Properties

Label 1785.2.a.s
Level 17851785
Weight 22
Character orbit 1785.a
Self dual yes
Analytic conductor 14.25314.253
Analytic rank 00
Dimension 22
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1785,2,Mod(1,1785)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1785, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1785.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1785=35717 1785 = 3 \cdot 5 \cdot 7 \cdot 17
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1785.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 14.253296760814.2532967608
Analytic rank: 00
Dimension: 22
Coefficient field: Q(17)\Q(\sqrt{17})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2x4 x^{2} - x - 4 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of β=12(1+17)\beta = \frac{1}{2}(1 + \sqrt{17}). We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+βq2+q3+(β+2)q4q5+βq6q7+(β+4)q8+q9βq10+(2β2)q11+(β+2)q12+(β+1)q13βq14++(2β2)q99+O(q100) q + \beta q^{2} + q^{3} + (\beta + 2) q^{4} - q^{5} + \beta q^{6} - q^{7} + (\beta + 4) q^{8} + q^{9} - \beta q^{10} + (2 \beta - 2) q^{11} + (\beta + 2) q^{12} + (\beta + 1) q^{13} - \beta q^{14} + \cdots + (2 \beta - 2) q^{99} +O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+q2+2q3+5q42q5+q62q7+9q8+2q9q102q11+5q12+3q13q142q15+3q162q17+q186q195q202q21+2q99+O(q100) 2 q + q^{2} + 2 q^{3} + 5 q^{4} - 2 q^{5} + q^{6} - 2 q^{7} + 9 q^{8} + 2 q^{9} - q^{10} - 2 q^{11} + 5 q^{12} + 3 q^{13} - q^{14} - 2 q^{15} + 3 q^{16} - 2 q^{17} + q^{18} - 6 q^{19} - 5 q^{20} - 2 q^{21}+ \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.

comment: embeddings in the coefficient field
 
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
−1.56155
2.56155
−1.56155 1.00000 0.438447 −1.00000 −1.56155 −1.00000 2.43845 1.00000 1.56155
1.2 2.56155 1.00000 4.56155 −1.00000 2.56155 −1.00000 6.56155 1.00000 −2.56155
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 1 -1
55 +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 1785.2.a.s 2
3.b odd 2 1 5355.2.a.v 2
5.b even 2 1 8925.2.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1785.2.a.s 2 1.a even 1 1 trivial
5355.2.a.v 2 3.b odd 2 1
8925.2.a.be 2 5.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(1785))S_{2}^{\mathrm{new}}(\Gamma_0(1785)):

T22T24 T_{2}^{2} - T_{2} - 4 Copy content Toggle raw display
T112+2T1116 T_{11}^{2} + 2T_{11} - 16 Copy content Toggle raw display
T1323T132 T_{13}^{2} - 3T_{13} - 2 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2T4 T^{2} - T - 4 Copy content Toggle raw display
33 (T1)2 (T - 1)^{2} Copy content Toggle raw display
55 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
77 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
1111 T2+2T16 T^{2} + 2T - 16 Copy content Toggle raw display
1313 T23T2 T^{2} - 3T - 2 Copy content Toggle raw display
1717 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
1919 T2+6T8 T^{2} + 6T - 8 Copy content Toggle raw display
2323 T25T32 T^{2} - 5T - 32 Copy content Toggle raw display
2929 T26T8 T^{2} - 6T - 8 Copy content Toggle raw display
3131 T29T+16 T^{2} - 9T + 16 Copy content Toggle raw display
3737 T2+3T2 T^{2} + 3T - 2 Copy content Toggle raw display
4141 T23T2 T^{2} - 3T - 2 Copy content Toggle raw display
4343 T2 T^{2} Copy content Toggle raw display
4747 T2+3T104 T^{2} + 3T - 104 Copy content Toggle raw display
5353 T2+14T+32 T^{2} + 14T + 32 Copy content Toggle raw display
5959 T24T64 T^{2} - 4T - 64 Copy content Toggle raw display
6161 T2+9T18 T^{2} + 9T - 18 Copy content Toggle raw display
6767 T24T64 T^{2} - 4T - 64 Copy content Toggle raw display
7171 T2+6T8 T^{2} + 6T - 8 Copy content Toggle raw display
7373 T214T+32 T^{2} - 14T + 32 Copy content Toggle raw display
7979 T2+2T16 T^{2} + 2T - 16 Copy content Toggle raw display
8383 T2+11T76 T^{2} + 11T - 76 Copy content Toggle raw display
8989 (T2)2 (T - 2)^{2} Copy content Toggle raw display
9797 (T6)2 (T - 6)^{2} Copy content Toggle raw display
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