Embedded newform 43.9.h.a.18.20

• Label: 43.9.h.a.18.20
• Level: $43$
• Weight: $9$
• Character: 43.18
• 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			18.20
Character	\(\chi\)	\(=\)	43.18
Dual form			43.9.h.a.12.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.04661 - 10.4794i) q^{2} +(-42.0771 + 61.7157i) q^{3} +(75.2641 + 94.3782i) q^{4} +(-348.351 - 375.433i) q^{5} +(434.397 + 752.397i) q^{6} +(-3116.27 - 1799.18i) q^{7} +(4271.80 - 975.010i) q^{8} +(358.651 + 913.828i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.04661	−	10.4794i	0.315413	−	0.654962i	−0.681639	−	0.731688i	\(-0.738733\pi\)
							0.997052	+	0.0767267i	\(0.0244469\pi\)
\(3\)	−42.0771	+	61.7157i	−0.519470	+	0.761923i	−0.992906	−	0.118906i	\(-0.962061\pi\)
							0.473435	+	0.880829i	\(0.343014\pi\)
\(5\)	−348.351	−	375.433i	−0.557362	−	0.600694i	0.390017	−	0.920808i	\(-0.372469\pi\)
							−0.947379	+	0.320114i	\(0.896279\pi\)
\(7\)	−3116.27	−	1799.18i	−1.29790	−	0.749346i	−0.317863	−	0.948137i	\(-0.602965\pi\)
							−0.980042	+	0.198791i	\(0.936299\pi\)
\(11\)	8149.30	−	10218.9i	0.556608	−	0.697965i	−0.421318	−	0.906913i	\(-0.638433\pi\)
							0.977927	+	0.208948i	\(0.0670040\pi\)
\(13\)	41025.5	+	12654.7i	1.43642	+	0.443076i	0.912635	−	0.408776i	\(-0.134044\pi\)
							0.523782	+	0.851852i	\(0.324521\pi\)
\(17\)	53655.0	+	49784.6i	0.642413	+	0.596072i	0.932606	−	0.360896i	\(-0.117529\pi\)
							−0.290193	+	0.956968i	\(0.593719\pi\)
\(19\)	122722.	+	48164.7i	0.941687	+	0.369585i	0.786004	−	0.618222i	\(-0.212147\pi\)
							0.155684	+	0.987807i	\(0.450242\pi\)
\(23\)	208155.	+	31374.3i	0.743834	+	0.112115i	0.510015	−	0.860165i	\(-0.329640\pi\)
							0.233818	+	0.972280i	\(0.424878\pi\)
\(29\)	−82894.2	−	121583.i	−0.117201	−	0.171903i	0.763155	−	0.646215i	\(-0.223649\pi\)
							−0.880357	+	0.474312i	\(0.842697\pi\)
\(31\)	−130657.	−	1.74349e6i	−0.141477	−	1.88787i	−0.392293	−	0.919840i	\(-0.628318\pi\)
							0.250817	−	0.968035i	\(-0.419301\pi\)
\(37\)	−32835.0	+	18957.3i	−0.0175198	+	0.0101151i	−0.508734	−	0.860924i	\(-0.669886\pi\)
							0.491214	+	0.871039i	\(0.336553\pi\)
\(41\)	3.52647e6	+	1.69826e6i	1.24797	+	0.600992i	0.936968	−	0.349416i	\(-0.113620\pi\)
							0.311005	+	0.950408i	\(0.399334\pi\)
\(43\)	2.77108e6	+	2.00233e6i	0.810540	+	0.585683i
							\(47\)	−2.91175e6	−	3.65122e6i	−0.596709	−	0.748249i	0.388152	−	0.921595i	\(-0.373114\pi\)
−0.984861	+	0.173346i	\(0.944542\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.18.20	yes	336
43.12	odd	42	inner	43.9.h.a.12.20	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.20	✓	336	43.12	odd	42	inner
43.9.h.a.18.20	yes	336	1.1	even	1	trivial