
// Make newform 6300.2.a.k in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6300_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_6300_2_a_k();".
// 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_6300_2_a_k();".
// 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_6300_a();"
function MakeCharacter_6300_a()
    N := 6300;
    order := 1;
    char_gens := [3151, 2801, 3277, 3601];
    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_6300_a_Hecke(Kf)
    return MakeCharacter_6300_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], [0], [0], [-1], [1], [-2], [0], [6], [-1], [-1], [-2], [-7], [8], [-1], [2], [14], [-10], [0], [-3], [9], [0], [1], [-2], [-2], [-10], [12], [2], [4], [9], [9], [-11], [8], [6], [-4], [21], [3], [-4], [16], [18], [18], [4], [12], [24], [-5], [-11], [24], [-4], [12], [-26], [2], [-9], [12], [10], [20], [-22], [9], [-14], [14], [2], [-17], [10], [18], [20], [12], [-6], [-15], [5], [-18], [-23], [-30], [30], [29], [28], [15], [5], [-14], [-25], [26], [-1], [14], [14], [31], [24], [4], [38], [-40], [25], [-33], [18], [8], [-30], [2], [23], [13], [-44], [6], [-14], [-30], [-16], [-13], [37], [-3], [-10], [39], [25], [22], [36], [8], [-5], [-8], [-22], [-23], [-13], [-10], [-35], [3], [44], [14], [2], [36], [18], [34], [-38], [-7], [-24], [-6], [-10], [-20], [-28], [-4], [47], [-4], [-28], [-43], [-24], [-44], [42], [4], [36], [-51], [30], [-18], [9], [-11], [28], [-32], [16], [32], [6], [-31], [-34], [26], [-51], [16], [44], [11], [9], [-12], [48], [-6], [20], [-43], [-32], [-42], [51], [2], [-35], [-10], [59], [36], [15], [-26], [-17], [10], [-4], [-56], [-31], [-10], [-34], [34], [-57], [40], [-3], [10], [39], [-42], [7], [-42], [-47], [36], [-42], [-21], [35], [-22], [37], [16], [-43], [1], [22], [2], [-50], [-10], [2], [-20], [-28], [0], [46], [-44], [-51], [-4], [13], [44], [-1], [24], [-20], [-68], [-5], [-54], [-9], [-61], [-67], [-40], [-7], [-19], [54], [-26], [-56], [7], [-28], [-71], [41], [10], [-44], [-7], [26], [50], [-40], [21], [49], [-32], [15], [62], [-14], [47], [-6], [0], [22], [-14], [-60], [24], [-2], [0], [72], [28], [-54], [76], [53], [52], [22], [38], [-48], [-51], [-46], [42], [32], [-32], [-51], [20], [23], [-13], [-68], [60], [32], [33], [36], [-36], [-75], [2], [-37], [-36], [-4], [32], [78], [28], [-16], [-51], [11], [-63], [-44], [-70], [53], [-6], [-45], [-32], [43], [60], [-20], [78], [0], [66], [52], [-54], [5], [69], [25], [2], [-37], [-70], [-81], [2], [-48], [-26], [-45], [4], [-33], [80], [26], [12], [-44], [-17], [-24], [50], [78], [-83], [-39], [26], [81], [58], [-57], [-81], [78], [-25], [30], [-24], [-36], [-2], [61], [-6], [-14], [16], [-46], [3], [-12], [-59], [31], [45], [-56], [-49], [-6], [-32], [88], [-13], [-18], [2], [15], [6], [78], [96], [-54], [52], [-85], [34], [48], [-11], [6], [10], [38], [3], [51], [45], [18], [-21], [30], [-22], [47], [24], [65], [36], [14], [44], [48], [-15], [81], [81], [28], [-36], [35], [-78], [-37], [8], [49], [54], [9], [-2], [25], [85], [-44], [-102], [0], [51], [-38], [57], [-51], [-6], [54], [-42], [-29], [-30], [48], [6], [66], [-53], [84], [-43], [-51], [62], [-64], [-84], [92], [-34], [-34], [-100], [-72], [-96], [-50], [68], [49], [6], [52], [-6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6300_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6300_2_a_k();".
// 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_6300_2_a_k(:prec:=1)
    chi := MakeCharacter_6300_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_6300_2_a_k();".
// 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_6300_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6300_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-1, 1]>,<13,R![2, 1]>,<17,R![0, 1]>,<37,R![7, 1]>],Snew);
    return Vf;
end function;