Properties

Label 2940.2.bb.d
Level $2940$
Weight $2$
Character orbit 2940.bb
Analytic conductor $23.476$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2940,2,Mod(949,2940)]
 
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([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2940.949");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.bb (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: \(23.4760181943\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - \zeta_{12}^{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} + (\zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{5} + ( - \zeta_{12}^{2} + 1) q^{9} + 4 \zeta_{12}^{2} q^{11} + (\zeta_{12}^{3} - 2) q^{15} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{17} - 4 \zeta_{12} q^{23} + ( - 4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 4 \zeta_{12}) q^{25} - \zeta_{12}^{3} q^{27} + 6 q^{29} - 4 \zeta_{12}^{2} q^{31} + 4 \zeta_{12} q^{33} - 8 \zeta_{12} q^{37} - 10 q^{41} + 4 \zeta_{12}^{3} q^{43} + (2 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12}) q^{45} - 4 \zeta_{12} q^{47} + (4 \zeta_{12}^{2} - 4) q^{51} + (12 \zeta_{12}^{3} - 12 \zeta_{12}) q^{53} + ( - 8 \zeta_{12}^{3} - 4) q^{55} + 4 \zeta_{12}^{2} q^{59} + (2 \zeta_{12}^{2} - 2) q^{61} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{67} - 4 q^{69} + (8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{73} + ( - 4 \zeta_{12}^{2} + 3 \zeta_{12} + 4) q^{75} + (12 \zeta_{12}^{2} - 12) q^{79} - \zeta_{12}^{2} q^{81} + 4 \zeta_{12}^{3} q^{83} + ( - 4 \zeta_{12}^{3} + 8) q^{85} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{87} + (10 \zeta_{12}^{2} - 10) q^{89} - 4 \zeta_{12} q^{93} - 8 \zeta_{12}^{3} q^{97} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} + 2 q^{9} + 8 q^{11} - 8 q^{15} + 6 q^{25} + 24 q^{29} - 8 q^{31} - 40 q^{41} + 2 q^{45} - 8 q^{51} - 16 q^{55} + 8 q^{59} - 4 q^{61} - 16 q^{69} + 8 q^{75} - 24 q^{79} - 2 q^{81} + 32 q^{85} - 20 q^{89} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 −0.866025 0.500000i 0 1.23205 1.86603i 0 0 0 0.500000 + 0.866025i 0
949.2 0 0.866025 + 0.500000i 0 −2.23205 + 0.133975i 0 0 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 1.23205 + 1.86603i 0 0 0 0.500000 0.866025i 0
1549.2 0 0.866025 0.500000i 0 −2.23205 0.133975i 0 0 0 0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2940.2.bb.d 4
5.b even 2 1 inner 2940.2.bb.d 4
7.b odd 2 1 2940.2.bb.e 4
7.c even 3 1 60.2.d.a 2
7.c even 3 1 inner 2940.2.bb.d 4
7.d odd 6 1 2940.2.k.c 2
7.d odd 6 1 2940.2.bb.e 4
21.h odd 6 1 180.2.d.a 2
28.g odd 6 1 240.2.f.b 2
35.c odd 2 1 2940.2.bb.e 4
35.i odd 6 1 2940.2.k.c 2
35.i odd 6 1 2940.2.bb.e 4
35.j even 6 1 60.2.d.a 2
35.j even 6 1 inner 2940.2.bb.d 4
35.l odd 12 1 300.2.a.a 1
35.l odd 12 1 300.2.a.d 1
56.k odd 6 1 960.2.f.c 2
56.p even 6 1 960.2.f.f 2
63.g even 3 1 1620.2.r.c 4
63.h even 3 1 1620.2.r.c 4
63.j odd 6 1 1620.2.r.d 4
63.n odd 6 1 1620.2.r.d 4
84.n even 6 1 720.2.f.c 2
105.o odd 6 1 180.2.d.a 2
105.x even 12 1 900.2.a.a 1
105.x even 12 1 900.2.a.h 1
112.u odd 12 1 3840.2.d.b 2
112.u odd 12 1 3840.2.d.be 2
112.w even 12 1 3840.2.d.o 2
112.w even 12 1 3840.2.d.r 2
140.p odd 6 1 240.2.f.b 2
140.w even 12 1 1200.2.a.a 1
140.w even 12 1 1200.2.a.s 1
168.s odd 6 1 2880.2.f.l 2
168.v even 6 1 2880.2.f.p 2
280.bf even 6 1 960.2.f.f 2
280.bi odd 6 1 960.2.f.c 2
280.br even 12 1 4800.2.a.bf 1
280.br even 12 1 4800.2.a.bk 1
280.bt odd 12 1 4800.2.a.bj 1
280.bt odd 12 1 4800.2.a.bn 1
315.r even 6 1 1620.2.r.c 4
315.v odd 6 1 1620.2.r.d 4
315.bo even 6 1 1620.2.r.c 4
315.br odd 6 1 1620.2.r.d 4
420.ba even 6 1 720.2.f.c 2
420.bp odd 12 1 3600.2.a.d 1
420.bp odd 12 1 3600.2.a.bm 1
560.cr even 12 1 3840.2.d.o 2
560.cr even 12 1 3840.2.d.r 2
560.cs odd 12 1 3840.2.d.b 2
560.cs odd 12 1 3840.2.d.be 2
840.cg odd 6 1 2880.2.f.l 2
840.cv even 6 1 2880.2.f.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.d.a 2 7.c even 3 1
60.2.d.a 2 35.j even 6 1
180.2.d.a 2 21.h odd 6 1
180.2.d.a 2 105.o odd 6 1
240.2.f.b 2 28.g odd 6 1
240.2.f.b 2 140.p odd 6 1
300.2.a.a 1 35.l odd 12 1
300.2.a.d 1 35.l odd 12 1
720.2.f.c 2 84.n even 6 1
720.2.f.c 2 420.ba even 6 1
900.2.a.a 1 105.x even 12 1
900.2.a.h 1 105.x even 12 1
960.2.f.c 2 56.k odd 6 1
960.2.f.c 2 280.bi odd 6 1
960.2.f.f 2 56.p even 6 1
960.2.f.f 2 280.bf even 6 1
1200.2.a.a 1 140.w even 12 1
1200.2.a.s 1 140.w even 12 1
1620.2.r.c 4 63.g even 3 1
1620.2.r.c 4 63.h even 3 1
1620.2.r.c 4 315.r even 6 1
1620.2.r.c 4 315.bo even 6 1
1620.2.r.d 4 63.j odd 6 1
1620.2.r.d 4 63.n odd 6 1
1620.2.r.d 4 315.v odd 6 1
1620.2.r.d 4 315.br odd 6 1
2880.2.f.l 2 168.s odd 6 1
2880.2.f.l 2 840.cg odd 6 1
2880.2.f.p 2 168.v even 6 1
2880.2.f.p 2 840.cv even 6 1
2940.2.k.c 2 7.d odd 6 1
2940.2.k.c 2 35.i odd 6 1
2940.2.bb.d 4 1.a even 1 1 trivial
2940.2.bb.d 4 5.b even 2 1 inner
2940.2.bb.d 4 7.c even 3 1 inner
2940.2.bb.d 4 35.j even 6 1 inner
2940.2.bb.e 4 7.b odd 2 1
2940.2.bb.e 4 7.d odd 6 1
2940.2.bb.e 4 35.c odd 2 1
2940.2.bb.e 4 35.i odd 6 1
3600.2.a.d 1 420.bp odd 12 1
3600.2.a.bm 1 420.bp odd 12 1
3840.2.d.b 2 112.u odd 12 1
3840.2.d.b 2 560.cs odd 12 1
3840.2.d.o 2 112.w even 12 1
3840.2.d.o 2 560.cr even 12 1
3840.2.d.r 2 112.w even 12 1
3840.2.d.r 2 560.cr even 12 1
3840.2.d.be 2 112.u odd 12 1
3840.2.d.be 2 560.cs odd 12 1
4800.2.a.bf 1 280.br even 12 1
4800.2.a.bj 1 280.bt odd 12 1
4800.2.a.bk 1 280.br even 12 1
4800.2.a.bn 1 280.bt odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2940, [\chi])\):

\( T_{11}^{2} - 4T_{11} + 16 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{31}^{2} + 4T_{31} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} - T^{2} + 10 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T - 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$41$ \( (T + 10)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$53$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
$59$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T + 144)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
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