Embedded newform 43.3.h.a.5.5

• Label: 43.3.h.a.5.5
• Level: $43$
• Weight: $3$
• Character: 43.5
• Analytic conductor: $1.172$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.17166513675\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(6\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			5.5
Character	\(\chi\)	\(=\)	43.5
Dual form			43.3.h.a.26.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.81675 + 0.414660i) q^{2} +(-1.31342 + 4.25801i) q^{3} +(-0.475255 - 0.228871i) q^{4} +(2.73180 - 1.07215i) q^{5} +(-4.15178 + 7.19109i) q^{6} +(6.14989 - 3.55064i) q^{7} +(-6.59618 - 5.26028i) q^{8} +(-8.96941 - 6.11524i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{25}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.81675	+	0.414660i	0.908373	+	0.207330i	0.651085	−	0.759005i	\(-0.274314\pi\)
							0.257288	+	0.966335i	\(0.417171\pi\)
\(3\)	−1.31342	+	4.25801i	−0.437807	+	1.41934i	0.421011	+	0.907056i	\(0.361675\pi\)
							−0.858818	+	0.512281i	\(0.828801\pi\)
\(5\)	2.73180	−	1.07215i	0.546361	−	0.214431i	−0.0760744	−	0.997102i	\(-0.524239\pi\)
							0.622435	+	0.782671i	\(0.286143\pi\)
\(7\)	6.14989	−	3.55064i	0.878556	−	0.507235i	0.00837424	−	0.999965i	\(-0.497334\pi\)
							0.870182	+	0.492730i	\(0.164001\pi\)
\(11\)	8.79534	−	4.23561i	0.799577	−	0.385056i	0.0109592	−	0.999940i	\(-0.496512\pi\)
							0.788617	+	0.614884i	\(0.210797\pi\)
\(13\)	−11.1026	−	1.67345i	−0.854048	−	0.128727i	−0.292596	−	0.956236i	\(-0.594519\pi\)
							−0.561452	+	0.827509i	\(0.689757\pi\)
\(17\)	−10.9550	+	27.9128i	−0.644410	+	1.64193i	0.116593	+	0.993180i	\(0.462803\pi\)
							−0.761002	+	0.648749i	\(0.775292\pi\)
\(19\)	−4.77980	−	7.01068i	−0.251569	−	0.368983i	0.679593	−	0.733589i	\(-0.262156\pi\)
							−0.931162	+	0.364606i	\(0.881204\pi\)
\(23\)	2.22503	−	29.6910i	0.0967405	−	1.29091i	−0.711762	−	0.702420i	\(-0.752103\pi\)
							0.808503	−	0.588492i	\(-0.200278\pi\)
\(29\)	−5.32250	−	17.2551i	−0.183535	−	0.595005i	−0.999849	−	0.0173963i	\(-0.994462\pi\)
							0.816314	−	0.577608i	\(-0.196014\pi\)
\(31\)	39.5644	+	36.7104i	1.27627	+	1.18421i	0.972872	+	0.231345i	\(0.0743126\pi\)
							0.303401	+	0.952863i	\(0.401878\pi\)
\(37\)	−1.51734	−	0.876034i	−0.0410091	−	0.0236766i	0.479355	−	0.877621i	\(-0.340871\pi\)
							−0.520364	+	0.853944i	\(0.674204\pi\)
\(41\)	14.7823	−	64.7655i	0.360544	−	1.57965i	−0.391273	−	0.920275i	\(-0.627965\pi\)
							0.751817	−	0.659372i	\(-0.229178\pi\)
\(43\)	−16.1440	+	39.8544i	−0.375442	+	0.926846i
							\(47\)	31.9846	+	15.4030i	0.680523	+	0.327722i	0.742013	−	0.670385i	\(-0.233871\pi\)
−0.0614906	+	0.998108i	\(0.519585\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.5.5	✓	72
3.2	odd	2		387.3.bn.b.91.2		72
43.26	odd	42	inner	43.3.h.a.26.5	yes	72
129.26	even	42		387.3.bn.b.370.2		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.5.5	✓	72	1.1	even	1	trivial
43.3.h.a.26.5	yes	72	43.26	odd	42	inner
387.3.bn.b.91.2		72	3.2	odd	2
387.3.bn.b.370.2		72	129.26	even	42