Embedded newform 40.6.k.a.3.9

• Label: 40.6.k.a.3.9
• Level: $40$
• Weight: $6$
• Character: 40.3
• Analytic conductor: $6.415$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	40.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41535279252\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3.9
Character	\(\chi\)	\(=\)	40.3
Dual form			40.6.k.a.27.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.81045 + 4.18096i) q^{2} +(10.0657 + 10.0657i) q^{3} +(-2.96087 - 31.8627i) q^{4} +(23.9125 + 50.5291i) q^{5} +(-80.4392 + 3.72940i) q^{6} +(96.6855 + 96.6855i) q^{7} +(144.499 + 109.032i) q^{8} -40.3635i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 2 q^{2} - 4 q^{3} + 176 q^{6} + 244 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(21\)	\(31\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.81045	+	4.18096i	−0.673600	+	0.739096i
							\(3\)	10.0657	+	10.0657i	0.645715	+	0.645715i	0.951954	−	0.306240i	\(-0.0990709\pi\)
−0.306240	+	0.951954i	\(0.599071\pi\)
\(5\)	23.9125	+	50.5291i	0.427760	+	0.903892i
							\(7\)	96.6855	+	96.6855i	0.745790	+	0.745790i	0.973686	−	0.227896i	\(-0.0731845\pi\)
−0.227896	+	0.973686i	\(0.573185\pi\)
\(11\)	−283.587			−0.706650			−0.353325	−	0.935501i	\(-0.614949\pi\)
							−0.353325	+	0.935501i	\(0.614949\pi\)
\(13\)	−699.519	+	699.519i	−1.14800	+	1.14800i	−0.161052	+	0.986946i	\(0.551489\pi\)
							−0.986946	+	0.161052i	\(0.948511\pi\)
\(17\)	399.025	−	399.025i	0.334871	−	0.334871i	−0.519562	−	0.854433i	\(-0.673905\pi\)
							0.854433	+	0.519562i	\(0.173905\pi\)
\(19\)			338.083i			0.214852i	0.994213	+	0.107426i	\(0.0342609\pi\)
							−0.994213	+	0.107426i	\(0.965739\pi\)
\(23\)	594.687	−	594.687i	0.234406	−	0.234406i	−0.580123	−	0.814529i	\(-0.696995\pi\)
							0.814529	+	0.580123i	\(0.196995\pi\)
\(29\)	5582.41			1.23261			0.616306	−	0.787507i	\(-0.288628\pi\)
							0.616306	+	0.787507i	\(0.288628\pi\)
\(31\)			144.182i			0.0269468i	0.999909	+	0.0134734i	\(0.00428884\pi\)
							−0.999909	+	0.0134734i	\(0.995711\pi\)
\(37\)	11418.4	+	11418.4i	1.37120	+	1.37120i	0.858674	+	0.512523i	\(0.171289\pi\)
							0.512523	+	0.858674i	\(0.328711\pi\)
\(41\)	11892.0			1.10483			0.552413	−	0.833570i	\(-0.313707\pi\)
							0.552413	+	0.833570i	\(0.313707\pi\)
\(43\)	−8345.23	−	8345.23i	−0.688283	−	0.688283i	0.273569	−	0.961852i	\(-0.411796\pi\)
							−0.961852	+	0.273569i	\(0.911796\pi\)
\(47\)	−17955.5	−	17955.5i	−1.18564	−	1.18564i	−0.978260	−	0.207383i	\(-0.933505\pi\)
							−0.207383	−	0.978260i	\(-0.566495\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.6.k.a.3.9	yes	56
4.3	odd	2		160.6.o.a.143.10		56
5.2	odd	4	inner	40.6.k.a.27.7	yes	56
8.3	odd	2	inner	40.6.k.a.3.7	✓	56
8.5	even	2		160.6.o.a.143.9		56
20.7	even	4		160.6.o.a.47.9		56
40.27	even	4	inner	40.6.k.a.27.9	yes	56
40.37	odd	4		160.6.o.a.47.10		56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.k.a.3.7	✓	56	8.3	odd	2	inner
40.6.k.a.3.9	yes	56	1.1	even	1	trivial
40.6.k.a.27.7	yes	56	5.2	odd	4	inner
40.6.k.a.27.9	yes	56	40.27	even	4	inner
160.6.o.a.47.9		56	20.7	even	4
160.6.o.a.47.10		56	40.37	odd	4
160.6.o.a.143.9		56	8.5	even	2
160.6.o.a.143.10		56	4.3	odd	2