Properties

Label 2700.2.a.w
Level $2700$
Weight $2$
Character orbit 2700.a
Self dual yes
Analytic conductor $21.560$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2700,2,Mod(1,2700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2700, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2700.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2700.a (trivial)

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: yes
Analytic conductor: \(21.5596085457\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 540)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{7} - \beta q^{11} + 3 q^{13} - 2 \beta q^{17} + 3 q^{19} + \beta q^{23} + 3 \beta q^{29} - 2 q^{31} + q^{37} - \beta q^{41} + 10 q^{43} + 2 \beta q^{47} - 6 q^{49} + 3 \beta q^{53} - 2 \beta q^{59} - q^{61} + 11 q^{67} + 3 \beta q^{71} + 13 q^{73} - \beta q^{77} - 3 q^{79} - 5 \beta q^{83} - 4 \beta q^{89} + 3 q^{91} - q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{7} + 6 q^{13} + 6 q^{19} - 4 q^{31} + 2 q^{37} + 20 q^{43} - 12 q^{49} - 2 q^{61} + 22 q^{67} + 26 q^{73} - 6 q^{79} + 6 q^{91} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.16228
−3.16228
0 0 0 0 0 1.00000 0 0 0
1.2 0 0 0 0 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.2.a.w 2
3.b odd 2 1 inner 2700.2.a.w 2
5.b even 2 1 2700.2.a.v 2
5.c odd 4 2 540.2.d.c 4
15.d odd 2 1 2700.2.a.v 2
15.e even 4 2 540.2.d.c 4
20.e even 4 2 2160.2.f.l 4
45.k odd 12 4 1620.2.r.g 8
45.l even 12 4 1620.2.r.g 8
60.l odd 4 2 2160.2.f.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.d.c 4 5.c odd 4 2
540.2.d.c 4 15.e even 4 2
1620.2.r.g 8 45.k odd 12 4
1620.2.r.g 8 45.l even 12 4
2160.2.f.l 4 20.e even 4 2
2160.2.f.l 4 60.l odd 4 2
2700.2.a.v 2 5.b even 2 1
2700.2.a.v 2 15.d odd 2 1
2700.2.a.w 2 1.a even 1 1 trivial
2700.2.a.w 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2700))\):

\( T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 10 \) Copy content Toggle raw display
\( T_{13} - 3 \) Copy content Toggle raw display
\( T_{17}^{2} - 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 10 \) Copy content Toggle raw display
$13$ \( (T - 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 40 \) Copy content Toggle raw display
$19$ \( (T - 3)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 10 \) Copy content Toggle raw display
$29$ \( T^{2} - 90 \) Copy content Toggle raw display
$31$ \( (T + 2)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 10 \) Copy content Toggle raw display
$43$ \( (T - 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 40 \) Copy content Toggle raw display
$53$ \( T^{2} - 90 \) Copy content Toggle raw display
$59$ \( T^{2} - 40 \) Copy content Toggle raw display
$61$ \( (T + 1)^{2} \) Copy content Toggle raw display
$67$ \( (T - 11)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 90 \) Copy content Toggle raw display
$73$ \( (T - 13)^{2} \) Copy content Toggle raw display
$79$ \( (T + 3)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 250 \) Copy content Toggle raw display
$89$ \( T^{2} - 160 \) Copy content Toggle raw display
$97$ \( (T + 1)^{2} \) Copy content Toggle raw display
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