Properties

Label 1521.1.bd.c
Level $1521$
Weight $1$
Character orbit 1521.bd
Analytic conductor $0.759$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
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] = [1521,1,Mod(19,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1521.19");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1521.bd (of order \(12\), degree \(4\), 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: \(0.759077884215\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 117)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.6591.1
Artin image: $D_4:C_{12}$
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{4} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{4} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{16} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{19} - \zeta_{12}^{3} q^{25} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{28} + (\zeta_{12}^{3} + 1) q^{31} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{37} + \zeta_{12}^{5} q^{49} + \zeta_{12}^{3} q^{64} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{67} + (\zeta_{12}^{3} - 1) q^{73} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{76} + (\zeta_{12}^{4} - \zeta_{12}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{7} + 2 q^{16} + 2 q^{19} - 2 q^{28} + 4 q^{31} - 2 q^{37} + 2 q^{67} - 4 q^{73} + 2 q^{76} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-\zeta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 0 0.866025 0.500000i 0 0 −0.366025 + 1.36603i 0 0 0
1333.1 0 0 −0.866025 + 0.500000i 0 0 1.36603 + 0.366025i 0 0 0
1432.1 0 0 −0.866025 0.500000i 0 0 1.36603 0.366025i 0 0 0
1441.1 0 0 0.866025 + 0.500000i 0 0 −0.366025 1.36603i 0 0 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 \(\Q(\sqrt{-3}) \)
13.c even 3 1 inner
13.d odd 4 1 inner
13.f odd 12 1 inner
39.f even 4 1 inner
39.i odd 6 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.1.bd.c 4
3.b odd 2 1 CM 1521.1.bd.c 4
13.b even 2 1 1521.1.bd.b 4
13.c even 3 1 117.1.j.a 2
13.c even 3 1 inner 1521.1.bd.c 4
13.d odd 4 1 1521.1.bd.b 4
13.d odd 4 1 inner 1521.1.bd.c 4
13.e even 6 1 1521.1.j.b 2
13.e even 6 1 1521.1.bd.b 4
13.f odd 12 1 117.1.j.a 2
13.f odd 12 1 1521.1.j.b 2
13.f odd 12 1 1521.1.bd.b 4
13.f odd 12 1 inner 1521.1.bd.c 4
39.d odd 2 1 1521.1.bd.b 4
39.f even 4 1 1521.1.bd.b 4
39.f even 4 1 inner 1521.1.bd.c 4
39.h odd 6 1 1521.1.j.b 2
39.h odd 6 1 1521.1.bd.b 4
39.i odd 6 1 117.1.j.a 2
39.i odd 6 1 inner 1521.1.bd.c 4
39.k even 12 1 117.1.j.a 2
39.k even 12 1 1521.1.j.b 2
39.k even 12 1 1521.1.bd.b 4
39.k even 12 1 inner 1521.1.bd.c 4
52.j odd 6 1 1872.1.bd.a 2
52.l even 12 1 1872.1.bd.a 2
65.n even 6 1 2925.1.s.a 2
65.o even 12 1 2925.1.t.b 2
65.q odd 12 1 2925.1.t.a 2
65.q odd 12 1 2925.1.t.b 2
65.s odd 12 1 2925.1.s.a 2
65.t even 12 1 2925.1.t.a 2
117.f even 3 1 1053.1.bb.a 4
117.h even 3 1 1053.1.bb.a 4
117.k odd 6 1 1053.1.bb.a 4
117.u odd 6 1 1053.1.bb.a 4
117.w odd 12 1 1053.1.bb.a 4
117.x even 12 1 1053.1.bb.a 4
117.bb odd 12 1 1053.1.bb.a 4
117.bc even 12 1 1053.1.bb.a 4
156.p even 6 1 1872.1.bd.a 2
156.v odd 12 1 1872.1.bd.a 2
195.x odd 6 1 2925.1.s.a 2
195.bc odd 12 1 2925.1.t.a 2
195.bh even 12 1 2925.1.s.a 2
195.bl even 12 1 2925.1.t.a 2
195.bl even 12 1 2925.1.t.b 2
195.bn odd 12 1 2925.1.t.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.1.j.a 2 13.c even 3 1
117.1.j.a 2 13.f odd 12 1
117.1.j.a 2 39.i odd 6 1
117.1.j.a 2 39.k even 12 1
1053.1.bb.a 4 117.f even 3 1
1053.1.bb.a 4 117.h even 3 1
1053.1.bb.a 4 117.k odd 6 1
1053.1.bb.a 4 117.u odd 6 1
1053.1.bb.a 4 117.w odd 12 1
1053.1.bb.a 4 117.x even 12 1
1053.1.bb.a 4 117.bb odd 12 1
1053.1.bb.a 4 117.bc even 12 1
1521.1.j.b 2 13.e even 6 1
1521.1.j.b 2 13.f odd 12 1
1521.1.j.b 2 39.h odd 6 1
1521.1.j.b 2 39.k even 12 1
1521.1.bd.b 4 13.b even 2 1
1521.1.bd.b 4 13.d odd 4 1
1521.1.bd.b 4 13.e even 6 1
1521.1.bd.b 4 13.f odd 12 1
1521.1.bd.b 4 39.d odd 2 1
1521.1.bd.b 4 39.f even 4 1
1521.1.bd.b 4 39.h odd 6 1
1521.1.bd.b 4 39.k even 12 1
1521.1.bd.c 4 1.a even 1 1 trivial
1521.1.bd.c 4 3.b odd 2 1 CM
1521.1.bd.c 4 13.c even 3 1 inner
1521.1.bd.c 4 13.d odd 4 1 inner
1521.1.bd.c 4 13.f odd 12 1 inner
1521.1.bd.c 4 39.f even 4 1 inner
1521.1.bd.c 4 39.i odd 6 1 inner
1521.1.bd.c 4 39.k even 12 1 inner
1872.1.bd.a 2 52.j odd 6 1
1872.1.bd.a 2 52.l even 12 1
1872.1.bd.a 2 156.p even 6 1
1872.1.bd.a 2 156.v odd 12 1
2925.1.s.a 2 65.n even 6 1
2925.1.s.a 2 65.s odd 12 1
2925.1.s.a 2 195.x odd 6 1
2925.1.s.a 2 195.bh even 12 1
2925.1.t.a 2 65.q odd 12 1
2925.1.t.a 2 65.t even 12 1
2925.1.t.a 2 195.bc odd 12 1
2925.1.t.a 2 195.bl even 12 1
2925.1.t.b 2 65.o even 12 1
2925.1.t.b 2 65.q odd 12 1
2925.1.t.b 2 195.bl even 12 1
2925.1.t.b 2 195.bn odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1521, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{7}^{4} - 2T_{7}^{3} + 2T_{7}^{2} - 4T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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