Embedded newform 33.7.g.a.13.1

• Label: 33.7.g.a.13.1
• 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			13.1
Character	\(\chi\)	\(=\)	33.13
Dual form			33.7.g.a.28.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-14.5443 - 4.72574i) q^{2} +(-12.6113 + 9.16267i) q^{3} +(137.428 + 99.8469i) q^{4} +(-61.8586 - 190.381i) q^{5} +(226.723 - 73.6669i) q^{6} +(168.802 - 232.336i) q^{7} +(-951.651 - 1309.84i) q^{8} +(75.0911 - 231.107i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 204 q^{4} - 224 q^{5} + 720 q^{7} - 3200 q^{8} - 2916 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.5443	−	4.72574i	−1.81804	−	0.590717i	−0.999876	−	0.0157312i	\(-0.994992\pi\)
							−0.818163	−	0.574986i	\(-0.805008\pi\)
\(3\)	−12.6113	+	9.16267i	−0.467086	+	0.339358i
							\(5\)	−61.8586	−	190.381i	−0.494869	−	1.52305i	−0.817161	−	0.576409i	\(-0.804454\pi\)
0.322293	−	0.946640i	\(-0.395546\pi\)
\(7\)	168.802	−	232.336i	0.492133	−	0.677363i	−0.488646	−	0.872482i	\(-0.662509\pi\)
							0.980780	+	0.195118i	\(0.0625091\pi\)
\(11\)	1156.29	+	659.201i	0.868740	+	0.495268i
							\(13\)	−1528.30	−	496.576i	−0.695632	−	0.226024i	−0.0602054	−	0.998186i	\(-0.519176\pi\)
−0.635426	+	0.772162i	\(0.719176\pi\)
\(17\)	−5702.23	+	1852.77i	−1.16064	+	0.377115i	−0.825142	−	0.564925i	\(-0.808905\pi\)
							−0.335499	+	0.942041i	\(0.608905\pi\)
\(19\)	−2881.69	−	3966.31i	−0.420133	−	0.578263i	0.545520	−	0.838098i	\(-0.316332\pi\)
							−0.965653	+	0.259834i	\(0.916332\pi\)
\(23\)	−2015.30			−0.165637			−0.0828184	−	0.996565i	\(-0.526392\pi\)
							−0.0828184	+	0.996565i	\(0.526392\pi\)
\(29\)	−8229.50	+	11326.9i	−0.337427	+	0.464428i	−0.943688	−	0.330838i	\(-0.892669\pi\)
							0.606261	+	0.795266i	\(0.292669\pi\)
\(31\)	−8129.83	+	25021.0i	−0.272895	+	0.839886i	0.716873	+	0.697204i	\(0.245573\pi\)
							−0.989769	+	0.142682i	\(0.954427\pi\)
\(37\)	23402.4	+	17002.8i	0.462014	+	0.335673i	0.794321	−	0.607499i	\(-0.207827\pi\)
							−0.332307	+	0.943171i	\(0.607827\pi\)
\(41\)	−23228.5	−	31971.3i	−0.337031	−	0.463884i	0.606540	−	0.795053i	\(-0.292557\pi\)
							−0.943571	+	0.331169i	\(0.892557\pi\)
\(43\)		−	83698.8i		−	1.05272i	−0.850261	−	0.526361i	\(-0.823556\pi\)
							0.850261	−	0.526361i	\(-0.176444\pi\)
\(47\)	−109601.	+	79629.5i	−1.05565	+	0.766974i	−0.973278	−	0.229628i	\(-0.926249\pi\)
							−0.0823705	+	0.996602i	\(0.526249\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.g.a.13.1	✓	48
3.2	odd	2		99.7.k.b.46.12		48
11.6	odd	10	inner	33.7.g.a.28.1	yes	48
33.17	even	10		99.7.k.b.28.12		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.g.a.13.1	✓	48	1.1	even	1	trivial
33.7.g.a.28.1	yes	48	11.6	odd	10	inner
99.7.k.b.28.12		48	33.17	even	10
99.7.k.b.46.12		48	3.2	odd	2