
// Make newform 6400.2.a.c in Magma, downloaded from the LMFDB on 27 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6400_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_6400_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_6400_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_6400_a();"
function MakeCharacter_6400_a()
    N := 6400;
    order := 1;
    char_gens := [4351, 4101, 5377];
    v := [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_6400_a_Hecke(Kf)
    return MakeCharacter_6400_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], [-1], [0], [-4], [3], [0], [1], [7], [-4], [-8], [4], [4], [-3], [-8], [0], [-12], [8], [4], [9], [16], [11], [4], [1], [-13], [14], [12], [16], [-13], [-20], [-9], [-12], [8], [-3], [-13], [8], [-20], [-20], [9], [-16], [-16], [-9], [12], [12], [-7], [-12], [24], [-15], [-8], [-24], [-8], [10], [-24], [-15], [13], [18], [-4], [-16], [4], [8], [-6], [-19], [-20], [7], [-12], [6], [-8], [-13], [-9], [-5], [-4], [18], [-28], [-8], [-20], [29], [4], [-12], [8], [-15], [21], [31], [-8], [-28], [-31], [-8], [-19], [-23], [-3], [-8], [-8], [40], [28], [-12], [24], [8], [8], [12], [-3], [-13], [32], [-1], [-16], [24], [37], [-8], [31], [5], [23], [36], [19], [32], [4], [42], [-40], [20], [14], [-24], [24], [-36], [1], [-40], [14], [-48], [-13], [-9], [-16], [-40], [-20], [-40], [32], [-24], [12], [40], [32], [-27], [-39], [8], [8], [12], [-10], [-8], [-36], [24], [-51], [32], [16], [28], [-27], [-5], [-56], [8], [2], [-25], [-24], [-40], [8], [-20], [-34], [-5], [-48], [-8], [-27], [32], [-45], [-31], [36], [4], [24], [1], [56], [-19], [4], [-12], [59], [24], [-29], [-35], [-20], [0], [-4], [-24], [-33], [28], [59], [16], [24], [20], [-1], [-19], [8], [31], [5], [1], [-60], [63], [26], [-41], [44], [15], [36], [12], [-8], [64], [-17], [24], [-52], [-44], [-56], [10], [40], [-2], [4], [8], [27], [-28], [29], [52], [-47], [-12], [36], [-28], [-16], [-7], [20], [-9], [28], [-45], [40], [0], [-27], [52], [65], [28], [-21], [-56], [-12], [65], [-24], [-51], [56], [-9], [45], [4], [24], [55], [-16], [-76], [-24], [5], [-56], [48], [79], [-60], [-37], [-24], [-63], [-40], [-51], [4], [-27], [-28], [57], [8], [64], [9], [-33], [-28], [-38], [27], [-44], [-64], [49], [-45], [-40], [-14], [-8], [11], [-16], [10], [-73], [-24], [-4], [80], [8], [-27], [-32], [-18], [-8], [-16], [34], [20], [8], [3], [-11], [56], [36], [28], [-8], [-40], [-71], [13], [-36], [-32], [-41], [-19], [-23], [24], [-20], [4], [-4], [36], [-20], [-7], [-8], [72], [-22], [-25], [80], [31], [71], [-56], [-75], [-48], [-28], [-42], [62], [89], [56], [76], [32], [48], [-60], [36], [-65], [35], [72], [-56], [-30], [43], [12], [-64], [85], [-12], [-32], [-28], [8], [-64], [5], [-72], [-36], [-55], [22], [12], [-80], [-4], [-77], [-48], [-45], [-50], [24], [20], [-69], [-60], [24], [-7], [-53], [8], [16], [-27], [8], [83], [72], [-40], [80], [-68], [55], [-20], [15], [-15], [70], [-20], [61], [8], [-33], [56], [-52], [64], [-64], [-88], [20], [31], [24], [53], [33], [36], [-11], [24], [93], [-29], [-52], [24], [9], [-64], [-11], [64], [72], [44], [30], [-24], [-23], [55], [-64], [-51], [8], [-37], [-52], [92], [4], [-47], [84], [24], [-100], [36], [-35], [-51], [-104], [-72], [-61], [91], [12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6400_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6400_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_6400_2_a_c(:prec:=1)
    chi := MakeCharacter_6400_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_6400_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_6400_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<7,R![4, 1]>,<11,R![-3, 1]>,<13,R![0, 1]>,<17,R![-1, 1]>,<29,R![8, 1]>,<31,R![-4, 1]>],Snew);
    return Vf;
end function;