
// Make newform 6900.2.a.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6900_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_6900_2_a_e();".
// 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_6900_2_a_e();".
// 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_6900_a();"
function MakeCharacter_6900_a()
    N := 6900;
    order := 1;
    char_gens := [3451, 4601, 277, 1201];
    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_6900_a_Hecke(Kf)
    return MakeCharacter_6900_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], [0], [-2], [-6], [2], [1], [6], [-4], [-8], [6], [-8], [-12], [6], [-6], [-10], [-8], [-6], [-2], [-10], [12], [6], [-8], [-6], [-8], [12], [2], [6], [-8], [6], [6], [20], [-6], [8], [4], [-20], [0], [24], [-6], [2], [0], [-14], [0], [-10], [8], [16], [12], [-10], [-24], [-6], [-10], [-12], [-12], [24], [-6], [32], [-14], [-6], [28], [6], [28], [-18], [28], [-12], [-28], [-32], [-12], [2], [-36], [-12], [-8], [-20], [2], [-24], [-6], [-14], [-30], [14], [-12], [-34], [24], [16], [-40], [24], [18], [-32], [-6], [40], [12], [12], [-8], [-30], [20], [24], [6], [-6], [-20], [2], [-44], [30], [36], [-42], [26], [34], [-36], [12], [-30], [-10], [-8], [4], [-18], [26], [-34], [-18], [-32], [-12], [-24], [24], [14], [46], [-6], [-36], [-16], [42], [-34], [6], [28], [-20], [44], [-48], [14], [16], [-6], [38], [42], [-32], [-42], [30], [44], [30], [40], [-12], [2], [-24], [34], [48], [8], [24], [22], [-30], [-20], [-24], [28], [12], [26], [42], [4], [18], [24], [-54], [-8], [-12], [54], [24], [-52], [46], [-34], [-60], [0], [2], [-12], [-44], [8], [-42], [32], [42], [-8], [26], [-8], [12], [-14], [24], [24], [6], [-2], [-44], [-34], [-30], [-14], [60], [-22], [-18], [12], [42], [-58], [64], [18], [-24], [30], [8], [-14], [38], [12], [24], [-22], [24], [-18], [-4], [-2], [-66], [-44], [-12], [-6], [-10], [-32], [42], [0], [48], [-58], [26], [-42], [16], [24], [-10], [-18], [54], [-8], [6], [-38], [-10], [-22], [-42], [28], [24], [50], [-42], [30], [18], [36], [8], [40], [14], [24], [54], [-32], [-24], [62], [72], [-32], [-18], [72], [-22], [-24], [54], [-70], [4], [-12], [-62], [16], [12], [38], [28], [-12], [26], [-66], [-54], [-80], [36], [-10], [76], [28], [-22], [22], [16], [-72], [74], [-34], [24], [60], [26], [-48], [2], [76], [30], [-68], [54], [-40], [-6], [42], [60], [36], [72], [-50], [-66], [38], [12], [-18], [-44], [4], [78], [-70], [-12], [-22], [-38], [36], [14], [-60], [-38], [-24], [-54], [-30], [-20], [48], [38], [-54], [18], [-68], [-6], [2], [52], [54], [16], [90], [38], [14], [-92], [-48], [-18], [14], [24], [-28], [24], [-46], [48], [74], [6], [86], [-92], [34], [-54], [42], [86], [18], [6], [-58], [28], [24], [6], [32], [-2], [-42], [4], [-10], [-12], [24], [-12], [-48], [84], [16], [-30], [60], [72], [52], [82], [-72], [-56], [62], [-66], [20], [-60], [6], [14], [-2], [-54], [-12], [40], [66], [82], [-30], [54], [-32], [48], [-22], [36], [-16], [-38], [52], [-72], [2], [-12], [6], [28], [96], [76], [-70], [-102], [14], [-90], [62], [-24], [4], [-78], [-78], [92], [-8], [-102], [52], [78], [70], [12], [36], [-58], [-32], [-54], [66], [-32], [-42], [24], [6], [-92], [12], [78], [-14], [-60], [36], [-78], [20], [30]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6900_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6900_2_a_e();".
// 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_6900_2_a_e(:prec:=1)
    chi := MakeCharacter_6900_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_6900_2_a_e();".
// 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_6900_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6900_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;