Embedded newform 43.7.h.a.3.3

• Label: 43.7.h.a.3.3
• 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			3.3
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.20719 + 10.8128i) q^{2} +(3.59499 + 0.269407i) q^{3} +(-49.8995 - 62.5720i) q^{4} +(-53.9072 + 174.763i) q^{5} +(-21.6329 + 37.4692i) q^{6} +(85.2807 - 49.2368i) q^{7} +(187.588 - 42.8158i) q^{8} +(-708.006 - 106.715i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.20719	+	10.8128i	−0.650899	+	1.35161i	0.270399	+	0.962748i	\(0.412844\pi\)
							−0.921298	+	0.388858i	\(0.872870\pi\)
\(3\)	3.59499	+	0.269407i	0.133148	+	0.00997805i	0.141137	−	0.989990i	\(-0.454924\pi\)
							−0.00798947	+	0.999968i	\(0.502543\pi\)
\(5\)	−53.9072	+	174.763i	−0.431257	+	1.39810i	0.435868	+	0.900011i	\(0.356442\pi\)
							−0.867125	+	0.498091i	\(0.834035\pi\)
\(7\)	85.2807	−	49.2368i	0.248632	−	0.143548i	−0.370506	−	0.928830i	\(-0.620816\pi\)
							0.619138	+	0.785283i	\(0.287482\pi\)
\(11\)	−88.2039	+	110.604i	−0.0662689	+	0.0830985i	−0.813861	−	0.581059i	\(-0.802638\pi\)
							0.747592	+	0.664158i	\(0.231210\pi\)
\(13\)	46.5875	−	43.2269i	0.0212050	−	0.0196754i	−0.669499	−	0.742813i	\(-0.733491\pi\)
							0.690704	+	0.723137i	\(0.257301\pi\)
\(17\)	4928.91	−	1520.37i	1.00324	−	0.309458i	0.250741	−	0.968054i	\(-0.419326\pi\)
							0.752496	+	0.658596i	\(0.228850\pi\)
\(19\)	−571.191	−	3789.61i	−0.0832762	−	0.552501i	−0.991097	−	0.133143i	\(-0.957493\pi\)
							0.907821	−	0.419359i	\(-0.137745\pi\)
\(23\)	629.593	+	1604.18i	0.0517459	+	0.131846i	0.954409	−	0.298500i	\(-0.0964864\pi\)
							−0.902664	+	0.430347i	\(0.858391\pi\)
\(29\)	−24051.4	+	1802.40i	−0.986158	+	0.0739024i	−0.558035	−	0.829818i	\(-0.688444\pi\)
							−0.428124	+	0.903720i	\(0.640825\pi\)
\(31\)	−11116.2	+	7578.90i	−0.373140	+	0.254402i	−0.735341	−	0.677698i	\(-0.762978\pi\)
							0.362201	+	0.932100i	\(0.382025\pi\)
\(37\)	−67818.3	−	39154.9i	−1.33888	−	0.773003i	−0.352238	−	0.935910i	\(-0.614579\pi\)
							−0.986641	+	0.162908i	\(0.947913\pi\)
\(41\)	−59704.4	−	28752.1i	−0.866272	−	0.417175i	−0.0526807	−	0.998611i	\(-0.516777\pi\)
							−0.813592	+	0.581437i	\(0.802491\pi\)
\(43\)	−73686.3	+	29861.3i	−0.926790	+	0.375581i
							\(47\)	72726.1	+	91195.7i	0.700482	+	0.878376i	0.997059	−	0.0766323i	\(-0.0244168\pi\)
−0.296578	+	0.955009i	\(0.595845\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.3.3	✓	252
43.29	odd	42	inner	43.7.h.a.29.3	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.3	✓	252	1.1	even	1	trivial
43.7.h.a.29.3	yes	252	43.29	odd	42	inner