Properties

Label 405.4.a.d
Level 405405
Weight 44
Character orbit 405.a
Self dual yes
Analytic conductor 23.89623.896
Analytic rank 11
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] = [405,4,Mod(1,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 405=345 405 = 3^{4} \cdot 5
Weight: k k == 4 4
Character orbit: [χ][\chi] == 405.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: 23.895773552323.8957735523
Analytic rank: 11
Dimension: 22
Coefficient field: Q(33)\Q(\sqrt{33})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x8 x^{2} - x - 8 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 45)
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=12(1+33)\beta = \frac{1}{2}(1 + \sqrt{33}). We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβq2+βq4+5q5+(β+5)q7+(7β8)q85βq10+(5β16)q11+(8β60)q13+(4β+8)q14+(7β56)q16++(319β+72)q98+O(q100) q - \beta q^{2} + \beta q^{4} + 5 q^{5} + ( - \beta + 5) q^{7} + (7 \beta - 8) q^{8} - 5 \beta q^{10} + ( - 5 \beta - 16) q^{11} + (8 \beta - 60) q^{13} + ( - 4 \beta + 8) q^{14} + ( - 7 \beta - 56) q^{16} + \cdots + (319 \beta + 72) q^{98} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2qq2+q4+10q5+9q79q85q1037q11112q13+12q14119q16+77q17+35q19+5q20+101q22267q23+50q2576q2612q28++463q98+O(q100) 2 q - q^{2} + q^{4} + 10 q^{5} + 9 q^{7} - 9 q^{8} - 5 q^{10} - 37 q^{11} - 112 q^{13} + 12 q^{14} - 119 q^{16} + 77 q^{17} + 35 q^{19} + 5 q^{20} + 101 q^{22} - 267 q^{23} + 50 q^{25} - 76 q^{26} - 12 q^{28}+ \cdots + 463 q^{98}+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
3.37228
−2.37228
−3.37228 0 3.37228 5.00000 0 1.62772 15.6060 0 −16.8614
1.2 2.37228 0 −2.37228 5.00000 0 7.37228 −24.6060 0 11.8614
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
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 405.4.a.d 2
3.b odd 2 1 405.4.a.e 2
5.b even 2 1 2025.4.a.l 2
9.c even 3 2 45.4.e.a 4
9.d odd 6 2 135.4.e.a 4
15.d odd 2 1 2025.4.a.j 2
45.j even 6 2 225.4.e.a 4
45.k odd 12 4 225.4.k.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.a 4 9.c even 3 2
135.4.e.a 4 9.d odd 6 2
225.4.e.a 4 45.j even 6 2
225.4.k.a 8 45.k odd 12 4
405.4.a.d 2 1.a even 1 1 trivial
405.4.a.e 2 3.b odd 2 1
2025.4.a.j 2 15.d odd 2 1
2025.4.a.l 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T22+T28 T_{2}^{2} + T_{2} - 8 acting on S4new(Γ0(405))S_{4}^{\mathrm{new}}(\Gamma_0(405)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+T8 T^{2} + T - 8 Copy content Toggle raw display
33 T2 T^{2} Copy content Toggle raw display
55 (T5)2 (T - 5)^{2} Copy content Toggle raw display
77 T29T+12 T^{2} - 9T + 12 Copy content Toggle raw display
1111 T2+37T+136 T^{2} + 37T + 136 Copy content Toggle raw display
1313 T2+112T+2608 T^{2} + 112T + 2608 Copy content Toggle raw display
1717 T277T3674 T^{2} - 77T - 3674 Copy content Toggle raw display
1919 T235T4850 T^{2} - 35T - 4850 Copy content Toggle raw display
2323 T2+267T+13458 T^{2} + 267T + 13458 Copy content Toggle raw display
2929 T2325T+13858 T^{2} - 325T + 13858 Copy content Toggle raw display
3131 (T+6)2 (T + 6)^{2} Copy content Toggle raw display
3737 T2+638T+100936 T^{2} + 638T + 100936 Copy content Toggle raw display
4141 T2238T15539 T^{2} - 238T - 15539 Copy content Toggle raw display
4343 T2+97T60092 T^{2} + 97T - 60092 Copy content Toggle raw display
4747 T2+901T+201952 T^{2} + 901T + 201952 Copy content Toggle raw display
5353 T2224T69956 T^{2} - 224T - 69956 Copy content Toggle raw display
5959 T2+85T18002 T^{2} + 85T - 18002 Copy content Toggle raw display
6161 T2+247T298454 T^{2} + 247T - 298454 Copy content Toggle raw display
6767 T2+606T187503 T^{2} + 606T - 187503 Copy content Toggle raw display
7171 T2394T61016 T^{2} - 394T - 61016 Copy content Toggle raw display
7373 T2+811T182276 T^{2} + 811T - 182276 Copy content Toggle raw display
7979 T2+840T+128748 T^{2} + 840T + 128748 Copy content Toggle raw display
8383 T2+387T1655532 T^{2} + 387 T - 1655532 Copy content Toggle raw display
8989 T2+1065T535050 T^{2} + 1065 T - 535050 Copy content Toggle raw display
9797 T2+1031T+124162 T^{2} + 1031 T + 124162 Copy content Toggle raw display
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