Embedded newform 33.7.g.a.13.5

• Label: 33.7.g.a.13.5
• 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			13.5
Character	\(\chi\)	\(=\)	33.13
Dual form			33.7.g.a.28.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.17091 - 1.68013i) q^{2} +(-12.6113 + 9.16267i) q^{3} +(-27.8616 - 20.2427i) q^{4} +(-25.6501 - 78.9429i) q^{5} +(80.6065 - 26.1906i) q^{6} +(-29.3745 + 40.4305i) q^{7} +(314.591 + 432.997i) q^{8} +(75.0911 - 231.107i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 204 q^{4} - 224 q^{5} + 720 q^{7} - 3200 q^{8} - 2916 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.17091	−	1.68013i	−0.646364	−	0.210016i	−0.0325536	−	0.999470i	\(-0.510364\pi\)
							−0.613810	+	0.789454i	\(0.710364\pi\)
\(3\)	−12.6113	+	9.16267i	−0.467086	+	0.339358i
							\(5\)	−25.6501	−	78.9429i	−0.205201	−	0.631543i	−0.999705	−	0.0242847i	\(-0.992269\pi\)
0.794504	−	0.607259i	\(-0.207731\pi\)
\(7\)	−29.3745	+	40.4305i	−0.0856399	+	0.117873i	−0.849687	−	0.527287i	\(-0.823209\pi\)
							0.764047	+	0.645160i	\(0.223209\pi\)
\(11\)	−670.917	+	1149.54i	−0.504070	+	0.863663i
							\(13\)	3717.62	+	1207.93i	1.69214	+	0.549808i	0.987203	−	0.159467i	\(-0.0509775\pi\)
0.704932	+	0.709275i	\(0.250977\pi\)
\(17\)	3270.08	−	1062.51i	0.665598	−	0.216266i	0.0433188	−	0.999061i	\(-0.486207\pi\)
							0.622279	+	0.782795i	\(0.286207\pi\)
\(19\)	3325.61	+	4577.31i	0.484854	+	0.667344i	0.979429	−	0.201791i	\(-0.0646761\pi\)
							−0.494575	+	0.869135i	\(0.664676\pi\)
\(23\)	−15468.0			−1.27131			−0.635655	−	0.771973i	\(-0.719270\pi\)
							−0.635655	+	0.771973i	\(0.719270\pi\)
\(29\)	−19515.6	+	26861.0i	−0.800182	+	1.10136i	0.192583	+	0.981281i	\(0.438314\pi\)
							−0.992765	+	0.120075i	\(0.961686\pi\)
\(31\)	7257.14	−	22335.2i	0.243602	−	0.749729i	−0.752262	−	0.658864i	\(-0.771037\pi\)
							0.995863	−	0.0908642i	\(-0.0289629\pi\)
\(37\)	52722.9	+	38305.5i	1.04087	+	0.756233i	0.970454	−	0.241285i	\(-0.0775690\pi\)
							0.0704109	+	0.997518i	\(0.477569\pi\)
\(41\)	22722.3	+	31274.6i	0.329686	+	0.453774i	0.941394	−	0.337310i	\(-0.109517\pi\)
							−0.611707	+	0.791084i	\(0.709517\pi\)
\(43\)			56820.2i			0.714657i	0.933979	+	0.357328i	\(0.116312\pi\)
							−0.933979	+	0.357328i	\(0.883688\pi\)
\(47\)	31042.3	−	22553.5i	0.298992	−	0.217231i	−0.428167	−	0.903700i	\(-0.640840\pi\)
							0.727159	+	0.686469i	\(0.240840\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.g.a.13.5	✓	48
3.2	odd	2		99.7.k.b.46.8		48
11.6	odd	10	inner	33.7.g.a.28.5	yes	48
33.17	even	10		99.7.k.b.28.8		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.g.a.13.5	✓	48	1.1	even	1	trivial
33.7.g.a.28.5	yes	48	11.6	odd	10	inner
99.7.k.b.28.8		48	33.17	even	10
99.7.k.b.46.8		48	3.2	odd	2