Embedded newform 43.5.h.a.29.3

• Label: 43.5.h.a.29.3
• Level: $43$
• Weight: $5$
• Character: 43.29
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			29.3
Character	\(\chi\)	\(=\)	43.29
Dual form			43.5.h.a.3.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.82198 - 5.85989i) q^{2} +(-0.311439 + 0.0233391i) q^{3} +(-16.3990 + 20.5636i) q^{4} +(13.6957 + 44.4004i) q^{5} +(1.01564 + 1.75914i) q^{6} +(19.2501 + 11.1141i) q^{7} +(65.3235 + 14.9097i) q^{8} +(-79.9988 + 12.0579i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{41}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.82198	−	5.85989i	−0.705494	−	1.46497i	−0.877350	−	0.479851i	\(-0.840691\pi\)
							0.171856	−	0.985122i	\(-0.445024\pi\)
\(3\)	−0.311439	+	0.0233391i	−0.0346044	+	0.00259324i	−0.0920210	−	0.995757i	\(-0.529333\pi\)
							0.0574167	+	0.998350i	\(0.481714\pi\)
\(5\)	13.6957	+	44.4004i	0.547829	+	1.77602i	0.626846	+	0.779144i	\(0.284346\pi\)
							−0.0790169	+	0.996873i	\(0.525178\pi\)
\(7\)	19.2501	+	11.1141i	0.392859	+	0.226817i	0.683398	−	0.730046i	\(-0.260501\pi\)
							−0.290539	+	0.956863i	\(0.593835\pi\)
\(11\)	29.6396	+	37.1669i	0.244956	+	0.307165i	0.889076	−	0.457759i	\(-0.151348\pi\)
							−0.644121	+	0.764924i	\(0.722776\pi\)
\(13\)	6.21016	+	5.76219i	0.0367465	+	0.0340958i	0.698332	−	0.715774i	\(-0.253926\pi\)
							−0.661586	+	0.749869i	\(0.730116\pi\)
\(17\)	206.586	+	63.7233i	0.714830	+	0.220496i	0.630776	−	0.775965i	\(-0.282737\pi\)
							0.0840545	+	0.996461i	\(0.473213\pi\)
\(19\)	−78.7786	+	522.662i	−0.218223	+	1.44782i	0.564493	+	0.825438i	\(0.309072\pi\)
							−0.782716	+	0.622379i	\(0.786166\pi\)
\(23\)	253.456	−	645.795i	0.479122	−	1.22078i	−0.463253	−	0.886226i	\(-0.653318\pi\)
							0.942375	−	0.334558i	\(-0.108587\pi\)
\(29\)	−1010.10	−	75.6968i	−1.20107	−	0.0900081i	−0.540878	−	0.841101i	\(-0.681908\pi\)
							−0.660197	+	0.751093i	\(0.729527\pi\)
\(31\)	738.096	+	503.225i	0.768050	+	0.523647i	0.882801	−	0.469747i	\(-0.155655\pi\)
							−0.114752	+	0.993394i	\(0.536607\pi\)
\(37\)	−143.548	+	82.8776i	−0.104856	+	0.0605388i	−0.551511	−	0.834168i	\(-0.685949\pi\)
							0.446655	+	0.894706i	\(0.352615\pi\)
\(41\)	2521.08	−	1214.09i	1.49975	−	0.722242i	0.509362	−	0.860553i	\(-0.329882\pi\)
							0.990389	+	0.138311i	\(0.0441673\pi\)
\(43\)	16.3230	+	1848.93i	0.00882799	+	0.999961i
							\(47\)	−571.675	+	716.857i	−0.258793	+	0.324517i	−0.894206	−	0.447656i	\(-0.852259\pi\)
0.635412	+	0.772173i	\(0.280830\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.29.3	yes	168
43.3	odd	42	inner	43.5.h.a.3.3	✓	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.3.3	✓	168	43.3	odd	42	inner
43.5.h.a.29.3	yes	168	1.1	even	1	trivial