Embedded newform 43.5.h.a.3.10

• Label: 43.5.h.a.3.10
• Level: $43$
• Weight: $5$
• Character: 43.3
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			3.10
Character	\(\chi\)	\(=\)	43.3
Dual form			43.5.h.a.29.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.19998 - 2.49178i) q^{2} +(-0.581777 - 0.0435981i) q^{3} +(5.20683 + 6.52916i) q^{4} +(5.37775 - 17.4342i) q^{5} +(-0.806755 + 1.39734i) q^{6} +(61.1701 - 35.3166i) q^{7} +(65.6584 - 14.9861i) q^{8} +(-79.7587 - 12.0217i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.19998	−	2.49178i	0.299994	−	0.622944i	−0.695420	−	0.718603i	\(-0.744782\pi\)
							0.995414	+	0.0956594i	\(0.0304960\pi\)
\(3\)	−0.581777	−	0.0435981i	−0.0646419	−	0.00484424i	0.0423699	−	0.999102i	\(-0.486509\pi\)
							−0.107012	+	0.994258i	\(0.534128\pi\)
\(5\)	5.37775	−	17.4342i	0.215110	−	0.697369i	−0.782129	−	0.623116i	\(-0.785866\pi\)
							0.997239	−	0.0742534i	\(-0.0236574\pi\)
\(7\)	61.1701	−	35.3166i	1.24837	−	0.720747i	0.277586	−	0.960701i	\(-0.410466\pi\)
							0.970784	+	0.239954i	\(0.0771323\pi\)
\(11\)	0.426520	−	0.534839i	0.00352496	−	0.00442015i	−0.780066	−	0.625697i	\(-0.784814\pi\)
							0.783591	+	0.621277i	\(0.213386\pi\)
\(13\)	64.2397	−	59.6057i	0.380117	−	0.352697i	−0.466856	−	0.884333i	\(-0.654613\pi\)
							0.846972	+	0.531637i	\(0.178423\pi\)
\(17\)	−113.350	+	34.9638i	−0.392214	+	0.120982i	−0.484588	−	0.874742i	\(-0.661031\pi\)
							0.0923742	+	0.995724i	\(0.470554\pi\)
\(19\)	83.7472	+	555.626i	0.231987	+	1.53913i	0.736924	+	0.675975i	\(0.236277\pi\)
							−0.504938	+	0.863156i	\(0.668485\pi\)
\(23\)	−0.0772625	−	0.196862i	−0.000146054	−	0.000372139i	0.930801	−	0.365527i	\(-0.119111\pi\)
							−0.930947	+	0.365155i	\(0.881016\pi\)
\(29\)	−1485.31	+	111.308i	−1.76612	+	0.132353i	−0.917778	−	0.397094i	\(-0.870019\pi\)
							−0.848344	+	0.529446i	\(0.822400\pi\)
\(31\)	−1143.82	+	779.844i	−1.19024	+	0.811492i	−0.985888	−	0.167406i	\(-0.946461\pi\)
							−0.204352	+	0.978897i	\(0.565509\pi\)
\(37\)	202.745	+	117.055i	0.148097	+	0.0855040i	0.572218	−	0.820102i	\(-0.306083\pi\)
							−0.424120	+	0.905606i	\(0.639417\pi\)
\(41\)	−142.369	−	68.5614i	−0.0846932	−	0.0407861i	0.391058	−	0.920366i	\(-0.372109\pi\)
							−0.475751	+	0.879580i	\(0.657824\pi\)
\(43\)	1243.84	−	1368.09i	0.672710	−	0.739907i
							\(47\)	−2003.01	−	2511.69i	−0.906749	−	1.13703i	−0.990080	−	0.140505i	\(-0.955127\pi\)
0.0833312	−	0.996522i	\(-0.473444\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.3.10	✓	168
43.29	odd	42	inner	43.5.h.a.29.10	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.3.10	✓	168	1.1	even	1	trivial
43.5.h.a.29.10	yes	168	43.29	odd	42	inner