
// Make newform 7605.2.a.bw in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7605_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_7605_2_a_bw();".
// 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_7605_2_a_bw();".
// 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 := [-2, -6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, 0, 1]];
    Rf_basisdens := [1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_7605_a();"
function MakeCharacter_7605_a()
    N := 7605;
    order := 1;
    char_gens := [6761, 1522, 6931];
    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_7605_a_Hecke(Kf)
    return MakeCharacter_7605_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, 0, 0], [1, 0, 0], [-1, -1, 0], [0, -2, 0], [0, 0, 0], [0, -1, 2], [-4, 0, -1], [0, 0, -2], [-2, -1, 1], [-1, 2, -3], [-2, 0, 2], [0, -1, -1], [3, -1, 1], [-4, 3, 0], [0, 2, 0], [-2, -1, -1], [-1, -2, 1], [-3, 3, -3], [-4, 1, -3], [-7, 3, 0], [-1, 2, -3], [-6, 0, -2], [10, 1, -1], [-5, 3, 1], [8, -4, 0], [3, 7, -1], [-8, 1, -2], [-9, -4, 2], [6, 2, 0], [-11, 3, 0], [-6, 3, 1], [-2, 7, -2], [5, -6, -2], [-10, -2, 2], [0, 4, 3], [-5, -1, -1], [3, -1, 4], [-8, 4, -2], [-6, -3, 2], [2, 1, 5], [18, -2, 3], [4, 5, -3], [-17, -3, 2], [6, -6, -2], [1, 0, -2], [-1, -6, -2], [-8, 6, 0], [10, 3, 2], [-14, 4, 1], [4, 10, -6], [18, -2, 0], [-10, 0, -1], [-2, 4, 0], [2, -1, -2], [8, 3, -4], [-10, 7, -1], [-5, -10, 2], [-2, -8, 0], [-2, 1, -1], [-7, -13, 4], [8, -1, 10], [9, 1, 1], [2, -7, 9], [-9, -5, 5], [2, -2, 6], [-1, -10, 2], [9, -1, -5], [8, -2, 8], [1, 2, -3], [12, -6, 8], [-16, -1, 5], [1, 1, -5], [-11, -9, 4], [1, 8, 5], [-2, -11, 0], [2, -4, -6], [-11, -1, 1], [-4, -10, 2], [-9, -10, 3], [-10, -3, 1], [7, -10, 9], [14, 14, -4], [-11, 9, 0], [19, 6, -1], [4, 1, -2], [6, 10, -6], [-1, 3, 1], [10, -5, 7], [3, 3, 5], [8, 9, -4], [-2, -11, 11], [-6, -4, 5], [-14, -13, 7], [-12, 10, 5], [6, -8, -6], [-16, 6, 6], [14, -3, 5], [-24, 2, 2], [-9, -6, -4], [-7, -5, 7], [2, -10, 12], [-2, 4, -4], [0, 9, 1], [4, -10, 1], [-18, 2, -4], [12, 15, -8], [2, -2, -2], [16, -7, -7], [6, 4, 5], [-10, -8, 3], [-3, -5, -1], [12, -6, 0], [3, 2, -4], [17, -12, -1], [0, 0, 6], [-33, -9, 4], [0, -4, 2], [-16, 3, -4], [0, 2, -2], [13, -4, 4], [-3, -3, -4], [6, 0, 4], [-30, -3, 4], [-29, 2, -9], [-8, -10, 4], [-13, 12, -4], [-22, -13, 5], [-13, -9, 9], [-29, 3, -9], [-4, 6, 0], [-10, 9, 0], [32, 10, -5], [0, 2, 1], [14, 2, -2], [-30, 2, 3], [8, -10, -10], [15, 13, 0], [10, -7, -2], [-20, 2, -2], [-1, -4, -4], [-18, -12, 6], [12, 4, 4], [-10, -3, -12], [-19, 8, 7], [24, 20, -8], [7, 1, -2], [20, 1, 6], [3, 2, -5], [-10, -2, 2], [-10, 2, -4], [24, 2, 0], [-11, 17, -1], [-10, -7, 6], [18, 0, 9], [-32, -6, 2], [12, 20, -9], [16, 0, -2], [-2, 16, 0], [-6, 5, 3], [-4, -7, -4], [-12, 2, -6], [-24, -10, 8], [26, -4, -4], [-20, 2, -10], [4, -2, 4], [-8, 8, 3], [-7, 5, 4], [23, -2, 3], [22, 5, 4], [-24, 3, 3], [-22, -18, 7], [-20, -2, -4], [-31, -7, -2], [4, -6, 5], [20, -7, -1], [25, -2, -2], [2, 8, -10], [-15, -5, -5], [15, -16, 2], [0, 2, -10], [12, 25, -11], [-3, -7, -11], [-8, 3, -6], [-6, -3, 0], [28, 0, 0], [-17, 13, 4], [32, -6, -4], [-13, 10, -17], [32, -3, 5], [-5, 3, 6], [-2, -13, 0], [11, 20, -7], [22, -1, -7], [-8, -5, 4], [-24, 16, -4], [13, -16, -11], [-3, -3, -3], [30, -15, -2], [-2, -1, 8], [-14, -23, 5], [5, -10, -1], [-1, 5, -15], [1, 0, -1], [6, -12, -12], [-30, -3, 4], [-7, 22, 0], [10, -6, 0], [10, 28, -8], [8, -6, -4], [-39, 1, -6], [-8, 8, -14], [-33, -3, 7], [2, -9, 2], [22, 10, 6], [26, -8, 1], [-13, 1, -8], [-2, -2, -6], [38, 1, 8], [20, 11, -2], [13, 6, -1], [17, -2, 5], [6, 12, -2], [29, 15, -3], [26, 12, 4], [23, 0, 0], [-16, -4, -2], [40, 2, -8], [9, 5, 2], [-36, 9, -3], [-5, 1, 4], [-41, -6, 9], [-24, -16, 12], [-40, 1, -9], [41, -9, -8], [-2, 10, 6], [11, -22, -1], [16, -3, 18], [30, -3, -3], [-20, 10, -12], [-36, -6, 10], [56, 2, 3], [36, 2, -6], [-7, -10, -3], [6, -24, 4], [44, -2, 2], [31, -9, 9], [-6, 4, 10], [21, 20, 1], [0, -16, 6], [17, 3, 18], [28, 17, 1], [2, 11, -2], [-19, 8, -9], [-16, -9, 0], [10, 7, -7], [-14, -24, 7], [26, -8, 21], [42, 9, -4], [-44, 6, -3], [-8, -6, 10], [16, 23, -6], [7, -12, -12], [25, 25, -12], [4, 1, 18], [-1, -2, 8], [6, -5, 17], [-34, -5, 1], [0, 2, -16], [14, -14, -8], [7, -10, 4], [-17, -1, -9], [22, -18, 14], [0, -24, 5], [-9, 15, -16], [-20, -24, 18], [-14, 16, 2], [-14, 10, -2], [9, 24, -16], [-18, 0, -8], [22, 5, 10], [-8, -4, 0], [-36, 10, 10], [17, 2, 0], [-24, 6, -18], [-34, 8, -10], [-19, -11, 7], [4, -17, 2], [47, 4, -4], [54, 3, -5], [-26, -19, 3], [-18, 19, 12], [-16, -10, 12], [14, 8, 8], [-23, 11, -20], [-6, -9, 15], [36, -14, -4], [-8, -20, 10], [54, -6, 2], [-1, 25, -10], [-41, 7, -7], [0, 1, 4], [16, 12, -4], [28, 7, -4], [39, -16, -5], [25, 9, -10], [0, -36, 6], [2, 18, 2], [-12, 21, 5], [-6, 24, -2], [-6, 15, -20], [-38, -8, 4], [8, -13, 11], [-41, 11, -14], [-36, 11, 6], [21, 2, 10], [-2, 19, -17], [-12, 6, -20], [-41, -11, 3], [-4, -25, 11], [-7, -6, -1], [23, -11, 12], [4, -27, 1], [-5, -13, -11], [-12, -18, 14], [-3, -4, 7], [27, 20, -2], [4, 14, 4], [48, -3, -8], [-32, 9, 6], [10, -8, -13], [36, 5, -8], [-19, 12, 4], [8, 13, 0], [11, -8, -9], [-22, 10, -18], [13, -30, 1], [-18, -28, 22], [-22, 22, -10], [-23, -17, 1], [8, -18, -11], [-22, 12, 8], [6, 9, -9], [7, -14, -16], [-6, 25, 10], [16, -20, -10], [-15, 4, -5], [12, -14, 12], [-52, -13, 1], [0, 10, -8], [17, -10, 2], [-14, -2, -20], [2, 4, 14], [-29, 15, -7], [-34, 20, -5], [36, -6, -4], [14, -17, 15], [-40, -1, 5], [-58, 2, -6], [-20, -10, -6], [-14, -10, 22], [-32, 0, 8], [-8, -20, 18], [-76, -3, -1], [31, 11, -8], [-11, -39, 18], [-20, -5, 4], [33, 21, -4], [58, -4, 1], [38, -1, 9], [-43, -16, 8], [12, -26, -8], [-62, 16, 4], [24, 12, -3], [-16, 22, 1], [-36, 13, -9], [-34, 8, -2], [17, -5, 11], [-36, 21, -7], [41, -11, 27], [22, -12, 16], [24, 9, 2], [-4, -24, -4], [2, 19, -4], [-5, -14, -14], [-8, 23, -10], [16, 0, -19], [5, -9, -5], [-41, -5, 2], [-6, -6, -14], [42, -6, 21], [-72, 1, -12], [10, -34, 2], [-2, 20, -15], [28, -7, -15], [35, -27, 6], [-24, 2, -11], [-12, -3, 9], [48, -22, 1], [-18, 3, -9], [5, -6, 21], [-2, 31, -4], [-5, -31, 7], [8, 31, -24], [-20, -15, -7], [41, 10, -2], [63, -1, 13], [-24, -1, -1], [1, 1, -5], [26, -10, -4], [25, -17, -4], [-28, 5, 2], [32, 13, 10], [-4, 20, -12], [-23, -7, -5], [-4, 19, -21], [-8, 12, -2], [-29, 9, -8], [8, -1, 4], [-12, 7, -6], [-12, 1, -23], [-1, 27, -8], [-8, 13, -12], [56, -14, 0], [35, -29, -2], [-62, -1, 0], [10, -1, -10], [14, -22, 20], [-33, 14, 10], [32, -2, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7605_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7605_2_a_bw();".
// 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_7605_2_a_bw(:prec:=3)
    chi := MakeCharacter_7605_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_7605_2_a_bw();".
// 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_7605_2_a_bw( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7605_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, -6, 0, 1]>,<7,R![-3, -3, 3, 1]>,<11,R![16, -24, 0, 1]>],Snew);
    return Vf;
end function;