Embedded newform 43.9.d.a.7.2

• Label: 43.9.d.a.7.2
• 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			7.2
Character	\(\chi\)	\(=\)	43.7
Dual form			43.9.d.a.37.27

$q$-expansion

\(f(q)\)	\(=\)	\(q-29.0831i q^{2} +(-96.3585 + 55.6326i) q^{3} -589.829 q^{4} +(-695.509 + 401.552i) q^{5} +(1617.97 + 2802.41i) q^{6} +(-1511.66 - 872.755i) q^{7} +9708.78i q^{8} +(2909.48 - 5039.36i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 84 q^{3} - 6474 q^{4} - 3 q^{5} + 1791 q^{6} - 5031 q^{7} + 57858 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	29.0831i		−	1.81770i	−0.417128	−	0.908848i	\(-0.636963\pi\)
							0.417128	−	0.908848i	\(-0.363037\pi\)
\(3\)	−96.3585	+	55.6326i	−1.18961	+	0.686822i	−0.958218	−	0.286038i	\(-0.907662\pi\)
							−0.231393	+	0.972860i	\(0.574328\pi\)
\(5\)	−695.509	+	401.552i	−1.11281	+	0.642484i	−0.939557	−	0.342393i	\(-0.888763\pi\)
							−0.173258	+	0.984877i	\(0.555429\pi\)
\(7\)	−1511.66	−	872.755i	−0.629594	−	0.363497i	0.151001	−	0.988534i	\(-0.451750\pi\)
							−0.780595	+	0.625037i	\(0.785084\pi\)
\(11\)	−26954.6			−1.84103			−0.920517	−	0.390702i	\(-0.872232\pi\)
							−0.920517	+	0.390702i	\(0.872232\pi\)
\(13\)	16529.0	−	28629.1i	0.578728	−	1.00239i	−0.416898	−	0.908953i	\(-0.636883\pi\)
							0.995626	−	0.0934324i	\(-0.0297839\pi\)
\(17\)	35163.2	−	60904.5i	0.421011	−	0.729212i	−0.575028	−	0.818134i	\(-0.695009\pi\)
							0.996039	+	0.0889219i	\(0.0283422\pi\)
\(19\)	−187235.	+	108100.i	−1.43672	+	0.829491i	−0.997620	−	0.0689466i	\(-0.978036\pi\)
							−0.439101	+	0.898438i	\(0.644703\pi\)
\(23\)	116619.	+	201989.i	0.416732	+	0.721801i	0.995609	−	0.0936145i	\(-0.0298421\pi\)
							−0.578877	+	0.815415i	\(0.696509\pi\)
\(29\)	−776854.	−	448517.i	−1.09837	−	0.634142i	−0.162575	−	0.986696i	\(-0.551980\pi\)
							−0.935792	+	0.352554i	\(0.885313\pi\)
\(31\)	121950.	+	211224.i	0.132049	+	0.228716i	0.924466	−	0.381264i	\(-0.124511\pi\)
							−0.792417	+	0.609979i	\(0.791178\pi\)
\(37\)	−583266.	+	336749.i	−0.311214	+	0.179680i	−0.647470	−	0.762091i	\(-0.724173\pi\)
							0.336255	+	0.941771i	\(0.390839\pi\)
\(41\)	3.35153e6			1.18606			0.593031	−	0.805180i	\(-0.297931\pi\)
							0.593031	+	0.805180i	\(0.297931\pi\)
\(43\)	3.36896e6	−	581636.i	0.985422	−	0.170129i
							\(47\)	−6.37851e6			−1.30716			−0.653578	−	0.756859i	\(-0.726733\pi\)
−0.653578	+	0.756859i	\(0.726733\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.d.a.7.2	✓	56
43.37	odd	6	inner	43.9.d.a.37.27	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.d.a.7.2	✓	56	1.1	even	1	trivial
43.9.d.a.37.27	yes	56	43.37	odd	6	inner