Embedded newform 33.4.f.a.8.6

• Label: 33.4.f.a.8.6
• 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			8.6
Character	\(\chi\)	\(=\)	33.8
Dual form			33.4.f.a.29.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.315290 - 0.229071i) q^{2} +(-3.41455 + 3.91674i) q^{3} +(-2.42520 + 7.46400i) q^{4} +(-2.50207 + 3.44380i) q^{5} +(-0.179357 + 2.01708i) q^{6} +(11.0922 + 3.60409i) q^{7} +(1.90859 + 5.87403i) q^{8} +(-3.68176 - 26.7478i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 3 q^{3} - 38 q^{4} + 45 q^{6} - 10 q^{7} - 65 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.315290	−	0.229071i	0.111472	−	0.0809890i	−0.530653	−	0.847589i	\(-0.678053\pi\)
							0.642125	+	0.766600i	\(0.278053\pi\)
\(3\)	−3.41455	+	3.91674i	−0.657130	+	0.753778i
							\(5\)	−2.50207	+	3.44380i	−0.223792	+	0.308023i	−0.906118	−	0.423025i	\(-0.860968\pi\)
0.682326	+	0.731048i	\(0.260968\pi\)
\(7\)	11.0922	+	3.60409i	0.598924	+	0.194602i	0.592761	−	0.805379i	\(-0.298038\pi\)
							0.00616375	+	0.999981i	\(0.498038\pi\)
\(11\)	25.5137	+	26.0778i	0.699333	+	0.714796i
							\(13\)	−19.1608	−	26.3725i	−0.408788	−	0.562648i	0.554134	−	0.832427i	\(-0.313049\pi\)
−0.962922	+	0.269779i	\(0.913049\pi\)
\(17\)	76.8025	+	55.8003i	1.09573	+	0.796091i	0.980357	−	0.197232i	\(-0.0631951\pi\)
							0.115369	+	0.993323i	\(0.463195\pi\)
\(19\)	55.1422	−	17.9168i	0.665815	−	0.216336i	0.0434404	−	0.999056i	\(-0.486168\pi\)
							0.622374	+	0.782720i	\(0.286168\pi\)
\(23\)		−	78.7828i		−	0.714232i	−0.934060	−	0.357116i	\(-0.883760\pi\)
							0.934060	−	0.357116i	\(-0.116240\pi\)
\(29\)	−49.8875	+	153.538i	−0.319444	+	0.983148i	0.654442	+	0.756112i	\(0.272904\pi\)
							−0.973886	+	0.227036i	\(0.927096\pi\)
\(31\)	35.3967	−	25.7172i	0.205079	−	0.148998i	−0.480506	−	0.876992i	\(-0.659547\pi\)
							0.685584	+	0.727993i	\(0.259547\pi\)
\(37\)	−27.2555	+	83.8838i	−0.121102	+	0.372714i	−0.993171	−	0.116670i	\(-0.962778\pi\)
							0.872069	+	0.489383i	\(0.162778\pi\)
\(41\)	−146.823	−	451.876i	−0.559267	−	1.72125i	−0.684399	−	0.729108i	\(-0.739935\pi\)
							0.125131	−	0.992140i	\(-0.460065\pi\)
\(43\)		−	285.559i		−	1.01273i	−0.862320	−	0.506364i	\(-0.830989\pi\)
							0.862320	−	0.506364i	\(-0.169011\pi\)
\(47\)	525.710	−	170.813i	1.63155	−	0.530121i	0.656920	−	0.753961i	\(-0.271859\pi\)
							0.974626	+	0.223839i	\(0.0718591\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.4.f.a.8.6	yes	40
3.2	odd	2	inner	33.4.f.a.8.5	✓	40
11.2	odd	10		363.4.d.d.362.22		40
11.7	odd	10	inner	33.4.f.a.29.5	yes	40
11.9	even	5		363.4.d.d.362.20		40
33.2	even	10		363.4.d.d.362.19		40
33.20	odd	10		363.4.d.d.362.21		40
33.29	even	10	inner	33.4.f.a.29.6	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.f.a.8.5	✓	40	3.2	odd	2	inner
33.4.f.a.8.6	yes	40	1.1	even	1	trivial
33.4.f.a.29.5	yes	40	11.7	odd	10	inner
33.4.f.a.29.6	yes	40	33.29	even	10	inner
363.4.d.d.362.19		40	33.2	even	10
363.4.d.d.362.20		40	11.9	even	5
363.4.d.d.362.21		40	33.20	odd	10
363.4.d.d.362.22		40	11.2	odd	10