Embedded newform 43.7.h.a.3.2

• Label: 43.7.h.a.3.2
• 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			3.2
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.27248 + 13.0249i) q^{2} +(36.1802 + 2.71133i) q^{3} +(-90.4018 - 113.360i) q^{4} +(22.1660 - 71.8603i) q^{5} +(-262.254 + 454.238i) q^{6} +(414.759 - 239.462i) q^{7} +(1141.53 - 260.547i) q^{8} +(580.795 + 87.5408i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.27248	+	13.0249i	−0.784060	+	1.62812i	−0.00595638	+	0.999982i	\(0.501896\pi\)
							−0.778104	+	0.628136i	\(0.783818\pi\)
\(3\)	36.1802	+	2.71133i	1.34001	+	0.100420i	0.725353	−	0.688378i	\(-0.241677\pi\)
							0.614654	+	0.788797i	\(0.289296\pi\)
\(5\)	22.1660	−	71.8603i	0.177328	−	0.574882i	−0.822652	−	0.568545i	\(-0.807507\pi\)
							0.999980	−	0.00633745i	\(-0.00201729\pi\)
\(7\)	414.759	−	239.462i	1.20921	−	0.698138i	0.246624	−	0.969111i	\(-0.420679\pi\)
							0.962587	+	0.270973i	\(0.0873453\pi\)
\(11\)	846.924	−	1062.01i	0.636306	−	0.797903i	−0.354229	−	0.935159i	\(-0.615257\pi\)
							0.990536	+	0.137256i	\(0.0438282\pi\)
\(13\)	1571.59	−	1458.22i	0.715334	−	0.663733i	−0.236050	−	0.971741i	\(-0.575853\pi\)
							0.951384	+	0.308008i	\(0.0996624\pi\)
\(17\)	−5328.85	+	1643.73i	−1.08464	+	0.334568i	−0.785008	−	0.619486i	\(-0.787341\pi\)
							−0.299635	+	0.954054i	\(0.596865\pi\)
\(19\)	1499.51	+	9948.62i	0.218620	+	1.45045i	0.781498	+	0.623907i	\(0.214456\pi\)
							−0.562878	+	0.826540i	\(0.690306\pi\)
\(23\)	−6761.32	−	17227.6i	−0.555710	−	1.41592i	−0.882268	−	0.470748i	\(-0.843984\pi\)
							0.326558	−	0.945177i	\(-0.394111\pi\)
\(29\)	−18213.1	+	1364.88i	−0.746774	+	0.0559630i	−0.442681	−	0.896679i	\(-0.645972\pi\)
							−0.304093	+	0.952642i	\(0.598353\pi\)
\(31\)	−3344.82	+	2280.46i	−0.112276	+	0.0765487i	−0.618156	−	0.786055i	\(-0.712120\pi\)
							0.505880	+	0.862604i	\(0.331168\pi\)
\(37\)	42707.1	+	24657.0i	0.843131	+	0.486782i	0.858327	−	0.513103i	\(-0.171504\pi\)
							−0.0151965	+	0.999885i	\(0.504837\pi\)
\(41\)	52146.9	+	25112.6i	0.756619	+	0.364368i	0.772091	−	0.635512i	\(-0.219211\pi\)
							−0.0154724	+	0.999880i	\(0.504925\pi\)
\(43\)	75488.3	+	24957.6i	0.949455	+	0.313904i
							\(47\)	69445.3	+	87081.7i	0.668882	+	0.838751i	0.994278	−	0.106826i	\(-0.0340689\pi\)
−0.325396	+	0.945578i	\(0.605498\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.3.2	✓	252
43.29	odd	42	inner	43.7.h.a.29.2	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.2	✓	252	1.1	even	1	trivial
43.7.h.a.29.2	yes	252	43.29	odd	42	inner