Embedded newform 43.7.d.a.7.6

• Label: 43.7.d.a.7.6
• 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			7.6
Character	\(\chi\)	\(=\)	43.7
Dual form			43.7.d.a.37.16

$q$-expansion

\(f(q)\)	\(=\)	\(q-7.84004i q^{2} +(36.4833 - 21.0637i) q^{3} +2.53376 q^{4} +(-116.673 + 67.3609i) q^{5} +(-165.140 - 286.031i) q^{6} +(-385.031 - 222.298i) q^{7} -521.627i q^{8} +(522.856 - 905.613i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 3 q^{3} - 1194 q^{4} - 3 q^{5} + 111 q^{6} - 147 q^{7} + 3852 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	7.84004i		−	0.980005i	−0.871721	−	0.490003i	\(-0.836996\pi\)
							0.871721	−	0.490003i	\(-0.163004\pi\)
\(3\)	36.4833	−	21.0637i	1.35123	−	0.780136i	0.362812	−	0.931862i	\(-0.381817\pi\)
							0.988423	+	0.151726i	\(0.0484833\pi\)
\(5\)	−116.673	+	67.3609i	−0.933381	+	0.538888i	−0.887879	−	0.460077i	\(-0.847822\pi\)
							−0.0455014	+	0.998964i	\(0.514489\pi\)
\(7\)	−385.031	−	222.298i	−1.12254	−	0.648098i	−0.180491	−	0.983577i	\(-0.557769\pi\)
							−0.942048	+	0.335478i	\(0.891102\pi\)
\(11\)	−327.603			−0.246133			−0.123067	−	0.992398i	\(-0.539273\pi\)
							−0.123067	+	0.992398i	\(0.539273\pi\)
\(13\)	1051.90	−	1821.95i	0.478790	−	0.829288i	−0.520914	−	0.853609i	\(-0.674409\pi\)
							0.999704	+	0.0243205i	\(0.00774223\pi\)
\(17\)	2727.39	−	4723.97i	0.555136	−	0.961525i	−0.442756	−	0.896642i	\(-0.645999\pi\)
							0.997893	−	0.0648826i	\(-0.0206673\pi\)
\(19\)	3872.49	−	2235.78i	0.564585	−	0.325963i	−0.190399	−	0.981707i	\(-0.560978\pi\)
							0.754984	+	0.655743i	\(0.227645\pi\)
\(23\)	8465.56	+	14662.8i	0.695780	+	1.20513i	0.969917	+	0.243436i	\(0.0782745\pi\)
							−0.274137	+	0.961691i	\(0.588392\pi\)
\(29\)	12578.3	+	7262.09i	0.515737	+	0.297761i	0.735189	−	0.677863i	\(-0.237094\pi\)
							−0.219452	+	0.975623i	\(0.570427\pi\)
\(31\)	25798.6	+	44684.4i	0.865986	+	1.49993i	0.866065	+	0.499931i	\(0.166641\pi\)
							−7.95439e−5		1.00000i	\(0.500025\pi\)
\(37\)	70836.6	−	40897.5i	1.39847	−	0.807405i	0.404235	−	0.914655i	\(-0.367538\pi\)
							0.994232	+	0.107250i	\(0.0342045\pi\)
\(41\)	−18379.1			−0.266669			−0.133335	−	0.991071i	\(-0.542569\pi\)
							−0.133335	+	0.991071i	\(0.542569\pi\)
\(43\)	−23774.6	−	75869.2i	−0.299025	−	0.954245i
							\(47\)	−55823.4			−0.537678			−0.268839	−	0.963185i	\(-0.586640\pi\)
−0.268839	+	0.963185i	\(0.586640\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.d.a.7.6	✓	42
43.37	odd	6	inner	43.7.d.a.37.16	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.d.a.7.6	✓	42	1.1	even	1	trivial
43.7.d.a.37.16	yes	42	43.37	odd	6	inner