Embedded newform 43.7.h.a.19.13

• Label: 43.7.h.a.19.13
• 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			19.13
Character	\(\chi\)	\(=\)	43.19
Dual form			43.7.h.a.34.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.14235 + 2.50594i) q^{2} +(2.73114 + 18.1199i) q^{3} +(-10.6467 - 46.6463i) q^{4} +(-26.7382 - 39.2177i) q^{5} +(-36.8253 + 63.7833i) q^{6} +(258.095 - 149.011i) q^{7} +(195.045 - 405.015i) q^{8} +(375.740 - 115.900i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.14235	+	2.50594i	0.392794	+	0.313243i	0.799894	−	0.600141i	\(-0.204889\pi\)
							−0.407101	+	0.913383i	\(0.633460\pi\)
\(3\)	2.73114	+	18.1199i	0.101153	+	0.671109i	0.980732	+	0.195356i	\(0.0625862\pi\)
							−0.879579	+	0.475753i	\(0.842176\pi\)
\(5\)	−26.7382	−	39.2177i	−0.213905	−	0.313741i	0.704258	−	0.709945i	\(-0.251280\pi\)
							−0.918163	+	0.396203i	\(0.870328\pi\)
\(7\)	258.095	−	149.011i	0.752465	−	0.434436i	−0.0741191	−	0.997249i	\(-0.523614\pi\)
							0.826584	+	0.562814i	\(0.190281\pi\)
\(11\)	153.595	−	672.945i	0.115398	−	0.505593i	−0.883884	−	0.467707i	\(-0.845080\pi\)
							0.999282	−	0.0378865i	\(-0.0120625\pi\)
\(13\)	103.298	+	1378.42i	0.0470178	+	0.627409i	0.970084	+	0.242769i	\(0.0780556\pi\)
							−0.923066	+	0.384641i	\(0.874325\pi\)
\(17\)	6688.31	+	4560.01i	1.36135	+	0.928152i	0.999991	−	0.00427554i	\(-0.00136095\pi\)
							0.361358	+	0.932427i	\(0.382313\pi\)
\(19\)	2978.54	−	9656.19i	0.434253	−	1.40781i	−0.429120	−	0.903247i	\(-0.641176\pi\)
							0.863373	−	0.504566i	\(-0.168348\pi\)
\(23\)	−15407.7	−	14296.3i	−1.26636	−	1.17501i	−0.975924	−	0.218110i	\(-0.930011\pi\)
							−0.290431	−	0.956896i	\(-0.593799\pi\)
\(29\)	1922.00	−	12751.6i	0.0788059	−	0.522843i	−0.914245	−	0.405161i	\(-0.867215\pi\)
							0.993051	−	0.117682i	\(-0.0375464\pi\)
\(31\)	19541.0	+	49789.6i	0.655936	+	1.67130i	0.738111	+	0.674679i	\(0.235718\pi\)
							−0.0821752	+	0.996618i	\(0.526187\pi\)
\(37\)	−65259.9	−	37677.8i	−1.28837	−	0.743842i	−0.310008	−	0.950734i	\(-0.600332\pi\)
							−0.978364	+	0.206892i	\(0.933665\pi\)
\(41\)	−30346.0	+	38052.7i	−0.440301	+	0.552120i	−0.951623	−	0.307270i	\(-0.900585\pi\)
							0.511322	+	0.859389i	\(0.329156\pi\)
\(43\)	845.497	+	79502.5i	0.0106343	+	0.999943i
							\(47\)	−18315.1	−	80243.5i	−0.176407	−	0.772888i	−0.983271	−	0.182151i	\(-0.941694\pi\)
0.806864	−	0.590737i	\(-0.201163\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.19.13	✓	252
43.34	odd	42	inner	43.7.h.a.34.13	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.19.13	✓	252	1.1	even	1	trivial
43.7.h.a.34.13	yes	252	43.34	odd	42	inner