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Properties

Label 2760.1.cm.f
Level $2760$
Weight $1$
Character orbit 2760.cm
Analytic conductor $1.377$
Analytic rank $0$
Dimension $20$
Projective image $D_{22}$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2760,1,Mod(29,2760)]
 
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([0, 11, 11, 11, 18]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2760.29");
 
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.cm (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: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{44})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) 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.

\(f(q)\) \(=\) \( q - \zeta_{44}^{5} q^{2} - \zeta_{44}^{21} q^{3} + \zeta_{44}^{10} q^{4} + \zeta_{44}^{13} q^{5} - \zeta_{44}^{4} q^{6} - \zeta_{44}^{15} q^{8} - \zeta_{44}^{20} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{44}^{5} q^{2} - \zeta_{44}^{21} q^{3} + \zeta_{44}^{10} q^{4} + \zeta_{44}^{13} q^{5} - \zeta_{44}^{4} q^{6} - \zeta_{44}^{15} q^{8} - \zeta_{44}^{20} q^{9} - \zeta_{44}^{18} q^{10} + \zeta_{44}^{9} q^{12} + \zeta_{44}^{12} q^{15} + \zeta_{44}^{20} q^{16} + (\zeta_{44}^{17} + \zeta_{44}^{11}) q^{17} - \zeta_{44}^{3} q^{18} + ( - \zeta_{44}^{14} + \zeta_{44}^{2}) q^{19} - \zeta_{44} q^{20} + \zeta_{44}^{7} q^{23} - \zeta_{44}^{14} q^{24} - \zeta_{44}^{4} q^{25} - \zeta_{44}^{19} q^{27} - \zeta_{44}^{17} q^{30} + (\zeta_{44}^{18} + \zeta_{44}^{6}) q^{31} + \zeta_{44}^{3} q^{32} + ( - \zeta_{44}^{16} + 1) q^{34} + \zeta_{44}^{8} q^{36} + (\zeta_{44}^{19} - \zeta_{44}^{7}) q^{38} + \zeta_{44}^{6} q^{40} + \zeta_{44}^{11} q^{45} - \zeta_{44}^{12} q^{46} + (\zeta_{44}^{15} - \zeta_{44}^{7}) q^{47} + \zeta_{44}^{19} q^{48} - \zeta_{44}^{10} q^{49} + \zeta_{44}^{9} q^{50} + (\zeta_{44}^{16} + \zeta_{44}^{10}) q^{51} + ( - \zeta_{44}^{9} - \zeta_{44}) q^{53} - \zeta_{44}^{2} q^{54} + ( - \zeta_{44}^{13} + \zeta_{44}) q^{57} - q^{60} + ( - \zeta_{44}^{16} + \zeta_{44}^{8}) q^{61} + ( - \zeta_{44}^{11} + \zeta_{44}) q^{62} - \zeta_{44}^{8} q^{64} + (\zeta_{44}^{21} - \zeta_{44}^{5}) q^{68} + \zeta_{44}^{6} q^{69} - \zeta_{44}^{13} q^{72} - \zeta_{44}^{3} q^{75} + (\zeta_{44}^{12} + \zeta_{44}^{2}) q^{76} + (\zeta_{44}^{8} + \zeta_{44}^{4}) q^{79} - \zeta_{44}^{11} q^{80} - \zeta_{44}^{18} q^{81} + (\zeta_{44}^{21} - \zeta_{44}^{19}) q^{83} + ( - \zeta_{44}^{8} - \zeta_{44}^{2}) q^{85} - \zeta_{44}^{16} q^{90} + \zeta_{44}^{17} q^{92} + (\zeta_{44}^{17} + \zeta_{44}^{5}) q^{93} + ( - \zeta_{44}^{20} + \zeta_{44}^{12}) q^{94} + (\zeta_{44}^{15} + \zeta_{44}^{5}) q^{95} + \zeta_{44}^{2} q^{96} + \zeta_{44}^{15} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{4} + 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{4} + 2 q^{6} + 2 q^{9} - 2 q^{10} - 2 q^{15} - 2 q^{16} - 2 q^{24} + 2 q^{25} + 4 q^{31} + 22 q^{34} - 2 q^{36} + 2 q^{40} + 2 q^{46} - 2 q^{49} - 2 q^{54} - 20 q^{60} + 2 q^{64} + 2 q^{69} - 4 q^{79} - 2 q^{81} + 2 q^{90} + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(1201\) \(1381\) \(1657\) \(1841\) \(2071\)
\(\chi(n)\) \(\zeta_{44}^{4}\) \(-1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
0.281733 + 0.959493i
−0.281733 − 0.959493i
0.540641 − 0.841254i
−0.540641 + 0.841254i
−0.909632 + 0.415415i
0.909632 − 0.415415i
−0.909632 − 0.415415i
0.909632 + 0.415415i
0.540641 + 0.841254i
−0.540641 − 0.841254i
−0.755750 + 0.654861i
0.755750 − 0.654861i
0.989821 + 0.142315i
−0.989821 − 0.142315i
0.989821 − 0.142315i
−0.989821 + 0.142315i
0.281733 − 0.959493i
−0.281733 + 0.959493i
−0.755750 − 0.654861i
0.755750 + 0.654861i
−0.989821 − 0.142315i 0.281733 − 0.959493i 0.959493 + 0.281733i −0.540641 − 0.841254i −0.415415 + 0.909632i 0 −0.909632 − 0.415415i −0.841254 − 0.540641i 0.415415 + 0.909632i
29.2 0.989821 + 0.142315i −0.281733 + 0.959493i 0.959493 + 0.281733i 0.540641 + 0.841254i −0.415415 + 0.909632i 0 0.909632 + 0.415415i −0.841254 − 0.540641i 0.415415 + 0.909632i
269.1 −0.281733 − 0.959493i 0.540641 + 0.841254i −0.841254 + 0.540641i 0.909632 − 0.415415i 0.654861 − 0.755750i 0 0.755750 + 0.654861i −0.415415 + 0.909632i −0.654861 − 0.755750i
269.2 0.281733 + 0.959493i −0.540641 − 0.841254i −0.841254 + 0.540641i −0.909632 + 0.415415i 0.654861 − 0.755750i 0 −0.755750 − 0.654861i −0.415415 + 0.909632i −0.654861 − 0.755750i
509.1 −0.540641 − 0.841254i −0.909632 − 0.415415i −0.415415 + 0.909632i −0.755750 − 0.654861i 0.142315 + 0.989821i 0 0.989821 − 0.142315i 0.654861 + 0.755750i −0.142315 + 0.989821i
509.2 0.540641 + 0.841254i 0.909632 + 0.415415i −0.415415 + 0.909632i 0.755750 + 0.654861i 0.142315 + 0.989821i 0 −0.989821 + 0.142315i 0.654861 + 0.755750i −0.142315 + 0.989821i
629.1 −0.540641 + 0.841254i −0.909632 + 0.415415i −0.415415 − 0.909632i −0.755750 + 0.654861i 0.142315 − 0.989821i 0 0.989821 + 0.142315i 0.654861 − 0.755750i −0.142315 − 0.989821i
629.2 0.540641 − 0.841254i 0.909632 − 0.415415i −0.415415 − 0.909632i 0.755750 − 0.654861i 0.142315 − 0.989821i 0 −0.989821 − 0.142315i 0.654861 − 0.755750i −0.142315 − 0.989821i
749.1 −0.281733 + 0.959493i 0.540641 − 0.841254i −0.841254 − 0.540641i 0.909632 + 0.415415i 0.654861 + 0.755750i 0 0.755750 − 0.654861i −0.415415 − 0.909632i −0.654861 + 0.755750i
749.2 0.281733 − 0.959493i −0.540641 + 0.841254i −0.841254 − 0.540641i −0.909632 − 0.415415i 0.654861 + 0.755750i 0 −0.755750 + 0.654861i −0.415415 − 0.909632i −0.654861 + 0.755750i
869.1 −0.909632 + 0.415415i −0.755750 − 0.654861i 0.654861 − 0.755750i 0.989821 + 0.142315i 0.959493 + 0.281733i 0 −0.281733 + 0.959493i 0.142315 + 0.989821i −0.959493 + 0.281733i
869.2 0.909632 − 0.415415i 0.755750 + 0.654861i 0.654861 − 0.755750i −0.989821 − 0.142315i 0.959493 + 0.281733i 0 0.281733 − 0.959493i 0.142315 + 0.989821i −0.959493 + 0.281733i
1589.1 −0.755750 − 0.654861i 0.989821 − 0.142315i 0.142315 + 0.989821i −0.281733 + 0.959493i −0.841254 − 0.540641i 0 0.540641 − 0.841254i 0.959493 − 0.281733i 0.841254 − 0.540641i
1589.2 0.755750 + 0.654861i −0.989821 + 0.142315i 0.142315 + 0.989821i 0.281733 − 0.959493i −0.841254 − 0.540641i 0 −0.540641 + 0.841254i 0.959493 − 0.281733i 0.841254 − 0.540641i
1829.1 −0.755750 + 0.654861i 0.989821 + 0.142315i 0.142315 − 0.989821i −0.281733 − 0.959493i −0.841254 + 0.540641i 0 0.540641 + 0.841254i 0.959493 + 0.281733i 0.841254 + 0.540641i
1829.2 0.755750 − 0.654861i −0.989821 − 0.142315i 0.142315 − 0.989821i 0.281733 + 0.959493i −0.841254 + 0.540641i 0 −0.540641 − 0.841254i 0.959493 + 0.281733i 0.841254 + 0.540641i
2189.1 −0.989821 + 0.142315i 0.281733 + 0.959493i 0.959493 − 0.281733i −0.540641 + 0.841254i −0.415415 − 0.909632i 0 −0.909632 + 0.415415i −0.841254 + 0.540641i 0.415415 − 0.909632i
2189.2 0.989821 − 0.142315i −0.281733 − 0.959493i 0.959493 − 0.281733i 0.540641 − 0.841254i −0.415415 − 0.909632i 0 0.909632 − 0.415415i −0.841254 + 0.540641i 0.415415 − 0.909632i
2309.1 −0.909632 − 0.415415i −0.755750 + 0.654861i 0.654861 + 0.755750i 0.989821 − 0.142315i 0.959493 − 0.281733i 0 −0.281733 − 0.959493i 0.142315 − 0.989821i −0.959493 − 0.281733i
2309.2 0.909632 + 0.415415i 0.755750 − 0.654861i 0.654861 + 0.755750i −0.989821 + 0.142315i 0.959493 − 0.281733i 0 0.281733 + 0.959493i 0.142315 − 0.989821i −0.959493 − 0.281733i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
184.p even 22 1 inner
552.r odd 22 1 inner
920.bf even 22 1 inner
2760.cm 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.cm.f yes 20
3.b odd 2 1 inner 2760.1.cm.f yes 20
5.b even 2 1 inner 2760.1.cm.f yes 20
8.b even 2 1 2760.1.cm.e ✓ 20
15.d odd 2 1 CM 2760.1.cm.f yes 20
23.c even 11 1 2760.1.cm.e ✓ 20
24.h odd 2 1 2760.1.cm.e ✓ 20
40.f even 2 1 2760.1.cm.e ✓ 20
69.h odd 22 1 2760.1.cm.e ✓ 20
115.j even 22 1 2760.1.cm.e ✓ 20
120.i odd 2 1 2760.1.cm.e ✓ 20
184.p even 22 1 inner 2760.1.cm.f yes 20
345.p odd 22 1 2760.1.cm.e ✓ 20
552.r odd 22 1 inner 2760.1.cm.f yes 20
920.bf even 22 1 inner 2760.1.cm.f yes 20
2760.cm odd 22 1 inner 2760.1.cm.f yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.1.cm.e ✓ 20 8.b even 2 1
2760.1.cm.e ✓ 20 23.c even 11 1
2760.1.cm.e ✓ 20 24.h odd 2 1
2760.1.cm.e ✓ 20 40.f even 2 1
2760.1.cm.e ✓ 20 69.h odd 22 1
2760.1.cm.e ✓ 20 115.j even 22 1
2760.1.cm.e ✓ 20 120.i odd 2 1
2760.1.cm.e ✓ 20 345.p odd 22 1
2760.1.cm.f yes 20 1.a even 1 1 trivial
2760.1.cm.f yes 20 3.b odd 2 1 inner
2760.1.cm.f yes 20 5.b even 2 1 inner
2760.1.cm.f yes 20 15.d odd 2 1 CM
2760.1.cm.f yes 20 184.p even 22 1 inner
2760.1.cm.f yes 20 552.r odd 22 1 inner
2760.1.cm.f yes 20 920.bf even 22 1 inner
2760.1.cm.f yes 20 2760.cm odd 22 1 inner

Hecke kernels

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

\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{19}^{10} + 11T_{19}^{6} - 11T_{19}^{5} + 22T_{19}^{2} + 33T_{19} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - T^{18} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{20} - T^{18} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{20} - T^{18} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} \) Copy content Toggle raw display
$17$ \( T^{20} + 11 T^{18} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( (T^{10} + 11 T^{6} + \cdots + 11)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} - T^{18} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{20} \) Copy content Toggle raw display
$31$ \( (T^{10} - 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} \) Copy content Toggle raw display
$41$ \( T^{20} \) Copy content Toggle raw display
$43$ \( T^{20} \) Copy content Toggle raw display
$47$ \( (T^{10} - 11 T^{8} + \cdots - 11)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} - 4 T^{18} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{20} \) Copy content Toggle raw display
$61$ \( (T^{10} + 11 T^{4} + \cdots + 11)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} \) Copy content Toggle raw display
$71$ \( T^{20} \) Copy content Toggle raw display
$73$ \( T^{20} \) Copy content Toggle raw display
$79$ \( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} - 4 T^{18} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{20} \) Copy content Toggle raw display
$97$ \( T^{20} \) Copy content Toggle raw display
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