Embedded newform 43.9.d.a.7.3

• Label: 43.9.d.a.7.3
• 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			7.3
Character	\(\chi\)	\(=\)	43.7
Dual form			43.9.d.a.37.26

$q$-expansion

\(f(q)\)	\(=\)	\(q-26.9022i q^{2} +(74.3127 - 42.9044i) q^{3} -467.727 q^{4} +(-575.075 + 332.020i) q^{5} +(-1154.22 - 1999.17i) q^{6} +(-1517.67 - 876.227i) q^{7} +5695.91i q^{8} +(401.081 - 694.692i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 84 q^{3} - 6474 q^{4} - 3 q^{5} + 1791 q^{6} - 5031 q^{7} + 57858 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	26.9022i		−	1.68139i	−0.541512	−	0.840693i	\(-0.682148\pi\)
							0.541512	−	0.840693i	\(-0.317852\pi\)
\(3\)	74.3127	−	42.9044i	0.917440	−	0.529684i	0.0346227	−	0.999400i	\(-0.488977\pi\)
							0.882818	+	0.469716i	\(0.155644\pi\)
\(5\)	−575.075	+	332.020i	−0.920120	+	0.531231i	−0.883673	−	0.468104i	\(-0.844937\pi\)
							−0.0364464	+	0.999336i	\(0.511604\pi\)
\(7\)	−1517.67	−	876.227i	−0.632099	−	0.364942i	0.149466	−	0.988767i	\(-0.452245\pi\)
							−0.781564	+	0.623825i	\(0.785578\pi\)
\(11\)	9944.56			0.679227			0.339613	−	0.940565i	\(-0.389704\pi\)
							0.339613	+	0.940565i	\(0.389704\pi\)
\(13\)	−7032.18	+	12180.1i	−0.246216	+	0.426459i	−0.962473	−	0.271378i	\(-0.912521\pi\)
							0.716257	+	0.697837i	\(0.245854\pi\)
\(17\)	−21770.5	+	37707.7i	−0.260659	+	0.451475i	−0.966417	−	0.256978i	\(-0.917273\pi\)
							0.705758	+	0.708453i	\(0.250607\pi\)
\(19\)	−5161.47	+	2979.98i	−0.0396058	+	0.0228664i	−0.519672	−	0.854366i	\(-0.673946\pi\)
							0.480066	+	0.877232i	\(0.340613\pi\)
\(23\)	−66369.3	−	114955.i	−0.237168	−	0.410787i	0.722733	−	0.691128i	\(-0.242886\pi\)
							−0.959900	+	0.280341i	\(0.909552\pi\)
\(29\)	−185365.	−	107021.i	−0.262082	−	0.151313i	0.363202	−	0.931710i	\(-0.381683\pi\)
							−0.625284	+	0.780397i	\(0.715017\pi\)
\(31\)	−335094.	−	580400.i	−0.362844	−	0.628465i	0.625583	−	0.780157i	\(-0.284861\pi\)
							−0.988428	+	0.151692i	\(0.951528\pi\)
\(37\)	1.32852e6	−	767021.i	0.708861	−	0.409261i	−0.101778	−	0.994807i	\(-0.532453\pi\)
							0.810639	+	0.585546i	\(0.199120\pi\)
\(41\)	−5.33956e6			−1.88960			−0.944800	−	0.327647i	\(-0.893744\pi\)
							−0.944800	+	0.327647i	\(0.893744\pi\)
\(43\)	−2.40928e6	−	2.42560e6i	−0.704716	−	0.709489i
							\(47\)	−2.73323e6			−0.560125			−0.280062	−	0.959982i	\(-0.590355\pi\)
−0.280062	+	0.959982i	\(0.590355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.d.a.7.3	✓	56
43.37	odd	6	inner	43.9.d.a.37.26	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.d.a.7.3	✓	56	1.1	even	1	trivial
43.9.d.a.37.26	yes	56	43.37	odd	6	inner