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Properties

Label 720.2.z.e
Level $720$
Weight $2$
Character orbit 720.z
Analytic conductor $5.749$
Analytic rank $0$
Dimension $6$
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(163,720)]
 
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, 3, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.163");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.z (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: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{3} - \beta_{2}) q^{4} + (2 \beta_{3} - 1) q^{5} + (\beta_{5} + \beta_{4} + \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots - 1) q^{8} + ( - \beta_{4} + 2 \beta_1) q^{10}+ \cdots + (4 \beta_{5} + 8 \beta_{3} - 3 \beta_1 - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{4} - 6 q^{5} - 2 q^{7} - 6 q^{8} + 4 q^{10} + 2 q^{11} + 6 q^{14} + 10 q^{16} + 2 q^{17} - 10 q^{19} - 10 q^{20} - 14 q^{22} - 10 q^{23} - 18 q^{25} - 26 q^{28} + 6 q^{29} - 10 q^{32} - 26 q^{34}+ \cdots - 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 3\nu^{3} + 2\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - \nu^{3} + 2\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + \nu^{4} - 2\nu^{3} + 3\nu^{2} - 2\nu + 5 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - \beta_{4} + 2\beta_{3} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{4} - 3\beta_{3} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(-\beta_{3}\) \(-1\) \(-\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
0.264658 − 1.38923i
1.40680 + 0.144584i
−0.671462 + 1.24464i
0.264658 + 1.38923i
1.40680 − 0.144584i
−0.671462 − 1.24464i
−1.38923 − 0.264658i 0 1.85991 + 0.735342i −1.00000 + 2.00000i 0 −3.24914 + 3.24914i −2.38923 − 1.51380i 0 1.91855 − 2.51380i
163.2 0.144584 − 1.40680i 0 −1.95819 − 0.406803i −1.00000 + 2.00000i 0 2.10278 − 2.10278i −0.855416 + 2.69597i 0 2.66902 + 1.69597i
163.3 1.24464 + 0.671462i 0 1.09828 + 1.67146i −1.00000 + 2.00000i 0 0.146365 − 0.146365i 0.244644 + 2.81783i 0 −2.58757 + 1.81783i
667.1 −1.38923 + 0.264658i 0 1.85991 − 0.735342i −1.00000 − 2.00000i 0 −3.24914 − 3.24914i −2.38923 + 1.51380i 0 1.91855 + 2.51380i
667.2 0.144584 + 1.40680i 0 −1.95819 + 0.406803i −1.00000 − 2.00000i 0 2.10278 + 2.10278i −0.855416 − 2.69597i 0 2.66902 − 1.69597i
667.3 1.24464 − 0.671462i 0 1.09828 − 1.67146i −1.00000 − 2.00000i 0 0.146365 + 0.146365i 0.244644 − 2.81783i 0 −2.58757 − 1.81783i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.s 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.z.e 6
3.b odd 2 1 240.2.y.d ✓ 6
5.c odd 4 1 720.2.bd.e 6
12.b even 2 1 960.2.y.d 6
15.e even 4 1 240.2.bc.d yes 6
16.f odd 4 1 720.2.bd.e 6
24.f even 2 1 1920.2.y.h 6
24.h odd 2 1 1920.2.y.g 6
48.i odd 4 1 960.2.bc.d 6
48.i odd 4 1 1920.2.bc.h 6
48.k even 4 1 240.2.bc.d yes 6
48.k even 4 1 1920.2.bc.g 6
60.l odd 4 1 960.2.bc.d 6
80.s even 4 1 inner 720.2.z.e 6
120.q odd 4 1 1920.2.bc.h 6
120.w even 4 1 1920.2.bc.g 6
240.z odd 4 1 240.2.y.d ✓ 6
240.bb even 4 1 960.2.y.d 6
240.bd odd 4 1 1920.2.y.g 6
240.bf even 4 1 1920.2.y.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.y.d ✓ 6 3.b odd 2 1
240.2.y.d ✓ 6 240.z odd 4 1
240.2.bc.d yes 6 15.e even 4 1
240.2.bc.d yes 6 48.k even 4 1
720.2.z.e 6 1.a even 1 1 trivial
720.2.z.e 6 80.s even 4 1 inner
720.2.bd.e 6 5.c odd 4 1
720.2.bd.e 6 16.f odd 4 1
960.2.y.d 6 12.b even 2 1
960.2.y.d 6 240.bb even 4 1
960.2.bc.d 6 48.i odd 4 1
960.2.bc.d 6 60.l odd 4 1
1920.2.y.g 6 24.h odd 2 1
1920.2.y.g 6 240.bd odd 4 1
1920.2.y.h 6 24.f even 2 1
1920.2.y.h 6 240.bf even 4 1
1920.2.bc.g 6 48.k even 4 1
1920.2.bc.g 6 120.w even 4 1
1920.2.bc.h 6 48.i odd 4 1
1920.2.bc.h 6 120.q odd 4 1

Hecke kernels

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

\( T_{7}^{6} + 2T_{7}^{5} + 2T_{7}^{4} - 32T_{7}^{3} + 196T_{7}^{2} - 56T_{7} + 8 \) Copy content Toggle raw display
\( T_{11}^{6} - 2T_{11}^{5} + 2T_{11}^{4} + 32T_{11}^{3} + 196T_{11}^{2} + 56T_{11} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{4} + 2 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$11$ \( T^{6} - 2 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 2 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$19$ \( T^{6} + 10 T^{5} + \cdots + 14792 \) Copy content Toggle raw display
$23$ \( T^{6} + 10 T^{5} + \cdots + 14792 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T + 2)^{3} \) Copy content Toggle raw display
$31$ \( T^{6} + 60 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$37$ \( T^{6} + 144 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$41$ \( (T^{2} + 16)^{3} \) Copy content Toggle raw display
$43$ \( T^{6} + 124 T^{4} + \cdots + 18496 \) Copy content Toggle raw display
$47$ \( T^{6} + 10 T^{5} + \cdots + 412232 \) Copy content Toggle raw display
$53$ \( (T^{3} - 6 T^{2} + \cdots + 344)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 6 T^{5} + \cdots + 35912 \) Copy content Toggle raw display
$61$ \( T^{6} + 14 T^{5} + \cdots + 4232 \) Copy content Toggle raw display
$67$ \( T^{6} + 380 T^{4} + \cdots + 1459264 \) Copy content Toggle raw display
$71$ \( (T^{3} - 8 T^{2} + \cdots + 512)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 10 T^{5} + \cdots + 42632 \) Copy content Toggle raw display
$79$ \( (T^{3} - 8 T^{2} + \cdots + 512)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 20 T^{2} + \cdots + 704)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 14 T^{2} + \cdots + 184)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 10 T^{5} + \cdots + 1338248 \) Copy content Toggle raw display
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