Embedded newform 33.3.h.b.26.4

• Label: 33.3.h.b.26.4
• Level: $33$
• Weight: $3$
• Character: 33.26
• 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			26.4
Root			\(2.10855 - 2.90217i\) of defining polynomial
Character	\(\chi\)	\(=\)	33.26
Dual form			33.3.h.b.14.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.10855 - 2.90217i) q^{2} +(-2.00253 + 2.23380i) q^{3} +(-2.74053 - 8.43448i) q^{4} +(1.22635 + 1.68793i) q^{5} +(2.26043 + 10.5218i) q^{6} +(2.73883 + 8.42924i) q^{7} +(-16.6100 - 5.39692i) q^{8} +(-0.979734 - 8.94651i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 10 q^{3} + 8 q^{4} - 33 q^{6} + 6 q^{7} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.10855	−	2.90217i	1.05427	−	1.45108i	0.169229	−	0.985577i	\(-0.445872\pi\)
							0.885045	−	0.465506i	\(-0.154128\pi\)
\(3\)	−2.00253	+	2.23380i	−0.667511	+	0.744600i
							\(5\)	1.22635	+	1.68793i	0.245270	+	0.337586i	0.913848	−	0.406057i	\(-0.133097\pi\)
−0.668578	+	0.743642i	\(0.733097\pi\)
\(7\)	2.73883	+	8.42924i	0.391261	+	1.20418i	0.931836	+	0.362881i	\(0.118207\pi\)
							−0.540575	+	0.841296i	\(0.681793\pi\)
\(11\)	−10.8108	−	2.03154i	−0.982798	−	0.184685i
							\(13\)	1.33068	+	0.966792i	0.102360	+	0.0743686i	0.637788	−	0.770212i	\(-0.279850\pi\)
−0.535428	+	0.844581i	\(0.679850\pi\)
\(17\)	−7.30235	−	10.0508i	−0.429550	−	0.591225i	0.538300	−	0.842753i	\(-0.319067\pi\)
							−0.967850	+	0.251529i	\(0.919067\pi\)
\(19\)	3.26497	−	10.0485i	0.171841	−	0.528871i	−0.827635	−	0.561267i	\(-0.810314\pi\)
							0.999475	+	0.0323966i	\(0.0103140\pi\)
\(23\)			20.3378i			0.884251i	0.896953	+	0.442126i	\(0.145775\pi\)
							−0.896953	+	0.442126i	\(0.854225\pi\)
\(29\)	−11.0405	+	3.58727i	−0.380707	+	0.123699i	−0.493118	−	0.869963i	\(-0.664143\pi\)
							0.112411	+	0.993662i	\(0.464143\pi\)
\(31\)	18.9215	+	13.7473i	0.610371	+	0.443460i	0.849545	−	0.527516i	\(-0.176877\pi\)
							−0.239174	+	0.970977i	\(0.576877\pi\)
\(37\)	2.23911	+	6.89128i	0.0605166	+	0.186251i	0.976745	−	0.214407i	\(-0.0687817\pi\)
							−0.916228	+	0.400657i	\(0.868782\pi\)
\(41\)	36.9741	+	12.0136i	0.901806	+	0.293015i	0.722982	−	0.690866i	\(-0.242771\pi\)
							0.178824	+	0.983881i	\(0.442771\pi\)
\(43\)	−15.8444			−0.368474			−0.184237	−	0.982882i	\(-0.558981\pi\)
							−0.184237	+	0.982882i	\(0.558981\pi\)
\(47\)	43.0910	+	14.0011i	0.916831	+	0.297896i	0.729166	−	0.684337i	\(-0.239908\pi\)
							0.187665	+	0.982233i	\(0.439908\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.3.h.b.26.4	yes	16
3.2	odd	2	inner	33.3.h.b.26.1	yes	16
11.2	odd	10		363.3.h.n.269.4		16
11.3	even	5	inner	33.3.h.b.14.1	✓	16
11.4	even	5		363.3.h.o.251.4		16
11.5	even	5		363.3.b.m.122.8		8
11.6	odd	10		363.3.b.l.122.1		8
11.7	odd	10		363.3.h.n.251.1		16
11.8	odd	10		363.3.h.j.245.4		16
11.9	even	5		363.3.h.o.269.1		16
11.10	odd	2		363.3.h.j.323.1		16
33.2	even	10		363.3.h.n.269.1		16
33.5	odd	10		363.3.b.m.122.1		8
33.8	even	10		363.3.h.j.245.1		16
33.14	odd	10	inner	33.3.h.b.14.4	yes	16
33.17	even	10		363.3.b.l.122.8		8
33.20	odd	10		363.3.h.o.269.4		16
33.26	odd	10		363.3.h.o.251.1		16
33.29	even	10		363.3.h.n.251.4		16
33.32	even	2		363.3.h.j.323.4		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.h.b.14.1	✓	16	11.3	even	5	inner
33.3.h.b.14.4	yes	16	33.14	odd	10	inner
33.3.h.b.26.1	yes	16	3.2	odd	2	inner
33.3.h.b.26.4	yes	16	1.1	even	1	trivial
363.3.b.l.122.1		8	11.6	odd	10
363.3.b.l.122.8		8	33.17	even	10
363.3.b.m.122.1		8	33.5	odd	10
363.3.b.m.122.8		8	11.5	even	5
363.3.h.j.245.1		16	33.8	even	10
363.3.h.j.245.4		16	11.8	odd	10
363.3.h.j.323.1		16	11.10	odd	2
363.3.h.j.323.4		16	33.32	even	2
363.3.h.n.251.1		16	11.7	odd	10
363.3.h.n.251.4		16	33.29	even	10
363.3.h.n.269.1		16	33.2	even	10
363.3.h.n.269.4		16	11.2	odd	10
363.3.h.o.251.1		16	33.26	odd	10
363.3.h.o.251.4		16	11.4	even	5
363.3.h.o.269.1		16	11.9	even	5
363.3.h.o.269.4		16	33.20	odd	10