Properties

Label 1425.1.t.a
Level $1425$
Weight $1$
Character orbit 1425.t
Analytic conductor $0.711$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1425,1,Mod(26,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, 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("1425.26");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1425.t (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: \(0.711167643002\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1083.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} + q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} + q^{7} - \zeta_{6} q^{9} - q^{12} - \zeta_{6} q^{13} + \zeta_{6}^{2} q^{16} + q^{19} - \zeta_{6}^{2} q^{21} - q^{27} - \zeta_{6} q^{28} - q^{31} + \zeta_{6}^{2} q^{36} + q^{37} - q^{39} + \zeta_{6}^{2} q^{43} + \zeta_{6} q^{48} + \zeta_{6}^{2} q^{52} - \zeta_{6}^{2} q^{57} + \zeta_{6} q^{61} - \zeta_{6} q^{63} + q^{64} - \zeta_{6} q^{67} + \zeta_{6}^{2} q^{73} - \zeta_{6} q^{76} - \zeta_{6}^{2} q^{79} + \zeta_{6}^{2} q^{81} - q^{84} - \zeta_{6} q^{91} + \zeta_{6}^{2} q^{93} - \zeta_{6}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - q^{4} + 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - q^{4} + 2 q^{7} - q^{9} - 2 q^{12} - q^{13} - q^{16} + 2 q^{19} + q^{21} - 2 q^{27} - q^{28} - 2 q^{31} - q^{36} + 2 q^{37} - 2 q^{39} - q^{43} + q^{48} - q^{52} + q^{57} + q^{61} - q^{63} + 2 q^{64} - q^{67} - q^{73} - q^{76} + q^{79} - q^{81} - 2 q^{84} - q^{91} - q^{93} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0 1.00000 0 −0.500000 + 0.866025i 0
1151.1 0 0.500000 0.866025i −0.500000 0.866025i 0 0 1.00000 0 −0.500000 0.866025i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
19.c even 3 1 inner
57.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1425.1.t.a 2
3.b odd 2 1 CM 1425.1.t.a 2
5.b even 2 1 57.1.h.a 2
5.c odd 4 2 1425.1.o.a 4
15.d odd 2 1 57.1.h.a 2
15.e even 4 2 1425.1.o.a 4
19.c even 3 1 inner 1425.1.t.a 2
20.d odd 2 1 912.1.bl.a 2
35.c odd 2 1 2793.1.bf.a 2
35.i odd 6 1 2793.1.n.b 2
35.i odd 6 1 2793.1.bi.a 2
35.j even 6 1 2793.1.n.a 2
35.j even 6 1 2793.1.bi.b 2
40.e odd 2 1 3648.1.bl.a 2
40.f even 2 1 3648.1.bl.b 2
45.h odd 6 1 1539.1.j.a 2
45.h odd 6 1 1539.1.n.a 2
45.j even 6 1 1539.1.j.a 2
45.j even 6 1 1539.1.n.a 2
57.h odd 6 1 inner 1425.1.t.a 2
60.h even 2 1 912.1.bl.a 2
95.d odd 2 1 1083.1.h.a 2
95.h odd 6 1 1083.1.b.a 1
95.h odd 6 1 1083.1.h.a 2
95.i even 6 1 57.1.h.a 2
95.i even 6 1 1083.1.b.b 1
95.m odd 12 2 1425.1.o.a 4
95.o odd 18 6 1083.1.l.b 6
95.p even 18 6 1083.1.l.a 6
105.g even 2 1 2793.1.bf.a 2
105.o odd 6 1 2793.1.n.a 2
105.o odd 6 1 2793.1.bi.b 2
105.p even 6 1 2793.1.n.b 2
105.p even 6 1 2793.1.bi.a 2
120.i odd 2 1 3648.1.bl.b 2
120.m even 2 1 3648.1.bl.a 2
285.b even 2 1 1083.1.h.a 2
285.n odd 6 1 57.1.h.a 2
285.n odd 6 1 1083.1.b.b 1
285.q even 6 1 1083.1.b.a 1
285.q even 6 1 1083.1.h.a 2
285.v even 12 2 1425.1.o.a 4
285.bd odd 18 6 1083.1.l.a 6
285.bf even 18 6 1083.1.l.b 6
380.p odd 6 1 912.1.bl.a 2
665.q odd 6 1 2793.1.bi.a 2
665.t even 6 1 2793.1.bi.b 2
665.bd even 6 1 2793.1.n.a 2
665.bh odd 6 1 2793.1.bf.a 2
665.bo odd 6 1 2793.1.n.b 2
760.z even 6 1 3648.1.bl.b 2
760.bm odd 6 1 3648.1.bl.a 2
855.s even 6 1 1539.1.n.a 2
855.ba odd 6 1 1539.1.j.a 2
855.bk even 6 1 1539.1.j.a 2
855.br odd 6 1 1539.1.n.a 2
1140.bn even 6 1 912.1.bl.a 2
1995.bl even 6 1 2793.1.bf.a 2
1995.bw even 6 1 2793.1.n.b 2
1995.bz odd 6 1 2793.1.n.a 2
1995.dh even 6 1 2793.1.bi.a 2
1995.di odd 6 1 2793.1.bi.b 2
2280.cf odd 6 1 3648.1.bl.b 2
2280.co even 6 1 3648.1.bl.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.1.h.a 2 5.b even 2 1
57.1.h.a 2 15.d odd 2 1
57.1.h.a 2 95.i even 6 1
57.1.h.a 2 285.n odd 6 1
912.1.bl.a 2 20.d odd 2 1
912.1.bl.a 2 60.h even 2 1
912.1.bl.a 2 380.p odd 6 1
912.1.bl.a 2 1140.bn even 6 1
1083.1.b.a 1 95.h odd 6 1
1083.1.b.a 1 285.q even 6 1
1083.1.b.b 1 95.i even 6 1
1083.1.b.b 1 285.n odd 6 1
1083.1.h.a 2 95.d odd 2 1
1083.1.h.a 2 95.h odd 6 1
1083.1.h.a 2 285.b even 2 1
1083.1.h.a 2 285.q even 6 1
1083.1.l.a 6 95.p even 18 6
1083.1.l.a 6 285.bd odd 18 6
1083.1.l.b 6 95.o odd 18 6
1083.1.l.b 6 285.bf even 18 6
1425.1.o.a 4 5.c odd 4 2
1425.1.o.a 4 15.e even 4 2
1425.1.o.a 4 95.m odd 12 2
1425.1.o.a 4 285.v even 12 2
1425.1.t.a 2 1.a even 1 1 trivial
1425.1.t.a 2 3.b odd 2 1 CM
1425.1.t.a 2 19.c even 3 1 inner
1425.1.t.a 2 57.h odd 6 1 inner
1539.1.j.a 2 45.h odd 6 1
1539.1.j.a 2 45.j even 6 1
1539.1.j.a 2 855.ba odd 6 1
1539.1.j.a 2 855.bk even 6 1
1539.1.n.a 2 45.h odd 6 1
1539.1.n.a 2 45.j even 6 1
1539.1.n.a 2 855.s even 6 1
1539.1.n.a 2 855.br odd 6 1
2793.1.n.a 2 35.j even 6 1
2793.1.n.a 2 105.o odd 6 1
2793.1.n.a 2 665.bd even 6 1
2793.1.n.a 2 1995.bz odd 6 1
2793.1.n.b 2 35.i odd 6 1
2793.1.n.b 2 105.p even 6 1
2793.1.n.b 2 665.bo odd 6 1
2793.1.n.b 2 1995.bw even 6 1
2793.1.bf.a 2 35.c odd 2 1
2793.1.bf.a 2 105.g even 2 1
2793.1.bf.a 2 665.bh odd 6 1
2793.1.bf.a 2 1995.bl even 6 1
2793.1.bi.a 2 35.i odd 6 1
2793.1.bi.a 2 105.p even 6 1
2793.1.bi.a 2 665.q odd 6 1
2793.1.bi.a 2 1995.dh even 6 1
2793.1.bi.b 2 35.j even 6 1
2793.1.bi.b 2 105.o odd 6 1
2793.1.bi.b 2 665.t even 6 1
2793.1.bi.b 2 1995.di odd 6 1
3648.1.bl.a 2 40.e odd 2 1
3648.1.bl.a 2 120.m even 2 1
3648.1.bl.a 2 760.bm odd 6 1
3648.1.bl.a 2 2280.co even 6 1
3648.1.bl.b 2 40.f even 2 1
3648.1.bl.b 2 120.i odd 2 1
3648.1.bl.b 2 760.z even 6 1
3648.1.bl.b 2 2280.cf odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(1425, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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