
// Make newform 7605.2.a.cf 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_cf();".
// 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_cf();".
// 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, 4, -3, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-2, -1, 1, 0], [1, -3, -1, 1]];
    Rf_basisdens := [1, 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, 0, 0], [1, 0, 0, 0], [3, 0, 1, -1], [0, -2, -1, 2], [0, 0, 0, 0], [0, 0, -2, 1], [4, 0, 1, 0], [4, -2, 0, -1], [-1, -2, -2, 0], [2, 0, -2, 0], [-1, -2, -1, 3], [-2, 0, 1, 0], [0, 0, 2, -1], [-4, 4, 2, 0], [2, 2, 0, 0], [-4, 2, -1, 0], [7, -2, -2, 2], [7, 0, -1, 1], [2, 0, 3, -6], [0, -2, 4, -2], [0, -6, 0, 2], [4, 0, -2, -2], [0, 8, -1, -2], [1, 2, -1, -3], [-1, 6, 4, -2], [-12, -2, 2, 0], [8, 0, -2, 1], [6, 2, 6, -4], [6, -4, -2, 5], [-2, -2, 0, 3], [0, 2, -2, -2], [-1, 4, -5, 7], [3, 4, -2, 4], [4, -4, -1, -2], [2, 2, 0, 4], [4, 2, 6, -6], [-1, 6, -7, 5], [-3, 2, -1, 5], [4, 6, 8, -3], [-7, -2, 8, -2], [10, -2, 2, -6], [-5, -4, 2, 6], [13, 2, -5, 1], [7, -2, 3, -1], [5, 10, -2, -2], [-5, 0, 0, -2], [7, -2, 3, 5], [-7, 2, -3, -3], [-12, 0, -4, 4], [8, -2, 6, 2], [12, -4, -2, -4], [8, 10, 9, -8], [-7, -2, -4, 4], [14, -6, 2, -1], [0, 0, 8, -3], [-13, 8, 0, -2], [10, -12, -5, 6], [-10, -6, -12, 9], [8, -4, 8, 0], [-2, -4, 2, -5], [-11, -4, 3, -1], [6, 2, 2, -4], [-4, 2, -2, 2], [8, 0, 6, 6], [-2, 4, 0, 2], [12, -12, -3, -2], [-2, 2, -8, 8], [2, 6, -6, -3], [-4, 6, -9, -2], [-5, 0, -9, 7], [-12, 12, -2, -8], [4, 0, -6, -5], [-8, 4, 0, -7], [16, 0, 5, -8], [9, 0, 1, -1], [6, -2, 8, -2], [9, -12, -3, 7], [16, -6, -3, 0], [-2, 12, -7, -2], [-21, 2, -4, -2], [12, -4, 0, 0], [-10, 2, 5, 6], [-4, 10, 0, 3], [-5, 2, 0, 6], [14, -2, 12, -2], [4, 0, 13, -2], [7, 0, -7, -9], [-6, 4, -5, 0], [-6, 6, -10, -2], [-6, 12, 0, -6], [6, -6, -5, 12], [-11, 12, -1, 1], [-17, 10, -2, 4], [26, -4, -2, -4], [-2, 8, 12, -11], [22, -12, -7, -2], [22, 4, 12, -8], [-8, 16, -8, -5], [0, -6, 6, -4], [-6, -6, -8, 10], [-1, 8, 9, -15], [0, 10, 6, -7], [1, -14, -2, 12], [12, -4, 6, -12], [-8, 14, 0, -6], [17, 4, 13, -9], [4, -6, -4, 6], [-4, 14, -8, -2], [-9, -18, -6, 10], [-2, 12, 2, 3], [-13, 2, -7, 7], [-3, -4, -23, 5], [4, 4, -6, 4], [-12, -4, -13, 2], [11, -12, -4, -4], [13, -2, -1, 1], [22, 4, 4, -7], [-18, -4, -12, 5], [-7, 6, -4, 2], [10, 4, 5, -8], [16, 10, -2, -5], [4, 22, 2, -6], [5, 2, -9, 11], [-4, -4, 1, 2], [6, 4, 0, -16], [-10, -4, -5, 14], [-21, 6, 4, -6], [-10, 14, 12, 2], [-6, 0, -12, 6], [28, 6, -3, 4], [9, -18, -11, 5], [-15, -4, -10, 14], [24, 6, 4, -11], [-14, -2, 11, 4], [-26, -2, 9, -2], [3, -20, 1, 11], [3, 20, 13, -15], [8, -8, 2, 9], [-9, 12, 8, 6], [8, 8, 10, -8], [2, 2, -1, -12], [2, -6, 4, -7], [6, 18, 6, -16], [23, -16, 0, -2], [32, -6, 7, -4], [4, -2, 12, 0], [-4, 12, -18, 6], [4, 14, 10, -18], [-14, -2, 6, 12], [-13, 4, -1, 9], [-5, 0, -12, -8], [-12, 12, -10, 16], [28, -6, -2, -3], [-10, -2, -16, -1], [-32, -8, -6, 8], [-13, 16, -8, 12], [6, 6, 13, -10], [-28, 10, 2, 2], [8, 2, -14, 4], [1, -6, 27, -5], [-26, 8, 6, -1], [20, 4, -2, 2], [3, 6, -20, 12], [-5, -10, -13, 13], [10, -6, 10, -18], [-5, 8, -20, 10], [-6, -12, 4, 3], [18, 12, 4, -4], [-12, 6, 18, -6], [-2, 22, 8, -4], [-6, 26, -7, -4], [25, -12, 2, 14], [-23, 0, -1, -7], [-4, -8, 0, -8], [-9, 8, 12, -2], [-2, -4, -9, 18], [-18, 2, 14, -8], [-22, -10, -10, 15], [9, 14, 6, 0], [4, 16, 6, -12], [12, 4, -4, 8], [12, -4, -2, 2], [-16, -2, -12, -8], [-1, 6, -13, -11], [5, -10, -16, 0], [-20, 12, -14, 6], [0, 8, 2, 10], [10, 8, 5, 6], [-8, 20, 5, 2], [8, -14, 4, 17], [-37, -2, 13, -5], [4, -10, -8, -2], [-26, 8, -5, 0], [-6, 4, -6, -5], [-14, 16, -16, -9], [8, -14, -2, 12], [16, -10, 12, -17], [-36, 14, 4, 12], [16, -18, -10, 8], [-42, 26, 15, 6], [-15, 14, -6, -12], [-9, 8, -19, 27], [14, -14, 2, -6], [8, 8, -15, 4], [30, 2, -2, -11], [20, 2, -4, -4], [-6, -26, -8, 17], [-36, 14, 3, 10], [-1, 26, 8, -12], [10, -8, -18, 9], [-2, 2, 20, 10], [16, -26, 4, -5], [-7, 12, -1, 11], [24, 16, 3, -8], [-28, -20, -24, 24], [-8, 8, 26, 0], [-47, 10, 8, 2], [-15, 16, 5, -11], [32, 2, -12, 2], [-33, 12, 2, -16], [4, 12, -18, 12], [-8, 26, 6, -24], [-37, -8, 3, 7], [14, -4, -16, -11], [-14, 20, 0, -16], [-34, -12, 12, -7], [-39, 2, -6, 6], [22, 4, 14, 1], [-20, -6, -12, 0], [14, 8, -16, 5], [-15, 4, 28, -4], [38, -28, -7, -2], [-14, -30, 4, 2], [-8, 10, -2, 8], [46, -2, 2, 0], [-4, 24, 15, -6], [-13, -22, -7, 13], [-41, 12, 8, -4], [-7, 0, -10, 2], [-25, 12, -11, 7], [25, -6, -14, 4], [18, -14, 0, -5], [34, 14, 29, -22], [-23, 14, 9, -17], [0, -16, -24, 8], [9, -10, 13, -7], [-26, 16, -3, 14], [-10, 8, -15, 28], [12, -2, -8, 15], [-9, -18, -5, 15], [2, 14, -1, 4], [34, -10, -2, -16], [15, 16, 6, -22], [0, 2, -2, -2], [12, -20, 13, -8], [-1, -24, 6, 0], [3, 8, 19, -13], [30, -14, 0, -4], [-19, 10, -1, -5], [-4, -14, -6, 0], [-34, -8, -28, 27], [-28, 24, -4, -12], [-2, 22, -8, -1], [25, -18, -5, -5], [-5, 0, 24, 4], [-20, -14, 21, -6], [4, 0, -8, -12], [37, 4, -7, 3], [2, -18, 10, -13], [-6, 0, 18, 8], [34, -22, -18, -8], [19, -12, -17, 19], [9, -14, 18, 14], [0, 2, 2, -3], [3, -10, 19, 15], [-11, -14, -27, 25], [4, -4, -12, -8], [0, -14, 9, 0], [-19, 14, 28, 0], [8, 4, -24, 13], [-6, 4, 11, -22], [18, 0, 16, -30], [12, -16, 27, -16], [6, -18, -14, 12], [-12, 20, 16, -8], [0, 14, 18, -2], [28, -10, -8, 2], [-38, 8, -5, -12], [21, 10, 14, -12], [17, -2, 36, -18], [30, -4, 6, -15], [5, -22, -1, -7], [-28, 4, -27, 10], [-56, -4, 0, -3], [-6, 0, 16, -4], [20, 2, 18, -14], [-16, 2, 2, 21], [-5, -4, 8, 8], [-23, 12, 13, -1], [-18, -6, -26, 17], [-10, -12, 24, 2], [-25, 14, -10, 10], [36, -26, -6, 8], [21, -26, 4, -12], [-13, -4, 5, 19], [-2, -16, -16, 14], [14, -24, -6, 28], [34, -4, -3, 8], [0, 22, 10, -6], [24, -10, 22, -1], [-22, -26, -1, 10], [10, -40, -6, 24], [32, 0, 24, -17], [13, -16, -21, 5], [-15, 34, 4, -12], [-14, 0, -9, 28], [-14, -10, 8, -4], [-17, -8, -11, -1], [21, -10, 2, 28], [-36, -16, -34, 24], [22, -8, 4, 12], [-7, -16, 28, 0], [-33, 12, 9, -1], [-10, 20, -20, 6], [3, 18, 10, -10], [16, 0, 18, -4], [29, -10, 19, -13], [-46, 2, -12, 9], [26, 2, 20, -15], [14, 2, 35, -24], [0, -10, -10, -14], [-11, 8, -10, -4], [11, -8, 31, -9], [8, -4, -11, 26], [18, -18, 22, -30], [-4, 2, -3, -22], [39, -2, 1, 1], [10, -2, -29, 4], [20, -22, 10, 0], [12, -22, 0, 2], [-52, 16, 2, -1], [-8, 24, -16, 8], [-7, -12, 24, -6], [-23, 14, 17, 5], [4, 4, 28, -20], [-26, 24, -12, -4], [45, 10, -15, 1], [4, 18, 11, -6], [22, 6, -12, -5], [4, 10, -12, 4], [7, -14, 15, 17], [-6, 26, 15, 0], [-28, -4, 6, 19], [-9, -10, -22, 8], [2, -22, 24, 8], [-12, 14, -5, -20], [0, 38, -15, -10], [-12, -16, 14, 10], [14, 26, 26, -14], [17, 2, 7, 17], [-13, 12, -12, -4], [26, -10, -8, -17], [12, -8, 19, -16], [4, -22, 10, -11], [34, 0, -12, -9], [5, 6, -9, -1], [23, -10, -19, 17], [-14, 24, 10, -20], [27, -40, -8, -2], [-15, 28, 22, -20], [40, -16, 18, 14], [10, -2, 6, -18], [39, -20, -10, -14], [18, 14, -16, 23], [16, -4, -22, 12], [-7, 2, -18, 12], [-3, -12, -9, -15], [35, 12, 0, 6], [-12, -34, -10, 11], [-32, 20, 4, 0], [27, -6, 11, -33], [-2, -14, 34, -10], [-48, 10, 8, 2], [42, -16, 3, 20], [33, -24, -5, 9], [-20, 10, -5, 2], [0, -10, 2, -10], [36, 24, 14, -16], [-14, -10, -22, 29], [12, -20, -3, 6], [11, -16, 27, 9], [-6, -10, 4, 12], [16, -16, -10, 11], [-18, -18, -17, 34], [6, -2, -6, -17], [2, -34, -5, 0], [-22, 14, -20, -26], [28, 18, -10, -10], [38, -20, 7, -8], [-30, 10, 19, 14], [18, 8, 12, -29], [-13, 6, 13, -33], [-36, -2, 8, 6], [10, -22, -1, 2], [-7, 18, 4, -4], [-25, -8, 17, -19], [42, -22, 1, 4], [-2, -30, -18, 12], [10, -20, -31, 8], [2, -18, 4, 16], [4, 2, 44, -21], [-26, 22, -24, -19], [23, 6, 0, -16], [0, -6, 12, -1], [34, 18, 28, -14], [-38, -4, -13, 32], [-6, -10, -16, 26], [-31, -22, -27, 9], [14, -18, 20, -13], [-51, -2, 8, -16], [-28, 10, 12, -30], [-3, 0, 5, 3], [44, 0, 28, -36], [-23, 28, -3, 3], [23, -46, 1, 15], [34, 18, -8, 14], [-48, 16, 20, 4], [14, 0, 31, -6], [9, -18, 24, 6]];
    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_cf();".
// 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_cf(:prec:=4)
    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_cf();".
// 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_cf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7605_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -4, -3, 2, 1]>,<7,R![-11, -22, 30, -10, 1]>,<11,R![33, 0, -30, 0, 1]>],Snew);
    return Vf;
end function;