Properties

Label 4800.2.f.bf
Level $4800$
Weight $2$
Character orbit 4800.f
Analytic conductor $38.328$
Analytic rank $0$
Dimension $2$
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] = [4800,2,Mod(3649,4800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4800.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.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: \(38.3281929702\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} - q^{9} + 4 q^{11} + 2 i q^{13} - 2 i q^{17} + 4 q^{19} - i q^{27} - 2 q^{29} + 4 i q^{33} - 10 i q^{37} - 2 q^{39} + 10 q^{41} - 4 i q^{43} - 8 i q^{47} + 7 q^{49} + 2 q^{51} + 10 i q^{53} + 4 i q^{57} - 4 q^{59} + 2 q^{61} + 12 i q^{67} - 8 q^{71} + 10 i q^{73} + q^{81} - 12 i q^{83} - 2 i q^{87} + 6 q^{89} - 2 i q^{97} - 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 8 q^{11} + 8 q^{19} - 4 q^{29} - 4 q^{39} + 20 q^{41} + 14 q^{49} + 4 q^{51} - 8 q^{59} + 4 q^{61} - 16 q^{71} + 2 q^{81} + 12 q^{89} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(4351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3649.1
1.00000i
1.00000i
0 1.00000i 0 0 0 0 0 −1.00000 0
3649.2 0 1.00000i 0 0 0 0 0 −1.00000 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 4800.2.f.bf 2
4.b odd 2 1 4800.2.f.c 2
5.b even 2 1 inner 4800.2.f.bf 2
5.c odd 4 1 960.2.a.l 1
5.c odd 4 1 4800.2.a.t 1
8.b even 2 1 75.2.b.b 2
8.d odd 2 1 1200.2.f.h 2
15.e even 4 1 2880.2.a.y 1
20.d odd 2 1 4800.2.f.c 2
20.e even 4 1 960.2.a.a 1
20.e even 4 1 4800.2.a.bz 1
24.f even 2 1 3600.2.f.e 2
24.h odd 2 1 225.2.b.b 2
40.e odd 2 1 1200.2.f.h 2
40.f even 2 1 75.2.b.b 2
40.i odd 4 1 15.2.a.a 1
40.i odd 4 1 75.2.a.b 1
40.k even 4 1 240.2.a.d 1
40.k even 4 1 1200.2.a.e 1
60.l odd 4 1 2880.2.a.bc 1
80.i odd 4 1 3840.2.k.m 2
80.j even 4 1 3840.2.k.r 2
80.s even 4 1 3840.2.k.r 2
80.t odd 4 1 3840.2.k.m 2
120.i odd 2 1 225.2.b.b 2
120.m even 2 1 3600.2.f.e 2
120.q odd 4 1 720.2.a.c 1
120.q odd 4 1 3600.2.a.u 1
120.w even 4 1 45.2.a.a 1
120.w even 4 1 225.2.a.b 1
280.s even 4 1 735.2.a.c 1
280.s even 4 1 3675.2.a.j 1
280.bt odd 12 2 735.2.i.e 2
280.bv even 12 2 735.2.i.d 2
360.br even 12 2 405.2.e.c 2
360.bu odd 12 2 405.2.e.f 2
440.t even 4 1 1815.2.a.d 1
440.t even 4 1 9075.2.a.g 1
520.bg odd 4 1 2535.2.a.j 1
680.bi odd 4 1 4335.2.a.c 1
760.t even 4 1 5415.2.a.j 1
840.bp odd 4 1 2205.2.a.i 1
920.x even 4 1 7935.2.a.d 1
1320.bn odd 4 1 5445.2.a.c 1
1560.bq even 4 1 7605.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 40.i odd 4 1
45.2.a.a 1 120.w even 4 1
75.2.a.b 1 40.i odd 4 1
75.2.b.b 2 8.b even 2 1
75.2.b.b 2 40.f even 2 1
225.2.a.b 1 120.w even 4 1
225.2.b.b 2 24.h odd 2 1
225.2.b.b 2 120.i odd 2 1
240.2.a.d 1 40.k even 4 1
405.2.e.c 2 360.br even 12 2
405.2.e.f 2 360.bu odd 12 2
720.2.a.c 1 120.q odd 4 1
735.2.a.c 1 280.s even 4 1
735.2.i.d 2 280.bv even 12 2
735.2.i.e 2 280.bt odd 12 2
960.2.a.a 1 20.e even 4 1
960.2.a.l 1 5.c odd 4 1
1200.2.a.e 1 40.k even 4 1
1200.2.f.h 2 8.d odd 2 1
1200.2.f.h 2 40.e odd 2 1
1815.2.a.d 1 440.t even 4 1
2205.2.a.i 1 840.bp odd 4 1
2535.2.a.j 1 520.bg odd 4 1
2880.2.a.y 1 15.e even 4 1
2880.2.a.bc 1 60.l odd 4 1
3600.2.a.u 1 120.q odd 4 1
3600.2.f.e 2 24.f even 2 1
3600.2.f.e 2 120.m even 2 1
3675.2.a.j 1 280.s even 4 1
3840.2.k.m 2 80.i odd 4 1
3840.2.k.m 2 80.t odd 4 1
3840.2.k.r 2 80.j even 4 1
3840.2.k.r 2 80.s even 4 1
4335.2.a.c 1 680.bi odd 4 1
4800.2.a.t 1 5.c odd 4 1
4800.2.a.bz 1 20.e even 4 1
4800.2.f.c 2 4.b odd 2 1
4800.2.f.c 2 20.d odd 2 1
4800.2.f.bf 2 1.a even 1 1 trivial
4800.2.f.bf 2 5.b even 2 1 inner
5415.2.a.j 1 760.t even 4 1
5445.2.a.c 1 1320.bn odd 4 1
7605.2.a.g 1 1560.bq even 4 1
7935.2.a.d 1 920.x even 4 1
9075.2.a.g 1 440.t even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4800, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 4 \) Copy content Toggle raw display
\( T_{19} - 4 \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 100 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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