
// Make newform 7436.2.a.b in Magma, downloaded from the LMFDB on 08 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7436_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_7436_2_a_b();".
// 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_7436_2_a_b();".
// 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_7436_a();"
function MakeCharacter_7436_a()
    N := 7436;
    order := 1;
    char_gens := [3719, 6085, 3213];
    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_7436_a_Hecke(Kf)
    return MakeCharacter_7436_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], [-1], [0], [2], [-4], [4], [-6], [4], [8], [-12], [-12], [-4], [6], [4], [-10], [-4], [4], [4], [-8], [12], [-16], [-8], [-14], [-4], [12], [-4], [-2], [-8], [12], [-8], [12], [-20], [-12], [-14], [4], [-12], [10], [16], [22], [20], [-20], [-20], [-4], [-4], [-28], [-20], [24], [6], [-20], [12], [24], [-18], [24], [-14], [-12], [2], [20], [-4], [-4], [4], [4], [6], [32], [-4], [2], [-12], [20], [16], [20], [-12], [2], [-36], [-28], [-10], [16], [-24], [-28], [-8], [0], [-4], [-34], [-32], [24], [0], [36], [20], [-36], [-8], [-12], [36], [20], [-20], [-16], [-24], [6], [20], [-12], [28], [-4], [-28], [-10], [-4], [8], [-4], [-4], [12], [-26], [-32], [-28], [-24], [-28], [-20], [18], [-4], [4], [18], [-36], [16], [-14], [-18], [-28], [28], [-10], [-8], [36], [12], [4], [-28], [12], [-28], [-10], [-20], [-4], [-24], [36], [-18], [54], [-4], [36], [-4], [36], [34], [44], [4], [54], [8], [-44], [28], [30], [16], [8], [-40], [36], [-16], [24], [-38], [4], [-12], [22], [-20], [40], [40], [-36], [36], [-2], [20], [26], [12], [32], [48], [36], [-12], [-6], [-28], [24], [-24], [26], [-52], [-12], [6], [36], [-36], [54], [-10], [-20], [-4], [36], [62], [28], [64], [0], [-44], [22], [-12], [-2], [20], [-24], [4], [-32], [-8], [14], [36], [30], [20], [-52], [-12], [8], [14], [38], [68], [-28], [60], [48], [8], [18], [52], [-8], [-50], [-12], [-24], [4], [44], [-30], [58], [-20], [-48], [-60], [-66], [52], [28], [42], [32], [-68], [-56], [12], [-64], [12], [44], [52], [-28], [-64], [-68], [-56], [12], [36], [60], [24], [52], [-44], [-36], [74], [10], [52], [-6], [12], [-42], [76], [24], [-12], [-28], [22], [-48], [56], [40], [-56], [-28], [-78], [18], [-52], [40], [-72], [-34], [4], [-84], [-12], [-4], [-12], [-16], [-20], [-72], [-36], [-28], [52], [82], [52], [-52], [30], [-18], [0], [-4], [-60], [26], [54], [20], [-42], [-44], [68], [-26], [-60], [16], [-24], [24], [64], [-24], [14], [-36], [-2], [64], [48], [14], [48], [68], [-10], [60], [44], [40], [34], [-28], [8], [54], [76], [12], [-14], [-60], [-4], [0], [-14], [20], [18], [56], [36], [-36], [-4], [-16], [36], [-16], [16], [48], [54], [60], [20], [72], [76], [58], [68], [-28], [54], [-36], [-32], [72], [24], [-46], [-90], [-84], [12], [-18], [36], [92], [-86], [-60], [-4], [-64], [-42], [-12], [36], [26], [-56], [40], [68], [78], [40], [-34], [12], [-24], [-52], [-14], [-70], [16], [80], [4], [-60], [4], [-20], [-28], [-54], [-84], [4], [-40], [-56], [4], [16], [-52], [-22], [-52], [-6], [-32], [76], [12], [-62], [36], [64], [-12], [-72], [32], [-36], [-60], [-60], [-18], [94], [16], [52], [-58], [-82], [-20], [28], [80], [-40], [-14], [-28], [20], [20], [-16], [-72], [-80], [44], [-92], [-64]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7436_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7436_2_a_b();".
// 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_7436_2_a_b(:prec:=1)
    chi := MakeCharacter_7436_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_7436_2_a_b();".
// 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_7436_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7436_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;