
// Make newform 1350.2.a.h in Magma, downloaded from the LMFDB on 08 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1350_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_1350_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_1350_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_1350_a();"
function MakeCharacter_1350_a()
    N := 1350;
    order := 1;
    char_gens := [1001, 1027];
    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_1350_a_Hecke( : Kf := Kf)
    return MakeCharacter_1350_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]];
    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], [1], [3], [4], [0], [2], [-6], [-6], [5], [-2], [6], [10], [6], [9], [-12], [8], [-14], [0], [7], [8], [-3], [18], [1], [3], [4], [9], [2], [-6], [7], [15], [6], [-4], [-3], [17], [4], [-20], [-6], [15], [9], [-16], [12], [-5], [9], [-7], [-22], [-8], [-12], [14], [18], [-30], [-10], [0], [12], [-30], [18], [-25], [-8], [-24], [-14], [-6], [16], [6], [19], [-3], [-10], [22], [3], [-10], [6], [-18], [-17], [-32], [20], [-24], [21], [-20], [-12], [23], [-12], [8], [-18], [-29], [-19], [-12], [18], [1], [21], [13], [-27], [-6], [16], [-39], [14], [-18], [15], [-36], [16], [20], [-8], [27], [3], [12], [-4], [-38], [-3], [18], [42], [35], [-32], [34], [-42], [-28], [-25], [-42], [4], [0], [39], [21], [14], [19], [42], [-36], [44], [-9], [44], [-36], [1], [22], [-16], [12], [41], [-2], [-48], [-31], [-18], [-32], [39], [-18], [2], [-6], [31], [0], [2], [-24], [10], [-48], [-40], [-36], [22], [36], [-2], [12], [-26], [-6], [29], [12], [-11], [33], [51], [-36], [-23], [27], [-42], [6], [47], [28], [11], [-6], [-15], [-58], [-30], [-26], [-31], [6], [26], [9], [-47], [-64], [55], [-33], [-14], [36], [12], [-15], [-38], [-8], [59], [54], [-2], [-57], [2], [57], [-36], [12], [23], [-62], [-12], [36], [-39], [-55], [-32], [-19], [9], [-18], [20], [21], [30], [-40], [7], [42], [-35], [-21], [24], [-55], [-32], [-42], [18], [-6], [14], [-4], [24], [-20], [57], [8], [24], [54], [-8], [12], [-14], [56], [-40], [-12], [10], [60], [14], [18], [-39], [-36], [60], [-34], [49], [-16], [-24], [12], [43], [-57], [14], [36], [76], [-72], [30], [17], [-75], [63], [-52], [-8], [9], [-65], [1], [-21], [-58], [16], [12], [-28], [18], [-6], [4], [-6], [32], [16], [-53], [59], [31], [79], [-51], [8], [47], [12], [-60], [77], [-30], [26], [22], [-18], [61], [3], [-64], [0], [-42], [45], [-60], [69], [10], [-66], [17], [-39], [63], [-26], [22], [27], [-25], [-51], [-22], [-47], [60], [86], [24], [16], [-84], [-9], [30], [-68], [-72], [-70], [39], [-12], [46], [-30], [-28], [67], [27], [-11], [24], [47], [74], [64], [-48], [63], [-46], [-6], [17], [24], [-16], [9], [2], [-78], [41], [61], [46], [-48], [-93], [8], [33], [36], [74], [82], [-42], [54], [-40], [-11], [-66], [-89], [14], [18], [-30], [-63], [-78], [42], [64], [-24], [36], [-15], [-74], [-62], [78], [-29], [-7], [-69], [20], [30], [-33], [-31], [52], [96], [-54], [-11], [54], [82], [21], [30], [88], [-48], [-4], [54], [-28], [4], [-74], [12], [17], [-30], [36], [-26], [12], [-14], [-16], [30], [-22], [33], [-4], [-90], [40], [-24], [-45], [-43], [-26], [12], [4], [21], [1], [87], [81], [44], [-41], [63], [-18], [-8], [-54], [-6], [66], [-2], [60], [-57], [43], [57], [-39], [-54], [56], [12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1350_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1350_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_1350_2_a_h(:prec:=1)
    chi := MakeCharacter_1350_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_1350_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_1350_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1350_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-1, 1]>,<11,R![-3, 1]>,<13,R![-4, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;