Properties

Label 2940.1.be.b
Level $2940$
Weight $1$
Character orbit 2940.be
Analytic conductor $1.467$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -20, -84, 105
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2940,1,Mod(1979,2940)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2940, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 3, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2940.1979");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2940.be (of order \(6\), degree \(2\), not 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: \(1.46725113714\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-5}, \sqrt{-21})\)
Artin image: $D_4:C_6$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12}^{5} q^{2} + \zeta_{12} q^{3} - \zeta_{12}^{4} q^{4} - \zeta_{12}^{2} q^{5} + q^{6} - \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12}^{5} q^{2} + \zeta_{12} q^{3} - \zeta_{12}^{4} q^{4} - \zeta_{12}^{2} q^{5} + q^{6} - \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} - \zeta_{12} q^{10} - \zeta_{12}^{5} q^{12} - \zeta_{12}^{3} q^{15} - \zeta_{12}^{2} q^{16} + \zeta_{12} q^{18} - q^{20} - \zeta_{12}^{5} q^{23} - \zeta_{12}^{4} q^{24} + \zeta_{12}^{4} q^{25} + \zeta_{12}^{3} q^{27} - \zeta_{12}^{2} q^{30} - \zeta_{12} q^{32} + q^{36} + \zeta_{12}^{5} q^{40} - q^{41} - \zeta_{12}^{4} q^{45} - 2 \zeta_{12}^{4} q^{46} - \zeta_{12}^{3} q^{48} + \zeta_{12}^{3} q^{50} + \zeta_{12}^{2} q^{54} - \zeta_{12} q^{60} - q^{64} + 2 q^{69} - \zeta_{12}^{5} q^{72} + \zeta_{12}^{5} q^{75} + \zeta_{12}^{4} q^{80} + \zeta_{12}^{4} q^{81} + 2 \zeta_{12}^{5} q^{82} + \zeta_{12}^{2} q^{89} - \zeta_{12}^{3} q^{90} - 2 \zeta_{12}^{3} q^{92} - \zeta_{12}^{2} q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{9} - 2 q^{16} - 4 q^{20} + 2 q^{24} - 2 q^{25} - 2 q^{30} + 4 q^{36} - 8 q^{41} + 2 q^{45} + 4 q^{46} + 2 q^{54} - 4 q^{64} + 8 q^{69} - 2 q^{80} - 2 q^{81} + 4 q^{89} - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1979.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 0 1.00000i 0.500000 0.866025i 0.866025 0.500000i
1979.2 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 0 1.00000i 0.500000 0.866025i −0.866025 + 0.500000i
2579.1 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i −0.500000 0.866025i 1.00000 0 1.00000i 0.500000 + 0.866025i 0.866025 + 0.500000i
2579.2 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −0.500000 0.866025i 1.00000 0 1.00000i 0.500000 + 0.866025i −0.866025 0.500000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
105.g even 2 1 RM by \(\Q(\sqrt{105}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
7.c even 3 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
28.g odd 6 1 inner
35.j even 6 1 inner
84.j odd 6 1 inner
105.p even 6 1 inner
140.p odd 6 1 inner
420.o odd 2 1 inner
420.be odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2940.1.be.b 4
3.b odd 2 1 2940.1.be.c 4
4.b odd 2 1 inner 2940.1.be.b 4
5.b even 2 1 inner 2940.1.be.b 4
7.b odd 2 1 2940.1.be.c 4
7.c even 3 1 420.1.o.b yes 2
7.c even 3 1 inner 2940.1.be.b 4
7.d odd 6 1 420.1.o.a 2
7.d odd 6 1 2940.1.be.c 4
12.b even 2 1 2940.1.be.c 4
15.d odd 2 1 2940.1.be.c 4
20.d odd 2 1 CM 2940.1.be.b 4
21.c even 2 1 inner 2940.1.be.b 4
21.g even 6 1 420.1.o.b yes 2
21.g even 6 1 inner 2940.1.be.b 4
21.h odd 6 1 420.1.o.a 2
21.h odd 6 1 2940.1.be.c 4
28.d even 2 1 2940.1.be.c 4
28.f even 6 1 420.1.o.a 2
28.f even 6 1 2940.1.be.c 4
28.g odd 6 1 420.1.o.b yes 2
28.g odd 6 1 inner 2940.1.be.b 4
35.c odd 2 1 2940.1.be.c 4
35.i odd 6 1 420.1.o.a 2
35.i odd 6 1 2940.1.be.c 4
35.j even 6 1 420.1.o.b yes 2
35.j even 6 1 inner 2940.1.be.b 4
35.k even 12 1 2100.1.m.b 1
35.k even 12 1 2100.1.m.c 1
35.l odd 12 1 2100.1.m.a 1
35.l odd 12 1 2100.1.m.d 1
60.h even 2 1 2940.1.be.c 4
84.h odd 2 1 CM 2940.1.be.b 4
84.j odd 6 1 420.1.o.b yes 2
84.j odd 6 1 inner 2940.1.be.b 4
84.n even 6 1 420.1.o.a 2
84.n even 6 1 2940.1.be.c 4
105.g even 2 1 RM 2940.1.be.b 4
105.o odd 6 1 420.1.o.a 2
105.o odd 6 1 2940.1.be.c 4
105.p even 6 1 420.1.o.b yes 2
105.p even 6 1 inner 2940.1.be.b 4
105.w odd 12 1 2100.1.m.a 1
105.w odd 12 1 2100.1.m.d 1
105.x even 12 1 2100.1.m.b 1
105.x even 12 1 2100.1.m.c 1
140.c even 2 1 2940.1.be.c 4
140.p odd 6 1 420.1.o.b yes 2
140.p odd 6 1 inner 2940.1.be.b 4
140.s even 6 1 420.1.o.a 2
140.s even 6 1 2940.1.be.c 4
140.w even 12 1 2100.1.m.a 1
140.w even 12 1 2100.1.m.d 1
140.x odd 12 1 2100.1.m.b 1
140.x odd 12 1 2100.1.m.c 1
420.o odd 2 1 inner 2940.1.be.b 4
420.ba even 6 1 420.1.o.a 2
420.ba even 6 1 2940.1.be.c 4
420.be odd 6 1 420.1.o.b yes 2
420.be odd 6 1 inner 2940.1.be.b 4
420.bp odd 12 1 2100.1.m.b 1
420.bp odd 12 1 2100.1.m.c 1
420.br even 12 1 2100.1.m.a 1
420.br even 12 1 2100.1.m.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.1.o.a 2 7.d odd 6 1
420.1.o.a 2 21.h odd 6 1
420.1.o.a 2 28.f even 6 1
420.1.o.a 2 35.i odd 6 1
420.1.o.a 2 84.n even 6 1
420.1.o.a 2 105.o odd 6 1
420.1.o.a 2 140.s even 6 1
420.1.o.a 2 420.ba even 6 1
420.1.o.b yes 2 7.c even 3 1
420.1.o.b yes 2 21.g even 6 1
420.1.o.b yes 2 28.g odd 6 1
420.1.o.b yes 2 35.j even 6 1
420.1.o.b yes 2 84.j odd 6 1
420.1.o.b yes 2 105.p even 6 1
420.1.o.b yes 2 140.p odd 6 1
420.1.o.b yes 2 420.be odd 6 1
2100.1.m.a 1 35.l odd 12 1
2100.1.m.a 1 105.w odd 12 1
2100.1.m.a 1 140.w even 12 1
2100.1.m.a 1 420.br even 12 1
2100.1.m.b 1 35.k even 12 1
2100.1.m.b 1 105.x even 12 1
2100.1.m.b 1 140.x odd 12 1
2100.1.m.b 1 420.bp odd 12 1
2100.1.m.c 1 35.k even 12 1
2100.1.m.c 1 105.x even 12 1
2100.1.m.c 1 140.x odd 12 1
2100.1.m.c 1 420.bp odd 12 1
2100.1.m.d 1 35.l odd 12 1
2100.1.m.d 1 105.w odd 12 1
2100.1.m.d 1 140.w even 12 1
2100.1.m.d 1 420.br even 12 1
2940.1.be.b 4 1.a even 1 1 trivial
2940.1.be.b 4 4.b odd 2 1 inner
2940.1.be.b 4 5.b even 2 1 inner
2940.1.be.b 4 7.c even 3 1 inner
2940.1.be.b 4 20.d odd 2 1 CM
2940.1.be.b 4 21.c even 2 1 inner
2940.1.be.b 4 21.g even 6 1 inner
2940.1.be.b 4 28.g odd 6 1 inner
2940.1.be.b 4 35.j even 6 1 inner
2940.1.be.b 4 84.h odd 2 1 CM
2940.1.be.b 4 84.j odd 6 1 inner
2940.1.be.b 4 105.g even 2 1 RM
2940.1.be.b 4 105.p even 6 1 inner
2940.1.be.b 4 140.p odd 6 1 inner
2940.1.be.b 4 420.o odd 2 1 inner
2940.1.be.b 4 420.be odd 6 1 inner
2940.1.be.c 4 3.b odd 2 1
2940.1.be.c 4 7.b odd 2 1
2940.1.be.c 4 7.d odd 6 1
2940.1.be.c 4 12.b even 2 1
2940.1.be.c 4 15.d odd 2 1
2940.1.be.c 4 21.h odd 6 1
2940.1.be.c 4 28.d even 2 1
2940.1.be.c 4 28.f even 6 1
2940.1.be.c 4 35.c odd 2 1
2940.1.be.c 4 35.i odd 6 1
2940.1.be.c 4 60.h even 2 1
2940.1.be.c 4 84.n even 6 1
2940.1.be.c 4 105.o odd 6 1
2940.1.be.c 4 140.c even 2 1
2940.1.be.c 4 140.s even 6 1
2940.1.be.c 4 420.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2940, [\chi])\):

\( T_{17} \) Copy content Toggle raw display
\( T_{23}^{4} - 4T_{23}^{2} + 16 \) Copy content Toggle raw display
\( T_{41} + 2 \) Copy content Toggle raw display
\( T_{983} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T + 2)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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