Embedded newform 43.7.h.a.3.9

• Label: 43.7.h.a.3.9
• 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			3.9
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.910519 + 1.89071i) q^{2} +(23.8619 + 1.78820i) q^{3} +(37.1576 + 46.5942i) q^{4} +(47.7974 - 154.955i) q^{5} +(-25.1077 + 43.4878i) q^{6} +(360.408 - 208.082i) q^{7} +(-252.868 + 57.7154i) q^{8} +(-154.664 - 23.3118i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.910519	+	1.89071i	−0.113815	+	0.236339i	−0.950093	−	0.311966i	\(-0.899012\pi\)
							0.836278	+	0.548305i	\(0.184727\pi\)
\(3\)	23.8619	+	1.78820i	0.883775	+	0.0662298i	0.508880	−	0.860838i	\(-0.330060\pi\)
							0.374895	+	0.927067i	\(0.377679\pi\)
\(5\)	47.7974	−	154.955i	0.382379	−	1.23964i	−0.536094	−	0.844158i	\(-0.680101\pi\)
							0.918473	−	0.395484i	\(-0.129423\pi\)
\(7\)	360.408	−	208.082i	1.05075	−	0.606652i	0.127893	−	0.991788i	\(-0.459179\pi\)
							0.922860	+	0.385136i	\(0.125845\pi\)
\(11\)	423.929	−	531.590i	0.318504	−	0.399391i	−0.596646	−	0.802504i	\(-0.703500\pi\)
							0.915150	+	0.403113i	\(0.132072\pi\)
\(13\)	−556.338	+	516.206i	−0.253226	+	0.234960i	−0.796563	−	0.604555i	\(-0.793351\pi\)
							0.543337	+	0.839515i	\(0.317161\pi\)
\(17\)	6478.19	−	1998.26i	1.31858	−	0.406728i	0.445927	−	0.895069i	\(-0.352874\pi\)
							0.872653	+	0.488341i	\(0.162398\pi\)
\(19\)	0.579777	+	3.84657i	8.45280e−5	+	0.000560806i	0.988873	−	0.148762i	\(-0.0475288\pi\)
							−0.988789	+	0.149323i	\(0.952291\pi\)
\(23\)	7158.30	+	18239.1i	0.588337	+	1.49906i	0.846193	+	0.532876i	\(0.178889\pi\)
							−0.257856	+	0.966183i	\(0.583016\pi\)
\(29\)	8768.33	−	657.096i	0.359520	−	0.0269423i	0.106254	−	0.994339i	\(-0.466114\pi\)
							0.253266	+	0.967397i	\(0.418495\pi\)
\(31\)	−29293.2	+	19971.8i	−0.983291	+	0.670397i	−0.944384	−	0.328846i	\(-0.893340\pi\)
							−0.0389078	+	0.999243i	\(0.512388\pi\)
\(37\)	−55204.9	−	31872.6i	−1.08987	−	0.629234i	−0.156325	−	0.987706i	\(-0.549965\pi\)
							−0.933541	+	0.358472i	\(0.883298\pi\)
\(41\)	−87081.4	−	41936.2i	−1.26350	−	0.608468i	−0.322399	−	0.946604i	\(-0.604489\pi\)
							−0.941097	+	0.338136i	\(0.890204\pi\)
\(43\)	22129.5	+	76365.2i	0.278334	+	0.960484i
							\(47\)	68801.3	+	86274.1i	0.662678	+	0.830972i	0.993632	−	0.112675i	\(-0.0359418\pi\)
−0.330954	+	0.943647i	\(0.607370\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.3.9	✓	252
43.29	odd	42	inner	43.7.h.a.29.9	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.9	✓	252	1.1	even	1	trivial
43.7.h.a.29.9	yes	252	43.29	odd	42	inner