Embedded newform 43.7.h.a.18.15

• Label: 43.7.h.a.18.15
• Level: $43$
• Weight: $7$
• Character: 43.18
• 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			18.15
Character	\(\chi\)	\(=\)	43.18
Dual form			43.7.h.a.12.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.00335 - 6.23652i) q^{2} +(14.9475 - 21.9239i) q^{3} +(10.0293 + 12.5763i) q^{4} +(96.6422 + 104.156i) q^{5} +(-91.8363 - 159.065i) q^{6} +(401.591 + 231.859i) q^{7} +(540.456 - 123.355i) q^{8} +(9.10321 + 23.1946i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.00335	−	6.23652i	0.375419	−	0.779565i	−0.624580	−	0.780961i	\(-0.714730\pi\)
							0.999999	+	0.00139564i	\(0.000444248\pi\)
\(3\)	14.9475	−	21.9239i	0.553609	−	0.811996i	−0.442804	−	0.896619i	\(-0.646016\pi\)
							0.996413	+	0.0846229i	\(0.0269685\pi\)
\(5\)	96.6422	+	104.156i	0.773138	+	0.833244i	0.989720	−	0.143020i	\(-0.0456813\pi\)
							−0.216582	+	0.976264i	\(0.569491\pi\)
\(7\)	401.591	+	231.859i	1.17082	+	0.675973i	0.953873	−	0.300211i	\(-0.0970571\pi\)
							0.216946	+	0.976184i	\(0.430390\pi\)
\(11\)	−377.667	+	473.580i	−0.283747	+	0.355808i	−0.903195	−	0.429230i	\(-0.858785\pi\)
							0.619448	+	0.785038i	\(0.287356\pi\)
\(13\)	−3636.33	−	1121.66i	−1.65513	−	0.510541i	−0.680199	−	0.733028i	\(-0.738107\pi\)
							−0.974936	+	0.222486i	\(0.928583\pi\)
\(17\)	−3856.18	−	3578.02i	−0.784894	−	0.728275i	0.182043	−	0.983291i	\(-0.441729\pi\)
							−0.966937	+	0.255015i	\(0.917919\pi\)
\(19\)	−648.455	−	254.500i	−0.0945407	−	0.0371045i	0.317599	−	0.948225i	\(-0.397123\pi\)
							−0.412140	+	0.911121i	\(0.635219\pi\)
\(23\)	321.883	+	48.5160i	0.0264554	+	0.00398751i	0.162257	−	0.986749i	\(-0.448123\pi\)
							−0.135801	+	0.990736i	\(0.543361\pi\)
\(29\)	−20341.8	−	29836.0i	−0.834058	−	1.22334i	−0.972485	−	0.232965i	\(-0.925157\pi\)
							0.138427	−	0.990373i	\(-0.455795\pi\)
\(31\)	−1493.55	−	19930.0i	−0.0501341	−	0.668994i	−0.964496	−	0.264096i	\(-0.914926\pi\)
							0.914362	−	0.404897i	\(-0.132693\pi\)
\(37\)	−46174.1	+	26658.6i	−0.911577	+	0.526299i	−0.880938	−	0.473231i	\(-0.843088\pi\)
							−0.0306390	+	0.999531i	\(0.509754\pi\)
\(41\)	57250.4	+	27570.4i	0.830668	+	0.400028i	0.800365	−	0.599512i	\(-0.204639\pi\)
							0.0303020	+	0.999541i	\(0.490353\pi\)
\(43\)	−46928.0	−	64180.4i	−0.590238	−	0.807230i
							\(47\)	−18452.6	−	23138.8i	−0.177731	−	0.222868i	0.684984	−	0.728558i	\(-0.259809\pi\)
−0.862715	+	0.505690i	\(0.831238\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.18.15	yes	252
43.12	odd	42	inner	43.7.h.a.12.15	✓	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.15	✓	252	43.12	odd	42	inner
43.7.h.a.18.15	yes	252	1.1	even	1	trivial