Embedded newform 33.7.h.a.5.3

• Label: 33.7.h.a.5.3
• Level: $33$
• Weight: $7$
• Character: 33.5
• Analytic conductor: $7.592$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	33.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.59178475946\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			5.3
Character	\(\chi\)	\(=\)	33.5
Dual form			33.7.h.a.20.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.7448 + 3.81612i) q^{2} +(-9.61413 + 25.2303i) q^{3} +(71.6006 - 52.0208i) q^{4} +(-165.956 - 53.9225i) q^{5} +(16.6343 - 333.014i) q^{6} +(-453.911 + 329.785i) q^{7} +(-177.861 + 244.804i) q^{8} +(-544.137 - 485.135i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 33 q^{3} + 634 q^{4} + 109 q^{6} + 402 q^{7} - 2471 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.7448	+	3.81612i	−1.46810	+	0.477015i	−0.930534	−	0.366206i	\(-0.880656\pi\)
							−0.537567	+	0.843221i	\(0.680656\pi\)
\(3\)	−9.61413	+	25.2303i	−0.356079	+	0.934456i
							\(5\)	−165.956	−	53.9225i	−1.32765	−	0.431380i	−0.442536	−	0.896751i	\(-0.645921\pi\)
−0.885116	+	0.465371i	\(0.845921\pi\)
\(7\)	−453.911	+	329.785i	−1.32335	+	0.961474i	−0.323471	+	0.946238i	\(0.604850\pi\)
							−0.999884	+	0.0152354i	\(0.995150\pi\)
\(11\)	1159.28	+	653.929i	0.870987	+	0.491307i
							\(13\)	−127.105	−	391.189i	−0.0578539	−	0.178056i	0.917953	−	0.396688i	\(-0.129841\pi\)
−0.975807	+	0.218632i	\(0.929841\pi\)
\(17\)	2005.78	+	651.717i	0.408260	+	0.132652i	0.505945	−	0.862566i	\(-0.331144\pi\)
							−0.0976853	+	0.995217i	\(0.531144\pi\)
\(19\)	2965.00	+	2154.20i	0.432278	+	0.314068i	0.782559	−	0.622576i	\(-0.213914\pi\)
							−0.350281	+	0.936645i	\(0.613914\pi\)
\(23\)			9647.72i			0.792942i	0.918047	+	0.396471i	\(0.129765\pi\)
							−0.918047	+	0.396471i	\(0.870235\pi\)
\(29\)	−26393.1	−	36327.0i	−1.08217	−	1.48948i	−0.857105	−	0.515142i	\(-0.827739\pi\)
							−0.225069	−	0.974343i	\(-0.572261\pi\)
\(31\)	−8199.22	−	25234.6i	−0.275225	−	0.847054i	−0.989160	−	0.146843i	\(-0.953089\pi\)
							0.713935	−	0.700212i	\(-0.246911\pi\)
\(37\)	−18264.0	+	13269.6i	−0.360571	+	0.261970i	−0.753290	−	0.657688i	\(-0.771534\pi\)
							0.392719	+	0.919658i	\(0.371534\pi\)
\(41\)	23526.0	−	32380.8i	0.341348	−	0.469825i	−0.603487	−	0.797373i	\(-0.706222\pi\)
							0.944834	+	0.327548i	\(0.106222\pi\)
\(43\)	114880.			1.44490			0.722452	−	0.691421i	\(-0.243015\pi\)
							0.722452	+	0.691421i	\(0.243015\pi\)
\(47\)	−69282.5	+	95359.2i	−0.667314	+	0.918478i	−0.999696	−	0.0246665i	\(-0.992148\pi\)
							0.332382	+	0.943145i	\(0.392148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.7.h.a.5.3	✓	88
3.2	odd	2	inner	33.7.h.a.5.20	yes	88
11.9	even	5	inner	33.7.h.a.20.20	yes	88
33.20	odd	10	inner	33.7.h.a.20.3	yes	88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.7.h.a.5.3	✓	88	1.1	even	1	trivial
33.7.h.a.5.20	yes	88	3.2	odd	2	inner
33.7.h.a.20.3	yes	88	33.20	odd	10	inner
33.7.h.a.20.20	yes	88	11.9	even	5	inner