Embedded newform 43.7.h.a.3.10

• Label: 43.7.h.a.3.10
• Level: $43$
• Weight: $7$
• Character: 43.3
• 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			3.10
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.806603 + 1.67493i) q^{2} +(-16.6260 - 1.24594i) q^{3} +(37.7486 + 47.3352i) q^{4} +(-23.7056 + 76.8517i) q^{5} +(15.4974 - 26.8423i) q^{6} +(289.941 - 167.398i) q^{7} +(-225.726 + 51.5205i) q^{8} +(-445.988 - 67.2218i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.806603	+	1.67493i	−0.100825	+	0.209366i	−0.945279	−	0.326262i	\(-0.894211\pi\)
							0.844454	+	0.535628i	\(0.179925\pi\)
\(3\)	−16.6260	−	1.24594i	−0.615776	−	0.0461460i	−0.236809	−	0.971556i	\(-0.576102\pi\)
							−0.378967	+	0.925410i	\(0.623721\pi\)
\(5\)	−23.7056	+	76.8517i	−0.189645	+	0.614814i	0.809954	+	0.586493i	\(0.199492\pi\)
							−0.999599	+	0.0283203i	\(0.990984\pi\)
\(7\)	289.941	−	167.398i	0.845310	−	0.488040i	−0.0137555	−	0.999905i	\(-0.504379\pi\)
							0.859066	+	0.511865i	\(0.171045\pi\)
\(11\)	−36.3774	+	45.6158i	−0.0273309	+	0.0342719i	−0.795309	−	0.606204i	\(-0.792691\pi\)
							0.767978	+	0.640476i	\(0.221263\pi\)
\(13\)	−2474.56	+	2296.06i	−1.12634	+	1.04509i	−0.127756	+	0.991806i	\(0.540777\pi\)
							−0.998580	+	0.0532809i	\(0.983032\pi\)
\(17\)	−7109.83	+	2193.09i	−1.44715	+	0.446385i	−0.916097	−	0.400956i	\(-0.868678\pi\)
							−0.531049	+	0.847341i	\(0.678202\pi\)
\(19\)	−1128.06	−	7484.19i	−0.164464	−	1.09115i	−0.906935	−	0.421270i	\(-0.861584\pi\)
							0.742471	−	0.669878i	\(-0.233654\pi\)
\(23\)	−3524.13	−	8979.34i	−0.289647	−	0.738007i	−0.999436	−	0.0335931i	\(-0.989305\pi\)
							0.709789	−	0.704414i	\(-0.248790\pi\)
\(29\)	−24570.7	+	1841.32i	−1.00745	+	0.0754980i	−0.568220	−	0.822876i	\(-0.692368\pi\)
							−0.439230	+	0.898374i	\(0.644749\pi\)
\(31\)	−20523.8	+	13992.9i	−0.688927	+	0.469703i	−0.856493	−	0.516158i	\(-0.827362\pi\)
							0.167566	+	0.985861i	\(0.446409\pi\)
\(37\)	75584.9	+	43638.9i	1.49221	+	0.861527i	0.999960	−	0.00892726i	\(-0.00284167\pi\)
							0.492249	+	0.870455i	\(0.336175\pi\)
\(41\)	45983.7	+	22144.6i	0.667195	+	0.321304i	0.736643	−	0.676282i	\(-0.236410\pi\)
							−0.0694486	+	0.997586i	\(0.522124\pi\)
\(43\)	−6895.84	−	79207.4i	−0.0867325	−	0.996232i
							\(47\)	112495.	+	141064.i	1.08352	+	1.35869i	0.928734	+	0.370747i	\(0.120898\pi\)
0.154789	+	0.987948i	\(0.450530\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.3.10	✓	252
43.29	odd	42	inner	43.7.h.a.29.10	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.10	✓	252	1.1	even	1	trivial
43.7.h.a.29.10	yes	252	43.29	odd	42	inner