LMFDB - Newform orbit 700.1.u.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,1,Mod(51,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("700.51");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$700 = 2^{2} \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	700.u (of order $6$ , degree $2$ , 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:	$0.349345508843$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{2} + 1$ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 140)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.980.1
Artin image:	$S_3\times C_{12}$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{24} - \cdots)$

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $q$ -expansion and trace form are shown below.

$f(q)$	$=$	$q - \zeta_{12} q^{2} - \zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} - q^{6} - \zeta_{12}^{5} q^{7} - \zeta_{12}^{3} q^{8} + \zeta_{12} q^{12} - q^{14} + \zeta_{12}^{4} q^{16} - \zeta_{12}^{4} q^{21} - \zeta_{12} q^{23} + \cdots + \zeta_{12}^{5} q^{98} +O(q^{100})$ q - z * q^2 - z^5 * q^3 + z^2 * q^4 - q^6 - z^5 * q^7 - z^3 * q^8 + z * q^12 - q^14 + z^4 * q^16 - z^4 * q^21 - z * q^23 - z^2 * q^24 + z^3 * q^27 + z * q^28 + q^29 - z^5 * q^32 - q^41 + z^5 * q^42 + z^3 * q^43 + z^2 * q^46 - 2z q^47 + z^3 * q^48 - z^4 * q^49 - z^4 * q^54 - z^2 * q^56 - z * q^58 - z^4 * q^61 - q^64 + z^5 * q^67 - q^69 + z^2 * q^81 + z * q^82 + z^3 * q^83 + q^84 - z^4 * q^86 - z^5 * q^87 + z^4 * q^89 - z^3 * q^92 + 2z^2 q^94 - z^4 * q^96 + z^5 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{4} - 4 q^{6} - 4 q^{14} - 2 q^{16} + 2 q^{21} - 2 q^{24} + 4 q^{29} - 4 q^{41} + 2 q^{46} + 2 q^{49} + 2 q^{54} - 2 q^{56} + 2 q^{61} - 4 q^{64} - 4 q^{69} + 2 q^{81} + 4 q^{84} + 2 q^{86} - 2 q^{89}+ \cdots + 2 q^{96}+O(q^{100})$ 4 * q + 2 * q^4 - 4 * q^6 - 4 * q^14 - 2 * q^16 + 2 * q^21 - 2 * q^24 + 4 * q^29 - 4 * q^41 + 2 * q^46 + 2 * q^49 + 2 * q^54 - 2 * q^56 + 2 * q^61 - 4 * q^64 - 4 * q^69 + 2 * q^81 + 4 * q^84 + 2 * q^86 - 2 * q^89 + 4 * q^94 + 2 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$ .

$n$	$101$	$351$	$477$
$\chi(n)$	$\zeta_{12}^{4}$	$-1$	$1$

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.

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}

51.1

0.866025	+	0.500000i
−0.866025	−	0.500000i
0.866025	−	0.500000i
−0.866025	+	0.500000i

−0.866025

−

0.500000i

0.866025

−

0.500000i

0.500000

0.866025i

−1.00000

0.866025

−

0.500000i

−

1.00000i

51.2

0.866025

0.500000i

−0.866025

0.500000i

0.500000

0.866025i

−1.00000

−0.866025

0.500000i

1.00000i

151.1

−0.866025

0.500000i

0.866025

0.500000i

0.500000

−

0.866025i

−1.00000

0.866025

0.500000i

1.00000i

151.2

0.866025

−

0.500000i

−0.866025

−

0.500000i

0.500000

−

0.866025i

−1.00000

−0.866025

−

0.500000i

−

1.00000i

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
7.c	even	3	1	inner
28.g	odd	6	1	inner
35.j	even	6	1	inner
140.p	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	700.1.u.a		4
4.b	odd	2	1	inner	700.1.u.a		4
5.b	even	2	1	inner	700.1.u.a		4
5.c	odd	4	1		140.1.p.a	✓	2
5.c	odd	4	1		140.1.p.b	yes	2
7.c	even	3	1	inner	700.1.u.a		4
15.e	even	4	1		1260.1.ci.a		2
15.e	even	4	1		1260.1.ci.b		2
20.d	odd	2	1	CM	700.1.u.a		4
20.e	even	4	1		140.1.p.a	✓	2
20.e	even	4	1		140.1.p.b	yes	2
28.g	odd	6	1	inner	700.1.u.a		4
35.f	even	4	1		980.1.p.a		2
35.f	even	4	1		980.1.p.b		2
35.j	even	6	1	inner	700.1.u.a		4
35.k	even	12	1		980.1.f.a		1
35.k	even	12	1		980.1.f.d		1
35.k	even	12	1		980.1.p.a		2
35.k	even	12	1		980.1.p.b		2
35.l	odd	12	1		140.1.p.a	✓	2
35.l	odd	12	1		140.1.p.b	yes	2
35.l	odd	12	1		980.1.f.b		1
35.l	odd	12	1		980.1.f.c		1
40.i	odd	4	1		2240.1.bt.a		2
40.i	odd	4	1		2240.1.bt.b		2
40.k	even	4	1		2240.1.bt.a		2
40.k	even	4	1		2240.1.bt.b		2
60.l	odd	4	1		1260.1.ci.a		2
60.l	odd	4	1		1260.1.ci.b		2
105.x	even	12	1		1260.1.ci.a		2
105.x	even	12	1		1260.1.ci.b		2
140.j	odd	4	1		980.1.p.a		2
140.j	odd	4	1		980.1.p.b		2
140.p	odd	6	1	inner	700.1.u.a		4
140.w	even	12	1		140.1.p.a	✓	2
140.w	even	12	1		140.1.p.b	yes	2
140.w	even	12	1		980.1.f.b		1
140.w	even	12	1		980.1.f.c		1
140.x	odd	12	1		980.1.f.a		1
140.x	odd	12	1		980.1.f.d		1
140.x	odd	12	1		980.1.p.a		2
140.x	odd	12	1		980.1.p.b		2
280.br	even	12	1		2240.1.bt.a		2
280.br	even	12	1		2240.1.bt.b		2
280.bt	odd	12	1		2240.1.bt.a		2
280.bt	odd	12	1		2240.1.bt.b		2
420.bp	odd	12	1		1260.1.ci.a		2
420.bp	odd	12	1		1260.1.ci.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.1.p.a	✓	2	5.c	odd	4	1
140.1.p.a	✓	2	20.e	even	4	1
140.1.p.a	✓	2	35.l	odd	12	1
140.1.p.a	✓	2	140.w	even	12	1
140.1.p.b	yes	2	5.c	odd	4	1
140.1.p.b	yes	2	20.e	even	4	1
140.1.p.b	yes	2	35.l	odd	12	1
140.1.p.b	yes	2	140.w	even	12	1
700.1.u.a		4	1.a	even	1	1	trivial
700.1.u.a		4	4.b	odd	2	1	inner
700.1.u.a		4	5.b	even	2	1	inner
700.1.u.a		4	7.c	even	3	1	inner
700.1.u.a		4	20.d	odd	2	1	CM
700.1.u.a		4	28.g	odd	6	1	inner
700.1.u.a		4	35.j	even	6	1	inner
700.1.u.a		4	140.p	odd	6	1	inner
980.1.f.a		1	35.k	even	12	1
980.1.f.a		1	140.x	odd	12	1
980.1.f.b		1	35.l	odd	12	1
980.1.f.b		1	140.w	even	12	1
980.1.f.c		1	35.l	odd	12	1
980.1.f.c		1	140.w	even	12	1
980.1.f.d		1	35.k	even	12	1
980.1.f.d		1	140.x	odd	12	1
980.1.p.a		2	35.f	even	4	1
980.1.p.a		2	35.k	even	12	1
980.1.p.a		2	140.j	odd	4	1
980.1.p.a		2	140.x	odd	12	1
980.1.p.b		2	35.f	even	4	1
980.1.p.b		2	35.k	even	12	1
980.1.p.b		2	140.j	odd	4	1
980.1.p.b		2	140.x	odd	12	1
1260.1.ci.a		2	15.e	even	4	1
1260.1.ci.a		2	60.l	odd	4	1
1260.1.ci.a		2	105.x	even	12	1
1260.1.ci.a		2	420.bp	odd	12	1
1260.1.ci.b		2	15.e	even	4	1
1260.1.ci.b		2	60.l	odd	4	1
1260.1.ci.b		2	105.x	even	12	1
1260.1.ci.b		2	420.bp	odd	12	1
2240.1.bt.a		2	40.i	odd	4	1
2240.1.bt.a		2	40.k	even	4	1
2240.1.bt.a		2	280.br	even	12	1
2240.1.bt.a		2	280.bt	odd	12	1
2240.1.bt.b		2	40.i	odd	4	1
2240.1.bt.b		2	40.k	even	4	1
2240.1.bt.b		2	280.br	even	12	1
2240.1.bt.b		2	280.bt	odd	12	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(700, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4} - T^{2} + 1$ T^4 - T^2 + 1
$3$	$T^{4} - T^{2} + 1$ T^4 - T^2 + 1
$5$	$T^{4}$ T^4
$7$	$T^{4} - T^{2} + 1$ T^4 - T^2 + 1
$11$	$T^{4}$ T^4
$13$	$T^{4}$ T^4
$17$	$T^{4}$ T^4
$19$	$T^{4}$ T^4
$23$	$T^{4} - T^{2} + 1$ T^4 - T^2 + 1
$29$	$(T - 1)^{4}$ (T - 1)^4
$31$	$T^{4}$ T^4
$37$	$T^{4}$ T^4
$41$	$(T + 1)^{4}$ (T + 1)^4
$43$	$(T^{2} + 1)^{2}$ (T^2 + 1)^2
$47$	$T^{4} - 4T^{2} + 16$ T^4 - 4*T^2 + 16
$53$	$T^{4}$ T^4
$59$	$T^{4}$ T^4
$61$	$(T^{2} - T + 1)^{2}$ (T^2 - T + 1)^2
$67$	$T^{4} - T^{2} + 1$ T^4 - T^2 + 1
$71$	$T^{4}$ T^4
$73$	$T^{4}$ T^4
$79$	$T^{4}$ T^4
$83$	$(T^{2} + 1)^{2}$ (T^2 + 1)^2
$89$	$(T^{2} + T + 1)^{2}$ (T^2 + T + 1)^2
$97$	$T^{4}$ T^4
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Database

Properties

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Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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	700.1.u.a

Level	$700$
Weight	$1$
Character orbit	700.u
Analytic conductor	$0.349$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{3}$
CM discriminant	-20
Inner twists	$8$