
// Make newform 1575.2.a.d in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1575_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_1575_2_a_d();".
// 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_1575_2_a_d();".
// 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_1575_a();"
function MakeCharacter_1575_a()
    N := 1575;
    order := 1;
    char_gens := [1226, 127, 451];
    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_1575_a_Hecke(Kf)
    return MakeCharacter_1575_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], [0], [0], [1], [0], [-4], [-2], [0], [0], [8], [-4], [-8], [4], [-8], [12], [-6], [8], [10], [-8], [-16], [-12], [-8], [-16], [-12], [-4], [-12], [-8], [4], [2], [18], [-8], [8], [-6], [8], [-16], [-16], [-12], [8], [-12], [6], [-16], [18], [0], [-16], [-10], [-4], [20], [24], [8], [14], [22], [0], [-14], [-24], [-14], [-24], [-4], [28], [16], [8], [16], [-18], [-16], [0], [12], [18], [-12], [-24], [12], [10], [14], [32], [24], [0], [-4], [4], [-8], [-4], [24], [-10], [24], [6], [32], [-20], [12], [20], [-16], [-16], [-12], [24], [0], [0], [-24], [32], [-20], [12], [36], [4], [16], [-30], [40], [2], [-24], [-24], [-12], [-20], [0], [2], [-16], [22], [-24], [8], [22], [40], [-8], [8], [-16], [28], [-46], [0], [-30], [-24], [50], [-20], [40], [-8], [-10], [-48], [40], [-52], [4], [-8], [-24], [-8], [-20], [30], [18], [-32], [38], [0], [16], [32], [-8], [-36], [-6], [0], [-36], [6], [-8], [24], [40], [28], [24], [-4], [8], [-48], [-8], [-4], [-44], [-36], [-36], [-6], [8], [56], [-18], [36], [-32], [28], [-14], [-18], [0], [6], [16], [-8], [-44], [-36], [-28], [-24], [56], [-26], [8], [24], [-40], [-54], [56], [60], [32], [64], [-26], [32], [-20], [4], [-4], [-4], [36], [42], [30], [8], [18], [-60], [48], [4], [20], [34], [8], [54], [4], [12], [16], [-8], [8], [20], [40], [24], [-32], [-58], [24], [-44], [-8], [-62], [-10], [52], [-32], [72], [-48], [-26], [26], [16], [24], [-48], [24], [24], [-56], [-48], [-16], [-4], [-50], [6], [-48], [-32], [36], [-56], [-72], [-46], [30], [-48], [56], [24], [4], [8], [64], [-12], [-16], [70], [-54], [64], [58], [32], [18], [-36], [72], [68], [-50], [52], [18], [-32], [-48], [-12], [-24], [-38], [-42], [40], [-68], [-64], [12], [8], [-36], [2], [-10], [-24], [4], [32], [60], [-50], [-32], [-48], [-32], [58], [68], [-4], [-48], [72], [22], [-56], [-16], [4], [-52], [14], [56], [-32], [-44], [18], [24], [-68], [52], [-8], [-84], [6], [48], [-16], [36], [24], [24], [-56], [48], [26], [40], [16], [-4], [-72], [80], [-72], [44], [-8], [6], [-46], [44], [-64], [-48], [38], [46], [-30], [-36], [-32], [4], [-48], [-50], [46], [58], [40], [-80], [6], [60], [-48], [66], [-32], [-34], [-72], [80], [-66], [-80], [16], [48], [-24], [-22], [-58], [-16], [8], [2], [96], [-64], [28], [40], [48], [32], [-56], [-54], [-8], [-10], [0], [-40], [48], [16], [68], [-32], [-8], [-64], [28], [36], [-44], [20], [-54], [-8], [-50], [80], [-76], [8], [92], [-24], [-12], [-50], [34], [-48], [16], [0], [8], [44], [-36], [-44], [24], [-90], [34], [24], [-6], [-36], [20], [-80], [-48], [-16], [-8], [-12], [-42], [-4], [-4], [-8], [-4], [-80], [-72], [-30], [-44], [-24], [28], [-72], [-24], [-36], [54], [12], [72], [48], [96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1575_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1575_2_a_d();".
// 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_1575_2_a_d(:prec:=1)
    chi := MakeCharacter_1575_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_1575_2_a_d();".
// 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_1575_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1575_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<11,R![0, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;