
// Make newform 7605.2.a.ci in Magma, downloaded from the LMFDB on 15 May 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_ci();".
// 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_ci();".
// 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 := [6, 0, -6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, 0, 1, 0], [0, -4, 0, 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], [0, 1, 1, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, 1, 0, 1], [3, 0, 1, 2], [0, -2, -2, 1], [3, 1, -1, 1], [3, 0, 0, -2], [6, -2, 0, 0], [-3, -1, -3, 0], [4, -1, 1, 2], [-6, -1, 0, 1], [0, 4, 0, -2], [3, 3, -3, 0], [-8, 0, -1, 0], [-3, 1, -4, 1], [3, -1, 3, -3], [6, -3, -1, 0], [-2, -2, 3, -2], [0, -2, 0, -3], [-3, 1, 3, 3], [9, 1, 0, -1], [6, 2, 2, -1], [10, 3, -1, 0], [-6, 1, -4, -2], [6, -6, -3, -2], [0, 0, 2, 0], [5, 1, 2, -5], [3, 3, -1, 3], [0, -5, 0, 1], [-1, 4, 0, -2], [-6, -2, 0, -1], [3, 10, -1, -2], [2, 1, 3, 4], [6, -1, -3, 0], [6, 6, -6, 4], [-6, -7, 8, -1], [3, -1, -5, -4], [1, -4, -5, -4], [-3, -5, 3, -2], [12, 1, -1, -2], [6, -6, 6, 3], [-7, 4, 0, 4], [-5, 6, -2, 0], [-6, -8, 2, -4], [0, -7, 0, 3], [-9, -10, 5, 0], [-6, -2, 0, 1], [6, -2, 0, -5], [9, 2, -1, 6], [0, -10, -2, -1], [0, -1, 0, 2], [6, 3, 10, -3], [-3, -1, 5, 2], [6, 6, 1, 0], [10, 8, 6, -4], [-15, 3, 3, -2], [5, -3, -6, 3], [-6, 7, 0, -1], [3, -3, 0, 3], [3, -5, -13, -2], [10, 7, -3, -2], [0, -6, 0, -5], [6, 4, 5, -6], [-4, 5, -3, 8], [-6, -4, 6, -4], [21, 2, -4, 0], [0, 8, -6, -3], [3, -1, -9, 0], [2, 3, -5, -6], [1, -5, 0, -5], [3, 0, -2, 4], [-12, 5, -12, -3], [0, 2, 2, 5], [3, 13, 2, -3], [-6, -4, -6, 0], [3, -4, -12, 0], [-3, 3, 13, 0], [3, 6, -2, 4], [30, -2, 0, -3], [-7, -9, 12, -3], [-8, 0, 3, -6], [-6, -9, 2, 3], [-6, 8, 0, 0], [9, -7, 4, 1], [-9, 3, -3, -7], [9, -3, -2, -5], [6, -1, -12, 2], [-3, -1, 15, -1], [-3, 4, -5, 5], [-15, -1, 1, -1], [9, -4, 5, 2], [12, -8, 10, 4], [0, -4, 0, -1], [-21, 5, -5, 2], [8, -6, -8, 0], [-12, 6, -9, -10], [-4, -3, -11, -6], [12, -4, -12, 1], [0, 4, 2, -2], [15, -11, 5, 4], [-13, -10, 9, -4], [18, 0, 10, -6], [-6, -13, 0, -1], [0, 2, 18, 0], [15, -5, -7, -8], [7, -8, 3, 4], [11, 0, -3, -3], [-3, -11, -2, 7], [6, -2, 12, -1], [6, -4, 11, 8], [21, -4, -10, -2], [0, 10, 0, -5], [-18, -1, -5, 2], [18, -6, 0, -3], [-6, -9, 10, -9], [-6, -2, 6, 4], [12, 6, -7, 0], [7, 1, 0, -5], [-18, -2, -6, -5], [12, 3, 6, 6], [3, 18, -4, -4], [6, -12, 6, -6], [-6, 2, 1, -6], [-3, 5, 1, -7], [-4, -3, 13, 6], [3, -1, -4, 1], [-6, -4, 6, 0], [6, -3, -12, -2], [13, 0, 9, 6], [13, -18, -7, -3], [-30, 0, 0, 5], [33, 6, 1, -2], [-24, -6, -6, 5], [12, 3, 3, 0], [-6, 7, -6, -8], [-12, 16, 0, -2], [-6, -4, -7, -8], [0, 0, -6, -2], [10, -8, -12, -2], [6, 5, 6, 1], [-16, -16, -3, 2], [6, -6, -6, -1], [12, 5, -3, 14], [-18, -9, 4, 6], [-2, -2, -9, 4], [12, -14, -6, 4], [0, -4, 16, 0], [-12, -4, 0, 8], [8, 3, 15, 0], [6, 3, 16, -3], [-1, -24, -3, 9], [12, -12, 4, 12], [7, -6, -13, 6], [-12, 8, 12, 8], [16, -8, -12, -2], [3, 13, 9, -2], [12, -11, -12, -8], [-6, 4, -4, 19], [0, -8, 6, 4], [6, 8, 4, -7], [-6, 8, 6, -2], [-24, 0, 6, -3], [-19, -18, 1, 6], [-7, 3, -8, -9], [-3, -2, 10, -14], [12, -9, -2, 9], [3, -5, 3, -2], [-15, 2, 1, -4], [12, -8, 0, -11], [18, 3, 3, -8], [25, -14, 7, 10], [-15, -11, -5, 10], [-18, 16, 3, 0], [6, 0, 12, -14], [-8, -9, 15, 0], [-17, 14, 0, -4], [24, -8, 8, 8], [-21, -5, 17, -2], [-14, 9, -7, 0], [-24, -3, -6, -5], [-6, -5, -12, -8], [6, -6, 10, 15], [13, -3, -12, 3], [12, 12, 4, 2], [3, -14, 12, 0], [9, -15, -3, -8], [-1, -19, 0, -7], [24, 7, 0, 5], [8, 4, -9, 10], [-9, -3, 9, -2], [12, -9, 2, -6], [12, 8, 12, 2], [21, 10, 2, -4], [-10, 19, -1, -2], [-6, 1, 0, -14], [0, -19, -6, -7], [21, -5, 3, 3], [-4, 4, -21, -2], [-9, -3, -22, 1], [20, 12, -11, 0], [-6, -6, 24, 0], [0, 15, -16, 0], [-6, -4, 3, -8], [6, 4, 2, 7], [-18, -10, 12, -3], [-16, 4, -22, -8], [-7, 3, 8, 3], [0, 8, -4, -10], [4, -3, -1, 18], [0, 13, -6, -8], [6, -18, -12, 1], [9, 12, -15, 6], [-11, -15, 6, -15], [24, -12, -14, -6], [24, 3, 18, 2], [18, 13, -6, 6], [-4, 2, 9, -10], [15, 4, -12, -2], [6, -16, 0, 3], [-21, 11, -4, 1], [12, 6, -12, 3], [-29, 14, -12, -10], [-30, 6, -12, -9], [12, 18, 12, -6], [11, 15, -16, 9], [15, 5, 15, 14], [1, 9, 6, -3], [-4, -20, 9, -2], [6, -4, 8, 10], [-15, 13, -1, -8], [23, 17, -16, 5], [12, 8, 18, -8], [-15, 0, -26, -4], [-6, 3, 24, 7], [-15, 11, 1, 20], [18, 2, 6, 5], [0, 12, 24, 2], [-37, 0, -5, 0], [22, 6, -6, -6], [-21, 8, 6, 2], [30, 14, 0, -8], [-24, -14, 20, -8], [-21, -11, 2, -3], [24, -6, -6, -2], [-15, -10, -2, 6], [6, -2, -14, 4], [6, 7, 7, -2], [-39, 1, -15, 1], [24, -7, -18, 0], [-10, -12, 27, 0], [0, 7, -26, -8], [21, 7, -9, 12], [-11, 0, 9, -6], [15, 0, 9, -1], [-18, 15, 14, 0], [-27, 6, -3, 11], [-10, 10, 2, 4], [36, 1, -12, -5], [6, 6, 7, -6], [23, 7, 12, 7], [-42, -3, -6, 2], [-5, 8, 24, -4], [-15, -19, 15, -10], [-15, 9, 27, 2], [-12, 8, -12, -12], [-36, 16, -4, -5], [-7, 16, 12, -8], [-9, -9, -4, 1], [6, -4, 0, -6], [7, 0, -15, -6], [25, -9, 12, -3], [12, -18, -2, 6], [24, 0, 6, -6], [-24, -16, -2, 10], [-12, 4, -5, -4], [48, -2, 10, 1], [12, -9, -6, -3], [-36, 0, -6, -4], [-6, 32, 4, -1], [-6, -14, -15, -8], [12, -4, 4, -16], [-12, 6, -6, 0], [-32, -3, -15, 18], [30, 1, -12, 7], [36, -8, 17, 4], [-9, -11, 3, -17], [15, 27, -9, 0], [-12, 7, 2, 1], [30, 0, 6, 4], [12, -8, 6, -6], [53, -3, 10, -3], [-15, -1, -19, -22], [-46, 16, 0, -2], [18, 4, 28, -11], [-24, 2, 34, 2], [0, -1, 5, -14], [10, -1, 19, 2], [24, 7, -18, 9], [34, 16, 6, -2], [-24, 3, -10, 6], [2, 14, 15, 14], [0, 7, -5, 12], [18, 14, 12, -7], [38, -24, -14, -6], [-9, 7, 15, -3], [-28, 0, -18, 6], [-6, 5, -6, -10], [18, 10, -12, 6], [9, -33, -1, -3], [-23, 15, 6, -9], [24, -11, 6, 7], [1, 18, -14, 12], [39, -21, -15, -3], [30, 6, 18, -3], [-3, 1, -8, 5], [-9, 1, 25, 4], [-26, 6, 5, 12], [-18, -19, 9, -2], [-3, -11, -1, -17], [-9, 19, 0, 15], [6, 4, -24, -5], [10, -4, 1, 2], [30, -14, 1, 4], [-12, 0, -10, -12], [24, -1, -18, 8], [18, -5, 32, -2], [15, -12, -1, 0], [-18, 7, -16, 10], [25, 0, 10, 0], [54, 7, 6, -6], [-3, -14, 2, -2], [12, -8, 6, 14], [15, -20, -12, -8], [-12, -6, 30, -4], [-24, 28, 14, 6], [-2, -9, 11, 0], [-39, 6, 1, 5], [30, 20, -22, -1], [21, -11, 3, 2], [19, -6, 18, 12], [6, 27, 12, -6], [-24, 0, 8, 15], [4, 6, 5, -18], [0, 24, 2, 2], [3, 3, -9, 25], [24, 10, -8, 4], [0, 0, 3, 8], [0, 4, -18, 0], [12, 0, 6, -9], [-22, 21, -5, -6], [-13, 6, -39, -6], [42, 22, -2, -2], [33, -21, 15, 15], [9, 37, -3, -6], [30, -14, -30, -5], [36, 10, -24, -2], [-30, 8, -16, -18], [-30, 14, -14, -10], [0, 22, 2, 1], [15, 11, -9, 9], [-35, 7, 0, 1], [17, 7, 6, -11], [12, -15, 12, -6], [0, -1, -7, -8], [-17, -14, -3, 22], [-9, -3, -9, 18], [31, 18, 6, 6], [36, 6, 6, -9], [0, 14, 14, -7], [-17, 18, 7, -6], [-7, 10, 5, -5], [15, -7, -21, -2], [24, -4, -18, 11], [3, 3, 16, -1], [9, 21, 3, 24], [-26, -3, -3, -24], [18, 16, 24, -7], [-42, 5, -12, -19], [12, 10, 0, -4], [-36, -15, -18, 4], [-12, -2, -7, 20], [-6, 17, -6, 14], [-3, 6, 13, 0], [32, 25, -1, -2], [12, 17, -13, 6], [30, -8, 4, -2], [21, -12, 19, 8], [-24, 5, 30, 5], [-18, 0, 0, -1], [19, 20, -23, 11], [-9, 33, -9, -18], [19, 9, -16, -3], [-33, 18, 13, 8], [-9, 5, 9, 8], [31, 10, 9, 10], [27, -23, 9, -1], [-9, -6, -20, -4], [18, 9, 0, 9], [21, -25, 2, 1], [18, 1, -12, -13], [-33, 23, -21, -20], [-47, 8, 28, 2], [27, 25, 14, -3], [9, -23, -15, 10], [-9, 9, 24, 3], [24, -26, 6, -3], [55, -5, -12, -5], [18, -25, 6, 17], [-6, 15, -8, -6], [-2, 36, 2, 12], [16, -15, 9, 6], [15, 9, -15, 3], [24, 38, -12, -4], [-23, 13, 18, 13], [-48, 7, -24, -2], [42, -31, -18, -7], [15, -9, 5, 0], [-36, 13, -5, 0], [-42, 1, 6, -4], [-6, -4, 18, -4], [36, -15, -13, -20], [12, -1, 18, 22], [-18, -9, 28, -9], [-6, 28, 24, -5], [42, 8, 9, -16], [-18, 6, -10, 21]];
    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_ci();".
// 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_ci(: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_ci();".
// 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_ci( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7605_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![6, 0, -6, 0, 1]>,<7,R![-3, -12, -12, 0, 1]>,<11,R![24, 0, -12, 0, 1]>],Snew);
    return Vf;
end function;