Properties

Label 720.4.f.b
Level $720$
Weight $4$
Character orbit 720.f
Analytic conductor $42.481$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (5 \beta - 5) q^{5} + 2 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (5 \beta - 5) q^{5} + 2 \beta q^{7} - 28 q^{11} - 8 \beta q^{13} + 54 \beta q^{17} + 32 q^{19} - 14 \beta q^{23} + ( - 50 \beta - 75) q^{25} - 238 q^{29} + 180 q^{31} + ( - 10 \beta - 40) q^{35} + 20 \beta q^{37} - 422 q^{41} - 138 \beta q^{43} - 30 \beta q^{47} + 327 q^{49} - 110 \beta q^{53} + ( - 140 \beta + 140) q^{55} + 804 q^{59} - 358 q^{61} + (40 \beta + 160) q^{65} - 442 \beta q^{67} - 64 q^{71} - 76 \beta q^{73} - 56 \beta q^{77} - 932 q^{79} - 646 \beta q^{83} + ( - 270 \beta - 1080) q^{85} - 1146 q^{89} + 64 q^{91} + (160 \beta - 160) q^{95} - 412 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{5} - 56 q^{11} + 64 q^{19} - 150 q^{25} - 476 q^{29} + 360 q^{31} - 80 q^{35} - 844 q^{41} + 654 q^{49} + 280 q^{55} + 1608 q^{59} - 716 q^{61} + 320 q^{65} - 128 q^{71} - 1864 q^{79} - 2160 q^{85} - 2292 q^{89} + 128 q^{91} - 320 q^{95}+O(q^{100}) \) 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
1.00000i
1.00000i
0 0 0 −5.00000 10.0000i 0 4.00000i 0 0 0
289.2 0 0 0 −5.00000 + 10.0000i 0 4.00000i 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.b 2
3.b odd 2 1 240.4.f.e 2
4.b odd 2 1 360.4.f.a 2
5.b even 2 1 inner 720.4.f.b 2
12.b even 2 1 120.4.f.c 2
15.d odd 2 1 240.4.f.e 2
15.e even 4 1 1200.4.a.l 1
15.e even 4 1 1200.4.a.z 1
20.d odd 2 1 360.4.f.a 2
20.e even 4 1 1800.4.a.o 1
20.e even 4 1 1800.4.a.u 1
24.f even 2 1 960.4.f.b 2
24.h odd 2 1 960.4.f.a 2
60.h even 2 1 120.4.f.c 2
60.l odd 4 1 600.4.a.f 1
60.l odd 4 1 600.4.a.k 1
120.i odd 2 1 960.4.f.a 2
120.m even 2 1 960.4.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.f.c 2 12.b even 2 1
120.4.f.c 2 60.h even 2 1
240.4.f.e 2 3.b odd 2 1
240.4.f.e 2 15.d odd 2 1
360.4.f.a 2 4.b odd 2 1
360.4.f.a 2 20.d odd 2 1
600.4.a.f 1 60.l odd 4 1
600.4.a.k 1 60.l odd 4 1
720.4.f.b 2 1.a even 1 1 trivial
720.4.f.b 2 5.b even 2 1 inner
960.4.f.a 2 24.h odd 2 1
960.4.f.a 2 120.i odd 2 1
960.4.f.b 2 24.f even 2 1
960.4.f.b 2 120.m even 2 1
1200.4.a.l 1 15.e even 4 1
1200.4.a.z 1 15.e even 4 1
1800.4.a.o 1 20.e even 4 1
1800.4.a.u 1 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 \(S_{4}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} + 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 10T + 125 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 28)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 256 \) Copy content Toggle raw display
$17$ \( T^{2} + 11664 \) Copy content Toggle raw display
$19$ \( (T - 32)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 784 \) Copy content Toggle raw display
$29$ \( (T + 238)^{2} \) Copy content Toggle raw display
$31$ \( (T - 180)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1600 \) Copy content Toggle raw display
$41$ \( (T + 422)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 76176 \) Copy content Toggle raw display
$47$ \( T^{2} + 3600 \) Copy content Toggle raw display
$53$ \( T^{2} + 48400 \) Copy content Toggle raw display
$59$ \( (T - 804)^{2} \) Copy content Toggle raw display
$61$ \( (T + 358)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 781456 \) Copy content Toggle raw display
$71$ \( (T + 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 23104 \) Copy content Toggle raw display
$79$ \( (T + 932)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1669264 \) Copy content Toggle raw display
$89$ \( (T + 1146)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 678976 \) Copy content Toggle raw display
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