
// Make newform 507.4.b.c in Magma, downloaded from the LMFDB on 26 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_507_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_507_b_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_507_4_b_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_507_4_b_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, 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_507_b();"
function MakeCharacter_507_b()
    N := 507;
    order := 2;
    char_gens := [170, 340];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_507_b_Hecke();"
function MakeCharacter_507_b_Hecke(Kf)
    N := 507;
    order := 2;
    char_gens := [170, 340];
    char_values := [[1, 0], [-1, 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 := 4;
    raw_aps := [[0, 3], [3, 0], [0, -9], [0, -2], [0, -30], [0, 0], [111, 0], [0, -46], [6, 0], [-105, 0], [0, -100], [0, -17], [0, -231], [514, 0], [0, 162], [639, 0], [0, -600], [233, 0], [0, 926], [0, -930], [0, 253], [-1324, 0], [0, 810], [0, -498], [0, 1358], [357, 0], [-1118, 0], [714, 0], [0, 2006], [-1119, 0], [604, 0], [-1584, 0], [0, -717], [-820, 0], [0, -1749], [0, 370], [-2611, 0], [0, 1636], [0, -264], [-1410, 0], [474, 0], [-2249, 0], [3444, 0], [0, 4273], [0, -1986], [2386, 0], [-1600, 0], [0, -3832], [0, -1398], [0, -4466], [1638, 0], [0, -594], [0, -2303], [-6324, 0], [-7833, 0], [-3030, 0], [-534, 0], [0, 3688], [-1865, 0], [0, -2997], [4114, 0], [0, 4665], [0, -1502], [-2106, 0], [-3898, 0], [0, 9351], [0, -9172], [11089, 0], [9762, 0], [0, 8290], [0, 12405], [0, 1098], [-5734, 0], [-8971, 0], [0, 7244], [0, -6312], [-3627, 0], [0, 3898], [0, 5703], [0, 6311], [-2328, 0], [0, 2045], [0, 5034], [-4283, 0], [1306, 0], [-5796, 0], [0, -2706], [0, -829], [0, -5493], [0, 15346], [9594, 0], [0, 12840], [0, -14086], [-11694, 0], [0, -3688], [-4746, 0], [0, -14505], [5085, 0], [-10882, 0], [0, 4699], [8270, 0], [0, 22785], [11928, 0], [7962, 0], [-20618, 0], [0, -3493], [0, 10416], [0, -2061], [12456, 0], [-781, 0], [19304, 0], [0, 12041], [0, 9717], [0, 21040], [0, 5068], [-10185, 0], [0, 25928], [-23160, 0], [16626, 0], [-14808, 0], [0, -4853], [16165, 0], [-25686, 0], [0, 19056], [0, -16390], [27846, 0], [0, 12283], [-25512, 0], [-6110, 0], [0, -27127], [0, 880], [0, -21876], [-11798, 0], [-8074, 0], [0, -19554], [0, 14030], [0, 36042], [0, -28628], [37434, 0], [37569, 0], [0, 5516], [0, 8778], [3088, 0], [0, -13176], [2359, 0], [0, 2676], [0, -2477], [17199, 0], [24338, 0], [0, 25146], [0, 18089], [15099, 0], [-33488, 0], [-39768, 0], [-32156, 0], [11520, 0], [4952, 0], [0, 8781], [50039, 0], [0, -50670], [0, -42384], [50538, 0], [0, -6886], [9060, 0], [0, -28311], [0, -4284], [-2458, 0], [24101, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_507_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_507_4_b_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_507_4_b_c(:prec:=2)
    chi := MakeCharacter_507_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_507_4_b_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_507_4_b_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_507_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![9, 0, 1]>,<5,R![81, 0, 1]>],Snew);
    return Vf;
end function;