Properties

Label 480.1.c.a
Level 480480
Weight 11
Character orbit 480.c
Analytic conductor 0.2400.240
Analytic rank 00
Dimension 44
Projective image D4D_{4}
CM discriminant -20
Inner twists 88

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [480,1,Mod(449,480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(480, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("480.449");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: N N == 480=2535 480 = 2^{5} \cdot 3 \cdot 5
Weight: k k == 1 1
Character orbit: [χ][\chi] == 480.c (of order 22, degree 11, 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: 0.2395512060640.239551206064
Analytic rank: 00
Dimension: 44
Coefficient field: Q(ζ8)\Q(\zeta_{8})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+1 x^{4} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: yes
Projective image: D4D_{4}
Projective field: Galois closure of 4.2.3600.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The qq-expansion and trace form are shown below.

f(q)f(q) == qζ8q3+ζ82q5+(ζ83+ζ8)q7+ζ82q9ζ83q15+(ζ82+1)q21+(ζ83+ζ8)q23q25++(ζ83+ζ8)q83+O(q100) q - \zeta_{8} q^{3} + \zeta_{8}^{2} q^{5} + (\zeta_{8}^{3} + \zeta_{8}) q^{7} + \zeta_{8}^{2} q^{9} - \zeta_{8}^{3} q^{15} + ( - \zeta_{8}^{2} + 1) q^{21} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{23} - q^{25} + \cdots + ( - \zeta_{8}^{3} + \zeta_{8}) q^{83} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+4q214q254q454q494q694q81+O(q100) 4 q + 4 q^{21} - 4 q^{25} - 4 q^{45} - 4 q^{49} - 4 q^{69} - 4 q^{81}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/480Z)×\left(\mathbb{Z}/480\mathbb{Z}\right)^\times.

nn 3131 9797 161161 421421
χ(n)\chi(n) 11 1-1 1-1 11

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
449.1
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0 −0.707107 0.707107i 0 1.00000i 0 1.41421i 0 1.00000i 0
449.2 0 −0.707107 + 0.707107i 0 1.00000i 0 1.41421i 0 1.00000i 0
449.3 0 0.707107 0.707107i 0 1.00000i 0 1.41421i 0 1.00000i 0
449.4 0 0.707107 + 0.707107i 0 1.00000i 0 1.41421i 0 1.00000i 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by Q(5)\Q(\sqrt{-5})
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.1.c.a 4
3.b odd 2 1 inner 480.1.c.a 4
4.b odd 2 1 inner 480.1.c.a 4
5.b even 2 1 inner 480.1.c.a 4
5.c odd 4 2 2400.1.l.a 4
8.b even 2 1 960.1.c.a 4
8.d odd 2 1 960.1.c.a 4
12.b even 2 1 inner 480.1.c.a 4
15.d odd 2 1 inner 480.1.c.a 4
15.e even 4 2 2400.1.l.a 4
16.e even 4 1 3840.1.i.a 4
16.e even 4 1 3840.1.i.b 4
16.f odd 4 1 3840.1.i.a 4
16.f odd 4 1 3840.1.i.b 4
20.d odd 2 1 CM 480.1.c.a 4
20.e even 4 2 2400.1.l.a 4
24.f even 2 1 960.1.c.a 4
24.h odd 2 1 960.1.c.a 4
40.e odd 2 1 960.1.c.a 4
40.f even 2 1 960.1.c.a 4
48.i odd 4 1 3840.1.i.a 4
48.i odd 4 1 3840.1.i.b 4
48.k even 4 1 3840.1.i.a 4
48.k even 4 1 3840.1.i.b 4
60.h even 2 1 inner 480.1.c.a 4
60.l odd 4 2 2400.1.l.a 4
80.k odd 4 1 3840.1.i.a 4
80.k odd 4 1 3840.1.i.b 4
80.q even 4 1 3840.1.i.a 4
80.q even 4 1 3840.1.i.b 4
120.i odd 2 1 960.1.c.a 4
120.m even 2 1 960.1.c.a 4
240.t even 4 1 3840.1.i.a 4
240.t even 4 1 3840.1.i.b 4
240.bm odd 4 1 3840.1.i.a 4
240.bm odd 4 1 3840.1.i.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.1.c.a 4 1.a even 1 1 trivial
480.1.c.a 4 3.b odd 2 1 inner
480.1.c.a 4 4.b odd 2 1 inner
480.1.c.a 4 5.b even 2 1 inner
480.1.c.a 4 12.b even 2 1 inner
480.1.c.a 4 15.d odd 2 1 inner
480.1.c.a 4 20.d odd 2 1 CM
480.1.c.a 4 60.h even 2 1 inner
960.1.c.a 4 8.b even 2 1
960.1.c.a 4 8.d odd 2 1
960.1.c.a 4 24.f even 2 1
960.1.c.a 4 24.h odd 2 1
960.1.c.a 4 40.e odd 2 1
960.1.c.a 4 40.f even 2 1
960.1.c.a 4 120.i odd 2 1
960.1.c.a 4 120.m even 2 1
2400.1.l.a 4 5.c odd 4 2
2400.1.l.a 4 15.e even 4 2
2400.1.l.a 4 20.e even 4 2
2400.1.l.a 4 60.l odd 4 2
3840.1.i.a 4 16.e even 4 1
3840.1.i.a 4 16.f odd 4 1
3840.1.i.a 4 48.i odd 4 1
3840.1.i.a 4 48.k even 4 1
3840.1.i.a 4 80.k odd 4 1
3840.1.i.a 4 80.q even 4 1
3840.1.i.a 4 240.t even 4 1
3840.1.i.a 4 240.bm odd 4 1
3840.1.i.b 4 16.e even 4 1
3840.1.i.b 4 16.f odd 4 1
3840.1.i.b 4 48.i odd 4 1
3840.1.i.b 4 48.k even 4 1
3840.1.i.b 4 80.k odd 4 1
3840.1.i.b 4 80.q even 4 1
3840.1.i.b 4 240.t even 4 1
3840.1.i.b 4 240.bm odd 4 1

Hecke kernels

This newform subspace is the entire newspace S1new(480,[χ])S_{1}^{\mathrm{new}}(480, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4 T^{4} Copy content Toggle raw display
33 T4+1 T^{4} + 1 Copy content Toggle raw display
55 (T2+1)2 (T^{2} + 1)^{2} Copy content Toggle raw display
77 (T2+2)2 (T^{2} + 2)^{2} Copy content Toggle raw display
1111 T4 T^{4} Copy content Toggle raw display
1313 T4 T^{4} Copy content Toggle raw display
1717 T4 T^{4} Copy content Toggle raw display
1919 T4 T^{4} Copy content Toggle raw display
2323 (T22)2 (T^{2} - 2)^{2} Copy content Toggle raw display
2929 T4 T^{4} Copy content Toggle raw display
3131 T4 T^{4} Copy content Toggle raw display
3737 T4 T^{4} Copy content Toggle raw display
4141 (T2+4)2 (T^{2} + 4)^{2} Copy content Toggle raw display
4343 (T2+2)2 (T^{2} + 2)^{2} Copy content Toggle raw display
4747 (T22)2 (T^{2} - 2)^{2} Copy content Toggle raw display
5353 T4 T^{4} Copy content Toggle raw display
5959 T4 T^{4} Copy content Toggle raw display
6161 T4 T^{4} Copy content Toggle raw display
6767 (T2+2)2 (T^{2} + 2)^{2} Copy content Toggle raw display
7171 T4 T^{4} Copy content Toggle raw display
7373 T4 T^{4} Copy content Toggle raw display
7979 T4 T^{4} Copy content Toggle raw display
8383 (T22)2 (T^{2} - 2)^{2} Copy content Toggle raw display
8989 T4 T^{4} Copy content Toggle raw display
9797 T4 T^{4} Copy content Toggle raw display
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