Embedded newform 43.7.h.a.3.12

• Label: 43.7.h.a.3.12
• 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			3.12
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.05720 - 2.19531i) q^{2} +(-37.2864 - 2.79423i) q^{3} +(36.2017 + 45.3954i) q^{4} +(67.3700 - 218.408i) q^{5} +(-45.5535 + 78.9010i) q^{6} +(-230.704 + 133.197i) q^{7} +(289.962 - 66.1820i) q^{8} +(661.611 + 99.7219i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.05720	−	2.19531i	0.132150	−	0.274413i	−0.824385	−	0.566030i	\(-0.808479\pi\)
							0.956535	+	0.291616i	\(0.0941931\pi\)
\(3\)	−37.2864	−	2.79423i	−1.38098	−	0.103490i	−0.636572	−	0.771217i	\(-0.719648\pi\)
							−0.744406	+	0.667727i	\(0.767267\pi\)
\(5\)	67.3700	−	218.408i	0.538960	−	1.74726i	−0.117671	−	0.993053i	\(-0.537543\pi\)
							0.656631	−	0.754212i	\(-0.271981\pi\)
\(7\)	−230.704	+	133.197i	−0.672606	+	0.388329i	−0.797063	−	0.603896i	\(-0.793614\pi\)
							0.124457	+	0.992225i	\(0.460281\pi\)
\(11\)	−1427.55	+	1790.10i	−1.07254	+	1.34493i	−0.137456	+	0.990508i	\(0.543893\pi\)
							−0.935087	+	0.354418i	\(0.884679\pi\)
\(13\)	−825.797	+	766.228i	−0.375875	+	0.348761i	−0.845377	−	0.534171i	\(-0.820624\pi\)
							0.469502	+	0.882932i	\(0.344434\pi\)
\(17\)	−3041.86	+	938.288i	−0.619144	+	0.190981i	−0.588431	−	0.808548i	\(-0.700254\pi\)
							−0.0307134	+	0.999528i	\(0.509778\pi\)
\(19\)	192.837	+	1279.39i	0.0281145	+	0.186527i	0.998480	−	0.0551141i	\(-0.0175523\pi\)
							−0.970366	+	0.241642i	\(0.922314\pi\)
\(23\)	2887.59	+	7357.45i	0.237329	+	0.604705i	0.999067	−	0.0431800i	\(-0.0137489\pi\)
							−0.761738	+	0.647885i	\(0.775654\pi\)
\(29\)	−7708.60	+	577.680i	−0.316069	+	0.0236861i	−0.231820	−	0.972759i	\(-0.574468\pi\)
							−0.0842487	+	0.996445i	\(0.526849\pi\)
\(31\)	−22037.9	+	15025.2i	−0.739750	+	0.504353i	−0.873598	−	0.486649i	\(-0.838219\pi\)
							0.133847	+	0.991002i	\(0.457267\pi\)
\(37\)	−85952.6	−	49624.7i	−1.69689	−	0.979700i	−0.948680	−	0.316239i	\(-0.897580\pi\)
							−0.748211	−	0.663461i	\(-0.769087\pi\)
\(41\)	78900.8	+	37996.6i	1.14480	+	0.551307i	0.907468	−	0.420122i	\(-0.138013\pi\)
							0.237332	+	0.971429i	\(0.423727\pi\)
\(43\)	51884.6	−	60244.1i	0.652579	−	0.757721i
							\(47\)	−46978.1	−	58908.7i	−0.452483	−	0.567396i	0.502302	−	0.864692i	\(-0.332486\pi\)
−0.954785	+	0.297296i	\(0.903915\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.3.12	✓	252
43.29	odd	42	inner	43.7.h.a.29.12	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.12	✓	252	1.1	even	1	trivial
43.7.h.a.29.12	yes	252	43.29	odd	42	inner