Properties

Label 1045.2.a.f
Level 10451045
Weight 22
Character orbit 1045.a
Self dual yes
Analytic conductor 8.3448.344
Analytic rank 11
Dimension 66
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] = [1045,2,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1045=51119 1045 = 5 \cdot 11 \cdot 19
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1045.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: 8.344367011228.34436701122
Analytic rank: 11
Dimension: 66
Coefficient field: 6.6.7281497.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x62x55x4+7x3+6x22x1 x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 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 a basis 1,β1,,β51,\beta_1,\ldots,\beta_{5} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == qβ4q2+β3q3+(β5+β4)q4q5+β2q6+(β3β1+1)q7+(β5β4β11)q8+(β5β3+β1)q9++(β5+β3β1)q99+O(q100) q - \beta_{4} q^{2} + \beta_{3} q^{3} + (\beta_{5} + \beta_{4}) q^{4} - q^{5} + \beta_{2} q^{6} + ( - \beta_{3} - \beta_1 + 1) q^{7} + ( - \beta_{5} - \beta_{4} - \beta_1 - 1) q^{8} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{9}+ \cdots + (\beta_{5} + \beta_{3} - \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q2q2q3+4q46q5+5q712q8+q9+2q106q11+q125q138q14+q15+4q16+q17+6q186q194q2021q21+q99+O(q100) 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9} + 2 q^{10} - 6 q^{11} + q^{12} - 5 q^{13} - 8 q^{14} + q^{15} + 4 q^{16} + q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20} - 21 q^{21}+ \cdots - q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x62x55x4+7x3+6x22x1 x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2ν2 \nu^{2} - \nu - 2 Copy content Toggle raw display
β3\beta_{3}== ν52ν44ν3+6ν2+2ν1 \nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 2\nu - 1 Copy content Toggle raw display
β4\beta_{4}== ν52ν45ν3+7ν2+5ν1 \nu^{5} - 2\nu^{4} - 5\nu^{3} + 7\nu^{2} + 5\nu - 1 Copy content Toggle raw display
β5\beta_{5}== ν5+3ν4+3ν311ν2+4 -\nu^{5} + 3\nu^{4} + 3\nu^{3} - 11\nu^{2} + 4 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+β1+2 \beta_{2} + \beta _1 + 2 Copy content Toggle raw display
ν3\nu^{3}== β4+β3+β2+4β1+2 -\beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 2 Copy content Toggle raw display
ν4\nu^{4}== β5β4+2β3+6β2+7β1+9 \beta_{5} - \beta_{4} + 2\beta_{3} + 6\beta_{2} + 7\beta _1 + 9 Copy content Toggle raw display
ν5\nu^{5}== 2β56β4+9β3+10β2+22β1+15 2\beta_{5} - 6\beta_{4} + 9\beta_{3} + 10\beta_{2} + 22\beta _1 + 15 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
0.497517
−1.79049
−0.748369
1.77015
2.59744
−0.326248
−2.51246 0.895533 4.31247 −1.00000 −2.24999 −0.393051 −5.80998 −2.19802 2.51246
1.2 −2.23198 −1.34246 2.98176 −1.00000 2.99635 4.13295 −2.19127 −1.19780 2.23198
1.3 −0.412130 1.67805 −1.83015 −1.00000 −0.691575 0.0703171 1.57852 −0.184142 0.412130
1.4 0.205229 −3.10246 −1.95788 −1.00000 −0.636714 2.33231 −0.812271 6.62527 −0.205229
1.5 1.21244 1.77266 −0.529980 −1.00000 2.14925 −3.37010 −3.06746 0.142317 −1.21244
1.6 1.73890 −0.901323 1.02379 −1.00000 −1.56731 2.22757 −1.69754 −2.18762 −1.73890
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
55 +1 +1
1111 +1 +1
1919 +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 1045.2.a.f 6
3.b odd 2 1 9405.2.a.z 6
5.b even 2 1 5225.2.a.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.f 6 1.a even 1 1 trivial
5225.2.a.l 6 5.b even 2 1
9405.2.a.z 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T26+2T256T248T23+11T22+3T21 T_{2}^{6} + 2T_{2}^{5} - 6T_{2}^{4} - 8T_{2}^{3} + 11T_{2}^{2} + 3T_{2} - 1 acting on S2new(Γ0(1045))S_{2}^{\mathrm{new}}(\Gamma_0(1045)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T6+2T5+1 T^{6} + 2 T^{5} + \cdots - 1 Copy content Toggle raw display
33 T6+T5+10 T^{6} + T^{5} + \cdots - 10 Copy content Toggle raw display
55 (T+1)6 (T + 1)^{6} Copy content Toggle raw display
77 T65T5++2 T^{6} - 5 T^{5} + \cdots + 2 Copy content Toggle raw display
1111 (T+1)6 (T + 1)^{6} Copy content Toggle raw display
1313 T6+5T5++2 T^{6} + 5 T^{5} + \cdots + 2 Copy content Toggle raw display
1717 T6T5+1360 T^{6} - T^{5} + \cdots - 1360 Copy content Toggle raw display
1919 (T+1)6 (T + 1)^{6} Copy content Toggle raw display
2323 T64T5+8912 T^{6} - 4 T^{5} + \cdots - 8912 Copy content Toggle raw display
2929 T6+9T5++3074 T^{6} + 9 T^{5} + \cdots + 3074 Copy content Toggle raw display
3131 T6+21T5+20 T^{6} + 21 T^{5} + \cdots - 20 Copy content Toggle raw display
3737 T6+3T5+592 T^{6} + 3 T^{5} + \cdots - 592 Copy content Toggle raw display
4141 T6+23T5+4210 T^{6} + 23 T^{5} + \cdots - 4210 Copy content Toggle raw display
4343 T67T5+326 T^{6} - 7 T^{5} + \cdots - 326 Copy content Toggle raw display
4747 T6+18T5+620 T^{6} + 18 T^{5} + \cdots - 620 Copy content Toggle raw display
5353 T6+17T5+4084 T^{6} + 17 T^{5} + \cdots - 4084 Copy content Toggle raw display
5959 T6+29T5+54968 T^{6} + 29 T^{5} + \cdots - 54968 Copy content Toggle raw display
6161 T617T5++2294 T^{6} - 17 T^{5} + \cdots + 2294 Copy content Toggle raw display
6767 T68T5+73006 T^{6} - 8 T^{5} + \cdots - 73006 Copy content Toggle raw display
7171 T6+12T5+5912 T^{6} + 12 T^{5} + \cdots - 5912 Copy content Toggle raw display
7373 T62T5+4000 T^{6} - 2 T^{5} + \cdots - 4000 Copy content Toggle raw display
7979 T63T5++28264 T^{6} - 3 T^{5} + \cdots + 28264 Copy content Toggle raw display
8383 T6+11T5+82582 T^{6} + 11 T^{5} + \cdots - 82582 Copy content Toggle raw display
8989 T6+22T5++3286 T^{6} + 22 T^{5} + \cdots + 3286 Copy content Toggle raw display
9797 T6+2T5+967268 T^{6} + 2 T^{5} + \cdots - 967268 Copy content Toggle raw display
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