
// Make newform 6300.2.a.w in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6300_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_6300_2_a_w();".
// 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_6300_2_a_w();".
// 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_6300_a();"
function MakeCharacter_6300_a()
    N := 6300;
    order := 1;
    char_gens := [3151, 2801, 3277, 3601];
    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_6300_a_Hecke(Kf)
    return MakeCharacter_6300_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], [0], [0], [1], [-2], [6], [-4], [-4], [2], [2], [0], [-2], [0], [4], [12], [-6], [8], [6], [8], [-14], [2], [12], [-4], [0], [2], [-16], [16], [18], [-2], [10], [-12], [-4], [-2], [4], [6], [-8], [-14], [16], [4], [16], [18], [-6], [18], [10], [-22], [-16], [-4], [-8], [0], [10], [14], [6], [14], [24], [-24], [18], [20], [24], [22], [10], [-4], [0], [-4], [-4], [-10], [18], [-28], [6], [6], [22], [12], [2], [16], [-22], [4], [8], [-22], [18], [18], [38], [0], [-6], [-30], [-26], [-24], [38], [14], [-10], [-40], [-4], [-8], [28], [16], [42], [-20], [-24], [36], [-12], [28], [-30], [-24], [-38], [8], [-30], [-24], [38], [32], [12], [-6], [-46], [0], [-6], [38], [-20], [20], [10], [-44], [12], [6], [-2], [22], [34], [-24], [30], [-28], [50], [38], [0], [-16], [6], [0], [30], [0], [42], [-32], [-38], [-8], [12], [0], [-10], [28], [-2], [20], [18], [-14], [-44], [-14], [-48], [44], [-14], [-46], [-36], [-44], [-12], [-12], [18], [16], [-48], [-2], [-28], [10], [-22], [-20], [-36], [30], [-48], [56], [-38], [14], [0], [38], [-50], [42], [-6], [24], [-4], [48], [-38], [40], [-34], [-36], [-20], [38], [-20], [-62], [-24], [-6], [-44], [-42], [24], [-46], [18], [52], [-8], [-30], [-28], [-34], [62], [0], [-36], [30], [-24], [-2], [-18], [-60], [0], [8], [42], [6], [-52], [-2], [20], [48], [4], [-36], [50], [8], [-20], [2], [14], [22], [16], [-42], [-16], [36], [50], [32], [-58], [-48], [66], [-30], [-44], [-8], [6], [4], [0], [-2], [66], [18], [-40], [-10], [44], [16], [-30], [-64], [-36], [48], [-32], [8], [18], [-70], [36], [-42], [-50], [-16], [-50], [26], [36], [40], [-50], [-76], [-74], [10], [-22], [-60], [-44], [-42], [-28], [-64], [54], [-42], [28], [18], [8], [26], [48], [-22], [66], [46], [48], [24], [-60], [-48], [-10], [-36], [-62], [-18], [2], [-16], [-24], [-18], [84], [54], [32], [34], [0], [8], [-36], [-24], [36], [-10], [-18], [-44], [66], [8], [70], [-50], [-46], [34], [22], [24], [50], [50], [4], [-62], [-34], [-36], [-10], [10], [-54], [-4], [-22], [-24], [56], [-62], [38], [-40], [-4], [-58], [86], [38], [-10], [72], [32], [-32], [-8], [30], [40], [86], [56], [10], [-42], [-12], [48], [78], [30], [-46], [52], [-72], [-48], [-76], [-38], [-18], [-24], [-74], [-28], [-16], [12], [-70], [38], [-22], [40], [46], [-26], [-52], [10], [84], [-36], [94], [58], [76], [-90], [62], [-80], [-34], [60], [22], [38], [-44], [-14], [72], [-38], [32], [-6], [28], [64], [80], [-82], [32], [-28], [62], [36], [-2], [-68], [-94], [-66], [-8], [-20], [48], [-62], [-58], [-6], [-28], [16], [-84], [8], [14], [78], [52], [16], [62], [-58], [14], [84], [58], [-20], [46], [8], [-60], [-92], [34], [10], [62], [84], [-26], [16], [22], [-54], [84], [76]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6300_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6300_2_a_w();".
// 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_6300_2_a_w(:prec:=1)
    chi := MakeCharacter_6300_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_6300_2_a_w();".
// 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_6300_2_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6300_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![2, 1]>,<13,R![-6, 1]>,<17,R![4, 1]>,<37,R![2, 1]>],Snew);
    return Vf;
end function;