
// Make newform 2475.2.a.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2475_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_2475_2_a_b();".
// 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_2475_2_a_b();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_2475_a();"
function MakeCharacter_2475_a()
    N := 2475;
    order := 1;
    char_gens := [551, 2377, 2026];
    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_2475_a_Hecke(Kf)
    return MakeCharacter_2475_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], [0], [0], [-3], [1], [2], [-3], [-1], [-1], [6], [4], [1], [-5], [4], [-3], [-10], [11], [14], [2], [-5], [2], [5], [-8], [-10], [-17], [-11], [2], [-18], [-12], [-18], [-5], [12], [-6], [0], [7], [-16], [10], [10], [-10], [9], [-1], [-25], [19], [4], [-23], [-18], [4], [-22], [8], [11], [-9], [-20], [4], [12], [-12], [18], [-24], [-3], [22], [-21], [-23], [-13], [32], [-12], [-1], [6], [4], [6], [10], [8], [-6], [-28], [4], [6], [-16], [0], [-6], [-6], [-38], [6], [-23], [9], [36], [2], [-35], [31], [-8], [28], [34], [24], [36], [-18], [-26], [28], [24], [-20], [-26], [42], [7], [32], [-33], [-22], [-18], [37], [36], [-43], [-9], [14], [-27], [-22], [40], [16], [28], [34], [34], [-16], [14], [13], [4], [6], [35], [4], [14], [-47], [-40], [23], [35], [-24], [28], [-6], [1], [14], [40], [-42], [-22], [-28], [-44], [43], [-22], [-15], [-21], [-18], [-42], [48], [-2], [-40], [-40], [-39], [-26], [-4], [-52], [0], [2], [-14], [2], [-53], [-39], [32], [18], [5], [27], [21], [20], [21], [-46], [49], [-34], [50], [44], [-12], [-4], [-46], [8], [26], [26], [44], [15], [-18], [-1], [14], [-54], [-4], [25], [-30], [-36], [34], [54], [-36], [10], [-40], [-66], [36], [-8], [6], [2], [-22], [14], [-46], [-31], [-4], [7], [8], [-37], [28], [-45], [-70], [20], [8], [9], [-8], [66], [-50], [-36], [37], [-30], [51], [-47], [-49], [3], [-6], [0], [-32], [28], [-20], [16], [-64], [56], [-33], [17], [-66], [29], [41], [-73], [-3], [26], [-42], [19], [15], [61], [27], [53], [7], [-22], [39], [-21], [-18], [-22], [75], [-4], [36], [-10], [-37], [-24], [-38], [-30], [-6], [-23], [12], [2], [32], [65], [30], [46], [-32], [20], [28], [24], [56], [79], [33], [-25], [-74], [-70], [-23], [-58], [-20], [13], [28], [0], [-26], [-24], [-2], [68], [-2], [20], [15], [-21], [-59], [38], [-69], [-12], [73], [63], [72], [34], [62], [30], [-82], [-50], [-31], [22], [27], [11], [68], [42], [78], [7], [1], [-37], [20], [46], [-54], [-29], [86], [78], [60], [39], [-16], [47], [58], [-83], [-22], [27], [12], [6], [-59], [-4], [-62], [34], [-75], [-20], [44], [-55], [-60], [-41], [65], [-66], [50], [-75], [53], [-13], [68], [-38], [-48], [-66], [-34], [27], [-12], [-6], [26], [13], [-11], [48], [25], [-16], [33], [81], [38], [17], [68], [-34], [-45], [-49], [-44], [-20], [-79], [3], [12], [-36], [76], [26], [74], [5], [21], [45], [-12], [74], [32], [78], [-34], [56], [66], [49], [-18], [-28], [65], [-32], [69], [-47], [-92], [15], [-84], [-9], [0], [24], [-34], [33], [74], [18], [9], [-50], [42], [28], [-42], [3], [-25], [-2], [-66], [-80], [-83], [-88], [51], [97], [-95], [-22], [44], [90], [62], [30], [-22], [28], [2], [29], [6], [35], [-94], [9], [-69], [6], [-30]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2475_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2475_2_a_b();".
// 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_2475_2_a_b(:prec:=1)
    chi := MakeCharacter_2475_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_2475_2_a_b();".
// 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_2475_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2475_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<7,R![3, 1]>,<29,R![-6, 1]>],Snew);
    return Vf;
end function;