
// Make newform 7200.2.a.bl in Magma, downloaded from the LMFDB on 20 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_bl();".
// 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_bl();".
// 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], [-6], [2], [-6], [-4], [8], [0], [8], [-2], [6], [4], [4], [6], [-6], [-6], [0], [-4], [-12], [8], [-12], [-14], [-8], [-8], [-6], [-12], [2], [-2], [-22], [-10], [6], [16], [4], [8], [-22], [-8], [16], [-14], [2], [-22], [-20], [-12], [6], [-8], [16], [10], [-20], [14], [-22], [20], [-10], [14], [6], [24], [0], [0], [-2], [2], [8], [-26], [-12], [12], [-28], [18], [4], [-20], [-4], [-14], [-6], [0], [10], [10], [8], [-12], [-36], [38], [18], [-26], [-26], [-2], [32], [-28], [-8], [-20], [18], [-16], [20], [14], [28], [-28], [-10], [14], [-36], [4], [0], [26], [24], [2], [12], [14], [4], [2], [-28], [16], [20], [-2], [32], [34], [22], [-34], [-38], [-24], [40], [-2], [-48], [-12], [10], [30], [-6], [28], [18], [-36], [-20], [0], [-10], [8], [-26], [-2], [20], [-12], [32], [-18], [18], [2], [26], [-24], [-30], [-34], [16], [24], [22], [-12], [-38], [-8], [6], [18], [-8], [24], [10], [26], [-32], [48], [-44], [-32], [48], [34], [8], [-20], [20], [6], [42], [14], [-30], [12], [-48], [10], [54], [-22], [-30], [6], [-48], [-4], [16], [-38], [16], [-60], [46], [62], [-34], [46], [22], [54], [-12], [-20], [42], [-56], [-14], [-28], [-20], [4], [56], [-28], [12], [-50], [-2], [50], [38], [12], [68], [32], [10], [-18], [2], [-18], [-40], [-20], [42], [20], [28], [20], [-54], [-4], [-4], [-26], [50], [-30], [-52], [30], [-10], [0], [6], [-10], [12], [-26], [6], [-24], [-34], [-70], [10], [-56], [-8], [-46], [16], [-48], [-34], [-2], [18], [-52], [-28], [12], [-18], [18], [2], [60], [2], [26], [28], [-16], [30], [-2], [-48], [6], [14], [-34], [-30], [-32], [-42], [4], [-22], [60], [30], [14], [-42], [-76], [12], [-42], [-40], [-30], [22], [4], [52], [48], [-4], [-6], [-60], [34], [-58], [70], [8], [16], [44], [-18], [60], [68], [40], [-30], [48], [6], [76], [12], [2], [18], [-42], [36], [32], [74], [-30], [52], [20], [10], [-40], [44], [-4], [-32], [-60], [78], [-36], [74], [8], [48], [58], [76], [20], [86], [-30], [-4], [52], [-42], [-36], [52], [24], [14], [50], [70], [56], [-84], [60], [90], [30], [-66], [16], [-4], [-24], [-28], [-74], [-54], [-38], [-54], [-86], [2], [-40], [-16], [-2], [-6], [54], [76], [12], [-22], [-40], [-80], [80], [-6], [58], [-30], [96], [18], [-18], [-24], [10], [-74], [-68], [-14], [-32], [16], [-18], [-86], [-22], [-14], [40], [-28], [-96], [-40], [38], [42], [72], [8], [34], [-72], [36], [14], [-70], [58], [-64], [-48], [-72], [70], [68], [88], [50], [6], [66], [64], [-8], [-96], [88], [-74], [100], [68], [22], [-26], [50], [6], [-64], [-88], [94], [-46], [-24], [-10], [-8], [78], [52], [28], [-76], [80], [-88], [98], [-18], [-16], [100], [-62], [52], [54], [-4], [-82], [-92], [-26], [-20], [88]];
    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_bl();".
// 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_bl(: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_bl();".
// 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_bl( : 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![6, 1]>,<13,R![-2, 1]>,<17,R![6, 1]>,<19,R![4, 1]>,<23,R![-8, 1]>],Snew);
    return Vf;
end function;