Embedded newform 43.5.h.a.3.6

• Label: 43.5.h.a.3.6
• Level: $43$
• Weight: $5$
• Character: 43.3
• 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			3.6
Character	\(\chi\)	\(=\)	43.3
Dual form			43.5.h.a.29.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.833611 + 1.73101i) q^{2} +(-5.15016 - 0.385951i) q^{3} +(7.67434 + 9.62332i) q^{4} +(0.609730 - 1.97669i) q^{5} +(4.96132 - 8.59325i) q^{6} +(-32.4341 + 18.7258i) q^{7} +(-53.0252 + 12.1027i) q^{8} +(-53.7201 - 8.09700i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.833611	+	1.73101i	−0.208403	+	0.432753i	−0.978802	−	0.204809i	\(-0.934343\pi\)
							0.770399	+	0.637562i	\(0.220057\pi\)
\(3\)	−5.15016	−	0.385951i	−0.572240	−	0.0428835i	−0.214532	−	0.976717i	\(-0.568823\pi\)
							−0.357708	+	0.933833i	\(0.616442\pi\)
\(5\)	0.609730	−	1.97669i	0.0243892	−	0.0790678i	−0.942560	−	0.334037i	\(-0.891589\pi\)
							0.966949	+	0.254969i	\(0.0820652\pi\)
\(7\)	−32.4341	+	18.7258i	−0.661919	+	0.382159i	−0.793008	−	0.609211i	\(-0.791486\pi\)
							0.131089	+	0.991371i	\(0.458153\pi\)
\(11\)	−114.909	+	144.091i	−0.949661	+	1.19084i	0.0318621	+	0.999492i	\(0.489856\pi\)
							−0.981523	+	0.191345i	\(0.938715\pi\)
\(13\)	50.0054	−	46.3983i	0.295890	−	0.274546i	−0.518231	−	0.855240i	\(-0.673409\pi\)
							0.814122	+	0.580694i	\(0.197219\pi\)
\(17\)	134.060	−	41.3520i	0.463875	−	0.143087i	−0.0540032	−	0.998541i	\(-0.517198\pi\)
							0.517878	+	0.855454i	\(0.326722\pi\)
\(19\)	−13.7485	−	91.2151i	−0.0380844	−	0.252673i	0.961685	−	0.274158i	\(-0.0883990\pi\)
							−0.999769	+	0.0214841i	\(0.993161\pi\)
\(23\)	173.109	+	441.074i	0.327238	+	0.833789i	0.995920	+	0.0902405i	\(0.0287636\pi\)
							−0.668682	+	0.743549i	\(0.733141\pi\)
\(29\)	−488.210	+	36.5863i	−0.580511	+	0.0435033i	−0.361750	−	0.932275i	\(-0.617821\pi\)
							−0.218761	+	0.975778i	\(0.570202\pi\)
\(31\)	638.029	−	435.001i	0.663922	−	0.452654i	−0.183893	−	0.982946i	\(-0.558870\pi\)
							0.847815	+	0.530292i	\(0.177918\pi\)
\(37\)	1177.74	+	679.968i	0.860291	+	0.496689i	0.864110	−	0.503303i	\(-0.167882\pi\)
							−0.00381843	+	0.999993i	\(0.501215\pi\)
\(41\)	−2551.56	−	1228.76i	−1.51788	−	0.730972i	−0.525113	−	0.851032i	\(-0.675977\pi\)
							−0.992767	+	0.120060i	\(0.961691\pi\)
\(43\)	−191.524	+	1839.05i	−0.103582	+	0.994621i
							\(47\)	1229.36	+	1541.57i	0.556525	+	0.697861i	0.977911	−	0.209021i	\(-0.0670276\pi\)
−0.421386	+	0.906881i	\(0.638456\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.3.6	✓	168
43.29	odd	42	inner	43.5.h.a.29.6	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.3.6	✓	168	1.1	even	1	trivial
43.5.h.a.29.6	yes	168	43.29	odd	42	inner