Embedded newform 43.9.h.a.3.15

• Label: 43.9.h.a.3.15
• 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			3.15
Character	\(\chi\)	\(=\)	43.3
Dual form			43.9.h.a.29.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.864743 - 1.79566i) q^{2} +(1.37697 + 0.103190i) q^{3} +(157.137 + 197.043i) q^{4} +(286.537 - 928.929i) q^{5} +(1.37602 - 2.38334i) q^{6} +(941.195 - 543.399i) q^{7} +(987.128 - 225.306i) q^{8} +(-6485.83 - 977.582i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.864743	−	1.79566i	0.0540464	−	0.112229i	−0.872196	−	0.489156i	\(-0.837305\pi\)
							0.926243	+	0.376927i	\(0.123019\pi\)
\(3\)	1.37697	+	0.103190i	0.0169997	+	0.00127395i	0.0832272	−	0.996531i	\(-0.473477\pi\)
							−0.0662275	+	0.997805i	\(0.521096\pi\)
\(5\)	286.537	−	928.929i	0.458459	−	1.48629i	−0.371485	−	0.928439i	\(-0.621151\pi\)
							0.829943	−	0.557848i	\(-0.188373\pi\)
\(7\)	941.195	−	543.399i	0.392001	−	0.226322i	−0.291026	−	0.956715i	\(-0.593997\pi\)
							0.683027	+	0.730393i	\(0.260663\pi\)
\(11\)	−7182.96	+	9007.15i	−0.490606	+	0.615201i	−0.964082	−	0.265606i	\(-0.914428\pi\)
							0.473475	+	0.880807i	\(0.342999\pi\)
\(13\)	33675.0	−	31245.8i	1.17905	−	1.09400i	0.185268	−	0.982688i	\(-0.440685\pi\)
							0.993785	−	0.111314i	\(-0.0355060\pi\)
\(17\)	51966.0	−	16029.4i	0.622191	−	0.191921i	0.0323982	−	0.999475i	\(-0.489686\pi\)
							0.589793	+	0.807554i	\(0.299209\pi\)
\(19\)	−17421.6	−	115584.i	−0.133682	−	0.886921i	−0.950656	−	0.310247i	\(-0.899588\pi\)
							0.816974	−	0.576674i	\(-0.195650\pi\)
\(23\)	−183414.	−	467331.i	−0.655422	−	1.66999i	−0.739186	−	0.673502i	\(-0.764789\pi\)
							0.0837642	−	0.996486i	\(-0.473306\pi\)
\(29\)	524135.	−	39278.5i	0.741056	−	0.0555344i	0.301148	−	0.953577i	\(-0.402630\pi\)
							0.439908	+	0.898043i	\(0.355011\pi\)
\(31\)	707945.	−	482669.i	0.766572	−	0.522640i	−0.115755	−	0.993278i	\(-0.536929\pi\)
							0.882326	+	0.470638i	\(0.155976\pi\)
\(37\)	3.01661e6	+	1.74164e6i	1.60958	+	0.929291i	0.989464	+	0.144781i	\(0.0462479\pi\)
							0.620116	+	0.784510i	\(0.287085\pi\)
\(41\)	−3.60263e6	−	1.73494e6i	−1.27492	−	0.613971i	−0.330844	−	0.943686i	\(-0.607333\pi\)
							−0.944081	+	0.329714i	\(0.893048\pi\)
\(43\)	2.77353e6	−	1.99893e6i	0.811258	−	0.584688i
							\(47\)	−248949.	−	312172.i	−0.0510175	−	0.0639739i	0.755668	−	0.654955i	\(-0.227312\pi\)
−0.806686	+	0.590981i	\(0.798741\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.3.15	✓	336
43.29	odd	42	inner	43.9.h.a.29.15	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.3.15	✓	336	1.1	even	1	trivial
43.9.h.a.29.15	yes	336	43.29	odd	42	inner