Embedded newform 43.9.h.a.3.10

• Label: 43.9.h.a.3.10
• Level: $43$
• Weight: $9$
• Character: 43.3
• 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			3.10
Character	\(\chi\)	\(=\)	43.3
Dual form			43.9.h.a.29.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.92801 + 12.3096i) q^{2} +(-79.6637 - 5.96997i) q^{3} +(43.2276 + 54.2057i) q^{4} +(-75.1797 + 243.727i) q^{5} +(545.735 - 945.241i) q^{6} +(-916.434 + 529.103i) q^{7} +(-4333.45 + 989.083i) q^{8} +(-177.047 - 26.6855i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.92801	+	12.3096i	−0.370500	+	0.769352i	−0.999971	−	0.00767267i	\(-0.997558\pi\)
							0.629470	+	0.777025i	\(0.283272\pi\)
\(3\)	−79.6637	−	5.96997i	−0.983503	−	0.0737034i	−0.426740	−	0.904374i	\(-0.640338\pi\)
							−0.556763	+	0.830671i	\(0.687957\pi\)
\(5\)	−75.1797	+	243.727i	−0.120288	+	0.389963i	−0.995612	−	0.0935725i	\(-0.970171\pi\)
							0.875325	+	0.483535i	\(0.160647\pi\)
\(7\)	−916.434	+	529.103i	−0.381688	+	0.220368i	−0.678553	−	0.734552i	\(-0.737392\pi\)
							0.296864	+	0.954920i	\(0.404059\pi\)
\(11\)	−7545.68	+	9461.99i	−0.515380	+	0.646267i	−0.969621	−	0.244613i	\(-0.921339\pi\)
							0.454241	+	0.890879i	\(0.349911\pi\)
\(13\)	−14000.3	+	12990.4i	−0.490191	+	0.454831i	−0.886124	−	0.463449i	\(-0.846612\pi\)
							0.395933	+	0.918280i	\(0.370421\pi\)
\(17\)	74750.5	−	23057.5i	0.894991	−	0.276068i	0.187059	−	0.982349i	\(-0.440105\pi\)
							0.707932	+	0.706281i	\(0.249628\pi\)
\(19\)	−17904.0	−	118785.i	−0.137384	−	0.911481i	−0.946238	−	0.323471i	\(-0.895150\pi\)
							0.808854	−	0.588009i	\(-0.200088\pi\)
\(23\)	−40034.0	−	102005.i	−0.143060	−	0.364511i	0.841328	−	0.540526i	\(-0.181775\pi\)
							−0.984388	+	0.176015i	\(0.943679\pi\)
\(29\)	970316.	−	72715.1i	1.37190	−	0.102809i	0.631706	−	0.775208i	\(-0.282355\pi\)
							0.740189	+	0.672398i	\(0.234736\pi\)
\(31\)	−109861.	+	74902.3i	−0.118959	+	0.0811051i	−0.621338	−	0.783543i	\(-0.713411\pi\)
							0.502379	+	0.864648i	\(0.332458\pi\)
\(37\)	−2.66618e6	−	1.53932e6i	−1.42260	−	0.821338i	−0.426079	−	0.904686i	\(-0.640105\pi\)
							−0.996521	+	0.0833480i	\(0.973439\pi\)
\(41\)	−245001.	−	117986.i	−0.0867026	−	0.0417538i	0.390030	−	0.920802i	\(-0.372465\pi\)
							−0.476732	+	0.879048i	\(0.658179\pi\)
\(43\)	1.76483e6	−	2.92806e6i	0.516214	−	0.856459i
							\(47\)	−1.52479e6	−	1.91202e6i	−0.312477	−	0.391834i	0.600648	−	0.799514i	\(-0.294909\pi\)
−0.913125	+	0.407680i	\(0.866338\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.3.10	✓	336
43.29	odd	42	inner	43.9.h.a.29.10	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.3.10	✓	336	1.1	even	1	trivial
43.9.h.a.29.10	yes	336	43.29	odd	42	inner