Embedded newform 43.7.f.a.2.12

• Label: 43.7.f.a.2.12
• Level: $43$
• Weight: $7$
• Character: 43.2
• 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			2.12
Character	\(\chi\)	\(=\)	43.2
Dual form			43.7.f.a.22.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0322332 - 0.0669329i) q^{2} +(-17.7426 + 36.8430i) q^{3} +(39.8999 - 50.0329i) q^{4} +(157.975 + 36.0568i) q^{5} +3.03791 q^{6} -203.317i q^{7} +(-9.27029 - 2.11588i) q^{8} +(-588.078 - 737.427i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 7 q^{2} - 7 q^{3} + 725 q^{4} - 7 q^{5} + 244 q^{6} + 2233 q^{8} + 6312 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0322332	−	0.0669329i	−0.00402915	−	0.00836661i	0.898945	−	0.438062i	\(-0.144335\pi\)
							−0.902974	+	0.429696i	\(0.858621\pi\)
\(3\)	−17.7426	+	36.8430i	−0.657134	+	1.36455i	0.259857	+	0.965647i	\(0.416325\pi\)
							−0.916992	+	0.398907i	\(0.869390\pi\)
\(5\)	157.975	+	36.0568i	1.26380	+	0.288454i	0.801353	−	0.598192i	\(-0.204114\pi\)
							0.462448	+	0.886646i	\(0.346971\pi\)
\(7\)		−	203.317i		−	0.592761i	−0.955070	−	0.296380i	\(-0.904220\pi\)
							0.955070	−	0.296380i	\(-0.0957796\pi\)
\(11\)	1391.85	+	1745.33i	1.04572	+	1.31129i	0.948758	+	0.316002i	\(0.102341\pi\)
							0.0969608	+	0.995288i	\(0.469088\pi\)
\(13\)	−91.2481	+	399.784i	−0.0415331	+	0.181968i	−0.991440	−	0.130566i	\(-0.958321\pi\)
							0.949907	+	0.312534i	\(0.101178\pi\)
\(17\)	2061.63	+	9032.59i	0.419627	+	1.83851i	0.534555	+	0.845134i	\(0.320479\pi\)
							−0.114928	+	0.993374i	\(0.536664\pi\)
\(19\)	905.668	+	722.246i	0.132041	+	0.105299i	0.687283	−	0.726389i	\(-0.258803\pi\)
							−0.555242	+	0.831689i	\(0.687375\pi\)
\(23\)	−8856.37	−	11105.5i	−0.727901	−	0.912759i	0.270855	−	0.962620i	\(-0.412694\pi\)
							−0.998756	+	0.0498611i	\(0.984122\pi\)
\(29\)	16451.0	+	34160.8i	0.674524	+	1.40066i	0.904078	+	0.427368i	\(0.140559\pi\)
							−0.229553	+	0.973296i	\(0.573727\pi\)
\(31\)	6780.13	−	3265.14i	0.227590	−	0.109602i	−0.316614	−	0.948554i	\(-0.602546\pi\)
							0.544204	+	0.838953i	\(0.316832\pi\)
\(37\)		−	39555.7i		−	0.780915i	−0.920621	−	0.390458i	\(-0.872317\pi\)
							0.920621	−	0.390458i	\(-0.127683\pi\)
\(41\)	98988.1	−	47670.1i	1.43625	−	0.691663i	0.456105	−	0.889926i	\(-0.349244\pi\)
							0.980149	+	0.198262i	\(0.0635298\pi\)
\(43\)	−8976.81	−	78998.6i	−0.112906	−	0.993606i
							\(47\)	−86015.0	+	107859.i	−0.828477	+	1.03888i	0.170093	+	0.985428i	\(0.445593\pi\)
−0.998571	+	0.0534497i	\(0.982978\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.f.a.2.12	✓	126
43.22	odd	14	inner	43.7.f.a.22.12	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.f.a.2.12	✓	126	1.1	even	1	trivial
43.7.f.a.22.12	yes	126	43.22	odd	14	inner