Newform orbit 80.7.h.a

• Label: 80.7.h.a
• Level: $80$
• Weight: $7$
• Character orbit: 80.h
• Self dual: yes
• Analytic conductor: $18.404$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -20
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	80.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(18.4043266896\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
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 - 7*b * q^3 + 125 * q^5 - 99*b * q^7 + 251 * q^9 - 875*b * q^15 + 13860 * q^21 + 4221*b * q^23 + 15625 * q^25 + 3346*b * q^27 + 44858 * q^29 - 12375*b * q^35 + 74338 * q^41 - 35343*b * q^43 + 31375 * q^45 + 46053*b * q^47 + 78371 * q^49 - 452342 * q^61 - 24849*b * q^63 + 134505*b * q^67 - 590940 * q^69 - 109375*b * q^75 - 651419 * q^81 + 36729*b * q^83 - 314006*b * q^87 + 511058 * q^89
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 250 q^{5} + 502 q^{9} + 27720 q^{21} + 31250 q^{25} + 89716 q^{29} + 148676 q^{41} + 62750 q^{45} + 156742 q^{49} - 904684 q^{61} - 1181880 q^{69} - 1302838 q^{81} + 1022116 q^{89}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\)	\(17\)	\(21\)	\(31\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

79.1

1.61803
−0.618034

0

−31.3050

0

125.000

0

−442.741

0

251.000

0

79.2

0

31.3050

0

125.000

0

442.741

0

251.000

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	80.7.h.a	✓	2
3.b	odd	2	1		720.7.j.a		2
4.b	odd	2	1	inner	80.7.h.a	✓	2
5.b	even	2	1	inner	80.7.h.a	✓	2
5.c	odd	4	2		400.7.b.b		2
8.b	even	2	1		320.7.h.c		2
8.d	odd	2	1		320.7.h.c		2
12.b	even	2	1		720.7.j.a		2
15.d	odd	2	1		720.7.j.a		2
20.d	odd	2	1	CM	80.7.h.a	✓	2
20.e	even	4	2		400.7.b.b		2
40.e	odd	2	1		320.7.h.c		2
40.f	even	2	1		320.7.h.c		2
60.h	even	2	1		720.7.j.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
80.7.h.a	✓	2	1.a	even	1	1	trivial
80.7.h.a	✓	2	4.b	odd	2	1	inner
80.7.h.a	✓	2	5.b	even	2	1	inner
80.7.h.a	✓	2	20.d	odd	2	1	CM
320.7.h.c		2	8.b	even	2	1
320.7.h.c		2	8.d	odd	2	1
320.7.h.c		2	40.e	odd	2	1
320.7.h.c		2	40.f	even	2	1
400.7.b.b		2	5.c	odd	4	2
400.7.b.b		2	20.e	even	4	2
720.7.j.a		2	3.b	odd	2	1
720.7.j.a		2	12.b	even	2	1
720.7.j.a		2	15.d	odd	2	1
720.7.j.a		2	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 980 \) acting on \(S_{7}^{\mathrm{new}}(80, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 980 \)
$5$	\( (T - 125)^{2} \)
$7$	\( T^{2} - 196020 \)
$11$	\( T^{2} \)