Embedded newform 43.9.f.a.22.12

• Label: 43.9.f.a.22.12
• 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			22.12
Character	\(\chi\)	\(=\)	43.22
Dual form			43.9.f.a.2.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.74733 + 7.78141i) q^{2} +(23.1343 + 48.0389i) q^{3} +(113.106 + 141.830i) q^{4} +(-1134.95 + 259.044i) q^{5} -460.503 q^{6} -3469.08i q^{7} +(-3683.04 + 840.631i) q^{8} +(2318.17 - 2906.90i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.74733	+	7.78141i	−0.234208	+	0.486338i	−0.984637	−	0.174613i	\(-0.944133\pi\)
							0.750429	+	0.660951i	\(0.229847\pi\)
\(3\)	23.1343	+	48.0389i	0.285609	+	0.593073i	0.993575	−	0.113173i	\(-0.0361014\pi\)
							−0.707966	+	0.706246i	\(0.750387\pi\)
\(5\)	−1134.95	+	259.044i	−1.81592	+	0.414471i	−0.989008	−	0.147865i	\(-0.952760\pi\)
							−0.826909	+	0.562336i	\(0.809903\pi\)
\(7\)		−	3469.08i		−	1.44485i	−0.691451	−	0.722424i	\(-0.743028\pi\)
							0.691451	−	0.722424i	\(-0.256972\pi\)
\(11\)	3830.44	−	4803.22i	0.261624	−	0.328067i	−0.633618	−	0.773646i	\(-0.718431\pi\)
							0.895242	+	0.445579i	\(0.147002\pi\)
\(13\)	−115.856	−	507.598i	−0.00405644	−	0.0177724i	0.972859	−	0.231398i	\(-0.0743299\pi\)
							−0.976916	+	0.213625i	\(0.931473\pi\)
\(17\)	−15894.9	+	69640.1i	−0.190310	+	0.833804i	0.786138	+	0.618051i	\(0.212078\pi\)
							−0.976448	+	0.215752i	\(0.930780\pi\)
\(19\)	41766.5	−	33307.7i	0.320489	−	0.255582i	−0.450009	−	0.893024i	\(-0.648579\pi\)
							0.770498	+	0.637442i	\(0.220008\pi\)
\(23\)	57084.2	−	71581.3i	0.203988	−	0.255793i	−0.669306	−	0.742987i	\(-0.733408\pi\)
							0.873294	+	0.487194i	\(0.161980\pi\)
\(29\)	11148.1	−	23149.3i	0.0157619	−	0.0327300i	−0.892939	−	0.450177i	\(-0.851361\pi\)
							0.908701	+	0.417447i	\(0.137075\pi\)
\(31\)	−1.53774e6	−	740535.i	−1.66508	−	0.801860i	−0.998402	−	0.0565149i	\(-0.982001\pi\)
							−0.666678	−	0.745345i	\(-0.732285\pi\)
\(37\)		−	2.51256e6i		−	1.34063i	−0.742075	−	0.670317i	\(-0.766158\pi\)
							0.742075	−	0.670317i	\(-0.233842\pi\)
\(41\)	−1.47521e6	−	710424.i	−0.522058	−	0.251410i	0.154246	−	0.988032i	\(-0.450705\pi\)
							−0.676304	+	0.736623i	\(0.736419\pi\)
\(43\)	−524572.	−	3.37832e6i	−0.153437	−	0.988158i
							\(47\)	−3.16054e6	−	3.96320e6i	−0.647695	−	0.812184i	0.344247	−	0.938879i	\(-0.388134\pi\)
−0.991942	+	0.126696i	\(0.959563\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.22.12	yes	174
43.2	odd	14	inner	43.9.f.a.2.12	✓	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.12	✓	174	43.2	odd	14	inner
43.9.f.a.22.12	yes	174	1.1	even	1	trivial