Embedded newform 43.7.d.a.7.20

• Label: 43.7.d.a.7.20
• Level: $43$
• Weight: $7$
• Character: 43.7
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(21\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.20
Character	\(\chi\)	\(=\)	43.7
Dual form			43.7.d.a.37.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+14.0830i q^{2} +(-28.7233 + 16.5834i) q^{3} -134.330 q^{4} +(147.495 - 85.1564i) q^{5} +(-233.544 - 404.510i) q^{6} +(-245.226 - 141.581i) q^{7} -990.457i q^{8} +(185.520 - 321.330i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 3 q^{3} - 1194 q^{4} - 3 q^{5} + 111 q^{6} - 147 q^{7} + 3852 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			14.0830i			1.76037i	0.474629	+	0.880186i	\(0.342582\pi\)
							−0.474629	+	0.880186i	\(0.657418\pi\)
\(3\)	−28.7233	+	16.5834i	−1.06383	+	0.614201i	−0.926488	−	0.376323i	\(-0.877188\pi\)
							−0.137339	+	0.990524i	\(0.543855\pi\)
\(5\)	147.495	−	85.1564i	1.17996	−	0.681251i	0.223956	−	0.974599i	\(-0.428103\pi\)
							0.956006	+	0.293348i	\(0.0947694\pi\)
\(7\)	−245.226	−	141.581i	−0.714944	−	0.412773i	0.0979447	−	0.995192i	\(-0.468773\pi\)
							−0.812889	+	0.582419i	\(0.802107\pi\)
\(11\)	−907.989			−0.682186			−0.341093	−	0.940030i	\(-0.610797\pi\)
							−0.341093	+	0.940030i	\(0.610797\pi\)
\(13\)	1023.00	−	1771.88i	0.465633	−	0.806500i	−0.533597	−	0.845739i	\(-0.679160\pi\)
							0.999230	+	0.0392388i	\(0.0124933\pi\)
\(17\)	−3803.60	+	6588.04i	−0.774192	+	1.34094i	0.161056	+	0.986945i	\(0.448510\pi\)
							−0.935248	+	0.353994i	\(0.884823\pi\)
\(19\)	5714.09	−	3299.03i	0.833079	−	0.480979i	−0.0218265	−	0.999762i	\(-0.506948\pi\)
							0.854906	+	0.518783i	\(0.173615\pi\)
\(23\)	−8931.65	−	15470.1i	−0.734088	−	1.27148i	−0.955122	−	0.296212i	\(-0.904277\pi\)
							0.221034	−	0.975266i	\(-0.429057\pi\)
\(29\)	19886.7	+	11481.6i	0.815395	+	0.470769i	0.848826	−	0.528672i	\(-0.177310\pi\)
							−0.0334308	+	0.999441i	\(0.510643\pi\)
\(31\)	−23828.4	−	41272.1i	−0.799853	−	1.38539i	−0.919711	−	0.392596i	\(-0.871577\pi\)
							0.119858	−	0.992791i	\(-0.461756\pi\)
\(37\)	44454.8	−	25666.0i	0.877635	−	0.506703i	0.00775705	−	0.999970i	\(-0.497531\pi\)
							0.869878	+	0.493267i	\(0.164197\pi\)
\(41\)	−57173.8			−0.829555			−0.414778	−	0.909923i	\(-0.636141\pi\)
							−0.414778	+	0.909923i	\(0.636141\pi\)
\(43\)	−24225.9	+	75726.3i	−0.304702	+	0.952448i
							\(47\)	−146686.			−1.41284			−0.706422	−	0.707791i	\(-0.749692\pi\)
−0.706422	+	0.707791i	\(0.749692\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.d.a.7.20	✓	42
43.37	odd	6	inner	43.7.d.a.37.2	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.d.a.7.20	✓	42	1.1	even	1	trivial
43.7.d.a.37.2	yes	42	43.37	odd	6	inner