Embedded newform 40.3.i.a.13.2

• Label: 40.3.i.a.13.2
• Level: $40$
• Weight: $3$
• Character: 40.13
• Analytic conductor: $1.090$
• Analytic rank: $0$
• Dimension: $20$
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.08992105744\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(10\) over \(\Q(i)\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial:	\( x^{20} - x^{18} - 3x^{16} + 11x^{14} + x^{12} - 40x^{10} + 4x^{8} + 176x^{6} - 192x^{4} - 256x^{2} + 1024 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 2^{20} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			13.2
Root			\(1.39859 + 0.209644i\) of defining polynomial
Character	\(\chi\)	\(=\)	40.13
Dual form			40.3.i.a.37.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.60823 - 1.18894i) q^{2} +(-2.52630 + 2.52630i) q^{3} +(1.17282 + 3.82420i) q^{4} +(3.09141 + 3.92978i) q^{5} +(7.06650 - 1.05925i) q^{6} +(-5.20520 + 5.20520i) q^{7} +(2.66059 - 7.54462i) q^{8} -3.76437i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} - 16 q^{6} - 4 q^{7} + 4 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.60823	−	1.18894i	−0.804116	−	0.594472i
							\(3\)	−2.52630	+	2.52630i	−0.842100	+	0.842100i	−0.989132	−	0.147032i	\(-0.953028\pi\)
0.147032	+	0.989132i	\(0.453028\pi\)
\(5\)	3.09141	+	3.92978i	0.618282	+	0.785957i
							\(7\)	−5.20520	+	5.20520i	−0.743600	+	0.743600i	−0.973269	−	0.229669i	\(-0.926236\pi\)
0.229669	+	0.973269i	\(0.426236\pi\)
\(11\)		−	2.49236i		−	0.226579i	−0.993562	−	0.113289i	\(-0.963861\pi\)
							0.993562	−	0.113289i	\(-0.0361387\pi\)
\(13\)	−6.65988	+	6.65988i	−0.512298	+	0.512298i	−0.915230	−	0.402932i	\(-0.867991\pi\)
							0.402932	+	0.915230i	\(0.367991\pi\)
\(17\)	21.9428	−	21.9428i	1.29075	−	1.29075i	0.356434	−	0.934320i	\(-0.383992\pi\)
							0.934320	−	0.356434i	\(-0.116008\pi\)
\(19\)	17.5567			0.924039			0.462020	−	0.886870i	\(-0.347125\pi\)
							0.462020	+	0.886870i	\(0.347125\pi\)
\(23\)	20.1791	+	20.1791i	0.877354	+	0.877354i	0.993260	−	0.115906i	\(-0.0369772\pi\)
							−0.115906	+	0.993260i	\(0.536977\pi\)
\(29\)	1.04754			0.0361220			0.0180610	−	0.999837i	\(-0.494251\pi\)
							0.0180610	+	0.999837i	\(0.494251\pi\)
\(31\)	−2.47263			−0.0797623			−0.0398812	−	0.999204i	\(-0.512698\pi\)
							−0.0398812	+	0.999204i	\(0.512698\pi\)
\(37\)	19.2786	+	19.2786i	0.521043	+	0.521043i	0.917886	−	0.396843i	\(-0.129894\pi\)
							−0.396843	+	0.917886i	\(0.629894\pi\)
\(41\)	−43.9414			−1.07174			−0.535871	−	0.844300i	\(-0.680017\pi\)
							−0.535871	+	0.844300i	\(0.680017\pi\)
\(43\)	32.9394	−	32.9394i	0.766032	−	0.766032i	−0.211373	−	0.977405i	\(-0.567794\pi\)
							0.977405	+	0.211373i	\(0.0677936\pi\)
\(47\)	33.7079	−	33.7079i	0.717189	−	0.717189i	−0.250840	−	0.968029i	\(-0.580707\pi\)
							0.968029	+	0.250840i	\(0.0807067\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.3.i.a.13.2	✓	20
3.2	odd	2		360.3.u.b.253.9		20
4.3	odd	2		160.3.m.a.113.8		20
5.2	odd	4	inner	40.3.i.a.37.8	yes	20
5.3	odd	4		200.3.i.b.157.3		20
5.4	even	2		200.3.i.b.93.9		20
8.3	odd	2		160.3.m.a.113.3		20
8.5	even	2	inner	40.3.i.a.13.8	yes	20
15.2	even	4		360.3.u.b.37.3		20
20.3	even	4		800.3.m.b.657.8		20
20.7	even	4		160.3.m.a.17.3		20
20.19	odd	2		800.3.m.b.593.3		20
24.5	odd	2		360.3.u.b.253.3		20
40.3	even	4		800.3.m.b.657.3		20
40.13	odd	4		200.3.i.b.157.9		20
40.19	odd	2		800.3.m.b.593.8		20
40.27	even	4		160.3.m.a.17.8		20
40.29	even	2		200.3.i.b.93.3		20
40.37	odd	4	inner	40.3.i.a.37.2	yes	20
120.77	even	4		360.3.u.b.37.9		20

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.3.i.a.13.2	✓	20	1.1	even	1	trivial
40.3.i.a.13.8	yes	20	8.5	even	2	inner
40.3.i.a.37.2	yes	20	40.37	odd	4	inner
40.3.i.a.37.8	yes	20	5.2	odd	4	inner
160.3.m.a.17.3		20	20.7	even	4
160.3.m.a.17.8		20	40.27	even	4
160.3.m.a.113.3		20	8.3	odd	2
160.3.m.a.113.8		20	4.3	odd	2
200.3.i.b.93.3		20	40.29	even	2
200.3.i.b.93.9		20	5.4	even	2
200.3.i.b.157.3		20	5.3	odd	4
200.3.i.b.157.9		20	40.13	odd	4
360.3.u.b.37.3		20	15.2	even	4
360.3.u.b.37.9		20	120.77	even	4
360.3.u.b.253.3		20	24.5	odd	2
360.3.u.b.253.9		20	3.2	odd	2
800.3.m.b.593.3		20	20.19	odd	2
800.3.m.b.593.8		20	40.19	odd	2
800.3.m.b.657.3		20	40.3	even	4
800.3.m.b.657.8		20	20.3	even	4