Embedded newform 43.9.d.a.7.7

• Label: 43.9.d.a.7.7
• Level: $43$
• Weight: $9$
• Character: 43.7
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.7
Character	\(\chi\)	\(=\)	43.7
Dual form			43.9.d.a.37.22

$q$-expansion

\(f(q)\)	\(=\)	\(q-18.7653i q^{2} +(-129.819 + 74.9511i) q^{3} -96.1366 q^{4} +(343.803 - 198.495i) q^{5} +(1406.48 + 2436.10i) q^{6} +(1031.87 + 595.751i) q^{7} -2999.89i q^{8} +(7954.84 - 13778.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 84 q^{3} - 6474 q^{4} - 3 q^{5} + 1791 q^{6} - 5031 q^{7} + 57858 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	18.7653i		−	1.17283i	−0.810010	−	0.586416i	\(-0.800539\pi\)
							0.810010	−	0.586416i	\(-0.199461\pi\)
\(3\)	−129.819	+	74.9511i	−1.60271	+	0.925322i	−0.611762	+	0.791042i	\(0.709539\pi\)
							−0.990943	+	0.134280i	\(0.957128\pi\)
\(5\)	343.803	−	198.495i	0.550085	−	0.317592i	−0.199071	−	0.979985i	\(-0.563792\pi\)
							0.749156	+	0.662393i	\(0.230459\pi\)
\(7\)	1031.87	+	595.751i	0.429767	+	0.248126i	0.699247	−	0.714880i	\(-0.253519\pi\)
							−0.269480	+	0.963006i	\(0.586852\pi\)
\(11\)	2338.44			0.159718			0.0798591	−	0.996806i	\(-0.474553\pi\)
							0.0798591	+	0.996806i	\(0.474553\pi\)
\(13\)	−19167.1	+	33198.4i	−0.671095	+	1.16237i	0.306500	+	0.951871i	\(0.400842\pi\)
							−0.977594	+	0.210499i	\(0.932491\pi\)
\(17\)	27917.4	−	48354.4i	0.334256	−	0.578949i	−0.649085	−	0.760715i	\(-0.724848\pi\)
							0.983342	+	0.181767i	\(0.0581816\pi\)
\(19\)	−17104.2	+	9875.09i	−0.131246	+	0.0757751i	−0.564186	−	0.825648i	\(-0.690810\pi\)
							0.432939	+	0.901423i	\(0.357476\pi\)
\(23\)	−155511.	−	269352.i	−0.555710	−	0.962519i	−0.997848	−	0.0655714i	\(-0.979113\pi\)
							0.442137	−	0.896947i	\(-0.354220\pi\)
\(29\)	−1.03431e6	−	597162.i	−1.46238	−	0.844307i	−0.463261	−	0.886222i	\(-0.653321\pi\)
							−0.999121	+	0.0419156i	\(0.986654\pi\)
\(31\)	−219940.	−	380948.i	−0.238154	−	0.412495i	0.722030	−	0.691861i	\(-0.243209\pi\)
							−0.960185	+	0.279366i	\(0.909876\pi\)
\(37\)	162249.	−	93674.5i	0.0865715	−	0.0499821i	−0.456089	−	0.889934i	\(-0.650750\pi\)
							0.542661	+	0.839952i	\(0.317417\pi\)
\(41\)	−3.34346e6			−1.18321			−0.591603	−	0.806229i	\(-0.701505\pi\)
							−0.591603	+	0.806229i	\(0.701505\pi\)
\(43\)	−1.62301e6	+	3.00899e6i	−0.474730	+	0.880131i
							\(47\)	−1.89249e6			−0.387831			−0.193916	−	0.981018i	\(-0.562119\pi\)
−0.193916	+	0.981018i	\(0.562119\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.d.a.7.7	✓	56
43.37	odd	6	inner	43.9.d.a.37.22	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.d.a.7.7	✓	56	1.1	even	1	trivial
43.9.d.a.37.22	yes	56	43.37	odd	6	inner