Embedded newform 43.7.d.a.37.18

• Label: 43.7.d.a.37.18
• Level: $43$
• Weight: $7$
• Character: 43.37
• 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			37.18
Character	\(\chi\)	\(=\)	43.37
Dual form			43.7.d.a.7.4

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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	\(0.231056\pi\)
							−0.747913	+	0.663797i	\(0.768944\pi\)
\(3\)	−24.0449	−	13.8823i	−0.890551	−	0.514160i	−0.0164282	−	0.999865i	\(-0.505229\pi\)
							−0.874123	+	0.485705i	\(0.838563\pi\)
\(5\)	200.632	+	115.835i	1.60506	+	0.926681i	0.990454	+	0.137845i	\(0.0440176\pi\)
							0.614604	+	0.788836i	\(0.289316\pi\)
\(7\)	396.423	−	228.875i	1.15575	−	0.667274i	0.205470	−	0.978663i	\(-0.434128\pi\)
							0.950282	+	0.311389i	\(0.100794\pi\)
\(11\)	−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	0.966509	+	0.256633i	\(0.0826133\pi\)
							−0.261003	+	0.965338i	\(0.584053\pi\)
\(17\)	3592.69	+	6222.72i	0.731262	+	1.26658i	0.956344	+	0.292243i	\(0.0944017\pi\)
							−0.225082	+	0.974340i	\(0.572265\pi\)
\(19\)	−4533.72	−	2617.54i	−0.660988	−	0.381622i	0.131665	−	0.991294i	\(-0.457968\pi\)
							−0.792653	+	0.609673i	\(0.791301\pi\)
\(23\)	4973.83	−	8614.93i	0.408797	−	0.708057i	−0.585958	−	0.810341i	\(-0.699282\pi\)
							0.994755	+	0.102284i	\(0.0326151\pi\)
\(29\)	21617.6	−	12481.0i	0.886368	−	0.511745i	0.0136155	−	0.999907i	\(-0.495666\pi\)
							0.872753	+	0.488162i	\(0.162333\pi\)
\(31\)	−10811.5	+	18726.1i	−0.362913	+	0.628584i	−0.988439	−	0.151619i	\(-0.951551\pi\)
							0.625526	+	0.780203i	\(0.284884\pi\)
\(37\)	−9193.26	−	5307.73i	−0.181495	−	0.104786i	0.406500	−	0.913651i	\(-0.366749\pi\)
							−0.587995	+	0.808865i	\(0.700082\pi\)
\(41\)	−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.37.18	yes	42
43.7	odd	6	inner	43.7.d.a.7.4	✓	42

    

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