Properties

Label 1400.1.m.a
Level $1400$
Weight $1$
Character orbit 1400.m
Self dual yes
Analytic conductor $0.699$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -7, -56, 8
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1400,1,Mod(1301,1400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1400.1301");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1400.m (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.698691017686\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{2}, \sqrt{-7})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.9800.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{7} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + q^{7} + q^{8} - q^{9} + q^{14} + q^{16} - q^{18} - 2 q^{23} + q^{28} + q^{32} - q^{36} - 2 q^{46} + q^{49} + q^{56} - q^{63} + q^{64} - 2 q^{71} - q^{72} - 2 q^{79} + q^{81} - 2 q^{92} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1301.1
0
1.00000 0 1.00000 0 0 1.00000 1.00000 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
8.b even 2 1 RM by \(\Q(\sqrt{2}) \)
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.1.m.a 1
5.b even 2 1 56.1.h.a 1
5.c odd 4 2 1400.1.c.a 2
7.b odd 2 1 CM 1400.1.m.a 1
8.b even 2 1 RM 1400.1.m.a 1
15.d odd 2 1 504.1.l.a 1
20.d odd 2 1 224.1.h.a 1
35.c odd 2 1 56.1.h.a 1
35.f even 4 2 1400.1.c.a 2
35.i odd 6 2 392.1.j.a 2
35.j even 6 2 392.1.j.a 2
40.e odd 2 1 224.1.h.a 1
40.f even 2 1 56.1.h.a 1
40.i odd 4 2 1400.1.c.a 2
56.h odd 2 1 CM 1400.1.m.a 1
60.h even 2 1 2016.1.l.a 1
80.k odd 4 2 1792.1.c.a 1
80.q even 4 2 1792.1.c.b 1
105.g even 2 1 504.1.l.a 1
105.o odd 6 2 3528.1.bw.a 2
105.p even 6 2 3528.1.bw.a 2
120.i odd 2 1 504.1.l.a 1
120.m even 2 1 2016.1.l.a 1
140.c even 2 1 224.1.h.a 1
140.p odd 6 2 1568.1.n.a 2
140.s even 6 2 1568.1.n.a 2
280.c odd 2 1 56.1.h.a 1
280.n even 2 1 224.1.h.a 1
280.s even 4 2 1400.1.c.a 2
280.ba even 6 2 1568.1.n.a 2
280.bf even 6 2 392.1.j.a 2
280.bi odd 6 2 1568.1.n.a 2
280.bk odd 6 2 392.1.j.a 2
420.o odd 2 1 2016.1.l.a 1
560.be even 4 2 1792.1.c.a 1
560.bf odd 4 2 1792.1.c.b 1
840.b odd 2 1 2016.1.l.a 1
840.u even 2 1 504.1.l.a 1
840.cb even 6 2 3528.1.bw.a 2
840.cg odd 6 2 3528.1.bw.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.1.h.a 1 5.b even 2 1
56.1.h.a 1 35.c odd 2 1
56.1.h.a 1 40.f even 2 1
56.1.h.a 1 280.c odd 2 1
224.1.h.a 1 20.d odd 2 1
224.1.h.a 1 40.e odd 2 1
224.1.h.a 1 140.c even 2 1
224.1.h.a 1 280.n even 2 1
392.1.j.a 2 35.i odd 6 2
392.1.j.a 2 35.j even 6 2
392.1.j.a 2 280.bf even 6 2
392.1.j.a 2 280.bk odd 6 2
504.1.l.a 1 15.d odd 2 1
504.1.l.a 1 105.g even 2 1
504.1.l.a 1 120.i odd 2 1
504.1.l.a 1 840.u even 2 1
1400.1.c.a 2 5.c odd 4 2
1400.1.c.a 2 35.f even 4 2
1400.1.c.a 2 40.i odd 4 2
1400.1.c.a 2 280.s even 4 2
1400.1.m.a 1 1.a even 1 1 trivial
1400.1.m.a 1 7.b odd 2 1 CM
1400.1.m.a 1 8.b even 2 1 RM
1400.1.m.a 1 56.h odd 2 1 CM
1568.1.n.a 2 140.p odd 6 2
1568.1.n.a 2 140.s even 6 2
1568.1.n.a 2 280.ba even 6 2
1568.1.n.a 2 280.bi odd 6 2
1792.1.c.a 1 80.k odd 4 2
1792.1.c.a 1 560.be even 4 2
1792.1.c.b 1 80.q even 4 2
1792.1.c.b 1 560.bf odd 4 2
2016.1.l.a 1 60.h even 2 1
2016.1.l.a 1 120.m even 2 1
2016.1.l.a 1 420.o odd 2 1
2016.1.l.a 1 840.b odd 2 1
3528.1.bw.a 2 105.o odd 6 2
3528.1.bw.a 2 105.p even 6 2
3528.1.bw.a 2 840.cb even 6 2
3528.1.bw.a 2 840.cg odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{23} + 2 \) Copy content Toggle raw display
\( T_{113} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T + 2 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T + 2 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T + 2 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
show more
show less