Embedded newform 43.7.h.a.12.16

• Label: 43.7.h.a.12.16
• Level: $43$
• Weight: $7$
• Character: 43.12
• 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			12.16
Character	\(\chi\)	\(=\)	43.12
Dual form			43.7.h.a.18.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.87752 + 8.05175i) q^{2} +(-6.20528 - 9.10147i) q^{3} +(-9.89222 + 12.4045i) q^{4} +(38.7075 - 41.7168i) q^{5} +(49.2217 - 85.2545i) q^{6} +(32.6490 - 18.8499i) q^{7} +(419.378 + 95.7204i) q^{8} +(222.002 - 565.653i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.87752	+	8.05175i	0.484690	+	1.00647i	0.989673	+	0.143340i	\(0.0457843\pi\)
							−0.504983	+	0.863129i	\(0.668501\pi\)
\(3\)	−6.20528	−	9.10147i	−0.229825	−	0.337091i	0.693953	−	0.720021i	\(-0.255868\pi\)
							−0.923778	+	0.382929i	\(0.874915\pi\)
\(5\)	38.7075	−	41.7168i	0.309660	−	0.333735i	−0.558977	−	0.829183i	\(-0.688806\pi\)
							0.868637	+	0.495449i	\(0.164996\pi\)
\(7\)	32.6490	−	18.8499i	0.0951865	−	0.0549559i	−0.451651	−	0.892195i	\(-0.649165\pi\)
							0.546838	+	0.837239i	\(0.315831\pi\)
\(11\)	602.384	+	755.366i	0.452580	+	0.567518i	0.954810	−	0.297216i	\(-0.0960583\pi\)
							−0.502230	+	0.864734i	\(0.667487\pi\)
\(13\)	2612.47	−	805.840i	1.18911	−	0.366791i	0.363801	−	0.931477i	\(-0.381479\pi\)
							0.825306	+	0.564686i	\(0.191003\pi\)
\(17\)	−385.470	+	357.664i	−0.0784592	+	0.0727995i	−0.718428	−	0.695602i	\(-0.755138\pi\)
							0.639968	+	0.768401i	\(0.278947\pi\)
\(19\)	−2968.05	+	1164.87i	−0.432723	+	0.169831i	−0.571697	−	0.820465i	\(-0.693715\pi\)
							0.138974	+	0.990296i	\(0.455620\pi\)
\(23\)	−449.415	+	67.7384i	−0.0369372	+	0.00556738i	−0.167485	−	0.985875i	\(-0.553565\pi\)
							0.130548	+	0.991442i	\(0.458326\pi\)
\(29\)	26790.5	−	39294.4i	1.09847	−	1.61115i	0.365612	−	0.930767i	\(-0.380860\pi\)
							0.732854	−	0.680386i	\(-0.238188\pi\)
\(31\)	2714.53	−	36222.9i	0.0911193	−	1.21590i	−0.744890	−	0.667187i	\(-0.767498\pi\)
							0.836009	−	0.548715i	\(-0.184883\pi\)
\(37\)	−15899.6	−	9179.61i	−0.313892	−	0.181225i	0.334775	−	0.942298i	\(-0.391340\pi\)
							−0.648667	+	0.761073i	\(0.724673\pi\)
\(41\)	−34816.3	+	16766.7i	−0.505163	+	0.243274i	−0.669066	−	0.743203i	\(-0.733306\pi\)
							0.163903	+	0.986476i	\(0.447592\pi\)
\(43\)	−65566.1	+	44971.7i	−0.824658	+	0.565632i
							\(47\)	−24968.6	+	31309.6i	−0.240492	+	0.301567i	−0.887400	−	0.461001i	\(-0.847490\pi\)
0.646908	+	0.762568i	\(0.276062\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.12.16	✓	252
43.18	odd	42	inner	43.7.h.a.18.16	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.16	✓	252	1.1	even	1	trivial
43.7.h.a.18.16	yes	252	43.18	odd	42	inner