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Properties

Label 3963.1.bi.a
Level $3963$
Weight $1$
Character orbit 3963.bi
Analytic conductor $1.978$
Analytic rank $0$
Dimension $20$
Projective image $D_{44}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3963,1,Mod(71,3963)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3963, base_ring=CyclotomicField(44))
 
chi = DirichletCharacter(H, H._module([22, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3963.71");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3963 = 3 \cdot 1321 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3963.bi (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: \(1.97779464506\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q - \zeta_{44}^{20} q^{3} - q^{4} + ( - \zeta_{44}^{15} - \zeta_{44}^{2}) q^{7} - \zeta_{44}^{18} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{44}^{20} q^{3} - q^{4} + ( - \zeta_{44}^{15} - \zeta_{44}^{2}) q^{7} - \zeta_{44}^{18} q^{9} + \zeta_{44}^{20} q^{12} + (\zeta_{44}^{21} + \zeta_{44}^{8}) q^{13} + q^{16} + ( - \zeta_{44}^{17} - \zeta_{44}^{10}) q^{19} + ( - \zeta_{44}^{13} - 1) q^{21} + \zeta_{44}^{14} q^{25} - \zeta_{44}^{16} q^{27} + (\zeta_{44}^{15} + \zeta_{44}^{2}) q^{28} + ( - \zeta_{44}^{11} + \zeta_{44}^{3}) q^{31} + \zeta_{44}^{18} q^{36} + ( - \zeta_{44}^{11} - \zeta_{44}^{9}) q^{37} + (\zeta_{44}^{19} + \zeta_{44}^{6}) q^{39} + (\zeta_{44}^{21} - \zeta_{44}^{5}) q^{43} - \zeta_{44}^{20} q^{48} + (\zeta_{44}^{17} + \cdots + \zeta_{44}^{4}) q^{49} + \cdots + (\zeta_{44}^{21} + \zeta_{44}^{4}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} - 20 q^{4} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} - 20 q^{4} - 2 q^{7} - 2 q^{9} - 2 q^{12} - 2 q^{13} + 20 q^{16} - 2 q^{19} - 20 q^{21} + 2 q^{25} + 2 q^{27} + 2 q^{28} + 2 q^{36} + 2 q^{39} + 2 q^{48} + 2 q^{52} + 2 q^{57} - 2 q^{61} - 2 q^{63} - 20 q^{64} - 2 q^{67} - 20 q^{73} - 2 q^{75} + 2 q^{76} - 2 q^{79} - 2 q^{81} + 20 q^{84} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3963\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(1322\)
\(\chi(n)\) \(-\zeta_{44}^{7}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
0.909632 − 0.415415i
−0.281733 + 0.959493i
0.540641 − 0.841254i
−0.755750 + 0.654861i
0.909632 + 0.415415i
−0.281733 − 0.959493i
0.281733 + 0.959493i
−0.909632 − 0.415415i
0.755750 − 0.654861i
−0.540641 + 0.841254i
0.281733 − 0.959493i
−0.909632 + 0.415415i
−0.989821 + 0.142315i
−0.755750 − 0.654861i
−0.540641 − 0.841254i
−0.989821 − 0.142315i
0.989821 + 0.142315i
0.540641 + 0.841254i
0.755750 + 0.654861i
0.989821 − 0.142315i
0 0.654861 + 0.755750i −1.00000 0 0 −1.64468 + 0.898064i 0 −0.142315 + 0.989821i 0
80.1 0 −0.841254 + 0.540641i −1.00000 0 0 1.75089 + 0.125226i 0 0.415415 − 0.909632i 0
155.1 0 −0.415415 + 0.909632i −1.00000 0 0 1.17116 + 1.56449i 0 −0.654861 − 0.755750i 0
548.1 0 0.142315 + 0.989821i −1.00000 0 0 −0.424047 + 1.94931i 0 −0.959493 + 0.281733i 0
614.1 0 0.654861 − 0.755750i −1.00000 0 0 −1.64468 − 0.898064i 0 −0.142315 − 0.989821i 0
644.1 0 −0.841254 − 0.540641i −1.00000 0 0 1.75089 − 0.125226i 0 0.415415 + 0.909632i 0
677.1 0 −0.841254 − 0.540641i −1.00000 0 0 −0.0683785 − 0.956056i 0 0.415415 + 0.909632i 0
707.1 0 0.654861 − 0.755750i −1.00000 0 0 0.334961 − 0.613435i 0 −0.142315 − 0.989821i 0
773.1 0 0.142315 + 0.989821i −1.00000 0 0 0.139418 + 0.0303285i 0 −0.959493 + 0.281733i 0
1166.1 0 −0.415415 + 0.909632i −1.00000 0 0 −0.340335 + 0.254771i 0 −0.654861 − 0.755750i 0
1241.1 0 −0.841254 + 0.540641i −1.00000 0 0 −0.0683785 + 0.956056i 0 0.415415 − 0.909632i 0
1250.1 0 0.654861 + 0.755750i −1.00000 0 0 0.334961 + 0.613435i 0 −0.142315 + 0.989821i 0
1376.1 0 0.959493 + 0.281733i −1.00000 0 0 −1.50013 − 0.559521i 0 0.841254 + 0.540641i 0
1685.1 0 0.142315 − 0.989821i −1.00000 0 0 −0.424047 − 1.94931i 0 −0.959493 − 0.281733i 0
2267.1 0 −0.415415 − 0.909632i −1.00000 0 0 −0.340335 − 0.254771i 0 −0.654861 + 0.755750i 0
2618.1 0 0.959493 − 0.281733i −1.00000 0 0 −1.50013 + 0.559521i 0 0.841254 − 0.540641i 0
2666.1 0 0.959493 − 0.281733i −1.00000 0 0 −0.418852 − 1.12299i 0 0.841254 − 0.540641i 0
3017.1 0 −0.415415 − 0.909632i −1.00000 0 0 1.17116 − 1.56449i 0 −0.654861 + 0.755750i 0
3599.1 0 0.142315 − 0.989821i −1.00000 0 0 0.139418 − 0.0303285i 0 −0.959493 − 0.281733i 0
3908.1 0 0.959493 + 0.281733i −1.00000 0 0 −0.418852 + 1.12299i 0 0.841254 + 0.540641i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
1321.r even 44 1 inner
3963.bi odd 44 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3963.1.bi.a ✓ 20
3.b odd 2 1 CM 3963.1.bi.a ✓ 20
1321.r even 44 1 inner 3963.1.bi.a ✓ 20
3963.bi odd 44 1 inner 3963.1.bi.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3963.1.bi.a ✓ 20 1.a even 1 1 trivial
3963.1.bi.a ✓ 20 3.b odd 2 1 CM
3963.1.bi.a ✓ 20 1321.r even 44 1 inner
3963.1.bi.a ✓ 20 3963.bi odd 44 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3963, [\chi])\).

Hecke characteristic polynomials

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