Embedded newform 33.4.f.a.17.5

• Label: 33.4.f.a.17.5
• Level: $33$
• Weight: $4$
• Character: 33.17
• Analytic conductor: $1.947$
• Analytic rank: $0$
• Dimension: $40$
• 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			17.5
Character	\(\chi\)	\(=\)	33.17
Dual form			33.4.f.a.2.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.325060 - 1.00043i) q^{2} +(-1.99228 + 4.79904i) q^{3} +(5.57694 - 4.05188i) q^{4} +(11.6880 + 3.79766i) q^{5} +(5.44873 + 0.433161i) q^{6} +(13.1668 + 18.1225i) q^{7} +(-12.6746 - 9.20865i) q^{8} +(-19.0617 - 19.1221i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 3 q^{3} - 38 q^{4} + 45 q^{6} - 10 q^{7} - 65 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.325060	−	1.00043i	−0.114926	−	0.353706i	0.877005	−	0.480480i	\(-0.159538\pi\)
							−0.991932	+	0.126774i	\(0.959538\pi\)
\(3\)	−1.99228	+	4.79904i	−0.383414	+	0.923577i
							\(5\)	11.6880	+	3.79766i	1.04541	+	0.339673i	0.780864	−	0.624702i	\(-0.214779\pi\)
0.264542	+	0.964374i	\(0.414779\pi\)
\(7\)	13.1668	+	18.1225i	0.710938	+	0.978522i	0.999776	+	0.0211444i	\(0.00673097\pi\)
							−0.288838	+	0.957378i	\(0.593269\pi\)
\(11\)	12.4720	−	34.2848i	0.341859	−	0.939751i
							\(13\)	−75.4132	+	24.5032i	−1.60891	+	0.522767i	−0.969289	−	0.245923i	\(-0.920909\pi\)
−0.639622	+	0.768690i	\(0.720909\pi\)
\(17\)	−10.2954	+	31.6859i	−0.146882	+	0.452056i	−0.997248	−	0.0741349i	\(-0.976380\pi\)
							0.850366	+	0.526191i	\(0.176380\pi\)
\(19\)	21.2305	−	29.2213i	0.256348	−	0.352833i	−0.661374	−	0.750057i	\(-0.730026\pi\)
							0.917722	+	0.397224i	\(0.130026\pi\)
\(23\)		−	134.233i		−	1.21693i	−0.793579	−	0.608467i	\(-0.791785\pi\)
							0.793579	−	0.608467i	\(-0.208215\pi\)
\(29\)	−87.6969	+	63.7155i	−0.561548	+	0.407989i	−0.832025	−	0.554738i	\(-0.812819\pi\)
							0.270477	+	0.962726i	\(0.412819\pi\)
\(31\)	9.67702	+	29.7828i	0.0560660	+	0.172553i	0.975168	−	0.221466i	\(-0.0710842\pi\)
							−0.919102	+	0.394019i	\(0.871084\pi\)
\(37\)	−6.44457	+	4.68225i	−0.0286346	+	0.0208043i	−0.602011	−	0.798488i	\(-0.705633\pi\)
							0.573376	+	0.819292i	\(0.305633\pi\)
\(41\)	6.01493	+	4.37011i	0.0229116	+	0.0166462i	0.599182	−	0.800613i	\(-0.295493\pi\)
							−0.576271	+	0.817259i	\(0.695493\pi\)
\(43\)		−	74.3056i		−	0.263523i	−0.991281	−	0.131762i	\(-0.957937\pi\)
							0.991281	−	0.131762i	\(-0.0420633\pi\)
\(47\)	−301.340	+	414.759i	−0.935211	+	1.28721i	0.0225810	+	0.999745i	\(0.492812\pi\)
							−0.957792	+	0.287462i	\(0.907188\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.4.f.a.17.5	yes	40
3.2	odd	2	inner	33.4.f.a.17.6	yes	40
11.2	odd	10	inner	33.4.f.a.2.6	yes	40
11.3	even	5		363.4.d.d.362.18		40
11.8	odd	10		363.4.d.d.362.24		40
33.2	even	10	inner	33.4.f.a.2.5	✓	40
33.8	even	10		363.4.d.d.362.17		40
33.14	odd	10		363.4.d.d.362.23		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.f.a.2.5	✓	40	33.2	even	10	inner
33.4.f.a.2.6	yes	40	11.2	odd	10	inner
33.4.f.a.17.5	yes	40	1.1	even	1	trivial
33.4.f.a.17.6	yes	40	3.2	odd	2	inner
363.4.d.d.362.17		40	33.8	even	10
363.4.d.d.362.18		40	11.3	even	5
363.4.d.d.362.23		40	33.14	odd	10
363.4.d.d.362.24		40	11.8	odd	10