Properties

Label 208.6.i.b
Level 208208
Weight 66
Character orbit 208.i
Analytic conductor 33.36033.360
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] = [208,6,Mod(81,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("208.81");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 208=2413 208 = 2^{4} \cdot 13
Weight: k k == 6 6
Character orbit: [χ][\chi] == 208.i (of order 33, degree 22, not 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: 33.359834521133.3598345211
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,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 26 2^{6}
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+(β5+2β3)q3+(β6β4+β12)q5+(2β7β6+16)q7+(7β7+7β67β5+11)q9++(4978β6+1112β4++17512)q99+O(q100) q + (\beta_{5} + 2 \beta_{3}) q^{3} + ( - \beta_{6} - \beta_{4} + \beta_1 - 2) q^{5} + ( - 2 \beta_{7} - \beta_{6} + \cdots - 16) q^{7} + (7 \beta_{7} + 7 \beta_{6} - 7 \beta_{5} + \cdots - 11) q^{9}+ \cdots + (4978 \beta_{6} + 1112 \beta_{4} + \cdots + 17512) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q8q320q568q730q9+480q11234q13992q15+60q17520q191804q216644q2313572q25+10240q271236q2924704q31+19454q33++146928q99+O(q100) 8 q - 8 q^{3} - 20 q^{5} - 68 q^{7} - 30 q^{9} + 480 q^{11} - 234 q^{13} - 992 q^{15} + 60 q^{17} - 520 q^{19} - 1804 q^{21} - 6644 q^{23} - 13572 q^{25} + 10240 q^{27} - 1236 q^{29} - 24704 q^{31} + 19454 q^{33}+ \cdots + 146928 q^{99}+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}== (174777ν71087478ν6+11735389ν55093863ν4+780153066ν3++180172599494)/5480995715 ( 174777 \nu^{7} - 1087478 \nu^{6} + 11735389 \nu^{5} - 5093863 \nu^{4} + 780153066 \nu^{3} + \cdots + 180172599494 ) / 5480995715 Copy content Toggle raw display
β2\beta_{2}== (33699079ν764388071ν6+2342156938ν56542587651ν4+207840771072)/526175588640 ( 33699079 \nu^{7} - 64388071 \nu^{6} + 2342156938 \nu^{5} - 6542587651 \nu^{4} + \cdots - 207840771072 ) / 526175588640 Copy content Toggle raw display
β3\beta_{3}== (33699079ν764388071ν6+2342156938ν56542587651ν4+207840771072)/526175588640 ( 33699079 \nu^{7} - 64388071 \nu^{6} + 2342156938 \nu^{5} - 6542587651 \nu^{4} + \cdots - 207840771072 ) / 526175588640 Copy content Toggle raw display
β4\beta_{4}== (1278708ν7699108ν685858756ν5+37267852ν46036679024ν3+18421442051)/5480995715 ( - 1278708 \nu^{7} - 699108 \nu^{6} - 85858756 \nu^{5} + 37267852 \nu^{4} - 6036679024 \nu^{3} + \cdots - 18421442051 ) / 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== (β3+β2)/4 ( -\beta_{3} + \beta_{2} ) / 4 Copy content Toggle raw display
ν2\nu^{2}== (4β7β4141β3+β2+4β1137)/4 ( -4\beta_{7} - \beta_{4} - 141\beta_{3} + \beta_{2} + 4\beta _1 - 137 ) / 4 Copy content Toggle raw display
ν3\nu^{3}== (16β665β4+95)/4 ( -16\beta_{6} - 65\beta_{4} + 95 ) / 4 Copy content Toggle raw display
ν4\nu^{4}== (276β716β5+9149β333β2)/4 ( 276\beta_{7} - 16\beta_{5} + 9149\beta_{3} - 33\beta_{2} ) / 4 Copy content Toggle raw display
ν5\nu^{5}== (128β7+1120β61120β5+4321β411743β34321β2+11615)/4 ( - 128 \beta_{7} + 1120 \beta_{6} - 1120 \beta_{5} + 4321 \beta_{4} - 11743 \beta_{3} - 4321 \beta_{2} + \cdots - 11615 ) / 4 Copy content Toggle raw display
ν6\nu^{6}== (608β663β418532β1+597065)/4 ( 608\beta_{6} - 63\beta_{4} - 18532\beta _1 + 597065 ) / 4 Copy content Toggle raw display
ν7\nu^{7}== (18176β7+74736β5+1109599β3+288513β2)/4 ( 18176\beta_{7} + 74736\beta_{5} + 1109599\beta_{3} + 288513\beta_{2} ) / 4 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/208Z)×\left(\mathbb{Z}/208\mathbb{Z}\right)^\times.

nn 5353 7979 145145
χ(n)\chi(n) 11 11 1β3-1 - \beta_{3}

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}
81.1
1.09142 + 1.89039i
−4.21857 7.30677i
3.95652 + 6.85289i
−0.329369 0.570484i
1.09142 1.89039i
−4.21857 + 7.30677i
3.95652 6.85289i
−0.329369 + 0.570484i
0 −12.4354 21.5388i 0 44.5576 0 27.1222 46.9770i 0 −187.780 + 325.244i 0
81.2 0 −2.52313 4.37020i 0 −9.82751 0 −17.3506 + 30.0522i 0 108.768 188.391i 0
81.3 0 1.64205 + 2.84411i 0 −58.6393 0 −61.9973 + 107.382i 0 116.107 201.104i 0
81.4 0 9.31652 + 16.1367i 0 13.9092 0 18.2258 31.5679i 0 −52.0950 + 90.2311i 0
113.1 0 −12.4354 + 21.5388i 0 44.5576 0 27.1222 + 46.9770i 0 −187.780 325.244i 0
113.2 0 −2.52313 + 4.37020i 0 −9.82751 0 −17.3506 30.0522i 0 108.768 + 188.391i 0
113.3 0 1.64205 2.84411i 0 −58.6393 0 −61.9973 107.382i 0 116.107 + 201.104i 0
113.4 0 9.31652 16.1367i 0 13.9092 0 18.2258 + 31.5679i 0 −52.0950 90.2311i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.4
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 208.6.i.b 8
4.b odd 2 1 13.6.c.a 8
12.b even 2 1 117.6.g.b 8
13.c even 3 1 inner 208.6.i.b 8
52.i odd 6 1 169.6.a.c 4
52.j odd 6 1 13.6.c.a 8
52.j odd 6 1 169.6.a.d 4
52.l even 12 2 169.6.b.c 8
156.p even 6 1 117.6.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.6.c.a 8 4.b odd 2 1
13.6.c.a 8 52.j odd 6 1
117.6.g.b 8 12.b even 2 1
117.6.g.b 8 156.p even 6 1
169.6.a.c 4 52.i odd 6 1
169.6.a.d 4 52.j odd 6 1
169.6.b.c 8 52.l even 12 2
208.6.i.b 8 1.a even 1 1 trivial
208.6.i.b 8 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T38+8T37+533T361912T35+219641T34+308600T33++58982400 T_{3}^{8} + 8 T_{3}^{7} + 533 T_{3}^{6} - 1912 T_{3}^{5} + 219641 T_{3}^{4} + 308600 T_{3}^{3} + \cdots + 58982400 acting on S6new(208,[χ])S_{6}^{\mathrm{new}}(208, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8 T^{8} Copy content Toggle raw display
33 T8+8T7++58982400 T^{8} + 8 T^{7} + \cdots + 58982400 Copy content Toggle raw display
55 (T4+10T3++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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