Properties

Label 1960.1.i.a
Level $1960$
Weight $1$
Character orbit 1960.i
Self dual yes
Analytic conductor $0.978$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -40
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1960,1,Mod(99,1960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1960.99");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1960.i (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.978167424761\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1960.1
Artin image: $D_6$
Artin field: Galois closure of 6.2.19208000.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{4} - q^{5} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - q^{5} - q^{8} + q^{9} + q^{10} - q^{11} + q^{13} + q^{16} - q^{18} - q^{19} - q^{20} + q^{22} + q^{23} + q^{25} - q^{26} - q^{32} + q^{36} + q^{37} + q^{38} + q^{40} - q^{41} - q^{44} - q^{45} - q^{46} + q^{47} - q^{50} + q^{52} + q^{53} + q^{55} + 2 q^{59} + q^{64} - q^{65} - q^{72} - q^{74} - q^{76} - q^{80} + q^{81} + q^{82} + q^{88} + 2 q^{89} + q^{90} + q^{92} - q^{94} + q^{95} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0
−1.00000 0 1.00000 −1.00000 0 0 −1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.1.i.a 1
5.b even 2 1 1960.1.i.d 1
7.b odd 2 1 1960.1.i.b 1
7.c even 3 2 280.1.bi.b yes 2
7.d odd 6 2 1960.1.bi.b 2
8.d odd 2 1 1960.1.i.d 1
21.h odd 6 2 2520.1.ef.a 2
28.g odd 6 2 1120.1.by.b 2
35.c odd 2 1 1960.1.i.c 1
35.i odd 6 2 1960.1.bi.a 2
35.j even 6 2 280.1.bi.a 2
35.l odd 12 4 1400.1.ba.a 4
40.e odd 2 1 CM 1960.1.i.a 1
56.e even 2 1 1960.1.i.c 1
56.k odd 6 2 280.1.bi.a 2
56.m even 6 2 1960.1.bi.a 2
56.p even 6 2 1120.1.by.a 2
105.o odd 6 2 2520.1.ef.b 2
140.p odd 6 2 1120.1.by.a 2
168.v even 6 2 2520.1.ef.b 2
280.n even 2 1 1960.1.i.b 1
280.ba even 6 2 1960.1.bi.b 2
280.bf even 6 2 1120.1.by.b 2
280.bi odd 6 2 280.1.bi.b yes 2
280.br even 12 4 1400.1.ba.a 4
840.cv even 6 2 2520.1.ef.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.bi.a 2 35.j even 6 2
280.1.bi.a 2 56.k odd 6 2
280.1.bi.b yes 2 7.c even 3 2
280.1.bi.b yes 2 280.bi odd 6 2
1120.1.by.a 2 56.p even 6 2
1120.1.by.a 2 140.p odd 6 2
1120.1.by.b 2 28.g odd 6 2
1120.1.by.b 2 280.bf even 6 2
1400.1.ba.a 4 35.l odd 12 4
1400.1.ba.a 4 280.br even 12 4
1960.1.i.a 1 1.a even 1 1 trivial
1960.1.i.a 1 40.e odd 2 1 CM
1960.1.i.b 1 7.b odd 2 1
1960.1.i.b 1 280.n even 2 1
1960.1.i.c 1 35.c odd 2 1
1960.1.i.c 1 56.e even 2 1
1960.1.i.d 1 5.b even 2 1
1960.1.i.d 1 8.d odd 2 1
1960.1.bi.a 2 35.i odd 6 2
1960.1.bi.a 2 56.m even 6 2
1960.1.bi.b 2 7.d odd 6 2
1960.1.bi.b 2 280.ba even 6 2
2520.1.ef.a 2 21.h odd 6 2
2520.1.ef.a 2 840.cv even 6 2
2520.1.ef.b 2 105.o odd 6 2
2520.1.ef.b 2 168.v even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1960, [\chi])\):

\( T_{13} - 1 \) Copy content Toggle raw display
\( T_{19} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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