Properties

Label 720.4.f.c
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: \( 1 \)
Twist minimal: no (minimal twist has level 30)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 11 i - 2) q^{5} + 2 i q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 11 i - 2) q^{5} + 2 i q^{7} + 70 q^{11} - 54 i q^{13} + 22 i q^{17} + 24 q^{19} + 100 i q^{23} + (44 i - 117) q^{25} + 216 q^{29} - 208 q^{31} + ( - 4 i + 22) q^{35} - 254 i q^{37} + 206 q^{41} + 292 i q^{43} - 320 i q^{47} + 339 q^{49} - 402 i q^{53} + ( - 770 i - 140) q^{55} + 370 q^{59} - 550 q^{61} + (108 i - 594) q^{65} - 728 i q^{67} - 540 q^{71} - 604 i q^{73} + 140 i q^{77} + 792 q^{79} - 404 i q^{83} + ( - 44 i + 242) q^{85} - 938 q^{89} + 108 q^{91} + ( - 264 i - 48) q^{95} + 56 i q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} + 140 q^{11} + 48 q^{19} - 234 q^{25} + 432 q^{29} - 416 q^{31} + 44 q^{35} + 412 q^{41} + 678 q^{49} - 280 q^{55} + 740 q^{59} - 1100 q^{61} - 1188 q^{65} - 1080 q^{71} + 1584 q^{79} + 484 q^{85} - 1876 q^{89} + 216 q^{91} - 96 q^{95}+O(q^{100}) \) 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 −2.00000 11.0000i 0 2.00000i 0 0 0
289.2 0 0 0 −2.00000 + 11.0000i 0 2.00000i 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.c 2
3.b odd 2 1 240.4.f.d 2
4.b odd 2 1 90.4.c.a 2
5.b even 2 1 inner 720.4.f.c 2
12.b even 2 1 30.4.c.a 2
15.d odd 2 1 240.4.f.d 2
15.e even 4 1 1200.4.a.h 1
15.e even 4 1 1200.4.a.bc 1
20.d odd 2 1 90.4.c.a 2
20.e even 4 1 450.4.a.e 1
20.e even 4 1 450.4.a.p 1
24.f even 2 1 960.4.f.c 2
24.h odd 2 1 960.4.f.d 2
60.h even 2 1 30.4.c.a 2
60.l odd 4 1 150.4.a.d 1
60.l odd 4 1 150.4.a.f 1
120.i odd 2 1 960.4.f.d 2
120.m even 2 1 960.4.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.c.a 2 12.b even 2 1
30.4.c.a 2 60.h even 2 1
90.4.c.a 2 4.b odd 2 1
90.4.c.a 2 20.d odd 2 1
150.4.a.d 1 60.l odd 4 1
150.4.a.f 1 60.l odd 4 1
240.4.f.d 2 3.b odd 2 1
240.4.f.d 2 15.d odd 2 1
450.4.a.e 1 20.e even 4 1
450.4.a.p 1 20.e even 4 1
720.4.f.c 2 1.a even 1 1 trivial
720.4.f.c 2 5.b even 2 1 inner
960.4.f.c 2 24.f even 2 1
960.4.f.c 2 120.m even 2 1
960.4.f.d 2 24.h odd 2 1
960.4.f.d 2 120.i odd 2 1
1200.4.a.h 1 15.e even 4 1
1200.4.a.bc 1 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 \(S_{4}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} - 70 \) 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
$5$ \( T^{2} + 4T + 125 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T - 70)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2916 \) Copy content Toggle raw display
$17$ \( T^{2} + 484 \) Copy content Toggle raw display
$19$ \( (T - 24)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 10000 \) Copy content Toggle raw display
$29$ \( (T - 216)^{2} \) Copy content Toggle raw display
$31$ \( (T + 208)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 64516 \) Copy content Toggle raw display
$41$ \( (T - 206)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 85264 \) Copy content Toggle raw display
$47$ \( T^{2} + 102400 \) Copy content Toggle raw display
$53$ \( T^{2} + 161604 \) Copy content Toggle raw display
$59$ \( (T - 370)^{2} \) Copy content Toggle raw display
$61$ \( (T + 550)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 529984 \) Copy content Toggle raw display
$71$ \( (T + 540)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 364816 \) Copy content Toggle raw display
$79$ \( (T - 792)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 163216 \) Copy content Toggle raw display
$89$ \( (T + 938)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 3136 \) Copy content Toggle raw display
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