
// Make newform 4312.2.a.a in Magma, downloaded from the LMFDB on 15 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_4312_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_4312_2_a_a();".
// 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_4312_2_a_a();".
// 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_4312_a();"
function MakeCharacter_4312_a()
    N := 4312;
    order := 1;
    char_gens := [1079, 2157, 3433, 3137];
    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_4312_a_Hecke( : Kf := Kf)
    return MakeCharacter_4312_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], [-2], [-4], [0], [1], [-2], [-2], [-8], [0], [-6], [-2], [-10], [-10], [-8], [-6], [-6], [6], [2], [-12], [-8], [2], [-12], [8], [8], [-12], [-10], [-6], [-4], [10], [18], [8], [-4], [10], [-16], [-2], [4], [-20], [-20], [0], [6], [-12], [16], [-16], [26], [22], [2], [-4], [2], [12], [24], [6], [16], [-2], [10], [-8], [-8], [8], [20], [-26], [-2], [-4], [18], [-28], [-30], [32], [-2], [-12], [14], [0], [-14], [24], [-36], [-22], [-18], [28], [-18], [-6], [-28], [-18], [-38], [30], [2], [-20], [-28], [-20], [4], [22], [42], [-6], [-8], [18], [-20], [-8], [24], [-4], [36], [0], [12], [28], [14], [-28], [6], [0], [38], [-12], [-28], [-30], [-30], [-48], [-2], [40], [-30], [10], [2], [-32], [-38], [34], [14], [34], [-40], [12], [-42], [-2], [12], [-34], [-6], [-38], [46], [6], [42], [28], [-8], [-32], [-54], [-38], [-6], [-24], [-8], [-20], [34], [-28], [30], [32], [48], [4], [-34], [6], [18], [-14], [16], [-46], [-12], [36], [-4], [-52], [-8], [-8], [44], [-30], [-54], [20], [18], [44], [26], [-2], [-6], [-32], [14], [2], [-8], [-28], [-20], [-20], [14], [-38], [-12], [-64], [-10], [44], [62], [-16], [0], [-6], [6], [56], [24], [-10], [-14], [-10], [-64], [12], [-36], [52], [-12], [16], [56], [22], [46], [50], [-44], [-58], [40], [-8], [18], [-6], [-12], [-46], [36], [34], [-42], [62], [-48], [16], [-42], [-12], [-40], [44], [-6], [0], [2], [-6], [4], [-50], [-32], [-12], [-22], [-52], [0], [-28], [8], [22], [28], [-64], [42], [26], [-60], [-28], [6], [60], [62], [12], [32], [38], [-38], [6], [-16], [-26], [-42], [-12], [-16], [-26], [-30], [0], [0], [-30], [-44], [46], [-24], [28], [46], [-44], [12], [50], [38], [24], [-14], [-30], [-52], [-16], [10], [28], [28], [4], [-64], [-22], [-54], [36], [54], [10], [40], [-16], [-8], [-84], [-2], [-32], [16], [-34], [-50], [-6], [30], [-6], [-30], [-42], [0], [6], [-14], [62], [32], [72], [24], [34], [10], [-44], [60], [-20], [10], [60], [20], [-24], [14], [-88], [-54], [62], [-4], [-12], [-46], [-46], [32], [-44], [30], [-12], [-42], [-14], [-40], [10], [-48], [20], [-14], [-52], [-46], [-26], [54], [-24], [-32], [48], [2], [6], [-26], [-56], [-76], [-10], [-14], [-42], [80], [66], [36], [-40], [-92], [56], [-48], [8], [-34], [78], [84], [54], [46], [-62], [-36], [-46], [24], [22], [78], [-48], [-16], [82], [74], [54], [-16], [-22], [52], [22], [-32], [6], [-32], [-6], [10], [60], [-2], [-18], [22], [-100], [-82], [52], [42], [16], [26], [48], [-4], [-28], [-54], [2], [80], [-4], [-14], [-24], [-50], [44], [-40], [-28], [10], [-6], [-62], [-40], [24], [-74], [96], [46], [-6], [14], [-34], [66], [-78], [20], [-8], [38], [84], [-24], [-98], [16], [-8], [-78], [-22], [-24], [-12], [-16], [0], [84], [78], [-34], [-60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4312_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4312_2_a_a();".
// 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_4312_2_a_a(:prec:=1)
    chi := MakeCharacter_4312_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_4312_2_a_a();".
// 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_4312_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4312_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![4, 1]>,<13,R![2, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;