
// Make newform 9360.2.a.bg in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9360_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_9360_2_a_bg();".
// 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_9360_2_a_bg();".
// 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_9360_a();"
function MakeCharacter_9360_a()
    N := 9360;
    order := 1;
    char_gens := [8191, 2341, 2081, 5617, 5761];
    v := [1, 1, 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_9360_a_Hecke(Kf)
    return MakeCharacter_9360_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+1)]];
    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], [1], [-2], [0], [1], [0], [-2], [-6], [0], [4], [2], [6], [4], [0], [6], [0], [-10], [-8], [0], [8], [-8], [-12], [-6], [8], [12], [-8], [12], [-16], [12], [-8], [-18], [-18], [-8], [-6], [-20], [2], [-20], [12], [6], [18], [-22], [24], [20], [-6], [16], [4], [-14], [12], [-16], [-24], [-24], [2], [18], [-12], [6], [-12], [4], [-22], [-6], [4], [-6], [16], [0], [14], [30], [-26], [-34], [24], [-16], [-18], [0], [16], [-22], [10], [-36], [-12], [-22], [6], [14], [-18], [-4], [24], [-34], [-8], [-24], [18], [32], [-18], [-14], [-12], [-24], [-2], [6], [-14], [6], [-18], [18], [4], [20], [4], [6], [0], [-30], [16], [-4], [36], [-6], [-24], [38], [-32], [26], [-18], [-14], [-8], [-42], [40], [-18], [-6], [-18], [-28], [26], [-6], [36], [-2], [36], [8], [36], [28], [26], [34], [-12], [-32], [14], [54], [-22], [42], [4], [-18], [-18], [22], [-30], [40], [-36], [-34], [-24], [26], [-12], [-20], [36], [-34], [18], [52], [18], [-44], [-36], [-56], [-30], [26], [-6], [36], [-48], [-50], [18], [-6], [-12], [-32], [-46], [-46], [-42], [36], [-40], [12], [-28], [-32], [42], [46], [-6], [-20], [50], [-2], [-18], [14], [42], [60], [-24], [-34], [16], [14], [-24], [2], [-36], [4], [42], [48], [36], [14], [-46], [-30], [6], [54], [-8], [14], [14], [24], [18], [-20], [-24], [-18], [-44], [38], [24], [52], [-12], [-48], [26], [-68], [42], [-60], [54], [50], [40], [30], [22], [0], [26], [48], [-12], [-44], [24], [50], [28], [28], [-30], [76], [12], [-58], [54], [18], [36], [12], [-44], [28], [20], [-6], [-72], [-38], [-12], [46], [-42], [-22], [42], [-24], [-70], [-66], [60], [14], [28], [54], [56], [-44], [-36], [56], [14], [42], [-32], [-18], [-30], [-32], [42], [26], [76], [-28], [-56], [50], [-50], [60], [-76], [14], [-30], [30], [40], [-42], [80], [16], [60], [-34], [42], [40], [30], [0], [60], [66], [36], [-10], [-12], [64], [42], [30], [28], [14], [18], [40], [0], [52], [-28], [12], [26], [0], [38], [66], [-42], [18], [76], [0], [74], [-72], [72], [-16], [54], [16], [32], [-60], [-38], [42], [14], [34], [-8], [-42], [6], [-76], [-54], [-8], [84], [-2], [-36], [20], [78], [-10], [-56], [-10], [-36], [-30], [64], [-6], [-66], [-34], [16], [-72], [90], [58], [92], [-6], [28], [-70], [-48], [-48], [72], [24], [-72], [-10], [54], [-42], [-12], [52], [-34], [-54], [-38], [-70], [30], [16], [-12], [-48], [40], [38], [-12], [0], [92], [-54], [14], [-102], [30], [34], [-78], [58], [0], [64], [-10], [88], [78], [50], [30], [48], [-20], [24], [-58], [-68], [78], [28], [-90], [92], [36], [10], [-78], [-90], [-80], [26], [-78], [-68], [-36], [-34], [6], [72], [40], [50], [-72], [24], [-8], [-54], [6], [-48], [-10], [-48], [-30], [80], [-42], [-36], [-6], [-38], [-72], [-10], [-36], [-56], [12], [-22], [-42], [-58], [104], [52], [-68], [-108], [-90], [68], [-48], [-10], [12], [-68], [-24], [-106], [-58], [-20], [24], [36], [-6], [8], [0], [80], [-102], [50], [42], [16], [-8], [42], [26], [4], [56], [-104], [-60], [30], [76], [-50], [-12], [24], [38], [-102], [98], [-24], [-20], [-78], [6], [14], [30], [50], [36], [82], [-12], [-40], [-84], [-62], [16], [2], [-42], [62], [-54], [-54], [104], [16], [42], [16], [40], [102], [-110], [42], [10], [-70], [96], [-30], [-32], [-58], [-44], [36], [72], [-16], [66], [-56], [-28], [36], [-106], [-36], [28], [-70], [46], [-78], [78], [98], [-18], [86], [-18], [36], [-72], [-56], [66], [-32], [-30], [-46], [-60], [110], [-42], [-106], [76], [72], [114], [100], [-48], [-18], [-26], [28], [12], [22], [18], [-18], [-20], [-42], [90], [-60], [14], [-92]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9360_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9360_2_a_bg();".
// 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_9360_2_a_bg(:prec:=1)
    chi := MakeCharacter_9360_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(4027) - 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_9360_2_a_bg();".
// 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_9360_2_a_bg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9360_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![0, 1]>,<17,R![0, 1]>,<19,R![2, 1]>,<31,R![-4, 1]>],Snew);
    return Vf;
end function;