Properties

Label 720.4.f.i
Level $720$
Weight $4$
Character orbit 720.f
Analytic conductor $42.481$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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: \(\Q(i, \sqrt{129})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 65x^{2} + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 - 5) q^{5} + ( - 3 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 - 5) q^{5} + ( - 3 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{7} + (\beta_{3} + \beta_{2} + 36) q^{11} + ( - \beta_{3} + \beta_{2} - 13 \beta_1) q^{13} + ( - 5 \beta_{3} + 5 \beta_{2} + 6 \beta_1) q^{17} + ( - 8 \beta_{3} - 8 \beta_{2} - 32) q^{19} + (8 \beta_{3} - 8 \beta_{2} + \beta_1) q^{23} + (11 \beta_{3} + 11 \beta_{2} + \cdots - 11) q^{25}+ \cdots + ( - 32 \beta_{3} + 32 \beta_{2} - 254 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 22 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 22 q^{5} + 148 q^{11} - 160 q^{19} + 100 q^{29} + 24 q^{31} + 796 q^{35} - 688 q^{41} - 3276 q^{49} - 1072 q^{55} - 1332 q^{59} + 488 q^{61} - 820 q^{65} + 616 q^{71} + 2104 q^{79} + 2148 q^{85} + 1368 q^{89} - 1352 q^{91} + 2944 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 65x^{2} + 1024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 33\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 32\nu^{2} + \nu + 1056 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 32\nu^{2} - \nu + 1056 ) / 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{3} - 2\beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} - 66 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -66\beta_{3} + 66\beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display

Character values

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

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

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
5.17891i
5.17891i
6.17891i
6.17891i
0 0 0 −11.1789 0.178908i 0 33.0735i 0 0 0
289.2 0 0 0 −11.1789 + 0.178908i 0 33.0735i 0 0 0
289.3 0 0 0 0.178908 11.1789i 0 35.0735i 0 0 0
289.4 0 0 0 0.178908 + 11.1789i 0 35.0735i 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.i 4
3.b odd 2 1 240.4.f.g 4
4.b odd 2 1 360.4.f.d 4
5.b even 2 1 inner 720.4.f.i 4
12.b even 2 1 120.4.f.d 4
15.d odd 2 1 240.4.f.g 4
15.e even 4 1 1200.4.a.bo 2
15.e even 4 1 1200.4.a.bq 2
20.d odd 2 1 360.4.f.d 4
20.e even 4 1 1800.4.a.bl 2
20.e even 4 1 1800.4.a.bn 2
24.f even 2 1 960.4.f.n 4
24.h odd 2 1 960.4.f.o 4
60.h even 2 1 120.4.f.d 4
60.l odd 4 1 600.4.a.t 2
60.l odd 4 1 600.4.a.v 2
120.i odd 2 1 960.4.f.o 4
120.m even 2 1 960.4.f.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.f.d 4 12.b even 2 1
120.4.f.d 4 60.h even 2 1
240.4.f.g 4 3.b odd 2 1
240.4.f.g 4 15.d odd 2 1
360.4.f.d 4 4.b odd 2 1
360.4.f.d 4 20.d odd 2 1
600.4.a.t 2 60.l odd 4 1
600.4.a.v 2 60.l odd 4 1
720.4.f.i 4 1.a even 1 1 trivial
720.4.f.i 4 5.b even 2 1 inner
960.4.f.n 4 24.f even 2 1
960.4.f.n 4 120.m even 2 1
960.4.f.o 4 24.h odd 2 1
960.4.f.o 4 120.i odd 2 1
1200.4.a.bo 2 15.e even 4 1
1200.4.a.bq 2 15.e even 4 1
1800.4.a.bl 2 20.e even 4 1
1800.4.a.bn 2 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 \(S_{4}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{4} + 2324T_{7}^{2} + 1345600 \) Copy content Toggle raw display
\( T_{11}^{2} - 74T_{11} + 1240 \) 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
$5$ \( T^{4} + 22 T^{3} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{4} + 2324 T^{2} + 1345600 \) Copy content Toggle raw display
$11$ \( (T^{2} - 74 T + 1240)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 5060 T^{2} + 5161984 \) Copy content Toggle raw display
$17$ \( T^{4} + 9492 T^{2} + 2903616 \) Copy content Toggle raw display
$19$ \( (T^{2} + 80 T - 6656)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 17312 T^{2} + 61716736 \) Copy content Toggle raw display
$29$ \( (T^{2} - 50 T - 5696)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12 T - 41760)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 6453872896 \) Copy content Toggle raw display
$41$ \( (T^{2} + 344 T + 24940)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 11664)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 27742233600 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 10072129600 \) Copy content Toggle raw display
$59$ \( (T^{2} + 666 T - 198840)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 244 T - 117212)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 8276632576 \) Copy content Toggle raw display
$71$ \( (T^{2} - 308 T - 162560)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 97443865600 \) Copy content Toggle raw display
$79$ \( (T^{2} - 1052 T + 251392)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 96800 T^{2} + 311451904 \) Copy content Toggle raw display
$89$ \( (T^{2} - 684 T - 180252)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 510230204416 \) Copy content Toggle raw display
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