Embedded newform 33.4.f.a.8.10

• Label: 33.4.f.a.8.10
• Level: $33$
• Weight: $4$
• Character: 33.8
• Analytic conductor: $1.947$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.94706303019\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(10\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			8.10
Character	\(\chi\)	\(=\)	33.8
Dual form			33.4.f.a.29.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.83133 - 2.78363i) q^{2} +(-4.74741 - 2.11237i) q^{3} +(4.45840 - 13.7215i) q^{4} +(-2.11757 + 2.91459i) q^{5} +(-24.0689 + 5.12183i) q^{6} +(28.5187 + 9.26628i) q^{7} +(-9.40652 - 28.9503i) q^{8} +(18.0758 + 20.0566i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 3 q^{3} - 38 q^{4} + 45 q^{6} - 10 q^{7} - 65 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.83133	−	2.78363i	1.35458	−	0.984160i	0.355812	−	0.934558i	\(-0.384205\pi\)
							0.998769	−	0.0496026i	\(-0.0157955\pi\)
\(3\)	−4.74741	−	2.11237i	−0.913639	−	0.406525i
							\(5\)	−2.11757	+	2.91459i	−0.189402	+	0.260689i	−0.893149	−	0.449762i	\(-0.851509\pi\)
0.703747	+	0.710451i	\(0.251509\pi\)
\(7\)	28.5187	+	9.26628i	1.53986	+	0.500332i	0.951339	−	0.308147i	\(-0.0997089\pi\)
							0.588525	+	0.808479i	\(0.299709\pi\)
\(11\)	−32.5614	+	16.4546i	−0.892513	+	0.451022i
							\(13\)	−23.3577	−	32.1491i	−0.498328	−	0.685890i	0.483569	−	0.875306i	\(-0.339340\pi\)
−0.981897	+	0.189417i	\(0.939340\pi\)
\(17\)	−18.3339	−	13.3204i	−0.261566	−	0.190039i	0.449271	−	0.893395i	\(-0.351684\pi\)
							−0.710837	+	0.703357i	\(0.751684\pi\)
\(19\)	−74.0299	+	24.0538i	−0.893874	+	0.290437i	−0.719706	−	0.694279i	\(-0.755724\pi\)
							−0.174168	+	0.984716i	\(0.555724\pi\)
\(23\)			39.4710i			0.357838i	0.983864	+	0.178919i	\(0.0572600\pi\)
							−0.983864	+	0.178919i	\(0.942740\pi\)
\(29\)	60.7406	−	186.940i	0.388939	−	1.19703i	−0.544643	−	0.838668i	\(-0.683335\pi\)
							0.933582	−	0.358364i	\(-0.116665\pi\)
\(31\)	−125.194	+	90.9588i	−0.725339	+	0.526990i	−0.888086	−	0.459678i	\(-0.847965\pi\)
							0.162746	+	0.986668i	\(0.447965\pi\)
\(37\)	103.703	−	319.166i	0.460776	−	1.41812i	−0.403442	−	0.915005i	\(-0.632186\pi\)
							0.864218	−	0.503117i	\(-0.167814\pi\)
\(41\)	−67.1986	−	206.816i	−0.255967	−	0.787787i	−0.993638	−	0.112625i	\(-0.964074\pi\)
							0.737670	−	0.675161i	\(-0.235926\pi\)
\(43\)			193.093i			0.684801i	0.939554	+	0.342400i	\(0.111240\pi\)
							−0.939554	+	0.342400i	\(0.888760\pi\)
\(47\)	−188.238	+	61.1622i	−0.584198	+	0.189818i	−0.586181	−	0.810180i	\(-0.699369\pi\)
							0.00198212	+	0.999998i	\(0.499369\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.4.f.a.8.10	yes	40
3.2	odd	2	inner	33.4.f.a.8.1	✓	40
11.2	odd	10		363.4.d.d.362.37		40
11.7	odd	10	inner	33.4.f.a.29.1	yes	40
11.9	even	5		363.4.d.d.362.3		40
33.2	even	10		363.4.d.d.362.4		40
33.20	odd	10		363.4.d.d.362.38		40
33.29	even	10	inner	33.4.f.a.29.10	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.f.a.8.1	✓	40	3.2	odd	2	inner
33.4.f.a.8.10	yes	40	1.1	even	1	trivial
33.4.f.a.29.1	yes	40	11.7	odd	10	inner
33.4.f.a.29.10	yes	40	33.29	even	10	inner
363.4.d.d.362.3		40	11.9	even	5
363.4.d.d.362.4		40	33.2	even	10
363.4.d.d.362.37		40	11.2	odd	10
363.4.d.d.362.38		40	33.20	odd	10