
// Make newform 5712.2.a.q in Magma, downloaded from the LMFDB on 14 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5712_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_5712_2_a_q();".
// 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_5712_2_a_q();".
// 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_5712_a();"
function MakeCharacter_5712_a()
    N := 5712;
    order := 1;
    char_gens := [2143, 1429, 3809, 3265, 2689];
    v := [1, 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_5712_a_Hecke( : Kf := Kf)
    return MakeCharacter_5712_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], [1], [-2], [1], [0], [-6], [1], [0], [8], [-6], [8], [10], [-6], [-12], [0], [-10], [8], [6], [-12], [0], [-6], [8], [-16], [2], [2], [-6], [-16], [-16], [-6], [-6], [-8], [4], [-22], [4], [-18], [16], [18], [-8], [8], [-2], [-4], [-10], [-16], [-6], [18], [24], [-8], [8], [4], [10], [-30], [-24], [2], [0], [-6], [16], [-18], [-24], [-6], [-22], [4], [-14], [32], [0], [26], [-6], [36], [2], [24], [-22], [-6], [-16], [-8], [-2], [8], [-8], [6], [-18], [2], [34], [12], [-34], [16], [-22], [-16], [-4], [34], [-38], [-6], [16], [-8], [-40], [-8], [20], [-8], [-24], [2], [10], [16], [26], [8], [-18], [-24], [-6], [-32], [18], [8], [26], [0], [26], [0], [14], [-14], [-4], [16], [-30], [36], [24], [42], [-12], [10], [-6], [30], [-24], [-36], [-10], [50], [8], [-40], [-14], [4], [16], [32], [30], [-22], [26], [-14], [4], [42], [-6], [-12], [2], [16], [8], [-38], [-40], [14], [-54], [56], [-8], [-22], [-30], [4], [-16], [0], [8], [16], [-30], [-22], [30], [32], [42], [8], [24], [-30], [48], [56], [30], [-6], [-2], [-28], [26], [24], [10], [8], [42], [-16], [10], [-16], [-54], [-8], [60], [-6], [18], [64], [-14], [-54], [-64], [-14], [-16], [2], [-48], [-20], [34], [-32], [-6], [66], [-30], [2], [24], [26], [16], [50], [-46], [-32], [-70], [56], [36], [-14], [-32], [10], [-30], [-16], [-16], [0], [-38], [-32], [18], [40], [-58], [14], [56], [2], [-8], [32], [-10], [58], [-16], [8], [-8], [14], [52], [40], [74], [16], [-24], [34], [66], [0], [-16], [-32], [56], [48], [-50], [18], [-24], [-16], [-76], [20], [0], [-2], [-30], [32], [42], [26], [20], [-22], [44], [22], [-14], [8], [-60], [30], [-50], [-14], [48], [78], [2], [-56], [-42], [-58], [-28], [-70], [32], [2], [48], [-44], [-26], [-38], [48], [-40], [-24], [56], [26], [-20], [80], [26], [18], [-64], [-22], [2], [76], [-54], [36], [10], [-50], [-64], [2], [4], [32], [50], [62], [-64], [8], [-56], [42], [-68], [-10], [-24], [38], [32], [66], [-62], [36], [-48], [-6], [-48], [16], [82], [18], [-52], [82], [-54], [-48], [10], [42], [-56], [4], [-48], [42], [2], [2], [80], [-16], [8], [-76], [-34], [-30], [-70], [-16], [-34], [-70], [30], [56], [6], [-64], [22], [-76], [-88], [54], [-64], [82], [-90], [8], [-58], [-30], [16], [52], [-78], [24], [-62], [26], [-8], [-48], [-16], [-86], [-2], [40], [34], [-28], [-20], [-72], [70], [-64], [50], [76], [-8], [34], [2], [34], [30], [58], [-16], [58], [-20], [0], [-32], [2], [-64], [-88], [-94], [-90], [36], [-72], [-40], [2], [48], [-86], [16], [78], [22], [-62], [-80], [-38], [90], [-32], [-98], [2], [16], [-60], [-46], [-10], [44], [-48], [58], [70], [72], [96], [-14], [-80], [70], [-106], [72], [72], [-6], [-54], [0], [-46], [-56], [104]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5712_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5712_2_a_q();".
// 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_5712_2_a_q(:prec:=1)
    chi := MakeCharacter_5712_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_5712_2_a_q();".
// 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_5712_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5712_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<11,R![0, 1]>,<13,R![6, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;