
// Make newform 7200.2.a.l in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7200_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_7200_2_a_l();".
// 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_7200_2_a_l();".
// 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_7200_a();"
function MakeCharacter_7200_a()
    N := 7200;
    order := 1;
    char_gens := [6751, 901, 6401, 577];
    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_7200_a_Hecke(Kf)
    return MakeCharacter_7200_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], [-2], [-4], [6], [2], [-8], [6], [2], [-4], [-2], [10], [-2], [2], [2], [0], [2], [-6], [-12], [-10], [8], [10], [6], [-10], [-14], [2], [6], [-14], [-6], [6], [4], [2], [16], [-10], [-20], [-10], [6], [-14], [-6], [0], [-18], [-4], [-2], [-22], [16], [-4], [2], [-18], [-10], [2], [-8], [-18], [-20], [26], [-2], [-18], [-28], [22], [10], [-26], [18], [-22], [28], [6], [2], [-4], [14], [-18], [-2], [-30], [16], [-18], [-10], [-24], [14], [-26], [-26], [-18], [14], [0], [10], [-20], [22], [16], [18], [-22], [6], [-6], [-38], [6], [24], [38], [-20], [40], [14], [18], [6], [22], [-10], [-14], [-46], [-30], [18], [-4], [38], [-10], [18], [-24], [-26], [-2], [22], [-30], [-8], [-36], [18], [30], [42], [-30], [-16], [10], [-10], [26], [34], [-12], [-18], [-10], [-40], [-2], [6], [-16], [-42], [-4], [-10], [-10], [-30], [-38], [-14], [-30], [6], [20], [38], [-6], [-50], [2], [24], [-10], [18], [16], [46], [-10], [-14], [-2], [2], [2], [-28], [-56], [-6], [-34], [-38], [38], [58], [22], [36], [42], [46], [44], [-18], [50], [26], [24], [-26], [12], [-26], [0], [54], [28], [50], [-38], [-14], [30], [20], [62], [42], [-18], [-22], [30], [-66], [46], [36], [38], [-14], [4], [-66], [30], [-14], [14], [-34], [-30], [14], [-30], [68], [-34], [-46], [32], [10], [-16], [-46], [70], [12], [-18], [38], [-46], [38], [-56], [70], [38], [-34], [-30], [-6], [58], [-56], [-50], [42], [-66], [-54], [-6], [16], [-42], [36], [38], [24], [20], [26], [22], [42], [22], [66], [-24], [60], [26], [52], [26], [-66], [-14], [-32], [-10], [60], [-64], [6], [54], [-18], [58], [-18], [42], [-8], [-50], [-30], [-6], [6], [-14], [-34], [22], [-50], [34], [-24], [18], [38], [14], [18], [22], [-38], [62], [0], [-26], [26], [-18], [46], [22], [-44], [-66], [-12], [-38], [30], [-6], [4], [30], [42], [40], [10], [-66], [30], [42], [60], [22], [30], [-4], [34], [-24], [58], [-2], [42], [-8], [18], [-76], [-58], [78], [14], [-64], [-34], [78], [-38], [-50], [46], [66], [-22], [40], [-44], [-58], [-22], [-44], [78], [-2], [-38], [66], [-2], [40], [-34], [66], [-62], [-62], [58], [-72], [42], [68], [54], [-62], [74], [-58], [-42], [-50], [-54], [6], [-4], [-62], [56], [26], [-14], [36], [-86], [-4], [-58], [-18], [10], [6], [90], [-56], [-44], [-86], [54], [-34], [38], [-78], [96], [-62], [-2], [-54], [-14], [6], [-60], [-40], [6], [54], [-20], [22], [24], [-60], [78], [42], [94], [-6], [18], [-58], [18], [-40], [-42], [-28], [-82], [46], [2], [22], [-6], [-80], [42], [-12], [62], [72], [-58], [4], [-42], [66], [18], [-82], [-62], [70], [12], [-82], [-2], [54], [48], [62], [18], [-6], [60], [-26], [-70], [-48], [-98], [10], [-90], [94], [22], [-30], [-80], [54], [-6], [-14], [-62], [-68], [96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7200_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7200_2_a_l();".
// 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_7200_2_a_l(:prec:=1)
    chi := MakeCharacter_7200_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_7200_2_a_l();".
// 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_7200_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![4, 1]>,<13,R![-6, 1]>,<17,R![-2, 1]>,<19,R![8, 1]>,<23,R![-6, 1]>],Snew);
    return Vf;
end function;