
// Make newform 3080.2.a.i in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3080_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_3080_2_a_i();".
// 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_3080_2_a_i();".
// 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, 0, 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_3080_a();"
function MakeCharacter_3080_a()
    N := 3080;
    order := 1;
    char_gens := [2311, 1541, 617, 2201, 2521];
    v := [1, 1, 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_3080_a_Hecke(Kf)
    return MakeCharacter_3080_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, 0], [0, 1], [1, 0], [1, 0], [-1, 0], [-2, 1], [-2, -3], [0, -4], [-2, 0], [-6, -2], [-6, -1], [0, 2], [0, -3], [-2, 0], [-4, 3], [4, 6], [6, 1], [-8, 3], [-2, -8], [4, 6], [-6, 3], [10, 2], [12, -4], [-6, -2], [2, 8], [0, -1], [-12, 1], [0, 8], [-10, 2], [2, 8], [4, -4], [4, 8], [-12, -6], [0, -6], [6, 0], [10, -2], [-2, 10], [10, 8], [-8, 10], [-10, -1], [0, -8], [-6, 0], [-4, -4], [-4, 6], [-16, 2], [-6, 11], [-12, -8], [0, -3], [-4, 12], [-2, -2], [4, 0], [0, 4], [-16, 9], [-2, -5], [-2, -2], [-4, 4], [2, -8], [4, 2], [-12, -14], [-6, -6], [-4, 0], [6, -7], [-16, -2], [-10, 15], [2, 8], [-2, -12], [12, 8], [0, -16], [-2, 0], [0, -13], [-10, 16], [-2, -10], [-20, -9], [-20, 2], [24, 10], [-8, -9], [2, -20], [-18, -10], [10, -8], [-16, -3], [14, -13], [12, -16], [-6, -10], [2, -18], [16, -14], [2, 22], [24, 4], [-12, -6], [20, 9], [-22, 0], [12, 19], [-8, 0], [-2, 20], [26, -6], [20, 14], [12, -4], [-10, 6], [26, -8], [4, -26], [-10, -8], [-4, 8], [-16, -14], [-28, -6], [42, -2], [-4, 16], [-22, -16], [12, -17], [6, -7], [8, -14], [-16, 17], [-16, 10], [8, 8], [-12, -6], [34, 5], [-24, -8], [-20, -8], [28, -7], [-12, -19], [6, 8], [18, -6], [-18, -6], [24, 8], [-18, 13], [14, -18], [-10, 3], [2, -18], [14, -8], [18, 13], [-12, -7], [-22, -15], [4, -4], [4, 12], [16, 18], [42, 4], [8, 13], [-20, 3], [6, -12], [-40, 2], [22, 18], [-26, 16], [-8, 10], [-6, 28], [22, -2], [26, -12], [6, 4], [-10, 29], [42, 5], [-46, -1], [-34, 11], [22, -18], [-16, -8], [18, -8], [-14, 18], [-12, -14], [-6, -24], [36, 10], [16, 12], [-18, 0], [-6, -11], [4, -29], [30, 6], [32, 6], [-22, 0], [14, 17], [2, 20], [-44, 3], [-4, -26], [6, 31], [-18, -24], [38, -4], [36, -12], [14, 30], [26, 18], [16, 20], [14, -29], [18, 30], [-34, -18], [20, 8], [-24, -8], [40, 7], [-30, -10], [-16, -8], [2, -36], [22, -29], [-38, 2], [26, 20], [-24, 10], [-32, 17], [34, 6], [8, -38], [14, 14], [12, -16], [40, -10], [-18, -2], [2, 12], [-18, -6], [-10, 2], [0, -18], [-6, -23], [-32, 6], [-14, 24], [-8, -6], [6, -2], [-28, 1], [-2, 11], [2, -28], [-2, -7], [-16, 24], [-30, -12], [-26, 17], [-8, -20], [34, 6], [2, 2], [48, -1], [-4, 0], [-2, -2], [54, 0], [40, 7], [10, -16], [38, -12], [-6, 42], [-32, -8], [-28, -8], [-30, -10], [20, 26], [10, 6], [-26, -30], [-8, -28], [-44, 10], [-10, 6], [-32, -10], [-32, -16], [-32, -8], [-18, 12], [-8, -27], [-48, 0], [-18, -18], [40, 6], [8, -10], [6, 9], [26, 8], [-40, -6], [-8, -3], [24, -12], [-18, -25], [28, -2], [-16, 29], [-42, -17], [12, -20], [-44, 16], [-16, -6], [-32, 13], [-2, -4], [-22, -26], [-38, 9], [52, 12], [-28, -4], [-28, 4], [26, -24], [-50, 15], [2, 12], [4, -20], [-16, 17], [-38, 23], [14, -6], [14, 37], [12, 12], [10, -2], [-18, 44], [12, -40], [-2, 36], [2, 16], [-6, -46], [8, -12], [22, -5], [-24, -3], [30, -30], [-6, 16], [6, 40], [20, 38], [-20, -2], [20, -10], [-20, -20], [56, 5], [28, -24], [-32, -30], [48, -2], [28, 18], [6, -49], [44, 3], [40, 12], [40, -1], [-8, 24], [8, -26], [-4, 30], [-44, -27], [-10, -9], [-14, -36], [-28, -30], [16, 6], [-14, -11], [-40, 12], [2, 22], [-30, 8], [-20, 18], [28, -10], [62, -2], [-14, -4], [-12, 36], [-44, -2], [-4, 32], [-54, 0], [-6, 14], [18, 34], [-22, 44], [60, -5], [10, 33], [8, -8], [-38, 10], [22, 34], [-20, -12], [50, -8], [8, 23], [22, 42], [-20, 10], [-26, -20], [0, 10], [-36, 15], [42, -12], [20, 36], [-26, 0], [-8, -2], [-24, -2], [4, 30], [-2, 22], [48, 13], [-76, 12], [-34, 27], [-26, 0], [20, -28], [50, 4], [4, -34], [16, -3], [16, -20], [-22, 0], [12, -24], [62, -8], [26, 32], [8, 50], [30, -6], [-32, 44], [56, -18], [-54, -8], [20, 32], [-70, -8], [2, 35], [-22, 7], [8, 52], [40, 6], [14, -22], [8, 34], [12, -31], [-34, -2], [30, 38], [16, -11], [68, -10], [46, 15], [20, -32], [-14, 52], [-4, 14], [26, -31], [-52, 8], [-30, -36], [-16, 10], [-84, 6], [42, -27], [58, -26], [-22, 45], [-28, -9], [26, -47], [18, 20], [16, 6], [42, 4], [-52, -14], [-48, 12], [-12, -59], [40, 0], [22, 44], [16, -4], [0, -15], [-48, 0], [-30, 8], [44, 38], [68, -13], [40, -20], [-28, -32], [84, -2], [-18, -4], [28, -2], [-26, 30], [-12, -39], [6, -20], [8, -16], [6, 44], [72, 23], [28, 40], [22, 0], [-26, 36], [-20, 9], [10, -49], [26, -11], [-56, 12], [62, -6], [22, -10], [28, 16], [34, 22], [8, -44], [-36, -41], [54, 28], [16, 8], [14, -60], [14, -27], [-30, -16], [8, -54], [14, -36], [-6, 28], [6, 4], [-10, -22], [-38, 33], [36, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3080_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3080_2_a_i();".
// 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_3080_2_a_i(:prec:=2)
    chi := MakeCharacter_3080_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_3080_2_a_i();".
// 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_3080_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3080_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 0, 1]>,<13,R![2, 4, 1]>,<17,R![-14, 4, 1]>],Snew);
    return Vf;
end function;