Embedded newform 43.3.h.a.12.4

• Label: 43.3.h.a.12.4
• Level: $43$
• Weight: $3$
• Character: 43.12
• 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			12.4
Character	\(\chi\)	\(=\)	43.12
Dual form			43.3.h.a.18.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.185466 + 0.385124i) q^{2} +(-2.30187 - 3.37622i) q^{3} +(2.38004 - 2.98447i) q^{4} +(-0.146154 + 0.157516i) q^{5} +(0.873345 - 1.51268i) q^{6} +(0.871816 - 0.503343i) q^{7} +(3.25776 + 0.743563i) q^{8} +(-2.81219 + 7.16534i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 14 q^{2} - 14 q^{3} + 12 q^{4} - 11 q^{5} + 2 q^{6} - 30 q^{7} - 42 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.185466	+	0.385124i	0.0927331	+	0.192562i	0.942176	−	0.335118i	\(-0.108776\pi\)
							−0.849443	+	0.527680i	\(0.823062\pi\)
\(3\)	−2.30187	−	3.37622i	−0.767289	−	1.12541i	−0.988653	−	0.150215i	\(-0.952003\pi\)
							0.221365	−	0.975191i	\(-0.428949\pi\)
\(5\)	−0.146154	+	0.157516i	−0.0292308	+	0.0315033i	−0.747498	−	0.664264i	\(-0.768745\pi\)
							0.718267	+	0.695767i	\(0.244936\pi\)
\(7\)	0.871816	−	0.503343i	0.124545	−	0.0719061i	−0.436433	−	0.899737i	\(-0.643758\pi\)
							0.560978	+	0.827831i	\(0.310425\pi\)
\(11\)	10.2871	+	12.8996i	0.935190	+	1.17269i	0.984760	+	0.173920i	\(0.0556435\pi\)
							−0.0495699	+	0.998771i	\(0.515785\pi\)
\(13\)	−8.32173	+	2.56691i	−0.640133	+	0.197455i	−0.597796	−	0.801648i	\(-0.703957\pi\)
							−0.0423374	+	0.999103i	\(0.513480\pi\)
\(17\)	−11.3523	+	10.5334i	−0.667784	+	0.619613i	−0.939412	−	0.342789i	\(-0.888628\pi\)
							0.271628	+	0.962402i	\(0.412438\pi\)
\(19\)	18.4023	−	7.22236i	0.968541	−	0.380124i	0.172260	−	0.985051i	\(-0.444893\pi\)
							0.796281	+	0.604927i	\(0.206798\pi\)
\(23\)	31.0766	−	4.68404i	1.35115	−	0.203654i	0.566743	−	0.823894i	\(-0.308203\pi\)
							0.784412	+	0.620241i	\(0.212965\pi\)
\(29\)	−3.63652	+	5.33380i	−0.125397	+	0.183924i	−0.883804	−	0.467858i	\(-0.845026\pi\)
							0.758406	+	0.651782i	\(0.225978\pi\)
\(31\)	1.13743	−	15.1779i	0.0366912	−	0.489610i	−0.948343	−	0.317248i	\(-0.897241\pi\)
							0.985034	−	0.172362i	\(-0.0551398\pi\)
\(37\)	−47.5485	−	27.4521i	−1.28509	−	0.741949i	−0.307319	−	0.951607i	\(-0.599432\pi\)
							−0.977775	+	0.209657i	\(0.932765\pi\)
\(41\)	−38.6775	+	18.6261i	−0.943354	+	0.454295i	−0.841351	−	0.540489i	\(-0.818239\pi\)
							−0.102003	+	0.994784i	\(0.532525\pi\)
\(43\)	−26.1913	−	34.1030i	−0.609101	−	0.793093i
							\(47\)	−46.0657	+	57.7646i	−0.980122	+	1.22903i	−0.00670864	+	0.999977i	\(0.502135\pi\)
−0.973413	+	0.229056i	\(0.926436\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.3.h.a.12.4	✓	72
3.2	odd	2		387.3.bn.b.55.3		72
43.18	odd	42	inner	43.3.h.a.18.4	yes	72
129.104	even	42		387.3.bn.b.190.3		72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.3.h.a.12.4	✓	72	1.1	even	1	trivial
43.3.h.a.18.4	yes	72	43.18	odd	42	inner
387.3.bn.b.55.3		72	3.2	odd	2
387.3.bn.b.190.3		72	129.104	even	42