Embedded newform 43.6.e.a.4.11

• Label: 43.6.e.a.4.11
• Level: $43$
• Weight: $6$
• Character: 43.4
• Analytic conductor: $6.897$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89650425196\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(18\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			4.11
Character	\(\chi\)	\(=\)	43.4
Dual form			43.6.e.a.11.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.240368 - 0.301412i) q^{2} +(8.08269 + 10.1354i) q^{3} +(7.08760 + 31.0528i) q^{4} +(33.3160 + 16.0441i) q^{5} +4.99775 q^{6} -195.731 q^{7} +(22.1783 + 10.6805i) q^{8} +(16.6767 - 73.0652i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 3 q^{2} - 9 q^{3} - 331 q^{4} + 67 q^{5} - 20 q^{6} - 624 q^{7} + 589 q^{8} - 579 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.240368	−	0.301412i	0.0424915	−	0.0532826i	−0.760132	−	0.649769i	\(-0.774866\pi\)
							0.802623	+	0.596486i	\(0.203437\pi\)
\(3\)	8.08269	+	10.1354i	0.518505	+	0.650185i	0.970291	−	0.241942i	\(-0.0777843\pi\)
							−0.451786	+	0.892126i	\(0.649213\pi\)
\(5\)	33.3160	+	16.0441i	0.595975	+	0.287006i	0.707453	−	0.706760i	\(-0.249844\pi\)
							−0.111478	+	0.993767i	\(0.535558\pi\)
\(7\)	−195.731			−1.50978			−0.754890	−	0.655852i	\(-0.772310\pi\)
							−0.754890	+	0.655852i	\(0.772310\pi\)
\(11\)	−118.569	+	519.484i	−0.295453	+	1.29446i	0.581365	+	0.813643i	\(0.302519\pi\)
							−0.876818	+	0.480822i	\(0.840338\pi\)
\(13\)	330.557	+	159.188i	0.542486	+	0.261247i	0.684997	−	0.728546i	\(-0.259803\pi\)
							−0.142511	+	0.989793i	\(0.545518\pi\)
\(17\)	1704.28	−	820.737i	1.43027	−	0.688782i	0.451222	−	0.892412i	\(-0.350988\pi\)
							0.979048	+	0.203630i	\(0.0652739\pi\)
\(19\)	168.830	+	739.694i	0.107292	+	0.470076i	0.999818	+	0.0190803i	\(0.00607383\pi\)
							−0.892526	+	0.450996i	\(0.851069\pi\)
\(23\)	−707.903	+	3101.53i	−0.279032	+	1.22252i	0.619988	+	0.784612i	\(0.287138\pi\)
							−0.899020	+	0.437908i	\(0.855720\pi\)
\(29\)	1603.49	−	2010.72i	0.354056	−	0.443973i	−0.572627	−	0.819816i	\(-0.694076\pi\)
							0.926683	+	0.375844i	\(0.122647\pi\)
\(31\)	4485.88	−	5625.12i	0.838385	−	1.05130i	−0.159558	−	0.987189i	\(-0.551007\pi\)
							0.997943	−	0.0641130i	\(-0.0204218\pi\)
\(37\)	7266.79			0.872646			0.436323	−	0.899790i	\(-0.356280\pi\)
							0.436323	+	0.899790i	\(0.356280\pi\)
\(41\)	7512.66	−	9420.58i	0.697966	−	0.875222i	−0.298904	−	0.954283i	\(-0.596621\pi\)
							0.996870	+	0.0790616i	\(0.0251924\pi\)
\(43\)	653.714	+	12107.1i	0.0539159	+	0.998545i
							\(47\)	−1117.67	−	4896.84i	−0.0738022	−	0.323349i	0.924528	−	0.381114i	\(-0.124460\pi\)
−0.998330	+	0.0577656i	\(0.981602\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.6.e.a.4.11	✓	108
43.11	even	7	inner	43.6.e.a.11.11	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.e.a.4.11	✓	108	1.1	even	1	trivial
43.6.e.a.11.11	yes	108	43.11	even	7	inner