
// Make newform 7742.2.a.i in Magma, downloaded from the LMFDB on 27 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_7742_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_7742_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_7742_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_7742_a();"
function MakeCharacter_7742_a()
    N := 7742;
    order := 1;
    char_gens := [2845, 4901];
    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_7742_a_Hecke( : Kf := Kf)
    return MakeCharacter_7742_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], [-2], [2], [0], [-4], [-2], [2], [0], [0], [8], [-8], [4], [10], [-2], [0], [-8], [-14], [0], [8], [8], [-6], [-1], [-12], [-6], [-10], [14], [8], [-14], [4], [-14], [8], [0], [-18], [-22], [-16], [-16], [-12], [-20], [-16], [20], [-4], [18], [-12], [14], [12], [12], [14], [16], [-14], [4], [18], [0], [22], [-6], [2], [0], [-22], [20], [10], [-18], [-4], [-24], [6], [-12], [10], [-6], [-10], [-10], [36], [-28], [6], [-8], [-32], [32], [-6], [-24], [10], [-2], [-10], [2], [-22], [-14], [32], [2], [-8], [22], [-10], [-10], [-12], [20], [12], [8], [-16], [6], [-40], [-36], [-40], [-6], [-40], [-38], [-32], [14], [40], [6], [16], [-42], [-2], [-6], [-32], [30], [12], [-16], [-42], [-10], [40], [18], [20], [8], [46], [-6], [16], [46], [18], [16], [-2], [48], [16], [24], [24], [-22], [-26], [48], [0], [2], [38], [-38], [14], [-24], [24], [42], [-40], [-30], [-4], [-22], [20], [40], [16], [30], [2], [-16], [34], [54], [-58], [-32], [-48], [0], [24], [18], [-18], [38], [-26], [2], [8], [32], [-26], [-24], [8], [22], [30], [-22], [-14], [18], [40], [26], [36], [-26], [-34], [48], [24], [62], [-8], [8], [4], [-18], [0], [-48], [-30], [34], [22], [-24], [-58], [-6], [-20], [0], [12], [18], [30], [68], [-54], [64], [-54], [40], [-34], [-6], [-38], [-30], [48], [-20], [26], [10], [-18], [-24], [-4], [-6], [16], [26], [-28], [-34], [-28], [-72], [30], [-8], [-10], [32], [16], [-8], [-26], [72], [8], [6], [-34], [14], [16], [-38], [-2], [0], [-14], [32], [6], [32], [-24], [-66], [-16], [4], [50], [-64], [32], [42], [38], [-36], [-32], [42], [28], [14], [-8], [22], [-40], [-38], [12], [58], [72], [-28], [54], [0], [-62], [-80], [78], [54], [-32], [8], [68], [56], [-66], [32], [-2], [-72], [4], [-66], [66], [36], [8], [-76], [44], [46], [-44], [-52], [-74], [-36], [48], [42], [-66], [-48], [-30], [30], [76], [-40], [40], [-44], [40], [-38], [-26], [30], [32], [-54], [32], [78], [72], [28], [-64], [-52], [-8], [44], [-42], [22], [4], [-26], [-40], [12], [14], [18], [54], [-70], [26], [-56], [66], [2], [52], [-74], [12], [-30], [22], [-58], [0], [-68], [-90], [32], [48], [46], [-2], [-16], [-82], [-2], [18], [48], [-22], [-18], [62], [-10], [36], [0], [16], [-10], [82], [-48], [-46], [54], [12], [78], [-18], [44], [18], [-18], [-16], [-36], [64], [-38], [-48], [-4], [22], [-2], [-20], [48], [-54], [-48], [-76], [28], [88], [58], [54], [-22], [-84], [-62], [0], [18], [-72], [96], [80], [-36], [44], [32], [66], [36], [-40], [-8], [-40], [30], [8], [2], [-100], [42], [-84], [-54], [40], [-86], [60], [-88], [-58], [14], [20], [-54], [-26], [34], [-74], [70], [90], [40], [-36], [-4], [-78], [8], [90], [-62], [48], [44], [50], [-36], [80], [-22], [-50], [56]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7742_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7742_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_7742_2_a_i(:prec:=1)
    chi := MakeCharacter_7742_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_7742_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_7742_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7742_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![-2, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;