Newform orbit 320.3.h.c

• Label: 320.3.h.c
• Level: $320$
• Weight: $3$
• Character orbit: 320.h
• Self dual: yes
• Analytic conductor: $8.719$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -20
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	320.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(8.71936845953\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{3} - 5 q^{5} - 3 \beta q^{7} + 11 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{5} + 22 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(257\)	\(261\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

319.1

1.61803
−0.618034

0

−4.47214

0

−5.00000

0

−13.4164

0

11.0000

0

319.2

0

4.47214

0

−5.00000

0

13.4164

0

11.0000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by \(\Q(\sqrt{-5}) \)
4.b	odd	2	1	inner
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	320.3.h.c		2
4.b	odd	2	1	inner	320.3.h.c		2
5.b	even	2	1	inner	320.3.h.c		2
5.c	odd	4	2		1600.3.b.d		2
8.b	even	2	1		80.3.h.a	✓	2
8.d	odd	2	1		80.3.h.a	✓	2
16.e	even	4	2		1280.3.e.e		4
16.f	odd	4	2		1280.3.e.e		4
20.d	odd	2	1	CM	320.3.h.c		2
20.e	even	4	2		1600.3.b.d		2
24.f	even	2	1		720.3.j.a		2
24.h	odd	2	1		720.3.j.a		2
40.e	odd	2	1		80.3.h.a	✓	2
40.f	even	2	1		80.3.h.a	✓	2
40.i	odd	4	2		400.3.b.b		2
40.k	even	4	2		400.3.b.b		2
80.k	odd	4	2		1280.3.e.e		4
80.q	even	4	2		1280.3.e.e		4
120.i	odd	2	1		720.3.j.a		2
120.m	even	2	1		720.3.j.a		2
120.q	odd	4	2		3600.3.e.o		2
120.w	even	4	2		3600.3.e.o		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
80.3.h.a	✓	2	8.b	even	2	1
80.3.h.a	✓	2	8.d	odd	2	1
80.3.h.a	✓	2	40.e	odd	2	1
80.3.h.a	✓	2	40.f	even	2	1
320.3.h.c		2	1.a	even	1	1	trivial
320.3.h.c		2	4.b	odd	2	1	inner
320.3.h.c		2	5.b	even	2	1	inner
320.3.h.c		2	20.d	odd	2	1	CM
400.3.b.b		2	40.i	odd	4	2
400.3.b.b		2	40.k	even	4	2
720.3.j.a		2	24.f	even	2	1
720.3.j.a		2	24.h	odd	2	1
720.3.j.a		2	120.i	odd	2	1
720.3.j.a		2	120.m	even	2	1
1280.3.e.e		4	16.e	even	4	2
1280.3.e.e		4	16.f	odd	4	2
1280.3.e.e		4	80.k	odd	4	2
1280.3.e.e		4	80.q	even	4	2
1600.3.b.d		2	5.c	odd	4	2
1600.3.b.d		2	20.e	even	4	2
3600.3.e.o		2	120.q	odd	4	2
3600.3.e.o		2	120.w	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 20 \) acting on \(S_{3}^{\mathrm{new}}(320, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 20 \)
$5$	\( (T + 5)^{2} \)
$7$	\( T^{2} - 180 \)
$11$	\( T^{2} \)