Embedded newform 43.7.f.a.22.20

• Label: 43.7.f.a.22.20
• Level: $43$
• Weight: $7$
• Character: 43.22
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $126$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(126\)
Relative dimension:	\(21\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			22.20
Character	\(\chi\)	\(=\)	43.22
Dual form			43.7.f.a.2.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.84606 - 12.1395i) q^{2} +(12.7263 + 26.4263i) q^{3} +(-73.2871 - 91.8991i) q^{4} +(119.457 - 27.2654i) q^{5} +395.200 q^{6} -91.2949i q^{7} +(-703.346 + 160.534i) q^{8} +(-81.8698 + 102.661i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 7 q^{2} - 7 q^{3} + 725 q^{4} - 7 q^{5} + 244 q^{6} + 2233 q^{8} + 6312 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.84606	−	12.1395i	0.730758	−	1.51743i	−0.120515	−	0.992712i	\(-0.538455\pi\)
							0.851273	−	0.524723i	\(-0.175831\pi\)
\(3\)	12.7263	+	26.4263i	0.471343	+	0.978753i	0.992147	+	0.125077i	\(0.0399178\pi\)
							−0.520804	+	0.853676i	\(0.674368\pi\)
\(5\)	119.457	−	27.2654i	0.955659	−	0.218123i	0.283868	−	0.958863i	\(-0.408382\pi\)
							0.671791	+	0.740740i	\(0.265525\pi\)
\(7\)		−	91.2949i		−	0.266166i	−0.991105	−	0.133083i	\(-0.957512\pi\)
							0.991105	−	0.133083i	\(-0.0424877\pi\)
\(11\)	986.877	−	1237.50i	0.741455	−	0.929755i	−0.257881	−	0.966177i	\(-0.583024\pi\)
							0.999337	+	0.0364212i	\(0.0115958\pi\)
\(13\)	−206.444	−	904.490i	−0.0939663	−	0.411693i	0.905966	−	0.423351i	\(-0.139146\pi\)
							−0.999932	+	0.0116577i	\(0.996289\pi\)
\(17\)	−1065.17	+	4666.82i	−0.216806	+	0.949891i	0.743014	+	0.669276i	\(0.233396\pi\)
							−0.959820	+	0.280615i	\(0.909462\pi\)
\(19\)	−1425.32	+	1136.65i	−0.207802	+	0.165717i	−0.721861	−	0.692038i	\(-0.756713\pi\)
							0.514058	+	0.857755i	\(0.328141\pi\)
\(23\)	−8252.74	+	10348.6i	−0.678289	+	0.850547i	−0.995195	−	0.0979095i	\(-0.968784\pi\)
							0.316907	+	0.948457i	\(0.397356\pi\)
\(29\)	−4817.08	+	10002.8i	−0.197510	+	0.410134i	−0.976076	−	0.217428i	\(-0.930233\pi\)
							0.778566	+	0.627563i	\(0.215947\pi\)
\(31\)	40254.5	+	19385.5i	1.35123	+	0.650718i	0.962663	−	0.270703i	\(-0.0872561\pi\)
							0.388566	+	0.921421i	\(0.372970\pi\)
\(37\)		−	18047.7i		−	0.356300i	−0.984003	−	0.178150i	\(-0.942989\pi\)
							0.984003	−	0.178150i	\(-0.0570112\pi\)
\(41\)	4598.53	+	2214.53i	0.0667217	+	0.0321315i	0.466947	−	0.884285i	\(-0.345354\pi\)
							−0.400225	+	0.916417i	\(0.631068\pi\)
\(43\)	−49340.2	+	62345.1i	−0.620576	+	0.784146i
							\(47\)	−88607.8	−	111111.i	−0.853451	−	1.07019i	−0.996753	−	0.0805144i	\(-0.974344\pi\)
0.143303	−	0.989679i	\(-0.454228\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.f.a.22.20	yes	126
43.2	odd	14	inner	43.7.f.a.2.20	✓	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.f.a.2.20	✓	126	43.2	odd	14	inner
43.7.f.a.22.20	yes	126	1.1	even	1	trivial