
// Make newform 6900.2.a.g in Magma, downloaded from the LMFDB on 15 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6900_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_6900_2_a_g();".
// 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_6900_2_a_g();".
// 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_6900_a();"
function MakeCharacter_6900_a()
    N := 6900;
    order := 1;
    char_gens := [3451, 4601, 277, 1201];
    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_6900_a_Hecke(Kf)
    return MakeCharacter_6900_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], [0], [0], [0], [6], [-2], [6], [1], [6], [-4], [-4], [6], [-4], [4], [-6], [14], [6], [4], [-2], [-10], [-14], [-4], [18], [-12], [18], [-4], [-12], [18], [-6], [8], [10], [18], [-4], [-18], [-16], [0], [12], [16], [0], [14], [2], [0], [18], [8], [2], [-8], [-24], [12], [30], [-8], [14], [-10], [-4], [-12], [-16], [10], [-8], [10], [-10], [24], [18], [28], [18], [24], [-4], [-20], [-20], [-12], [34], [-12], [-20], [12], [-8], [6], [8], [-10], [18], [38], [-2], [-20], [-10], [-8], [-28], [-16], [-8], [-14], [-4], [-6], [-32], [28], [-12], [-16], [6], [28], [-16], [22], [22], [-8], [2], [-28], [-22], [28], [10], [-10], [-14], [36], [-36], [6], [22], [-32], [16], [-14], [-10], [-22], [-46], [-44], [-12], [24], [24], [-34], [6], [6], [-44], [48], [-18], [38], [10], [-8], [16], [20], [-16], [-30], [-4], [-30], [30], [46], [-28], [-30], [-42], [-44], [38], [40], [-12], [2], [24], [-46], [32], [8], [48], [-2], [14], [-36], [32], [-8], [-44], [38], [10], [-32], [6], [-32], [6], [8], [-20], [34], [-24], [-20], [-50], [30], [4], [-48], [34], [44], [40], [-32], [-54], [-16], [-30], [12], [34], [-24], [4], [58], [0], [48], [-30], [-2], [56], [-50], [-18], [34], [-44], [-50], [-10], [-68], [14], [-18], [-4], [-42], [-56], [-54], [-32], [26], [-10], [-44], [48], [-2], [-16], [30], [20], [-34], [22], [-64], [-36], [30], [46], [8], [2], [48], [48], [-42], [-50], [-2], [-44], [-56], [70], [66], [-6], [36], [-70], [-6], [-46], [-66], [-18], [-24], [32], [-6], [34], [34], [-34], [12], [40], [40], [14], [-8], [66], [-32], [8], [-6], [-48], [12], [66], [-72], [2], [-40], [-70], [-6], [-32], [44], [34], [52], [-36], [6], [56], [-76], [-58], [-6], [-18], [12], [-36], [38], [-80], [16], [-18], [-50], [-56], [0], [26], [-26], [-8], [28], [-42], [48], [-22], [12], [18], [32], [2], [-48], [34], [30], [36], [-84], [-24], [14], [-30], [10], [76], [-30], [28], [-16], [-14], [-50], [-76], [-50], [-62], [-4], [38], [12], [66], [-80], [-10], [14], [68], [-16], [-74], [54], [46], [-32], [-6], [-26], [-32], [-34], [-40], [6], [6], [-30], [12], [8], [-54], [46], [-24], [-44], [24], [-34], [-40], [82], [-38], [-58], [88], [-22], [-2], [-14], [-54], [22], [-54], [-42], [12], [-40], [34], [80], [-2], [54], [-40], [-74], [-84], [-72], [-4], [-24], [84], [-92], [-54], [84], [72], [-20], [58], [-88], [44], [6], [90], [-68], [12], [10], [-54], [30], [54], [-4], [92], [78], [74], [30], [-70], [16], [64], [-66], [-36], [24], [2], [-16], [-32], [10], [60], [-94], [-68], [-80], [-80], [78], [102], [2], [-34], [94], [48], [88], [-34], [-6], [-52], [-12], [18], [32], [90], [30], [28], [-12], [34], [28], [-6], [78], [80], [58], [-24], [50], [-48], [-92], [26], [-38], [84], [76], [-62], [44], [66]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6900_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6900_2_a_g();".
// 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_6900_2_a_g(:prec:=1)
    chi := MakeCharacter_6900_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_6900_2_a_g();".
// 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_6900_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6900_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;