Properties

Label 1425.1.o.a
Level $1425$
Weight $1$
Character orbit 1425.o
Analytic conductor $0.711$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1425,1,Mod(524,1425)]
 
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, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1425.524");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1425.o (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: \(0.711167643002\)
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: \( x^{4} - x^{2} + 1 \) 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: $S_3\times C_{12}$
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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{4} - \zeta_{12}^{3} q^{7} - \zeta_{12}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{4} - \zeta_{12}^{3} q^{7} - \zeta_{12}^{4} q^{9} + \zeta_{12}^{3} q^{12} - \zeta_{12} q^{13} - \zeta_{12}^{2} q^{16} - q^{19} + \zeta_{12}^{2} q^{21} + \zeta_{12}^{3} q^{27} - \zeta_{12} q^{28} - q^{31} - \zeta_{12}^{2} q^{36} - \zeta_{12}^{3} q^{37} + q^{39} - \zeta_{12}^{5} q^{43} + \zeta_{12} q^{48} + \zeta_{12}^{5} q^{52} - \zeta_{12}^{5} q^{57} - \zeta_{12}^{4} q^{61} - \zeta_{12} q^{63} - q^{64} + \zeta_{12} q^{67} - \zeta_{12}^{5} q^{73} + \zeta_{12}^{4} q^{76} - \zeta_{12}^{2} q^{79} - \zeta_{12}^{2} q^{81} + q^{84} + \zeta_{12}^{4} q^{91} - \zeta_{12}^{5} q^{93} - 2 \zeta_{12}^{5} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{9} - 2 q^{16} - 4 q^{19} + 2 q^{21} - 4 q^{31} - 2 q^{36} + 4 q^{39} + 2 q^{61} - 4 q^{64} - 2 q^{76} - 2 q^{79} - 2 q^{81} + 4 q^{84} - 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{12}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
524.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 −0.866025 0.500000i 0.500000 + 0.866025i 0 0 1.00000i 0 0.500000 + 0.866025i 0
524.2 0 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0 1.00000i 0 0.500000 + 0.866025i 0
824.1 0 −0.866025 + 0.500000i 0.500000 0.866025i 0 0 1.00000i 0 0.500000 0.866025i 0
824.2 0 0.866025 0.500000i 0.500000 0.866025i 0 0 1.00000i 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 \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
15.d odd 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner
95.i even 6 1 inner
285.n 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.o.a 4
3.b odd 2 1 CM 1425.1.o.a 4
5.b even 2 1 inner 1425.1.o.a 4
5.c odd 4 1 57.1.h.a 2
5.c odd 4 1 1425.1.t.a 2
15.d odd 2 1 inner 1425.1.o.a 4
15.e even 4 1 57.1.h.a 2
15.e even 4 1 1425.1.t.a 2
19.c even 3 1 inner 1425.1.o.a 4
20.e even 4 1 912.1.bl.a 2
35.f even 4 1 2793.1.bf.a 2
35.k even 12 1 2793.1.n.b 2
35.k even 12 1 2793.1.bi.a 2
35.l odd 12 1 2793.1.n.a 2
35.l odd 12 1 2793.1.bi.b 2
40.i odd 4 1 3648.1.bl.b 2
40.k even 4 1 3648.1.bl.a 2
45.k odd 12 1 1539.1.j.a 2
45.k odd 12 1 1539.1.n.a 2
45.l even 12 1 1539.1.j.a 2
45.l even 12 1 1539.1.n.a 2
57.h odd 6 1 inner 1425.1.o.a 4
60.l odd 4 1 912.1.bl.a 2
95.g even 4 1 1083.1.h.a 2
95.i even 6 1 inner 1425.1.o.a 4
95.l even 12 1 1083.1.b.a 1
95.l even 12 1 1083.1.h.a 2
95.m odd 12 1 57.1.h.a 2
95.m odd 12 1 1083.1.b.b 1
95.m odd 12 1 1425.1.t.a 2
95.q odd 36 6 1083.1.l.a 6
95.r even 36 6 1083.1.l.b 6
105.k odd 4 1 2793.1.bf.a 2
105.w odd 12 1 2793.1.n.b 2
105.w odd 12 1 2793.1.bi.a 2
105.x even 12 1 2793.1.n.a 2
105.x even 12 1 2793.1.bi.b 2
120.q odd 4 1 3648.1.bl.a 2
120.w even 4 1 3648.1.bl.b 2
285.j odd 4 1 1083.1.h.a 2
285.n odd 6 1 inner 1425.1.o.a 4
285.v even 12 1 57.1.h.a 2
285.v even 12 1 1083.1.b.b 1
285.v even 12 1 1425.1.t.a 2
285.w odd 12 1 1083.1.b.a 1
285.w odd 12 1 1083.1.h.a 2
285.bi even 36 6 1083.1.l.a 6
285.bj odd 36 6 1083.1.l.b 6
380.v even 12 1 912.1.bl.a 2
665.bw odd 12 1 2793.1.bi.b 2
665.by even 12 1 2793.1.bi.a 2
665.cc odd 12 1 2793.1.n.a 2
665.cg even 12 1 2793.1.n.b 2
665.ci even 12 1 2793.1.bf.a 2
760.br odd 12 1 3648.1.bl.b 2
760.bw even 12 1 3648.1.bl.a 2
855.bw odd 12 1 1539.1.n.a 2
855.bx even 12 1 1539.1.n.a 2
855.cb odd 12 1 1539.1.j.a 2
855.ci even 12 1 1539.1.j.a 2
1140.bu odd 12 1 912.1.bl.a 2
1995.dr odd 12 1 2793.1.bi.a 2
1995.dt even 12 1 2793.1.bi.b 2
1995.dx odd 12 1 2793.1.bf.a 2
1995.ei odd 12 1 2793.1.n.b 2
1995.em even 12 1 2793.1.n.a 2
2280.dj odd 12 1 3648.1.bl.a 2
2280.ds even 12 1 3648.1.bl.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.1.h.a 2 5.c odd 4 1
57.1.h.a 2 15.e even 4 1
57.1.h.a 2 95.m odd 12 1
57.1.h.a 2 285.v even 12 1
912.1.bl.a 2 20.e even 4 1
912.1.bl.a 2 60.l odd 4 1
912.1.bl.a 2 380.v even 12 1
912.1.bl.a 2 1140.bu odd 12 1
1083.1.b.a 1 95.l even 12 1
1083.1.b.a 1 285.w odd 12 1
1083.1.b.b 1 95.m odd 12 1
1083.1.b.b 1 285.v even 12 1
1083.1.h.a 2 95.g even 4 1
1083.1.h.a 2 95.l even 12 1
1083.1.h.a 2 285.j odd 4 1
1083.1.h.a 2 285.w odd 12 1
1083.1.l.a 6 95.q odd 36 6
1083.1.l.a 6 285.bi even 36 6
1083.1.l.b 6 95.r even 36 6
1083.1.l.b 6 285.bj odd 36 6
1425.1.o.a 4 1.a even 1 1 trivial
1425.1.o.a 4 3.b odd 2 1 CM
1425.1.o.a 4 5.b even 2 1 inner
1425.1.o.a 4 15.d odd 2 1 inner
1425.1.o.a 4 19.c even 3 1 inner
1425.1.o.a 4 57.h odd 6 1 inner
1425.1.o.a 4 95.i even 6 1 inner
1425.1.o.a 4 285.n odd 6 1 inner
1425.1.t.a 2 5.c odd 4 1
1425.1.t.a 2 15.e even 4 1
1425.1.t.a 2 95.m odd 12 1
1425.1.t.a 2 285.v even 12 1
1539.1.j.a 2 45.k odd 12 1
1539.1.j.a 2 45.l even 12 1
1539.1.j.a 2 855.cb odd 12 1
1539.1.j.a 2 855.ci even 12 1
1539.1.n.a 2 45.k odd 12 1
1539.1.n.a 2 45.l even 12 1
1539.1.n.a 2 855.bw odd 12 1
1539.1.n.a 2 855.bx even 12 1
2793.1.n.a 2 35.l odd 12 1
2793.1.n.a 2 105.x even 12 1
2793.1.n.a 2 665.cc odd 12 1
2793.1.n.a 2 1995.em even 12 1
2793.1.n.b 2 35.k even 12 1
2793.1.n.b 2 105.w odd 12 1
2793.1.n.b 2 665.cg even 12 1
2793.1.n.b 2 1995.ei odd 12 1
2793.1.bf.a 2 35.f even 4 1
2793.1.bf.a 2 105.k odd 4 1
2793.1.bf.a 2 665.ci even 12 1
2793.1.bf.a 2 1995.dx odd 12 1
2793.1.bi.a 2 35.k even 12 1
2793.1.bi.a 2 105.w odd 12 1
2793.1.bi.a 2 665.by even 12 1
2793.1.bi.a 2 1995.dr odd 12 1
2793.1.bi.b 2 35.l odd 12 1
2793.1.bi.b 2 105.x even 12 1
2793.1.bi.b 2 665.bw odd 12 1
2793.1.bi.b 2 1995.dt even 12 1
3648.1.bl.a 2 40.k even 4 1
3648.1.bl.a 2 120.q odd 4 1
3648.1.bl.a 2 760.bw even 12 1
3648.1.bl.a 2 2280.dj odd 12 1
3648.1.bl.b 2 40.i odd 4 1
3648.1.bl.b 2 120.w even 4 1
3648.1.bl.b 2 760.br odd 12 1
3648.1.bl.b 2 2280.ds even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1425, [\chi])\).

Hecke characteristic polynomials

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