Embedded newform 33.3.h.b.5.3

• Label: 33.3.h.b.5.3
• Level: $33$
• Weight: $3$
• Character: 33.5
• Analytic conductor: $0.899$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.899184872389\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(4\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	\( x^{16} - 12x^{14} + 180x^{12} - 2562x^{10} + 25179x^{8} - 96398x^{6} + 239275x^{4} - 536393x^{2} + 1771561 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			5.3
Root			\(1.90610 - 0.619331i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.5
Dual form			33.3.h.b.20.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.90610 - 0.619331i) q^{2} +(-1.63362 - 2.51621i) q^{3} +(0.0135968 - 0.00987866i) q^{4} +(5.21596 + 1.69477i) q^{5} +(-4.67221 - 3.78440i) q^{6} +(-4.52308 + 3.28621i) q^{7} +(-4.69235 + 6.45847i) q^{8} +(-3.66258 + 8.22104i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 10 q^{3} + 8 q^{4} - 33 q^{6} + 6 q^{7} - 28 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.90610	−	0.619331i	0.953052	−	0.309666i	0.209097	−	0.977895i	\(-0.432948\pi\)
							0.743955	+	0.668229i	\(0.232948\pi\)
\(3\)	−1.63362	−	2.51621i	−0.544540	−	0.838735i
							\(5\)	5.21596	+	1.69477i	1.04319	+	0.338953i	0.779993	−	0.625788i	\(-0.215223\pi\)
0.263198	+	0.964742i	\(0.415223\pi\)
\(7\)	−4.52308	+	3.28621i	−0.646155	+	0.469459i	−0.861959	−	0.506978i	\(-0.830763\pi\)
							0.215804	+	0.976437i	\(0.430763\pi\)
\(11\)	2.23693	−	10.7702i	0.203357	−	0.979105i
							\(13\)	−3.00265	−	9.24122i	−0.230973	−	0.710863i	−0.997630	−	0.0688058i	\(-0.978081\pi\)
0.766657	−	0.642057i	\(-0.221919\pi\)
\(17\)	16.9969	+	5.52262i	0.999817	+	0.324860i	0.762792	−	0.646643i	\(-0.223828\pi\)
							0.237024	+	0.971504i	\(0.423828\pi\)
\(19\)	−15.0954	−	10.9674i	−0.794493	−	0.577233i	0.114801	−	0.993389i	\(-0.463377\pi\)
							−0.909293	+	0.416156i	\(0.863377\pi\)
\(23\)		−	12.3649i		−	0.537603i	−0.963196	−	0.268801i	\(-0.913372\pi\)
							0.963196	−	0.268801i	\(-0.0866275\pi\)
\(29\)	−1.45613	−	2.00420i	−0.0502115	−	0.0691102i	0.783173	−	0.621804i	\(-0.213600\pi\)
							−0.833384	+	0.552694i	\(0.813600\pi\)
\(31\)	15.2132	+	46.8213i	0.490747	+	1.51037i	0.823481	+	0.567344i	\(0.192029\pi\)
							−0.332733	+	0.943021i	\(0.607971\pi\)
\(37\)	31.8192	−	23.1180i	0.859979	−	0.624811i	−0.0679003	−	0.997692i	\(-0.521630\pi\)
							0.927879	+	0.372881i	\(0.121630\pi\)
\(41\)	−33.2237	+	45.7285i	−0.810335	+	1.11533i	0.180937	+	0.983495i	\(0.442087\pi\)
							−0.991272	+	0.131835i	\(0.957913\pi\)
\(43\)	−43.9060			−1.02107			−0.510534	−	0.859857i	\(-0.670552\pi\)
							−0.510534	+	0.859857i	\(0.670552\pi\)
\(47\)	33.9646	−	46.7482i	0.722651	−	0.994643i	−0.276781	−	0.960933i	\(-0.589268\pi\)
							0.999432	−	0.0337102i	\(-0.0107323\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.3.h.b.5.3	yes	16
3.2	odd	2	inner	33.3.h.b.5.2	✓	16
11.2	odd	10		363.3.h.j.251.3		16
11.3	even	5		363.3.b.m.122.3		8
11.4	even	5		363.3.h.o.245.3		16
11.5	even	5		363.3.h.o.323.2		16
11.6	odd	10		363.3.h.n.323.3		16
11.7	odd	10		363.3.h.n.245.2		16
11.8	odd	10		363.3.b.l.122.6		8
11.9	even	5	inner	33.3.h.b.20.2	yes	16
11.10	odd	2		363.3.h.j.269.2		16
33.2	even	10		363.3.h.j.251.2		16
33.5	odd	10		363.3.h.o.323.3		16
33.8	even	10		363.3.b.l.122.3		8
33.14	odd	10		363.3.b.m.122.6		8
33.17	even	10		363.3.h.n.323.2		16
33.20	odd	10	inner	33.3.h.b.20.3	yes	16
33.26	odd	10		363.3.h.o.245.2		16
33.29	even	10		363.3.h.n.245.3		16
33.32	even	2		363.3.h.j.269.3		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.h.b.5.2	✓	16	3.2	odd	2	inner
33.3.h.b.5.3	yes	16	1.1	even	1	trivial
33.3.h.b.20.2	yes	16	11.9	even	5	inner
33.3.h.b.20.3	yes	16	33.20	odd	10	inner
363.3.b.l.122.3		8	33.8	even	10
363.3.b.l.122.6		8	11.8	odd	10
363.3.b.m.122.3		8	11.3	even	5
363.3.b.m.122.6		8	33.14	odd	10
363.3.h.j.251.2		16	33.2	even	10
363.3.h.j.251.3		16	11.2	odd	10
363.3.h.j.269.2		16	11.10	odd	2
363.3.h.j.269.3		16	33.32	even	2
363.3.h.n.245.2		16	11.7	odd	10
363.3.h.n.245.3		16	33.29	even	10
363.3.h.n.323.2		16	33.17	even	10
363.3.h.n.323.3		16	11.6	odd	10
363.3.h.o.245.2		16	33.26	odd	10
363.3.h.o.245.3		16	11.4	even	5
363.3.h.o.323.2		16	11.5	even	5
363.3.h.o.323.3		16	33.5	odd	10