Embedded newform 43.4.g.a.24.7

• Label: 43.4.g.a.24.7
• Level: $43$
• Weight: $4$
• Character: 43.24
• Analytic conductor: $2.537$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.53708213025\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(10\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			24.7
Character	\(\chi\)	\(=\)	43.24
Dual form			43.4.g.a.9.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.789787 - 0.990362i) q^{2} +(-6.14934 + 0.926863i) q^{3} +(1.42311 + 6.23507i) q^{4} +(-11.5199 + 7.85413i) q^{5} +(-3.93874 + 6.82209i) q^{6} +(-0.966032 - 1.67322i) q^{7} +(16.4291 + 7.91186i) q^{8} +(11.1548 - 3.44080i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 12 q^{2} - 9 q^{3} - 92 q^{4} + 5 q^{5} - 22 q^{6} - 54 q^{7} + 2 q^{8} + 201 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.789787	−	0.990362i	0.279232	−	0.350146i	−0.622362	−	0.782730i	\(-0.713827\pi\)
							0.901594	+	0.432584i	\(0.142398\pi\)
\(3\)	−6.14934	+	0.926863i	−1.18344	+	0.178375i	−0.711132	−	0.703059i	\(-0.751817\pi\)
							−0.472308	+	0.881433i	\(0.656579\pi\)
\(5\)	−11.5199	+	7.85413i	−1.03037	+	0.702495i	−0.955588	−	0.294705i	\(-0.904779\pi\)
							−0.0747828	+	0.997200i	\(0.523826\pi\)
\(7\)	−0.966032	−	1.67322i	−0.0521608	−	0.0903452i	0.838766	−	0.544492i	\(-0.183278\pi\)
							−0.890927	+	0.454147i	\(0.849944\pi\)
\(11\)	−7.67566	+	33.6293i	−0.210391	+	0.921783i	0.753911	+	0.656977i	\(0.228165\pi\)
							−0.964302	+	0.264806i	\(0.914692\pi\)
\(13\)	−3.66052	−	48.8462i	−0.0780958	−	1.04212i	−0.889291	−	0.457341i	\(-0.848802\pi\)
							0.811196	−	0.584775i	\(-0.198817\pi\)
\(17\)	40.7603	+	27.7899i	0.581519	+	0.396473i	0.818047	−	0.575152i	\(-0.195057\pi\)
							−0.236528	+	0.971625i	\(0.576009\pi\)
\(19\)	−58.3151	−	17.9878i	−0.704127	−	0.217194i	−0.0780429	−	0.996950i	\(-0.524867\pi\)
							−0.626084	+	0.779756i	\(0.715343\pi\)
\(23\)	138.188	+	128.220i	1.25279	+	1.16242i	0.979708	+	0.200429i	\(0.0642336\pi\)
							0.273082	+	0.961991i	\(0.411957\pi\)
\(29\)	10.8304	+	1.63242i	0.0693500	+	0.0104528i	0.183626	−	0.982996i	\(-0.441217\pi\)
							−0.114276	+	0.993449i	\(0.536455\pi\)
\(31\)	26.3549	+	67.1512i	0.152693	+	0.389055i	0.986668	−	0.162747i	\(-0.0520355\pi\)
							−0.833975	+	0.551802i	\(0.813940\pi\)
\(37\)	−144.912	+	250.995i	−0.643876	+	1.11523i	0.340684	+	0.940178i	\(0.389341\pi\)
							−0.984560	+	0.175048i	\(0.943992\pi\)
\(41\)	32.8220	−	41.1575i	0.125023	−	0.156774i	−0.715381	−	0.698735i	\(-0.753747\pi\)
							0.840403	+	0.541961i	\(0.182318\pi\)
\(43\)	−263.036	+	101.584i	−0.932850	+	0.360264i
							\(47\)	110.186	+	482.758i	0.341965	+	1.49825i	0.794921	+	0.606713i	\(0.207512\pi\)
−0.452956	+	0.891533i	\(0.649631\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.g.a.24.7	yes	120
43.3	odd	42		1849.4.a.k.1.27		60
43.9	even	21	inner	43.4.g.a.9.7	✓	120
43.40	even	21		1849.4.a.l.1.34		60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.g.a.9.7	✓	120	43.9	even	21	inner
43.4.g.a.24.7	yes	120	1.1	even	1	trivial
1849.4.a.k.1.27		60	43.3	odd	42
1849.4.a.l.1.34		60	43.40	even	21