Properties

Label 720.2.x.f
Level $720$
Weight $2$
Character orbit 720.x
Analytic conductor $5.749$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,2,Mod(127,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.127");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.x (of order \(4\), 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: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2} - 1) q^{5} + (\beta_{7} + \beta_{6} + \beta_{4}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2} - 1) q^{5} + (\beta_{7} + \beta_{6} + \beta_{4}) q^{7} + \beta_{6} q^{11} + 2 \beta_{3} q^{13} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{17} + (\beta_{7} + \beta_{5} - 2 \beta_{4}) q^{19} + (\beta_{6} + 2 \beta_{5} - \beta_{4}) q^{23} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{25} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{29} + 6 \beta_{6} q^{31} + (3 \beta_{6} - 2 \beta_{5} - 2 \beta_{4}) q^{35} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{37} + (2 \beta_{3} - 2 \beta_1 + 6) q^{41} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4}) q^{43} + (4 \beta_{7} - \beta_{6} - \beta_{4}) q^{47} + ( - 4 \beta_{3} - 3 \beta_{2} - 4 \beta_1) q^{49} + ( - 2 \beta_{2} + 2) q^{53} + ( - \beta_{6} - \beta_{5} - \beta_{4}) q^{55} + 7 \beta_{4} q^{59} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{61} + ( - 2 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{65} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4}) q^{67} + ( - 2 \beta_{7} - 4 \beta_{6} + 2 \beta_{5}) q^{71} + ( - 3 \beta_{2} + 3) q^{73} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{77} + ( - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4}) q^{79} + (\beta_{6} - \beta_{4}) q^{83} + (4 \beta_{3} + 4 \beta_{2} - 6) q^{85} - 2 \beta_{2} q^{89} + ( - 2 \beta_{7} - 6 \beta_{6} + 2 \beta_{5}) q^{91} + ( - 2 \beta_{7} + \beta_{6} + \cdots + 5 \beta_{4}) q^{95}+ \cdots + (3 \beta_{2} + 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} - 16 q^{17} - 32 q^{37} + 48 q^{41} + 16 q^{53} - 16 q^{61} + 24 q^{73} - 16 q^{77} - 48 q^{85} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{24}^{7} - \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{4} + 2\zeta_{24}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{6} - 2\zeta_{24}^{4} + 2\zeta_{24}^{2} + 1 \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{6} + \beta_{4} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{7} + \beta_{5} + 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( -\beta_{7} + \beta_{5} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{6} - \beta_{4} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{6} + \beta_{4} + 2\beta_{3} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−0.258819 0.965926i
−0.965926 0.258819i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.258819 0.965926i
0.965926 0.258819i
0 0 0 −2.22474 + 0.224745i 0 −0.317837 + 0.317837i 0 0 0
127.2 0 0 0 −2.22474 + 0.224745i 0 0.317837 0.317837i 0 0 0
127.3 0 0 0 0.224745 2.22474i 0 −3.14626 + 3.14626i 0 0 0
127.4 0 0 0 0.224745 2.22474i 0 3.14626 3.14626i 0 0 0
703.1 0 0 0 −2.22474 0.224745i 0 −0.317837 0.317837i 0 0 0
703.2 0 0 0 −2.22474 0.224745i 0 0.317837 + 0.317837i 0 0 0
703.3 0 0 0 0.224745 + 2.22474i 0 −3.14626 3.14626i 0 0 0
703.4 0 0 0 0.224745 + 2.22474i 0 3.14626 + 3.14626i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.x.f 8
3.b odd 2 1 240.2.w.b 8
4.b odd 2 1 inner 720.2.x.f 8
5.b even 2 1 3600.2.x.n 8
5.c odd 4 1 inner 720.2.x.f 8
5.c odd 4 1 3600.2.x.n 8
12.b even 2 1 240.2.w.b 8
15.d odd 2 1 1200.2.w.b 8
15.e even 4 1 240.2.w.b 8
15.e even 4 1 1200.2.w.b 8
20.d odd 2 1 3600.2.x.n 8
20.e even 4 1 inner 720.2.x.f 8
20.e even 4 1 3600.2.x.n 8
24.f even 2 1 960.2.w.d 8
24.h odd 2 1 960.2.w.d 8
60.h even 2 1 1200.2.w.b 8
60.l odd 4 1 240.2.w.b 8
60.l odd 4 1 1200.2.w.b 8
120.q odd 4 1 960.2.w.d 8
120.w even 4 1 960.2.w.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.w.b 8 3.b odd 2 1
240.2.w.b 8 12.b even 2 1
240.2.w.b 8 15.e even 4 1
240.2.w.b 8 60.l odd 4 1
720.2.x.f 8 1.a even 1 1 trivial
720.2.x.f 8 4.b odd 2 1 inner
720.2.x.f 8 5.c odd 4 1 inner
720.2.x.f 8 20.e even 4 1 inner
960.2.w.d 8 24.f even 2 1
960.2.w.d 8 24.h odd 2 1
960.2.w.d 8 120.q odd 4 1
960.2.w.d 8 120.w even 4 1
1200.2.w.b 8 15.d odd 2 1
1200.2.w.b 8 15.e even 4 1
1200.2.w.b 8 60.h even 2 1
1200.2.w.b 8 60.l odd 4 1
3600.2.x.n 8 5.b even 2 1
3600.2.x.n 8 5.c odd 4 1
3600.2.x.n 8 20.d odd 2 1
3600.2.x.n 8 20.e even 4 1

Hecke kernels

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

\( T_{7}^{8} + 392T_{7}^{4} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 2 \) Copy content Toggle raw display
\( T_{13}^{4} + 144 \) Copy content Toggle raw display
\( T_{17}^{4} + 8T_{17}^{3} + 32T_{17}^{2} - 32T_{17} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 4 T^{3} + 8 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 392T^{4} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 40 T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 2336 T^{4} + 160000 \) Copy content Toggle raw display
$29$ \( (T^{4} + 116 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 16 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 12)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + 6272 T^{4} + 4096 \) Copy content Toggle raw display
$47$ \( T^{8} + 23072 T^{4} + 71639296 \) Copy content Toggle raw display
$53$ \( (T^{2} - 4 T + 8)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 98)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 20)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + 6272 T^{4} + 4096 \) Copy content Toggle raw display
$71$ \( (T^{4} + 160 T^{2} + 256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 6 T + 18)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 112 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 6 T + 18)^{4} \) Copy content Toggle raw display
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