
// Make newform 8372.2.a.c in Magma, downloaded from the LMFDB on 21 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8372_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_8372_2_a_c();".
// 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_8372_2_a_c();".
// 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_8372_a();"
function MakeCharacter_8372_a()
    N := 8372;
    order := 1;
    char_gens := [4187, 5981, 1289, 8009];
    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_8372_a_Hecke(Kf)
    return MakeCharacter_8372_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+1)]];
    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], [1], [5], [-1], [5], [0], [1], [5], [3], [-3], [5], [2], [-8], [-2], [-3], [-7], [-3], [-2], [13], [8], [8], [10], [-6], [12], [-7], [12], [-14], [6], [-15], [-14], [11], [-20], [6], [-6], [10], [-2], [-21], [8], [-17], [-21], [-20], [-4], [18], [-1], [-12], [11], [0], [6], [7], [-20], [-22], [-1], [8], [22], [6], [-20], [-14], [-25], [-7], [34], [-4], [12], [-9], [26], [-20], [-22], [-33], [-10], [1], [25], [17], [-10], [-20], [12], [16], [-21], [6], [-10], [25], [27], [31], [-19], [34], [25], [-4], [-11], [5], [30], [-19], [28], [-34], [41], [-4], [-19], [23], [5], [37], [20], [27], [3], [5], [12], [-26], [13], [-9], [-31], [35], [40], [-22], [5], [-18], [-4], [8], [10], [28], [14], [18], [-26], [-10], [-19], [-15], [-24], [27], [-10], [10], [-42], [-24], [4], [-4], [-31], [14], [-2], [-1], [2], [34], [-50], [-54], [-21], [25], [-6], [44], [-28], [-16], [-34], [10], [42], [-2], [12], [-34], [10], [13], [24], [-10], [12], [34], [18], [-18], [12], [34], [12], [32], [-15], [-3], [30], [1], [58], [-7], [52], [-55], [34], [24], [-23], [36], [61], [-52], [-62], [25], [-10], [-32], [-21], [-49], [39], [-48], [11], [7], [0], [-50], [-47], [56], [-16], [0], [30], [16], [42], [-13], [-8], [38], [-42], [-27], [-62], [41], [2], [-10], [12], [4], [-21], [8], [-42], [27], [20], [-6], [62], [0], [-26], [-48], [6], [65], [-6], [-42], [50], [-52], [29], [46], [-42], [-26], [47], [-40], [22], [-17], [53], [-47], [-42], [0], [18], [46], [54], [0], [-12], [-12], [-34], [-32], [-20], [-18], [48], [65], [-38], [53], [66], [-58], [-64], [63], [-26], [32], [-32], [-38], [-46], [-58], [41], [-10], [-48], [57], [-62], [-2], [47], [-3], [-50], [-31], [-63], [8], [-67], [-32], [14], [-56], [19], [-18], [-22], [11], [-52], [-36], [-75], [33], [4], [-71], [63], [10], [-53], [-48], [-12], [6], [-36], [12], [-80], [-77], [22], [53], [30], [-34], [68], [26], [3], [-50], [-40], [0], [22], [35], [-82], [-60], [47], [-4], [55], [36], [41], [-17], [-90], [81], [-66], [58], [-15], [28], [-31], [18], [76], [34], [-19], [-20], [-23], [32], [62], [0], [90], [72], [-4], [-12], [-12], [-7], [38], [2], [13], [-46], [-94], [-49], [-46], [69], [9], [46], [-28], [10], [-17], [51], [-72], [-78], [81], [92], [84], [-48], [-54], [-57], [-48], [33], [-8], [-47], [20], [-12], [-23], [69], [49], [48], [60], [-16], [48], [-78], [-61], [-77], [-60], [-48], [-38], [11], [68], [4], [-30], [90], [38], [-36], [4], [68], [28], [9], [-81], [-6], [-63], [-30], [-52], [33], [-14], [76], [83], [22], [50], [21], [49], [31], [62], [21], [38], [-62], [18], [20], [-1], [-62], [-95], [-64], [-100], [-44], [-72], [-88], [-14], [56], [29], [-66], [-74], [-90], [-56], [61], [-63], [66], [-102], [43], [42]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8372_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8372_2_a_c();".
// 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_8372_2_a_c(:prec:=1)
    chi := MakeCharacter_8372_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_8372_2_a_c();".
// 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_8372_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8372_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![-4, 1]>],Snew);
    return Vf;
end function;