\)
							−0.931335	+	0.364164i	\(0.881355\pi\)
\(47\)	4348.04	−	13381.9i	0.0418793	−	0.128891i	−0.927931	−	0.372752i	\(-0.878414\pi\)
							0.969810	+	0.243861i	\(0.0784141\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.g.a.7.10	✓	48
3.2	odd	2		99.7.k.b.73.3		48
11.8	odd	10	inner	33.7.g.a.19.10	yes	48
33.8	even	10		99.7.k.b.19.3		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.g.a.7.10	✓	48	1.1	even	1	trivial
33.7.g.a.19.10	yes	48	11.8	odd	10	inner
99.7.k.b.19.3		48	33.8	even	10
99.7.k.b.73.3		48	3.2	odd	2