Properties

Label 7605.2.a.y
Level 76057605
Weight 22
Character orbit 7605.a
Self dual yes
Analytic conductor 60.72660.726
Analytic rank 11
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] = [7605,2,Mod(1,7605)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) 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("7605.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 7605=325132 7605 = 3^{2} \cdot 5 \cdot 13^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 7605.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,4,-2,0,-2,-12,0,2,0] 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: yes
Analytic conductor: 60.726230737260.7262307372
Analytic rank: 11
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: x23 x^{2} - 3 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 195)
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=3\beta = \sqrt{3}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β1)q2+(2β+2)q4q5+(2β1)q7+(2β6)q8+(β+1)q102βq11+(β5)q14+(4β+8)q16+(β+5)q17++(2β+6)q98+O(q100) q + (\beta - 1) q^{2} + ( - 2 \beta + 2) q^{4} - q^{5} + ( - 2 \beta - 1) q^{7} + (2 \beta - 6) q^{8} + ( - \beta + 1) q^{10} - 2 \beta q^{11} + (\beta - 5) q^{14} + ( - 4 \beta + 8) q^{16} + (\beta + 5) q^{17} + \cdots + (2 \beta + 6) q^{98} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q2q2+4q42q52q712q8+2q1010q14+16q16+10q17+4q194q2012q22+8q23+2q25+20q28+2q294q3116q32++12q98+O(q100) 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 2 q^{7} - 12 q^{8} + 2 q^{10} - 10 q^{14} + 16 q^{16} + 10 q^{17} + 4 q^{19} - 4 q^{20} - 12 q^{22} + 8 q^{23} + 2 q^{25} + 20 q^{28} + 2 q^{29} - 4 q^{31} - 16 q^{32}+ \cdots + 12 q^{98}+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
−1.73205
1.73205
−2.73205 0 5.46410 −1.00000 0 2.46410 −9.46410 0 2.73205
1.2 0.732051 0 −1.46410 −1.00000 0 −4.46410 −2.53590 0 −0.732051
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 1 -1
55 +1 +1
1313 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 7605.2.a.y 2
3.b odd 2 1 2535.2.a.s 2
13.b even 2 1 7605.2.a.bk 2
13.f odd 12 2 585.2.bu.a 4
39.d odd 2 1 2535.2.a.n 2
39.k even 12 2 195.2.bb.a 4
195.bc odd 12 2 975.2.w.f 4
195.bh even 12 2 975.2.bc.h 4
195.bn odd 12 2 975.2.w.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bb.a 4 39.k even 12 2
585.2.bu.a 4 13.f odd 12 2
975.2.w.a 4 195.bn odd 12 2
975.2.w.f 4 195.bc odd 12 2
975.2.bc.h 4 195.bh even 12 2
2535.2.a.n 2 39.d odd 2 1
2535.2.a.s 2 3.b odd 2 1
7605.2.a.y 2 1.a even 1 1 trivial
7605.2.a.bk 2 13.b even 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(7605))S_{2}^{\mathrm{new}}(\Gamma_0(7605)):

T22+2T22 T_{2}^{2} + 2T_{2} - 2 Copy content Toggle raw display
T72+2T711 T_{7}^{2} + 2T_{7} - 11 Copy content Toggle raw display
T11212 T_{11}^{2} - 12 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+2T2 T^{2} + 2T - 2 Copy content Toggle raw display
33 T2 T^{2} Copy content Toggle raw display
55 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
77 T2+2T11 T^{2} + 2T - 11 Copy content Toggle raw display
1111 T212 T^{2} - 12 Copy content Toggle raw display
1313 T2 T^{2} Copy content Toggle raw display
1717 T210T+22 T^{2} - 10T + 22 Copy content Toggle raw display
1919 T24T8 T^{2} - 4T - 8 Copy content Toggle raw display
2323 T28T+4 T^{2} - 8T + 4 Copy content Toggle raw display
2929 T22T2 T^{2} - 2T - 2 Copy content Toggle raw display
3131 T2+4T23 T^{2} + 4T - 23 Copy content Toggle raw display
3737 (T+4)2 (T + 4)^{2} Copy content Toggle raw display
4141 T2+14T+46 T^{2} + 14T + 46 Copy content Toggle raw display
4343 T2+4T+1 T^{2} + 4T + 1 Copy content Toggle raw display
4747 T210T2 T^{2} - 10T - 2 Copy content Toggle raw display
5353 T248 T^{2} - 48 Copy content Toggle raw display
5959 T2+18T+78 T^{2} + 18T + 78 Copy content Toggle raw display
6161 T22T11 T^{2} - 2T - 11 Copy content Toggle raw display
6767 T2+18T+69 T^{2} + 18T + 69 Copy content Toggle raw display
7171 T2+22T+118 T^{2} + 22T + 118 Copy content Toggle raw display
7373 T210T83 T^{2} - 10T - 83 Copy content Toggle raw display
7979 T2+10T23 T^{2} + 10T - 23 Copy content Toggle raw display
8383 T2+12T+24 T^{2} + 12T + 24 Copy content Toggle raw display
8989 T2+6T+6 T^{2} + 6T + 6 Copy content Toggle raw display
9797 T226T+157 T^{2} - 26T + 157 Copy content Toggle raw display
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