
// Make newform 1848.2.a.g in Magma, downloaded from the LMFDB on 20 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1848_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_1848_2_a_g();".
// 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_1848_2_a_g();".
// 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_1848_a();"
function MakeCharacter_1848_a()
    N := 1848;
    order := 1;
    char_gens := [463, 925, 617, 1585, 673];
    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_1848_a_Hecke( : Kf := Kf)
    return MakeCharacter_1848_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], [-1], [-2], [2], [4], [0], [6], [-8], [-10], [-6], [-12], [-8], [-10], [12], [-2], [-4], [0], [-6], [0], [-12], [10], [-14], [-10], [16], [-12], [-10], [18], [16], [20], [10], [12], [-18], [-24], [-10], [12], [24], [-2], [-4], [-2], [8], [-14], [-2], [16], [12], [24], [-12], [-18], [-6], [0], [2], [12], [-14], [-24], [6], [0], [-2], [10], [-4], [6], [-12], [0], [-22], [-18], [20], [18], [-12], [-34], [18], [24], [-8], [-2], [-12], [8], [6], [6], [18], [10], [4], [-10], [0], [34], [8], [36], [2], [26], [14], [-24], [-12], [0], [16], [-12], [-36], [-24], [22], [10], [-36], [-10], [12], [6], [20], [-38], [-28], [2], [44], [-30], [-16], [42], [32], [14], [42], [28], [32], [2], [4], [16], [-34], [-4], [30], [-14], [-42], [-44], [-28], [22], [-10], [8], [16], [-2], [12], [-24], [-40], [-10], [42], [-46], [-18], [-12], [38], [-54], [-4], [-50], [32], [-12], [38], [0], [6], [10], [-4], [-24], [38], [2], [-4], [8], [-28], [-8], [40], [2], [-38], [-18], [-36], [-6], [-24], [-20], [18], [-48], [56], [-10], [2], [14], [-12], [22], [24], [10], [-56], [-54], [-44], [-42], [40], [-34], [-24], [-12], [38], [-6], [-8], [46], [-58], [-28], [58], [16], [34], [4], [28], [6], [-36], [-38], [-14], [-50], [18], [24], [-58], [-16], [-50], [-14], [44], [-10], [24], [-20], [-6], [60], [34], [30], [64], [60], [-72], [58], [32], [-30], [-32], [-50], [-66], [8], [2], [-56], [-60], [-50], [74], [-24], [-8], [68], [46], [4], [-16], [-38], [-68], [-32], [2], [-18], [36], [48], [-52], [28], [-32], [62], [-30], [56], [-8], [20], [20], [48], [38], [18], [-48], [-22], [14], [60], [22], [-52], [-34], [42], [64], [-4], [-26], [14], [-30], [-44], [-18], [-54], [36], [-2], [-58], [-68], [-6], [64], [2], [16], [4], [22], [-22], [-12], [16], [48], [56], [-10], [-20], [-8], [-14], [30], [16], [50], [30], [-12], [-54], [-4], [86], [-34], [-24], [14], [-4], [-12], [-6], [54], [80], [-36], [-60], [-62], [84], [-10], [-48], [14], [64], [-42], [34], [44], [-24], [-38], [20], [-32], [-62], [-62], [-60], [26], [30], [-56], [-22], [2], [-4], [-52], [48], [-66], [-10], [30], [-64], [52], [20], [12], [-34], [-78], [-22], [-32], [38], [58], [86], [32], [78], [-20], [78], [84], [-48], [30], [-76], [-54], [78], [-64], [-50], [-54], [40], [-36], [-46], [-64], [-66], [-70], [-72], [84], [-92], [-86], [-66], [88], [42], [-4], [44], [-32], [78], [-40], [78], [20], [16], [-46], [-30], [-86], [-90], [74], [-72], [50], [-12], [32], [40], [62], [36], [-56], [-62], [46], [4], [-92], [0], [-54], [32], [-54], [20], [78], [-2], [66], [-32], [-70], [-42], [72], [-66], [50], [20], [84], [34], [62], [68], [-100], [-86], [22], [32], [-64], [18], [40], [-18], [-26], [-24], [-84], [58], [54], [12], [-22], [-100], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1848_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1848_2_a_g();".
// 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_1848_2_a_g(:prec:=1)
    chi := MakeCharacter_1848_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_1848_2_a_g();".
// 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_1848_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1848_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;