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Properties

Label 1.54.a.a
Level 11
Weight 5454
Character orbit 1.a
Self dual yes
Analytic conductor 17.79017.790
Analytic rank 11
Dimension 44
CM no
Inner twists 11

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1,54,Mod(1,1)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 54, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1.1"); S:= CuspForms(chi, 54); N := Newforms(S);
 
Level: N N == 1 1
Weight: k k == 54 54
Character orbit: [χ][\chi] == 1.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 17.790310760817.7903107608
Analytic rank: 11
Dimension: 44
Coefficient field: Q[x]/(x4−⋯ )\mathbb{Q}[x]/(x^{4} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4−x3−2315873743412x2−421178019174503472x+612167648493870378955584 x^{4} - x^{3} - 2315873743412x^{2} - 421178019174503472x + 612167648493870378955584 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 227⋅38⋅53⋅7⋅13 2^{27}\cdot 3^{8}\cdot 5^{3}\cdot 7\cdot 13
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

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

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β1−17119080)q2+(−β2+8136β1−262102751820)q3+(β3+374β2+⋯+19 ⁣⋯ ⁣32)q4+(−600β3+⋯−11 ⁣⋯ ⁣50)q5+⋯+(−12 ⁣⋯ ⁣16β3+⋯+50 ⁣⋯ ⁣96)q99+O(q100) q + (\beta_1 - 17119080) q^{2} + ( - \beta_{2} + 8136 \beta_1 - 262102751820) q^{3} + (\beta_{3} + 374 \beta_{2} + \cdots + 19\!\cdots\!32) q^{4} + ( - 600 \beta_{3} + \cdots - 11\!\cdots\!50) q^{5}+ \cdots + ( - 12\!\cdots\!16 \beta_{3} + \cdots + 50\!\cdots\!96) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 4q−68476320q2−1048411007280q3+78 ⁣⋯ ⁣28q4−45 ⁣⋯ ⁣00q5+36 ⁣⋯ ⁣28q6−22 ⁣⋯ ⁣00q7+13 ⁣⋯ ⁣40q8+11 ⁣⋯ ⁣12q9+23 ⁣⋯ ⁣00q10+⋯+20 ⁣⋯ ⁣84q99+O(q100) 4 q - 68476320 q^{2} - 1048411007280 q^{3} + 78\!\cdots\!28 q^{4} - 45\!\cdots\!00 q^{5} + 36\!\cdots\!28 q^{6} - 22\!\cdots\!00 q^{7} + 13\!\cdots\!40 q^{8} + 11\!\cdots\!12 q^{9} + 23\!\cdots\!00 q^{10}+ \cdots + 20\!\cdots\!84 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4−x3−2315873743412x2−421178019174503472x+612167648493870378955584 x^{4} - x^{3} - 2315873743412x^{2} - 421178019174503472x + 612167648493870378955584 : Copy content Toggle raw display

β1\beta_{1}== 96ν−24 96\nu - 24 Copy content Toggle raw display
β2\beta_{2}== (−ν3+611437ν2+1620402960440ν−392120600840009328)/119308 ( -\nu^{3} + 611437\nu^{2} + 1620402960440\nu - 392120600840009328 ) / 119308 Copy content Toggle raw display
β3\beta_{3}== (187ν3+435432545ν2−452992885700072ν−563273827738686130416)/59654 ( 187\nu^{3} + 435432545\nu^{2} - 452992885700072\nu - 563273827738686130416 ) / 59654 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (β1+24)/96 ( \beta _1 + 24 ) / 96 Copy content Toggle raw display
ν2\nu^{2}== (β3+374β2+26188788β1+10671546209644800)/9216 ( \beta_{3} + 374\beta_{2} + 26188788\beta _1 + 10671546209644800 ) / 9216 Copy content Toggle raw display
ν3\nu^{3}== (611437β3−870865090β2+171571478170596β1+2911198475853482464512)/9216 ( 611437\beta_{3} - 870865090\beta_{2} + 171571478170596\beta _1 + 2911198475853482464512 ) / 9216 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−1.26509e6
−708531.
447512.
1.52611e6
−1.38568e8 −5.95414e12 1.01938e16 −5.35900e18 8.25051e20 2.55363e22 −1.64427e23 1.60685e25 7.42585e26
1.2 −8.51381e7 6.54009e12 −1.75871e15 2.26338e17 −5.56810e20 −3.24923e22 9.16589e23 2.33895e25 −1.92700e25
1.3 2.58420e7 −2.97907e12 −8.33939e15 5.38136e18 −7.69850e19 2.33436e22 −4.48271e23 −1.05084e25 1.39065e26
1.4 1.29387e8 1.34470e12 7.73392e15 −4.81259e18 1.73988e20 −1.66125e22 −1.64746e23 −1.75750e25 −6.22689e26
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1.54.a.a ✓ 4
3.b odd 2 1 9.54.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.54.a.a ✓ 4 1.a even 1 1 trivial
9.54.a.b 4 3.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace S54new(Γ0(1))S_{54}^{\mathrm{new}}(\Gamma_0(1)).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4+⋯+39 ⁣⋯ ⁣76 T^{4} + \cdots + 39\!\cdots\!76 Copy content Toggle raw display
33 T4+⋯+15 ⁣⋯ ⁣76 T^{4} + \cdots + 15\!\cdots\!76 Copy content Toggle raw display
55 T4+⋯+31 ⁣⋯ ⁣00 T^{4} + \cdots + 31\!\cdots\!00 Copy content Toggle raw display
77 T4+⋯+32 ⁣⋯ ⁣96 T^{4} + \cdots + 32\!\cdots\!96 Copy content Toggle raw display
1111 T4+⋯+36 ⁣⋯ ⁣76 T^{4} + \cdots + 36\!\cdots\!76 Copy content Toggle raw display
1313 T4+⋯+14 ⁣⋯ ⁣56 T^{4} + \cdots + 14\!\cdots\!56 Copy content Toggle raw display
1717 T4+⋯−32 ⁣⋯ ⁣64 T^{4} + \cdots - 32\!\cdots\!64 Copy content Toggle raw display
1919 T4+⋯−62 ⁣⋯ ⁣00 T^{4} + \cdots - 62\!\cdots\!00 Copy content Toggle raw display
2323 T4+⋯+24 ⁣⋯ ⁣36 T^{4} + \cdots + 24\!\cdots\!36 Copy content Toggle raw display
2929 T4+⋯+28 ⁣⋯ ⁣00 T^{4} + \cdots + 28\!\cdots\!00 Copy content Toggle raw display
3131 T4+⋯+31 ⁣⋯ ⁣96 T^{4} + \cdots + 31\!\cdots\!96 Copy content Toggle raw display
3737 T4+⋯+39 ⁣⋯ ⁣16 T^{4} + \cdots + 39\!\cdots\!16 Copy content Toggle raw display
4141 T4+⋯−72 ⁣⋯ ⁣44 T^{4} + \cdots - 72\!\cdots\!44 Copy content Toggle raw display
4343 T4+⋯+79 ⁣⋯ ⁣96 T^{4} + \cdots + 79\!\cdots\!96 Copy content Toggle raw display
4747 T4+⋯+82 ⁣⋯ ⁣56 T^{4} + \cdots + 82\!\cdots\!56 Copy content Toggle raw display
5353 T4+⋯−80 ⁣⋯ ⁣24 T^{4} + \cdots - 80\!\cdots\!24 Copy content Toggle raw display
5959 T4+⋯−37 ⁣⋯ ⁣00 T^{4} + \cdots - 37\!\cdots\!00 Copy content Toggle raw display
6161 T4+⋯+18 ⁣⋯ ⁣76 T^{4} + \cdots + 18\!\cdots\!76 Copy content Toggle raw display
6767 T4+⋯−18 ⁣⋯ ⁣64 T^{4} + \cdots - 18\!\cdots\!64 Copy content Toggle raw display
7171 T4+⋯+83 ⁣⋯ ⁣36 T^{4} + \cdots + 83\!\cdots\!36 Copy content Toggle raw display
7373 T4+⋯+80 ⁣⋯ ⁣36 T^{4} + \cdots + 80\!\cdots\!36 Copy content Toggle raw display
7979 T4+⋯+39 ⁣⋯ ⁣00 T^{4} + \cdots + 39\!\cdots\!00 Copy content Toggle raw display
8383 T4+⋯+36 ⁣⋯ ⁣16 T^{4} + \cdots + 36\!\cdots\!16 Copy content Toggle raw display
8989 T4+⋯−28 ⁣⋯ ⁣00 T^{4} + \cdots - 28\!\cdots\!00 Copy content Toggle raw display
9797 T4+⋯+21 ⁣⋯ ⁣56 T^{4} + \cdots + 21\!\cdots\!56 Copy content Toggle raw display
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