Properties

Label 46.8.a.c
Level 4646
Weight 88
Character orbit 46.a
Self dual yes
Analytic conductor 14.37014.370
Analytic rank 11
Dimension 33
CM no
Inner twists 11

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [46,8,Mod(1,46)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(46, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("46.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: N N == 46=223 46 = 2 \cdot 23
Weight: k k == 8 8
Character orbit: [χ][\chi] == 46.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 14.369711172314.3697111723
Analytic rank: 11
Dimension: 33
Coefficient field: 3.3.285765.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x3x2121x+46 x^{3} - x^{2} - 121x + 46 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 2 2
Twist minimal: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

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,β21,\beta_1,\beta_2 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+8q2+(β2+2β15)q3+64q4+(15β29β1182)q5+(8β2+16β140)q6+(49β263β1456)q7+512q8++(382074β2+4860β1+4504032)q99+O(q100) q + 8 q^{2} + ( - \beta_{2} + 2 \beta_1 - 5) q^{3} + 64 q^{4} + (15 \beta_{2} - 9 \beta_1 - 182) q^{5} + ( - 8 \beta_{2} + 16 \beta_1 - 40) q^{6} + ( - 49 \beta_{2} - 63 \beta_1 - 456) q^{7} + 512 q^{8}+ \cdots + ( - 382074 \beta_{2} + 4860 \beta_1 + 4504032) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 3q+24q212q3+192q4570q596q61382q7+1536q83759q94560q1011792q11768q1212q1311056q1412990q15+12288q1625584q17++13899030q99+O(q100) 3 q + 24 q^{2} - 12 q^{3} + 192 q^{4} - 570 q^{5} - 96 q^{6} - 1382 q^{7} + 1536 q^{8} - 3759 q^{9} - 4560 q^{10} - 11792 q^{11} - 768 q^{12} - 12 q^{13} - 11056 q^{14} - 12990 q^{15} + 12288 q^{16} - 25584 q^{17}+ \cdots + 13899030 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x3x2121x+46 x^{3} - x^{2} - 121x + 46 : Copy content Toggle raw display

β1\beta_{1}== (ν2+5ν81)/5 ( \nu^{2} + 5\nu - 81 ) / 5 Copy content Toggle raw display
β2\beta_{2}== (ν25ν81)/5 ( \nu^{2} - 5\nu - 81 ) / 5 Copy content Toggle raw display
ν\nu== (β2+β1)/2 ( -\beta_{2} + \beta_1 ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (5β2+5β1+162)/2 ( 5\beta_{2} + 5\beta _1 + 162 ) / 2 Copy content Toggle raw display

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}
1.1
−10.7048
0.379427
11.3254
8.00000 −30.3959 64.0000 115.226 −243.167 −1058.60 512.000 −1263.09 921.805
1.2 8.00000 −20.0329 64.0000 −288.133 −160.263 1349.86 512.000 −1785.68 −2305.07
1.3 8.00000 38.4288 64.0000 −397.092 307.430 −1673.26 512.000 −710.227 −3176.74
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1
2323 +1 +1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 46.8.a.c 3
3.b odd 2 1 414.8.a.d 3
4.b odd 2 1 368.8.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.8.a.c 3 1.a even 1 1 trivial
368.8.a.b 3 4.b odd 2 1
414.8.a.d 3 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T33+12T321329T323400 T_{3}^{3} + 12T_{3}^{2} - 1329T_{3} - 23400 acting on S8new(Γ0(46))S_{8}^{\mathrm{new}}(\Gamma_0(46)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T8)3 (T - 8)^{3} Copy content Toggle raw display
33 T3+12T2+23400 T^{3} + 12 T^{2} + \cdots - 23400 Copy content Toggle raw display
55 T3+570T2+13183600 T^{3} + 570 T^{2} + \cdots - 13183600 Copy content Toggle raw display
77 T3+2391036528 T^{3} + \cdots - 2391036528 Copy content Toggle raw display
1111 T3+77567772648 T^{3} + \cdots - 77567772648 Copy content Toggle raw display
1313 T3+164166584566 T^{3} + \cdots - 164166584566 Copy content Toggle raw display
1717 T3+14898856912840 T^{3} + \cdots - 14898856912840 Copy content Toggle raw display
1919 T3+556058270392 T^{3} + \cdots - 556058270392 Copy content Toggle raw display
2323 (T+12167)3 (T + 12167)^{3} Copy content Toggle raw display
2929 T3+33 ⁣ ⁣02 T^{3} + \cdots - 33\!\cdots\!02 Copy content Toggle raw display
3131 T3+829548394660080 T^{3} + \cdots - 829548394660080 Copy content Toggle raw display
3737 T3+18 ⁣ ⁣12 T^{3} + \cdots - 18\!\cdots\!12 Copy content Toggle raw display
4141 T3+19 ⁣ ⁣74 T^{3} + \cdots - 19\!\cdots\!74 Copy content Toggle raw display
4343 T3+14 ⁣ ⁣60 T^{3} + \cdots - 14\!\cdots\!60 Copy content Toggle raw display
4747 T3++23 ⁣ ⁣00 T^{3} + \cdots + 23\!\cdots\!00 Copy content Toggle raw display
5353 T3++11 ⁣ ⁣72 T^{3} + \cdots + 11\!\cdots\!72 Copy content Toggle raw display
5959 T3++19 ⁣ ⁣72 T^{3} + \cdots + 19\!\cdots\!72 Copy content Toggle raw display
6161 T3+23 ⁣ ⁣88 T^{3} + \cdots - 23\!\cdots\!88 Copy content Toggle raw display
6767 T3++84 ⁣ ⁣80 T^{3} + \cdots + 84\!\cdots\!80 Copy content Toggle raw display
7171 T3+54 ⁣ ⁣20 T^{3} + \cdots - 54\!\cdots\!20 Copy content Toggle raw display
7373 T3++16 ⁣ ⁣82 T^{3} + \cdots + 16\!\cdots\!82 Copy content Toggle raw display
7979 T3+43 ⁣ ⁣12 T^{3} + \cdots - 43\!\cdots\!12 Copy content Toggle raw display
8383 T3+77 ⁣ ⁣88 T^{3} + \cdots - 77\!\cdots\!88 Copy content Toggle raw display
8989 T3++49 ⁣ ⁣96 T^{3} + \cdots + 49\!\cdots\!96 Copy content Toggle raw display
9797 T3++59 ⁣ ⁣72 T^{3} + \cdots + 59\!\cdots\!72 Copy content Toggle raw display
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