Embedded newform 43.7.h.a.3.19

• Label: 43.7.h.a.3.19
• Level: $43$
• Weight: $7$
• Character: 43.3
• 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			3.19
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.75579 - 11.9520i) q^{2} +(-1.56239 - 0.117085i) q^{3} +(-69.8182 - 87.5493i) q^{4} +(36.5881 - 118.616i) q^{5} +(-10.3922 + 17.9998i) q^{6} +(348.105 - 200.979i) q^{7} +(-620.528 + 141.631i) q^{8} +(-718.430 - 108.286i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.75579	−	11.9520i	0.719473	−	1.49400i	−0.143984	−	0.989580i	\(-0.545991\pi\)
							0.863457	−	0.504422i	\(-0.168294\pi\)
\(3\)	−1.56239	−	0.117085i	−0.0578663	−	0.00433648i	0.0457655	−	0.998952i	\(-0.485427\pi\)
							−0.103632	+	0.994616i	\(0.533046\pi\)
\(5\)	36.5881	−	118.616i	0.292705	−	0.948927i	−0.683107	−	0.730318i	\(-0.739372\pi\)
							0.975813	−	0.218609i	\(-0.0701519\pi\)
\(7\)	348.105	−	200.979i	1.01488	−	0.585944i	0.102266	−	0.994757i	\(-0.467391\pi\)
							0.912618	+	0.408814i	\(0.134057\pi\)
\(11\)	−1063.89	+	1334.08i	−0.799320	+	1.00232i	0.200424	+	0.979709i	\(0.435768\pi\)
							−0.999744	+	0.0226065i	\(0.992804\pi\)
\(13\)	−1117.09	+	1036.51i	−0.508463	+	0.471784i	−0.892189	−	0.451662i	\(-0.850831\pi\)
							0.383726	+	0.923447i	\(0.374641\pi\)
\(17\)	7377.33	−	2275.61i	1.50159	−	0.463180i	0.568461	−	0.822710i	\(-0.307539\pi\)
							0.933134	+	0.359530i	\(0.117063\pi\)
\(19\)	368.191	+	2442.78i	0.0536799	+	0.356143i	0.999505	+	0.0314658i	\(0.0100175\pi\)
							−0.945825	+	0.324677i	\(0.894744\pi\)
\(23\)	−7078.80	−	18036.5i	−0.581803	−	1.48241i	−0.854054	−	0.520185i	\(-0.825863\pi\)
							0.272250	−	0.962226i	\(-0.412232\pi\)
\(29\)	12002.4	−	899.456i	0.492124	−	0.0368796i	0.173643	−	0.984809i	\(-0.444446\pi\)
							0.318481	+	0.947929i	\(0.396827\pi\)
\(31\)	26049.7	−	17760.4i	0.874417	−	0.596167i	−0.0407829	−	0.999168i	\(-0.512985\pi\)
							0.915200	+	0.403001i	\(0.132033\pi\)
\(37\)	−14981.5	−	8649.57i	−0.295767	−	0.170761i	0.344773	−	0.938686i	\(-0.387956\pi\)
							−0.640540	+	0.767925i	\(0.721289\pi\)
\(41\)	82125.3	+	39549.4i	1.19159	+	0.573837i	0.921266	−	0.388933i	\(-0.127156\pi\)
							0.270320	+	0.962771i	\(0.412871\pi\)
\(43\)	−14258.4	−	78218.0i	−0.179335	−	0.983788i
							\(47\)	84426.0	+	105867.i	0.813173	+	1.01969i	0.999309	+	0.0371625i	\(0.0118319\pi\)
−0.186136	+	0.982524i	\(0.559597\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.3.19	✓	252
43.29	odd	42	inner	43.7.h.a.29.19	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.19	✓	252	1.1	even	1	trivial
43.7.h.a.29.19	yes	252	43.29	odd	42	inner