Properties

Label 2520.2.bi.q
Level $2520$
Weight $2$
Character orbit 2520.bi
Analytic conductor $20.122$
Analytic rank $0$
Dimension $6$
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] = [2520,2,Mod(361,2520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("2520.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.bi (of order \(3\), degree \(2\), 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: \(20.1223013094\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.11337408.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 18x^{4} + 81x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1 + 1) q^{7} + ( - \beta_{4} - \beta_{2} - 2 \beta_1 + 1) q^{11} + ( - 2 \beta_{3} + \beta_1 + 1) q^{13} + ( - 2 \beta_{2} + 2) q^{17} + ( - 2 \beta_{5} + \beta_{4} - \beta_{2} - \beta_1) q^{19} + ( - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1) q^{23} + (\beta_{2} - 1) q^{25} + ( - 2 \beta_{4} - \beta_1 - 4) q^{29} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 4) q^{31} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{35} + (2 \beta_{5} - \beta_{4} - 3 \beta_{2} + \beta_1) q^{37} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 3) q^{41} + (\beta_{4} + \beta_{3} + 4) q^{43} + ( - 2 \beta_{5} + 3 \beta_{4} - 5 \beta_{2} - 3 \beta_1) q^{47} + ( - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 1) q^{49} + ( - 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{53} + ( - 2 \beta_{4} - \beta_1 + 1) q^{55} + ( - 8 \beta_{2} + 8) q^{59} + (2 \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1) q^{61} + ( - 2 \beta_{5} + \beta_{4} + \beta_{2} - \beta_1) q^{65} + ( - 3 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 2) q^{67} + ( - 4 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 4 \beta_1) q^{73} + (2 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 7) q^{77} + ( - 4 \beta_{5} + 2 \beta_{4} + 6 \beta_{2} - 2 \beta_1) q^{79} + (\beta_{4} + \beta_{3} + 10) q^{83} + 2 q^{85} + ( - 2 \beta_{5} + \beta_{4} - \beta_1) q^{89} + ( - 2 \beta_{5} - 10 \beta_{2} + \beta_1 + 9) q^{91} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{5} + 6 q^{7} + 3 q^{11} + 6 q^{13} + 6 q^{17} - 3 q^{19} + 3 q^{23} - 3 q^{25} - 24 q^{29} - 12 q^{31} + 3 q^{35} - 9 q^{37} - 18 q^{41} + 24 q^{43} - 15 q^{47} - 12 q^{49} + 9 q^{53} + 6 q^{55} + 24 q^{59} + 6 q^{61} + 3 q^{65} - 6 q^{67} + 39 q^{77} + 18 q^{79} + 60 q^{83} + 12 q^{85} + 24 q^{91} + 3 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 18x^{4} + 81x^{2} + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 9\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{2} + \nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 9\nu^{2} - 2\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + \nu^{4} + 15\nu^{3} + 11\nu^{2} + 48\nu + 12 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{3} - \beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 9\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} - 18\beta_{3} + 11\beta _1 + 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{5} - 4\beta_{4} - 4\beta_{3} - 60\beta_{2} + 87\beta _1 + 30 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\) \(1\) \(1\)

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
0.391571i
3.17656i
2.78499i
0.391571i
3.17656i
2.78499i
0 0 0 0.500000 + 0.866025i 0 −0.292113 + 2.62958i 0 0 0
361.2 0 0 0 0.500000 + 0.866025i 0 0.647140 2.56539i 0 0 0
361.3 0 0 0 0.500000 + 0.866025i 0 2.64497 0.0641892i 0 0 0
1801.1 0 0 0 0.500000 0.866025i 0 −0.292113 2.62958i 0 0 0
1801.2 0 0 0 0.500000 0.866025i 0 0.647140 + 2.56539i 0 0 0
1801.3 0 0 0 0.500000 0.866025i 0 2.64497 + 0.0641892i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.3
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 2520.2.bi.q 6
3.b odd 2 1 280.2.q.e 6
7.c even 3 1 inner 2520.2.bi.q 6
12.b even 2 1 560.2.q.l 6
15.d odd 2 1 1400.2.q.j 6
15.e even 4 2 1400.2.bh.i 12
21.c even 2 1 1960.2.q.w 6
21.g even 6 1 1960.2.a.v 3
21.g even 6 1 1960.2.q.w 6
21.h odd 6 1 280.2.q.e 6
21.h odd 6 1 1960.2.a.w 3
84.j odd 6 1 3920.2.a.cb 3
84.n even 6 1 560.2.q.l 6
84.n even 6 1 3920.2.a.cc 3
105.o odd 6 1 1400.2.q.j 6
105.o odd 6 1 9800.2.a.ce 3
105.p even 6 1 9800.2.a.cf 3
105.x even 12 2 1400.2.bh.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.e 6 3.b odd 2 1
280.2.q.e 6 21.h odd 6 1
560.2.q.l 6 12.b even 2 1
560.2.q.l 6 84.n even 6 1
1400.2.q.j 6 15.d odd 2 1
1400.2.q.j 6 105.o odd 6 1
1400.2.bh.i 12 15.e even 4 2
1400.2.bh.i 12 105.x even 12 2
1960.2.a.v 3 21.g even 6 1
1960.2.a.w 3 21.h odd 6 1
1960.2.q.w 6 21.c even 2 1
1960.2.q.w 6 21.g even 6 1
2520.2.bi.q 6 1.a even 1 1 trivial
2520.2.bi.q 6 7.c even 3 1 inner
3920.2.a.cb 3 84.j odd 6 1
3920.2.a.cc 3 84.n even 6 1
9800.2.a.ce 3 105.o odd 6 1
9800.2.a.cf 3 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2520, [\chi])\):

\( T_{11}^{6} - 3T_{11}^{5} + 33T_{11}^{4} - 16T_{11}^{3} + 708T_{11}^{2} - 1056T_{11} + 1936 \) Copy content Toggle raw display
\( T_{13}^{3} - 3T_{13}^{2} - 24T_{13} + 68 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} - 6 T^{5} + 24 T^{4} - 80 T^{3} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} + 33 T^{4} + \cdots + 1936 \) Copy content Toggle raw display
$13$ \( (T^{3} - 3 T^{2} - 24 T + 68)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{3} \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} + 33 T^{4} - 104 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{6} - 3 T^{5} + 24 T^{4} + 59 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$29$ \( (T^{3} + 12 T^{2} + 21 T - 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 12 T^{5} + 132 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$37$ \( T^{6} + 9 T^{5} + 81 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$41$ \( (T^{3} + 9 T^{2} - 45 T - 381)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 12 T^{2} + 39 T - 22)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 15 T^{5} + 321 T^{4} + \cdots + 2521744 \) Copy content Toggle raw display
$53$ \( T^{6} - 9 T^{5} + 153 T^{4} + \cdots + 389376 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T + 64)^{3} \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} + 123 T^{4} + \cdots + 295936 \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + 105 T^{4} - 430 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} + 108 T^{4} + 672 T^{3} + \cdots + 112896 \) Copy content Toggle raw display
$79$ \( T^{6} - 18 T^{5} + 324 T^{4} + \cdots + 589824 \) Copy content Toggle raw display
$83$ \( (T^{3} - 30 T^{2} + 291 T - 904)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 27 T^{4} - 84 T^{3} + \cdots + 1764 \) Copy content Toggle raw display
$97$ \( (T + 2)^{6} \) Copy content Toggle raw display
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