Embedded newform 33.7.g.a.13.6

• Label: 33.7.g.a.13.6
• Level: $33$
• Weight: $7$
• Character: 33.13
• 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			13.6
Character	\(\chi\)	\(=\)	33.13
Dual form			33.7.g.a.28.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.69988 - 0.877245i) q^{2} +(12.6113 - 9.16267i) q^{3} +(-45.2573 - 32.8813i) q^{4} +(71.7465 + 220.813i) q^{5} +(-42.0870 + 13.6749i) q^{6} +(166.082 - 228.592i) q^{7} +(200.136 + 275.463i) q^{8} +(75.0911 - 231.107i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 204 q^{4} - 224 q^{5} + 720 q^{7} - 3200 q^{8} - 2916 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{1}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.69988	−	0.877245i	−0.337485	−	0.109656i	0.135372	−	0.990795i	\(-0.456777\pi\)
							−0.472857	+	0.881139i	\(0.656777\pi\)
\(3\)	12.6113	−	9.16267i	0.467086	−	0.339358i
							\(5\)	71.7465	+	220.813i	0.573972	+	1.76650i	0.639654	+	0.768663i	\(0.279078\pi\)
−0.0656818	+	0.997841i	\(0.520922\pi\)
\(7\)	166.082	−	228.592i	0.484204	−	0.666450i	−0.495102	−	0.868835i	\(-0.664869\pi\)
							0.979306	+	0.202385i	\(0.0648692\pi\)
\(11\)	1246.83	−	465.807i	0.936762	−	0.349968i
							\(13\)	2220.54	+	721.498i	1.01072	+	0.328402i	0.767140	−	0.641480i	\(-0.221679\pi\)
0.243576	+	0.969882i	\(0.421679\pi\)
\(17\)	4246.49	−	1379.77i	0.864337	−	0.280840i	0.156898	−	0.987615i	\(-0.449851\pi\)
							0.707439	+	0.706775i	\(0.249851\pi\)
\(19\)	877.706	+	1208.06i	0.127964	+	0.176127i	0.868192	−	0.496229i	\(-0.165282\pi\)
							−0.740228	+	0.672356i	\(0.765282\pi\)
\(23\)	1556.53			0.127930			0.0639650	−	0.997952i	\(-0.479625\pi\)
							0.0639650	+	0.997952i	\(0.479625\pi\)
\(29\)	−23957.5	+	32974.7i	−0.982308	+	1.35203i	−0.0467313	+	0.998907i	\(0.514880\pi\)
							−0.935577	+	0.353123i	\(0.885120\pi\)
\(31\)	8709.54	−	26805.2i	0.292355	−	0.899776i	−0.691742	−	0.722144i	\(-0.743157\pi\)
							0.984097	−	0.177631i	\(-0.0568434\pi\)
\(37\)	−17148.9	−	12459.4i	−0.338556	−	0.245975i	0.405496	−	0.914097i	\(-0.367099\pi\)
							−0.744052	+	0.668121i	\(0.767099\pi\)
\(41\)	−8472.98	−	11662.1i	−0.122938	−	0.169209i	0.743112	−	0.669167i	\(-0.233349\pi\)
							−0.866050	+	0.499958i	\(0.833349\pi\)
\(43\)			30917.5i			0.388865i	0.980916	+	0.194433i	\(0.0622866\pi\)
							−0.980916	+	0.194433i	\(0.937713\pi\)
\(47\)	47847.6	−	34763.3i	0.460857	−	0.334833i	−0.333010	−	0.942923i	\(-0.608064\pi\)
							0.793868	+	0.608091i	\(0.208064\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.g.a.13.6	✓	48
3.2	odd	2		99.7.k.b.46.7		48
11.6	odd	10	inner	33.7.g.a.28.6	yes	48
33.17	even	10		99.7.k.b.28.7		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.g.a.13.6	✓	48	1.1	even	1	trivial
33.7.g.a.28.6	yes	48	11.6	odd	10	inner
99.7.k.b.28.7		48	33.17	even	10
99.7.k.b.46.7		48	3.2	odd	2