
// Make newform 1274.2.a.h in Magma, downloaded from the LMFDB on 30 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1274_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_1274_2_a_h();".
// 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_1274_2_a_h();".
// 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_1274_a();"
function MakeCharacter_1274_a()
    N := 1274;
    order := 1;
    char_gens := [885, 197];
    v := [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_1274_a_Hecke(Kf)
    return MakeCharacter_1274_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], [-3], [4], [0], [1], [1], [0], [6], [-7], [-4], [-7], [9], [3], [4], [-7], [0], [10], [-1], [1], [16], [-5], [11], [0], [6], [1], [5], [14], [-4], [-14], [-7], [-11], [-8], [-6], [-4], [9], [0], [-5], [-4], [0], [-2], [-6], [-15], [-8], [20], [-27], [4], [-10], [-21], [-24], [24], [-5], [6], [18], [-3], [-24], [24], [-9], [23], [-18], [-8], [-19], [26], [4], [-30], [-14], [21], [-7], [17], [-24], [26], [-25], [12], [-32], [-16], [-8], [15], [-36], [28], [12], [26], [-15], [19], [18], [-34], [-2], [12], [-2], [-26], [-12], [16], [4], [-24], [38], [16], [-5], [20], [-10], [18], [-27], [-10], [28], [9], [-11], [-35], [6], [-30], [-14], [-14], [-29], [26], [-6], [-49], [22], [-22], [-2], [7], [16], [8], [-36], [-18], [4], [7], [-39], [1], [32], [0], [-13], [-2], [26], [4], [24], [16], [-27], [54], [-45], [-21], [-14], [-32], [-15], [6], [28], [-46], [-3], [-12], [14], [-17], [-16], [-22], [-5], [28], [-45], [-18], [-14], [38], [-30], [36], [27], [15], [36], [50], [24], [-34], [-22], [15], [42], [40], [-57], [-5], [32], [-14], [-36], [18], [37], [-8], [-6], [-4], [23], [58], [44], [-59], [54], [-25], [46], [37], [4], [38], [-8], [0], [-52], [-27], [44], [21], [10], [22], [-2], [-66], [32], [34], [46], [32], [-21], [40], [28], [-10], [-22], [-18], [44], [-6], [42], [45], [-19], [-69], [35], [-64], [15], [35], [-35], [28], [22], [-9], [70], [-35], [-48], [10], [-45], [56], [-44], [-45], [-28], [-69], [40], [44], [0], [-53], [-5], [-11], [53], [37], [-12], [52], [-45], [47], [0], [14], [-41], [-6], [1], [14], [65], [51], [-25], [-50], [-2], [58], [41], [-8], [-42], [10], [27], [-7], [64], [-78], [22], [23], [75], [32], [-5], [79], [39], [-36], [-49], [67], [-21], [-40], [-26], [-24], [-52], [33], [-8], [-60], [82], [-42], [-62], [34], [36], [73], [-31], [-10], [-53], [16], [-42], [-23], [18], [70], [22], [6], [0], [13], [-25], [-86], [46], [65], [0], [-16], [52], [-4], [-50], [-30], [-14], [72], [66], [86], [-27], [70], [-82], [76], [12], [50], [34], [-31], [-65], [16], [-45], [2], [26], [-22], [13], [-24], [-80], [74], [-53], [-52], [-86], [36], [-49], [-74], [14], [61], [66], [22], [-54], [45], [14], [67], [-49], [88], [-46], [-28], [43], [15], [38], [16], [-21], [10], [54], [0], [39], [-78], [75], [-96], [-62], [-42], [5], [-67], [-11], [31], [36], [82], [-15], [42], [-85], [62], [36], [24], [-72], [60], [15], [31], [40], [-4], [36], [44], [72], [64], [-28], [96], [-54], [49], [-4], [64], [6], [-46], [-9], [28], [30], [-101], [51], [-92], [-30], [-54], [-16], [62], [-92], [-67], [48], [-92], [-35], [36], [-52], [-60], [86], [12], [86], [-44], [35], [-18], [-66], [56], [27], [-54], [26], [38], [-12], [36], [-38], [104], [-46], [36], [14], [62]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1274_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1274_2_a_h();".
// 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_1274_2_a_h(:prec:=1)
    chi := MakeCharacter_1274_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_1274_2_a_h();".
// 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_1274_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1274_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<5,R![-4, 1]>,<11,R![-1, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;