Embedded newform 43.9.f.a.2.12

• Label: 43.9.f.a.2.12
• Level: $43$
• Weight: $9$
• Character: 43.2
• 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			2.12
Character	\(\chi\)	\(=\)	43.2
Dual form			43.9.f.a.22.12

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.12	✓	174	1.1	even	1	trivial
43.9.f.a.22.12	yes	174	43.22	odd	14	inner