Properties

Label 45.22.a.g
Level 4545
Weight 2222
Character orbit 45.a
Self dual yes
Analytic conductor 125.765125.765
Analytic rank 00
Dimension 44
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] = [45,22,Mod(1,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: N N == 45=325 45 = 3^{2} \cdot 5
Weight: k k == 22 22
Character orbit: [χ][\chi] == 45.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: 125.764804929125.764804929
Analytic rank: 00
Dimension: 44
Coefficient field: Q[x]/(x4)\mathbb{Q}[x]/(x^{4} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x33512166x2+363520480x+2321089280000 x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 25345272 2^{5}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2}
Twist minimal: no (minimal twist has level 15)
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+(β1+224)q2+(β2+294β1290854)q49765625q5+(11β3908β2+58725955)q7+(22β339β2+18724672)q8++(9641431331840β3+49 ⁣ ⁣64)q98+O(q100) q + (\beta_1 + 224) q^{2} + (\beta_{2} + 294 \beta_1 - 290854) q^{4} - 9765625 q^{5} + ( - 11 \beta_{3} - 908 \beta_{2} + \cdots - 58725955) q^{7} + (22 \beta_{3} - 39 \beta_{2} + \cdots - 18724672) q^{8}+ \cdots + ( - 9641431331840 \beta_{3} + \cdots - 49\!\cdots\!64) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+897q21163123q439062500q5234577504q776855629q88759765625q1031491830256q1127017977768q13+2233308126672q1411324681311775q16+2946095028888q17+19 ⁣ ⁣63q98+O(q100) 4 q + 897 q^{2} - 1163123 q^{4} - 39062500 q^{5} - 234577504 q^{7} - 76855629 q^{8} - 8759765625 q^{10} - 31491830256 q^{11} - 27017977768 q^{13} + 2233308126672 q^{14} - 11324681311775 q^{16} + 2946095028888 q^{17}+ \cdots - 19\!\cdots\!63 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4x33512166x2+363520480x+2321089280000 x^{4} - x^{3} - 3512166x^{2} + 363520480x + 2321089280000 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2+154ν1756122 \nu^{2} + 154\nu - 1756122 Copy content Toggle raw display
β3\beta_{3}== (ν3+711ν22080768ν978048758)/22 ( \nu^{3} + 711\nu^{2} - 2080768\nu - 978048758 ) / 22 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2154β1+1756122 \beta_{2} - 154\beta _1 + 1756122 Copy content Toggle raw display
ν3\nu^{3}== 22β3711β2+2190262β1270553984 22\beta_{3} - 711\beta_{2} + 2190262\beta _1 - 270553984 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
−1711.97
−849.272
1069.36
1492.88
−1487.97 0 116903. −9.76562e6 0 −1.26833e9 2.94655e9 0 1.45310e10
1.2 −625.272 0 −1.70619e6 −9.76562e6 0 3.78633e8 2.37812e9 0 6.10617e9
1.3 1293.36 0 −424379. −9.76562e6 0 1.27960e9 −3.26124e9 0 −1.26304e10
1.4 1716.88 0 850541. −9.76562e6 0 −6.24477e8 −2.14029e9 0 −1.67665e10
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 1 -1
55 +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 45.22.a.g 4
3.b odd 2 1 15.22.a.e 4
15.d odd 2 1 75.22.a.h 4
15.e even 4 2 75.22.b.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.a.e 4 3.b odd 2 1
45.22.a.g 4 1.a even 1 1 trivial
75.22.a.h 4 15.d odd 2 1
75.22.b.h 8 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T24897T233210438T22+1891862624T2+2065963121664 T_{2}^{4} - 897T_{2}^{3} - 3210438T_{2}^{2} + 1891862624T_{2} + 2065963121664 acting on S22new(Γ0(45))S_{22}^{\mathrm{new}}(\Gamma_0(45)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4++2065963121664 T^{4} + \cdots + 2065963121664 Copy content Toggle raw display
33 T4 T^{4} Copy content Toggle raw display
55 (T+9765625)4 (T + 9765625)^{4} Copy content Toggle raw display
77 T4++38 ⁣ ⁣00 T^{4} + \cdots + 38\!\cdots\!00 Copy content Toggle raw display
1111 T4++28 ⁣ ⁣64 T^{4} + \cdots + 28\!\cdots\!64 Copy content Toggle raw display
1313 T4++26 ⁣ ⁣84 T^{4} + \cdots + 26\!\cdots\!84 Copy content Toggle raw display
1717 T4++31 ⁣ ⁣56 T^{4} + \cdots + 31\!\cdots\!56 Copy content Toggle raw display
1919 T4++34 ⁣ ⁣00 T^{4} + \cdots + 34\!\cdots\!00 Copy content Toggle raw display
2323 T4+97 ⁣ ⁣00 T^{4} + \cdots - 97\!\cdots\!00 Copy content Toggle raw display
2929 T4+51 ⁣ ⁣00 T^{4} + \cdots - 51\!\cdots\!00 Copy content Toggle raw display
3131 T4++15 ⁣ ⁣00 T^{4} + \cdots + 15\!\cdots\!00 Copy content Toggle raw display
3737 T4++50 ⁣ ⁣00 T^{4} + \cdots + 50\!\cdots\!00 Copy content Toggle raw display
4141 T4+70 ⁣ ⁣00 T^{4} + \cdots - 70\!\cdots\!00 Copy content Toggle raw display
4343 T4+16 ⁣ ⁣04 T^{4} + \cdots - 16\!\cdots\!04 Copy content Toggle raw display
4747 T4+21 ⁣ ⁣44 T^{4} + \cdots - 21\!\cdots\!44 Copy content Toggle raw display
5353 T4++34 ⁣ ⁣00 T^{4} + \cdots + 34\!\cdots\!00 Copy content Toggle raw display
5959 T4++97 ⁣ ⁣00 T^{4} + \cdots + 97\!\cdots\!00 Copy content Toggle raw display
6161 T4++27 ⁣ ⁣76 T^{4} + \cdots + 27\!\cdots\!76 Copy content Toggle raw display
6767 T4++28 ⁣ ⁣56 T^{4} + \cdots + 28\!\cdots\!56 Copy content Toggle raw display
7171 T4+17 ⁣ ⁣44 T^{4} + \cdots - 17\!\cdots\!44 Copy content Toggle raw display
7373 T4+99 ⁣ ⁣00 T^{4} + \cdots - 99\!\cdots\!00 Copy content Toggle raw display
7979 T4+12 ⁣ ⁣00 T^{4} + \cdots - 12\!\cdots\!00 Copy content Toggle raw display
8383 T4+79 ⁣ ⁣56 T^{4} + \cdots - 79\!\cdots\!56 Copy content Toggle raw display
8989 T4++85 ⁣ ⁣00 T^{4} + \cdots + 85\!\cdots\!00 Copy content Toggle raw display
9797 T4+16 ⁣ ⁣24 T^{4} + \cdots - 16\!\cdots\!24 Copy content Toggle raw display
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