
// Make newform 3520.2.a.bn in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3520_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_3520_2_a_bn();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_3520_2_a_bn();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [-2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_3520_a();"
function MakeCharacter_3520_a()
    N := 3520;
    order := 1;
    char_gens := [2751, 1541, 2817, 321];
    v := [1, 1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_3520_a_Hecke(Kf)
    return MakeCharacter_3520_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0], [0, 1], [1, 0], [-2, 0], [-1, 0], [4, -1], [4, 1], [0, 0], [0, -1], [-2, 2], [0, 0], [2, 2], [6, 0], [6, 0], [0, 1], [-6, -2], [4, -2], [-2, 4], [-4, -3], [0, 4], [-4, 1], [4, 0], [6, 0], [-2, -4], [-2, 2], [2, -4], [4, 1], [-2, -2], [2, 2], [14, 2], [10, -2], [0, 4], [6, -6], [4, 0], [-6, 2], [-12, 0], [14, 0], [-8, -3], [-6, -6], [8, 5], [-4, -2], [-10, -4], [-8, 4], [-4, 1], [8, 1], [16, -2], [16, 0], [-8, -1], [-14, -4], [10, -4], [8, -5], [-12, -4], [6, 0], [-12, 0], [2, -4], [6, -6], [-6, -4], [-4, 4], [-4, -1], [6, 4], [-10, -8], [12, -1], [22, -2], [16, -4], [10, -4], [-10, 4], [-4, 4], [-12, -3], [6, -6], [-10, -6], [10, 4], [12, -4], [0, -3], [-16, -7], [-28, -2], [-20, 5], [2, -8], [-2, 4], [6, -4], [18, -6], [20, 2], [6, 0], [0, -4], [-2, -2], [-16, 0], [24, 1], [6, 8], [-8, 3], [10, -8], [8, -5], [12, 1], [-36, 0], [-16, -3], [12, -4], [-4, -2], [-6, 8], [2, 4], [14, 4], [2, 14], [-6, 0], [34, 0], [-24, 5], [-6, 2], [26, 2], [-12, 12], [-10, -6], [-12, 5], [12, -3], [0, -2], [-10, 12], [10, 10], [0, 9], [6, -2], [20, -2], [0, 0], [30, 0], [-24, 9], [-4, 11], [-6, -2], [-12, -12], [-22, -8], [-4, -3], [-20, -1], [-12, 7], [28, 4], [-10, 8], [-2, -8], [24, -2], [-28, 3], [-8, 9], [4, -12], [-10, -14], [-16, 0], [-2, -4], [-30, 0], [-2, -6], [2, -10], [-2, 6], [-10, -8], [6, -6], [-20, -12], [30, 4], [28, -3], [6, -10], [-18, 0], [32, -2], [24, -3], [-12, 13], [4, -10], [-12, 1], [24, 9], [-30, 8], [-40, 1], [10, 10], [36, -3], [24, -12], [-32, 0], [-6, 4], [24, -9], [18, -4], [-44, 1], [56, -1], [2, -6], [20, -4], [38, -2], [-36, 5], [-24, -12], [16, 9], [-26, 2], [18, 6], [28, -4], [-26, 4], [28, 4], [-16, 7], [-8, 6], [-30, -4], [-4, -12], [26, 8], [18, 12], [-30, -6], [44, -1], [-4, -8], [14, -12], [-8, -13], [48, -1], [30, 0], [8, -7], [36, 3], [2, -22], [24, -8], [-18, -6], [30, -8], [-20, -12], [2, 16], [34, -8], [18, -4], [30, 0], [42, 6], [8, -1], [6, -6], [6, -10], [-32, 0], [-38, 6], [6, 6], [-12, 2], [30, -4], [-40, -10], [42, -6], [-18, -10], [4, 4], [24, 3], [-26, -8], [-4, 1], [12, 9], [36, 4], [34, -8], [-58, 0], [34, -8], [-12, 3], [50, 2], [-34, 4], [20, -16], [58, 4], [20, 9], [-18, 2], [-22, -6], [30, 10], [8, -22], [-6, 6], [24, 8], [-34, 0], [36, 12], [-40, -8], [-18, 8], [-28, 7], [14, 16], [-34, 12], [-12, -1], [-20, 14], [-16, 4], [-24, 9], [-16, 12], [4, 9], [-10, 4], [-32, 11], [-20, 16], [8, -11], [-12, 12], [44, 0], [38, 10], [-44, -1], [-46, -8], [-36, -1], [18, 0], [8, 5], [-44, 8], [-34, 12], [-14, 20], [14, 16], [-32, -5], [22, 10], [-6, 0], [-2, -6], [20, 7], [-22, 8], [-28, 2], [22, 16], [26, -8], [14, 4], [12, -13], [-14, -24], [-36, 9], [-18, -18], [-40, 0], [24, -9], [8, 23], [-20, -7], [-18, -6], [-34, -12], [-48, 8], [-10, 10], [8, 12], [30, -12], [10, 12], [-14, 0], [16, 16], [-10, 18], [12, -9], [24, 18], [10, -10], [2, -16], [-16, 19], [32, -5], [-24, 0], [20, 3], [42, -2], [0, 12], [42, -2], [60, 8], [-2, -18], [32, 7], [24, -15], [32, 4], [12, -9], [20, 0], [10, 8], [4, 17], [-6, 24], [-24, -14], [8, -15], [34, 4], [34, 8], [18, 20], [-28, -5], [-6, -2], [-58, 6], [-28, -14], [52, 8], [62, 6], [42, 6], [-8, 20], [14, 10], [6, 24], [0, -21], [-24, 19], [-2, 24], [4, 6], [-8, 9], [54, 8], [-40, 1], [-10, -12], [6, 0], [12, -12], [-6, 6], [32, 16], [-16, 7], [10, 0], [-16, -1], [22, -8], [-2, 12], [6, 0], [-30, 8], [-30, -2], [-48, 0], [-54, 2], [8, -4], [66, 0], [-8, 3], [32, 4], [34, 2], [20, 8], [10, 0], [-66, 0], [34, 0], [14, -6], [-36, -15], [-24, -2], [16, 20], [48, 7], [-20, -11], [12, -21], [-14, -28], [8, -19], [16, 20], [-64, -9], [-10, 6], [-24, 25], [22, -22], [-2, 4], [-60, -4], [4, -18], [22, -2], [-54, -10], [-32, 4], [6, -12], [-12, -22], [-4, 16], [-32, -17], [26, 14], [64, 3], [-14, -8], [-30, 20], [-14, -18], [-60, -7], [0, -12], [-12, 9], [-8, -4], [-42, 0], [66, 6], [36, -13], [-82, -4], [-54, -10], [12, -6], [36, 9], [-48, 16], [-40, -9], [56, 0], [42, 16], [-20, 4], [22, 8], [-70, 6], [30, -2], [-26, -24], [42, 0], [-6, -30], [44, 12], [14, 28], [18, 20], [-28, -9], [-20, 10], [96, 3], [60, -5], [-40, 1], [44, -4], [20, -11], [6, -20], [-48, -20], [-4, 3], [-30, 12], [10, 16], [22, -4], [84, 3], [68, 7], [-72, -4], [-46, -12], [-30, 6], [-16, 13], [18, 10], [-36, 12], [-4, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3520_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3520_2_a_bn();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_3520_2_a_bn(:prec:=2)
    chi := MakeCharacter_3520_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_3520_2_a_bn();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_3520_2_a_bn( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-8, 0, 1]>,<7,R![2, 1]>,<13,R![8, -8, 1]>,<17,R![8, -8, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;