
// Make newform 1472.2.a.i in Magma, downloaded from the LMFDB on 15 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1472_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_1472_2_a_i();".
// 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_1472_2_a_i();".
// 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_1472_a();"
function MakeCharacter_1472_a()
    N := 1472;
    order := 1;
    char_gens := [1151, 645, 833];
    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_1472_a_Hecke( : Kf := Kf)
    return MakeCharacter_1472_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 := [[0], [0], [0], [4], [-6], [2], [6], [6], [1], [6], [0], [8], [6], [2], [-8], [8], [-4], [4], [-2], [-8], [6], [12], [-10], [10], [-18], [-6], [8], [-2], [12], [2], [8], [12], [-2], [-4], [-8], [-8], [24], [8], [-8], [-2], [0], [-12], [-4], [-22], [6], [20], [-20], [-16], [-14], [-28], [-6], [8], [14], [-10], [-10], [0], [-26], [16], [-10], [-6], [-22], [0], [-16], [16], [-2], [-10], [28], [-22], [-24], [-2], [34], [12], [-28], [8], [-2], [-24], [28], [2], [6], [-14], [-10], [-8], [-36], [14], [-16], [-24], [-18], [14], [-10], [16], [6], [8], [32], [40], [-32], [4], [-42], [14], [6], [-2], [28], [28], [10], [6], [22], [30], [-12], [-26], [48], [-26], [16], [28], [18], [10], [-40], [-10], [2], [-24], [42], [-26], [-12], [-2], [16], [-20], [44], [12], [-32], [-32], [-12], [-16], [16], [24], [-32], [-36], [50], [-34], [-44], [-22], [36], [-46], [20], [-30], [24], [-18], [54], [24], [2], [-22], [8], [-8], [6], [18], [-36], [-48], [34], [-20], [-44], [46], [10], [48], [32], [6], [16], [-46], [-18], [52], [24], [14], [-26], [-6], [-42], [-26], [8], [-54], [32], [26], [-16], [10], [-32], [16], [32], [22], [54], [-6], [36], [4], [-54], [-6], [22], [48], [22], [36], [-22], [50], [-18], [-14], [-38], [32], [-22], [-48], [-60], [48], [-10], [34], [-30], [-2], [-4], [4], [-50], [-8], [-18], [2], [-24], [50], [-48], [-42], [8], [42], [-56], [18], [18], [16], [-6], [24], [24], [-6], [-54], [48], [8], [56], [-58], [-46], [40], [38], [66], [-20], [-58], [36], [12], [24], [26], [-40], [-64], [-14], [18], [40], [8], [54], [26], [8], [12], [42], [-8], [-70], [18], [24], [-52], [-2], [-2], [-62], [-32], [-34], [62], [44], [-30], [10], [-60], [-30], [-14], [-74], [-10], [2], [-30], [72], [-30], [16], [-68], [34], [34], [-54], [0], [-8], [20], [20], [-40], [48], [-26], [-20], [-64], [30], [-48], [-26], [6], [-86], [-2], [40], [52], [-30], [40], [60], [14], [60], [-4], [4], [22], [50], [24], [-52], [-32], [58], [-8], [48], [-26], [28], [-60], [-54], [40], [8], [34], [6], [74], [-10], [18], [24], [-18], [-74], [34], [68], [48], [68], [-82], [-14], [32], [-44], [-86], [-72], [-16], [66], [58], [-64], [-62], [30], [-62], [52], [84], [0], [10], [24], [-76], [-80], [0], [-74], [-42], [-76], [-20], [74], [0], [-10], [42], [40], [-68], [-42], [72], [-30], [-92], [-42], [-54], [-12], [-26], [-16], [48], [-24], [20], [40], [-30], [-44], [4], [10], [30], [-54], [-36], [38], [32], [-58], [-26], [96], [80], [-82], [-58], [-84], [58], [6], [-20], [-4], [44], [46], [-32], [6], [-94], [42], [34], [-18], [60], [-34], [42], [0], [-28], [-50], [98], [24], [-10], [-30], [78], [-2], [-78], [54], [48], [-88], [-42], [68], [20], [88], [-48], [-60], [-14], [-38], [-6], [26], [52], [-56]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1472_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1472_2_a_i();".
// 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_1472_2_a_i(:prec:=1)
    chi := MakeCharacter_1472_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_1472_2_a_i();".
// 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_1472_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1472_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![0, 1]>,<7,R![-4, 1]>],Snew);
    return Vf;
end function;