
// Make newform 9075.2.a.n in Magma, downloaded from the LMFDB on 19 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_n();".
// 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_n();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
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 := [[1], [-1], [0], [2], [0], [-4], [6], [-6], [-4], [6], [8], [6], [-6], [6], [-8], [-6], [0], [4], [-12], [8], [-16], [2], [0], [10], [6], [14], [8], [-8], [0], [-6], [2], [16], [-14], [-22], [-6], [2], [-18], [4], [0], [-18], [12], [-26], [0], [-20], [-18], [16], [-22], [16], [-16], [-22], [-6], [-12], [20], [20], [-18], [24], [6], [14], [-4], [-22], [14], [18], [-14], [-28], [26], [30], [-28], [8], [8], [36], [-18], [8], [-24], [-24], [-20], [-36], [-6], [-14], [-10], [-4], [-28], [-6], [-12], [-18], [-26], [-16], [-30], [-8], [-22], [8], [-16], [28], [16], [-36], [-4], [16], [14], [-10], [14], [32], [-34], [2], [-12], [-18], [-26], [-14], [0], [-6], [-4], [24], [42], [-8], [-30], [4], [16], [-38], [-12], [20], [10], [40], [10], [-4], [-2], [-28], [28], [-14], [-6], [-40], [8], [44], [-34], [20], [0], [-10], [-50], [40], [-54], [22], [-26], [46], [50], [30], [-8], [-8], [-2], [12], [-24], [22], [-36], [36], [32], [46], [52], [-12], [-20], [32], [22], [14], [20], [38], [-4], [6], [-46], [0], [-30], [4], [-32], [-24], [20], [-22], [-40], [-10], [24], [16], [16], [-26], [62], [54], [2], [-12], [16], [32], [-26], [-42], [40], [-34], [20], [4], [4], [-12], [18], [48], [52], [-46], [12], [6], [16], [2], [14], [4], [-6], [10], [10], [28], [-12], [-42], [-24], [12], [70], [44], [44], [-66], [-48], [24], [16], [62], [30], [50], [-16], [-46], [24], [10], [-18], [-16], [-60], [-4], [18], [-12], [22], [-36], [26], [58], [-22], [38], [28], [28], [62], [-38], [-68], [12], [56], [-42], [56], [-78], [-78], [-32], [48], [-52], [-2], [-36], [12], [-22], [48], [6], [-70], [-20], [22], [74], [78], [-4], [-10], [-68], [32], [32], [14], [-4], [6], [30], [-46], [-58], [70], [-20], [-70], [30], [8], [-16], [36], [20], [16], [-56], [-24], [16], [4], [-36], [30], [68], [38], [-2], [-40], [26], [-54], [-8], [-6], [0], [-32], [-78], [0], [2], [28], [58], [4], [10], [-30], [40], [-36], [-6], [12], [2], [-44], [-32], [-56], [-78], [-30], [36], [72], [12], [24], [8], [-62], [-2], [-22], [74], [-30], [40], [18], [-78], [-20], [28], [32], [22], [0], [6], [-14], [40], [-34], [-32], [-10], [42], [-6], [-54], [-6], [-66], [-22], [32], [10], [36], [-58], [12], [96], [10], [-70], [-26], [-18], [58], [-12], [-66], [-24], [-36], [-18], [-8], [72], [82], [52], [28], [28], [-38], [10], [-38], [-32], [-88], [20], [-36], [-42], [-98], [-34], [12], [76], [52], [-10], [-56], [66], [-50], [18], [-2], [-82], [-56], [8], [18], [14], [48], [-62], [66], [-28], [-4], [12], [-16], [34], [-78], [-60], [54], [20], [-30], [10], [-50], [-62], [-14], [-102], [82], [-36], [76], [-28], [42], [-84], [-42], [-88], [66], [-4], [-48], [-82], [72], [-50], [-48], [-32], [-12], [38], [-14], [-44], [-102], [4], [-36]];
    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_n();".
// 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_n(:prec:=1)
    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_n();".
// 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_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9075_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<7,R![-2, 1]>,<13,R![4, 1]>,<17,R![-6, 1]>,<19,R![6, 1]>,<23,R![4, 1]>,<37,R![-6, 1]>],Snew);
    return Vf;
end function;