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Properties

Label 117.10.a.g
Level 117117
Weight 1010
Character orbit 117.a
Self dual yes
Analytic conductor 60.25960.259
Analytic rank 11
Dimension 88
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,10,Mod(1,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: N N == 117=32⋅13 117 = 3^{2} \cdot 13
Weight: k k == 10 10
Character orbit: [χ][\chi] == 117.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: 60.259192831260.2591928312
Analytic rank: 11
Dimension: 88
Coefficient field: Q[x]/(x8−⋯ )\mathbb{Q}[x]/(x^{8} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8−2613x6+2010020x4−399948096x2+6898597888 x^{8} - 2613x^{6} + 2010020x^{4} - 399948096x^{2} + 6898597888 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 211⋅34⋅7 2^{11}\cdot 3^{4}\cdot 7
Twist minimal: yes
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,…,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2+(β4+141)q4+(β3−7β1)q5+(β6−2β5−β4+1220)q7+(−β3+β2+50β1)q8+(−5β6+9β5+⋯−4812)q10+⋯+(35664β7+⋯+14268301β1)q98+O(q100) q + \beta_1 q^{2} + (\beta_{4} + 141) q^{4} + (\beta_{3} - 7 \beta_1) q^{5} + (\beta_{6} - 2 \beta_{5} - \beta_{4} + 1220) q^{7} + ( - \beta_{3} + \beta_{2} + 50 \beta_1) q^{8} + ( - 5 \beta_{6} + 9 \beta_{5} + \cdots - 4812) q^{10}+ \cdots + (35664 \beta_{7} + \cdots + 14268301 \beta_1) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q+1130q4+9756q7−38516q10−228488q13−314526q16−884340q19−1422584q22+1535856q25−1630900q28−12449540q31−6019872q34+6397904q37−13411388q40+⋯−3297114688q97+O(q100) 8 q + 1130 q^{4} + 9756 q^{7} - 38516 q^{10} - 228488 q^{13} - 314526 q^{16} - 884340 q^{19} - 1422584 q^{22} + 1535856 q^{25} - 1630900 q^{28} - 12449540 q^{31} - 6019872 q^{34} + 6397904 q^{37} - 13411388 q^{40}+ \cdots - 3297114688 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−2613x6+2010020x4−399948096x2+6898597888 x^{8} - 2613x^{6} + 2010020x^{4} - 399948096x^{2} + 6898597888 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (−13ν7+36753ν5−20387252ν3−4606636608ν)/9832192 ( -13\nu^{7} + 36753\nu^{5} - 20387252\nu^{3} - 4606636608\nu ) / 9832192 Copy content Toggle raw display
β3\beta_{3}== (−13ν7+36753ν5−30219444ν3+5953137600ν)/9832192 ( -13\nu^{7} + 36753\nu^{5} - 30219444\nu^{3} + 5953137600\nu ) / 9832192 Copy content Toggle raw display
β4\beta_{4}== ν2−653 \nu^{2} - 653 Copy content Toggle raw display
β5\beta_{5}== (−13ν6+36753ν4−26532372ν2+2208301472)/1229024 ( -13\nu^{6} + 36753\nu^{4} - 26532372\nu^{2} + 2208301472 ) / 1229024 Copy content Toggle raw display
β6\beta_{6}== (−93ν6+168385ν4−68324212ν2+2797233120)/1229024 ( -93\nu^{6} + 168385\nu^{4} - 68324212\nu^{2} + 2797233120 ) / 1229024 Copy content Toggle raw display
β7\beta_{7}== (−197ν7+462409ν5−303934644ν3+43614769984ν)/9832192 ( -197\nu^{7} + 462409\nu^{5} - 303934644\nu^{3} + 43614769984\nu ) / 9832192 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β4+653 \beta_{4} + 653 Copy content Toggle raw display
ν3\nu^{3}== −β3+β2+1074β1 -\beta_{3} + \beta_{2} + 1074\beta_1 Copy content Toggle raw display
ν4\nu^{4}== −13β6+93β5+1285β4+701591 -13\beta_{6} + 93\beta_{5} + 1285\beta_{4} + 701591 Copy content Toggle raw display
ν5\nu^{5}== −104β7−53β3+1629β2+1256654β1 -104\beta_{7} - 53\beta_{3} + 1629\beta_{2} + 1256654\beta_1 Copy content Toggle raw display
ν6\nu^{6}== −36753β6+168385β5+1591941β4+820633583 -36753\beta_{6} + 168385\beta_{5} + 1591941\beta_{4} + 820633583 Copy content Toggle raw display
ν7\nu^{7}== −294024β7+1418411β3+2280861β2+1514096862β1 -294024\beta_{7} + 1418411\beta_{3} + 2280861\beta_{2} + 1514096862\beta_1 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.

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
−36.0840
−31.9131
−16.5372
−4.36150
4.36150
16.5372
31.9131
36.0840
−36.0840 0 790.056 −548.189 0 −6372.32 −10033.4 0 19780.9
1.2 −31.9131 0 506.444 1635.60 0 9377.46 177.316 0 −52197.1
1.3 −16.5372 0 −238.522 −172.959 0 1545.12 12411.5 0 2860.24
1.4 −4.36150 0 −492.977 −2361.10 0 327.747 4383.21 0 10298.0
1.5 4.36150 0 −492.977 2361.10 0 327.747 −4383.21 0 10298.0
1.6 16.5372 0 −238.522 172.959 0 1545.12 −12411.5 0 2860.24
1.7 31.9131 0 506.444 −1635.60 0 9377.46 −177.316 0 −52197.1
1.8 36.0840 0 790.056 548.189 0 −6372.32 10033.4 0 19780.9
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 +1 +1
1313 +1 +1

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.10.a.g ✓ 8
3.b odd 2 1 inner 117.10.a.g ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.10.a.g ✓ 8 1.a even 1 1 trivial
117.10.a.g ✓ 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T28−2613T26+2010020T24−399948096T22+6898597888 T_{2}^{8} - 2613T_{2}^{6} + 2010020T_{2}^{4} - 399948096T_{2}^{2} + 6898597888 acting on S10new(Γ0(117))S_{10}^{\mathrm{new}}(\Gamma_0(117)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8+⋯+6898597888 T^{8} + \cdots + 6898597888 Copy content Toggle raw display
33 T8 T^{8} Copy content Toggle raw display
55 T8+⋯+13 ⁣⋯ ⁣00 T^{8} + \cdots + 13\!\cdots\!00 Copy content Toggle raw display
77 (T4+⋯−30260923880096)2 (T^{4} + \cdots - 30260923880096)^{2} Copy content Toggle raw display
1111 T8+⋯+39 ⁣⋯ ⁣88 T^{8} + \cdots + 39\!\cdots\!88 Copy content Toggle raw display
1313 (T+28561)8 (T + 28561)^{8} Copy content Toggle raw display
1717 T8+⋯+42 ⁣⋯ ⁣28 T^{8} + \cdots + 42\!\cdots\!28 Copy content Toggle raw display
1919 (T4+⋯+24 ⁣⋯ ⁣24)2 (T^{4} + \cdots + 24\!\cdots\!24)^{2} Copy content Toggle raw display
2323 T8+⋯+11 ⁣⋯ ⁣00 T^{8} + \cdots + 11\!\cdots\!00 Copy content Toggle raw display
2929 T8+⋯+18 ⁣⋯ ⁣00 T^{8} + \cdots + 18\!\cdots\!00 Copy content Toggle raw display
3131 (T4+⋯−22 ⁣⋯ ⁣60)2 (T^{4} + \cdots - 22\!\cdots\!60)^{2} Copy content Toggle raw display
3737 (T4+⋯+75 ⁣⋯ ⁣48)2 (T^{4} + \cdots + 75\!\cdots\!48)^{2} Copy content Toggle raw display
4141 T8+⋯+43 ⁣⋯ ⁣52 T^{8} + \cdots + 43\!\cdots\!52 Copy content Toggle raw display
4343 (T4+⋯+65 ⁣⋯ ⁣84)2 (T^{4} + \cdots + 65\!\cdots\!84)^{2} Copy content Toggle raw display
4747 T8+⋯+21 ⁣⋯ ⁣92 T^{8} + \cdots + 21\!\cdots\!92 Copy content Toggle raw display
5353 T8+⋯+22 ⁣⋯ ⁣92 T^{8} + \cdots + 22\!\cdots\!92 Copy content Toggle raw display
5959 T8+⋯+13 ⁣⋯ ⁣32 T^{8} + \cdots + 13\!\cdots\!32 Copy content Toggle raw display
6161 (T4+⋯−60 ⁣⋯ ⁣64)2 (T^{4} + \cdots - 60\!\cdots\!64)^{2} Copy content Toggle raw display
6767 (T4+⋯+36 ⁣⋯ ⁣00)2 (T^{4} + \cdots + 36\!\cdots\!00)^{2} Copy content Toggle raw display
7171 T8+⋯+26 ⁣⋯ ⁣00 T^{8} + \cdots + 26\!\cdots\!00 Copy content Toggle raw display
7373 (T4+⋯−57 ⁣⋯ ⁣28)2 (T^{4} + \cdots - 57\!\cdots\!28)^{2} Copy content Toggle raw display
7979 (T4+⋯+32 ⁣⋯ ⁣20)2 (T^{4} + \cdots + 32\!\cdots\!20)^{2} Copy content Toggle raw display
8383 T8+⋯+27 ⁣⋯ ⁣48 T^{8} + \cdots + 27\!\cdots\!48 Copy content Toggle raw display
8989 T8+⋯+58 ⁣⋯ ⁣28 T^{8} + \cdots + 58\!\cdots\!28 Copy content Toggle raw display
9797 (T4+⋯−45 ⁣⋯ ⁣20)2 (T^{4} + \cdots - 45\!\cdots\!20)^{2} Copy content Toggle raw display
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