Properties

Label 2760.1.co.a
Level $2760$
Weight $1$
Character orbit 2760.co
Analytic conductor $1.377$
Analytic rank $0$
Dimension $10$
Projective image $D_{22}$
CM discriminant -15
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2760,1,Mod(419,2760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2760, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 11, 11, 11, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2760.419");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2760.co (of order \(22\), degree \(10\), 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: \(1.37741943487\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{22}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{22} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{22}^{5} q^{2} - \zeta_{22}^{2} q^{3} + \zeta_{22}^{10} q^{4} - \zeta_{22}^{7} q^{5} + \zeta_{22}^{7} q^{6} + \zeta_{22}^{4} q^{8} + \zeta_{22}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{22}^{5} q^{2} - \zeta_{22}^{2} q^{3} + \zeta_{22}^{10} q^{4} - \zeta_{22}^{7} q^{5} + \zeta_{22}^{7} q^{6} + \zeta_{22}^{4} q^{8} + \zeta_{22}^{4} q^{9} - \zeta_{22} q^{10} + \zeta_{22} q^{12} + \zeta_{22}^{9} q^{15} - \zeta_{22}^{9} q^{16} + ( - \zeta_{22}^{10} + 1) q^{17} - \zeta_{22}^{9} q^{18} + (\zeta_{22}^{7} - \zeta_{22}^{5}) q^{19} + \zeta_{22}^{6} q^{20} - \zeta_{22}^{8} q^{23} - \zeta_{22}^{6} q^{24} - \zeta_{22}^{3} q^{25} - \zeta_{22}^{6} q^{27} + \zeta_{22}^{3} q^{30} + ( - \zeta_{22}^{10} + \zeta_{22}^{8}) q^{31} - \zeta_{22}^{3} q^{32} + ( - \zeta_{22}^{5} - \zeta_{22}^{4}) q^{34} - \zeta_{22}^{3} q^{36} + (\zeta_{22}^{10} + \zeta_{22}) q^{38} + q^{40} + q^{45} - \zeta_{22}^{2} q^{46} + ( - \zeta_{22}^{8} - \zeta_{22}^{3}) q^{47} - q^{48} + \zeta_{22}^{2} q^{49} + \zeta_{22}^{8} q^{50} + ( - \zeta_{22}^{2} - \zeta_{22}) q^{51} + (\zeta_{22}^{9} - \zeta_{22}^{4}) q^{53} - q^{54} + ( - \zeta_{22}^{9} + \zeta_{22}^{7}) q^{57} - \zeta_{22}^{8} q^{60} + ( - \zeta_{22}^{6} + \zeta_{22}) q^{61} + ( - \zeta_{22}^{4} + \zeta_{22}^{2}) q^{62} + \zeta_{22}^{8} q^{64} + (\zeta_{22}^{10} + \zeta_{22}^{9}) q^{68} + \zeta_{22}^{10} q^{69} + \zeta_{22}^{8} q^{72} + \zeta_{22}^{5} q^{75} + ( - \zeta_{22}^{6} + \zeta_{22}^{4}) q^{76} + (\zeta_{22}^{6} - \zeta_{22}^{3}) q^{79} - \zeta_{22}^{5} q^{80} + \zeta_{22}^{8} q^{81} + (\zeta_{22}^{6} - \zeta_{22}^{2}) q^{83} + ( - \zeta_{22}^{7} - \zeta_{22}^{6}) q^{85} - \zeta_{22}^{5} q^{90} + \zeta_{22}^{7} q^{92} + ( - \zeta_{22}^{10} - \zeta_{22}) q^{93} + (\zeta_{22}^{8} - \zeta_{22}^{2}) q^{94} + (\zeta_{22}^{3} - \zeta_{22}) q^{95} + \zeta_{22}^{5} q^{96} - \zeta_{22}^{7} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - q^{8} - q^{9} - q^{10} + q^{12} + q^{15} - q^{16} + 11 q^{17} - q^{18} - q^{20} + q^{23} + q^{24} - q^{25} + q^{27} + q^{30} - q^{32} - q^{36} + 10 q^{40} + 10 q^{45} + q^{46} - 10 q^{48} - q^{49} - q^{50} + 2 q^{53} - 10 q^{54} + q^{60} + 2 q^{61} - q^{64} - q^{69} - q^{72} + q^{75} - 2 q^{79} - q^{80} - q^{81} - q^{90} + q^{92} + q^{96} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1201\) \(1381\) \(1657\) \(1841\) \(2071\)
\(\chi(n)\) \(\zeta_{22}^{3}\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1
−0.415415 0.909632i
0.959493 + 0.281733i
−0.841254 0.540641i
−0.841254 + 0.540641i
0.959493 0.281733i
0.142315 0.989821i
−0.415415 + 0.909632i
0.142315 + 0.989821i
0.654861 + 0.755750i
0.654861 0.755750i
0.841254 0.540641i 0.654861 0.755750i 0.415415 0.909632i −0.142315 + 0.989821i 0.142315 0.989821i 0 −0.142315 0.989821i −0.142315 0.989821i 0.415415 + 0.909632i
539.1 −0.142315 0.989821i −0.841254 0.540641i −0.959493 + 0.281733i 0.415415 0.909632i −0.415415 + 0.909632i 0 0.415415 + 0.909632i 0.415415 + 0.909632i −0.959493 0.281733i
659.1 −0.959493 + 0.281733i −0.415415 0.909632i 0.841254 0.540641i −0.654861 0.755750i 0.654861 + 0.755750i 0 −0.654861 + 0.755750i −0.654861 + 0.755750i 0.841254 + 0.540641i
779.1 −0.959493 0.281733i −0.415415 + 0.909632i 0.841254 + 0.540641i −0.654861 + 0.755750i 0.654861 0.755750i 0 −0.654861 0.755750i −0.654861 0.755750i 0.841254 0.540641i
1019.1 −0.142315 + 0.989821i −0.841254 + 0.540641i −0.959493 0.281733i 0.415415 + 0.909632i −0.415415 0.909632i 0 0.415415 0.909632i 0.415415 0.909632i −0.959493 + 0.281733i
1259.1 −0.654861 + 0.755750i 0.959493 + 0.281733i −0.142315 0.989821i 0.841254 0.540641i −0.841254 + 0.540641i 0 0.841254 + 0.540641i 0.841254 + 0.540641i −0.142315 + 0.989821i
1739.1 0.841254 + 0.540641i 0.654861 + 0.755750i 0.415415 + 0.909632i −0.142315 0.989821i 0.142315 + 0.989821i 0 −0.142315 + 0.989821i −0.142315 + 0.989821i 0.415415 0.909632i
1859.1 −0.654861 0.755750i 0.959493 0.281733i −0.142315 + 0.989821i 0.841254 + 0.540641i −0.841254 0.540641i 0 0.841254 0.540641i 0.841254 0.540641i −0.142315 0.989821i
2219.1 0.415415 + 0.909632i 0.142315 0.989821i −0.654861 + 0.755750i −0.959493 + 0.281733i 0.959493 0.281733i 0 −0.959493 0.281733i −0.959493 0.281733i −0.654861 0.755750i
2459.1 0.415415 0.909632i 0.142315 + 0.989821i −0.654861 0.755750i −0.959493 0.281733i 0.959493 + 0.281733i 0 −0.959493 + 0.281733i −0.959493 + 0.281733i −0.654861 + 0.755750i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
184.j even 22 1 inner
2760.co odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2760.1.co.a 10
3.b odd 2 1 2760.1.co.d yes 10
5.b even 2 1 2760.1.co.d yes 10
8.d odd 2 1 2760.1.co.b yes 10
15.d odd 2 1 CM 2760.1.co.a 10
23.d odd 22 1 2760.1.co.b yes 10
24.f even 2 1 2760.1.co.c yes 10
40.e odd 2 1 2760.1.co.c yes 10
69.g even 22 1 2760.1.co.c yes 10
115.i odd 22 1 2760.1.co.c yes 10
120.m even 2 1 2760.1.co.b yes 10
184.j even 22 1 inner 2760.1.co.a 10
345.n even 22 1 2760.1.co.b yes 10
552.z odd 22 1 2760.1.co.d yes 10
920.bn even 22 1 2760.1.co.d yes 10
2760.co odd 22 1 inner 2760.1.co.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.1.co.a 10 1.a even 1 1 trivial
2760.1.co.a 10 15.d odd 2 1 CM
2760.1.co.a 10 184.j even 22 1 inner
2760.1.co.a 10 2760.co odd 22 1 inner
2760.1.co.b yes 10 8.d odd 2 1
2760.1.co.b yes 10 23.d odd 22 1
2760.1.co.b yes 10 120.m even 2 1
2760.1.co.b yes 10 345.n even 22 1
2760.1.co.c yes 10 24.f even 2 1
2760.1.co.c yes 10 40.e odd 2 1
2760.1.co.c yes 10 69.g even 22 1
2760.1.co.c yes 10 115.i odd 22 1
2760.1.co.d yes 10 3.b odd 2 1
2760.1.co.d yes 10 5.b even 2 1
2760.1.co.d yes 10 552.z odd 22 1
2760.1.co.d yes 10 920.bn even 22 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2760, [\chi])\):

\( T_{17}^{10} - 11 T_{17}^{9} + 55 T_{17}^{8} - 165 T_{17}^{7} + 330 T_{17}^{6} - 462 T_{17}^{5} + 462 T_{17}^{4} - 330 T_{17}^{3} + 165 T_{17}^{2} - 55 T_{17} + 11 \) Copy content Toggle raw display
\( T_{31}^{10} - 22T_{31}^{5} + 33T_{31}^{3} + 11T_{31}^{2} - 11T_{31} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} \) Copy content Toggle raw display
$17$ \( T^{10} - 11 T^{9} + 55 T^{8} - 165 T^{7} + \cdots + 11 \) Copy content Toggle raw display
$19$ \( T^{10} + 11 T^{6} - 11 T^{5} + 22 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$23$ \( T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{10} \) Copy content Toggle raw display
$31$ \( T^{10} - 22 T^{5} + 33 T^{3} + 11 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$37$ \( T^{10} \) Copy content Toggle raw display
$41$ \( T^{10} \) Copy content Toggle raw display
$43$ \( T^{10} \) Copy content Toggle raw display
$47$ \( T^{10} + 11 T^{8} + 44 T^{6} + 77 T^{4} + \cdots + 11 \) Copy content Toggle raw display
$53$ \( T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{10} \) Copy content Toggle raw display
$61$ \( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{10} \) Copy content Toggle raw display
$71$ \( T^{10} \) Copy content Toggle raw display
$73$ \( T^{10} \) Copy content Toggle raw display
$79$ \( T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{10} - 22 T^{5} + 33 T^{3} + 11 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$89$ \( T^{10} \) Copy content Toggle raw display
$97$ \( T^{10} \) Copy content Toggle raw display
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