Embedded newform 43.7.h.a.19.19

• Label: 43.7.h.a.19.19
• Level: $43$
• Weight: $7$
• Character: 43.19
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $252$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(252\)
Relative dimension:	\(21\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			19.19
Character	\(\chi\)	\(=\)	43.19
Dual form			43.7.h.a.34.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.2866 + 8.20330i) q^{2} +(-4.13834 - 27.4561i) q^{3} +(24.2789 + 106.373i) q^{4} +(86.4849 + 126.850i) q^{5} +(182.661 - 316.378i) q^{6} +(220.001 - 127.018i) q^{7} +(-257.507 + 534.719i) q^{8} +(-40.0987 + 12.3688i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{19}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.2866	+	8.20330i	1.28583	+	1.02541i	0.997698	+	0.0678201i	\(0.0216044\pi\)
							0.288128	+	0.957592i	\(0.406967\pi\)
\(3\)	−4.13834	−	27.4561i	−0.153272	−	1.01689i	−0.924733	−	0.380616i	\(-0.875712\pi\)
							0.771461	−	0.636276i	\(-0.219526\pi\)
\(5\)	86.4849	+	126.850i	0.691879	+	1.01480i	0.998025	+	0.0628155i	\(0.0200080\pi\)
							−0.306146	+	0.951984i	\(0.599040\pi\)
\(7\)	220.001	−	127.018i	0.641402	−	0.370314i	−0.143752	−	0.989614i	\(-0.545917\pi\)
							0.785155	+	0.619300i	\(0.212583\pi\)
\(11\)	−297.965	+	1305.47i	−0.223866	+	0.980820i	0.730672	+	0.682729i	\(0.239207\pi\)
							−0.954538	+	0.298091i	\(0.903650\pi\)
\(13\)	86.6334	+	1156.04i	0.0394326	+	0.526191i	0.981587	+	0.191013i	\(0.0611774\pi\)
							−0.942155	+	0.335178i	\(0.891204\pi\)
\(17\)	345.162	+	235.328i	0.0702549	+	0.0478990i	0.597938	−	0.801542i	\(-0.295987\pi\)
							−0.527683	+	0.849441i	\(0.676939\pi\)
\(19\)	−122.123	+	395.912i	−0.0178047	+	0.0577215i	−0.964039	−	0.265760i	\(-0.914377\pi\)
							0.946235	+	0.323481i	\(0.104853\pi\)
\(23\)	−11485.5	−	10657.0i	−0.943985	−	0.875890i	0.0484294	−	0.998827i	\(-0.484578\pi\)
							−0.992415	+	0.122936i	\(0.960769\pi\)
\(29\)	4028.19	−	26725.3i	0.165164	−	1.09579i	−0.740580	−	0.671968i	\(-0.765449\pi\)
							0.905744	−	0.423825i	\(-0.139313\pi\)
\(31\)	−8495.27	−	21645.6i	−0.285162	−	0.726581i	−0.999638	−	0.0269209i	\(-0.991430\pi\)
							0.714475	−	0.699661i	\(-0.246665\pi\)
\(37\)	−79093.0	−	45664.4i	−1.56147	−	0.901513i	−0.997109	−	0.0759832i	\(-0.975790\pi\)
							−0.564358	−	0.825530i	\(-0.690876\pi\)
\(41\)	−42903.4	+	53799.2i	−0.622501	+	0.780592i	−0.988694	−	0.149947i	\(-0.952090\pi\)
							0.366193	+	0.930539i	\(0.380661\pi\)
\(43\)	79124.3	+	7791.40i	0.995187	+	0.0979964i
							\(47\)	−13986.4	−	61278.4i	−0.134714	−	0.590220i	−0.996547	−	0.0830283i	\(-0.973541\pi\)
0.861833	−	0.507192i	\(-0.169316\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.19.19	✓	252
43.34	odd	42	inner	43.7.h.a.34.19	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.19.19	✓	252	1.1	even	1	trivial
43.7.h.a.34.19	yes	252	43.34	odd	42	inner