Embedded newform 43.7.h.a.3.7

• Label: 43.7.h.a.3.7
• Level: $43$
• Weight: $7$
• Character: 43.3
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $252$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(252\)
Relative dimension:	\(21\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			3.7
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.62546 + 5.45182i) q^{2} +(-13.6379 - 1.02202i) q^{3} +(17.0740 + 21.4102i) q^{4} +(29.1602 - 94.5350i) q^{5} +(41.3776 - 71.6680i) q^{6} +(-270.866 + 156.384i) q^{7} +(-539.110 + 123.048i) q^{8} +(-535.910 - 80.7755i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.62546	+	5.45182i	−0.328182	+	0.681478i	−0.998143	−	0.0609193i	\(-0.980597\pi\)
							0.669960	+	0.742397i	\(0.266311\pi\)
\(3\)	−13.6379	−	1.02202i	−0.505107	−	0.0378525i	−0.180260	−	0.983619i	\(-0.557694\pi\)
							−0.324847	+	0.945767i	\(0.605313\pi\)
\(5\)	29.1602	−	94.5350i	0.233281	−	0.756280i	−0.760930	−	0.648835i	\(-0.775257\pi\)
							0.994211	−	0.107445i	\(-0.0342671\pi\)
\(7\)	−270.866	+	156.384i	−0.789696	+	0.455931i	−0.839855	−	0.542810i	\(-0.817360\pi\)
							0.0501597	+	0.998741i	\(0.484027\pi\)
\(11\)	1164.50	−	1460.24i	0.874908	−	1.09710i	−0.119639	−	0.992817i	\(-0.538174\pi\)
							0.994548	−	0.104283i	\(-0.0332548\pi\)
\(13\)	2316.92	−	2149.78i	1.05458	−	0.978509i	0.0547814	−	0.998498i	\(-0.482554\pi\)
							0.999800	+	0.0199897i	\(0.00636334\pi\)
\(17\)	308.693	−	95.2193i	0.0628319	−	0.0193811i	−0.263180	−	0.964747i	\(-0.584771\pi\)
							0.326012	+	0.945366i	\(0.394295\pi\)
\(19\)	−472.485	−	3134.73i	−0.0688853	−	0.457024i	−0.996507	−	0.0835043i	\(-0.973389\pi\)
							0.927622	−	0.373520i	\(-0.121849\pi\)
\(23\)	−8278.49	−	21093.2i	−0.680405	−	1.73364i	−0.679534	−	0.733644i	\(-0.737818\pi\)
							−0.000870972	−	1.00000i	\(-0.500277\pi\)
\(29\)	−32028.0	+	2400.17i	−1.31322	+	0.0984120i	−0.712852	−	0.701314i	\(-0.752597\pi\)
							−0.600365	+	0.799726i	\(0.704978\pi\)
\(31\)	9173.44	−	6254.35i	0.307927	−	0.209941i	−0.399490	−	0.916737i	\(-0.630813\pi\)
							0.707417	+	0.706797i	\(0.249860\pi\)
\(37\)	−51438.9	−	29698.2i	−1.01551	−	0.586308i	−0.102713	−	0.994711i	\(-0.532752\pi\)
							−0.912802	+	0.408403i	\(0.866086\pi\)
\(41\)	62695.9	+	30192.8i	0.909678	+	0.438078i	0.829375	−	0.558693i	\(-0.188697\pi\)
							0.0803033	+	0.996770i	\(0.474411\pi\)
\(43\)	−60173.2	+	51966.8i	−0.756829	+	0.653612i
							\(47\)	−78287.3	−	98169.1i	−0.754045	−	0.945543i	0.245671	−	0.969353i	\(-0.420992\pi\)
−0.999716	+	0.0238101i	\(0.992420\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.3.7	✓	252
43.29	odd	42	inner	43.7.h.a.29.7	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.7	✓	252	1.1	even	1	trivial
43.7.h.a.29.7	yes	252	43.29	odd	42	inner