
// Make newform 3600.2.a.o in Magma, downloaded from the LMFDB on 12 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3600_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_3600_2_a_o();".
// 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_3600_2_a_o();".
// 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_3600_a();"
function MakeCharacter_3600_a()
    N := 3600;
    order := 1;
    char_gens := [3151, 901, 2801, 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_3600_a_Hecke( : Kf := Kf)
    return MakeCharacter_3600_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]];
    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], [2], [6], [-2], [0], [-4], [0], [8], [2], [-2], [4], [-8], [-6], [10], [2], [8], [12], [-4], [0], [-4], [10], [-8], [8], [14], [12], [10], [-6], [-2], [-18], [18], [20], [20], [8], [22], [-16], [12], [14], [10], [2], [12], [-4], [-22], [0], [-12], [-26], [-28], [-10], [14], [20], [22], [-18], [18], [-4], [0], [8], [2], [18], [-16], [-6], [-12], [12], [-4], [-2], [8], [-28], [12], [10], [14], [0], [-2], [6], [-20], [16], [-20], [2], [-22], [-10], [-10], [22], [32], [-4], [0], [36], [-30], [32], [-12], [-6], [12], [20], [18], [-18], [0], [-24], [0], [-22], [-16], [-38], [28], [18], [-44], [10], [8], [32], [12], [-6], [0], [2], [-22], [26], [-2], [-20], [-32], [-2], [24], [-48], [-26], [-50], [2], [36], [-2], [-4], [8], [-32], [30], [-40], [18], [-14], [-40], [-24], [-32], [2], [18], [-30], [54], [-32], [-2], [30], [-52], [8], [-6], [-28], [-30], [40], [-14], [-42], [-20], [36], [22], [18], [24], [12], [-12], [-48], [0], [-30], [-8], [28], [12], [-46], [38], [-18], [-42], [16], [8], [-18], [-10], [54], [-30], [-18], [32], [-44], [-40], [-30], [28], [-12], [34], [-10], [-62], [22], [-14], [18], [16], [60], [-58], [-16], [50], [52], [36], [-4], [-52], [-12], [12], [-6], [-18], [6], [18], [16], [-60], [48], [-18], [-50], [50], [18], [0], [-4], [50], [8], [52], [-12], [14], [12], [-20], [22], [38], [-22], [-8], [34], [-18], [-40], [30], [-46], [12], [30], [34], [40], [-22], [2], [46], [20], [-32], [18], [-56], [-28], [30], [-46], [-30], [12], [36], [-32], [34], [10], [-26], [60], [-42], [42], [40], [-4], [42], [-2], [-68], [70], [-46], [30], [-38], [8], [-22], [-68], [54], [-28], [70], [-14], [18], [40], [60], [-2], [64], [-66], [62], [68], [76], [-40], [-28], [14], [52], [-70], [-58], [-18], [36], [-32], [-8], [22], [-52], [-28], [-24], [-82], [40], [-50], [28], [12], [54], [42], [26], [-60], [48], [54], [50], [-12], [76], [-42], [40], [36], [-32], [32], [-68], [-10], [-20], [6], [36], [0], [18], [-36], [32], [-10], [10], [-68], [-4], [30], [88], [-68], [-72], [-86], [54], [-58], [-60], [-36], [32], [-26], [22], [-62], [40], [-44], [-12], [12], [-50], [-66], [-38], [-2], [6], [38], [40], [-32], [34], [-30], [-18], [-52], [-68], [-42], [28], [-48], [48], [54], [50], [54], [0], [42], [58], [-4], [62], [78], [-48], [50], [-72], [-64], [18], [-66], [42], [82], [-20], [-64], [0], [-72], [-78], [50], [-8], [-24], [-30], [-8], [-12], [74], [38], [-82], [60], [36], [-72], [-38], [84], [92], [50], [-6], [-30], [8], [-8], [-64], [40], [-10], [88], [-12], [-50], [34], [38], [18], [0], [-32], [42], [-102], [-16], [-90], [-104], [18], [36], [8], [92], [-32], [-40], [78], [58], [36], [-60], [22], [-88], [30], [-44], [18], [76], [-10], [48], [-40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3600_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3600_2_a_o();".
// 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_3600_2_a_o(:prec:=1)
    chi := MakeCharacter_3600_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_3600_2_a_o();".
// 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_3600_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![-2, 1]>,<13,R![-6, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;