Properties

Label 979.1.r.a
Level $979$
Weight $1$
Character orbit 979.r
Analytic conductor $0.489$
Analytic rank $0$
Dimension $10$
Projective image $D_{22}$
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(87,979)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(979, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 15]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("979.87");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 979 = 11 \cdot 89 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 979.r (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: \(0.488584647368\)
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: \(\Z[a_1, a_2, a_3]\)
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.

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

Character values

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

\(n\) \(90\) \(804\)
\(\chi(n)\) \(-1\) \(\zeta_{22}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
87.1
0.654861 + 0.755750i
0.142315 0.989821i
0.959493 + 0.281733i
0.959493 0.281733i
−0.841254 + 0.540641i
−0.415415 + 0.909632i
−0.415415 0.909632i
−0.841254 0.540641i
0.142315 + 0.989821i
0.654861 0.755750i
0 −0.584585 + 0.909632i 0.841254 0.540641i 1.10181 1.27155i 0 0 0 −0.0702757 0.153882i 0
263.1 0 −1.65486 + 0.755750i 0.415415 0.909632i 0.118239 + 0.822373i 0 0 0 1.51255 1.74557i 0
340.1 0 −1.14231 0.989821i −0.654861 0.755750i −1.25667 + 0.368991i 0 0 0 0.182822 + 1.27155i 0
406.1 0 −1.14231 + 0.989821i −0.654861 + 0.755750i −1.25667 0.368991i 0 0 0 0.182822 1.27155i 0
615.1 0 −1.95949 0.281733i −0.142315 0.989821i 0.239446 + 0.153882i 0 0 0 2.80075 + 0.822373i 0
648.1 0 −0.158746 + 0.540641i −0.959493 + 0.281733i 0.797176 + 1.74557i 0 0 0 0.574161 + 0.368991i 0
769.1 0 −0.158746 0.540641i −0.959493 0.281733i 0.797176 1.74557i 0 0 0 0.574161 0.368991i 0
901.1 0 −1.95949 + 0.281733i −0.142315 + 0.989821i 0.239446 0.153882i 0 0 0 2.80075 0.822373i 0
912.1 0 −1.65486 0.755750i 0.415415 + 0.909632i 0.118239 0.822373i 0 0 0 1.51255 + 1.74557i 0
934.1 0 −0.584585 0.909632i 0.841254 + 0.540641i 1.10181 + 1.27155i 0 0 0 −0.0702757 + 0.153882i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 87.1
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.f even 22 1 inner
979.r odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 979.1.r.a 10
11.b odd 2 1 CM 979.1.r.a 10
89.f even 22 1 inner 979.1.r.a 10
979.r odd 22 1 inner 979.1.r.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
979.1.r.a 10 1.a even 1 1 trivial
979.1.r.a 10 11.b odd 2 1 CM
979.1.r.a 10 89.f even 22 1 inner
979.1.r.a 10 979.r odd 22 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 11 T^{9} + 55 T^{8} + 165 T^{7} + \cdots + 11 \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + 4 T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{10} \) Copy content Toggle raw display
$17$ \( T^{10} \) Copy content Toggle raw display
$19$ \( T^{10} \) Copy content Toggle raw display
$23$ \( T^{10} + 11 T^{6} + 11 T^{5} + 22 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$29$ \( T^{10} \) Copy content Toggle raw display
$31$ \( T^{10} - 11 T^{7} + 33 T^{4} + 11 T^{3} + \cdots + 11 \) Copy content Toggle raw display
$37$ \( T^{10} + 11 T^{8} + 44 T^{6} + 77 T^{4} + \cdots + 11 \) Copy content Toggle raw display
$41$ \( T^{10} \) Copy content Toggle raw display
$43$ \( T^{10} \) Copy content Toggle raw display
$47$ \( T^{10} + 2 T^{9} + 4 T^{8} - 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( T^{10} - 2 T^{9} + 4 T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{10} + 11 T^{6} - 11 T^{5} + 22 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$61$ \( T^{10} \) 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
$83$ \( T^{10} \) Copy content Toggle raw display
$89$ \( T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{10} + 9 T^{9} + 37 T^{8} + 91 T^{7} + \cdots + 1 \) Copy content Toggle raw display
show more
show less