
// Make newform 7605.2.a.bd in Magma, downloaded from the LMFDB on 25 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_bd();".
// 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_bd();".
// 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 := [-4, -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_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], [-1, 0], [3, -1], [-1, 1], [0, 0], [-1, 1], [0, 2], [5, -1], [-2, -2], [-6, 0], [3, 3], [-1, -1], [-2, 2], [-6, -2], [3, -3], [-12, 0], [1, -3], [-4, 6], [-13, 1], [6, 0], [-3, 3], [-4, 8], [-3, -3], [1, -7], [0, 4], [-10, -2], [-7, 7], [2, 0], [-10, 2], [12, -4], [0, 0], [4, -2], [11, -3], [5, 5], [4, -6], [-12, 6], [7, -9], [-10, 2], [18, -2], [-10, 2], [21, -3], [-6, 10], [-17, -1], [-12, 6], [-8, 0], [-4, 4], [0, 6], [-16, 4], [-12, -2], [-15, -5], [-5, -7], [6, -12], [16, 4], [-18, 2], [8, -4], [-6, 2], [4, 6], [-2, -4], [6, -4], [-4, 0], [10, 4], [-5, -1], [18, -6], [24, 2], [-22, 8], [-24, -2], [-10, -8], [-3, -9], [8, 6], [2, 8], [4, 4], [-12, 4], [2, 4], [10, 8], [-24, 8], [2, 14], [7, -1], [-14, 4], [12, -2], [-18, 10], [-24, -6], [24, -8], [16, -10], [-1, 9], [5, -5], [7, 11], [21, -3], [15, -9], [11, 7], [-11, 7], [-3, -9], [9, 9], [-2, -10], [6, -16], [24, -4], [21, 1], [10, -14], [0, -12], [-24, 2], [10, -6], [14, -8], [-27, 3], [-38, 2], [1, -9], [-27, 1], [12, 8], [-14, -4], [6, -6], [15, -17], [-24, 0], [-11, 17], [30, -12], [24, -6], [2, 12], [8, -16], [3, 3], [19, 5], [-18, -6], [6, -2], [14, -8], [-14, 12], [-17, 1], [-12, -8], [-4, 10], [-8, -16], [-4, -2], [36, 0], [18, -2], [19, -1], [-18, -12], [12, -12], [-9, 9], [6, -8], [6, -20], [30, 0], [4, 2], [4, -6], [-3, 3], [28, -8], [6, -4], [-5, -17], [-40, 4], [-2, -2], [-30, 12], [-9, 21], [-3, -15], [5, -1], [-9, 5], [-6, 6], [-6, -16], [-6, -6], [-10, -18], [3, 1], [-18, 6], [-12, 0], [25, -9], [9, -19], [-12, -6], [29, 5], [6, -26], [27, 1], [-8, 6], [-4, -8], [-18, 24], [-12, -12], [-13, 9], [18, 4], [20, -2], [27, 9], [4, -8], [-20, -6], [2, 2], [-11, 5], [3, -11], [34, -10], [-54, 0], [-63, 1], [-24, 4], [-18, 8], [31, -21], [8, -16], [28, -6], [28, 10], [-26, -14], [42, 6], [-12, -6], [25, -15], [16, -6], [-11, 7], [-12, -10], [26, -22], [-7, 15], [-5, 7], [0, -12], [27, 9], [26, -20], [6, 12], [-32, -2], [-27, -9], [30, -12], [-15, -5], [1, 9], [-15, -3], [5, 11], [1, 7], [-48, -6], [-12, 8], [-19, 13], [-5, -3], [-6, 20], [-14, -2], [24, -4], [4, 16], [-32, -8], [10, -12], [42, 2], [30, -6], [-8, -8], [-4, -22], [-25, -9], [-22, -8], [2, -28], [-8, 14], [-35, 7], [-35, -3], [-15, -1], [-6, 14], [14, -14], [23, 1], [-24, 6], [3, 21], [20, 18], [20, 4], [12, -24], [-18, 2], [30, 12], [36, -14], [-4, 4], [16, -4], [48, -12], [-17, 17], [-30, 18], [-2, 0], [34, 4], [0, 20], [48, -18], [-37, 13], [-42, 12], [-16, 16], [-58, 0], [66, 0], [12, 12], [-23, -7], [41, 11], [-23, 23], [2, 0], [-17, 11], [18, 2], [-23, 21], [-4, 12], [-25, -11], [16, 18], [6, 24], [-12, 18], [17, -21], [41, 13], [-23, 13], [-15, -7], [38, 2], [-14, 0], [13, 9], [-15, 25], [8, 4], [-40, 6], [29, 9], [20, 8], [46, 0], [40, 14], [-64, 4], [37, 7], [-46, 0], [-32, 8], [-46, -12], [44, 10], [-34, 2], [-38, 8], [18, 4], [-38, -12], [4, -16], [-24, 0], [-28, 4], [-22, -4], [12, -16], [12, -18], [62, 2], [-40, -8], [-15, 15], [20, -16], [-5, -9], [-28, 6], [8, -14], [33, 15], [-77, 1], [52, 0], [27, -25], [-5, 5], [-41, 15], [-40, -8], [8, 6], [21, -29], [-18, 12], [36, 16], [-24, 24], [-50, -14], [-7, 9], [29, 11], [-13, 17], [-66, 0], [24, -40], [-16, -8], [-38, 24], [-24, 8], [71, -9], [32, -26], [27, 3], [-32, 14], [-17, -9], [-75, -5], [-46, 14], [-48, 18], [-51, -1], [-12, -16], [36, 0], [20, -26], [-6, -30], [4, 2], [-18, 36], [26, -24], [38, 6], [37, -7], [-11, 7], [-47, 25], [0, 4], [-38, 8], [-2, 26], [33, 13], [43, 3], [72, 0], [13, -5], [-48, 30], [-33, 3], [-13, -1], [60, -4], [-33, 31], [30, -30], [7, 13], [20, -32], [-6, -26], [22, -14], [6, 28], [6, 18], [84, -8], [-16, 4], [80, -8], [42, -4], [-36, -2], [-5, 27], [-35, -19], [-30, 6], [-36, 36], [-6, -18], [-4, 4], [-69, 5], [-36, 14], [5, -29], [48, -24], [-39, 21], [-58, 10], [12, -10], [-58, 8], [6, -12], [-36, 18], [18, 4], [50, -12], [-26, 10], [8, 6], [12, 18], [32, -18], [-7, 23], [40, -22], [-32, -2], [39, 25], [70, -6], [35, -23], [8, -6], [-66, 8], [4, 28], [31, -27], [6, -4], [-42, 24], [45, 11], [13, -27], [-32, -26], [-57, -1], [-49, -15], [26, -12], [-18, 16], [45, -27], [-75, 3], [60, 6], [-27, -5], [-39, -21], [35, -15], [32, 6], [-4, -4], [1, -5], [18, -30], [20, 2], [-5, 25], [18, -6], [15, -33], [24, 24], [30, 2], [42, -24], [-80, -2], [-51, -9], [15, 3], [18, -12], [-4, -8]];
    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_bd();".
// 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_bd(:prec:=2)
    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_bd();".
// 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_bd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7605_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-4, 1, 1]>,<7,R![2, -5, 1]>,<11,R![-4, 1, 1]>],Snew);
    return Vf;
end function;