LMFDB - Newform orbit 720.2.by.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,2,Mod(49,720)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("720.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	720.by (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.74922894553$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{24}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{24}^{7} - \zeta_{24}^{6} + \cdots - \zeta_{24}) q^{3}+ \cdots + (2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} + \cdots - 1) q^{9}+O(q^{10}) $ q + (-z^7 - z^6 + z^2 - z) * q^3 + (z^5 + z^4 + z^2 - 2z - 1) q^5 + (-z^7 + 2z^6 - 2z^5 + 2z^3 - 2z^2 + z) * q^7 + (2z^7 - 2z^5 + z^4 - 2z^3 - 1) q^9	$ q + ( - \zeta_{24}^{7} - \zeta_{24}^{6} + \cdots - \zeta_{24}) q^{3}+ \cdots + ( - 2 \zeta_{24}^{7} + 3 \zeta_{24}^{5} + \cdots - 3) q^{99}+O(q^{100}) $ q + (-z^7 - z^6 + z^2 - z) * q^3 + (z^5 + z^4 + z^2 - 2z - 1) q^5 + (-z^7 + 2z^6 - 2z^5 + 2z^3 - 2z^2 + z) * q^7 + (2z^7 - 2z^5 + z^4 - 2z^3 - 1) q^9 + (z^7 - 2z^5 - z^4 - 2z^3 + z) * q^11 + (z^7 - z^5 + z^3 + 2z) q^13 + (2z^7 - 2z^5 + 2z^4 - z^3 + 2z^2 + 2z) q^15 + (-4z^7 - z^6 - 2z^5 + 2z^3 - 2z) * q^17 + (2z^7 - z^5 - z^3 - z - 3) q^19 + (-3z^7 + z^5 + z^3 + 2z - 4) * q^21 + (-2z^7 + 2z^5 - 2z^3 - 2z^2 - 4z) q^23 + (2z^7 - z^6 - 4z^5 - 4z^3 + z^2 + 2z) * q^25 + (5z^6 + z^5 - z^3 - z) q^27 - 6z^4 q^29 + (z^7 + z^5 - 4z^4 + z^3 - 2z + 4) * q^31 + (4z^6 + 2z^5 - 2z^3 - 3z^2 - 2z) q^33 + (-4z^7 + z^6 + 2z^5 + 2z^3 + 2z - 5) * q^35 + 8z^6 q^37 + (-2z^7 + z^5 - 4z^4 + z^3 + z + 2) * q^39 + (-z^4 + 1) * q^41 + (z^7 - 5z^6 + 2z^5 - 2z^3 + 5z^2 - z) * q^43 + (-4z^7 + 3z^6 - 2z^5 - 3z^4 + z^2 + z + 4) * q^45 + (-z^7 - 2z^6 - 2z^5 + 2z^3 + 2z^2 + z) * q^47 + (4z^7 + 4z^5 - 3z^4 + 4z^3 - 8z + 3) q^49 + (3z^7 - 4z^5 + 3z^4 - 4z^3 + z - 8) * q^51 + (2z^7 - 6z^6 + z^5 - z^3 + z) * q^53 + (-4z^7 + 2z^6 + z^5 + 2z^3 + z + 4) q^55 + (4z^7 + 5z^6 + 2z^5 - 2z^3 - z^2 + 2z) q^57 + (5z^7 + 5z^5 + z^4 + 5z^3 - 10z - 1) * q^59 + (z^7 - 2z^5 - 2z^4 - 2z^3 + z) q^61 + (2z^7 + 2z^6 - 3z^5 + 3z^3 - 8z^2 + 5z) * q^63 + (3z^6 + 4z^5 - 3z^4 - 3z^2 - 2z) q^65 + (-z^7 + z^5 - z^3 + 7z^2 - 2z) * q^67 + (4z^7 + 8z^4 - 4z - 6) q^69 + (-2z^7 + z^5 + z^3 + z) q^71 + (-8z^7 - 5z^6 - 4z^5 + 4z^3 - 4z) q^73 + (-z^7 + 8z^6 + z^5 - z^4 - 3z^3 - 4z^2 - 4z + 1) * q^75 + (z^7 - z^5 + z^3 - 4z^2 + 2z) * q^77 + (3z^7 - 6z^5 - 6z^3 + 3z) * q^79 + (-4z^7 + 7z^4 + 4z) q^81 + (4z^6 - 4z^2) * q^83 + (z^7 - 4z^5 + 5z^4 + z^3 + 7z^2 + 8z - 5) * q^85 + (6z^7 + 6z^5 - 6z^3 - 6z^2) * q^87 + (-4z^7 + 2z^5 + 2z^3 + 2z - 8) * q^89 + (4z^7 - 2z^5 - 2z^3 - 2z + 6) * q^91 + (2z^7 - 6z^6 + 5z^5 - 5z^3 + 4z^2 - 3z) * q^93 + (-2z^7 - 3z^5 - 6z^4 - 2z^3 + 6z + 6) q^95 + (-13z^6 + 13z^2) * q^97 + (-2z^7 + 3z^5 + 8z^4 + 3z^3 - z - 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{5} - 4 q^{9}+O(q^{10}) $ 8 * q - 4 * q^5 - 4 * q^9	$ 8 q - 4 q^{5} - 4 q^{9} - 4 q^{11} + 8 q^{15} - 24 q^{19} - 32 q^{21} - 24 q^{29} + 16 q^{31} - 40 q^{35} + 4 q^{41} + 20 q^{45} + 12 q^{49} - 52 q^{51} + 32 q^{55} - 4 q^{59} - 8 q^{61} - 12 q^{65} - 16 q^{69} + 4 q^{75} + 28 q^{81} - 20 q^{85} - 64 q^{89} + 48 q^{91} + 24 q^{95} + 8 q^{99}+O(q^{100}) $ 8 * q - 4 * q^5 - 4 * q^9 - 4 * q^11 + 8 * q^15 - 24 * q^19 - 32 * q^21 - 24 * q^29 + 16 * q^31 - 40 * q^35 + 4 * q^41 + 20 * q^45 + 12 * q^49 - 52 * q^51 + 32 * q^55 - 4 * q^59 - 8 * q^61 - 12 * q^65 - 16 * q^69 + 4 * q^75 + 28 * q^81 - 20 * q^85 - 64 * q^89 + 48 * q^91 + 24 * q^95 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$-1$	$-1 + \zeta_{24}^{4}$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

49.1

−0.258819	+	0.965926i
0.258819	−	0.965926i
0.965926	+	0.258819i
−0.965926	−	0.258819i
−0.258819	−	0.965926i
0.258819	+	0.965926i
0.965926	−	0.258819i
−0.965926	+	0.258819i

−1.57313

−

0.724745i

−1.81431

−

1.30701i

3.85337

−

2.22474i

1.94949

2.28024i

49.2

−0.158919

1.72474i

−0.917738

2.03906i

−0.389270

0.224745i

−2.94949

−

0.548188i

49.3

0.158919

−

1.72474i

−1.30701

1.81431i

0.389270

−

0.224745i

−2.94949

−

0.548188i

49.4

1.57313

0.724745i

2.03906

0.917738i

−3.85337

2.22474i

1.94949

2.28024i

529.1

−1.57313

0.724745i

−1.81431

1.30701i

3.85337

2.22474i

1.94949

−

2.28024i

529.2

−0.158919

−

1.72474i

−0.917738

−

2.03906i

−0.389270

−

0.224745i

−2.94949

0.548188i

529.3

0.158919

1.72474i

−1.30701

−

1.81431i

0.389270

0.224745i

−2.94949

0.548188i

529.4

1.57313

−

0.724745i

2.03906

−

0.917738i

−3.85337

−

2.22474i

1.94949

−

2.28024i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
9.c	even	3	1	inner
45.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	720.2.by.c		8
3.b	odd	2	1		2160.2.by.d		8
4.b	odd	2	1		90.2.i.b	✓	8
5.b	even	2	1	inner	720.2.by.c		8
9.c	even	3	1	inner	720.2.by.c		8
9.d	odd	6	1		2160.2.by.d		8
12.b	even	2	1		270.2.i.b		8
15.d	odd	2	1		2160.2.by.d		8
20.d	odd	2	1		90.2.i.b	✓	8
20.e	even	4	1		450.2.e.k		4
20.e	even	4	1		450.2.e.n		4
36.f	odd	6	1		90.2.i.b	✓	8
36.f	odd	6	1		810.2.c.f		4
36.h	even	6	1		270.2.i.b		8
36.h	even	6	1		810.2.c.e		4
45.h	odd	6	1		2160.2.by.d		8
45.j	even	6	1	inner	720.2.by.c		8
60.h	even	2	1		270.2.i.b		8
60.l	odd	4	1		1350.2.e.j		4
60.l	odd	4	1		1350.2.e.m		4
180.n	even	6	1		270.2.i.b		8
180.n	even	6	1		810.2.c.e		4
180.p	odd	6	1		90.2.i.b	✓	8
180.p	odd	6	1		810.2.c.f		4
180.v	odd	12	1		1350.2.e.j		4
180.v	odd	12	1		1350.2.e.m		4
180.v	odd	12	1		4050.2.a.bm		2
180.v	odd	12	1		4050.2.a.bz		2
180.x	even	12	1		450.2.e.k		4
180.x	even	12	1		450.2.e.n		4
180.x	even	12	1		4050.2.a.bq		2
180.x	even	12	1		4050.2.a.bs		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
90.2.i.b	✓	8	4.b	odd	2	1
90.2.i.b	✓	8	20.d	odd	2	1
90.2.i.b	✓	8	36.f	odd	6	1
90.2.i.b	✓	8	180.p	odd	6	1
270.2.i.b		8	12.b	even	2	1
270.2.i.b		8	36.h	even	6	1
270.2.i.b		8	60.h	even	2	1
270.2.i.b		8	180.n	even	6	1
450.2.e.k		4	20.e	even	4	1
450.2.e.k		4	180.x	even	12	1
450.2.e.n		4	20.e	even	4	1
450.2.e.n		4	180.x	even	12	1
720.2.by.c		8	1.a	even	1	1	trivial
720.2.by.c		8	5.b	even	2	1	inner
720.2.by.c		8	9.c	even	3	1	inner
720.2.by.c		8	45.j	even	6	1	inner
810.2.c.e		4	36.h	even	6	1
810.2.c.e		4	180.n	even	6	1
810.2.c.f		4	36.f	odd	6	1
810.2.c.f		4	180.p	odd	6	1
1350.2.e.j		4	60.l	odd	4	1
1350.2.e.j		4	180.v	odd	12	1
1350.2.e.m		4	60.l	odd	4	1
1350.2.e.m		4	180.v	odd	12	1
2160.2.by.d		8	3.b	odd	2	1
2160.2.by.d		8	9.d	odd	6	1
2160.2.by.d		8	15.d	odd	2	1
2160.2.by.d		8	45.h	odd	6	1
4050.2.a.bm		2	180.v	odd	12	1
4050.2.a.bq		2	180.x	even	12	1
4050.2.a.bs		2	180.x	even	12	1
4050.2.a.bz		2	180.v	odd	12	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(720, [\chi])$:

$ T_{7}^{8} - 20T_{7}^{6} + 396T_{7}^{4} - 80T_{7}^{2} + 16 $

$ T_{11}^{4} + 2T_{11}^{3} + 9T_{11}^{2} - 10T_{11} + 25 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} + 2 T^{6} + \cdots + 81 $ T^8 + 2T^6 - 5T^4 + 18*T^2 + 81
$5$	$ T^{8} + 4 T^{7} + \cdots + 625 $ T^8 + 4T^7 + 8T^6 - 8T^5 - 41T^4 - 40T^3 + 200T^2 + 500*T + 625
$7$	$ T^{8} - 20 T^{6} + \cdots + 16 $ T^8 - 20T^6 + 396T^4 - 80*T^2 + 16
$11$	$ (T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 25)^{2} $ (T^4 + 2T^3 + 9T^2 - 10*T + 25)^2
$13$	$ (T^{4} - 6 T^{2} + 36)^{2} $ (T^4 - 6*T^2 + 36)^2
$17$	$ (T^{4} + 50 T^{2} + 529)^{2} $ (T^4 + 50*T^2 + 529)^2
$19$	$ (T^{2} + 6 T + 3)^{4} $ (T^2 + 6*T + 3)^4
$23$	$ T^{8} - 56 T^{6} + \cdots + 160000 $ T^8 - 56T^6 + 2736T^4 - 22400*T^2 + 160000
$29$	$ (T^{2} + 6 T + 36)^{4} $ (T^2 + 6*T + 36)^4
$31$	$ (T^{4} - 8 T^{3} + \cdots + 100)^{2} $ (T^4 - 8T^3 + 54T^2 - 80*T + 100)^2
$37$	$ (T^{2} + 64)^{4} $ (T^2 + 64)^4
$41$	$ (T^{2} - T + 1)^{4} $ (T^2 - T + 1)^4
$43$	$ T^{8} - 62 T^{6} + \cdots + 130321 $ T^8 - 62T^6 + 3483T^4 - 22382*T^2 + 130321
$47$	$ T^{8} - 20 T^{6} + \cdots + 16 $ T^8 - 20T^6 + 396T^4 - 80*T^2 + 16
$53$	$ (T^{4} + 84 T^{2} + 900)^{2} $ (T^4 + 84*T^2 + 900)^2
$59$	$ (T^{4} + 2 T^{3} + \cdots + 22201)^{2} $ (T^4 + 2T^3 + 153T^2 - 298*T + 22201)^2
$61$	$ (T^{4} + 4 T^{3} + 18 T^{2} + \cdots + 4)^{2} $ (T^4 + 4T^3 + 18T^2 - 8*T + 4)^2
$67$	$ T^{8} - 110 T^{6} + \cdots + 3418801 $ T^8 - 110T^6 + 10251T^4 - 203390*T^2 + 3418801
$71$	$ (T^{2} - 6)^{4} $ (T^2 - 6)^4
$73$	$ (T^{4} + 242 T^{2} + 5041)^{2} $ (T^4 + 242*T^2 + 5041)^2
$79$	$ (T^{4} + 54 T^{2} + 2916)^{2} $ (T^4 + 54*T^2 + 2916)^2
$83$	$ (T^{4} - 16 T^{2} + 256)^{2} $ (T^4 - 16*T^2 + 256)^2
$89$	$ (T^{2} + 16 T + 40)^{4} $ (T^2 + 16*T + 40)^4
$97$	$ (T^{4} - 169 T^{2} + 28561)^{2} $ (T^4 - 169*T^2 + 28561)^2
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.by (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{24}^{7} - \zeta_{24}^{6} + \cdots - \zeta_{24}) q^{3}+ \cdots + (2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} + \cdots - 1) q^{9}+O(q^{10}) \) q + (-z^7 - z^6 + z^2 - z) * q^3 + (z^5 + z^4 + z^2 - 2z - 1) q^5 + (-z^7 + 2z^6 - 2z^5 + 2z^3 - 2z^2 + z) * q^7 + (2z^7 - 2z^5 + z^4 - 2z^3 - 1) q^9	\( q + ( - \zeta_{24}^{7} - \zeta_{24}^{6} + \cdots - \zeta_{24}) q^{3}+ \cdots + ( - 2 \zeta_{24}^{7} + 3 \zeta_{24}^{5} + \cdots - 3) q^{99}+O(q^{100}) \) q + (-z^7 - z^6 + z^2 - z) * q^3 + (z^5 + z^4 + z^2 - 2z - 1) q^5 + (-z^7 + 2z^6 - 2z^5 + 2z^3 - 2z^2 + z) * q^7 + (2z^7 - 2z^5 + z^4 - 2z^3 - 1) q^9 + (z^7 - 2z^5 - z^4 - 2z^3 + z) * q^11 + (z^7 - z^5 + z^3 + 2z) q^13 + (2z^7 - 2z^5 + 2z^4 - z^3 + 2z^2 + 2z) q^15 + (-4z^7 - z^6 - 2z^5 + 2z^3 - 2z) * q^17 + (2z^7 - z^5 - z^3 - z - 3) q^19 + (-3z^7 + z^5 + z^3 + 2z - 4) * q^21 + (-2z^7 + 2z^5 - 2z^3 - 2z^2 - 4z) q^23 + (2z^7 - z^6 - 4z^5 - 4z^3 + z^2 + 2z) * q^25 + (5z^6 + z^5 - z^3 - z) q^27 - 6z^4 q^29 + (z^7 + z^5 - 4z^4 + z^3 - 2z + 4) * q^31 + (4z^6 + 2z^5 - 2z^3 - 3z^2 - 2z) q^33 + (-4z^7 + z^6 + 2z^5 + 2z^3 + 2z - 5) * q^35 + 8z^6 q^37 + (-2z^7 + z^5 - 4z^4 + z^3 + z + 2) * q^39 + (-z^4 + 1) * q^41 + (z^7 - 5z^6 + 2z^5 - 2z^3 + 5z^2 - z) * q^43 + (-4z^7 + 3z^6 - 2z^5 - 3z^4 + z^2 + z + 4) * q^45 + (-z^7 - 2z^6 - 2z^5 + 2z^3 + 2z^2 + z) * q^47 + (4z^7 + 4z^5 - 3z^4 + 4z^3 - 8z + 3) q^49 + (3z^7 - 4z^5 + 3z^4 - 4z^3 + z - 8) * q^51 + (2z^7 - 6z^6 + z^5 - z^3 + z) * q^53 + (-4z^7 + 2z^6 + z^5 + 2z^3 + z + 4) q^55 + (4z^7 + 5z^6 + 2z^5 - 2z^3 - z^2 + 2z) q^57 + (5z^7 + 5z^5 + z^4 + 5z^3 - 10z - 1) * q^59 + (z^7 - 2z^5 - 2z^4 - 2z^3 + z) q^61 + (2z^7 + 2z^6 - 3z^5 + 3z^3 - 8z^2 + 5z) * q^63 + (3z^6 + 4z^5 - 3z^4 - 3z^2 - 2z) q^65 + (-z^7 + z^5 - z^3 + 7z^2 - 2z) * q^67 + (4z^7 + 8z^4 - 4z - 6) q^69 + (-2z^7 + z^5 + z^3 + z) q^71 + (-8z^7 - 5z^6 - 4z^5 + 4z^3 - 4z) q^73 + (-z^7 + 8z^6 + z^5 - z^4 - 3z^3 - 4z^2 - 4z + 1) * q^75 + (z^7 - z^5 + z^3 - 4z^2 + 2z) * q^77 + (3z^7 - 6z^5 - 6z^3 + 3z) * q^79 + (-4z^7 + 7z^4 + 4z) q^81 + (4z^6 - 4z^2) * q^83 + (z^7 - 4z^5 + 5z^4 + z^3 + 7z^2 + 8z - 5) * q^85 + (6z^7 + 6z^5 - 6z^3 - 6z^2) * q^87 + (-4z^7 + 2z^5 + 2z^3 + 2z - 8) * q^89 + (4z^7 - 2z^5 - 2z^3 - 2z + 6) * q^91 + (2z^7 - 6z^6 + 5z^5 - 5z^3 + 4z^2 - 3z) * q^93 + (-2z^7 - 3z^5 - 6z^4 - 2z^3 + 6z + 6) q^95 + (-13z^6 + 13z^2) * q^97 + (-2z^7 + 3z^5 + 8z^4 + 3z^3 - z - 3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{5} - 4 q^{9}+O(q^{10}) \) 8 * q - 4 * q^5 - 4 * q^9	\( 8 q - 4 q^{5} - 4 q^{9} - 4 q^{11} + 8 q^{15} - 24 q^{19} - 32 q^{21} - 24 q^{29} + 16 q^{31} - 40 q^{35} + 4 q^{41} + 20 q^{45} + 12 q^{49} - 52 q^{51} + 32 q^{55} - 4 q^{59} - 8 q^{61} - 12 q^{65} - 16 q^{69} + 4 q^{75} + 28 q^{81} - 20 q^{85} - 64 q^{89} + 48 q^{91} + 24 q^{95} + 8 q^{99}+O(q^{100}) \) 8 * q - 4 * q^5 - 4 * q^9 - 4 * q^11 + 8 * q^15 - 24 * q^19 - 32 * q^21 - 24 * q^29 + 16 * q^31 - 40 * q^35 + 4 * q^41 + 20 * q^45 + 12 * q^49 - 52 * q^51 + 32 * q^55 - 4 * q^59 - 8 * q^61 - 12 * q^65 - 16 * q^69 + 4 * q^75 + 28 * q^81 - 20 * q^85 - 64 * q^89 + 48 * q^91 + 24 * q^95 + 8 * q^99

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	720.2.by.c

Level	$720$
Weight	$2$
Character orbit	720.by
Analytic conductor	$5.749$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(5.74922894553\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{24})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{4} + 1 \) x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

\(n\)	\(181\)	\(271\)	\(577\)	\(641\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(-1 + \zeta_{24}^{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 49.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} + 2 T^{6} + \cdots + 81 \) T^8 + 2T^6 - 5T^4 + 18*T^2 + 81
$5$	\( T^{8} + 4 T^{7} + \cdots + 625 \) T^8 + 4T^7 + 8T^6 - 8T^5 - 41T^4 - 40T^3 + 200T^2 + 500*T + 625
$7$	\( T^{8} - 20 T^{6} + \cdots + 16 \) T^8 - 20T^6 + 396T^4 - 80*T^2 + 16
$11$	\( (T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 25)^{2} \) (T^4 + 2T^3 + 9T^2 - 10*T + 25)^2
$13$	\( (T^{4} - 6 T^{2} + 36)^{2} \) (T^4 - 6*T^2 + 36)^2
$17$	\( (T^{4} + 50 T^{2} + 529)^{2} \) (T^4 + 50*T^2 + 529)^2
$19$	\( (T^{2} + 6 T + 3)^{4} \) (T^2 + 6*T + 3)^4
$23$	\( T^{8} - 56 T^{6} + \cdots + 160000 \) T^8 - 56T^6 + 2736T^4 - 22400*T^2 + 160000
$29$	\( (T^{2} + 6 T + 36)^{4} \) (T^2 + 6*T + 36)^4
$31$	\( (T^{4} - 8 T^{3} + \cdots + 100)^{2} \) (T^4 - 8T^3 + 54T^2 - 80*T + 100)^2
$37$	\( (T^{2} + 64)^{4} \) (T^2 + 64)^4
$41$	\( (T^{2} - T + 1)^{4} \) (T^2 - T + 1)^4
$43$	\( T^{8} - 62 T^{6} + \cdots + 130321 \) T^8 - 62T^6 + 3483T^4 - 22382*T^2 + 130321
$47$	\( T^{8} - 20 T^{6} + \cdots + 16 \) T^8 - 20T^6 + 396T^4 - 80*T^2 + 16
$53$	\( (T^{4} + 84 T^{2} + 900)^{2} \) (T^4 + 84*T^2 + 900)^2
$59$	\( (T^{4} + 2 T^{3} + \cdots + 22201)^{2} \) (T^4 + 2T^3 + 153T^2 - 298*T + 22201)^2
$61$	\( (T^{4} + 4 T^{3} + 18 T^{2} + \cdots + 4)^{2} \) (T^4 + 4T^3 + 18T^2 - 8*T + 4)^2
$67$	\( T^{8} - 110 T^{6} + \cdots + 3418801 \) T^8 - 110T^6 + 10251T^4 - 203390*T^2 + 3418801
$71$	\( (T^{2} - 6)^{4} \) (T^2 - 6)^4
$73$	\( (T^{4} + 242 T^{2} + 5041)^{2} \) (T^4 + 242*T^2 + 5041)^2
$79$	\( (T^{4} + 54 T^{2} + 2916)^{2} \) (T^4 + 54*T^2 + 2916)^2
$83$	\( (T^{4} - 16 T^{2} + 256)^{2} \) (T^4 - 16*T^2 + 256)^2
$89$	\( (T^{2} + 16 T + 40)^{4} \) (T^2 + 16*T + 40)^4
$97$	\( (T^{4} - 169 T^{2} + 28561)^{2} \) (T^4 - 169*T^2 + 28561)^2
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