Embedded newform 43.9.h.a.12.10

• Label: 43.9.h.a.12.10
• Level: $43$
• Weight: $9$
• Character: 43.12
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(28\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			12.10
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.82965 - 10.0289i) q^{2} +(11.1776 + 16.3945i) q^{3} +(82.3606 - 103.277i) q^{4} +(-812.927 + 876.127i) q^{5} +(110.435 - 191.278i) q^{6} +(876.930 - 506.296i) q^{7} +(-4211.67 - 961.286i) q^{8} +(2253.16 - 5740.96i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.82965	−	10.0289i	−0.301853	−	0.626805i	0.693777	−	0.720190i	\(-0.255945\pi\)
							−0.995630	+	0.0933854i	\(0.970231\pi\)
\(3\)	11.1776	+	16.3945i	0.137995	+	0.202401i	0.889015	−	0.457878i	\(-0.151390\pi\)
							−0.751020	+	0.660279i	\(0.770438\pi\)
\(5\)	−812.927	+	876.127i	−1.30068	+	1.40180i	−0.435627	+	0.900127i	\(0.643473\pi\)
							−0.865056	+	0.501675i	\(0.832717\pi\)
\(7\)	876.930	−	506.296i	0.365235	−	0.210869i	−0.306139	−	0.951987i	\(-0.599037\pi\)
							0.671375	+	0.741118i	\(0.265704\pi\)
\(11\)	9874.37	+	12382.1i	0.674433	+	0.845712i	0.994828	−	0.101572i	\(-0.0323872\pi\)
							−0.320395	+	0.947284i	\(0.603816\pi\)
\(13\)	8504.86	−	2623.40i	0.297779	−	0.0918525i	−0.142266	−	0.989828i	\(-0.545439\pi\)
							0.440045	+	0.897976i	\(0.354963\pi\)
\(17\)	89897.8	−	83412.9i	1.07635	−	0.998706i	0.0763504	−	0.997081i	\(-0.475673\pi\)
							0.999999	−	0.00162496i	\(-0.000517242\pi\)
\(19\)	−71186.7	+	27938.7i	−0.546242	+	0.214384i	−0.622383	−	0.782713i	\(-0.713835\pi\)
							0.0761412	+	0.997097i	\(0.475740\pi\)
\(23\)	419725.	−	63263.4i	1.49987	−	0.226069i	0.652791	−	0.757538i	\(-0.273598\pi\)
							0.847078	+	0.531469i	\(0.178360\pi\)
\(29\)	228030.	−	334458.i	0.322403	−	0.472879i	−0.630485	−	0.776202i	\(-0.717144\pi\)
							0.952888	+	0.303323i	\(0.0980962\pi\)
\(31\)	100427.	−	1.34011e6i	0.108744	−	1.45109i	−0.629735	−	0.776810i	\(-0.716836\pi\)
							0.738479	−	0.674277i	\(-0.235545\pi\)
\(37\)	−1.17398e6	−	677797.i	−0.626402	−	0.361654i	0.152955	−	0.988233i	\(-0.451121\pi\)
							−0.779358	+	0.626580i	\(0.784454\pi\)
\(41\)	2.42297e6	−	1.16684e6i	0.857459	−	0.412931i	0.0471183	−	0.998889i	\(-0.484996\pi\)
							0.810341	+	0.585959i	\(0.199282\pi\)
\(43\)	1.87747e6	+	2.85715e6i	0.549159	+	0.835718i
							\(47\)	−1.66207e6	+	2.08417e6i	−0.340610	+	0.427111i	−0.922405	−	0.386224i	\(-0.873779\pi\)
0.581795	+	0.813335i	\(0.302351\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.10	✓	336
43.18	odd	42	inner	43.9.h.a.18.10	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.10	✓	336	1.1	even	1	trivial
43.9.h.a.18.10	yes	336	43.18	odd	42	inner