Properties

Label 117.6.g.b
Level 117117
Weight 66
Character orbit 117.g
Analytic conductor 18.76518.765
Analytic rank 00
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] = [117,6,Mod(55,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.55");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 117=3213 117 = 3^{2} \cdot 13
Weight: k k == 6 6
Character orbit: [χ][\chi] == 117.g (of order 33, degree 22, minimal)

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: no
Analytic conductor: 18.764906918118.7649069181
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(ζ3)\Q(\zeta_{3})
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: x8x7+70x6133x5+4766x46777x3+16825x2+9696x+9216 x^{8} - x^{7} + 70x^{6} - 133x^{5} + 4766x^{4} - 6777x^{3} + 16825x^{2} + 9696x + 9216 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 22 2^{2}
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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+(β3+β1+1)q2+(β7+β4++β1)q4+(β6β42β2+2)q5+(2β7+β6+4β1)q7+(4β6+3β4+2)q8++(1049β7+502β6++8633β1)q98+O(q100) q + ( - \beta_{3} + \beta_1 + 1) q^{2} + ( - \beta_{7} + \beta_{4} + \cdots + \beta_1) q^{4} + (\beta_{6} - \beta_{4} - 2 \beta_{2} + 2) q^{5} + (2 \beta_{7} + \beta_{6} + \cdots - 4 \beta_1) q^{7} + ( - 4 \beta_{6} + 3 \beta_{4} + \cdots - 2) q^{8}+ \cdots + (1049 \beta_{7} + 502 \beta_{6} + \cdots + 8633 \beta_1) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+5q217q4+20q5+68q718q8+303q10+480q11234q13+1324q14+351q1660q17+520q194429q203926q226644q2313572q25++53863q98+O(q100) 8 q + 5 q^{2} - 17 q^{4} + 20 q^{5} + 68 q^{7} - 18 q^{8} + 303 q^{10} + 480 q^{11} - 234 q^{13} + 1324 q^{14} + 351 q^{16} - 60 q^{17} + 520 q^{19} - 4429 q^{20} - 3926 q^{22} - 6644 q^{23} - 13572 q^{25}+ \cdots + 53863 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8x7+70x6133x5+4766x46777x3+16825x2+9696x+9216 x^{8} - x^{7} + 70x^{6} - 133x^{5} + 4766x^{4} - 6777x^{3} + 16825x^{2} + 9696x + 9216 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (319677ν7+174777ν6+21464689ν59316963ν4+1509169756ν3++3235111584)/5480995715 ( 319677 \nu^{7} + 174777 \nu^{6} + 21464689 \nu^{5} - 9316963 \nu^{4} + 1509169756 \nu^{3} + \cdots + 3235111584 ) / 5480995715 Copy content Toggle raw display
β3\beta_{3}== (33699079ν764388071ν6+2342156938ν56542587651ν4++318334817568)/526175588640 ( 33699079 \nu^{7} - 64388071 \nu^{6} + 2342156938 \nu^{5} - 6542587651 \nu^{4} + \cdots + 318334817568 ) / 526175588640 Copy content Toggle raw display
β4\beta_{4}== (814131ν7737924ν6+54664767ν523727789ν4+3798492578ν3++192123818377)/5480995715 ( 814131 \nu^{7} - 737924 \nu^{6} + 54664767 \nu^{5} - 23727789 \nu^{4} + 3798492578 \nu^{3} + \cdots + 192123818377 ) / 5480995715 Copy content Toggle raw display
β5\beta_{5}== (163476431ν7+338880779ν612327003362ν5+33519877559ν4++1093886512128)/263087794320 ( - 163476431 \nu^{7} + 338880779 \nu^{6} - 12327003362 \nu^{5} + 33519877559 \nu^{4} + \cdots + 1093886512128 ) / 263087794320 Copy content Toggle raw display
β6\beta_{6}== (319677ν7+174777ν6+21464689ν59316963ν4+1424846745ν3++6608032024)/337292044 ( 319677 \nu^{7} + 174777 \nu^{6} + 21464689 \nu^{5} - 9316963 \nu^{4} + 1424846745 \nu^{3} + \cdots + 6608032024 ) / 337292044 Copy content Toggle raw display
β7\beta_{7}== (87076937ν7+166612553ν66060637334ν5+17508240653ν4++537815172096)/40475045280 ( - 87076937 \nu^{7} + 166612553 \nu^{6} - 6060637334 \nu^{5} + 17508240653 \nu^{4} + \cdots + 537815172096 ) / 40475045280 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β7+β435β3β2β1 -\beta_{7} + \beta_{4} - 35\beta_{3} - \beta_{2} - \beta_1 Copy content Toggle raw display
ν3\nu^{3}== 4β6+65β2+40 -4\beta_{6} + 65\beta_{2} + 40 Copy content Toggle raw display
ν4\nu^{4}== 69β74β5+2279β3+105β12279 69\beta_{7} - 4\beta_{5} + 2279\beta_{3} + 105\beta _1 - 2279 Copy content Toggle raw display
ν5\nu^{5}== 32β7+280β6280β5+32β44016β34385β24385β1 -32\beta_{7} + 280\beta_{6} - 280\beta_{5} + 32\beta_{4} - 4016\beta_{3} - 4385\beta_{2} - 4385\beta_1 Copy content Toggle raw display
ν6\nu^{6}== 152β64633β4+9329β2+153915 152\beta_{6} - 4633\beta_{4} + 9329\beta_{2} + 153915 Copy content Toggle raw display
ν7\nu^{7}== 4544β7+18684β5+349528β3+297601β1349528 4544\beta_{7} + 18684\beta_{5} + 349528\beta_{3} + 297601\beta _1 - 349528 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/117Z)×\left(\mathbb{Z}/117\mathbb{Z}\right)^\times.

nn 2828 9292
χ(n)\chi(n) β3-\beta_{3} 11

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}
55.1
−4.21857 7.30677i
−0.329369 0.570484i
1.09142 + 1.89039i
3.95652 + 6.85289i
−4.21857 + 7.30677i
−0.329369 + 0.570484i
1.09142 1.89039i
3.95652 6.85289i
−3.71857 6.44074i 0 −11.6555 + 20.1878i 9.82751 0 17.3506 30.0522i −64.6219 0 −36.5442 63.2965i
55.2 0.170631 + 0.295541i 0 15.9418 27.6120i −13.9092 0 −18.2258 + 31.5679i 21.8010 0 −2.37335 4.11076i
55.3 1.59142 + 2.75641i 0 10.9348 18.9396i −44.5576 0 −27.1222 + 46.9770i 171.458 0 −70.9097 122.819i
55.4 4.45652 + 7.71892i 0 −23.7211 + 41.0862i 58.6393 0 61.9973 107.382i −137.637 0 261.327 + 452.632i
100.1 −3.71857 + 6.44074i 0 −11.6555 20.1878i 9.82751 0 17.3506 + 30.0522i −64.6219 0 −36.5442 + 63.2965i
100.2 0.170631 0.295541i 0 15.9418 + 27.6120i −13.9092 0 −18.2258 31.5679i 21.8010 0 −2.37335 + 4.11076i
100.3 1.59142 2.75641i 0 10.9348 + 18.9396i −44.5576 0 −27.1222 46.9770i 171.458 0 −70.9097 + 122.819i
100.4 4.45652 7.71892i 0 −23.7211 41.0862i 58.6393 0 61.9973 + 107.382i −137.637 0 261.327 452.632i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.6.g.b 8
3.b odd 2 1 13.6.c.a 8
12.b even 2 1 208.6.i.b 8
13.c even 3 1 inner 117.6.g.b 8
39.h odd 6 1 169.6.a.c 4
39.i odd 6 1 13.6.c.a 8
39.i odd 6 1 169.6.a.d 4
39.k even 12 2 169.6.b.c 8
156.p even 6 1 208.6.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.6.c.a 8 3.b odd 2 1
13.6.c.a 8 39.i odd 6 1
117.6.g.b 8 1.a even 1 1 trivial
117.6.g.b 8 13.c even 3 1 inner
169.6.a.c 4 39.h odd 6 1
169.6.a.d 4 39.i odd 6 1
169.6.b.c 8 39.k even 12 2
208.6.i.b 8 12.b even 2 1
208.6.i.b 8 156.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T285T27+85T26164T25+4832T2414640T23+49504T2216704T2+5184 T_{2}^{8} - 5T_{2}^{7} + 85T_{2}^{6} - 164T_{2}^{5} + 4832T_{2}^{4} - 14640T_{2}^{3} + 49504T_{2}^{2} - 16704T_{2} + 5184 acting on S6new(117,[χ])S_{6}^{\mathrm{new}}(117, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T85T7++5184 T^{8} - 5 T^{7} + \cdots + 5184 Copy content Toggle raw display
33 T8 T^{8} Copy content Toggle raw display
55 (T410T3++357156)2 (T^{4} - 10 T^{3} + \cdots + 357156)^{2} Copy content Toggle raw display
77 T8++72383069214976 T^{8} + \cdots + 72383069214976 Copy content Toggle raw display
1111 T8++40 ⁣ ⁣56 T^{8} + \cdots + 40\!\cdots\!56 Copy content Toggle raw display
1313 T8++19 ⁣ ⁣01 T^{8} + \cdots + 19\!\cdots\!01 Copy content Toggle raw display
1717 T8++14 ⁣ ⁣09 T^{8} + \cdots + 14\!\cdots\!09 Copy content Toggle raw display
1919 T8++15 ⁣ ⁣96 T^{8} + \cdots + 15\!\cdots\!96 Copy content Toggle raw display
2323 T8++10 ⁣ ⁣16 T^{8} + \cdots + 10\!\cdots\!16 Copy content Toggle raw display
2929 T8++18 ⁣ ⁣61 T^{8} + \cdots + 18\!\cdots\!61 Copy content Toggle raw display
3131 (T4++4693845913600)2 (T^{4} + \cdots + 4693845913600)^{2} Copy content Toggle raw display
3737 T8++28 ⁣ ⁣09 T^{8} + \cdots + 28\!\cdots\!09 Copy content Toggle raw display
4141 T8++96 ⁣ ⁣41 T^{8} + \cdots + 96\!\cdots\!41 Copy content Toggle raw display
4343 T8++63 ⁣ ⁣04 T^{8} + \cdots + 63\!\cdots\!04 Copy content Toggle raw display
4747 (T4++24 ⁣ ⁣92)2 (T^{4} + \cdots + 24\!\cdots\!92)^{2} Copy content Toggle raw display
5353 (T4+38 ⁣ ⁣36)2 (T^{4} + \cdots - 38\!\cdots\!36)^{2} Copy content Toggle raw display
5959 T8++21 ⁣ ⁣44 T^{8} + \cdots + 21\!\cdots\!44 Copy content Toggle raw display
6161 T8++13 ⁣ ⁣61 T^{8} + \cdots + 13\!\cdots\!61 Copy content Toggle raw display
6767 T8++69 ⁣ ⁣56 T^{8} + \cdots + 69\!\cdots\!56 Copy content Toggle raw display
7171 T8++47 ⁣ ⁣56 T^{8} + \cdots + 47\!\cdots\!56 Copy content Toggle raw display
7373 (T4+34 ⁣ ⁣48)2 (T^{4} + \cdots - 34\!\cdots\!48)^{2} Copy content Toggle raw display
7979 (T4++43 ⁣ ⁣00)2 (T^{4} + \cdots + 43\!\cdots\!00)^{2} Copy content Toggle raw display
8383 (T4+30 ⁣ ⁣40)2 (T^{4} + \cdots - 30\!\cdots\!40)^{2} Copy content Toggle raw display
8989 T8++91 ⁣ ⁣36 T^{8} + \cdots + 91\!\cdots\!36 Copy content Toggle raw display
9797 T8++11 ⁣ ⁣16 T^{8} + \cdots + 11\!\cdots\!16 Copy content Toggle raw display
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