Properties

Label 294.4.e.d
Level 294294
Weight 44
Character orbit 294.e
Analytic conductor 17.34717.347
Analytic rank 00
Dimension 22
Inner twists 22

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [294,4,Mod(67,294)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("294.67"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: N N == 294=2372 294 = 2 \cdot 3 \cdot 7^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 294.e (of order 33, degree 22, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,3,-4,2,-12,0,16,-9,4,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 17.346561541717.3465615417
Analytic rank: 00
Dimension: 22
Coefficient field: Q(3)\Q(\sqrt{-3})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2x+1 x^{2} - x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 42)
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 primitive root of unity ζ6\zeta_{6}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q2ζ6q2+(3ζ6+3)q3+(4ζ64)q4+2ζ6q56q6+8q89ζ6q9+(4ζ6+4)q10+(8ζ6+8)q11+12ζ6q12+72q99+O(q100) q - 2 \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + (4 \zeta_{6} - 4) q^{4} + 2 \zeta_{6} q^{5} - 6 q^{6} + 8 q^{8} - 9 \zeta_{6} q^{9} + ( - 4 \zeta_{6} + 4) q^{10} + ( - 8 \zeta_{6} + 8) q^{11} + 12 \zeta_{6} q^{12} + \cdots - 72 q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q2+3q34q4+2q512q6+16q89q9+4q10+8q11+12q12+84q13+12q1516q162q1718q18124q1916q2032q22+144q99+O(q100) 2 q - 2 q^{2} + 3 q^{3} - 4 q^{4} + 2 q^{5} - 12 q^{6} + 16 q^{8} - 9 q^{9} + 4 q^{10} + 8 q^{11} + 12 q^{12} + 84 q^{13} + 12 q^{15} - 16 q^{16} - 2 q^{17} - 18 q^{18} - 124 q^{19} - 16 q^{20} - 32 q^{22}+ \cdots - 144 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/294Z)×\left(\mathbb{Z}/294\mathbb{Z}\right)^\times.

nn 197197 199199
χ(n)\chi(n) 11 ζ6-\zeta_{6}

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}
67.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 1.73205i 1.50000 2.59808i −2.00000 + 3.46410i 1.00000 + 1.73205i −6.00000 0 8.00000 −4.50000 7.79423i 2.00000 3.46410i
79.1 −1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i 1.00000 1.73205i −6.00000 0 8.00000 −4.50000 + 7.79423i 2.00000 + 3.46410i
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 294.4.e.d 2
3.b odd 2 1 882.4.g.r 2
7.b odd 2 1 294.4.e.a 2
7.c even 3 1 294.4.a.h 1
7.c even 3 1 inner 294.4.e.d 2
7.d odd 6 1 42.4.a.b 1
7.d odd 6 1 294.4.e.a 2
21.c even 2 1 882.4.g.s 2
21.g even 6 1 126.4.a.c 1
21.g even 6 1 882.4.g.s 2
21.h odd 6 1 882.4.a.d 1
21.h odd 6 1 882.4.g.r 2
28.f even 6 1 336.4.a.d 1
28.g odd 6 1 2352.4.a.ba 1
35.i odd 6 1 1050.4.a.d 1
35.k even 12 2 1050.4.g.n 2
56.j odd 6 1 1344.4.a.f 1
56.m even 6 1 1344.4.a.t 1
84.j odd 6 1 1008.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.b 1 7.d odd 6 1
126.4.a.c 1 21.g even 6 1
294.4.a.h 1 7.c even 3 1
294.4.e.a 2 7.b odd 2 1
294.4.e.a 2 7.d odd 6 1
294.4.e.d 2 1.a even 1 1 trivial
294.4.e.d 2 7.c even 3 1 inner
336.4.a.d 1 28.f even 6 1
882.4.a.d 1 21.h odd 6 1
882.4.g.r 2 3.b odd 2 1
882.4.g.r 2 21.h odd 6 1
882.4.g.s 2 21.c even 2 1
882.4.g.s 2 21.g even 6 1
1008.4.a.j 1 84.j odd 6 1
1050.4.a.d 1 35.i odd 6 1
1050.4.g.n 2 35.k even 12 2
1344.4.a.f 1 56.j odd 6 1
1344.4.a.t 1 56.m even 6 1
2352.4.a.ba 1 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(294,[χ])S_{4}^{\mathrm{new}}(294, [\chi]):

T522T5+4 T_{5}^{2} - 2T_{5} + 4 Copy content Toggle raw display
T1128T11+64 T_{11}^{2} - 8T_{11} + 64 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
33 T23T+9 T^{2} - 3T + 9 Copy content Toggle raw display
55 T22T+4 T^{2} - 2T + 4 Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T28T+64 T^{2} - 8T + 64 Copy content Toggle raw display
1313 (T42)2 (T - 42)^{2} Copy content Toggle raw display
1717 T2+2T+4 T^{2} + 2T + 4 Copy content Toggle raw display
1919 T2+124T+15376 T^{2} + 124T + 15376 Copy content Toggle raw display
2323 T2+76T+5776 T^{2} + 76T + 5776 Copy content Toggle raw display
2929 (T254)2 (T - 254)^{2} Copy content Toggle raw display
3131 T2+72T+5184 T^{2} + 72T + 5184 Copy content Toggle raw display
3737 T2+398T+158404 T^{2} + 398T + 158404 Copy content Toggle raw display
4141 (T+462)2 (T + 462)^{2} Copy content Toggle raw display
4343 (T212)2 (T - 212)^{2} Copy content Toggle raw display
4747 T2+264T+69696 T^{2} + 264T + 69696 Copy content Toggle raw display
5353 T2162T+26244 T^{2} - 162T + 26244 Copy content Toggle raw display
5959 T2+772T+595984 T^{2} + 772T + 595984 Copy content Toggle raw display
6161 T230T+900 T^{2} - 30T + 900 Copy content Toggle raw display
6767 T2764T+583696 T^{2} - 764T + 583696 Copy content Toggle raw display
7171 (T+236)2 (T + 236)^{2} Copy content Toggle raw display
7373 T2418T+174724 T^{2} - 418T + 174724 Copy content Toggle raw display
7979 T2+552T+304704 T^{2} + 552T + 304704 Copy content Toggle raw display
8383 (T+1036)2 (T + 1036)^{2} Copy content Toggle raw display
8989 T230T+900 T^{2} - 30T + 900 Copy content Toggle raw display
9797 (T1190)2 (T - 1190)^{2} Copy content Toggle raw display
show more
show less