Properties

Label 336.4.q.h
Level 336336
Weight 44
Character orbit 336.q
Analytic conductor 19.82519.825
Analytic rank 00
Dimension 44
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [336,4,Mod(193,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.193"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 336=2437 336 = 2^{4} \cdot 3 \cdot 7
Weight: k k == 4 4
Character orbit: [χ][\chi] == 336.q (of order 33, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-6,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 19.824641761919.8246417619
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q(3,19)\Q(\sqrt{-3}, \sqrt{-19})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4x34x25x+25 x^{4} - x^{3} - 4x^{2} - 5x + 25 Copy content Toggle raw display
Coefficient ring: Z[a1,,a19]\Z[a_1, \ldots, a_{19}]
Coefficient ring index: 32 3^{2}
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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+3β2q3+(β3+2β2+2β1+1)q5+(β36β23β1+3)q7+(9β29)q9+(2β3+25β2+1)q11++(9β3+9β2++234)q99+O(q100) q + 3 \beta_{2} q^{3} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{5} + (\beta_{3} - 6 \beta_{2} - 3 \beta_1 + 3) q^{7} + ( - 9 \beta_{2} - 9) q^{9} + ( - 2 \beta_{3} + 25 \beta_{2} + \cdots - 1) q^{11}+ \cdots + (9 \beta_{3} + 9 \beta_{2} + \cdots + 234) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q6q3+3q5+20q718q951q11+122q1318q1524q17+169q19+15q21192q2311q25+108q2778q2992q31153q33+294q35++918q99+O(q100) 4 q - 6 q^{3} + 3 q^{5} + 20 q^{7} - 18 q^{9} - 51 q^{11} + 122 q^{13} - 18 q^{15} - 24 q^{17} + 169 q^{19} + 15 q^{21} - 192 q^{23} - 11 q^{25} + 108 q^{27} - 78 q^{29} - 92 q^{31} - 153 q^{33} + 294 q^{35}+ \cdots + 918 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4x34x25x+25 x^{4} - x^{3} - 4x^{2} - 5x + 25 : Copy content Toggle raw display

β1\beta_{1}== (ν3+4ν2+56ν25)/20 ( \nu^{3} + 4\nu^{2} + 56\nu - 25 ) / 20 Copy content Toggle raw display
β2\beta_{2}== (ν3+4ν24ν25)/20 ( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20 Copy content Toggle raw display
β3\beta_{3}== (7ν3+2ν2+28ν+35)/10 ( -7\nu^{3} + 2\nu^{2} + 28\nu + 35 ) / 10 Copy content Toggle raw display
ν\nu== (β2+β1)/3 ( -\beta_{2} + \beta_1 ) / 3 Copy content Toggle raw display
ν2\nu^{2}== (β3+14β2+14)/3 ( \beta_{3} + 14\beta_{2} + 14 ) / 3 Copy content Toggle raw display
ν3\nu^{3}== (4β3+4β1+19)/3 ( -4\beta_{3} + 4\beta _1 + 19 ) / 3 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/336Z)×\left(\mathbb{Z}/336\mathbb{Z}\right)^\times.

nn 8585 113113 127127 241241
χ(n)\chi(n) 11 11 11 1β2-1 - \beta_{2}

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}
193.1
−1.63746 1.52274i
2.13746 + 0.656712i
−1.63746 + 1.52274i
2.13746 0.656712i
0 −1.50000 + 2.59808i 0 −4.91238 8.50848i 0 16.3248 + 8.74657i 0 −4.50000 7.79423i 0
193.2 0 −1.50000 + 2.59808i 0 6.41238 + 11.1066i 0 −6.32475 17.4068i 0 −4.50000 7.79423i 0
289.1 0 −1.50000 2.59808i 0 −4.91238 + 8.50848i 0 16.3248 8.74657i 0 −4.50000 + 7.79423i 0
289.2 0 −1.50000 2.59808i 0 6.41238 11.1066i 0 −6.32475 + 17.4068i 0 −4.50000 + 7.79423i 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.q.h 4
4.b odd 2 1 84.4.i.b 4
7.c even 3 1 inner 336.4.q.h 4
7.c even 3 1 2352.4.a.cb 2
7.d odd 6 1 2352.4.a.bp 2
12.b even 2 1 252.4.k.d 4
28.d even 2 1 588.4.i.i 4
28.f even 6 1 588.4.a.h 2
28.f even 6 1 588.4.i.i 4
28.g odd 6 1 84.4.i.b 4
28.g odd 6 1 588.4.a.g 2
84.h odd 2 1 1764.4.k.z 4
84.j odd 6 1 1764.4.a.p 2
84.j odd 6 1 1764.4.k.z 4
84.n even 6 1 252.4.k.d 4
84.n even 6 1 1764.4.a.x 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.b 4 4.b odd 2 1
84.4.i.b 4 28.g odd 6 1
252.4.k.d 4 12.b even 2 1
252.4.k.d 4 84.n even 6 1
336.4.q.h 4 1.a even 1 1 trivial
336.4.q.h 4 7.c even 3 1 inner
588.4.a.g 2 28.g odd 6 1
588.4.a.h 2 28.f even 6 1
588.4.i.i 4 28.d even 2 1
588.4.i.i 4 28.f even 6 1
1764.4.a.p 2 84.j odd 6 1
1764.4.a.x 2 84.n even 6 1
1764.4.k.z 4 84.h odd 2 1
1764.4.k.z 4 84.j odd 6 1
2352.4.a.bp 2 7.d odd 6 1
2352.4.a.cb 2 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T543T53+135T52+378T5+15876 T_{5}^{4} - 3T_{5}^{3} + 135T_{5}^{2} + 378T_{5} + 15876 acting on S4new(336,[χ])S_{4}^{\mathrm{new}}(336, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4 T^{4} Copy content Toggle raw display
33 (T2+3T+9)2 (T^{2} + 3 T + 9)^{2} Copy content Toggle raw display
55 T43T3++15876 T^{4} - 3 T^{3} + \cdots + 15876 Copy content Toggle raw display
77 T420T3++117649 T^{4} - 20 T^{3} + \cdots + 117649 Copy content Toggle raw display
1111 T4+51T3++272484 T^{4} + 51 T^{3} + \cdots + 272484 Copy content Toggle raw display
1313 (T261T2276)2 (T^{2} - 61 T - 2276)^{2} Copy content Toggle raw display
1717 T4+24T3++65028096 T^{4} + 24 T^{3} + \cdots + 65028096 Copy content Toggle raw display
1919 T4169T3++49168144 T^{4} - 169 T^{3} + \cdots + 49168144 Copy content Toggle raw display
2323 (T2+96T+9216)2 (T^{2} + 96 T + 9216)^{2} Copy content Toggle raw display
2929 (T2+39T36684)2 (T^{2} + 39 T - 36684)^{2} Copy content Toggle raw display
3131 T4+92T3++114682681 T^{4} + 92 T^{3} + \cdots + 114682681 Copy content Toggle raw display
3737 T4++1506681856 T^{4} + \cdots + 1506681856 Copy content Toggle raw display
4141 (T2174T140688)2 (T^{2} - 174 T - 140688)^{2} Copy content Toggle raw display
4343 (T2497T+15454)2 (T^{2} - 497 T + 15454)^{2} Copy content Toggle raw display
4747 T4180T3++611671824 T^{4} - 180 T^{3} + \cdots + 611671824 Copy content Toggle raw display
5353 T4++5356483344 T^{4} + \cdots + 5356483344 Copy content Toggle raw display
5959 T4++161150862096 T^{4} + \cdots + 161150862096 Copy content Toggle raw display
6161 T4++19408947856 T^{4} + \cdots + 19408947856 Copy content Toggle raw display
6767 T4++32767516324 T^{4} + \cdots + 32767516324 Copy content Toggle raw display
7171 (T21404T+361476)2 (T^{2} - 1404 T + 361476)^{2} Copy content Toggle raw display
7373 T4++165260136484 T^{4} + \cdots + 165260136484 Copy content Toggle raw display
7979 T4+182T3++38800441 T^{4} + 182 T^{3} + \cdots + 38800441 Copy content Toggle raw display
8383 (T2399T197334)2 (T^{2} - 399 T - 197334)^{2} Copy content Toggle raw display
8989 T4++11416495104 T^{4} + \cdots + 11416495104 Copy content Toggle raw display
9797 (T2841T+120262)2 (T^{2} - 841 T + 120262)^{2} Copy content Toggle raw display
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