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Properties

Label 968.1.x.a
Level $968$
Weight $1$
Character orbit 968.x
Analytic conductor $0.483$
Analytic rank $0$
Dimension $10$
Projective image $D_{11}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [968,1,Mod(67,968)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(968, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([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("968.67");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 968.x (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: \(0.483094932229\)
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.

\(f(q)\) \(=\) \( q + \zeta_{22}^{8} q^{2} + (\zeta_{22}^{6} - \zeta_{22}^{5}) q^{3} - \zeta_{22}^{5} q^{4} + ( - \zeta_{22}^{3} + \zeta_{22}^{2}) q^{6} + \zeta_{22}^{2} q^{8} + (\zeta_{22}^{10} - \zeta_{22} + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{22}^{8} q^{2} + (\zeta_{22}^{6} - \zeta_{22}^{5}) q^{3} - \zeta_{22}^{5} q^{4} + ( - \zeta_{22}^{3} + \zeta_{22}^{2}) q^{6} + \zeta_{22}^{2} q^{8} + (\zeta_{22}^{10} - \zeta_{22} + 1) q^{9} - \zeta_{22}^{9} q^{11} + (\zeta_{22}^{10} + 1) q^{12} + \zeta_{22}^{10} q^{16} + ( - \zeta_{22}^{9} - \zeta_{22}^{5}) q^{17} + ( - \zeta_{22}^{9} + \cdots - \zeta_{22}^{7}) q^{18} + \cdots + (\zeta_{22}^{10} + \cdots + \zeta_{22}^{8}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{8} + 8 q^{9} - q^{11} + 9 q^{12} - q^{16} - 2 q^{17} - 3 q^{18} - 2 q^{19} - q^{22} - 2 q^{24} - q^{25} - 4 q^{27} - q^{32} - 2 q^{33} - 2 q^{34} - 3 q^{36} - 2 q^{38} - 2 q^{41} - 2 q^{43} - q^{44} - 2 q^{48} - q^{49} - q^{50} + 7 q^{51} - 4 q^{54} + 7 q^{57} - 2 q^{59} - q^{64} + 9 q^{66} - 2 q^{67} - 2 q^{68} - 3 q^{72} - 2 q^{73} - 2 q^{75} + 9 q^{76} + 6 q^{81} - 2 q^{82} - 2 q^{83} - 2 q^{86} + 10 q^{88} - 2 q^{89} - 2 q^{96} - 2 q^{97} - q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(727\) \(849\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{22}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.415415 − 0.909632i
−0.841254 − 0.540641i
−0.841254 + 0.540641i
−0.415415 + 0.909632i
0.142315 + 0.989821i
0.654861 + 0.755750i
0.959493 + 0.281733i
0.959493 − 0.281733i
0.654861 − 0.755750i
0.142315 − 0.989821i
−0.959493 + 0.281733i 1.68251 0.841254 − 0.540641i 0 −1.61435 + 0.474017i 0 −0.654861 + 0.755750i 1.83083 0
155.1 −0.142315 − 0.989821i −1.91899 −0.959493 + 0.281733i 0 0.273100 + 1.89945i 0 0.415415 + 0.909632i 2.68251 0
331.1 −0.142315 + 0.989821i −1.91899 −0.959493 − 0.281733i 0 0.273100 − 1.89945i 0 0.415415 − 0.909632i 2.68251 0
419.1 −0.959493 − 0.281733i 1.68251 0.841254 + 0.540641i 0 −1.61435 − 0.474017i 0 −0.654861 − 0.755750i 1.83083 0
507.1 0.415415 − 0.909632i −1.30972 −0.654861 − 0.755750i 0 −0.544078 + 1.19136i 0 −0.959493 + 0.281733i 0.715370 0
595.1 0.841254 + 0.540641i 0.830830 0.415415 + 0.909632i 0 0.698939 + 0.449181i 0 −0.142315 + 0.989821i −0.309721 0
683.1 −0.654861 + 0.755750i −0.284630 −0.142315 − 0.989821i 0 0.186393 − 0.215109i 0 0.841254 + 0.540641i −0.918986 0
771.1 −0.654861 − 0.755750i −0.284630 −0.142315 + 0.989821i 0 0.186393 + 0.215109i 0 0.841254 − 0.540641i −0.918986 0
859.1 0.841254 − 0.540641i 0.830830 0.415415 − 0.909632i 0 0.698939 − 0.449181i 0 −0.142315 − 0.989821i −0.309721 0
947.1 0.415415 + 0.909632i −1.30972 −0.654861 + 0.755750i 0 −0.544078 − 1.19136i 0 −0.959493 − 0.281733i 0.715370 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
121.e even 11 1 inner
968.x odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.1.x.a ✓ 10
4.b odd 2 1 3872.1.bj.a 10
8.b even 2 1 3872.1.bj.a 10
8.d odd 2 1 CM 968.1.x.a ✓ 10
121.e even 11 1 inner 968.1.x.a ✓ 10
484.k odd 22 1 3872.1.bj.a 10
968.t even 22 1 3872.1.bj.a 10
968.x odd 22 1 inner 968.1.x.a ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
968.1.x.a ✓ 10 1.a even 1 1 trivial
968.1.x.a ✓ 10 8.d odd 2 1 CM
968.1.x.a ✓ 10 121.e even 11 1 inner
968.1.x.a ✓ 10 968.x odd 22 1 inner
3872.1.bj.a 10 4.b odd 2 1
3872.1.bj.a 10 8.b even 2 1
3872.1.bj.a 10 484.k odd 22 1
3872.1.bj.a 10 968.t even 22 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(968, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{5} + T^{4} - 4 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + T^{9} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{10} \) Copy content Toggle raw display
$17$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} \) Copy content Toggle raw display
$31$ \( T^{10} \) Copy content Toggle raw display
$37$ \( T^{10} \) Copy content Toggle raw display
$41$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{10} \) Copy content Toggle raw display
$53$ \( T^{10} \) Copy content Toggle raw display
$59$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{10} \) Copy content Toggle raw display
$67$ \( T^{10} + 2 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( T^{10} \) Copy content Toggle raw display
$73$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{10} \) Copy content Toggle raw display
$83$ \( T^{10} + 2 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{10} + 2 T^{9} + \cdots + 1 \) Copy content Toggle raw display
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