
// Make newform 3150.2.a.bs in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3150_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_3150_2_a_bs();".
// 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_3150_2_a_bs();".
// 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
        poly := [-6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_3150_a();"
function MakeCharacter_3150_a()
    N := 3150;
    order := 1;
    char_gens := [2801, 127, 451];
    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_3150_a_Hecke(Kf)
    return MakeCharacter_3150_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 := [[-1, 0], [0, 0], [0, 0], [1, 0], [0, 2], [-2, -1], [2, 0], [4, -1], [-2, 2], [-2, -2], [4, 2], [-2, 0], [6, -2], [-4, 2], [4, 2], [-6, -2], [4, -1], [6, -1], [8, 0], [6, -2], [2, -2], [2, 2], [0, -1], [10, 0], [-6, -4], [-6, 1], [-8, -2], [-8, 0], [2, 2], [10, 4], [10, 2], [4, 1], [8, -4], [4, -1], [-6, 4], [0, 8], [-6, 1], [12, -2], [0, -2], [6, -5], [-4, -4], [2, -5], [2, -6], [2, 8], [14, 2], [12, 2], [12, 0], [4, 0], [0, 3], [-2, -7], [20, 4], [16, -4], [6, -6], [4, 1], [6, -6], [0, 4], [-2, -7], [12, 0], [10, 2], [18, 0], [-16, 5], [6, 5], [8, 5], [-12, 0], [2, 8], [2, -10], [-4, -6], [10, -8], [24, 2], [-6, -1], [18, 2], [8, 8], [-32, 0], [-10, 6], [16, 6], [-12, 2], [-18, 2], [10, -3], [0, 12], [10, 10], [4, -1], [-14, 4], [-8, 4], [10, -4], [8, -12], [0, 4], [10, 0], [-10, -8], [-10, -3], [16, -8], [8, 1], [-24, 6], [22, -6], [0, -8], [16, -4], [-4, 0], [34, -1], [-18, -6], [16, 7], [-10, 8], [4, 6], [-2, 6], [-8, -9], [-24, -4], [-16, 2], [-22, 10], [-16, 7], [-6, -2], [18, -2], [-10, -12], [-4, -12], [18, 4], [10, 8], [-44, 1], [-18, -10], [-16, -4], [-8, -7], [-12, -14], [-30, 4], [-12, -2], [-14, -11], [8, 4], [-34, -1], [-16, 8], [4, 7], [-26, 6], [6, -14], [-20, 10], [-8, -6], [2, -17], [-20, 10], [-30, -6], [-32, -4], [42, 4], [26, -2], [-26, -6], [-6, 3], [-48, -1], [26, -11], [-28, -8], [-12, 11], [-18, 4], [0, 16], [4, -8], [-2, -17], [-32, -2], [2, 13], [-14, -16], [12, 17], [-16, -12], [10, 12], [-18, 4], [-32, 4], [-12, 6], [32, 4], [16, 8], [-22, -2], [-6, 14], [46, -2], [22, -1], [0, 18], [12, -4], [2, 14], [12, -1], [0, -12], [-28, 6], [46, -6], [-50, -3], [4, -16], [26, 5], [-36, 4], [-18, 5], [14, 6], [4, -20], [4, -6], [-6, -16], [32, 0], [-22, 10], [12, -12], [2, 17], [-50, 2], [-20, -13], [-26, 0], [30, -2], [10, -6], [-18, -3], [-6, -14], [-8, 13], [22, 12], [24, -14], [14, 10], [-12, 12], [-20, -12], [-18, -11], [-20, -12], [26, -10], [-4, -16], [-18, -12], [10, -2], [40, 4], [2, 12], [40, -2], [-42, 5], [-6, 24], [-12, 13], [-26, 17], [-12, -2], [-28, 6], [0, 0], [20, 3], [22, -16], [14, -9], [30, 6], [-48, -5], [32, 12], [-46, 2], [-24, 8], [54, 6], [-48, 0], [26, 0], [34, 2], [8, 8], [30, 0], [6, 2], [40, 3], [-18, 2], [30, -6], [-8, 12], [28, 0], [-12, 10], [14, -20], [-12, -17], [2, -10], [-18, -16], [40, 1], [-12, 16], [22, 2], [6, -10], [48, 8], [-12, 0], [-12, -18], [52, 5], [-16, 0], [30, 0], [54, -2], [-12, -22], [28, 0], [-12, 15], [-48, 2], [24, 8], [26, -2], [-30, -8], [-16, 12], [-22, -12], [6, -15], [-16, 4], [-10, -2], [8, -15], [-54, -1], [-10, 12], [-2, -6], [12, -10], [58, -7], [-34, 17], [26, 4], [28, -17], [-18, 12], [-10, 12], [-24, -8], [6, 20], [34, 7], [0, 2], [10, 6], [8, 8], [-6, -14], [24, 0], [-24, -6], [70, 0], [32, 0], [-36, 11], [12, 16], [-14, 14], [20, 18], [38, 11], [-16, -9], [0, -8], [68, 4], [-66, 2], [-52, -2], [-30, 0], [-6, -4], [0, -7], [-18, 16], [76, 3], [-22, -6], [14, 9], [-56, 12], [2, 1], [-12, -17], [8, -15], [26, 12], [22, -20], [30, 10], [0, -26], [40, -2], [-8, 4], [32, -10], [-70, 5], [24, 4], [22, 18], [-64, 0], [-6, 14], [18, -20], [-48, -2], [0, -12], [14, 4], [12, -3], [38, 10], [-2, 14], [30, 0], [28, 1], [16, -12], [-38, 9], [32, -12], [38, -8], [2, -10], [4, 14], [16, -23], [-18, -10], [22, -4], [22, 0], [-54, 4], [-8, -18], [-24, -5], [-8, -20], [-56, 7], [-6, 14], [-50, -6], [-14, 24], [-28, -6], [-62, -6], [14, -8], [-18, -3], [-14, -6], [30, -16], [8, -2], [2, -5], [36, -2], [-48, -16], [10, 3], [-36, -13], [-20, 12], [-38, -16], [0, 6], [14, -16], [-58, 6], [20, 10], [-36, 1], [12, -20], [6, -30], [-42, -10], [-6, -4], [-10, -2], [-44, 16], [16, 13], [30, 16], [-22, -19], [-10, -14], [-36, -8], [36, 8], [20, 15], [-24, 0], [-6, -16], [4, 12], [-62, -10], [-20, -25], [72, 4], [42, -22], [-34, 6], [22, -16], [-6, -9], [-34, 0], [0, 32], [0, 8], [-28, -3], [12, -24], [-14, -6], [38, -15], [44, 0], [12, -10], [20, 0], [-30, -1], [20, 20], [40, -13], [24, 16], [-24, 12], [-32, 18], [-46, 14], [20, 8], [-2, -10], [6, -9], [28, 12], [0, -8], [6, 4], [-6, -1], [-16, 2], [-30, 2], [-36, -24], [-64, -3], [60, -5], [54, -10], [6, 24], [4, 38], [-48, 20], [44, -4], [-30, -13], [6, 6], [68, 10], [42, -20], [12, 36], [6, -14], [-22, -5], [-8, 0], [4, -11], [-6, -20], [6, -11], [-16, -2], [-6, -16], [-60, 3], [-52, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3150_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3150_2_a_bs();".
// 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_3150_2_a_bs(:prec:=2)
    chi := MakeCharacter_3150_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_3150_2_a_bs();".
// 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_3150_2_a_bs( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3150_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-24, 0, 1]>,<13,R![-2, 4, 1]>,<17,R![-2, 1]>,<19,R![10, -8, 1]>,<29,R![-20, 4, 1]>],Snew);
    return Vf;
end function;