Properties

Label 540.2.n.b
Level $540$
Weight $2$
Character orbit 540.n
Analytic conductor $4.312$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [540,2,Mod(179,540)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(540, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("540.179");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 540.n (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: \(4.31192170915\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 180)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{5} + (3 \beta_{2} + 3) q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{5} + (3 \beta_{2} + 3) q^{7} + 2 \beta_{3} q^{8} + ( - \beta_{3} - 2 \beta_1 + 2) q^{10} + 3 \beta_1 q^{11} + ( - \beta_{3} + \beta_1) q^{13} + (3 \beta_{3} + 3 \beta_1) q^{14} + ( - 4 \beta_{2} - 4) q^{16} + \beta_{3} q^{17} + (3 \beta_{3} + 6 \beta_1) q^{19} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{20} + 6 \beta_{2} q^{22} + ( - 3 \beta_{2} - 6) q^{23} + (4 \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{25} + (4 \beta_{2} + 2) q^{26} - 6 q^{28} + (\beta_{2} - 1) q^{29} + ( - 3 \beta_{3} + 3 \beta_1) q^{31} + ( - 4 \beta_{3} - 4 \beta_1) q^{32} + ( - 2 \beta_{2} - 2) q^{34} + ( - 3 \beta_{3} - 6 \beta_{2} - 3) q^{35} + ( - \beta_{3} - 2 \beta_1) q^{37} + (6 \beta_{2} - 6) q^{38} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{40} + (3 \beta_{2} + 6) q^{41} + ( - 6 \beta_{2} - 6) q^{43} + 6 \beta_{3} q^{44} + ( - 3 \beta_{3} - 6 \beta_1) q^{46} + ( - 3 \beta_{2} + 3) q^{47} + 2 \beta_{2} q^{49} + (\beta_{3} - 4 \beta_{2} + \beta_1 - 8) q^{50} + (4 \beta_{3} + 2 \beta_1) q^{52} - 4 \beta_{3} q^{53} + ( - 3 \beta_{3} - 6 \beta_1 + 6) q^{55} - 6 \beta_1 q^{56} + (\beta_{3} - \beta_1) q^{58} + ( - 7 \beta_{2} - 7) q^{61} + (12 \beta_{2} + 6) q^{62} + 8 q^{64} + ( - 2 \beta_{2} - 3 \beta_1 + 2) q^{65} - 3 \beta_{2} q^{67} + ( - 2 \beta_{3} - 2 \beta_1) q^{68} + ( - 6 \beta_{3} + 6 \beta_{2} - 3 \beta_1 + 6) q^{70} - 3 \beta_{3} q^{71} + ( - 2 \beta_{3} - 4 \beta_1) q^{73} + ( - 2 \beta_{2} + 2) q^{74} + (6 \beta_{3} - 6 \beta_1) q^{76} + (9 \beta_{3} + 9 \beta_1) q^{77} + (4 \beta_{3} + 8 \beta_{2} + 4) q^{80} + (3 \beta_{3} + 6 \beta_1) q^{82} + (3 \beta_{2} - 3) q^{83} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{85} + ( - 6 \beta_{3} - 6 \beta_1) q^{86} + ( - 12 \beta_{2} - 12) q^{88} + ( - 10 \beta_{2} - 5) q^{89} + (3 \beta_{3} + 6 \beta_1) q^{91} + ( - 6 \beta_{2} + 6) q^{92} + ( - 3 \beta_{3} + 3 \beta_1) q^{94} + ( - 9 \beta_{3} + 6 \beta_{2} - 9 \beta_1 + 12) q^{95} + ( - 12 \beta_{3} - 6 \beta_1) q^{97} + 2 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 6 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 6 q^{5} + 6 q^{7} + 8 q^{10} - 8 q^{16} + 12 q^{20} - 12 q^{22} - 18 q^{23} + 2 q^{25} - 24 q^{28} - 6 q^{29} - 4 q^{34} - 36 q^{38} - 8 q^{40} + 18 q^{41} - 12 q^{43} + 18 q^{47} - 4 q^{49} - 24 q^{50} + 24 q^{55} - 14 q^{61} + 32 q^{64} + 12 q^{65} + 6 q^{67} + 12 q^{70} + 12 q^{74} - 18 q^{83} - 4 q^{85} - 24 q^{88} + 36 q^{92} + 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/540\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(461\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i 0 −1.00000 + 1.73205i −2.20711 + 0.358719i 0 1.50000 + 2.59808i 2.82843 0 2.00000 + 2.44949i
179.2 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −0.792893 2.09077i 0 1.50000 + 2.59808i −2.82843 0 2.00000 2.44949i
359.1 −0.707107 + 1.22474i 0 −1.00000 1.73205i −2.20711 0.358719i 0 1.50000 2.59808i 2.82843 0 2.00000 2.44949i
359.2 0.707107 1.22474i 0 −1.00000 1.73205i −0.792893 + 2.09077i 0 1.50000 2.59808i −2.82843 0 2.00000 + 2.44949i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
20.d odd 2 1 inner
180.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 540.2.n.b 4
3.b odd 2 1 180.2.n.a 4
4.b odd 2 1 540.2.n.a 4
5.b even 2 1 540.2.n.a 4
9.c even 3 1 180.2.n.a 4
9.d odd 6 1 inner 540.2.n.b 4
12.b even 2 1 180.2.n.b yes 4
15.d odd 2 1 180.2.n.b yes 4
15.e even 4 2 900.2.r.b 8
20.d odd 2 1 inner 540.2.n.b 4
36.f odd 6 1 180.2.n.b yes 4
36.h even 6 1 540.2.n.a 4
45.h odd 6 1 540.2.n.a 4
45.j even 6 1 180.2.n.b yes 4
45.k odd 12 2 900.2.r.b 8
60.h even 2 1 180.2.n.a 4
60.l odd 4 2 900.2.r.b 8
180.n even 6 1 inner 540.2.n.b 4
180.p odd 6 1 180.2.n.a 4
180.x even 12 2 900.2.r.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.n.a 4 3.b odd 2 1
180.2.n.a 4 9.c even 3 1
180.2.n.a 4 60.h even 2 1
180.2.n.a 4 180.p odd 6 1
180.2.n.b yes 4 12.b even 2 1
180.2.n.b yes 4 15.d odd 2 1
180.2.n.b yes 4 36.f odd 6 1
180.2.n.b yes 4 45.j even 6 1
540.2.n.a 4 4.b odd 2 1
540.2.n.a 4 5.b even 2 1
540.2.n.a 4 36.h even 6 1
540.2.n.a 4 45.h odd 6 1
540.2.n.b 4 1.a even 1 1 trivial
540.2.n.b 4 9.d odd 6 1 inner
540.2.n.b 4 20.d odd 2 1 inner
540.2.n.b 4 180.n even 6 1 inner
900.2.r.b 8 15.e even 4 2
900.2.r.b 8 45.k odd 12 2
900.2.r.b 8 60.l odd 4 2
900.2.r.b 8 180.x even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 3T_{7} + 9 \) acting on \(S_{2}^{\mathrm{new}}(540, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + 17 T^{2} + 30 T + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$13$ \( T^{4} - 6T^{2} + 36 \) Copy content Toggle raw display
$17$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 54)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 54T^{2} + 2916 \) Copy content Toggle raw display
$37$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 216 T^{2} + 46656 \) Copy content Toggle raw display
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