Properties

Label 1900.1.bf.a
Level $1900$
Weight $1$
Character orbit 1900.bf
Analytic conductor $0.948$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1900,1,Mod(657,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 8]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1900.657");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1900.bf (of order \(12\), degree \(4\), 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: \(0.948223524003\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 380)
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.0.180500.1

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{4} + \zeta_{12}) q^{3} - \zeta_{12}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{4} + \zeta_{12}) q^{3} - \zeta_{12}^{5} q^{9} + q^{11} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{13} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{17} - \zeta_{12}^{5} q^{19} + (\zeta_{12}^{3} - 1) q^{27} + \zeta_{12}^{5} q^{29} - q^{31} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{33} + \zeta_{12}^{3} q^{39} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{47} + \zeta_{12}^{3} q^{49} - \zeta_{12}^{2} q^{51} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{53} + ( - \zeta_{12}^{3} + 1) q^{57} - \zeta_{12} q^{59} - \zeta_{12}^{2} q^{61} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{67} + \zeta_{12}^{4} q^{71} + \zeta_{12} q^{79} + \zeta_{12}^{4} q^{81} + (\zeta_{12}^{3} - 1) q^{87} - \zeta_{12}^{5} q^{89} + (\zeta_{12}^{4} - \zeta_{12}) q^{93} - \zeta_{12}^{5} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{11} + 2 q^{13} + 2 q^{17} - 4 q^{31} + 2 q^{33} + 2 q^{47} - 4 q^{51} - 2 q^{53} + 4 q^{57} - 2 q^{61} - 2 q^{67} - 2 q^{71} - 2 q^{81} - 4 q^{87} - 2 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-\zeta_{12}^{3}\) \(-\zeta_{12}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
657.1
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 −0.366025 1.36603i 0 0 0 0 0 −0.866025 + 0.500000i 0
957.1 0 1.36603 + 0.366025i 0 0 0 0 0 0.866025 + 0.500000i 0
1493.1 0 1.36603 0.366025i 0 0 0 0 0 0.866025 0.500000i 0
1793.1 0 −0.366025 + 1.36603i 0 0 0 0 0 −0.866025 0.500000i 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
5.c odd 4 1 inner
19.c even 3 1 inner
95.m odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.1.bf.a 4
5.b even 2 1 380.1.x.a 4
5.c odd 4 1 380.1.x.a 4
5.c odd 4 1 inner 1900.1.bf.a 4
15.d odd 2 1 3420.1.es.b 4
15.e even 4 1 3420.1.es.b 4
19.c even 3 1 inner 1900.1.bf.a 4
20.d odd 2 1 1520.1.cr.b 4
20.e even 4 1 1520.1.cr.b 4
95.i even 6 1 380.1.x.a 4
95.m odd 12 1 380.1.x.a 4
95.m odd 12 1 inner 1900.1.bf.a 4
285.n odd 6 1 3420.1.es.b 4
285.v even 12 1 3420.1.es.b 4
380.p odd 6 1 1520.1.cr.b 4
380.v even 12 1 1520.1.cr.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.1.x.a 4 5.b even 2 1
380.1.x.a 4 5.c odd 4 1
380.1.x.a 4 95.i even 6 1
380.1.x.a 4 95.m odd 12 1
1520.1.cr.b 4 20.d odd 2 1
1520.1.cr.b 4 20.e even 4 1
1520.1.cr.b 4 380.p odd 6 1
1520.1.cr.b 4 380.v even 12 1
1900.1.bf.a 4 1.a even 1 1 trivial
1900.1.bf.a 4 5.c odd 4 1 inner
1900.1.bf.a 4 19.c even 3 1 inner
1900.1.bf.a 4 95.m odd 12 1 inner
3420.1.es.b 4 15.d odd 2 1
3420.1.es.b 4 15.e even 4 1
3420.1.es.b 4 285.n odd 6 1
3420.1.es.b 4 285.v even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1900, [\chi])\).

Hecke characteristic polynomials

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