
// Make newform 1470.2.a.m in Magma, downloaded from the LMFDB on 20 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1470_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_1470_2_a_m();".
// 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_1470_2_a_m();".
// 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_1470_a();"
function MakeCharacter_1470_a()
    N := 1470;
    order := 1;
    char_gens := [491, 1177, 1081];
    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_1470_a_Hecke( : Kf := Kf)
    return MakeCharacter_1470_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], [-1], [1], [0], [0], [-2], [6], [4], [0], [-6], [4], [2], [-6], [8], [12], [6], [12], [-2], [8], [0], [-14], [-16], [-12], [-6], [-14], [6], [16], [12], [14], [18], [8], [-12], [-6], [4], [18], [-16], [-2], [-16], [-12], [6], [0], [-2], [0], [-22], [-18], [4], [-28], [16], [12], [-2], [-6], [0], [-26], [-12], [6], [0], [-18], [-20], [-22], [-30], [-20], [6], [-20], [0], [10], [30], [-4], [-22], [12], [-26], [-18], [0], [-8], [26], [20], [36], [-30], [-2], [18], [22], [12], [-10], [24], [34], [28], [12], [-6], [2], [30], [-16], [-12], [24], [-16], [24], [-4], [-12], [-18], [-6], [-20], [38], [8], [30], [36], [-6], [-4], [34], [12], [-42], [-24], [-2], [-32], [2], [42], [28], [-16], [18], [4], [12], [-42], [0], [-26], [50], [30], [36], [-20], [-6], [14], [0], [-8], [-2], [44], [-24], [-40], [-22], [-30], [-50], [6], [28], [6], [-30], [4], [-30], [8], [-12], [46], [0], [-26], [-18], [4], [-48], [50], [42], [-40], [12], [-16], [0], [-16], [-54], [34], [-18], [-36], [-6], [56], [-12], [18], [36], [56], [-26], [2], [30], [0], [22], [24], [-46], [-20], [-30], [-28], [-30], [40], [46], [8], [36], [-22], [-18], [-24], [-42], [26], [-44], [-22], [-48], [-14], [-36], [-52], [30], [-36], [-18], [2], [26], [-42], [-36], [66], [-20], [22], [46], [36], [-18], [28], [12], [-54], [4], [2], [54], [-40], [12], [-48], [22], [56], [18], [48], [-42], [14], [4], [42], [56], [-12], [-34], [-18], [48], [-8], [24], [-22], [52], [32], [42], [-44], [-12], [-2], [-18], [72], [-48], [36], [-20], [-56], [-10], [-66], [-24], [16], [-12], [44], [-48], [50], [18], [-24], [-2], [-18], [48], [38], [-68], [54], [-38], [-16], [-12], [70], [-2], [-18], [-68], [-78], [42], [-40], [-18], [22], [56], [58], [-64], [-62], [-32], [-36], [-10], [26], [36], [60], [56], [60], [-74], [-20], [24], [26], [54], [-68], [42], [-30], [12], [-54], [60], [26], [-18], [28], [-18], [36], [-20], [-14], [-42], [80], [-36], [-4], [50], [-36], [-2], [0], [26], [12], [66], [66], [-88], [72], [70], [-36], [-24], [-14], [-54], [-20], [-22], [-66], [-64], [-6], [-74], [20], [-44], [48], [6], [-34], [54], [-44], [-36], [20], [36], [-58], [30], [22], [-56], [26], [42], [6], [8], [54], [72], [-2], [-40], [-24], [-42], [76], [2], [90], [64], [62], [-42], [48], [36], [-30], [-48], [-22], [-30], [-72], [-24], [76], [-46], [-90], [32], [-22], [24], [-92], [-24], [42], [52], [2], [-60], [-72], [-14], [-30], [-14], [-18], [90], [-16], [-30], [76], [-84], [-64], [22], [80], [-12], [50], [6], [-72], [28], [48], [26], [-92], [18], [-28], [-6], [-50], [66], [8], [-42], [6], [-68], [50], [42], [-44], [36], [-38], [6], [12], [-4], [50], [78], [48], [16], [78], [-36], [-6], [70], [-48], [84], [-38], [30], [60], [-78], [-20], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1470_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1470_2_a_m();".
// 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_1470_2_a_m(:prec:=1)
    chi := MakeCharacter_1470_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_1470_2_a_m();".
// 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_1470_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1470_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![0, 1]>,<13,R![2, 1]>,<17,R![-6, 1]>,<19,R![-4, 1]>,<31,R![-4, 1]>],Snew);
    return Vf;
end function;