
// Make newform 7623.2.a.bm in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7623_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_7623_2_a_bm();".
// 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_7623_2_a_bm();".
// 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_7623_a();"
function MakeCharacter_7623_a()
    N := 7623;
    order := 1;
    char_gens := [848, 4357, 4600];
    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_7623_a_Hecke(Kf)
    return MakeCharacter_7623_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, 0], [-1, 0], [-1, 0], [0, 0], [-1, 4], [4, -2], [3, -6], [-2, 6], [5, 0], [-4, 2], [-7, 0], [0, 4], [-2, 6], [1, 2], [6, -10], [5, -10], [-2, 0], [-11, -2], [0, -4], [-7, -4], [12, -4], [8, 2], [2, -4], [6, -6], [12, -10], [10, -12], [-3, 2], [-4, -2], [-2, -4], [4, -14], [-4, -8], [-16, 6], [-12, 4], [5, 0], [8, -10], [-8, 12], [-19, 6], [-14, 14], [-4, -4], [-4, 8], [-2, -2], [12, 2], [-6, 14], [-14, 4], [-10, -10], [2, 12], [-6, 0], [-2, 0], [6, -2], [-10, 8], [-1, -18], [-21, 8], [-1, 18], [7, 0], [-15, 18], [-6, -8], [-9, -6], [4, -4], [9, -4], [15, 2], [-16, 0], [16, -8], [0, 16], [16, -14], [-18, 10], [-4, 12], [2, -10], [28, 0], [-9, -12], [-27, -4], [16, -12], [-10, 16], [4, 14], [-25, 10], [-28, 8], [-16, -8], [12, -18], [26, -6], [-26, -8], [-5, -10], [-13, 0], [5, -6], [-8, 4], [-5, 10], [-16, 4], [-20, 0], [-14, 2], [-28, 0], [-11, -10], [-23, 10], [8, -6], [12, -8], [-7, 18], [-1, 2], [6, 6], [30, 0], [-21, 28], [13, -14], [32, -8], [0, 24], [-15, -4], [26, -14], [30, 0], [-20, -4], [16, 14], [-25, 34], [-8, 24], [28, 4], [-27, 0], [-13, -10], [0, 22], [-30, 4], [-28, 26], [32, -20], [28, -30], [-22, 42], [-21, -14], [40, -8], [5, -30], [22, -30], [10, 22], [2, -28], [-16, 34], [2, 0], [14, 16], [-25, 40], [-9, -2], [18, -10], [18, 16], [-18, 26], [9, 10], [41, 2], [-27, 40], [8, -32], [-9, 8], [3, -4], [5, 14], [5, 4], [-39, 28], [-19, -6], [-5, 4], [-11, 10], [21, 14], [16, -32], [5, -30], [22, -12], [10, 16], [-40, 0], [42, -2], [-12, -12], [-45, 16], [-21, -10], [12, 2], [28, 0], [22, 12], [-26, 22], [-3, 16], [30, -16], [-48, 20], [-2, -12], [25, -32], [34, -44], [-21, 18], [2, 10], [-20, 12], [-23, 10], [-22, 8], [18, 24], [23, -24], [-31, 22], [-10, 4], [16, -8], [-32, -14], [-6, -8], [-54, 8], [-6, 18], [-14, 4], [-31, -6], [45, 0], [17, -38], [-46, 6], [-30, 28], [32, -28], [-4, 0], [-13, 16], [8, -12], [-28, 14], [-36, 2], [12, -10], [-10, 8], [-5, -2], [17, -50], [5, -24], [21, 14], [9, -36], [-48, 22], [-30, 28], [-10, 6], [38, -18], [10, 0], [-27, 30], [18, 10], [-7, -16], [40, -40], [18, -32], [-24, 28], [27, -26], [-18, 36], [-6, -34], [-2, 8], [13, -40], [-13, 34], [-12, -12], [-30, 20], [-14, -28], [-24, 32], [-4, 2], [-40, 14], [0, 42], [-32, 10], [-63, 6], [22, 26], [-21, 30], [44, -12], [-42, 24], [21, 0], [34, -28], [-11, 6], [27, -30], [-6, 40], [57, 6], [38, -40], [17, -20], [-14, 26], [26, 4], [-8, 6], [29, -20], [4, -18], [-35, 6], [58, 6], [-4, 24], [40, 18], [10, 20], [-60, 8], [-8, -4], [56, 4], [57, -18], [-24, 28], [17, 14], [-26, 16], [-38, 30], [-10, 24], [18, 4], [-18, 44], [44, -8], [17, -20], [37, 10], [-3, 0], [-27, -32], [2, 28], [-44, -16], [14, -8], [-3, -12], [1, 12], [8, -6], [-42, 44], [7, -8], [8, -4], [38, -48], [40, 24], [-31, -2], [-30, 38], [36, -22], [29, -24], [-8, -26], [16, -48], [-10, -20], [20, -14], [-14, -18], [-2, 2], [32, -40], [-60, -14], [25, 16], [-37, 38], [38, -30], [-25, 8], [22, -28], [-12, -26], [16, 18], [28, -10], [-12, -32], [-15, 48], [38, 18], [28, -44], [-62, 34], [26, 12], [-70, -8], [-26, 62], [35, -26], [15, 12], [-37, 0], [14, -18], [22, -42], [-63, 30], [-5, -44], [-42, -22], [14, 2], [-26, 2], [38, 6], [-54, -4], [8, 4], [61, -48], [-4, -24], [-64, 24], [-66, 32], [-16, 52], [-16, 16], [30, 18], [-18, 36], [-25, -14], [3, 20], [-58, 20], [28, -68], [31, 16], [-28, 20], [12, -24], [-2, -18], [-9, 34], [69, 0], [4, 8], [-48, -20], [27, 6], [44, -10], [22, -38], [-3, 30], [-5, -20], [-30, 18], [0, 14], [36, -8], [-22, -28], [16, 2], [4, -38], [63, -2], [32, -22], [1, -22], [18, -12], [12, -28], [36, -28], [-26, -24], [-24, 4], [1, -36], [36, -36], [-29, 70], [-48, 6], [-28, 14], [49, 22], [68, 8], [-51, -12], [14, 14], [-52, 8], [32, -20], [56, 12], [21, -22], [16, -16], [23, -68], [0, -54], [10, -28], [34, -32], [-28, -18], [32, 6], [19, -10], [43, 4], [-12, 0], [67, -28], [31, 18], [-29, 62], [21, -40], [-26, 32], [14, -24], [-63, 12], [-6, 34], [-26, -4], [37, -48], [-44, 8], [-35, 2], [1, 22], [-42, 20], [16, 30], [-41, 46], [33, 4], [2, 28], [-10, 50], [74, -22], [-22, -20], [-28, 38], [-5, 30], [-47, 64], [36, 12], [13, 28], [-49, 28], [36, -30], [16, -38], [50, -56], [-38, -14], [-43, 62], [-77, 0], [-6, 16], [-66, -20], [-12, 4], [-18, -52], [-34, 44], [24, 44], [38, -70], [14, 36], [-53, 52], [-59, -2], [-20, 6], [-6, -4], [-36, 50], [-20, 30], [7, 0], [-4, 42], [-20, 30], [-58, 24], [35, -36], [8, 26], [45, 0], [26, -8], [80, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7623_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7623_2_a_bm();".
// 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_7623_2_a_bm(:prec:=2)
    chi := MakeCharacter_7623_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_7623_2_a_bm();".
// 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_7623_2_a_bm( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7623_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -1, 1]>,<5,R![1, 1]>,<13,R![-19, -2, 1]>],Snew);
    return Vf;
end function;