Properties

Label 1323.2.a.be
Level 13231323
Weight 22
Character orbit 1323.a
Self dual yes
Analytic conductor 10.56410.564
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,2,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1323.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1323=3372 1323 = 3^{3} \cdot 7^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1323.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: 10.564208187410.5642081874
Analytic rank: 00
Dimension: 44
Coefficient field: Q(2,5)\Q(\sqrt{2}, \sqrt{5})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x46x2+4 x^{4} - 6x^{2} + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 2 2
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,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2+(β3+1)q4+(β3+2)q5+2β2q8+(2β2+3β1)q10+(β2β1)q11β1q13+5q17+(β2+3β1)q19++(3β2β1)q97+O(q100) q + \beta_1 q^{2} + (\beta_{3} + 1) q^{4} + (\beta_{3} + 2) q^{5} + 2 \beta_{2} q^{8} + (2 \beta_{2} + 3 \beta_1) q^{10} + ( - \beta_{2} - \beta_1) q^{11} - \beta_1 q^{13} + 5 q^{17} + ( - \beta_{2} + 3 \beta_1) q^{19}+ \cdots + (3 \beta_{2} - \beta_1) q^{97}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+4q4+8q5+20q17+28q2016q22+16q2512q26+8q37+32q38+8q41+12q46+12q4748q58+20q594q6216q6436q67+20q68++20q89+O(q100) 4 q + 4 q^{4} + 8 q^{5} + 20 q^{17} + 28 q^{20} - 16 q^{22} + 16 q^{25} - 12 q^{26} + 8 q^{37} + 32 q^{38} + 8 q^{41} + 12 q^{46} + 12 q^{47} - 48 q^{58} + 20 q^{59} - 4 q^{62} - 16 q^{64} - 36 q^{67} + 20 q^{68}+ \cdots + 20 q^{89}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x46x2+4 x^{4} - 6x^{2} + 4 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν34ν)/2 ( \nu^{3} - 4\nu ) / 2 Copy content Toggle raw display
β3\beta_{3}== ν23 \nu^{2} - 3 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β3+3 \beta_{3} + 3 Copy content Toggle raw display
ν3\nu^{3}== 2β2+4β1 2\beta_{2} + 4\beta_1 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
−2.28825
−0.874032
0.874032
2.28825
−2.28825 0 3.23607 4.23607 0 0 −2.82843 0 −9.69316
1.2 −0.874032 0 −1.23607 −0.236068 0 0 2.82843 0 0.206331
1.3 0.874032 0 −1.23607 −0.236068 0 0 −2.82843 0 −0.206331
1.4 2.28825 0 3.23607 4.23607 0 0 2.82843 0 9.69316
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 1 -1
77 +1 +1

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.a.be yes 4
3.b odd 2 1 1323.2.a.bb 4
7.b odd 2 1 1323.2.a.bb 4
21.c even 2 1 inner 1323.2.a.be yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.2.a.bb 4 3.b odd 2 1
1323.2.a.bb 4 7.b odd 2 1
1323.2.a.be yes 4 1.a even 1 1 trivial
1323.2.a.be yes 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(1323))S_{2}^{\mathrm{new}}(\Gamma_0(1323)):

T246T22+4 T_{2}^{4} - 6T_{2}^{2} + 4 Copy content Toggle raw display
T524T51 T_{5}^{2} - 4T_{5} - 1 Copy content Toggle raw display
T1346T132+4 T_{13}^{4} - 6T_{13}^{2} + 4 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T46T2+4 T^{4} - 6T^{2} + 4 Copy content Toggle raw display
33 T4 T^{4} Copy content Toggle raw display
55 (T24T1)2 (T^{2} - 4 T - 1)^{2} Copy content Toggle raw display
77 T4 T^{4} Copy content Toggle raw display
1111 T414T2+4 T^{4} - 14T^{2} + 4 Copy content Toggle raw display
1313 T46T2+4 T^{4} - 6T^{2} + 4 Copy content Toggle raw display
1717 (T5)4 (T - 5)^{4} Copy content Toggle raw display
1919 T446T2+484 T^{4} - 46T^{2} + 484 Copy content Toggle raw display
2323 T436T2+4 T^{4} - 36T^{2} + 4 Copy content Toggle raw display
2929 T4126T2+3844 T^{4} - 126T^{2} + 3844 Copy content Toggle raw display
3131 T454T2+484 T^{4} - 54T^{2} + 484 Copy content Toggle raw display
3737 (T24T41)2 (T^{2} - 4 T - 41)^{2} Copy content Toggle raw display
4141 (T24T41)2 (T^{2} - 4 T - 41)^{2} Copy content Toggle raw display
4343 (T25)2 (T^{2} - 5)^{2} Copy content Toggle raw display
4747 (T26T11)2 (T^{2} - 6 T - 11)^{2} Copy content Toggle raw display
5353 (T210)2 (T^{2} - 10)^{2} Copy content Toggle raw display
5959 (T210T55)2 (T^{2} - 10 T - 55)^{2} Copy content Toggle raw display
6161 T414T2+4 T^{4} - 14T^{2} + 4 Copy content Toggle raw display
6767 (T2+18T+76)2 (T^{2} + 18 T + 76)^{2} Copy content Toggle raw display
7171 T4216T2+7744 T^{4} - 216T^{2} + 7744 Copy content Toggle raw display
7373 T4180T2+100 T^{4} - 180T^{2} + 100 Copy content Toggle raw display
7979 (T2+14T+29)2 (T^{2} + 14 T + 29)^{2} Copy content Toggle raw display
8383 (T24T1)2 (T^{2} - 4 T - 1)^{2} Copy content Toggle raw display
8989 (T210T20)2 (T^{2} - 10 T - 20)^{2} Copy content Toggle raw display
9797 T430T2+100 T^{4} - 30T^{2} + 100 Copy content Toggle raw display
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