Properties

Label 3864.1.dl.b
Level $3864$
Weight $1$
Character orbit 3864.dl
Analytic conductor $1.928$
Analytic rank $0$
Dimension $10$
Projective image $D_{11}$
CM discriminant -168
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3864,1,Mod(587,3864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3864, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 11, 11, 11, 20]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3864.587");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3864.dl (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: \(1.92838720881\)
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: \(D_{11}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{11} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{22}^{5} q^{2} + \zeta_{22}^{7} q^{3} + \zeta_{22}^{10} q^{4} + \zeta_{22} q^{6} + \zeta_{22}^{9} q^{7} + \zeta_{22}^{4} q^{8} - \zeta_{22}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{22}^{5} q^{2} + \zeta_{22}^{7} q^{3} + \zeta_{22}^{10} q^{4} + \zeta_{22} q^{6} + \zeta_{22}^{9} q^{7} + \zeta_{22}^{4} q^{8} - \zeta_{22}^{3} q^{9} - \zeta_{22}^{6} q^{12} + ( - \zeta_{22}^{8} + \zeta_{22}^{7}) q^{13} + \zeta_{22}^{3} q^{14} - \zeta_{22}^{9} q^{16} + (\zeta_{22}^{9} - \zeta_{22}^{4}) q^{17} + \zeta_{22}^{8} q^{18} - \zeta_{22}^{5} q^{21} + \zeta_{22}^{4} q^{23} - q^{24} - \zeta_{22}^{5} q^{25} + ( - \zeta_{22}^{2} + \zeta_{22}) q^{26} - \zeta_{22}^{10} q^{27} - \zeta_{22}^{8} q^{28} + (\zeta_{22}^{8} - \zeta_{22}^{5}) q^{29} + (\zeta_{22}^{5} + \zeta_{22}^{3}) q^{31} - \zeta_{22}^{3} q^{32} + (\zeta_{22}^{9} + \zeta_{22}^{3}) q^{34} + \zeta_{22}^{2} q^{36} + (\zeta_{22}^{4} - \zeta_{22}^{3}) q^{39} + (\zeta_{22}^{5} - 1) q^{41} + \zeta_{22}^{10} q^{42} + ( - \zeta_{22}^{3} + 1) q^{43} - \zeta_{22}^{9} q^{46} + \zeta_{22}^{5} q^{48} - \zeta_{22}^{7} q^{49} + \zeta_{22}^{10} q^{50} + ( - \zeta_{22}^{5} + 1) q^{51} + (\zeta_{22}^{7} - \zeta_{22}^{6}) q^{52} + (\zeta_{22}^{10} + \zeta_{22}^{8}) q^{53} - \zeta_{22}^{4} q^{54} - \zeta_{22}^{2} q^{56} + (\zeta_{22}^{10} + \zeta_{22}^{2}) q^{58} + ( - \zeta_{22}^{4} - 1) q^{59} + ( - \zeta_{22}^{6} - \zeta_{22}^{2}) q^{61} + ( - \zeta_{22}^{10} - \zeta_{22}^{8}) q^{62} + \zeta_{22} q^{63} + \zeta_{22}^{8} q^{64} + ( - \zeta_{22}^{7} - \zeta_{22}^{3}) q^{67} + ( - \zeta_{22}^{8} + \zeta_{22}^{3}) q^{68} - q^{69} + ( - \zeta_{22}^{7} - \zeta_{22}^{3}) q^{71} - \zeta_{22}^{7} q^{72} + \zeta_{22} q^{75} + ( - \zeta_{22}^{9} + \zeta_{22}^{8}) q^{78} + \zeta_{22}^{6} q^{81} + ( - \zeta_{22}^{10} + \zeta_{22}^{5}) q^{82} + (\zeta_{22}^{9} - \zeta_{22}^{8}) q^{83} + \zeta_{22}^{4} q^{84} + (\zeta_{22}^{8} - \zeta_{22}^{5}) q^{86} + ( - \zeta_{22}^{4} + \zeta_{22}) q^{87} + ( - \zeta_{22}^{8} - \zeta_{22}^{6}) q^{89} + (\zeta_{22}^{6} - \zeta_{22}^{5}) q^{91} - \zeta_{22}^{3} q^{92} + (\zeta_{22}^{10} - \zeta_{22}) q^{93} - \zeta_{22}^{10} q^{96} - \zeta_{22} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + q^{3} - q^{4} + q^{6} + q^{7} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} + q^{3} - q^{4} + q^{6} + q^{7} - q^{8} - q^{9} + q^{12} + 2 q^{13} + q^{14} - q^{16} + 2 q^{17} - q^{18} - q^{21} - q^{23} - 10 q^{24} - q^{25} + 2 q^{26} + q^{27} + q^{28} - 2 q^{29} + 2 q^{31} - q^{32} + 2 q^{34} - q^{36} - 2 q^{39} - 9 q^{41} - q^{42} + 9 q^{43} - q^{46} + q^{48} - q^{49} - q^{50} + 9 q^{51} + 2 q^{52} - 2 q^{53} + q^{54} + q^{56} - 2 q^{58} - 9 q^{59} + 2 q^{61} + 2 q^{62} + q^{63} - q^{64} - 2 q^{67} + 2 q^{68} - 10 q^{69} - 2 q^{71} - q^{72} + q^{75} - 2 q^{78} - q^{81} + 2 q^{82} + 2 q^{83} - q^{84} - 2 q^{86} + 2 q^{87} + 2 q^{89} - 2 q^{91} - q^{92} - 2 q^{93} + q^{96} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3864\mathbb{Z}\right)^\times\).

\(n\) \(967\) \(1289\) \(1933\) \(2761\) \(2857\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\) \(-\zeta_{22}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
587.1
−0.415415 0.909632i
0.959493 + 0.281733i
0.959493 0.281733i
0.654861 0.755750i
0.142315 + 0.989821i
0.654861 + 0.755750i
−0.841254 + 0.540641i
−0.415415 + 0.909632i
−0.841254 0.540641i
0.142315 0.989821i
0.841254 0.540641i 0.142315 0.989821i 0.415415 0.909632i 0 −0.415415 0.909632i 0.654861 + 0.755750i −0.142315 0.989821i −0.959493 0.281733i 0
923.1 −0.142315 0.989821i −0.415415 + 0.909632i −0.959493 + 0.281733i 0 0.959493 + 0.281733i −0.841254 + 0.540641i 0.415415 + 0.909632i −0.654861 0.755750i 0
1595.1 −0.142315 + 0.989821i −0.415415 0.909632i −0.959493 0.281733i 0 0.959493 0.281733i −0.841254 0.540641i 0.415415 0.909632i −0.654861 + 0.755750i 0
2099.1 0.415415 0.909632i 0.959493 + 0.281733i −0.654861 0.755750i 0 0.654861 0.755750i 0.142315 0.989821i −0.959493 + 0.281733i 0.841254 + 0.540641i 0
2267.1 −0.654861 0.755750i −0.841254 0.540641i −0.142315 + 0.989821i 0 0.142315 + 0.989821i 0.959493 + 0.281733i 0.841254 0.540641i 0.415415 + 0.909632i 0
2603.1 0.415415 + 0.909632i 0.959493 0.281733i −0.654861 + 0.755750i 0 0.654861 + 0.755750i 0.142315 + 0.989821i −0.959493 0.281733i 0.841254 0.540641i 0
2939.1 −0.959493 0.281733i 0.654861 0.755750i 0.841254 + 0.540641i 0 −0.841254 + 0.540641i −0.415415 0.909632i −0.654861 0.755750i −0.142315 0.989821i 0
3107.1 0.841254 + 0.540641i 0.142315 + 0.989821i 0.415415 + 0.909632i 0 −0.415415 + 0.909632i 0.654861 0.755750i −0.142315 + 0.989821i −0.959493 + 0.281733i 0
3275.1 −0.959493 + 0.281733i 0.654861 + 0.755750i 0.841254 0.540641i 0 −0.841254 0.540641i −0.415415 + 0.909632i −0.654861 + 0.755750i −0.142315 + 0.989821i 0
3443.1 −0.654861 + 0.755750i −0.841254 + 0.540641i −0.142315 0.989821i 0 0.142315 0.989821i 0.959493 0.281733i 0.841254 + 0.540641i 0.415415 0.909632i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 587.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
168.e odd 2 1 CM by \(\Q(\sqrt{-42}) \)
23.c even 11 1 inner
3864.dl odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3864.1.dl.b yes 10
3.b odd 2 1 3864.1.dl.c yes 10
7.b odd 2 1 3864.1.dl.a 10
8.d odd 2 1 3864.1.dl.d yes 10
21.c even 2 1 3864.1.dl.d yes 10
23.c even 11 1 inner 3864.1.dl.b yes 10
24.f even 2 1 3864.1.dl.a 10
56.e even 2 1 3864.1.dl.c yes 10
69.h odd 22 1 3864.1.dl.c yes 10
161.l odd 22 1 3864.1.dl.a 10
168.e odd 2 1 CM 3864.1.dl.b yes 10
184.k odd 22 1 3864.1.dl.d yes 10
483.v even 22 1 3864.1.dl.d yes 10
552.x even 22 1 3864.1.dl.a 10
1288.bo even 22 1 3864.1.dl.c yes 10
3864.dl odd 22 1 inner 3864.1.dl.b yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.1.dl.a 10 7.b odd 2 1
3864.1.dl.a 10 24.f even 2 1
3864.1.dl.a 10 161.l odd 22 1
3864.1.dl.a 10 552.x even 22 1
3864.1.dl.b yes 10 1.a even 1 1 trivial
3864.1.dl.b yes 10 23.c even 11 1 inner
3864.1.dl.b yes 10 168.e odd 2 1 CM
3864.1.dl.b yes 10 3864.dl odd 22 1 inner
3864.1.dl.c yes 10 3.b odd 2 1
3864.1.dl.c yes 10 56.e even 2 1
3864.1.dl.c yes 10 69.h odd 22 1
3864.1.dl.c yes 10 1288.bo even 22 1
3864.1.dl.d yes 10 8.d odd 2 1
3864.1.dl.d yes 10 21.c even 2 1
3864.1.dl.d yes 10 184.k odd 22 1
3864.1.dl.d yes 10 483.v even 22 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3864, [\chi])\):

\( T_{13}^{10} - 2T_{13}^{9} + 4T_{13}^{8} + 3T_{13}^{7} - 6T_{13}^{6} + 12T_{13}^{5} + 9T_{13}^{4} - 7T_{13}^{3} + 14T_{13}^{2} - 6T_{13} + 1 \) Copy content Toggle raw display
\( T_{17}^{10} - 2T_{17}^{9} + 4T_{17}^{8} + 3T_{17}^{7} - 6T_{17}^{6} + 12T_{17}^{5} + 9T_{17}^{4} - 7T_{17}^{3} + 14T_{17}^{2} - 6T_{17} + 1 \) 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
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{10} \) 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
$29$ \( T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{10} \) Copy content Toggle raw display
$41$ \( T^{10} + 9 T^{9} + 37 T^{8} + 91 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{10} - 9 T^{9} + 37 T^{8} - 91 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{10} \) Copy content Toggle raw display
$53$ \( T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{10} + 9 T^{9} + 37 T^{8} + 91 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{10} + 2 T^{9} + 4 T^{8} + 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{10} \) Copy content Toggle raw display
$79$ \( T^{10} \) Copy content Toggle raw display
$83$ \( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{10} \) Copy content Toggle raw display
show more
show less