Properties

Label 3200.2.a.br
Level 32003200
Weight 22
Character orbit 3200.a
Self dual yes
Analytic conductor 25.55225.552
Analytic rank 00
Dimension 33
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3200,2,Mod(1,3200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 3200=2752 3200 = 2^{7} \cdot 5^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 3200.a (trivial)

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: yes
Analytic conductor: 25.552128646825.5521286468
Analytic rank: 00
Dimension: 33
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x3x23x+1 x^{3} - x^{2} - 3x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 640)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

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

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,β21,\beta_1,\beta_2 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β11)q3+(β2+1)q7+(β2β1+1)q9+2β1q11+(β2+β1+2)q13+(2β1+2)q17+2β2q19+(β2+3β12)q21++(2β2+4β18)q99+O(q100) q + (\beta_1 - 1) q^{3} + (\beta_{2} + 1) q^{7} + (\beta_{2} - \beta_1 + 1) q^{9} + 2 \beta_1 q^{11} + (\beta_{2} + \beta_1 + 2) q^{13} + ( - 2 \beta_1 + 2) q^{17} + 2 \beta_{2} q^{19} + ( - \beta_{2} + 3 \beta_1 - 2) q^{21}+ \cdots + ( - 2 \beta_{2} + 4 \beta_1 - 8) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 3q2q3+4q7+3q9+2q11+8q13+4q17+2q194q21+8q238q276q298q31+20q33+8q37+4q39+2q4114q43+4q47+3q49+22q99+O(q100) 3 q - 2 q^{3} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 8 q^{13} + 4 q^{17} + 2 q^{19} - 4 q^{21} + 8 q^{23} - 8 q^{27} - 6 q^{29} - 8 q^{31} + 20 q^{33} + 8 q^{37} + 4 q^{39} + 2 q^{41} - 14 q^{43} + 4 q^{47} + 3 q^{49}+ \cdots - 22 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x3x23x+1 x^{3} - x^{2} - 3x + 1 : Copy content Toggle raw display

β1\beta_{1}== ν22 \nu^{2} - 2 Copy content Toggle raw display
β2\beta_{2}== ν2+2ν+2 -\nu^{2} + 2\nu + 2 Copy content Toggle raw display
ν\nu== (β2+β1)/2 ( \beta_{2} + \beta_1 ) / 2 Copy content Toggle raw display
ν2\nu^{2}== β1+2 \beta _1 + 2 Copy content Toggle raw display

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}
1.1
0.311108
−1.48119
2.17009
0 −2.90321 0 0 0 3.52543 0 5.42864 0
1.2 0 −0.806063 0 0 0 −2.15633 0 −2.35026 0
1.3 0 1.70928 0 0 0 2.63090 0 −0.0783777 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 +1 +1
55 1 -1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.a.br 3
4.b odd 2 1 3200.2.a.bs 3
5.b even 2 1 3200.2.a.bt 3
5.c odd 4 2 640.2.c.b yes 6
8.b even 2 1 3200.2.a.bu 3
8.d odd 2 1 3200.2.a.bp 3
20.d odd 2 1 3200.2.a.bq 3
20.e even 4 2 640.2.c.a 6
40.e odd 2 1 3200.2.a.bv 3
40.f even 2 1 3200.2.a.bo 3
40.i odd 4 2 640.2.c.c yes 6
40.k even 4 2 640.2.c.d yes 6
80.i odd 4 2 1280.2.f.k 6
80.j even 4 2 1280.2.f.l 6
80.s even 4 2 1280.2.f.i 6
80.t odd 4 2 1280.2.f.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.c.a 6 20.e even 4 2
640.2.c.b yes 6 5.c odd 4 2
640.2.c.c yes 6 40.i odd 4 2
640.2.c.d yes 6 40.k even 4 2
1280.2.f.i 6 80.s even 4 2
1280.2.f.j 6 80.t odd 4 2
1280.2.f.k 6 80.i odd 4 2
1280.2.f.l 6 80.j even 4 2
3200.2.a.bo 3 40.f even 2 1
3200.2.a.bp 3 8.d odd 2 1
3200.2.a.bq 3 20.d odd 2 1
3200.2.a.br 3 1.a even 1 1 trivial
3200.2.a.bs 3 4.b odd 2 1
3200.2.a.bt 3 5.b even 2 1
3200.2.a.bu 3 8.b even 2 1
3200.2.a.bv 3 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(3200))S_{2}^{\mathrm{new}}(\Gamma_0(3200)):

T33+2T324T34 T_{3}^{3} + 2T_{3}^{2} - 4T_{3} - 4 Copy content Toggle raw display
T734T724T7+20 T_{7}^{3} - 4T_{7}^{2} - 4T_{7} + 20 Copy content Toggle raw display
T1132T11220T11+8 T_{11}^{3} - 2T_{11}^{2} - 20T_{11} + 8 Copy content Toggle raw display
T1338T132+8T13+16 T_{13}^{3} - 8T_{13}^{2} + 8T_{13} + 16 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T3 T^{3} Copy content Toggle raw display
33 T3+2T2+4 T^{3} + 2 T^{2} + \cdots - 4 Copy content Toggle raw display
55 T3 T^{3} Copy content Toggle raw display
77 T34T2++20 T^{3} - 4 T^{2} + \cdots + 20 Copy content Toggle raw display
1111 T32T2++8 T^{3} - 2 T^{2} + \cdots + 8 Copy content Toggle raw display
1313 T38T2++16 T^{3} - 8 T^{2} + \cdots + 16 Copy content Toggle raw display
1717 T34T2++32 T^{3} - 4 T^{2} + \cdots + 32 Copy content Toggle raw display
1919 T32T2++104 T^{3} - 2 T^{2} + \cdots + 104 Copy content Toggle raw display
2323 T38T2+4 T^{3} - 8 T^{2} + \cdots - 4 Copy content Toggle raw display
2929 (T+2)3 (T + 2)^{3} Copy content Toggle raw display
3131 T3+8T2+128 T^{3} + 8 T^{2} + \cdots - 128 Copy content Toggle raw display
3737 T38T2++272 T^{3} - 8 T^{2} + \cdots + 272 Copy content Toggle raw display
4141 T32T2++184 T^{3} - 2 T^{2} + \cdots + 184 Copy content Toggle raw display
4343 T3+14T2+100 T^{3} + 14 T^{2} + \cdots - 100 Copy content Toggle raw display
4747 T34T2++116 T^{3} - 4 T^{2} + \cdots + 116 Copy content Toggle raw display
5353 T316T2+80 T^{3} - 16 T^{2} + \cdots - 80 Copy content Toggle raw display
5959 T3+2T2+104 T^{3} + 2 T^{2} + \cdots - 104 Copy content Toggle raw display
6161 T3+10T2+8 T^{3} + 10 T^{2} + \cdots - 8 Copy content Toggle raw display
6767 T3+10T2+604 T^{3} + 10 T^{2} + \cdots - 604 Copy content Toggle raw display
7171 T312T2++320 T^{3} - 12 T^{2} + \cdots + 320 Copy content Toggle raw display
7373 T320T2+1184 T^{3} - 20T^{2} + 1184 Copy content Toggle raw display
7979 T3+16T2+128 T^{3} + 16 T^{2} + \cdots - 128 Copy content Toggle raw display
8383 T3+30T2++524 T^{3} + 30 T^{2} + \cdots + 524 Copy content Toggle raw display
8989 T3+10T2+1096 T^{3} + 10 T^{2} + \cdots - 1096 Copy content Toggle raw display
9797 T328T2+608 T^{3} - 28 T^{2} + \cdots - 608 Copy content Toggle raw display
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