Embedded newform 43.9.h.a.12.21

• Label: 43.9.h.a.12.21
• 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			12.21
Character	\(\chi\)	\(=\)	43.12
Dual form			43.9.h.a.18.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.11071 + 12.6890i) q^{2} +(70.0091 + 102.685i) q^{3} +(35.9431 - 45.0712i) q^{4} +(610.716 - 658.195i) q^{5} +(-875.160 + 1515.82i) q^{6} +(-3108.29 + 1794.57i) q^{7} +(4306.59 + 982.951i) q^{8} +(-3245.83 + 8270.25i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 14 q^{2} + 70 q^{3} + 6460 q^{4} - 11 q^{5} - 1798 q^{6} + 5010 q^{7} + 5362 q^{8} - 40176 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.11071	+	12.6890i	0.381919	+	0.793063i	0.999976	+	0.00692991i	\(0.00220588\pi\)
							−0.618057	+	0.786133i	\(0.712080\pi\)
\(3\)	70.0091	+	102.685i	0.864310	+	1.26771i	0.962153	+	0.272509i	\(0.0878533\pi\)
							−0.0978430	+	0.995202i	\(0.531194\pi\)
\(5\)	610.716	−	658.195i	0.977145	−	1.05311i	−0.0214312	−	0.999770i	\(-0.506822\pi\)
							0.998576	−	0.0533416i	\(-0.0169872\pi\)
\(7\)	−3108.29	+	1794.57i	−1.29458	+	0.747427i	−0.979463	−	0.201625i	\(-0.935378\pi\)
							−0.315119	+	0.949052i	\(0.602045\pi\)
\(11\)	11515.8	+	14440.3i	0.786543	+	0.986293i	0.999956	+	0.00933130i	\(0.00297029\pi\)
							−0.213414	+	0.976962i	\(0.568458\pi\)
\(13\)	13207.0	−	4073.83i	0.462415	−	0.142636i	−0.0547895	−	0.998498i	\(-0.517449\pi\)
							0.517205	+	0.855862i	\(0.326973\pi\)
\(17\)	−103768.	+	96283.0i	−1.24242	+	1.15280i	−0.260105	+	0.965580i	\(0.583757\pi\)
							−0.982318	+	0.187220i	\(0.940052\pi\)
\(19\)	7237.42	−	2840.48i	0.0555353	−	0.0217960i	−0.337411	−	0.941358i	\(-0.609551\pi\)
							0.392946	+	0.919562i	\(0.371456\pi\)
\(23\)	347562.	−	52386.6i	1.24200	−	0.187201i	0.505015	−	0.863111i	\(-0.331487\pi\)
							0.736985	+	0.675909i	\(0.236249\pi\)
\(29\)	−157663.	+	231250.i	−0.222915	+	0.326956i	−0.921362	−	0.388706i	\(-0.872922\pi\)
							0.698447	+	0.715661i	\(0.253875\pi\)
\(31\)	109203.	−	1.45721e6i	0.118246	−	1.57788i	−0.550783	−	0.834648i	\(-0.685671\pi\)
							0.669029	−	0.743236i	\(-0.266710\pi\)
\(37\)	−3.00130e6	−	1.73280e6i	−1.60141	−	0.924573i	−0.991206	−	0.132328i	\(-0.957755\pi\)
							−0.610202	−	0.792246i	\(-0.708912\pi\)
\(41\)	1.28558e6	−	619103.i	0.454951	−	0.219093i	−0.192352	−	0.981326i	\(-0.561611\pi\)
							0.647302	+	0.762233i	\(0.275897\pi\)
\(43\)	−1.80849e6	−	2.90130e6i	−0.528984	−	0.848632i
							\(47\)	−67413.2	+	84533.4i	−0.0138151	+	0.0173236i	−0.788691	−	0.614790i	\(-0.789241\pi\)
0.774876	+	0.632113i	\(0.217812\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.h.a.12.21	✓	336
43.18	odd	42	inner	43.9.h.a.18.21	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.h.a.12.21	✓	336	1.1	even	1	trivial
43.9.h.a.18.21	yes	336	43.18	odd	42	inner