Properties

Label 2.44.a.b
Level 22
Weight 4444
Character orbit 2.a
Self dual yes
Analytic conductor 23.42223.422
Analytic rank 11
Dimension 22
CM no
Inner twists 11

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2,44,Mod(1,2)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 44, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2.1"); S:= CuspForms(chi, 44); N := Newforms(S);
 
Level: N N == 2 2
Weight: k k == 44 44
Character orbit: [χ][\chi] == 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,4194304] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 23.422079069123.4220790691
Analytic rank: 11
Dimension: 22
Coefficient field: Q[x]/(x2)\mathbb{Q}[x]/(x^{2} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2x397496384250 x^{2} - x - 397496384250 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 28335 2^{8}\cdot 3^{3}\cdot 5
Twist minimal: yes
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=172801589985537001\beta = 17280\sqrt{1589985537001}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+2097152q2+(β11170817028)q3+4398046511104q4+(19308β23660329199010)q5+(2097152β23 ⁣ ⁣56)q6+(87838422β11 ⁣ ⁣64)q7+92 ⁣ ⁣08q8++(62 ⁣ ⁣95β11 ⁣ ⁣56)q99+O(q100) q + 2097152 q^{2} + ( - \beta - 11170817028) q^{3} + 4398046511104 q^{4} + (19308 \beta - 23660329199010) q^{5} + ( - 2097152 \beta - 23\!\cdots\!56) q^{6} + (87838422 \beta - 11\!\cdots\!64) q^{7} + 92\!\cdots\!08 q^{8}+ \cdots + ( - 62\!\cdots\!95 \beta - 11\!\cdots\!56) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q+4194304q222341634056q3+8796093022208q447320658398020q546 ⁣ ⁣12q622 ⁣ ⁣28q7+18 ⁣ ⁣16q8+54 ⁣ ⁣14q999 ⁣ ⁣40q1041 ⁣ ⁣16q11+23 ⁣ ⁣12q99+O(q100) 2 q + 4194304 q^{2} - 22341634056 q^{3} + 8796093022208 q^{4} - 47320658398020 q^{5} - 46\!\cdots\!12 q^{6} - 22\!\cdots\!28 q^{7} + 18\!\cdots\!16 q^{8} + 54\!\cdots\!14 q^{9} - 99\!\cdots\!40 q^{10} - 41\!\cdots\!16 q^{11}+ \cdots - 23\!\cdots\!12 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.

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
630474.
−630473.
2.09715e6 −3.29600e10 4.39805e12 3.97045e14 −6.91221e16 1.80269e18 9.22337e18 7.58103e20 8.32663e20
1.2 2.09715e6 1.06183e10 4.39805e12 −4.44365e14 2.22683e16 −2.02515e18 9.22337e18 −2.15508e20 −9.31902e20
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 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 2.44.a.b 2
3.b odd 2 1 18.44.a.c 2
4.b odd 2 1 16.44.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.44.a.b 2 1.a even 1 1 trivial
16.44.a.b 2 4.b odd 2 1
18.44.a.c 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T32+22341634056T3349979984298584645616 T_{3}^{2} + 22341634056T_{3} - 349979984298584645616 acting on S44new(Γ0(2))S_{44}^{\mathrm{new}}(\Gamma_0(2)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2097152)2 (T - 2097152)^{2} Copy content Toggle raw display
33 T2+34 ⁣ ⁣16 T^{2} + \cdots - 34\!\cdots\!16 Copy content Toggle raw display
55 T2+17 ⁣ ⁣00 T^{2} + \cdots - 17\!\cdots\!00 Copy content Toggle raw display
77 T2+36 ⁣ ⁣04 T^{2} + \cdots - 36\!\cdots\!04 Copy content Toggle raw display
1111 T2++28 ⁣ ⁣64 T^{2} + \cdots + 28\!\cdots\!64 Copy content Toggle raw display
1313 T2++56 ⁣ ⁣04 T^{2} + \cdots + 56\!\cdots\!04 Copy content Toggle raw display
1717 T2+90 ⁣ ⁣24 T^{2} + \cdots - 90\!\cdots\!24 Copy content Toggle raw display
1919 T2++37 ⁣ ⁣00 T^{2} + \cdots + 37\!\cdots\!00 Copy content Toggle raw display
2323 T2+50 ⁣ ⁣56 T^{2} + \cdots - 50\!\cdots\!56 Copy content Toggle raw display
2929 T2++10 ⁣ ⁣00 T^{2} + \cdots + 10\!\cdots\!00 Copy content Toggle raw display
3131 T2+14 ⁣ ⁣56 T^{2} + \cdots - 14\!\cdots\!56 Copy content Toggle raw display
3737 T2+40 ⁣ ⁣24 T^{2} + \cdots - 40\!\cdots\!24 Copy content Toggle raw display
4141 T2++35 ⁣ ⁣04 T^{2} + \cdots + 35\!\cdots\!04 Copy content Toggle raw display
4343 T2++18 ⁣ ⁣84 T^{2} + \cdots + 18\!\cdots\!84 Copy content Toggle raw display
4747 T2++43 ⁣ ⁣96 T^{2} + \cdots + 43\!\cdots\!96 Copy content Toggle raw display
5353 T2++45 ⁣ ⁣24 T^{2} + \cdots + 45\!\cdots\!24 Copy content Toggle raw display
5959 T2+16 ⁣ ⁣00 T^{2} + \cdots - 16\!\cdots\!00 Copy content Toggle raw display
6161 T2+13 ⁣ ⁣16 T^{2} + \cdots - 13\!\cdots\!16 Copy content Toggle raw display
6767 T2++36 ⁣ ⁣16 T^{2} + \cdots + 36\!\cdots\!16 Copy content Toggle raw display
7171 T2+72 ⁣ ⁣96 T^{2} + \cdots - 72\!\cdots\!96 Copy content Toggle raw display
7373 T2+64 ⁣ ⁣16 T^{2} + \cdots - 64\!\cdots\!16 Copy content Toggle raw display
7979 T2++27 ⁣ ⁣00 T^{2} + \cdots + 27\!\cdots\!00 Copy content Toggle raw display
8383 T2+20 ⁣ ⁣16 T^{2} + \cdots - 20\!\cdots\!16 Copy content Toggle raw display
8989 T2+18 ⁣ ⁣00 T^{2} + \cdots - 18\!\cdots\!00 Copy content Toggle raw display
9797 T2+64 ⁣ ⁣64 T^{2} + \cdots - 64\!\cdots\!64 Copy content Toggle raw display
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