
// Make newform 7200.2.a.ca in Magma, downloaded from the LMFDB on 19 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_ca();".
// 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_ca();".
// 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], [4], [0], [4], [0], [8], [4], [6], [8], [-4], [-6], [-4], [-4], [12], [0], [-6], [-12], [-16], [0], [8], [-12], [10], [8], [-14], [12], [12], [-10], [8], [4], [-16], [0], [-8], [-14], [8], [4], [-4], [-4], [-4], [-16], [14], [16], [24], [-12], [-8], [-8], [-4], [-4], [2], [-8], [-16], [14], [-16], [24], [-12], [-6], [-24], [-4], [26], [-20], [20], [-12], [0], [-8], [12], [-8], [-16], [28], [10], [24], [0], [-28], [20], [8], [-12], [-30], [28], [30], [10], [16], [-14], [-16], [40], [-8], [20], [30], [-8], [-22], [-20], [-4], [32], [4], [32], [24], [-28], [6], [-10], [12], [-10], [36], [4], [-28], [-10], [-40], [-16], [28], [8], [32], [-26], [20], [-20], [8], [24], [-8], [34], [12], [12], [20], [0], [-30], [8], [-12], [36], [-8], [-42], [2], [-16], [-28], [-4], [-40], [-12], [8], [12], [6], [14], [28], [-12], [12], [-22], [-8], [-18], [44], [-36], [-26], [-32], [12], [-48], [-56], [36], [-28], [2], [-4], [12], [20], [-32], [24], [-2], [16], [10], [-20], [-48], [-12], [-16], [48], [-12], [24], [-28], [30], [-20], [0], [-42], [-48], [16], [-56], [10], [40], [18], [44], [-10], [4], [-32], [-4], [24], [36], [-2], [-12], [44], [-38], [-16], [-64], [20], [-40], [-10], [12], [56], [-2], [-44], [-8], [36], [-10], [56], [-28], [30], [-16], [-12], [56], [20], [-6], [8], [-40], [14], [-36], [-68], [-16], [-26], [4], [30], [-68], [36], [50], [24], [18], [-68], [12], [-2], [-24], [48], [-44], [-16], [20], [-8], [-56], [74], [-4], [12], [-46], [44], [48], [-16], [-44], [24], [-36], [-6], [-56], [48], [4], [32], [40], [4], [-36], [-2], [12], [-54], [-68], [-64], [-18], [-28], [-36], [56], [76], [60], [66], [76], [48], [-40], [54], [-42], [-20], [-12], [70], [-28], [-16], [-24], [16], [60], [12], [10], [-10], [-80], [4], [-8], [-68], [-66], [36], [80], [-40], [36], [-24], [78], [70], [12], [-8], [-48], [-36], [42], [-40], [4], [48], [-28], [40], [-28], [-40], [52], [8], [-40], [12], [-6], [-48], [4], [-44], [30], [-50], [-4], [-68], [38], [48], [32], [-16], [30], [-24], [8], [-42], [-20], [16], [-50], [56], [-84], [-36], [-36], [42], [36], [-8], [52], [-24], [-20], [70], [24], [22], [-12], [-28], [-80], [2], [-40], [-4], [0], [-78], [-28], [0], [4], [-88], [-88], [38], [60], [-14], [-24], [0], [48], [96], [84], [68], [70], [-52], [-80], [20], [32], [-84], [-4], [-10], [16], [-56], [4], [18], [8], [28], [0], [0], [64], [46], [72], [-6], [64], [52], [8], [56], [36], [-24], [92], [-20], [-100], [-46], [-12], [0], [-28], [-80], [-24], [-56], [-86], [40], [14], [-26], [-16], [-92], [48], [2], [-40], [-4], [62], [12], [-16], [32], [-36], [-76], [-8], [-80], [74], [-64], [52], [-48], [-44], [-86], [-28], [92], [-96], [88], [52], [20], [-26], [-56], [64]];
    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_ca();".
// 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_ca(: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_ca();".
// 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_ca( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, 1]>,<11,R![0, 1]>,<13,R![-4, 1]>,<17,R![0, 1]>,<19,R![-8, 1]>,<23,R![-4, 1]>],Snew);
    return Vf;
end function;