Properties

Label 720.2.q.g
Level $720$
Weight $2$
Character orbit 720.q
Analytic conductor $5.749$
Analytic rank $0$
Dimension $4$
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] = [720,2,Mod(241,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.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: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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 360)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2} + \beta_1) q^{3} + \beta_{2} q^{5} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} - 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2} + \beta_1) q^{3} + \beta_{2} q^{5} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} - 2 \beta_1 + 1) q^{9} + ( - \beta_{2} + \beta_1 + 1) q^{15} - 2 q^{17} + ( - 2 \beta_{3} + 4 \beta_1 + 2) q^{19} + ( - 4 \beta_{2} - 2 \beta_1 + 1) q^{21} + ( - \beta_{3} - 5 \beta_{2} - \beta_1) q^{23} + (\beta_{2} - 1) q^{25} + (\beta_{3} - 5) q^{27} + ( - 4 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 3) q^{29} + ( - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1) q^{31} + ( - \beta_{3} + 2 \beta_1 + 1) q^{35} - 6 q^{37} + (2 \beta_{3} + 5 \beta_{2} + 2 \beta_1) q^{41} + ( - 8 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 2) q^{43} + ( - 2 \beta_{3} + 1) q^{45} + (2 \beta_{3} + 7 \beta_{2} - \beta_1 - 7) q^{47} + ( - 2 \beta_{3} - 2 \beta_1) q^{49} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{51} + ( - 4 \beta_{3} + 8 \beta_1 - 2) q^{53} + ( - 4 \beta_{3} - 6 \beta_{2} + 8) q^{57} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{59} + ( - 3 \beta_{2} + 3) q^{61} + (\beta_{3} + 3 \beta_{2} - 3 \beta_1 - 8) q^{63} + ( - 5 \beta_{3} - \beta_{2} - 5 \beta_1) q^{67} + (2 \beta_{3} + 3 \beta_{2} - 6 \beta_1 - 7) q^{69} + (4 \beta_{3} - 8 \beta_1) q^{71} + ( - 4 \beta_{3} + 8 \beta_1 + 4) q^{73} + (\beta_{3} + 1) q^{75} + (4 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{79} + (4 \beta_{3} + 7 \beta_{2} - 4 \beta_1) q^{81} + (2 \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{83} - 2 \beta_{2} q^{85} + (5 \beta_{3} - 8 \beta_{2} - 4 \beta_1 + 7) q^{87} + ( - 4 \beta_{3} + 8 \beta_1 - 7) q^{89} + (4 \beta_{3} - 10 \beta_{2} + 4 \beta_1 + 2) q^{93} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{95} + (2 \beta_{2} - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{15} - 8 q^{17} + 8 q^{19} - 4 q^{21} - 10 q^{23} - 2 q^{25} - 20 q^{27} - 6 q^{29} + 12 q^{31} + 4 q^{35} - 24 q^{37} + 10 q^{41} + 4 q^{43} + 4 q^{45} - 14 q^{47} + 4 q^{51} - 8 q^{53} + 20 q^{57} - 12 q^{59} + 6 q^{61} - 26 q^{63} - 2 q^{67} - 22 q^{69} + 16 q^{73} + 4 q^{75} - 4 q^{79} + 14 q^{81} - 6 q^{83} - 4 q^{85} + 12 q^{87} - 28 q^{89} - 12 q^{93} + 4 q^{95} - 4 q^{97}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
0 −1.72474 0.158919i 0 0.500000 + 0.866025i 0 −0.724745 + 1.25529i 0 2.94949 + 0.548188i 0
241.2 0 0.724745 1.57313i 0 0.500000 + 0.866025i 0 1.72474 2.98735i 0 −1.94949 2.28024i 0
481.1 0 −1.72474 + 0.158919i 0 0.500000 0.866025i 0 −0.724745 1.25529i 0 2.94949 0.548188i 0
481.2 0 0.724745 + 1.57313i 0 0.500000 0.866025i 0 1.72474 + 2.98735i 0 −1.94949 + 2.28024i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.q.g 4
3.b odd 2 1 2160.2.q.g 4
4.b odd 2 1 360.2.q.c 4
9.c even 3 1 inner 720.2.q.g 4
9.c even 3 1 6480.2.a.bc 2
9.d odd 6 1 2160.2.q.g 4
9.d odd 6 1 6480.2.a.bl 2
12.b even 2 1 1080.2.q.c 4
36.f odd 6 1 360.2.q.c 4
36.f odd 6 1 3240.2.a.j 2
36.h even 6 1 1080.2.q.c 4
36.h even 6 1 3240.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.c 4 4.b odd 2 1
360.2.q.c 4 36.f odd 6 1
720.2.q.g 4 1.a even 1 1 trivial
720.2.q.g 4 9.c even 3 1 inner
1080.2.q.c 4 12.b even 2 1
1080.2.q.c 4 36.h even 6 1
2160.2.q.g 4 3.b odd 2 1
2160.2.q.g 4 9.d odd 6 1
3240.2.a.j 2 36.f odd 6 1
3240.2.a.o 2 36.h even 6 1
6480.2.a.bc 2 9.c even 3 1
6480.2.a.bl 2 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{4} - 2T_{7}^{3} + 9T_{7}^{2} + 10T_{7} + 25 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + T^{2} + 6 T + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + 9 T^{2} + 10 T + 25 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T + 2)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 10 T^{3} + 81 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$29$ \( T^{4} + 6 T^{3} + 51 T^{2} - 90 T + 225 \) Copy content Toggle raw display
$31$ \( T^{4} - 12 T^{3} + 132 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$37$ \( (T + 6)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 10 T^{3} + 99 T^{2} - 10 T + 1 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} + 108 T^{2} + \cdots + 8464 \) Copy content Toggle raw display
$47$ \( T^{4} + 14 T^{3} + 153 T^{2} + \cdots + 1849 \) Copy content Toggle raw display
$53$ \( (T^{2} + 4 T - 92)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} + 132 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$61$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} + 153 T^{2} + \cdots + 22201 \) Copy content Toggle raw display
$71$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 8 T - 80)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 4 T^{3} + 36 T^{2} - 80 T + 400 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} + 33 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$89$ \( (T^{2} + 14 T - 47)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
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