Properties

Label 2800.2.k.a
Level $2800$
Weight $2$
Character orbit 2800.k
Analytic conductor $22.358$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2800,2,Mod(2351,2800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2800.2351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.k (of order \(2\), degree \(1\), 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: \(22.3581125660\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{3} + ( - \beta + 2) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} + ( - \beta + 2) q^{7} + q^{9} - 3 \beta q^{11} - 2 \beta q^{13} - 8 q^{19} + (2 \beta - 4) q^{21} - 3 \beta q^{23} + 4 q^{27} - 9 q^{29} + 2 q^{31} + 6 \beta q^{33} + 7 q^{37} + 4 \beta q^{39} + 6 \beta q^{41} + \beta q^{43} + ( - 4 \beta + 1) q^{49} - 6 q^{53} + 16 q^{57} + 6 q^{59} + 2 \beta q^{61} + ( - \beta + 2) q^{63} - 7 \beta q^{67} + 6 \beta q^{69} - 3 \beta q^{71} + 6 \beta q^{73} + ( - 6 \beta - 9) q^{77} - 3 \beta q^{79} - 11 q^{81} + 6 q^{83} + 18 q^{87} + 6 \beta q^{89} + ( - 4 \beta - 6) q^{91} - 4 q^{93} + 2 \beta q^{97} - 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 4 q^{7} + 2 q^{9} - 16 q^{19} - 8 q^{21} + 8 q^{27} - 18 q^{29} + 4 q^{31} + 14 q^{37} + 2 q^{49} - 12 q^{53} + 32 q^{57} + 12 q^{59} + 4 q^{63} - 18 q^{77} - 22 q^{81} + 12 q^{83} + 36 q^{87} - 12 q^{91} - 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(351\) \(801\) \(2101\) \(2577\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2351.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −2.00000 0 0 0 2.00000 1.73205i 0 1.00000 0
2351.2 0 −2.00000 0 0 0 2.00000 + 1.73205i 0 1.00000 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
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2800.2.k.a 2
4.b odd 2 1 2800.2.k.f yes 2
5.b even 2 1 2800.2.k.d yes 2
5.c odd 4 2 2800.2.e.d 4
7.b odd 2 1 2800.2.k.f yes 2
20.d odd 2 1 2800.2.k.c yes 2
20.e even 4 2 2800.2.e.a 4
28.d even 2 1 inner 2800.2.k.a 2
35.c odd 2 1 2800.2.k.c yes 2
35.f even 4 2 2800.2.e.a 4
140.c even 2 1 2800.2.k.d yes 2
140.j odd 4 2 2800.2.e.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2800.2.e.a 4 20.e even 4 2
2800.2.e.a 4 35.f even 4 2
2800.2.e.d 4 5.c odd 4 2
2800.2.e.d 4 140.j odd 4 2
2800.2.k.a 2 1.a even 1 1 trivial
2800.2.k.a 2 28.d even 2 1 inner
2800.2.k.c yes 2 20.d odd 2 1
2800.2.k.c yes 2 35.c odd 2 1
2800.2.k.d yes 2 5.b even 2 1
2800.2.k.d yes 2 140.c even 2 1
2800.2.k.f yes 2 4.b odd 2 1
2800.2.k.f yes 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2800, [\chi])\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 27 \) Copy content Toggle raw display
\( T_{19} + 8 \) Copy content Toggle raw display
\( T_{37} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 27 \) Copy content Toggle raw display
$13$ \( T^{2} + 12 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 27 \) Copy content Toggle raw display
$29$ \( (T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T - 7)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 108 \) Copy content Toggle raw display
$43$ \( T^{2} + 3 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 12 \) Copy content Toggle raw display
$67$ \( T^{2} + 147 \) Copy content Toggle raw display
$71$ \( T^{2} + 27 \) Copy content Toggle raw display
$73$ \( T^{2} + 108 \) Copy content Toggle raw display
$79$ \( T^{2} + 27 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 108 \) Copy content Toggle raw display
$97$ \( T^{2} + 12 \) Copy content Toggle raw display
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