Embedded newform 33.5.h.a.5.6

• Label: 33.5.h.a.5.6
• Level: $33$
• Weight: $5$
• Character: 33.5
• Analytic conductor: $3.411$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	33.h (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.41120878177\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	33.5
Dual form			33.5.h.a.20.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.90239 + 0.618124i) q^{2} +(1.59869 - 8.85687i) q^{3} +(-9.70726 + 7.05274i) q^{4} +(-8.40855 - 2.73210i) q^{5} +(2.43332 + 17.8374i) q^{6} +(-61.9407 + 45.0026i) q^{7} +(32.9194 - 45.3097i) q^{8} +(-75.8884 - 28.3188i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 90 q^{4} - 83 q^{6} - 86 q^{7} + 232 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.90239	+	0.618124i	−0.475598	+	0.154531i	−0.537000	−	0.843582i	\(-0.680443\pi\)
							0.0614026	+	0.998113i	\(0.480443\pi\)
\(3\)	1.59869	−	8.85687i	0.177632	−	0.984097i
							\(5\)	−8.40855	−	2.73210i	−0.336342	−	0.109284i	0.135977	−	0.990712i	\(-0.456583\pi\)
−0.472319	+	0.881428i	\(0.656583\pi\)
\(7\)	−61.9407	+	45.0026i	−1.26410	+	0.918420i	−0.998951	−	0.0457884i	\(-0.985420\pi\)
							−0.265146	+	0.964208i	\(0.585420\pi\)
\(11\)	−110.634	−	49.0015i	−0.914330	−	0.404971i
							\(13\)	−14.9141	−	45.9010i	−0.0882493	−	0.271603i	0.897186	−	0.441652i	\(-0.145607\pi\)
−0.985436	+	0.170049i	\(0.945607\pi\)
\(17\)	186.116	+	60.4726i	0.643999	+	0.209248i	0.612766	−	0.790264i	\(-0.290057\pi\)
							0.0312325	+	0.999512i	\(0.490057\pi\)
\(19\)	298.193	+	216.650i	0.826020	+	0.600139i	0.918430	−	0.395582i	\(-0.129457\pi\)
							−0.0924106	+	0.995721i	\(0.529457\pi\)
\(23\)		−	418.729i		−	0.791548i	−0.918348	−	0.395774i	\(-0.870476\pi\)
							0.918348	−	0.395774i	\(-0.129524\pi\)
\(29\)	−798.795	−	1099.45i	−0.949816	−	1.30731i	−0.951609	−	0.307311i	\(-0.900571\pi\)
							0.00179349	−	0.999998i	\(-0.499429\pi\)
\(31\)	185.992	+	572.425i	0.193540	+	0.595656i	0.999991	+	0.00435377i	\(0.00138585\pi\)
							−0.806450	+	0.591302i	\(0.798614\pi\)
\(37\)	−1482.41	+	1077.03i	−1.08284	+	0.786728i	−0.978176	−	0.207779i	\(-0.933376\pi\)
							−0.104663	+	0.994508i	\(0.533376\pi\)
\(41\)	565.214	−	777.950i	0.336237	−	0.462790i	−0.607101	−	0.794625i	\(-0.707668\pi\)
							0.943338	+	0.331835i	\(0.107668\pi\)
\(43\)	435.575			0.235573			0.117787	−	0.993039i	\(-0.462420\pi\)
							0.117787	+	0.993039i	\(0.462420\pi\)
\(47\)	−501.345	+	690.043i	−0.226956	+	0.312378i	−0.907275	−	0.420538i	\(-0.861841\pi\)
							0.680319	+	0.732916i	\(0.261841\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.h.a.5.6	✓	56
3.2	odd	2	inner	33.5.h.a.5.9	yes	56
11.9	even	5	inner	33.5.h.a.20.9	yes	56
33.20	odd	10	inner	33.5.h.a.20.6	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.h.a.5.6	✓	56	1.1	even	1	trivial
33.5.h.a.5.9	yes	56	3.2	odd	2	inner
33.5.h.a.20.6	yes	56	33.20	odd	10	inner
33.5.h.a.20.9	yes	56	11.9	even	5	inner