
// Make newform 2394.2.a.i in Magma, downloaded from the LMFDB on 01 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_2394_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_2394_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_2394_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_2394_a();"
function MakeCharacter_2394_a()
    N := 2394;
    order := 1;
    char_gens := [533, 1711, 1009];
    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_2394_a_Hecke( : Kf := Kf)
    return MakeCharacter_2394_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], [-2], [-4], [0], [-1], [-2], [-6], [-4], [-6], [6], [0], [4], [10], [4], [-2], [-6], [-12], [-14], [-14], [-4], [14], [-8], [0], [12], [12], [-6], [-6], [6], [0], [2], [20], [-6], [18], [2], [8], [0], [-14], [-12], [0], [-26], [-6], [-22], [-16], [10], [-16], [20], [22], [18], [-2], [-12], [8], [-6], [18], [26], [0], [22], [-6], [4], [-14], [16], [-8], [-6], [6], [14], [-2], [-6], [-14], [8], [18], [8], [-10], [-26], [16], [6], [-22], [-18], [16], [8], [-18], [20], [0], [20], [-26], [-14], [22], [0], [16], [36], [-24], [-26], [-14], [-24], [0], [34], [-10], [16], [-14], [34], [-14], [44], [14], [44], [2], [-36], [-4], [-20], [-40], [16], [30], [18], [-4], [-4], [2], [-12], [24], [-18], [-20], [40], [-42], [-42], [32], [52], [-42], [-22], [40], [-40], [18], [-28], [-16], [-26], [-2], [-12], [-22], [-46], [-16], [18], [6], [-16], [-54], [8], [-28], [20], [0], [14], [-42], [-44], [-16], [2], [-32], [-16], [8], [-26], [-20], [-16], [8], [-42], [-6], [34], [58], [-60], [-36], [46], [-40], [26], [-38], [-46], [-16], [48], [-56], [-30], [-46], [16], [36], [20], [22], [0], [-34], [-60], [-52], [-30], [18], [10], [-56], [38], [52], [30], [52], [4], [-6], [-22], [-66], [58], [-18], [-22], [-10], [-48], [24], [50], [-64], [48], [28], [20], [-16], [-32], [-16], [-18], [-8], [-2], [-32], [8], [12], [24], [-12], [16], [-34], [-44], [-18], [34], [64], [46], [-24], [-44], [-26], [18], [-16], [36], [10], [-46], [0], [-10], [-10], [28], [20], [-14], [18], [54], [0], [64], [-28], [0], [-2], [-62], [24], [-24], [28], [-58], [50], [2], [-12], [-58], [52], [-60], [6], [-54], [4], [-18], [6], [78], [-48], [-6], [76], [-12], [-4], [34], [-12], [34], [-74], [40], [38], [42], [44], [72], [-16], [30], [14], [-22], [-20], [24], [-24], [-48], [-60], [36], [-6], [62], [74], [-32], [2], [-30], [-12], [-6], [-44], [-26], [-12], [-60], [0], [4], [-28], [-2], [42], [44], [72], [-56], [70], [16], [60], [-50], [66], [12], [38], [-30], [26], [-90], [8], [12], [12], [-66], [90], [-84], [-58], [78], [-38], [54], [-52], [-2], [72], [-12], [14], [-62], [-34], [64], [64], [8], [-4], [-38], [-50], [-66], [-72], [-10], [42], [30], [-18], [94], [44], [50], [-38], [-72], [16], [-44], [-70], [58], [-48], [86], [86], [84], [32], [86], [-36], [10], [92], [-4], [24], [84], [22], [36], [14], [90], [-42], [-28], [18], [54], [-56], [74], [60], [-82], [-86], [36], [20], [-2], [-74], [52], [38], [76], [-24], [28], [-98], [32], [-88], [-90], [-42], [18], [36], [72], [94], [68], [-30], [-10], [-58], [-68], [-58], [10], [-94], [-54], [-72], [-18], [94], [-16], [84], [56], [62], [-20], [-56], [38], [28], [-96], [40], [-48], [48], [30], [-88], [26], [-52], [0], [58], [60], [18], [20], [-84]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2394_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2394_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_2394_2_a_i(:prec:=1)
    chi := MakeCharacter_2394_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_2394_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_2394_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2394_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![2, 1]>,<13,R![4, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;