Properties

Label 2760.2.k.a
Level $2760$
Weight $2$
Character orbit 2760.k
Analytic conductor $22.039$
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] = [2760,2,Mod(2209,2760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2760, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2760.2209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2760.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: \(22.0387109579\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{3} + (i - 2) q^{5} + 2 i q^{7} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{3} + (i - 2) q^{5} + 2 i q^{7} - q^{9} - 4 q^{11} + (2 i + 1) q^{15} - 2 i q^{17} + 4 q^{19} + 2 q^{21} - i q^{23} + ( - 4 i + 3) q^{25} + i q^{27} - 6 q^{29} - 4 q^{31} + 4 i q^{33} + ( - 4 i - 2) q^{35} + 6 i q^{37} - 2 q^{41} + 2 i q^{43} + ( - i + 2) q^{45} + 3 q^{49} - 2 q^{51} - 6 i q^{53} + ( - 4 i + 8) q^{55} - 4 i q^{57} + 14 q^{59} + 10 q^{61} - 2 i q^{63} - 14 i q^{67} - q^{69} + 10 q^{71} + ( - 3 i - 4) q^{75} - 8 i q^{77} + 16 q^{79} + q^{81} - 16 i q^{83} + (4 i + 2) q^{85} + 6 i q^{87} + 8 q^{89} + 4 i q^{93} + (4 i - 8) q^{95} - 10 i q^{97} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} - 2 q^{9} - 8 q^{11} + 2 q^{15} + 8 q^{19} + 4 q^{21} + 6 q^{25} - 12 q^{29} - 8 q^{31} - 4 q^{35} - 4 q^{41} + 4 q^{45} + 6 q^{49} - 4 q^{51} + 16 q^{55} + 28 q^{59} + 20 q^{61} - 2 q^{69} + 20 q^{71} - 8 q^{75} + 32 q^{79} + 2 q^{81} + 4 q^{85} + 16 q^{89} - 16 q^{95} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times\).

\(n\) \(1201\) \(1381\) \(1657\) \(1841\) \(2071\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2209.1
1.00000i
1.00000i
0 1.00000i 0 −2.00000 + 1.00000i 0 2.00000i 0 −1.00000 0
2209.2 0 1.00000i 0 −2.00000 1.00000i 0 2.00000i 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2760.2.k.a 2
5.b even 2 1 inner 2760.2.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.k.a 2 1.a even 1 1 trivial
2760.2.k.a 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(2760, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 36 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 4 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T - 14)^{2} \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 196 \) Copy content Toggle raw display
$71$ \( (T - 10)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T - 8)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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