Embedded newform 43.7.d.a.7.4

• Label: 43.7.d.a.7.4
• Level: $43$
• Weight: $7$
• Character: 43.7
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(21\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.4
Character	\(\chi\)	\(=\)	43.7
Dual form			43.7.d.a.37.18

$q$-expansion

\(f(q)\)	\(=\)	\(q-10.6207i q^{2} +(-24.0449 + 13.8823i) q^{3} -48.8003 q^{4} +(200.632 - 115.835i) q^{5} +(147.441 + 255.375i) q^{6} +(396.423 + 228.875i) q^{7} -161.432i q^{8} +(20.9373 - 36.2644i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 3 q^{3} - 1194 q^{4} - 3 q^{5} + 111 q^{6} - 147 q^{7} + 3852 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	10.6207i		−	1.32759i	−0.747913	−	0.663797i	\(-0.768944\pi\)
							0.747913	−	0.663797i	\(-0.231056\pi\)
\(3\)	−24.0449	+	13.8823i	−0.890551	+	0.514160i	−0.874123	−	0.485705i	\(-0.838563\pi\)
							−0.0164282	+	0.999865i	\(0.505229\pi\)
\(5\)	200.632	−	115.835i	1.60506	−	0.926681i	0.614604	−	0.788836i	\(-0.289316\pi\)
							0.990454	−	0.137845i	\(-0.0440176\pi\)
\(7\)	396.423	+	228.875i	1.15575	+	0.667274i	0.950282	−	0.311389i	\(-0.100794\pi\)
							0.205470	+	0.978663i	\(0.434128\pi\)
\(11\)	−1243.46			−0.934231			−0.467116	−	0.884196i	\(-0.654707\pi\)
							−0.467116	+	0.884196i	\(0.654707\pi\)
\(13\)	1550.00	−	2684.67i	0.705505	−	1.22197i	−0.261003	−	0.965338i	\(-0.584053\pi\)
							0.966509	−	0.256633i	\(-0.0826133\pi\)
\(17\)	3592.69	−	6222.72i	0.731262	−	1.26658i	−0.225082	−	0.974340i	\(-0.572265\pi\)
							0.956344	−	0.292243i	\(-0.0944017\pi\)
\(19\)	−4533.72	+	2617.54i	−0.660988	+	0.381622i	−0.792653	−	0.609673i	\(-0.791301\pi\)
							0.131665	+	0.991294i	\(0.457968\pi\)
\(23\)	4973.83	+	8614.93i	0.408797	+	0.708057i	0.994755	−	0.102284i	\(-0.0326151\pi\)
							−0.585958	+	0.810341i	\(0.699282\pi\)
\(29\)	21617.6	+	12481.0i	0.886368	+	0.511745i	0.872753	−	0.488162i	\(-0.162333\pi\)
							0.0136155	+	0.999907i	\(0.495666\pi\)
\(31\)	−10811.5	−	18726.1i	−0.362913	−	0.628584i	0.625526	−	0.780203i	\(-0.284884\pi\)
							−0.988439	+	0.151619i	\(0.951551\pi\)
\(37\)	−9193.26	+	5307.73i	−0.181495	+	0.104786i	−0.587995	−	0.808865i	\(-0.700082\pi\)
							0.406500	+	0.913651i	\(0.366749\pi\)
\(41\)	−44305.1			−0.642839			−0.321420	−	0.946937i	\(-0.604160\pi\)
							−0.321420	+	0.946937i	\(0.604160\pi\)
\(43\)	−63329.9	−	48069.6i	−0.796533	−	0.604596i
							\(47\)	−2609.90			−0.0251379			−0.0125690	−	0.999921i	\(-0.504001\pi\)
−0.0125690	+	0.999921i	\(0.504001\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.d.a.7.4	✓	42
43.37	odd	6	inner	43.7.d.a.37.18	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.d.a.7.4	✓	42	1.1	even	1	trivial
43.7.d.a.37.18	yes	42	43.37	odd	6	inner