
// Make newform 624.4.q.b in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_624_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_624_q_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_624_4_q_b();".
// 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_624_4_q_b();".
// 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, -1, 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_624_q();"
function MakeCharacter_624_q()
    N := 624;
    order := 3;
    char_gens := [79, 469, 209, 145];
    v := [3, 3, 3, 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_624_q_Hecke();"
function MakeCharacter_624_q_Hecke(Kf)
    N := 624;
    order := 3;
    char_gens := [79, 469, 209, 145];
    char_values := [[1, 0], [1, 0], [1, 0], [0, -1]];
    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, 0], [3, -3], [-9, 0], [0, 2], [30, -30], [13, 39], [0, 111], [0, -46], [-6, 6], [105, -105], [100, 0], [-17, 17], [231, -231], [0, -514], [162, 0], [639, 0], [0, 600], [0, -233], [926, -926], [0, -930], [-253, 0], [1324, 0], [-810, 0], [-498, 498], [0, -1358], [357, -357], [-1118, 0], [714, -714], [2006, 0], [0, 1119], [-604, 604], [1584, 0], [0, -717], [0, -820], [0, 1749], [370, 0], [-2611, 0], [0, -1636], [264, -264], [0, -1410], [-474, 474], [2249, 0], [0, 3444], [4273, -4273], [1986, -1986], [0, -2386], [-1600, 1600], [-3832, 3832], [0, -1398], [4466, 0], [-1638, 0], [594, 0], [0, -2303], [0, 6324], [-7833, 7833], [-3030, 3030], [0, 534], [-3688, 3688], [0, -1865], [2997, 0], [-4114, 4114], [0, 4665], [-1502, 0], [-2106, 0], [-3898, 0], [9351, 0], [0, -9172], [-11089, 0], [0, 9762], [8290, -8290], [-12405, 12405], [1098, 0], [-5734, 5734], [0, 8971], [7244, -7244], [0, -6312], [3627, 0], [0, 3898], [5703, -5703], [0, -6311], [-2328, 2328], [2045, 0], [5034, -5034], [0, -4283], [-1306, 1306], [5796, 0], [0, -2706], [829, -829], [0, 5493], [15346, 0], [9594, 0], [-12840, 12840], [0, -14086], [11694, -11694], [3688, 0], [0, -4746], [14505, -14505], [5085, 0], [-10882, 10882], [-4699, 0], [-8270, 0], [22785, -22785], [0, -11928], [7962, -7962], [-20618, 0], [-3493, 0], [10416, -10416], [2061, 0], [-12456, 0], [781, -781], [0, 19304], [-12041, 12041], [0, -9717], [21040, 0], [0, -5068], [0, -10185], [0, 25928], [23160, -23160], [-16626, 16626], [0, -14808], [-4853, 4853], [16165, -16165], [-25686, 0], [0, -19056], [-16390, 16390], [-27846, 0], [0, 12283], [0, 25512], [-6110, 0], [-27127, 0], [-880, 880], [-21876, 21876], [11798, -11798], [8074, -8074], [0, -19554], [-14030, 14030], [0, -36042], [0, 28628], [0, 37434], [-37569, 37569], [-5516, 0], [-8778, 8778], [0, -3088], [-13176, 0], [2359, -2359], [0, -2676], [2477, 0], [-17199, 0], [-24338, 0], [-25146, 0], [0, -18089], [15099, -15099], [-33488, 0], [-39768, 39768], [32156, -32156], [-11520, 0], [0, 4952], [0, -8781], [50039, 0], [-50670, 0], [42384, -42384], [0, 50538], [6886, 0], [0, 9060], [28311, -28311], [-4284, 0], [-2458, 2458], [0, -24101]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_624_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_624_4_q_b();".
// 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_624_4_q_b(:prec:=2)
    chi := MakeCharacter_624_q();
    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_624_4_q_b();".
// 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_624_4_q_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_624_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![9, 1]>,<7,R![4, -2, 1]>],Snew);
    return Vf;
end function;