Embedded newform 33.4.f.a.8.5

• Label: 33.4.f.a.8.5
• 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			8.5
Character	\(\chi\)	\(=\)	33.8
Dual form			33.4.f.a.29.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.315290 + 0.229071i) q^{2} +(5.06463 + 1.16169i) q^{3} +(-2.42520 + 7.46400i) q^{4} +(2.50207 - 3.44380i) q^{5} +(-1.86294 + 0.793892i) q^{6} +(11.0922 + 3.60409i) q^{7} +(-1.90859 - 5.87403i) q^{8} +(24.3009 + 11.7671i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 3 q^{3} - 38 q^{4} + 45 q^{6} - 10 q^{7} - 65 q^{9}+O(q^{10}) \)

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	−0.642125	−	0.766600i	\(-0.721947\pi\)
							0.530653	+	0.847589i	\(0.321947\pi\)
\(3\)	5.06463	+	1.16169i	0.974688	+	0.223568i
							\(5\)	2.50207	−	3.44380i	0.223792	−	0.308023i	−0.682326	−	0.731048i	\(-0.739032\pi\)
0.906118	+	0.423025i	\(0.139032\pi\)
\(7\)	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	−0.115369	−	0.993323i	\(-0.536805\pi\)
							−0.980357	+	0.197232i	\(0.936805\pi\)
\(19\)	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	\(0.116240\pi\)
							−0.934060	+	0.357116i	\(0.883760\pi\)
\(29\)	49.8875	−	153.538i	0.319444	−	0.983148i	−0.654442	−	0.756112i	\(-0.727096\pi\)
							0.973886	−	0.227036i	\(-0.0729036\pi\)
\(31\)	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	\(0.260065\pi\)
							−0.125131	+	0.992140i	\(0.539935\pi\)
\(43\)		−	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	−0.974626	−	0.223839i	\(-0.928141\pi\)
							−0.656920	+	0.753961i	\(0.728141\pi\)

(See \(a_n\) instead)

Twists

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

    

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