Embedded newform 43.7.h.a.18.16

• Label: 43.7.h.a.18.16
• Level: $43$
• Weight: $7$
• Character: 43.18
• 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			18.16
Character	\(\chi\)	\(=\)	43.18
Dual form			43.7.h.a.12.16

$q$-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 \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.16	✓	252	43.12	odd	42	inner
43.7.h.a.18.16	yes	252	1.1	even	1	trivial