Embedded newform 43.7.h.a.18.17

• Label: 43.7.h.a.18.17
• 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			18.17
Character	\(\chi\)	\(=\)	43.18
Dual form			43.7.h.a.12.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.91090 - 8.12108i) q^{2} +(-30.1530 + 44.2263i) q^{3} +(-10.7533 - 13.4843i) q^{4} +(-104.403 - 112.520i) q^{5} +(241.240 + 417.840i) q^{6} +(148.599 + 85.7937i) q^{7} +(410.852 - 93.7743i) q^{8} +(-780.431 - 1988.50i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.91090	−	8.12108i	0.488863	−	1.01513i	−0.499961	−	0.866048i	\(-0.666652\pi\)
							0.988824	−	0.149087i	\(-0.0476333\pi\)
\(3\)	−30.1530	+	44.2263i	−1.11678	+	1.63801i	−0.449302	+	0.893380i	\(0.648327\pi\)
							−0.667475	+	0.744632i	\(0.732625\pi\)
\(5\)	−104.403	−	112.520i	−0.835228	−	0.900162i	0.160999	−	0.986955i	\(-0.448529\pi\)
							−0.996226	+	0.0867931i	\(0.972338\pi\)
\(7\)	148.599	+	85.7937i	0.433233	+	0.250127i	0.700723	−	0.713433i	\(-0.252861\pi\)
							−0.267490	+	0.963561i	\(0.586194\pi\)
\(11\)	1378.83	−	1729.00i	1.03594	−	1.29902i	0.0827727	−	0.996568i	\(-0.473622\pi\)
							0.953164	−	0.302455i	\(-0.0978061\pi\)
\(13\)	−1260.07	−	388.679i	−0.573539	−	0.176913i	−0.00560050	−	0.999984i	\(-0.501783\pi\)
							−0.567939	+	0.823071i	\(0.692259\pi\)
\(17\)	3903.64	+	3622.04i	0.794552	+	0.737237i	0.968901	−	0.247447i	\(-0.0795916\pi\)
							−0.174349	+	0.984684i	\(0.555782\pi\)
\(19\)	−5127.30	−	2012.32i	−0.747529	−	0.293384i	−0.0391670	−	0.999233i	\(-0.512470\pi\)
							−0.708362	+	0.705849i	\(0.750566\pi\)
\(23\)	−7171.21	−	1080.89i	−0.589398	−	0.0888375i	−0.152430	−	0.988314i	\(-0.548710\pi\)
							−0.436968	+	0.899477i	\(0.643948\pi\)
\(29\)	−17953.4	−	26332.8i	−0.736126	−	1.07970i	−0.993549	−	0.113400i	\(-0.963826\pi\)
							0.257423	−	0.966299i	\(-0.417127\pi\)
\(31\)	−1756.55	−	23439.6i	−0.0589625	−	0.786800i	−0.945806	−	0.324731i	\(-0.894726\pi\)
							0.886844	−	0.462069i	\(-0.152893\pi\)
\(37\)	−2691.91	+	1554.18i	−0.0531442	+	0.0306828i	−0.526337	−	0.850276i	\(-0.676435\pi\)
							0.473192	+	0.880959i	\(0.343102\pi\)
\(41\)	30047.2	+	14470.0i	0.435965	+	0.209950i	0.638976	−	0.769227i	\(-0.279358\pi\)
							−0.203011	+	0.979177i	\(0.565073\pi\)
\(43\)	28381.4	−	74268.8i	0.356967	−	0.934117i
							\(47\)	−5749.87	−	7210.11i	−0.0553815	−	0.0694462i	0.753369	−	0.657598i	\(-0.228427\pi\)
−0.808751	+	0.588151i	\(0.799856\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.18.17	yes	252
43.12	odd	42	inner	43.7.h.a.12.17	✓	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.12.17	✓	252	43.12	odd	42	inner
43.7.h.a.18.17	yes	252	1.1	even	1	trivial