Properties

Label 240.6.a.q
Level 240240
Weight 66
Character orbit 240.a
Self dual yes
Analytic conductor 38.49238.492
Analytic rank 00
Dimension 22
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] = [240,6,Mod(1,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 240=2435 240 = 2^{4} \cdot 3 \cdot 5
Weight: k k == 6 6
Character orbit: [χ][\chi] == 240.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: 38.492116755138.4921167551
Analytic rank: 00
Dimension: 22
Coefficient field: Q(409)\Q(\sqrt{409})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x102 x^{2} - x - 102 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 24 2^{4}
Twist minimal: no (minimal twist has level 15)
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 β=8409\beta = 8\sqrt{409}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+9q3+25q5+(β+56)q7+81q9+(2β124)q11+(β+438)q13+225q15+(5β+1018)q17+(13β732)q19+(9β+504)q21++(162β10044)q99+O(q100) q + 9 q^{3} + 25 q^{5} + ( - \beta + 56) q^{7} + 81 q^{9} + ( - 2 \beta - 124) q^{11} + ( - \beta + 438) q^{13} + 225 q^{15} + (5 \beta + 1018) q^{17} + (13 \beta - 732) q^{19} + ( - 9 \beta + 504) q^{21}+ \cdots + ( - 162 \beta - 10044) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+18q3+50q5+112q7+162q9248q11+876q13+450q15+2036q171464q19+1008q21+3216q23+1250q25+1458q27+1948q292672q312232q33+20088q99+O(q100) 2 q + 18 q^{3} + 50 q^{5} + 112 q^{7} + 162 q^{9} - 248 q^{11} + 876 q^{13} + 450 q^{15} + 2036 q^{17} - 1464 q^{19} + 1008 q^{21} + 3216 q^{23} + 1250 q^{25} + 1458 q^{27} + 1948 q^{29} - 2672 q^{31} - 2232 q^{33}+ \cdots - 20088 q^{99}+O(q^{100}) 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
10.6119
−9.61187
0 9.00000 0 25.0000 0 −105.790 0 81.0000 0
1.2 0 9.00000 0 25.0000 0 217.790 0 81.0000 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1
33 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 240.6.a.q 2
3.b odd 2 1 720.6.a.bd 2
4.b odd 2 1 15.6.a.c 2
8.b even 2 1 960.6.a.bf 2
8.d odd 2 1 960.6.a.bj 2
12.b even 2 1 45.6.a.e 2
20.d odd 2 1 75.6.a.h 2
20.e even 4 2 75.6.b.e 4
28.d even 2 1 735.6.a.g 2
60.h even 2 1 225.6.a.m 2
60.l odd 4 2 225.6.b.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.a.c 2 4.b odd 2 1
45.6.a.e 2 12.b even 2 1
75.6.a.h 2 20.d odd 2 1
75.6.b.e 4 20.e even 4 2
225.6.a.m 2 60.h even 2 1
225.6.b.g 4 60.l odd 4 2
240.6.a.q 2 1.a even 1 1 trivial
720.6.a.bd 2 3.b odd 2 1
735.6.a.g 2 28.d even 2 1
960.6.a.bf 2 8.b even 2 1
960.6.a.bj 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T72112T723040 T_{7}^{2} - 112T_{7} - 23040 acting on S6new(Γ0(240))S_{6}^{\mathrm{new}}(\Gamma_0(240)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 (T9)2 (T - 9)^{2} Copy content Toggle raw display
55 (T25)2 (T - 25)^{2} Copy content Toggle raw display
77 T2112T23040 T^{2} - 112T - 23040 Copy content Toggle raw display
1111 T2+248T89328 T^{2} + 248T - 89328 Copy content Toggle raw display
1313 T2876T+165668 T^{2} - 876T + 165668 Copy content Toggle raw display
1717 T22036T+381924 T^{2} - 2036 T + 381924 Copy content Toggle raw display
1919 T2+1464T3887920 T^{2} + 1464 T - 3887920 Copy content Toggle raw display
2323 T23216T+2350080 T^{2} - 3216 T + 2350080 Copy content Toggle raw display
2929 T21948T+529860 T^{2} - 1948 T + 529860 Copy content Toggle raw display
3131 T2+2672T1382400 T^{2} + 2672 T - 1382400 Copy content Toggle raw display
3737 T28668T49300220 T^{2} - 8668 T - 49300220 Copy content Toggle raw display
4141 T2+7628T15712860 T^{2} + 7628 T - 15712860 Copy content Toggle raw display
4343 T216440T+67149584 T^{2} - 16440 T + 67149584 Copy content Toggle raw display
4747 T219360T61495104 T^{2} - 19360 T - 61495104 Copy content Toggle raw display
5353 T2+14356T476289180 T^{2} + 14356 T - 476289180 Copy content Toggle raw display
5959 T2+1067881200 T^{2} + \cdots - 1067881200 Copy content Toggle raw display
6161 T220220T149182204 T^{2} - 20220 T - 149182204 Copy content Toggle raw display
6767 T212904T125898096 T^{2} - 12904 T - 125898096 Copy content Toggle raw display
7171 T240976T627281856 T^{2} - 40976 T - 627281856 Copy content Toggle raw display
7373 T259124T+836113700 T^{2} - 59124 T + 836113700 Copy content Toggle raw display
7979 T2++1448870400 T^{2} + \cdots + 1448870400 Copy content Toggle raw display
8383 T2++2064089232 T^{2} + \cdots + 2064089232 Copy content Toggle raw display
8989 T2+2895368220 T^{2} + \cdots - 2895368220 Copy content Toggle raw display
9797 T2+15772164476 T^{2} + \cdots - 15772164476 Copy content Toggle raw display
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