Embedded newform 43.5.h.a.29.6

• Label: 43.5.h.a.29.6
• 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			29.6
Character	\(\chi\)	\(=\)	43.29
Dual form			43.5.h.a.3.6

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.3.6	✓	168	43.3	odd	42	inner
43.5.h.a.29.6	yes	168	1.1	even	1	trivial