Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2940,2,Mod(361,2940)]
 
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, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2940.361");
 
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.q (of order \(3\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} + \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} + \zeta_{6} q^{5} - \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} - q^{13} - q^{15} + (4 \zeta_{6} - 4) q^{17} - \zeta_{6} q^{19} - 4 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + q^{27} + (5 \zeta_{6} - 5) q^{31} + 2 \zeta_{6} q^{33} + 5 \zeta_{6} q^{37} + ( - \zeta_{6} + 1) q^{39} - 2 q^{41} - 9 q^{43} + ( - \zeta_{6} + 1) q^{45} - 2 \zeta_{6} q^{47} - 4 \zeta_{6} q^{51} + (12 \zeta_{6} - 12) q^{53} + 2 q^{55} + q^{57} + (8 \zeta_{6} - 8) q^{59} - 14 \zeta_{6} q^{61} - \zeta_{6} q^{65} + (9 \zeta_{6} - 9) q^{67} + 4 q^{69} + 2 q^{71} + ( - \zeta_{6} + 1) q^{73} - \zeta_{6} q^{75} + 3 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 18 q^{83} - 4 q^{85} - 4 \zeta_{6} q^{89} - 5 \zeta_{6} q^{93} + ( - \zeta_{6} + 1) q^{95} - 10 q^{97} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + q^{5} - q^{9} + 2 q^{11} - 2 q^{13} - 2 q^{15} - 4 q^{17} - q^{19} - 4 q^{23} - q^{25} + 2 q^{27} - 5 q^{31} + 2 q^{33} + 5 q^{37} + q^{39} - 4 q^{41} - 18 q^{43} + q^{45} - 2 q^{47} - 4 q^{51} - 12 q^{53} + 4 q^{55} + 2 q^{57} - 8 q^{59} - 14 q^{61} - q^{65} - 9 q^{67} + 8 q^{69} + 4 q^{71} + q^{73} - q^{75} + 3 q^{79} - q^{81} + 36 q^{83} - 8 q^{85} - 4 q^{89} - 5 q^{93} + q^{95} - 20 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 0 0 −0.500000 0.866025i 0
961.1 0 −0.500000 0.866025i 0 0.500000 0.866025i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2940.2.q.h 2
7.b odd 2 1 420.2.q.a 2
7.c even 3 1 2940.2.a.h 1
7.c even 3 1 inner 2940.2.q.h 2
7.d odd 6 1 420.2.q.a 2
7.d odd 6 1 2940.2.a.d 1
21.c even 2 1 1260.2.s.d 2
21.g even 6 1 1260.2.s.d 2
21.g even 6 1 8820.2.a.j 1
21.h odd 6 1 8820.2.a.y 1
28.d even 2 1 1680.2.bg.a 2
28.f even 6 1 1680.2.bg.a 2
35.c odd 2 1 2100.2.q.a 2
35.f even 4 2 2100.2.bc.c 4
35.i odd 6 1 2100.2.q.a 2
35.k even 12 2 2100.2.bc.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.a 2 7.b odd 2 1
420.2.q.a 2 7.d odd 6 1
1260.2.s.d 2 21.c even 2 1
1260.2.s.d 2 21.g even 6 1
1680.2.bg.a 2 28.d even 2 1
1680.2.bg.a 2 28.f even 6 1
2100.2.q.a 2 35.c odd 2 1
2100.2.q.a 2 35.i odd 6 1
2100.2.bc.c 4 35.f even 4 2
2100.2.bc.c 4 35.k even 12 2
2940.2.a.d 1 7.d odd 6 1
2940.2.a.h 1 7.c even 3 1
2940.2.q.h 2 1.a even 1 1 trivial
2940.2.q.h 2 7.c even 3 1 inner
8820.2.a.j 1 21.g even 6 1
8820.2.a.y 1 21.h odd 6 1

Hecke kernels

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

\( T_{11}^{2} - 2T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} + 16 \) Copy content Toggle raw display
\( T_{31}^{2} + 5T_{31} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T + 9)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$61$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$67$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$71$ \( (T - 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$83$ \( (T - 18)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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