Properties

Label 720.4.x.d
Level $720$
Weight $4$
Character orbit 720.x
Analytic conductor $42.481$
Analytic rank $0$
Dimension $4$
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] = [720,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.127");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(42.4813752041\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{35})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 17x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 80)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (10 \beta_1 + 5) q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (10 \beta_1 + 5) q^{5} - \beta_{2} q^{7} + (5 \beta_{3} - 5 \beta_{2}) q^{11} + ( - 15 \beta_1 - 15) q^{13} + (65 \beta_1 - 65) q^{17} + (10 \beta_{3} + 10 \beta_{2}) q^{19} - 17 \beta_{3} q^{23} + (100 \beta_1 - 75) q^{25} + 4 \beta_1 q^{29} + ( - 15 \beta_{3} + 15 \beta_{2}) q^{31} + (10 \beta_{3} - 5 \beta_{2}) q^{35} + (115 \beta_1 - 115) q^{37} - 18 q^{41} - 31 \beta_{3} q^{43} - 47 \beta_{2} q^{47} - 273 \beta_1 q^{49} + ( - 265 \beta_1 - 265) q^{53} + (75 \beta_{3} + 25 \beta_{2}) q^{55} + (10 \beta_{3} + 10 \beta_{2}) q^{59} - 722 q^{61} + ( - 225 \beta_1 + 75) q^{65} - 5 \beta_{2} q^{67} + ( - 5 \beta_{3} + 5 \beta_{2}) q^{71} + ( - 135 \beta_1 - 135) q^{73} + (350 \beta_1 - 350) q^{77} + ( - 80 \beta_{3} - 80 \beta_{2}) q^{79} + 7 \beta_{3} q^{83} + ( - 325 \beta_1 - 975) q^{85} + 104 \beta_1 q^{89} + ( - 15 \beta_{3} + 15 \beta_{2}) q^{91} + ( - 50 \beta_{3} + 150 \beta_{2}) q^{95} + (615 \beta_1 - 615) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 20 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 20 q^{5} - 60 q^{13} - 260 q^{17} - 300 q^{25} - 460 q^{37} - 72 q^{41} - 1060 q^{53} - 2888 q^{61} + 300 q^{65} - 540 q^{73} - 1400 q^{77} - 3900 q^{85} - 2460 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 17x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 8\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 18\nu^{2} + 26\nu - 153 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} - 18\nu^{2} + 26\nu + 153 ) / 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} + 34 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 2\beta_{2} + 13\beta_1 \) Copy content Toggle raw display

Character values

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

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

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
2.95804 0.500000i
−2.95804 0.500000i
2.95804 + 0.500000i
−2.95804 + 0.500000i
0 0 0 5.00000 10.0000i 0 −5.91608 + 5.91608i 0 0 0
127.2 0 0 0 5.00000 10.0000i 0 5.91608 5.91608i 0 0 0
703.1 0 0 0 5.00000 + 10.0000i 0 −5.91608 5.91608i 0 0 0
703.2 0 0 0 5.00000 + 10.0000i 0 5.91608 + 5.91608i 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
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 720.4.x.d 4
3.b odd 2 1 80.4.n.b 4
4.b odd 2 1 inner 720.4.x.d 4
5.c odd 4 1 inner 720.4.x.d 4
12.b even 2 1 80.4.n.b 4
15.d odd 2 1 400.4.n.c 4
15.e even 4 1 80.4.n.b 4
15.e even 4 1 400.4.n.c 4
20.e even 4 1 inner 720.4.x.d 4
24.f even 2 1 320.4.n.e 4
24.h odd 2 1 320.4.n.e 4
60.h even 2 1 400.4.n.c 4
60.l odd 4 1 80.4.n.b 4
60.l odd 4 1 400.4.n.c 4
120.q odd 4 1 320.4.n.e 4
120.w even 4 1 320.4.n.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.4.n.b 4 3.b odd 2 1
80.4.n.b 4 12.b even 2 1
80.4.n.b 4 15.e even 4 1
80.4.n.b 4 60.l odd 4 1
320.4.n.e 4 24.f even 2 1
320.4.n.e 4 24.h odd 2 1
320.4.n.e 4 120.q odd 4 1
320.4.n.e 4 120.w even 4 1
400.4.n.c 4 15.d odd 2 1
400.4.n.c 4 15.e even 4 1
400.4.n.c 4 60.h even 2 1
400.4.n.c 4 60.l odd 4 1
720.4.x.d 4 1.a even 1 1 trivial
720.4.x.d 4 4.b odd 2 1 inner
720.4.x.d 4 5.c odd 4 1 inner
720.4.x.d 4 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{4} + 4900 \) Copy content Toggle raw display
\( T_{13}^{2} + 30T_{13} + 450 \) Copy content Toggle raw display
\( T_{17}^{2} + 130T_{17} + 8450 \) 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
$5$ \( (T^{2} - 10 T + 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 4900 \) Copy content Toggle raw display
$11$ \( (T^{2} + 3500)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 30 T + 450)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 130 T + 8450)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 14000)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 409252900 \) Copy content Toggle raw display
$29$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 31500)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 230 T + 26450)^{2} \) Copy content Toggle raw display
$41$ \( (T + 18)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 4525252900 \) Copy content Toggle raw display
$47$ \( T^{4} + 23910436900 \) Copy content Toggle raw display
$53$ \( (T^{2} + 530 T + 140450)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 14000)^{2} \) Copy content Toggle raw display
$61$ \( (T + 722)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 3062500 \) Copy content Toggle raw display
$71$ \( (T^{2} + 3500)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 270 T + 36450)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 896000)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 11764900 \) Copy content Toggle raw display
$89$ \( (T^{2} + 10816)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1230 T + 756450)^{2} \) Copy content Toggle raw display
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