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Properties

Label 720.4.f.j
Level $720$
Weight $4$
Character orbit 720.f
Analytic conductor $42.481$
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] = [720,4,Mod(289,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.f (of order \(2\), degree \(1\), 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: \(42.4813752041\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 21x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2} - 1) q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - 2 \beta_{3} - \beta_{2} - 20) q^{11} + ( - 4 \beta_{2} - \beta_1) q^{13} + ( - 3 \beta_{2} - 8 \beta_1) q^{17} + ( - 8 \beta_{3} - 4 \beta_{2} + 32) q^{19} + (3 \beta_{2} - \beta_1) q^{23} + (6 \beta_{3} - 4 \beta_{2} + 5 \beta_1 + 61) q^{25} + (14 \beta_{3} + 7 \beta_{2} - 166) q^{29} + (4 \beta_{3} + 2 \beta_{2} - 28) q^{31} + (10 \beta_{3} - 10 \beta_{2} + \cdots - 80) q^{35}+ \cdots + ( - 2 \beta_{2} - 122 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{5} - 84 q^{11} + 112 q^{19} + 256 q^{25} - 636 q^{29} - 104 q^{31} - 300 q^{35} + 816 q^{41} - 140 q^{49} + 864 q^{55} + 372 q^{59} + 680 q^{61} - 948 q^{65} - 72 q^{71} + 760 q^{79} - 508 q^{85} + 2232 q^{89} - 1944 q^{91} + 2784 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 21x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 32\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{3} + 34\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 15\nu^{2} - 17\nu + 160 ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + 2\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + \beta_{2} - 64 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16\beta_{2} - 17\beta_1 ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
2.70156i
− 2.70156i
− 3.70156i
3.70156i
0 0 0 −11.1047 − 1.29844i 0 16.2094i 0 0 0
289.2 0 0 0 −11.1047 + 1.29844i 0 − 16.2094i 0 0 0
289.3 0 0 0 8.10469 − 7.70156i 0 − 22.2094i 0 0 0
289.4 0 0 0 8.10469 + 7.70156i 0 22.2094i 0 0 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.4.f.j 4
3.b odd 2 1 240.4.f.f 4
4.b odd 2 1 45.4.b.b 4
5.b even 2 1 inner 720.4.f.j 4
12.b even 2 1 15.4.b.a ✓ 4
15.d odd 2 1 240.4.f.f 4
15.e even 4 1 1200.4.a.bn 2
15.e even 4 1 1200.4.a.bt 2
20.d odd 2 1 45.4.b.b 4
20.e even 4 1 225.4.a.i 2
20.e even 4 1 225.4.a.o 2
24.f even 2 1 960.4.f.q 4
24.h odd 2 1 960.4.f.p 4
60.h even 2 1 15.4.b.a ✓ 4
60.l odd 4 1 75.4.a.c 2
60.l odd 4 1 75.4.a.f 2
120.i odd 2 1 960.4.f.p 4
120.m even 2 1 960.4.f.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a ✓ 4 12.b even 2 1
15.4.b.a ✓ 4 60.h even 2 1
45.4.b.b 4 4.b odd 2 1
45.4.b.b 4 20.d odd 2 1
75.4.a.c 2 60.l odd 4 1
75.4.a.f 2 60.l odd 4 1
225.4.a.i 2 20.e even 4 1
225.4.a.o 2 20.e even 4 1
240.4.f.f 4 3.b odd 2 1
240.4.f.f 4 15.d odd 2 1
720.4.f.j 4 1.a even 1 1 trivial
720.4.f.j 4 5.b even 2 1 inner
960.4.f.p 4 24.h odd 2 1
960.4.f.p 4 120.i odd 2 1
960.4.f.q 4 24.f even 2 1
960.4.f.q 4 120.m even 2 1
1200.4.a.bn 2 15.e even 4 1
1200.4.a.bt 2 15.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 \(S_{4}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{4} + 756T_{7}^{2} + 129600 \) Copy content Toggle raw display
\( T_{11}^{2} + 42T_{11} + 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{4} + 756 T^{2} + 129600 \) Copy content Toggle raw display
$11$ \( (T^{2} + 42 T + 72)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 3780 T^{2} + 1327104 \) Copy content Toggle raw display
$17$ \( T^{4} + 7252 T^{2} + 2483776 \) Copy content Toggle raw display
$19$ \( (T^{2} - 56 T - 5120)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 2464 T^{2} + 6400 \) Copy content Toggle raw display
$29$ \( (T^{2} + 318 T + 7200)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 52 T - 800)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 106596 T^{2} + 41990400 \) Copy content Toggle raw display
$41$ \( (T^{2} - 408 T + 40140)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 8256266496 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 6186766336 \) Copy content Toggle raw display
$53$ \( T^{4} + 218644 T^{2} + 813390400 \) Copy content Toggle raw display
$59$ \( (T^{2} - 186 T + 8280)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 340 T - 65564)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 9419867136 \) Copy content Toggle raw display
$71$ \( (T^{2} + 36 T - 331776)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 104976000000 \) Copy content Toggle raw display
$79$ \( (T^{2} - 380 T - 886400)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 40558737664 \) Copy content Toggle raw display
$89$ \( (T^{2} - 1116 T + 98820)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 196199387136 \) Copy content Toggle raw display
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