
// Make newform 3150.2.m.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3150_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3150_m_Hecke();"
// 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_m_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_3150_2_m_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
        poly := [1, 0, 0, 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_m();"
function MakeCharacter_3150_m()
    N := 3150;
    order := 4;
    char_gens := [2801, 127, 451];
    v := [2, 1, 4];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3150_m_Hecke();"
function MakeCharacter_3150_m_Hecke(Kf)
    N := 3150;
    order := 4;
    char_gens := [2801, 127, 451];
    char_values := [[-1, 0, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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], [0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, 2, 2, 2], [-2, 0, 2, 2], [-3, 0, 3, -2], [0, -5, 0, -5], [-2, -4, -2, 0], [6, -2, 0, 2], [-6, -2, 0, 2], [1, -10, 1, 0], [0, -2, -2, -2], [5, 0, -5, -2], [-5, 0, 5, 2], [2, 4, 2, 0], [8, 0, 0, 0], [8, -1, 0, 1], [-1, 2, -1, 0], [0, -3, -8, -3], [-6, 0, 6, 0], [0, -8, -2, -8], [4, 8, 4, 0], [2, 8, 0, -8], [-4, -6, -4, 0], [0, 4, -10, 4], [2, 0, -2, -2], [-2, 0, 2, 10], [0, -10, -6, -10], [-4, -2, -4, 0], [4, -4, 4, 0], [0, 0, 6, 0], [-6, 0, 6, -10], [0, -3, 4, -3], [6, -8, 0, 8], [-2, -8, 0, 8], [4, -2, 4, 0], [-5, 0, 5, -14], [-3, 0, 3, -14], [-8, 0, -8, 0], [-2, -8, 0, 8], [-4, -5, 0, 5], [0, -1, -12, -1], [6, 0, -6, 0], [-6, 0, 6, -4], [0, -4, 6, -4], [12, -6, 0, 6], [-2, 0, 2, -14], [10, 0, -10, 8], [0, -5, 12, -5], [4, 2, 4, 0], [4, 15, 0, -15], [-2, -4, 0, 4], [0, 12, -10, 12], [3, 0, -3, 14], [-22, 0, -22, 0], [-10, -8, 0, 8], [10, -2, 0, 2], [-11, -10, -11, 0], [0, -5, 20, -5], [-18, 0, 18, 0], [-14, -12, -14, 0], [-16, 12, -16, 0], [0, 0, 16, 0], [-14, 0, 14, 8], [-8, 0, 8, 6], [-16, 10, 0, -10], [-10, 12, -10, 0], [12, 0, -12, -4], [0, -9, 4, -9], [3, -26, 3, 0], [-8, -9, 0, 9], [20, 0, 20, 0], [-7, 0, 7, 6], [0, -4, 0, -4], [-1, -6, -1, 0], [6, -8, 0, 8], [24, 2, 24, 0], [0, 5, 24, 5], [0, -8, -26, -8], [-20, 0, 0, 0], [14, 10, 0, -10], [0, 15, -8, 15], [20, 0, -20, -6], [0, 14, 6, 14], [-16, -12, -16, 0], [-28, -3, 0, 3], [16, -12, 16, 0], [0, 8, 26, 8], [14, 0, -14, 12], [4, 0, -4, -8], [4, 4, 0, -4], [-20, 0, -20, 0], [0, -12, -6, -12], [0, 0, -24, 0], [-19, -2, -19, 0], [-24, 4, 0, -4], [0, 12, -10, 12], [2, 0, -2, -40], [18, -6, 0, 6], [-11, -18, -11, 0], [-12, 0, 12, -14], [-6, -20, -6, 0], [8, -13, 0, 13], [-8, 12, 0, -12], [14, -4, 14, 0], [14, 0, -14, 0], [13, 2, 13, 0], [12, -5, 0, 5], [-18, 0, 0, 0], [-6, 26, -6, 0], [5, 0, -5, 2], [-26, 0, 26, -2], [0, 19, 8, 19], [-12, 12, 0, -12], [0, 23, 8, 23], [14, 0, -14, 20], [3, 0, -3, 30], [-4, 2, -4, 0], [2, -14, 0, 14], [-32, 3, 0, -3], [6, 0, -6, 28], [-16, 0, 16, -12], [6, -14, 6, 0], [4, -15, 0, 15], [0, -6, -18, -6], [0, 8, -6, 8], [-20, 18, 0, -18], [6, 6, 6, 0], [-8, 0, 8, 18], [0, 16, -20, 16], [4, 20, 4, 0], [28, 12, 0, -12], [-17, -10, -17, 0], [0, 4, 10, 4], [0, 10, -2, 10], [2, -48, 2, 0], [24, 16, 24, 0], [-16, 0, 16, -28], [-20, 21, 0, -21], [-28, 3, 0, -3], [0, -10, 26, -10], [26, 0, -26, 16], [8, 0, -8, 4], [0, 19, -8, 19], [-24, -2, 0, 2], [12, 0, -12, 22], [-3, 0, 3, 34], [0, -9, 28, -9], [4, -32, 4, 0], [7, 18, 7, 0], [0, -6, -34, -6], [29, 0, -29, -14], [19, 0, -19, 26], [7, -6, 7, 0], [0, -7, 20, -7], [0, -8, -20, -8], [18, -4, 0, 4], [24, 10, 24, 0], [0, 8, -2, 8], [-10, 0, 10, 2], [22, 26, 22, 0], [-24, 24, -24, 0], [0, -4, -30, -4], [20, 0, -20, -26], [23, -2, 23, 0], [-12, 8, 0, -8], [4, -46, 4, 0], [0, -8, 32, -8], [8, -36, 8, 0], [-22, -6, 0, 6], [-44, -7, 0, 7], [0, -5, 12, -5], [18, 0, -18, 32], [0, 6, 26, 6], [34, 8, 0, -8], [-8, 2, 0, -2], [0, 0, -18, 0], [6, 0, -6, -42], [0, -1, -8, -1], [6, 20, 6, 0], [0, 24, 8, 24], [23, 0, -23, -22], [-9, 0, 9, -6], [-14, 44, -14, 0], [48, 4, 0, -4], [-9, -14, -9, 0], [-14, 0, 14, 44], [0, 0, -4, 0], [0, 10, -44, 10], [0, 0, 0, 34], [-22, -6, -22, 0], [24, -24, 0, 24], [0, -8, 38, -8], [18, 0, -18, -10], [15, -34, 15, 0], [-8, -32, 0, 32], [-17, 0, 17, -22], [-1, 0, 1, 46], [-11, 14, -11, 0], [14, 24, 0, -24], [6, 12, 0, -12], [-6, -18, -6, 0], [0, -38, 2, -38], [46, 4, 0, -4], [14, 0, -14, 12], [0, 6, -46, 6], [16, 12, 16, 0], [-40, 1, 0, -1], [-4, -23, 0, 23], [-26, -4, -26, 0], [0, 40, -2, 40], [4, 0, -4, -16], [32, 0, -32, -12], [12, 26, 0, -26], [-10, -6, 0, 6], [-16, 40, -16, 0], [0, 6, 42, 6], [34, 0, -34, -8], [0, 6, 0, 0], [-42, 0, 0, 0], [0, -20, -2, -20], [20, -21, 0, 21], [10, 0, -10, -24], [14, 0, -14, 8], [0, 0, -26, 0], [41, 14, 41, 0], [-12, -17, 0, 17], [-8, 16, -8, 0], [0, -24, 38, -24], [7, 0, -7, -6], [0, -29, -24, -29], [24, -20, 0, 20], [0, -17, 8, -17], [12, 0, -12, 48], [7, 0, -7, 50], [0, -8, -18, -8], [-6, -8, -6, 0], [38, -24, 0, 24], [0, 26, -24, 26], [30, 2, 30, 0], [16, -3, 0, 3], [-8, 0, 8, -56], [0, -30, -14, -30], [25, -18, 25, 0], [8, -10, 0, 10], [-6, -38, -6, 0], [0, 8, -34, 8], [0, 10, 0, 10], [12, -12, 12, 0], [25, -18, 25, 0], [0, 2, -42, 2], [8, 0, -8, -36], [0, -26, 18, -26], [-30, 4, -30, 0], [2, 24, 0, -24], [-10, -22, 0, 22], [10, 52, 10, 0], [2, 0, -2, -16], [-8, 54, -8, 0], [34, 0, -34, 12], [-4, 0, 4, -60], [0, 21, 24, 21], [10, 0, -10, -14], [-13, 0, 13, 10], [0, -39, -4, -39], [66, 4, 0, -4], [0, 24, -22, 24], [-21, 0, 21, 30], [-30, -32, -30, 0], [12, -27, 0, 27], [-7, 2, -7, 0], [-14, 0, 14, -16], [0, -4, 42, -4], [-14, 24, -14, 0], [-16, 0, 16, 40], [18, 0, -18, 46], [0, 36, 22, 36], [56, 20, 0, -20], [0, 32, 0, 32], [-3, -30, -3, 0], [-42, 0, 0, 0], [7, 0, -7, 58], [24, -15, 0, 15], [14, -12, 14, 0], [0, -39, -8, -39], [0, 0, 0, 32], [22, 0, -22, 8], [0, -2, 54, -2], [-70, 0, 0, 0], [0, -20, 10, -20], [-10, 0, 10, 16], [8, 58, 8, 0], [0, 16, 46, 16], [25, 0, -25, 14], [20, -40, 0, 40], [-14, 14, 0, -14], [-26, 16, -26, 0], [2, 0, 0, 0], [-20, -36, -20, 0], [-8, 0, 8, -22], [-40, 0, 40, 18], [0, 4, 40, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3150_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3150_2_m_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_3150_2_m_c(:prec:=4)
    chi := MakeCharacter_3150_m();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_m_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_3150_2_m_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3150_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![16, 0, 24, 0, 1]>,<13,R![16, 32, 32, 8, 1]>,<17,R![196, 168, 72, 12, 1]>],Snew);
    return Vf;
end function;