Properties

Label 720.2.cx.a
Level $720$
Weight $2$
Character orbit 720.cx
Analytic conductor $5.749$
Analytic rank $0$
Dimension $8$
CM no
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(223,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 8, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.cx (of order \(12\), degree \(4\), 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: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{4} - 1) q^{3} + (\beta_{7} - 2 \beta_{4} + 2) q^{5} + (\beta_{7} + \beta_{2} - 1) q^{7} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{4} - 1) q^{3} + (\beta_{7} - 2 \beta_{4} + 2) q^{5} + (\beta_{7} + \beta_{2} - 1) q^{7} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{9} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 2) q^{11} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} + 4 \beta_{3} + 2 \beta_1 + 1) q^{13} + (2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{15} + ( - 2 \beta_{7} + 2 \beta_1 + 2) q^{17} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{19} + ( - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{21} + (2 \beta_{7} + \beta_{6} + 2 \beta_{4} - 3 \beta_1 + 1) q^{23} + ( - 3 \beta_{4} + 4 \beta_1) q^{25} + ( - 4 \beta_{7} + \beta_{5} + 4 \beta_1 + 4) q^{27} + ( - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 4 \beta_1) q^{29} + (2 \beta_{4} + 2) q^{31} + (3 \beta_{7} - 2 \beta_{4} - 2 \beta_{3} - 3 \beta_1 + 3) q^{33} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_1) q^{35} + (2 \beta_{5} + 2 \beta_{4} - 4 \beta_{2} - 2 \beta_1) q^{37} + (10 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 5 \beta_1) q^{39} + ( - \beta_{7} - \beta_{6} + \beta_{5} + 6 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 5) q^{41} + ( - 4 \beta_{7} + 3 \beta_{4} + 2 \beta_{3} + 3 \beta_1 - 4) q^{43} + (3 \beta_{7} + \beta_{6} + 3 \beta_{5} - \beta_{3} - 3 \beta_1 - 1) q^{45} + ( - 5 \beta_{7} - 3 \beta_{4} + \beta_{2} + 3 \beta_1 + 5) q^{47} + (3 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{49} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} - 4) q^{51} + ( - 3 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} + \beta_1 - 1) q^{53} + ( - 3 \beta_{7} - 3 \beta_{6} + \beta_{5} + 4 \beta_{4} + 3 \beta_{3} + \beta_1) q^{55} + (2 \beta_{7} + 4 \beta_{5} + 6 \beta_{4} + 4 \beta_1 - 2) q^{57} + (3 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} - 6 \beta_1) q^{59} + 5 \beta_{4} q^{61} + ( - 3 \beta_{7} + \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 3 \beta_1 - 4) q^{63} + ( - 2 \beta_{7} - 8 \beta_{6} - 4 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{65}+ \cdots + ( - 7 \beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 8 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 8 q^{5} - 6 q^{7} - 12 q^{11} + 2 q^{15} + 16 q^{17} + 20 q^{21} + 18 q^{23} - 12 q^{25} + 28 q^{27} + 24 q^{31} + 20 q^{33} - 8 q^{37} - 24 q^{41} - 24 q^{43} - 16 q^{45} + 30 q^{47} - 8 q^{51} - 16 q^{53} - 8 q^{57} + 20 q^{61} - 22 q^{63} + 42 q^{67} - 4 q^{75} + 16 q^{77} - 4 q^{81} - 18 q^{83} + 24 q^{85} + 16 q^{87} - 12 q^{93} - 24 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 13\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 4\nu^{4} + 8\nu^{2} + 4\nu - 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} - 8\nu^{2} + 4\nu + 3 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - \nu^{6} - 8\nu^{5} + 24\nu^{3} - \nu - 13 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} - 3\nu^{6} + 8\nu^{5} + 8\nu^{4} - 24\nu^{3} - 16\nu^{2} + 9\nu + 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 16\nu^{5} - 40\nu^{3} + 15\nu ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{2} + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{4} + 3\beta_{3} - 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{7} - 5\beta_{6} + 5\beta_{5} - 5\beta_{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{6} - 4\beta_{5} + 4\beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{3} - 13\beta_{2} + 16\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{1} - \beta_{7}\) \(-1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
−0.535233 0.309017i
1.40126 + 0.809017i
−1.40126 0.809017i
0.535233 + 0.309017i
−1.40126 + 0.809017i
0.535233 0.309017i
−0.535233 + 0.309017i
1.40126 0.809017i
0 −1.34425 1.09224i 0 0.133975 + 2.23205i 0 −2.71028 + 0.726216i 0 0.614017 + 2.93649i 0
223.2 0 1.71028 0.273784i 0 0.133975 + 2.23205i 0 0.344250 0.0922415i 0 2.85008 0.936492i 0
367.1 0 −1.09224 + 1.34425i 0 1.86603 + 1.23205i 0 −0.726216 2.71028i 0 −0.614017 2.93649i 0
367.2 0 −0.273784 1.71028i 0 1.86603 + 1.23205i 0 0.0922415 + 0.344250i 0 −2.85008 + 0.936492i 0
463.1 0 −1.09224 1.34425i 0 1.86603 1.23205i 0 −0.726216 + 2.71028i 0 −0.614017 + 2.93649i 0
463.2 0 −0.273784 + 1.71028i 0 1.86603 1.23205i 0 0.0922415 0.344250i 0 −2.85008 0.936492i 0
607.1 0 −1.34425 + 1.09224i 0 0.133975 2.23205i 0 −2.71028 0.726216i 0 0.614017 2.93649i 0
607.2 0 1.71028 + 0.273784i 0 0.133975 2.23205i 0 0.344250 + 0.0922415i 0 2.85008 + 0.936492i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
36.f odd 6 1 inner
180.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.cx.a 8
4.b odd 2 1 720.2.cx.b yes 8
5.c odd 4 1 inner 720.2.cx.a 8
9.c even 3 1 720.2.cx.b yes 8
20.e even 4 1 720.2.cx.b yes 8
36.f odd 6 1 inner 720.2.cx.a 8
45.k odd 12 1 720.2.cx.b yes 8
180.x even 12 1 inner 720.2.cx.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.cx.a 8 1.a even 1 1 trivial
720.2.cx.a 8 5.c odd 4 1 inner
720.2.cx.a 8 36.f odd 6 1 inner
720.2.cx.a 8 180.x even 12 1 inner
720.2.cx.b yes 8 4.b odd 2 1
720.2.cx.b yes 8 9.c even 3 1
720.2.cx.b yes 8 20.e even 4 1
720.2.cx.b yes 8 45.k odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 6T_{7}^{7} + 18T_{7}^{6} + 36T_{7}^{5} + 23T_{7}^{4} - 36T_{7}^{3} + 18T_{7}^{2} - 6T_{7} + 1 \) acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + 2 T^{6} - 8 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 4 T^{3} + 11 T^{2} - 20 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + 18 T^{6} + 36 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{4} + 6 T^{3} + 10 T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 900 T^{4} + 810000 \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T + 8)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 64 T^{2} + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 18 T^{7} + 162 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$29$ \( T^{8} - 62 T^{6} + 3843 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T + 12)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + 8 T^{2} - 112 T + 784)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + 123 T^{2} + 252 T + 441)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$47$ \( T^{8} - 30 T^{7} + 450 T^{6} + \cdots + 1500625 \) Copy content Toggle raw display
$53$ \( (T^{4} + 8 T^{3} + 32 T^{2} - 176 T + 484)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 64 T^{6} + 3612 T^{4} + \cdots + 234256 \) Copy content Toggle raw display
$61$ \( (T^{2} - 5 T + 25)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 42 T^{7} + 882 T^{6} + \cdots + 25411681 \) Copy content Toggle raw display
$71$ \( (T^{4} + 256 T^{2} + 14884)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 14400)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 64 T^{6} + 3612 T^{4} + \cdots + 234256 \) Copy content Toggle raw display
$83$ \( T^{8} + 18 T^{7} + \cdots + 141158161 \) Copy content Toggle raw display
$89$ \( (T^{4} + 482 T^{2} + 57121)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 6 T^{3} + 18 T^{2} - 108 T + 324)^{2} \) Copy content Toggle raw display
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