Embedded newform 33.7.g.a.13.8

• Label: 33.7.g.a.13.8
• 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			13.8
Character	\(\chi\)	\(=\)	33.13
Dual form			33.7.g.a.28.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.51299 + 1.46636i) q^{2} +(-12.6113 + 9.16267i) q^{3} +(-33.5602 - 24.3829i) q^{4} +(6.35291 + 19.5523i) q^{5} +(-70.3505 + 22.8583i) q^{6} +(355.989 - 489.977i) q^{7} +(-294.210 - 404.946i) q^{8} +(75.0911 - 231.107i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 204 q^{4} - 224 q^{5} + 720 q^{7} - 3200 q^{8} - 2916 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.51299	+	1.46636i	0.564123	+	0.183295i	0.577176	−	0.816620i	\(-0.304155\pi\)
							−0.0130522	+	0.999915i	\(0.504155\pi\)
\(3\)	−12.6113	+	9.16267i	−0.467086	+	0.339358i
							\(5\)	6.35291	+	19.5523i	0.0508233	+	0.156418i	0.973247	−	0.229761i	\(-0.0737946\pi\)
−0.922424	+	0.386179i	\(0.873795\pi\)
\(7\)	355.989	−	489.977i	1.03787	−	1.42851i	0.138999	−	0.990293i	\(-0.455611\pi\)
							0.898871	−	0.438213i	\(-0.144389\pi\)
\(11\)	−598.682	−	1188.76i	−0.449799	−	0.893130i
							\(13\)	654.054	+	212.515i	0.297703	+	0.0967297i	0.454060	−	0.890971i	\(-0.349975\pi\)
−0.156357	+	0.987701i	\(0.549975\pi\)
\(17\)	−3272.50	+	1063.30i	−0.666089	+	0.216425i	−0.622495	−	0.782624i	\(-0.713881\pi\)
							−0.0435942	+	0.999049i	\(0.513881\pi\)
\(19\)	−4160.65	−	5726.65i	−0.606598	−	0.834910i	0.389694	−	0.920944i	\(-0.372581\pi\)
							−0.996292	+	0.0860339i	\(0.972581\pi\)
\(23\)	864.656			0.0710657			0.0355328	−	0.999369i	\(-0.488687\pi\)
							0.0355328	+	0.999369i	\(0.488687\pi\)
\(29\)	−8642.05	+	11894.8i	−0.354342	+	0.487710i	−0.948562	−	0.316593i	\(-0.897461\pi\)
							0.594219	+	0.804303i	\(0.297461\pi\)
\(31\)	−5055.48	+	15559.2i	−0.169698	+	0.522277i	−0.999352	−	0.0360012i	\(-0.988538\pi\)
							0.829654	+	0.558279i	\(0.188538\pi\)
\(37\)	−5771.30	−	4193.10i	−0.113938	−	0.0827808i	0.529357	−	0.848399i	\(-0.322433\pi\)
							−0.643295	+	0.765618i	\(0.722433\pi\)
\(41\)	65245.5	+	89802.8i	0.946672	+	1.30298i	0.952990	+	0.303002i	\(0.0979887\pi\)
							−0.00631852	+	0.999980i	\(0.502011\pi\)
\(43\)		−	134833.i		−	1.69586i	−0.530107	−	0.847931i	\(-0.677848\pi\)
							0.530107	−	0.847931i	\(-0.322152\pi\)
\(47\)	19259.3	−	13992.7i	0.185502	−	0.134775i	−0.491159	−	0.871070i	\(-0.663427\pi\)
							0.676660	+	0.736295i	\(0.263427\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.g.a.13.8	✓	48
3.2	odd	2		99.7.k.b.46.5		48
11.6	odd	10	inner	33.7.g.a.28.8	yes	48
33.17	even	10		99.7.k.b.28.5		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.g.a.13.8	✓	48	1.1	even	1	trivial
33.7.g.a.28.8	yes	48	11.6	odd	10	inner
99.7.k.b.28.5		48	33.17	even	10
99.7.k.b.46.5		48	3.2	odd	2