
// Make newform 5376.2.a.g in Magma, downloaded from the LMFDB on 12 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5376_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_5376_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_5376_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_5376_a();"
function MakeCharacter_5376_a()
    N := 5376;
    order := 1;
    char_gens := [2815, 5125, 1793, 4609];
    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_5376_a_Hecke( : Kf := Kf)
    return MakeCharacter_5376_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], [-2], [2], [2], [0], [0], [8], [-8], [-4], [2], [10], [-4], [-12], [4], [-2], [2], [-8], [-6], [-16], [0], [14], [-10], [-2], [4], [-6], [-4], [14], [-8], [0], [-18], [-16], [-4], [-8], [-22], [14], [-24], [18], [-2], [-6], [16], [-2], [12], [-20], [-22], [-12], [-24], [26], [6], [8], [14], [-24], [-18], [-16], [18], [20], [8], [-26], [-4], [22], [-8], [-20], [6], [-4], [-6], [30], [-18], [-14], [6], [-24], [12], [-24], [26], [-12], [24], [10], [-26], [-26], [20], [24], [-8], [14], [0], [18], [34], [2], [10], [-32], [40], [4], [-16], [18], [10], [36], [18], [42], [-28], [16], [-10], [4], [-36], [26], [18], [34], [-16], [-34], [24], [26], [-12], [-44], [-46], [4], [-40], [-10], [-4], [24], [24], [10], [-18], [18], [42], [-14], [32], [0], [4], [-20], [20], [-10], [-10], [40], [24], [-52], [-30], [22], [30], [8], [-14], [-54], [-36], [12], [32], [-30], [6], [0], [-30], [34], [-8], [-24], [-52], [30], [-30], [24], [22], [-8], [-32], [-14], [26], [-50], [-38], [6], [8], [36], [-30], [56], [40], [38], [-2], [-6], [-46], [-54], [24], [-26], [-32], [6], [-2], [64], [-32], [-26], [48], [12], [-4], [22], [-24], [18], [16], [24], [-10], [52], [2], [-22], [-46], [-62], [-10], [54], [-18], [-4], [30], [36], [60], [56], [-2], [58], [4], [-2], [40], [-6], [42], [-8], [14], [30], [-32], [28], [-60], [-50], [24], [-62], [-8], [-8], [68], [40], [-30], [56], [-40], [44], [6], [64], [48], [-26], [-12], [60], [-40], [-58], [52], [24], [-42], [-20], [-6], [-52], [50], [36], [24], [40], [70], [-40], [28], [-24], [-6], [-56], [-64], [6], [48], [34], [54], [-66], [20], [-16], [6], [-2], [-72], [74], [-2], [22], [-6], [-76], [12], [22], [-18], [-48], [-54], [62], [58], [-56], [42], [16], [-10], [32], [-22], [0], [16], [16], [40], [38], [-28], [-48], [-22], [20], [-4], [54], [-68], [-32], [-62], [-12], [80], [66], [-20], [-54], [40], [-76], [-26], [48], [-24], [50], [2], [-62], [-42], [-58], [-80], [-48], [-56], [-48], [2], [-10], [0], [26], [36], [-16], [-14], [-6], [36], [-42], [-18], [-16], [-6], [22], [-2], [-44], [88], [-24], [40], [24], [-48], [-12], [70], [92], [4], [-34], [2], [40], [32], [-6], [-54], [24], [-48], [42], [50], [-30], [-48], [-34], [-76], [-10], [-60], [0], [32], [38], [-12], [-68], [-46], [-24], [40], [-42], [16], [-6], [-56], [-22], [-6], [-64], [98], [-90], [-8], [-88], [-76], [-24], [16], [48], [48], [50], [10], [54], [-62], [-6], [-40], [-78], [44], [-4], [40], [14], [54], [60], [14], [54], [-70], [68], [-8], [-10], [-80], [-102], [-74], [-64], [-50], [-14], [-32], [-78], [-22], [-52], [-56], [30], [76], [28], [42], [72], [22], [74], [-26], [-78], [24], [36], [-14], [68], [0], [38], [-64], [48], [-74], [-66], [-70], [-10], [-48], [-32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5376_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5376_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_5376_2_a_g(:prec:=1)
    chi := MakeCharacter_5376_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_5376_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_5376_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5376_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<11,R![2, 1]>,<13,R![-2, 1]>,<19,R![0, 1]>,<29,R![-8, 1]>],Snew);
    return Vf;
end function;