Properties

Label 208.2.p.a
Level 208208
Weight 22
Character orbit 208.p
Analytic conductor 1.6611.661
Analytic rank 00
Dimension 88
Inner twists 44

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [208,2,Mod(77,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("208.77"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 208=2413 208 = 2^{4} \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 208.p (of order 44, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 1.660888362041.66088836204
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(i)\Q(i)
Coefficient field: 8.0.959512576.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x8+7x4+81 x^{8} + 7x^{4} + 81 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 22 2^{2}
Twist minimal: yes
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

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,,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+(β4+β1)q2β3q32q4β4q5+(β7β5+β4+β1)q6+(β7+2β4β1)q7+(2β42β1)q8++(2β72β5++6β1)q99+O(q100) q + (\beta_{4} + \beta_1) q^{2} - \beta_{3} q^{3} - 2 q^{4} - \beta_{4} q^{5} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_1) q^{6} + (\beta_{7} + 2 \beta_{4} - \beta_1) q^{7} + ( - 2 \beta_{4} - 2 \beta_1) q^{8}+ \cdots + (2 \beta_{7} - 2 \beta_{5} + \cdots + 6 \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+4q316q4+8q108q128q13+32q16+24q1716q2244q2724q29+8q3012q35+24q3816q40+32q424q43+16q48+24q49+24q95+O(q100) 8 q + 4 q^{3} - 16 q^{4} + 8 q^{10} - 8 q^{12} - 8 q^{13} + 32 q^{16} + 24 q^{17} - 16 q^{22} - 44 q^{27} - 24 q^{29} + 8 q^{30} - 12 q^{35} + 24 q^{38} - 16 q^{40} + 32 q^{42} - 4 q^{43} + 16 q^{48} + 24 q^{49}+ \cdots - 24 q^{95}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8+7x4+81 x^{8} + 7x^{4} + 81 : Copy content Toggle raw display

β1\beta_{1}== (ν5+ν)/15 ( \nu^{5} + \nu ) / 15 Copy content Toggle raw display
β2\beta_{2}== (ν6+16ν2)/45 ( \nu^{6} + 16\nu^{2} ) / 45 Copy content Toggle raw display
β3\beta_{3}== (ν6+3ν4ν2+3)/15 ( -\nu^{6} + 3\nu^{4} - \nu^{2} + 3 ) / 15 Copy content Toggle raw display
β4\beta_{4}== (2ν7+13ν3)/135 ( -2\nu^{7} + 13\nu^{3} ) / 135 Copy content Toggle raw display
β5\beta_{5}== (ν7+16ν3+45ν)/45 ( \nu^{7} + 16\nu^{3} + 45\nu ) / 45 Copy content Toggle raw display
β6\beta_{6}== (ν63ν4ν23)/15 ( -\nu^{6} - 3\nu^{4} - \nu^{2} - 3 ) / 15 Copy content Toggle raw display
β7\beta_{7}== (ν716ν3+45ν)/45 ( -\nu^{7} - 16\nu^{3} + 45\nu ) / 45 Copy content Toggle raw display
ν\nu== (β7+β5)/2 ( \beta_{7} + \beta_{5} ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (β6+β3+6β2)/2 ( \beta_{6} + \beta_{3} + 6\beta_{2} ) / 2 Copy content Toggle raw display
ν3\nu^{3}== β7+β5+3β4 -\beta_{7} + \beta_{5} + 3\beta_{4} Copy content Toggle raw display
ν4\nu^{4}== (5β6+5β32)/2 ( -5\beta_{6} + 5\beta_{3} - 2 ) / 2 Copy content Toggle raw display
ν5\nu^{5}== (β7β5+30β1)/2 ( -\beta_{7} - \beta_{5} + 30\beta_1 ) / 2 Copy content Toggle raw display
ν6\nu^{6}== 8β68β33β2 -8\beta_{6} - 8\beta_{3} - 3\beta_{2} Copy content Toggle raw display
ν7\nu^{7}== (13β7+13β596β4)/2 ( -13\beta_{7} + 13\beta_{5} - 96\beta_{4} ) / 2 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) β2-\beta_{2} 11 1-1

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}
77.1
−1.52616 0.819051i
0.819051 + 1.52616i
1.52616 + 0.819051i
−0.819051 1.52616i
1.52616 0.819051i
−0.819051 + 1.52616i
−1.52616 + 0.819051i
0.819051 1.52616i
1.41421i −1.15831 1.15831i −2.00000 0.707107 + 0.707107i −1.63810 + 1.63810i −4.46653 2.82843i 0.316625i 1.00000 1.00000i
77.2 1.41421i 2.15831 + 2.15831i −2.00000 0.707107 + 0.707107i 3.05231 3.05231i 0.223888 2.82843i 6.31662i 1.00000 1.00000i
77.3 1.41421i −1.15831 1.15831i −2.00000 −0.707107 0.707107i 1.63810 1.63810i 4.46653 2.82843i 0.316625i 1.00000 1.00000i
77.4 1.41421i 2.15831 + 2.15831i −2.00000 −0.707107 0.707107i −3.05231 + 3.05231i −0.223888 2.82843i 6.31662i 1.00000 1.00000i
181.1 1.41421i −1.15831 + 1.15831i −2.00000 −0.707107 + 0.707107i 1.63810 + 1.63810i 4.46653 2.82843i 0.316625i 1.00000 + 1.00000i
181.2 1.41421i 2.15831 2.15831i −2.00000 −0.707107 + 0.707107i −3.05231 3.05231i −0.223888 2.82843i 6.31662i 1.00000 + 1.00000i
181.3 1.41421i −1.15831 + 1.15831i −2.00000 0.707107 0.707107i −1.63810 1.63810i −4.46653 2.82843i 0.316625i 1.00000 + 1.00000i
181.4 1.41421i 2.15831 2.15831i −2.00000 0.707107 0.707107i 3.05231 + 3.05231i 0.223888 2.82843i 6.31662i 1.00000 + 1.00000i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
16.e even 4 1 inner
208.p even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.2.p.a 8
4.b odd 2 1 832.2.p.a 8
13.b even 2 1 inner 208.2.p.a 8
16.e even 4 1 inner 208.2.p.a 8
16.f odd 4 1 832.2.p.a 8
52.b odd 2 1 832.2.p.a 8
208.o odd 4 1 832.2.p.a 8
208.p even 4 1 inner 208.2.p.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
208.2.p.a 8 1.a even 1 1 trivial
208.2.p.a 8 13.b even 2 1 inner
208.2.p.a 8 16.e even 4 1 inner
208.2.p.a 8 208.p even 4 1 inner
832.2.p.a 8 4.b odd 2 1
832.2.p.a 8 16.f odd 4 1
832.2.p.a 8 52.b odd 2 1
832.2.p.a 8 208.o odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T342T33+2T32+10T3+25 T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} + 10T_{3} + 25 acting on S2new(208,[χ])S_{2}^{\mathrm{new}}(208, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2+2)4 (T^{2} + 2)^{4} Copy content Toggle raw display
33 (T42T3+2T2++25)2 (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 25)^{2} Copy content Toggle raw display
55 (T4+1)2 (T^{4} + 1)^{2} Copy content Toggle raw display
77 (T420T2+1)2 (T^{4} - 20 T^{2} + 1)^{2} Copy content Toggle raw display
1111 (T4+16)2 (T^{4} + 16)^{2} Copy content Toggle raw display
1313 (T4+4T3++169)2 (T^{4} + 4 T^{3} + \cdots + 169)^{2} Copy content Toggle raw display
1717 (T3)8 (T - 3)^{8} Copy content Toggle raw display
1919 T8+1592T4+16 T^{8} + 1592T^{4} + 16 Copy content Toggle raw display
2323 (T4+40T2+4)2 (T^{4} + 40 T^{2} + 4)^{2} Copy content Toggle raw display
2929 (T4+12T3++16)2 (T^{4} + 12 T^{3} + \cdots + 16)^{2} Copy content Toggle raw display
3131 (T2+18)4 (T^{2} + 18)^{4} Copy content Toggle raw display
3737 T8+7586T4+390625 T^{8} + 7586 T^{4} + 390625 Copy content Toggle raw display
4141 T8 T^{8} Copy content Toggle raw display
4343 (T4+2T3++2401)2 (T^{4} + 2 T^{3} + \cdots + 2401)^{2} Copy content Toggle raw display
4747 (T4+124T2+1369)2 (T^{4} + 124 T^{2} + 1369)^{2} Copy content Toggle raw display
5353 (T4+12T3++4900)2 (T^{4} + 12 T^{3} + \cdots + 4900)^{2} Copy content Toggle raw display
5959 T8+20792T4+92236816 T^{8} + 20792 T^{4} + 92236816 Copy content Toggle raw display
6161 (T28T+32)4 (T^{2} - 8 T + 32)^{4} Copy content Toggle raw display
6767 (T4+1936)2 (T^{4} + 1936)^{2} Copy content Toggle raw display
7171 (T4180T2+81)2 (T^{4} - 180 T^{2} + 81)^{2} Copy content Toggle raw display
7373 (T4188T2+2500)2 (T^{4} - 188 T^{2} + 2500)^{2} Copy content Toggle raw display
7979 (T2+2T98)4 (T^{2} + 2 T - 98)^{4} Copy content Toggle raw display
8383 T8+82616T4+6250000 T^{8} + 82616 T^{4} + 6250000 Copy content Toggle raw display
8989 (T2198)4 (T^{2} - 198)^{4} Copy content Toggle raw display
9797 (T2+72)4 (T^{2} + 72)^{4} Copy content Toggle raw display
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