
// Make newform 9075.2.a.bh in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9075_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_9075_2_a_bh();".
// 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_9075_2_a_bh();".
// 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 := [-3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_9075_a();"
function MakeCharacter_9075_a()
    N := 9075;
    order := 1;
    char_gens := [3026, 727, 5326];
    v := [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_9075_a_Hecke(Kf)
    return MakeCharacter_9075_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, 1], [-1, 0], [0, 0], [2, 0], [0, 0], [2, -2], [0, 0], [-2, 2], [0, 4], [0, -2], [-4, 4], [-2, -4], [0, 2], [2, 4], [0, -4], [6, 4], [0, 4], [-2, 0], [-8, 0], [0, -8], [2, 6], [10, 2], [12, 2], [-6, -4], [10, 0], [0, -6], [-8, 0], [12, 2], [10, 0], [6, -4], [2, -4], [-12, 4], [18, 0], [-2, 6], [12, -2], [10, -6], [10, 4], [4, 8], [0, 6], [-12, 0], [0, -4], [2, -8], [-12, 4], [14, -6], [-12, 0], [-4, 12], [-2, -6], [4, 8], [12, -2], [-10, 8], [12, 0], [12, 0], [-14, -8], [12, 8], [6, -8], [24, 2], [6, -8], [22, 6], [26, -2], [0, 2], [2, 4], [0, 8], [14, 0], [12, 4], [-14, 4], [18, -4], [-4, -8], [2, -18], [12, -6], [22, 8], [-6, 4], [0, -12], [-20, 0], [-10, 6], [-4, 8], [0, -8], [18, 4], [-2, 0], [6, 8], [-14, -12], [24, -4], [2, 0], [12, 12], [-14, -8], [-2, 18], [-12, 0], [-6, 12], [2, -6], [-12, 14], [28, 0], [-12, -4], [-12, 0], [4, -16], [24, -4], [8, -12], [0, -18], [6, -8], [6, 0], [-22, 0], [-14, 8], [2, 0], [24, 12], [0, -6], [12, 10], [-14, -6], [-2, -12], [12, -4], [-12, -12], [-24, 8], [-2, -20], [2, 12], [-10, 6], [30, -4], [20, 0], [-28, -4], [-6, 4], [4, 24], [0, 16], [-6, 8], [-12, -16], [14, 0], [14, 10], [-12, -12], [12, -12], [20, -16], [-36, 2], [2, 8], [-12, -4], [-8, 0], [2, -30], [-14, 2], [36, 6], [20, -4], [-14, 12], [36, 2], [-2, 28], [18, 8], [2, -12], [18, -20], [12, -2], [-2, 6], [-24, -2], [-8, -24], [12, -6], [2, -16], [36, -12], [14, 6], [-24, 8], [20, 16], [24, -8], [-10, 14], [6, 4], [4, 0], [-12, -14], [-32, -8], [12, -4], [22, -6], [-18, -12], [-10, 6], [-24, 2], [12, -4], [-24, -12], [2, -12], [12, -8], [-6, 24], [-24, 0], [-28, -12], [38, 6], [-14, 0], [-18, -12], [-12, -8], [-46, 0], [0, -4], [26, 10], [8, -8], [-18, -12], [10, -6], [6, 16], [2, -20], [-26, 4], [-32, 0], [12, -24], [10, -12], [48, -4], [-24, -4], [30, -4], [-22, -6], [16, 0], [-14, 4], [-12, -20], [-26, -12], [-12, 2], [32, 20], [18, 20], [-24, -14], [6, 8], [-26, 12], [10, 8], [0, 12], [-12, 22], [-12, 2], [-14, -26], [34, 16], [10, -24], [-24, -20], [-6, 28], [8, -8], [-24, 2], [36, -2], [-40, 4], [38, 6], [-18, 12], [-56, 0], [-12, -4], [0, 12], [-46, 0], [-34, -4], [-48, -10], [24, -16], [-6, 0], [34, 12], [10, -30], [-42, -12], [-8, -16], [-48, -2], [22, -4], [18, -8], [-36, 4], [-10, 8], [24, 8], [46, -16], [-2, -10], [-14, -2], [36, 6], [-32, -8], [-12, -26], [-10, 32], [24, 8], [36, 16], [24, -16], [36, 0], [-14, 6], [16, 0], [-22, 0], [36, -16], [0, -12], [-20, 16], [12, 8], [34, 10], [36, 6], [-22, 6], [-48, 6], [-24, 24], [-22, -24], [24, 12], [24, -20], [26, 0], [-10, 12], [-30, 4], [-10, -34], [-10, -4], [-24, -22], [34, -12], [-34, 2], [-6, 20], [8, -20], [6, 28], [30, -20], [-34, -16], [12, -4], [-34, -8], [-8, 8], [-26, 20], [22, -30], [2, -6], [4, 0], [-36, 4], [-2, -8], [-2, -12], [-12, 36], [0, 10], [44, -20], [-12, 22], [22, -12], [-46, 0], [24, 32], [-14, 0], [12, -4], [-4, -12], [24, -18], [42, -20], [-36, -8], [36, 16], [36, -4], [14, 18], [-12, -10], [44, -12], [-6, 24], [-24, 4], [-34, -8], [14, -10], [24, -4], [34, -2], [36, 20], [56, -12], [34, -12], [12, 28], [-34, -24], [-48, -8], [38, 14], [48, -10], [18, 28], [0, -26], [40, 16], [24, 30], [-38, -24], [12, 0], [-24, 28], [22, 20], [-36, 26], [-26, -6], [-2, -20], [-12, 26], [-44, 24], [12, -16], [62, 8], [8, -36], [4, 0], [-24, 10], [12, -8], [58, 12], [54, 0], [-14, 2], [-12, 38], [10, -18], [12, -24], [14, 24], [-60, 0], [38, 24], [-22, -24], [22, -32], [-54, -20], [-60, -2], [8, 24], [-30, 0], [24, 16], [-58, 0], [52, 0], [24, 16], [18, -24], [-2, 14], [-2, 40], [-18, -44], [2, 12], [-74, 0], [-24, 0], [24, 32], [-24, 16], [48, -12], [-48, 8], [14, 10], [48, 22], [48, 4], [48, 16], [28, -8], [22, -4], [0, -44], [-46, 0], [46, -12], [-24, -28], [44, 8], [-60, -10], [48, 10], [-2, 42], [10, 12], [-24, 12], [-12, 40], [38, -34], [-36, 10], [50, -30], [-6, 36], [30, -32], [-70, -4], [0, 32], [-38, -26], [0, 16], [-4, 12], [-50, -24], [-46, -24], [0, 8], [14, -48], [-18, -4], [12, -8], [16, 0], [-60, -4], [-22, 18], [82, 6], [-30, 24], [44, 24], [24, 38], [22, -36], [54, -4], [50, 8], [6, 0], [36, 6], [-62, -6], [-74, 8], [48, 22], [16, 24], [12, 8], [38, 14], [0, -8], [12, 24], [10, -6], [-22, 22], [18, 48], [12, 20], [16, 8], [18, 20], [72, -10], [30, 12], [62, 18], [-24, -32], [0, -24], [-2, -16], [-6, -16], [-12, -8], [-24, 38], [-76, 0], [-60, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9075_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9075_2_a_bh();".
// 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_9075_2_a_bh(:prec:=2)
    chi := MakeCharacter_9075_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_9075_2_a_bh();".
// 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_9075_2_a_bh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9075_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-3, 0, 1]>,<7,R![-2, 1]>,<13,R![-8, -4, 1]>,<17,R![0, 1]>,<19,R![-8, 4, 1]>,<23,R![-48, 0, 1]>,<37,R![-44, 4, 1]>],Snew);
    return Vf;
end function;