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Properties

Label 2800.1.di.d
Level $2800$
Weight $1$
Character orbit 2800.di
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(499,2800)]
 
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, 6, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2800.499");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2800.di (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: \(1.39738203537\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.0.2508800.2

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{12}^{2} q^{2} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{3} + \zeta_{12}^{4} q^{4} + ( - \zeta_{12}^{3} + 1) q^{6} + \zeta_{12}^{5} q^{7} - q^{8} + \zeta_{12}^{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{2} q^{2} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{3} + \zeta_{12}^{4} q^{4} + ( - \zeta_{12}^{3} + 1) q^{6} + \zeta_{12}^{5} q^{7} - q^{8} + \zeta_{12}^{5} q^{9} + ( - \zeta_{12}^{5} + \zeta_{12}^{2}) q^{12} - \zeta_{12} q^{14} - \zeta_{12}^{2} q^{16} - \zeta_{12} q^{17} - \zeta_{12} q^{18} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{19} + (\zeta_{12}^{3} + 1) q^{21} + \zeta_{12}^{5} q^{23} + (\zeta_{12}^{4} + \zeta_{12}) q^{24} + ( - \zeta_{12}^{3} - 1) q^{27} - \zeta_{12}^{3} q^{28} - \zeta_{12} q^{31} - \zeta_{12}^{4} q^{32} - \zeta_{12}^{3} q^{34} - \zeta_{12}^{3} q^{36} + ( - \zeta_{12}^{4} + \zeta_{12}) q^{38} - \zeta_{12}^{3} q^{41} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{42} + ( - \zeta_{12}^{3} - 1) q^{43} - \zeta_{12} q^{46} - \zeta_{12}^{2} q^{47} + (\zeta_{12}^{3} - 1) q^{48} - \zeta_{12}^{4} q^{49} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{51} + (\zeta_{12}^{4} + \zeta_{12}) q^{53} - \zeta_{12}^{5} q^{56} + ( - \zeta_{12}^{3} - 2) q^{57} + (\zeta_{12}^{4} + \zeta_{12}) q^{59} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{61} - \zeta_{12}^{3} q^{62} - \zeta_{12}^{4} q^{63} + q^{64} - \zeta_{12}^{5} q^{68} + (\zeta_{12}^{3} + 1) q^{69} - q^{71} - \zeta_{12}^{5} q^{72} + (\zeta_{12}^{3} + 1) q^{76} + \zeta_{12}^{5} q^{79} + \zeta_{12}^{4} q^{81} - \zeta_{12}^{5} q^{82} + ( - \zeta_{12}^{3} + 1) q^{83} + (\zeta_{12}^{4} - \zeta_{12}) q^{84} + ( - \zeta_{12}^{5} - \zeta_{12}^{2}) q^{86} + \zeta_{12}^{5} q^{89} - \zeta_{12}^{3} q^{92} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{93} - \zeta_{12}^{4} q^{94} + (\zeta_{12}^{5} - \zeta_{12}^{2}) q^{96} + \zeta_{12}^{3} q^{97} + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{6} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{6} - 4 q^{8} + 2 q^{12} - 2 q^{16} - 2 q^{19} + 4 q^{21} - 2 q^{24} + 2 q^{32} + 2 q^{38} + 2 q^{42} - 4 q^{43} - 2 q^{47} - 4 q^{48} + 2 q^{49} + 2 q^{51} - 2 q^{53} - 8 q^{57} - 2 q^{59} - 2 q^{61} + 2 q^{63} + 4 q^{64} + 4 q^{69} - 4 q^{71} + 4 q^{76} - 2 q^{81} + 4 q^{83} - 2 q^{84} - 2 q^{86} + 2 q^{93} + 2 q^{94} - 2 q^{96} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
0.866025 − 0.500000i
−0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.500000 − 0.866025i −0.366025 + 1.36603i −0.500000 − 0.866025i 0 1.00000 + 1.00000i −0.866025 − 0.500000i −1.00000 −0.866025 − 0.500000i 0
1299.1 0.500000 + 0.866025i 1.36603 − 0.366025i −0.500000 + 0.866025i 0 1.00000 + 1.00000i 0.866025 − 0.500000i −1.00000 0.866025 − 0.500000i 0
1899.1 0.500000 − 0.866025i 1.36603 + 0.366025i −0.500000 − 0.866025i 0 1.00000 − 1.00000i 0.866025 + 0.500000i −1.00000 0.866025 + 0.500000i 0
2699.1 0.500000 + 0.866025i −0.366025 − 1.36603i −0.500000 + 0.866025i 0 1.00000 − 1.00000i −0.866025 + 0.500000i −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
80.k odd 4 1 inner
560.cs 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.di.d 4
5.b even 2 1 2800.1.di.a 4
5.c odd 4 1 2800.1.dc.b ✓ 4
5.c odd 4 1 2800.1.dc.c yes 4
7.c even 3 1 inner 2800.1.di.d 4
16.f odd 4 1 2800.1.di.a 4
35.j even 6 1 2800.1.di.a 4
35.l odd 12 1 2800.1.dc.b ✓ 4
35.l odd 12 1 2800.1.dc.c yes 4
80.j even 4 1 2800.1.dc.b ✓ 4
80.k odd 4 1 inner 2800.1.di.d 4
80.s even 4 1 2800.1.dc.c yes 4
112.u odd 12 1 2800.1.di.a 4
560.cf even 12 1 2800.1.dc.c yes 4
560.cs odd 12 1 inner 2800.1.di.d 4
560.db even 12 1 2800.1.dc.b ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2800.1.dc.b ✓ 4 5.c odd 4 1
2800.1.dc.b ✓ 4 35.l odd 12 1
2800.1.dc.b ✓ 4 80.j even 4 1
2800.1.dc.b ✓ 4 560.db even 12 1
2800.1.dc.c yes 4 5.c odd 4 1
2800.1.dc.c yes 4 35.l odd 12 1
2800.1.dc.c yes 4 80.s even 4 1
2800.1.dc.c yes 4 560.cf even 12 1
2800.1.di.a 4 5.b even 2 1
2800.1.di.a 4 16.f odd 4 1
2800.1.di.a 4 35.j even 6 1
2800.1.di.a 4 112.u odd 12 1
2800.1.di.d 4 1.a even 1 1 trivial
2800.1.di.d 4 7.c even 3 1 inner
2800.1.di.d 4 80.k odd 4 1 inner
2800.1.di.d 4 560.cs odd 12 1 inner

Hecke kernels

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

\( T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} - 4T_{3} + 4 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display

Hecke characteristic polynomials

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