Embedded newform 43.9.f.a.2.10

• Label: 43.9.f.a.2.10
• 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			2.10
Character	\(\chi\)	\(=\)	43.2
Dual form			43.9.f.a.22.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.51717 - 11.4565i) q^{2} +(38.3400 - 79.6139i) q^{3} +(58.8006 - 73.7336i) q^{4} +(121.203 + 27.6637i) q^{5} -1123.63 q^{6} -1965.00i q^{7} +(-4342.77 - 991.209i) q^{8} +(-777.697 - 975.201i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.51717	−	11.4565i	−0.344823	−	0.716033i	0.654371	−	0.756174i	\(-0.272934\pi\)
							−0.999194	+	0.0401410i	\(0.987219\pi\)
\(3\)	38.3400	−	79.6139i	0.473334	−	0.982888i	−0.518468	−	0.855097i	\(-0.673497\pi\)
							0.991801	−	0.127790i	\(-0.0407884\pi\)
\(5\)	121.203	+	27.6637i	0.193924	+	0.0442620i	0.318380	−	0.947963i	\(-0.396861\pi\)
							−0.124455	+	0.992225i	\(0.539718\pi\)
\(7\)		−	1965.00i		−	0.818410i	−0.912442	−	0.409205i	\(-0.865806\pi\)
							0.912442	−	0.409205i	\(-0.134194\pi\)
\(11\)	−1856.69	−	2328.22i	−0.126815	−	0.159021i	0.714371	−	0.699768i	\(-0.246713\pi\)
							−0.841185	+	0.540747i	\(0.818142\pi\)
\(13\)	11489.4	−	50338.3i	0.402275	−	1.76248i	−0.215873	−	0.976421i	\(-0.569260\pi\)
							0.618149	−	0.786061i	\(-0.287883\pi\)
\(17\)	23510.8	+	103008.i	0.281496	+	1.23331i	0.895876	+	0.444304i	\(0.146549\pi\)
							−0.614380	+	0.789010i	\(0.710594\pi\)
\(19\)	−61353.1	−	48927.5i	−0.470785	−	0.375438i	0.359167	−	0.933273i	\(-0.383061\pi\)
							−0.829952	+	0.557835i	\(0.811632\pi\)
\(23\)	132109.	+	165659.i	0.472086	+	0.591977i	0.959680	−	0.281093i	\(-0.0906971\pi\)
							−0.487595	+	0.873070i	\(0.662126\pi\)
\(29\)	129603.	+	269124.i	0.183242	+	0.380505i	0.972273	−	0.233847i	\(-0.0751316\pi\)
							−0.789032	+	0.614353i	\(0.789417\pi\)
\(31\)	−1.48369e6	+	714508.i	−1.60656	+	0.773678i	−0.999775	−	0.0211896i	\(-0.993255\pi\)
							−0.606783	+	0.794867i	\(0.707540\pi\)
\(37\)			2.13935e6i			1.14150i	0.821124	+	0.570750i	\(0.193347\pi\)
							−0.821124	+	0.570750i	\(0.806653\pi\)
\(41\)	2.75444e6	−	1.32647e6i	0.974762	−	0.469421i	0.122462	−	0.992473i	\(-0.460921\pi\)
							0.852300	+	0.523053i	\(0.175207\pi\)
\(43\)	3.35220e6	−	671508.i	0.980521	−	0.196416i
							\(47\)	1.93363e6	−	2.42469e6i	0.396261	−	0.496896i	−0.543175	−	0.839619i	\(-0.682778\pi\)
0.939436	+	0.342724i	\(0.111350\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.2.10	✓	174
43.22	odd	14	inner	43.9.f.a.22.10	yes	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.10	✓	174	1.1	even	1	trivial
43.9.f.a.22.10	yes	174	43.22	odd	14	inner