Properties

Label 2800.1.dc.a
Level $2800$
Weight $1$
Character orbit 2800.dc
Analytic conductor $1.397$
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] = [2800,1,Mod(51,2800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2800, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 0, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2800.51");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2800.dc (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: \(1.39738203537\)
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: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.0.2508800.3

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12}^{2} q^{2} + \zeta_{12}^{4} q^{4} + \zeta_{12}^{2} q^{7} + q^{8} + \zeta_{12}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12}^{2} q^{2} + \zeta_{12}^{4} q^{4} + \zeta_{12}^{2} q^{7} + q^{8} + \zeta_{12}^{5} q^{9} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{11} + ( - \zeta_{12}^{3} + 1) q^{13} - \zeta_{12}^{4} q^{14} - \zeta_{12}^{2} q^{16} + \zeta_{12}^{4} q^{17} + \zeta_{12} q^{18} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{19} + (\zeta_{12}^{3} - 1) q^{22} + \zeta_{12}^{2} q^{23} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{26} - q^{28} + \zeta_{12} q^{31} + \zeta_{12}^{4} q^{32} + q^{34} - \zeta_{12}^{3} q^{36} + (\zeta_{12}^{4} - \zeta_{12}) q^{38} - \zeta_{12}^{3} q^{41} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{44} - \zeta_{12}^{4} q^{46} - \zeta_{12}^{5} q^{47} + \zeta_{12}^{4} q^{49} + (\zeta_{12}^{4} + \zeta_{12}) q^{52} + \zeta_{12}^{2} q^{56} - \zeta_{12}^{3} q^{62} - \zeta_{12} q^{63} + q^{64} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{67} - \zeta_{12}^{2} q^{68} + q^{71} + \zeta_{12}^{5} q^{72} + \zeta_{12} q^{73} + (\zeta_{12}^{3} + 1) q^{76} + ( - \zeta_{12}^{3} + 1) q^{77} - \zeta_{12}^{5} q^{79} - \zeta_{12}^{4} q^{81} + \zeta_{12}^{5} q^{82} + (\zeta_{12}^{3} + 1) q^{83} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{88} + \zeta_{12}^{5} q^{89} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{91} - q^{92} - \zeta_{12} q^{94} + q^{97} + q^{98} + (\zeta_{12}^{3} + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} + 2 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{4} + 2 q^{7} + 4 q^{8} + 2 q^{11} + 4 q^{13} + 2 q^{14} - 2 q^{16} - 2 q^{17} - 2 q^{19} - 4 q^{22} + 2 q^{23} - 2 q^{26} - 4 q^{28} - 2 q^{32} + 4 q^{34} - 2 q^{38} + 2 q^{44} + 2 q^{46} - 2 q^{49} - 2 q^{52} + 2 q^{56} + 4 q^{64} + 2 q^{67} - 2 q^{68} + 4 q^{71} + 4 q^{76} + 4 q^{77} + 2 q^{81} + 4 q^{83} + 2 q^{88} + 2 q^{91} - 4 q^{92} + 4 q^{97} + 4 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(\zeta_{12}^{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 0.500000 0.866025i 1.00000 −0.866025 0.500000i 0
851.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 0.500000 + 0.866025i 1.00000 0.866025 0.500000i 0
1451.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 0.500000 0.866025i 1.00000 0.866025 + 0.500000i 0
2251.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 0.500000 + 0.866025i 1.00000 −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
7.c even 3 1 inner
16.f odd 4 1 inner
112.u odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.1.dc.a 4
5.b even 2 1 2800.1.dc.d yes 4
5.c odd 4 1 2800.1.di.b 4
5.c odd 4 1 2800.1.di.c 4
7.c even 3 1 inner 2800.1.dc.a 4
16.f odd 4 1 inner 2800.1.dc.a 4
35.j even 6 1 2800.1.dc.d yes 4
35.l odd 12 1 2800.1.di.b 4
35.l odd 12 1 2800.1.di.c 4
80.j even 4 1 2800.1.di.c 4
80.k odd 4 1 2800.1.dc.d yes 4
80.s even 4 1 2800.1.di.b 4
112.u odd 12 1 inner 2800.1.dc.a 4
560.cf even 12 1 2800.1.di.b 4
560.cs odd 12 1 2800.1.dc.d yes 4
560.db even 12 1 2800.1.di.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2800.1.dc.a 4 1.a even 1 1 trivial
2800.1.dc.a 4 7.c even 3 1 inner
2800.1.dc.a 4 16.f odd 4 1 inner
2800.1.dc.a 4 112.u odd 12 1 inner
2800.1.dc.d yes 4 5.b even 2 1
2800.1.dc.d yes 4 35.j even 6 1
2800.1.dc.d yes 4 80.k odd 4 1
2800.1.dc.d yes 4 560.cs odd 12 1
2800.1.di.b 4 5.c odd 4 1
2800.1.di.b 4 35.l odd 12 1
2800.1.di.b 4 80.s even 4 1
2800.1.di.b 4 560.cf even 12 1
2800.1.di.c 4 5.c odd 4 1
2800.1.di.c 4 35.l odd 12 1
2800.1.di.c 4 80.j even 4 1
2800.1.di.c 4 560.db even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2800, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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