Embedded newform 24.6.f.b.11.9

• Label: 24.6.f.b.11.9
• Level: $24$
• Weight: $6$
• Character: 24.11
• Analytic conductor: $3.849$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 24 = 2^{3} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	24.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.84921167551\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - 26*x^14 + 904*x^12 - 55232*x^10 + 1401856*x^8 - 56557568*x^6 + 947912704*x^4 - 27917287424*x^2 + 1099511627776
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{32}\cdot 3^{10} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			11.9
Root			\(2.35284 - 5.14433i\) of defining polynomial
Character	\(\chi\)	\(=\)	24.11
Dual form			24.6.f.b.11.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.35284 - 5.14433i) q^{2} +(8.05622 + 13.3453i) q^{3} +(-20.9283 - 24.2076i) q^{4} +95.9504 q^{5} +(87.6077 - 10.0445i) q^{6} -122.802i q^{7} +(-173.773 + 50.7054i) q^{8} +(-113.195 + 215.026i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 52 q^{4} + 264 q^{6} + 480 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\)	\(7\)	\(13\)	\(17\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.35284	−	5.14433i	0.415927	−	0.909398i
							\(3\)	8.05622	+	13.3453i	0.516807	+	0.856102i
\(5\)	95.9504			1.71641										0.858206	−	0.513305i	\(-0.171579\pi\)
							0.858206	+	0.513305i	\(0.171579\pi\)
\(7\)		−	122.802i		−	0.947240i	−0.880729	−	0.473620i	\(-0.842947\pi\)
							0.880729	−	0.473620i	\(-0.157053\pi\)
\(11\)		−	77.6307i		−	0.193443i	−0.995312	−	0.0967213i	\(-0.969164\pi\)
							0.995312	−	0.0967213i	\(-0.0308355\pi\)
\(13\)			267.974i			0.439778i	0.975525	+	0.219889i	\(0.0705696\pi\)
							−0.975525	+	0.219889i	\(0.929430\pi\)
\(17\)			958.891i			0.804724i	0.915481	+	0.402362i	\(0.131811\pi\)
							−0.915481	+	0.402362i	\(0.868189\pi\)
\(19\)	−1992.07			−1.26596			−0.632982	−	0.774167i	\(-0.718169\pi\)
							−0.632982	+	0.774167i	\(0.718169\pi\)
\(23\)	−2142.09			−0.844343			−0.422171	−	0.906516i	\(-0.638732\pi\)
							−0.422171	+	0.906516i	\(0.638732\pi\)
\(29\)	−1413.81			−0.312174			−0.156087	−	0.987743i	\(-0.549888\pi\)
							−0.156087	+	0.987743i	\(0.549888\pi\)
\(31\)		−	535.205i		−	0.100027i	−0.998749	−	0.0500134i	\(-0.984074\pi\)
							0.998749	−	0.0500134i	\(-0.0159264\pi\)
\(37\)		−	15204.5i		−	1.82586i	−0.408120	−	0.912928i	\(-0.633815\pi\)
							0.408120	−	0.912928i	\(-0.366185\pi\)
\(41\)			7477.78i			0.694725i	0.937731	+	0.347363i	\(0.112923\pi\)
							−0.937731	+	0.347363i	\(0.887077\pi\)
\(43\)	8815.59			0.727076			0.363538	−	0.931579i	\(-0.381569\pi\)
							0.363538	+	0.931579i	\(0.381569\pi\)
\(47\)	−4179.37			−0.275972			−0.137986	−	0.990434i	\(-0.544063\pi\)
							−0.137986	+	0.990434i	\(0.544063\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	24.6.f.b.11.9	yes	16
3.2	odd	2	inner	24.6.f.b.11.8	yes	16
4.3	odd	2		96.6.f.b.47.6		16
8.3	odd	2	inner	24.6.f.b.11.7	✓	16
8.5	even	2		96.6.f.b.47.5		16
12.11	even	2		96.6.f.b.47.7		16
24.5	odd	2		96.6.f.b.47.8		16
24.11	even	2	inner	24.6.f.b.11.10	yes	16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.6.f.b.11.7	✓	16	8.3	odd	2	inner
24.6.f.b.11.8	yes	16	3.2	odd	2	inner
24.6.f.b.11.9	yes	16	1.1	even	1	trivial
24.6.f.b.11.10	yes	16	24.11	even	2	inner
96.6.f.b.47.5		16	8.5	even	2
96.6.f.b.47.6		16	4.3	odd	2
96.6.f.b.47.7		16	12.11	even	2
96.6.f.b.47.8		16	24.5	odd	2