Embedded newform 43.3.h.a.26.1

• Label: 43.3.h.a.26.1
• Level: $43$
• Weight: $3$
• Character: 43.26
• 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			26.1
Character	\(\chi\)	\(=\)	43.26
Dual form			43.3.h.a.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.19097 + 0.728319i) q^{2} +(-0.352152 - 1.14165i) q^{3} +(6.04798 - 2.91256i) q^{4} +(2.28850 + 0.898170i) q^{5} +(1.95519 + 3.38649i) q^{6} +(10.0706 + 5.81424i) q^{7} +(-6.94183 + 5.53593i) q^{8} +(6.25680 - 4.26581i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{17}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.19097	+	0.728319i	−1.59549	+	0.364159i	−0.925659	−	0.378358i	\(-0.876489\pi\)
							−0.669827	+	0.742517i	\(0.733632\pi\)
\(3\)	−0.352152	−	1.14165i	−0.117384	−	0.380550i	0.877745	−	0.479127i	\(-0.159047\pi\)
							−0.995129	+	0.0985779i	\(0.968571\pi\)
\(5\)	2.28850	+	0.898170i	0.457700	+	0.179634i	0.582978	−	0.812488i	\(-0.301887\pi\)
							−0.125278	+	0.992122i	\(0.539982\pi\)
\(7\)	10.0706	+	5.81424i	1.43865	+	0.830606i	0.997756	−	0.0669531i	\(-0.0213278\pi\)
							0.440895	+	0.897559i	\(0.354661\pi\)
\(11\)	−8.52330	−	4.10460i	−0.774845	−	0.373146i	0.00429875	−	0.999991i	\(-0.498632\pi\)
							−0.779144	+	0.626845i	\(0.784346\pi\)
\(13\)	3.56642	−	0.537551i	0.274340	−	0.0413501i	−0.0104313	−	0.999946i	\(-0.503320\pi\)
							0.284771	+	0.958595i	\(0.408082\pi\)
\(17\)	2.56415	+	6.53334i	0.150832	+	0.384314i	0.986241	−	0.165315i	\(-0.0528642\pi\)
							−0.835409	+	0.549629i	\(0.814769\pi\)
\(19\)	10.4891	−	15.3847i	0.552060	−	0.809723i	−0.444218	−	0.895919i	\(-0.646519\pi\)
							0.996278	+	0.0861954i	\(0.0274709\pi\)
\(23\)	2.94151	+	39.2518i	0.127892	+	1.70660i	0.579985	+	0.814627i	\(0.303058\pi\)
							−0.452093	+	0.891971i	\(0.649323\pi\)
\(29\)	4.30310	−	13.9503i	0.148383	−	0.481045i	−0.850616	−	0.525788i	\(-0.823771\pi\)
							0.998999	+	0.0447429i	\(0.0142469\pi\)
\(31\)	−32.3496	+	30.0160i	−1.04354	+	0.968259i	−0.999526	−	0.0307717i	\(-0.990204\pi\)
							−0.0440090	+	0.999031i	\(0.514013\pi\)
\(37\)	12.4503	−	7.18816i	0.336494	−	0.194275i	−0.322227	−	0.946663i	\(-0.604431\pi\)
							0.658720	+	0.752388i	\(0.271098\pi\)
\(41\)	−16.7860	−	73.5444i	−0.409415	−	1.79377i	−0.586920	−	0.809645i	\(-0.699660\pi\)
							0.177504	−	0.984120i	\(-0.443198\pi\)
\(43\)	−42.9996	−	0.190850i	−0.999990	−	0.00443838i
							\(47\)	−52.2411	+	25.1580i	−1.11151	+	0.535277i	−0.897261	−	0.441501i	\(-0.854446\pi\)
−0.214253	+	0.976778i	\(0.568732\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.26.1	yes	72
3.2	odd	2		387.3.bn.b.370.6		72
43.5	odd	42	inner	43.3.h.a.5.1	✓	72
129.5	even	42		387.3.bn.b.91.6		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.5.1	✓	72	43.5	odd	42	inner
43.3.h.a.26.1	yes	72	1.1	even	1	trivial
387.3.bn.b.91.6		72	129.5	even	42
387.3.bn.b.370.6		72	3.2	odd	2