Embedded newform 43.9.f.a.22.6

• Label: 43.9.f.a.22.6
• Level: $43$
• Weight: $9$
• Character: 43.22
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $174$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(174\)
Relative dimension:	\(29\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			22.6
Character	\(\chi\)	\(=\)	43.22
Dual form			43.9.f.a.2.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.26628 + 19.2416i) q^{2} +(5.00685 + 10.3968i) q^{3} +(-124.763 - 156.448i) q^{4} +(-758.071 + 173.025i) q^{5} -246.447 q^{6} +4635.90i q^{7} +(-1163.81 + 265.632i) q^{8} +(4007.69 - 5025.49i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.26628	+	19.2416i	−0.579143	+	1.20260i	0.381385	+	0.924416i	\(0.375447\pi\)
							−0.960527	+	0.278186i	\(0.910267\pi\)
\(3\)	5.00685	+	10.3968i	0.0618129	+	0.128356i	0.929587	−	0.368604i	\(-0.120164\pi\)
							−0.867774	+	0.496960i	\(0.834450\pi\)
\(5\)	−758.071	+	173.025i	−1.21291	+	0.276840i	−0.780685	−	0.624924i	\(-0.785130\pi\)
							−0.432229	+	0.901764i	\(0.642273\pi\)
\(7\)			4635.90i			1.93082i	0.260733	+	0.965411i	\(0.416036\pi\)
							−0.260733	+	0.965411i	\(0.583964\pi\)
\(11\)	−14701.2	+	18434.7i	−1.00411	+	1.25911i	−0.0384623	+	0.999260i	\(0.512246\pi\)
							−0.965648	+	0.259854i	\(0.916325\pi\)
\(13\)	−5437.50	−	23823.3i	−0.190382	−	0.834119i	−0.976410	−	0.215927i	\(-0.930723\pi\)
							0.786027	−	0.618192i	\(-0.212134\pi\)
\(17\)	−19249.4	+	84337.1i	−0.230474	+	1.00977i	0.718775	+	0.695243i	\(0.244703\pi\)
							−0.949248	+	0.314528i	\(0.898154\pi\)
\(19\)	83361.1	−	66478.2i	0.639660	−	0.510112i	−0.249104	−	0.968477i	\(-0.580136\pi\)
							0.888764	+	0.458365i	\(0.151565\pi\)
\(23\)	110556.	−	138633.i	0.395067	−	0.495398i	−0.544023	−	0.839071i	\(-0.683099\pi\)
							0.939090	+	0.343672i	\(0.111671\pi\)
\(29\)	267216.	−	554880.i	0.377808	−	0.784526i	−0.622191	−	0.782866i	\(-0.713757\pi\)
							0.999999	−	0.00166044i	\(-0.000528535\pi\)
\(31\)	1.01562e6	+	489096.i	1.09973	+	0.529600i	0.893572	−	0.448920i	\(-0.148191\pi\)
							0.206153	+	0.978520i	\(0.433905\pi\)
\(37\)			2.37243e6i			1.26586i	0.774209	+	0.632930i	\(0.218148\pi\)
							−0.774209	+	0.632930i	\(0.781852\pi\)
\(41\)	−1.86646e6	−	898838.i	−0.660514	−	0.318087i	0.0734262	−	0.997301i	\(-0.476607\pi\)
							−0.733941	+	0.679214i	\(0.762321\pi\)
\(43\)	−268848.	−	3.40821e6i	−0.0786382	−	0.996903i
							\(47\)	−2.12977e6	−	2.67065e6i	−0.436457	−	0.547300i	0.514148	−	0.857701i	\(-0.328108\pi\)
−0.950606	+	0.310401i	\(0.899537\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.22.6	yes	174
43.2	odd	14	inner	43.9.f.a.2.6	✓	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.6	✓	174	43.2	odd	14	inner
43.9.f.a.22.6	yes	174	1.1	even	1	trivial