
// Make newform 3675.2.a.s in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3675_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_3675_2_a_s();".
// 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_3675_2_a_s();".
// 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_3675_a();"
function MakeCharacter_3675_a()
    N := 3675;
    order := 1;
    char_gens := [1226, 1177, 2551];
    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_3675_a_Hecke(Kf)
    return MakeCharacter_3675_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 := [[-1, 1], [-1, 0], [0, 0], [0, 0], [0, -2], [2, -2], [2, -4], [2, 2], [2, -4], [6, 0], [-2, -6], [-2, -4], [6, -4], [-8, 0], [0, 4], [0, -6], [8, 4], [0, 0], [0, 0], [0, 2], [14, 2], [-8, 0], [-4, -4], [6, 8], [2, 6], [2, -4], [12, 0], [6, 8], [6, 0], [0, 2], [-8, 8], [0, -4], [-16, -2], [-10, -6], [-2, 4], [0, 12], [2, 10], [12, 0], [0, 8], [-2, 16], [-4, -2], [8, 4], [-12, -2], [-10, 4], [12, 6], [2, 2], [-4, 4], [4, -12], [-12, 12], [-4, 8], [-12, 2], [12, 10], [-16, 4], [24, -4], [6, 4], [14, 8], [6, 4], [-14, 6], [-10, 8], [18, 4], [0, 0], [18, 4], [12, -12], [4, -4], [6, 6], [-8, 10], [-4, -4], [-10, -12], [6, -12], [16, 0], [14, -8], [4, 14], [-4, -12], [2, -16], [-4, 12], [0, 0], [-6, 4], [22, -2], [30, 0], [12, 0], [-12, -8], [-6, 0], [-8, 2], [-10, -2], [-22, -6], [-18, 4], [10, 16], [-22, -4], [2, -8], [-16, 16], [-4, -20], [-20, -4], [4, 24], [4, -10], [-12, -12], [-16, -8], [-34, -4], [-6, -20], [-8, 20], [18, 0], [16, -8], [-24, 2], [-20, -8], [22, 8], [-28, -8], [-10, 6], [20, 8], [-18, -16], [-36, 6], [28, -4], [20, 12], [-10, -24], [12, 18], [-34, 10], [-8, -8], [-6, 20], [12, 12], [8, -8], [0, 14], [12, -22], [-8, 8], [-6, 12], [10, -4], [22, 4], [2, 6], [-10, -16], [2, 0], [-4, 4], [16, 4], [-22, -18], [4, 24], [18, -8], [-32, 4], [10, 4], [-2, -8], [20, 12], [-10, 16], [-8, 4], [30, 16], [-2, -4], [-18, 18], [-22, 16], [8, -24], [6, 8], [8, -28], [-24, 16], [-2, 6], [-22, 16], [-26, 14], [-30, 12], [6, 24], [2, 32], [-28, 0], [24, 12], [28, -8], [12, 14], [16, -20], [-42, 0], [10, 18], [22, 16], [18, 4], [8, 10], [-28, -16], [28, -8], [12, -6], [-16, 4], [0, -12], [30, -18], [2, 32], [18, 12], [32, -10], [4, -4], [0, -6], [-10, 24], [-18, -18], [-6, 12], [4, -8], [18, 12], [8, 20], [12, -12], [-4, -8], [12, 0], [18, 24], [2, 4], [30, 0], [10, 28], [-18, 12], [-24, 12], [46, 0], [-4, -20], [-46, -10], [14, -16], [-20, 12], [-6, 24], [-6, 28], [-26, -8], [6, -24], [66, 0], [6, -12], [0, 8], [10, 16], [-14, -22], [-10, 18], [16, 12], [-12, 32], [30, 8], [22, -18], [-26, -8], [22, -28], [-2, 6], [2, -8], [-18, -4], [8, -24], [12, 12], [-24, 16], [-4, 36], [-20, -24], [-54, 8], [22, 32], [32, 14], [-46, -16], [-50, 10], [-10, -28], [-8, 16], [-20, -32], [-14, 8], [-2, -4], [48, -2], [36, -24], [12, -18], [-38, -12], [-22, 26], [-16, -4], [6, -28], [-20, -20], [24, -20], [36, 0], [12, -2], [-36, 14], [20, 20], [-30, -16], [-10, -22], [-64, 0], [-10, -16], [-30, 24], [48, 8], [-8, -28], [36, -8], [-4, -40], [10, 44], [38, -20], [-14, -16], [-18, -4], [36, -24], [62, 8], [0, 30], [2, 16], [0, 24], [22, 8], [-58, 6], [-16, 16], [-18, -12], [36, 24], [26, -6], [-18, -16], [26, 30], [-14, -32], [-22, -20], [0, 0], [0, -10], [20, -12], [-16, 24], [18, 6], [-8, 4], [-14, -34], [40, -4], [-18, -24], [34, 0], [6, 24], [16, -20], [-24, 36], [-48, 0], [32, -4], [-12, 36], [-8, -28], [4, 14], [50, -12], [32, -38], [46, -6], [22, -40], [10, -16], [36, -4], [-12, 26], [60, -8], [-38, -8], [74, -8], [-42, -18], [26, -8], [8, 28], [-8, 44], [-14, 42], [0, -34], [-8, 8], [-6, 12], [20, -16], [-10, 28], [66, -16], [12, 24], [40, 2], [10, -28], [0, 36], [-42, 8], [-2, 40], [-68, 0], [-62, -4], [64, 12], [-24, -12], [16, 14], [10, 30], [54, -16], [-10, 54], [34, 4], [-78, 0], [-32, -24], [60, -2], [8, -36], [-4, 40], [8, -24], [6, -52], [60, 10], [10, -24], [-8, -34], [10, -10], [-52, 16], [-20, 32], [-60, 16], [2, 16], [-18, -4], [-16, 12], [40, 12], [42, 0], [-44, -14], [54, 20], [-48, 12], [48, 6], [-36, 26], [20, -24], [-68, 0], [-36, 12], [18, -12], [-22, 2], [26, 32], [18, -8], [-32, 0], [-30, 24], [54, 8], [36, -28], [16, 4], [-88, 2], [58, -16], [-14, 40], [-66, -20], [-14, 16], [28, -22], [48, -12], [2, 36], [-30, 8], [28, 8], [38, -24], [36, -18], [38, 22], [-30, 16], [-66, -20], [74, -2], [62, 0], [-24, 36], [24, -10], [-18, -18], [18, -36], [-2, -22], [18, -20], [0, 10], [76, -8], [-12, 6], [-50, 26], [8, -40], [64, -4], [58, -18], [56, -8], [-56, 28], [22, 0], [6, 12], [60, -6], [-20, 28], [48, -6], [-34, -36], [22, -42], [18, 8], [20, 24], [30, 24], [-52, -4], [16, 26], [4, -24], [-22, 16], [6, -24], [-2, 30], [30, 0], [-6, -64], [44, 24], [0, -20], [-38, -38], [36, 26], [78, 12], [36, 12], [22, 24], [54, 12], [20, -46], [-64, 12], [10, 20], [-48, -28], [-22, 4], [-66, 6], [-30, -32], [60, 24], [6, -66], [-6, 12], [-54, 16], [14, 40], [-14, -66], [-48, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3675_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3675_2_a_s();".
// 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_3675_2_a_s(:prec:=2)
    chi := MakeCharacter_3675_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_3675_2_a_s();".
// 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_3675_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3675_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 2, 1]>,<11,R![-8, 0, 1]>,<13,R![-4, -4, 1]>],Snew);
    return Vf;
end function;