
// Make newform 2925.2.c.x in Magma, downloaded from the LMFDB on 15 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2925_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2925_c_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_2925_2_c_x();".
// 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_2925_2_c_x();".
// 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 := [18, -24, 16, 2, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [285, 6, -1, -6, 24, -1], [-147, 292, -136, 101, -11, -5], [-7, -36, 6, 36, -13, 6], [-267, 386, 154, 7, -28, 23], [267, -386, -23, -7, 28, -23]];
    Rf_basisdens := [1, 131, 393, 131, 393, 131];
    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_2925_c();"
function MakeCharacter_2925_c()
    N := 2925;
    order := 2;
    char_gens := [326, 352, 2251];
    v := [2, 1, 2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2925_c_Hecke();"
function MakeCharacter_2925_c_Hecke(Kf)
    N := 2925;
    order := 2;
    char_gens := [326, 352, 2251];
    char_values := [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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]];
    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, 0, 0, -1], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, -1], [1, 1, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0], [0, 0, 0, 0, -3, 2], [-3, -2, 0, 1, 0, 0], [0, 0, 1, 0, 1, 2], [-5, 0, 0, 2, 0, 0], [1, -3, 0, 1, 0, 0], [0, 0, -1, 0, 1, -2], [-1, -2, 0, 1, 0, 0], [0, 0, 3, 0, 1, 0], [0, 0, 4, 0, 4, -1], [0, 0, -1, 0, 2, 2], [2, 1, 0, -2, 0, 0], [-3, -2, 0, 2, 0, 0], [0, 0, 3, 0, -1, 3], [5, -2, 0, 1, 0, 0], [0, 0, -1, 0, -9, 0], [-3, -4, 0, 1, 0, 0], [0, 0, 0, 0, 6, -5], [-6, -2, 0, 0, 0, 0], [0, 0, 2, 0, 10, -2], [6, 0, 0, 3, 0, 0], [0, 0, -3, 0, 1, -4], [0, 0, 0, 0, 0, 6], [-3, -4, 0, 1, 0, 0], [0, 0, 4, 0, 4, 2], [0, 0, -1, 0, -5, 2], [3, -6, 0, 3, 0, 0], [0, 0, -2, 0, -8, -2], [4, 4, 0, -6, 0, 0], [6, 2, 0, -6, 0, 0], [8, 7, 0, 0, 0, 0], [0, 0, -1, 0, 4, 0], [0, 0, -1, 0, -3, -2], [0, 0, -7, 0, -7, 0], [0, 0, -1, 0, 2, 4], [-5, -4, 0, 5, 0, 0], [0, -2, 0, -7, 0, 0], [16, 4, 0, 2, 0, 0], [0, 0, 5, 0, 3, -8], [0, 0, -3, 0, 9, 6], [-14, 4, 0, 0, 0, 0], [1, -2, 0, 7, 0, 0], [0, 0, 3, 0, 1, -2], [0, 0, 5, 0, 11, -1], [6, 8, 0, -8, 0, 0], [0, 0, 0, 0, 0, -2], [8, 7, 0, -2, 0, 0], [-9, -8, 0, -1, 0, 0], [7, 0, 0, 5, 0, 0], [0, 0, 3, 0, -12, 6], [0, 0, 1, 0, 1, -8], [-23, -2, 0, 2, 0, 0], [-2, -9, 0, -2, 0, 0], [0, 0, 6, 0, 2, 6], [0, 4, 0, -6, 0, 0], [0, 0, 0, 0, 10, -4], [0, 0, 0, 0, 6, 0], [0, 0, 5, 0, 1, 8], [18, 0, 0, 0, 0, 0], [0, 0, 3, 0, 4, -12], [0, 0, 3, 0, 3, 0], [-17, -2, 0, -5, 0, 0], [0, 0, 1, 0, 24, 2], [0, 0, -6, 0, 0, 8], [9, -2, 0, 1, 0, 0], [0, 0, -1, 0, 23, -2], [8, -7, 0, -2, 0, 0], [0, 0, 4, 0, 0, -8], [0, 0, 3, 0, 16, 2], [-5, 11, 0, -3, 0, 0], [0, 0, 5, 0, -13, -10], [-8, 2, 0, -4, 0, 0], [0, 0, -4, 0, 4, 14], [-17, 2, 0, -1, 0, 0], [-11, -2, 0, 3, 0, 0], [-13, -12, 0, 1, 0, 0], [18, -4, 0, 2, 0, 0], [3, -2, 0, -9, 0, 0], [0, 0, -6, 0, -14, 10], [11, -2, 0, 5, 0, 0], [0, 0, -3, 0, 15, 4], [-2, 8, 0, 2, 0, 0], [0, 0, -5, 0, -9, 10], [-20, -4, 0, 2, 0, 0], [0, 0, 5, 0, 27, -1], [0, 0, 8, 0, 2, -6], [-13, 3, 0, 7, 0, 0], [0, 0, -5, 0, 15, -7], [16, 4, 0, 2, 0, 0], [-4, 9, 0, -4, 0, 0], [0, 0, 1, 0, -17, -2], [-9, 0, 0, 3, 0, 0], [-22, -4, 0, 4, 0, 0], [0, 0, 8, 0, 12, -2], [9, -4, 0, -1, 0, 0], [0, 0, -1, 0, -17, -8], [0, 0, -7, 0, 11, -4], [0, 0, -7, 0, 17, -2], [-10, -4, 0, -5, 0, 0], [11, -6, 0, -3, 0, 0], [0, 0, 6, 0, 26, 2], [0, 0, -3, 0, 15, -9], [0, 0, -10, 0, -4, -4], [-13, 2, 0, 7, 0, 0], [10, 0, 0, 1, 0, 0], [0, 0, -6, 0, -10, 6], [0, 0, 0, 0, -2, 4], [0, 0, 0, 0, -12, 10], [-17, -8, 0, 9, 0, 0], [-17, -4, 0, 1, 0, 0], [1, -10, 0, -4, 0, 0], [0, 0, -7, 0, 15, -2], [0, 0, -4, 0, 2, 14], [0, 0, -10, 0, -7, 14], [-7, 4, 0, -11, 0, 0], [11, 6, 0, 9, 0, 0], [0, 0, -5, 0, -4, 6], [0, 0, -6, 0, 12, -16], [0, 0, 5, 0, -13, 7], [14, -11, 0, 0, 0, 0], [-9, 16, 0, -6, 0, 0], [-1, -10, 0, -7, 0, 0], [1, -16, 0, -1, 0, 0], [0, 0, 7, 0, 3, 0], [0, 0, -2, 0, 38, -4], [3, -17, 0, 7, 0, 0], [0, 0, -5, 0, -11, 19], [1, 14, 0, -5, 0, 0], [0, 0, 6, 0, -1, -16], [-7, 12, 0, -5, 0, 0], [-4, -2, 0, -10, 0, 0], [0, 0, 6, 0, 6, -6], [0, 0, 5, 0, -5, 11], [0, 0, 8, 0, 5, 12], [10, 2, 0, 2, 0, 0], [4, -5, 0, 4, 0, 0], [-29, -2, 0, -1, 0, 0], [0, 0, -16, 0, -6, 6], [0, 0, -7, 0, 23, 5], [23, -10, 0, -4, 0, 0], [-31, 2, 0, 1, 0, 0], [0, 0, -4, 0, 12, 12], [0, 0, 4, 0, -14, -16], [-13, -4, 0, -1, 0, 0], [0, 0, -11, 0, -11, 5], [0, 0, -14, 0, 12, -10], [22, 10, 0, 11, 0, 0], [0, 0, 2, 0, 24, 6], [0, 0, -7, 0, -13, 8], [0, 0, 4, 0, -36, 4], [-3, -2, 0, 3, 0, 0], [6, 10, 0, -2, 0, 0], [-4, -8, 0, 10, 0, 0], [0, 0, 4, 0, -3, -2], [-1, -12, 0, 13, 0, 0], [0, 0, 5, 0, 35, -7], [0, 0, -15, 0, 6, -4], [0, 0, -7, 0, -35, 5], [18, -6, 0, -6, 0, 0], [0, 0, 2, 0, 14, -4], [0, 0, 3, 0, -21, 3], [9, 18, 0, -1, 0, 0], [0, 0, 0, 0, -1, -18], [-26, -6, 0, 0, 0, 0], [0, 0, 11, 0, -10, -4], [5, 6, 0, 13, 0, 0], [4, 10, 0, 10, 0, 0], [27, -6, 0, 3, 0, 0], [0, 0, 9, 0, 7, 2], [-12, -16, 0, -2, 0, 0], [-22, 4, 0, 1, 0, 0], [1, 13, 0, 1, 0, 0], [-8, 18, 0, -4, 0, 0], [0, 0, 8, 0, 24, -2], [8, 12, 0, 11, 0, 0], [0, 0, 3, 0, 5, -2], [-9, -6, 0, 9, 0, 0], [0, 0, -14, 0, 2, 6], [0, 0, 4, 0, -26, -2], [0, 0, 2, 0, -22, 5], [16, 22, 0, 8, 0, 0], [0, 0, -1, 0, 10, -8], [0, 0, 0, 0, -20, -1], [-11, -18, 0, -3, 0, 0], [19, 15, 0, 5, 0, 0], [0, 0, -12, 0, -38, 0], [0, 0, 10, 0, -32, 11], [-7, -14, 0, 3, 0, 0], [17, 16, 0, -5, 0, 0], [0, 0, -8, 0, -26, 16], [0, 0, -3, 0, 48, 2], [1, -10, 0, 1, 0, 0], [0, 0, 9, 0, 22, 0], [0, 0, -2, 0, -20, 16], [0, 0, -4, 0, 8, -14], [1, -8, 0, -13, 0, 0], [-16, 4, 0, 0, 0, 0], [0, 0, -6, 0, 20, 8], [16, 14, 0, 3, 0, 0], [-35, 1, 0, -7, 0, 0], [0, 0, 14, 0, 20, -16], [-8, 9, 0, 6, 0, 0], [0, 0, 3, 0, -21, 4], [-6, -12, 0, -18, 0, 0], [3, -10, 0, -1, 0, 0], [0, 0, 2, 0, 37, -6], [-4, -14, 0, -8, 0, 0], [0, 0, 2, 0, -12, -22], [0, 0, -4, 0, -40, -7], [41, 7, 0, -17, 0, 0], [-22, -6, 0, 12, 0, 0], [0, 0, 4, 0, -6, -24], [-18, 0, 0, -9, 0, 0], [0, 0, 4, 0, -32, 13], [0, 0, 14, 0, 14, -16], [-10, 6, 0, -6, 0, 0], [31, 10, 0, -21, 0, 0], [10, -20, 0, 14, 0, 0], [0, 0, 14, 0, 0, -9], [0, 0, 7, 0, 1, 0], [-7, -16, 0, -4, 0, 0], [0, 0, 0, 0, 30, -12], [-24, 12, 0, -6, 0, 0], [0, 0, -7, 0, -41, 2], [10, -5, 0, 2, 0, 0], [0, 0, -10, 0, 12, 4], [16, 18, 0, -18, 0, 0], [15, 5, 0, 11, 0, 0], [-10, 10, 0, -2, 0, 0], [0, 0, 1, 0, -19, -6], [0, 0, 3, 0, -27, -2], [-42, 8, 0, 4, 0, 0], [0, 0, -7, 0, -25, 2], [-6, 2, 0, 0, 0, 0], [-18, -20, 0, 6, 0, 0], [0, 0, -9, 0, -9, -12], [5, 20, 0, 9, 0, 0], [0, 0, 19, 0, 11, 8], [-29, 0, 0, 9, 0, 0], [0, 0, 7, 0, 13, 18], [33, -6, 0, 9, 0, 0], [0, 0, 5, 0, 19, 6], [51, 12, 0, -3, 0, 0], [-55, -1, 0, -1, 0, 0], [0, 0, 3, 0, -21, 4], [0, 0, 12, 0, 20, -8], [-13, 2, 0, 19, 0, 0], [0, 0, 7, 0, -23, -18], [-8, -18, 0, 9, 0, 0], [0, 0, -18, 0, -6, -2], [-32, 1, 0, -4, 0, 0], [-8, -4, 0, 22, 0, 0], [0, 0, -3, 0, 23, -15], [0, 0, -2, 0, 25, -26], [0, 0, 4, 0, 12, 10], [0, 0, -8, 0, -34, 0], [0, 0, 3, 0, -39, 0], [-6, 6, 0, 4, 0, 0], [0, 0, -12, 0, 16, 18], [0, 0, 4, 0, -38, -10], [17, 18, 0, -13, 0, 0], [3, -6, 0, -3, 0, 0], [9, -28, 0, 3, 0, 0], [0, 0, -12, 0, -20, 17], [0, 0, 3, 0, -24, 14], [50, -2, 0, -6, 0, 0], [0, 0, 2, 0, 22, -15], [0, 0, 2, 0, -66, 2], [-31, 2, 0, 5, 0, 0], [0, 0, -13, 0, -32, 2], [0, 0, -9, 0, -11, 13], [0, 0, -19, 0, -1, 14], [-17, -10, 0, 15, 0, 0], [-28, 12, 0, -12, 0, 0], [-10, -12, 0, 4, 0, 0], [0, 0, -6, 0, -18, 26], [-27, -10, 0, -7, 0, 0], [0, 0, 6, 0, 0, -6], [15, 0, 0, -13, 0, 0], [0, 0, -14, 0, 6, 7], [17, 18, 0, -5, 0, 0], [0, 0, 4, 0, 5, 12], [0, 0, -7, 0, -19, -6], [-21, -13, 0, 1, 0, 0], [30, -2, 0, -15, 0, 0], [36, 6, 0, -3, 0, 0], [0, 0, 11, 0, 5, -10], [0, 0, 2, 0, -4, 0], [-5, -13, 0, -1, 0, 0], [0, 0, -1, 0, -36, -2], [-17, -16, 0, 20, 0, 0], [-4, -30, 0, 6, 0, 0], [0, 0, -20, 0, 16, -14], [11, 20, 0, 7, 0, 0], [0, 0, 2, 0, 28, -3], [0, 0, -22, 0, -27, 12], [0, 0, -15, 0, -15, 26], [23, 6, 0, -1, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2925_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2925_2_c_x();".
// 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_2925_2_c_x(:prec:=6)
    chi := MakeCharacter_2925_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2925_2_c_x();".
// 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_2925_2_c_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2925_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![9, 0, 31, 0, 11, 0, 1]>,<7,R![9, 0, 39, 0, 19, 0, 1]>,<11,R![9, -11, -1, 1]>],Snew);
    return Vf;
end function;