Properties

Label 979.1.v.a
Level $979$
Weight $1$
Character orbit 979.v
Analytic conductor $0.489$
Analytic rank $0$
Dimension $20$
Projective image $D_{44}$
CM discriminant -11
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [979,1,Mod(10,979)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(979, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([22, 43]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("979.10");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 979 = 11 \cdot 89 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 979.v (of order \(44\), degree \(20\), 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: \(0.488584647368\)
Analytic rank: \(0\)
Dimension: \(20\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{44}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{44} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{44}^{20} - \zeta_{44}^{11}) q^{3} + \zeta_{44}^{12} q^{4} + ( - \zeta_{44}^{19} + \zeta_{44}^{17}) q^{5} + ( - \zeta_{44}^{18} - \zeta_{44}^{9} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{44}^{20} - \zeta_{44}^{11}) q^{3} + \zeta_{44}^{12} q^{4} + ( - \zeta_{44}^{19} + \zeta_{44}^{17}) q^{5} + ( - \zeta_{44}^{18} - \zeta_{44}^{9} - 1) q^{9} + \zeta_{44}^{4} q^{11} + (\zeta_{44}^{10} + \zeta_{44}) q^{12} + ( - \zeta_{44}^{17} + \zeta_{44}^{15} - \zeta_{44}^{8} + \zeta_{44}^{6}) q^{15} - \zeta_{44}^{2} q^{16} + (\zeta_{44}^{9} - \zeta_{44}^{7}) q^{20} + (\zeta_{44}^{16} - \zeta_{44}^{13}) q^{23} + ( - \zeta_{44}^{16} + \zeta_{44}^{14} - \zeta_{44}^{12}) q^{25} + (\zeta_{44}^{20} - \zeta_{44}^{16} + \zeta_{44}^{11} + \zeta_{44}^{7}) q^{27} + ( - \zeta_{44}^{8} + \zeta_{44}^{7}) q^{31} + ( - \zeta_{44}^{15} + \zeta_{44}^{2}) q^{33} + ( - \zeta_{44}^{21} - \zeta_{44}^{12} + \zeta_{44}^{8}) q^{36} + (\zeta_{44}^{10} - \zeta_{44}) q^{37} + \zeta_{44}^{16} q^{44} + (\zeta_{44}^{19} - \zeta_{44}^{17} - \zeta_{44}^{15} + \zeta_{44}^{13} - \zeta_{44}^{6} + \zeta_{44}^{4}) q^{45} + (\zeta_{44}^{20} - \zeta_{44}^{4}) q^{47} + (\zeta_{44}^{13} - 1) q^{48} + \zeta_{44}^{3} q^{49} + (\zeta_{44}^{17} - \zeta_{44}^{3}) q^{53} + (\zeta_{44}^{21} + \zeta_{44}) q^{55} + ( - \zeta_{44}^{10} - \zeta_{44}^{3}) q^{59} + ( - \zeta_{44}^{20} + \zeta_{44}^{18} + \zeta_{44}^{7} - \zeta_{44}^{5}) q^{60} - \zeta_{44}^{14} q^{64} + (\zeta_{44}^{15} + \zeta_{44}^{5}) q^{67} + (\zeta_{44}^{14} - \zeta_{44}^{11} + \zeta_{44}^{5} - \zeta_{44}^{2}) q^{69} + ( - \zeta_{44}^{21} - \zeta_{44}^{9}) q^{71} + ( - \zeta_{44}^{14} + \zeta_{44}^{12} - \zeta_{44}^{10} - \zeta_{44}^{5} + \zeta_{44}^{3} - \zeta_{44}) q^{75} + (\zeta_{44}^{21} - \zeta_{44}^{19}) q^{80} + (\zeta_{44}^{18} - \zeta_{44}^{14} + \zeta_{44}^{9} + \zeta_{44}^{5} + 1) q^{81} + \zeta_{44}^{18} q^{89} + ( - \zeta_{44}^{6} + \zeta_{44}^{3}) q^{92} + (\zeta_{44}^{19} - \zeta_{44}^{18} - \zeta_{44}^{6} + \zeta_{44}^{5}) q^{93} + (\zeta_{44}^{11} - \zeta_{44}^{3}) q^{97} + ( - \zeta_{44}^{13} - \zeta_{44}^{4} + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} - 2 q^{4} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} - 2 q^{4} - 22 q^{9} - 2 q^{11} + 2 q^{12} + 4 q^{15} - 2 q^{16} - 2 q^{23} + 6 q^{25} + 2 q^{31} + 2 q^{33} + 2 q^{37} - 2 q^{44} - 4 q^{45} - 20 q^{48} - 2 q^{59} + 4 q^{60} - 2 q^{64} - 6 q^{75} + 20 q^{81} + 2 q^{89} - 2 q^{92} - 4 q^{93} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/979\mathbb{Z}\right)^\times\).

\(n\) \(90\) \(804\)
\(\chi(n)\) \(-1\) \(-\zeta_{44}^{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−0.755750 + 0.654861i
0.540641 + 0.841254i
−0.755750 0.654861i
−0.281733 + 0.959493i
0.909632 + 0.415415i
0.909632 0.415415i
−0.989821 0.142315i
0.281733 + 0.959493i
0.989821 0.142315i
0.540641 0.841254i
−0.540641 + 0.841254i
−0.989821 + 0.142315i
−0.281733 0.959493i
0.989821 + 0.142315i
−0.909632 + 0.415415i
−0.909632 0.415415i
0.281733 0.959493i
0.755750 + 0.654861i
−0.540641 0.841254i
0.755750 0.654861i
0 0.142315 0.0101786i −0.654861 0.755750i −0.368991 1.25667i 0 0 0 −0.969672 + 0.139418i 0
21.1 0 −0.415415 + 0.0903680i 0.841254 0.540641i −1.27155 1.10181i 0 0 0 −0.745229 + 0.340335i 0
98.1 0 0.142315 + 0.0101786i −0.654861 + 0.755750i −0.368991 + 1.25667i 0 0 0 −0.969672 0.139418i 0
109.1 0 −0.841254 0.459359i −0.959493 0.281733i 1.74557 + 0.797176i 0 0 0 −0.0439442 0.0683785i 0
131.1 0 0.654861 + 0.244250i 0.415415 0.909632i 0.822373 0.118239i 0 0 0 −0.386565 0.334961i 0
142.1 0 0.654861 0.244250i 0.415415 + 0.909632i 0.822373 + 0.118239i 0 0 0 −0.386565 + 0.334961i 0
285.1 0 0.959493 + 0.718267i −0.142315 + 0.989821i −0.153882 0.239446i 0 0 0 0.122986 + 0.418852i 0
307.1 0 −0.841254 1.54064i −0.959493 + 0.281733i −1.74557 + 0.797176i 0 0 0 −1.12523 + 1.75089i 0
351.1 0 0.959493 + 1.28173i −0.142315 0.989821i 0.153882 0.239446i 0 0 0 −0.440479 + 1.50013i 0
373.1 0 −0.415415 0.0903680i 0.841254 + 0.540641i −1.27155 + 1.10181i 0 0 0 −0.745229 0.340335i 0
428.1 0 −0.415415 + 1.90963i 0.841254 + 0.540641i 1.27155 1.10181i 0 0 0 −2.56449 1.17116i 0
450.1 0 0.959493 0.718267i −0.142315 0.989821i −0.153882 + 0.239446i 0 0 0 0.122986 0.418852i 0
494.1 0 −0.841254 + 0.459359i −0.959493 + 0.281733i 1.74557 0.797176i 0 0 0 −0.0439442 + 0.0683785i 0
516.1 0 0.959493 1.28173i −0.142315 + 0.989821i 0.153882 + 0.239446i 0 0 0 −0.440479 1.50013i 0
659.1 0 0.654861 + 1.75575i 0.415415 + 0.909632i −0.822373 0.118239i 0 0 0 −1.89806 + 1.64468i 0
670.1 0 0.654861 1.75575i 0.415415 0.909632i −0.822373 + 0.118239i 0 0 0 −1.89806 1.64468i 0
692.1 0 −0.841254 + 1.54064i −0.959493 0.281733i −1.74557 0.797176i 0 0 0 −1.12523 1.75089i 0
703.1 0 0.142315 1.98982i −0.654861 + 0.755750i 0.368991 1.25667i 0 0 0 −2.94931 0.424047i 0
780.1 0 −0.415415 1.90963i 0.841254 0.540641i 1.27155 + 1.10181i 0 0 0 −2.56449 + 1.17116i 0
791.1 0 0.142315 + 1.98982i −0.654861 0.755750i 0.368991 + 1.25667i 0 0 0 −2.94931 + 0.424047i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
89.g even 44 1 inner
979.v odd 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 979.1.v.a 20
11.b odd 2 1 CM 979.1.v.a 20
89.g even 44 1 inner 979.1.v.a 20
979.v odd 44 1 inner 979.1.v.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
979.1.v.a 20 1.a even 1 1 trivial
979.1.v.a 20 11.b odd 2 1 CM
979.1.v.a 20 89.g even 44 1 inner
979.1.v.a 20 979.v odd 44 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(979, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 2 T^{19} + 13 T^{18} - 22 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{20} - 4 T^{18} + 16 T^{16} - 9 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{20} \) Copy content Toggle raw display
$11$ \( (T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} \) Copy content Toggle raw display
$23$ \( T^{20} + 2 T^{19} + 2 T^{18} - 4 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{20} \) Copy content Toggle raw display
$31$ \( T^{20} - 2 T^{19} + 2 T^{18} - 22 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{20} - 2 T^{19} + 2 T^{18} + 29 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{20} \) Copy content Toggle raw display
$43$ \( T^{20} \) Copy content Toggle raw display
$47$ \( (T^{10} + 11 T^{7} + 33 T^{4} - 11 T^{3} + \cdots + 11)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} - 4 T^{18} + 16 T^{16} - 42 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{20} + 2 T^{19} + 2 T^{18} - 4 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{20} \) Copy content Toggle raw display
$67$ \( T^{20} + 22 T^{12} + 462 T^{10} + \cdots + 121 \) Copy content Toggle raw display
$71$ \( T^{20} - 4 T^{18} + 16 T^{16} - 42 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$73$ \( T^{20} \) Copy content Toggle raw display
$79$ \( T^{20} \) Copy content Toggle raw display
$83$ \( T^{20} \) Copy content Toggle raw display
$89$ \( (T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + 11 T^{18} + 55 T^{16} + \cdots + 121 \) Copy content Toggle raw display
show more
show less