Properties

Label 5712.2.a.bq
Level 57125712
Weight 22
Character orbit 5712.a
Self dual yes
Analytic conductor 45.61145.611
Analytic rank 00
Dimension 22
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] = [5712,2,Mod(1,5712)] 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("5712.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5712, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: N N == 5712=243717 5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17
Weight: k k == 2 2
Character orbit: [χ][\chi] == 5712.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,-2,0,2,0,2,0,-2,0,-6,0,-2,0,-2,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 45.610549634645.6105496346
Analytic rank: 00
Dimension: 22
Coefficient field: Q(2)\Q(\sqrt{2})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x22 x^{2} - 2 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 357)
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 β=2\beta = \sqrt{2}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+q3+(β1)q5+q7+q9q11+(β3)q13+(β1)q15q17+(β+5)q19+q21+(2β+1)q23+(2β2)q25+q27+q99+O(q100) q + q^{3} + (\beta - 1) q^{5} + q^{7} + q^{9} - q^{11} + (\beta - 3) q^{13} + (\beta - 1) q^{15} - q^{17} + ( - \beta + 5) q^{19} + q^{21} + (2 \beta + 1) q^{23} + ( - 2 \beta - 2) q^{25} + q^{27}+ \cdots - q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+2q32q5+2q7+2q92q116q132q152q17+10q19+2q21+2q234q25+2q27+8q292q332q358q376q39+6q41+2q99+O(q100) 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 6 q^{13} - 2 q^{15} - 2 q^{17} + 10 q^{19} + 2 q^{21} + 2 q^{23} - 4 q^{25} + 2 q^{27} + 8 q^{29} - 2 q^{33} - 2 q^{35} - 8 q^{37} - 6 q^{39} + 6 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) 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
−1.41421
1.41421
0 1.00000 0 −2.41421 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 0.414214 0 1.00000 0 1.00000 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 1 -1
33 1 -1
77 1 -1
1717 +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 5712.2.a.bq 2
4.b odd 2 1 357.2.a.f 2
12.b even 2 1 1071.2.a.e 2
20.d odd 2 1 8925.2.a.bg 2
28.d even 2 1 2499.2.a.r 2
68.d odd 2 1 6069.2.a.g 2
84.h odd 2 1 7497.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
357.2.a.f 2 4.b odd 2 1
1071.2.a.e 2 12.b even 2 1
2499.2.a.r 2 28.d even 2 1
5712.2.a.bq 2 1.a even 1 1 trivial
6069.2.a.g 2 68.d odd 2 1
7497.2.a.s 2 84.h odd 2 1
8925.2.a.bg 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(5712))S_{2}^{\mathrm{new}}(\Gamma_0(5712)):

T52+2T51 T_{5}^{2} + 2T_{5} - 1 Copy content Toggle raw display
T11+1 T_{11} + 1 Copy content Toggle raw display
T132+6T13+7 T_{13}^{2} + 6T_{13} + 7 Copy content Toggle raw display
T19210T19+23 T_{19}^{2} - 10T_{19} + 23 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2 T^{2} Copy content Toggle raw display
33 (T1)2 (T - 1)^{2} Copy content Toggle raw display
55 T2+2T1 T^{2} + 2T - 1 Copy content Toggle raw display
77 (T1)2 (T - 1)^{2} Copy content Toggle raw display
1111 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
1313 T2+6T+7 T^{2} + 6T + 7 Copy content Toggle raw display
1717 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
1919 T210T+23 T^{2} - 10T + 23 Copy content Toggle raw display
2323 T22T7 T^{2} - 2T - 7 Copy content Toggle raw display
2929 T28T+8 T^{2} - 8T + 8 Copy content Toggle raw display
3131 T298 T^{2} - 98 Copy content Toggle raw display
3737 T2+8T2 T^{2} + 8T - 2 Copy content Toggle raw display
4141 T26T41 T^{2} - 6T - 41 Copy content Toggle raw display
4343 T26T+1 T^{2} - 6T + 1 Copy content Toggle raw display
4747 T28T+14 T^{2} - 8T + 14 Copy content Toggle raw display
5353 T24T14 T^{2} - 4T - 14 Copy content Toggle raw display
5959 T216T+46 T^{2} - 16T + 46 Copy content Toggle raw display
6161 T2+4T158 T^{2} + 4T - 158 Copy content Toggle raw display
6767 T24T28 T^{2} - 4T - 28 Copy content Toggle raw display
7171 (T+2)2 (T + 2)^{2} Copy content Toggle raw display
7373 T2+12T+28 T^{2} + 12T + 28 Copy content Toggle raw display
7979 T24T158 T^{2} - 4T - 158 Copy content Toggle raw display
8383 (T6)2 (T - 6)^{2} Copy content Toggle raw display
8989 T216T+56 T^{2} - 16T + 56 Copy content Toggle raw display
9797 T2+8T16 T^{2} + 8T - 16 Copy content Toggle raw display
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