
// Make newform 6300.2.a.n in Magma, downloaded from the LMFDB on 25 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_n();".
// 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_n();".
// 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], [4], [6], [2], [6], [2], [-6], [-2], [4], [-8], [4], [4], [6], [-4], [14], [-4], [0], [10], [0], [-16], [-8], [-10], [-16], [-16], [-6], [-14], [18], [16], [12], [18], [10], [-22], [16], [-10], [8], [24], [-2], [-4], [-10], [0], [-8], [10], [18], [-24], [16], [20], [2], [-26], [16], [10], [12], [6], [-10], [12], [-14], [28], [6], [-28], [-6], [12], [-12], [-14], [14], [-8], [-8], [18], [22], [-2], [0], [16], [24], [-28], [28], [-2], [10], [-22], [-10], [40], [-2], [16], [10], [-30], [-30], [-18], [-20], [0], [24], [12], [4], [12], [-12], [-36], [36], [-12], [44], [20], [10], [-20], [6], [40], [10], [-8], [-38], [0], [18], [0], [-34], [-40], [-44], [-34], [26], [8], [-2], [12], [-24], [-34], [-4], [14], [-12], [38], [26], [-2], [-38], [-6], [-44], [0], [-34], [16], [-54], [20], [-8], [24], [30], [-6], [4], [-34], [18], [34], [-10], [-24], [-22], [6], [32], [2], [6], [-46], [18], [44], [-36], [24], [12], [-24], [24], [36], [-24], [14], [-44], [6], [-14], [20], [24], [-18], [-8], [52], [18], [-34], [-54], [36], [-30], [-32], [8], [22], [-36], [20], [-18], [16], [-22], [-44], [-16], [-40], [-42], [-26], [-24], [28], [-4], [50], [20], [6], [2], [4], [0], [-42], [-6], [30], [-48], [-42], [16], [30], [42], [-30], [10], [64], [30], [54], [-46], [-70], [10], [20], [36], [-16], [-24], [-16], [-26], [-52], [68], [-2], [62], [-54], [22], [18], [8], [12], [30], [14], [32], [-16], [-44], [-40], [-50], [32], [18], [-20], [-16], [30], [10], [60], [-12], [54], [2], [-8], [34], [-34], [24], [-48], [-72], [20], [26], [-36], [-20], [-26], [-26], [-14], [-60], [-74], [-4], [58], [26], [24], [-46], [-62], [62], [18], [38], [6], [28], [4], [54], [26], [20], [14], [-44], [-2], [8], [58], [-34], [-46], [-36], [-24], [-40], [-32], [58], [-52], [-8], [84], [22], [-10], [64], [46], [48], [-18], [72], [64], [-24], [30], [-18], [76], [-60], [-58], [6], [44], [78], [-16], [-80], [-26], [-46], [-88], [-24], [-24], [18], [-54], [0], [-22], [2], [-28], [-16], [-50], [42], [2], [-76], [48], [64], [-66], [-70], [-48], [36], [38], [-46], [-26], [-58], [50], [-32], [12], [-72], [22], [-6], [-62], [-56], [-24], [94], [76], [80], [86], [36], [66], [-40], [20], [90], [-34], [-36], [-42], [0], [-10], [54], [-36], [-12], [26], [-46], [-8], [0], [90], [28], [36], [4], [-6], [12], [-30], [-4], [-70], [74], [-78], [18], [-12], [52], [8], [-6], [12], [2], [72], [-14], [68], [-98], [42], [-84], [-84], [-34], [-36], [92], [-70], [6], [-84], [44], [-64], [16], [14], [44], [-40], [-54], [-38], [26], [-4], [-54], [-36], [-58], [84], [6], [68], [-84], [14], [18], [-102], [-32], [16], [4], [40], [0], [14], [-84], [-42], [74], [-22], [-24], [26], [18], [-26], [-22], [-42], [-56]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6300_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-4, 1]>,<13,R![-6, 1]>,<17,R![-2, 1]>,<37,R![-4, 1]>],Snew);
    return Vf;
end function;