
// Make newform 8112.2.a.bi in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8112_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_8112_2_a_bi();".
// 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_8112_2_a_bi();".
// 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_8112_a();"
function MakeCharacter_8112_a()
    N := 8112;
    order := 1;
    char_gens := [5071, 6085, 2705, 3889];
    v := [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_8112_a_Hecke(Kf)
    return MakeCharacter_8112_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], [1], [4], [-2], [-4], [0], [2], [-2], [0], [-6], [-10], [-10], [-8], [-4], [-4], [-10], [-8], [-14], [2], [16], [10], [16], [0], [4], [2], [10], [8], [-12], [2], [6], [-12], [-4], [-8], [0], [0], [18], [2], [2], [-12], [2], [-12], [2], [8], [14], [12], [-4], [12], [14], [-4], [2], [-18], [-24], [2], [20], [30], [16], [-14], [-10], [22], [8], [16], [-8], [22], [-24], [-6], [12], [2], [-30], [-12], [-18], [16], [-12], [-4], [-18], [6], [12], [-26], [-18], [12], [-14], [20], [2], [-20], [-18], [-16], [4], [-12], [18], [-12], [22], [28], [-16], [-26], [-12], [-6], [-16], [-12], [-2], [-16], [-38], [-32], [-24], [-28], [6], [8], [30], [-8], [36], [-24], [-22], [4], [14], [48], [14], [-30], [26], [26], [24], [-46], [12], [-10], [22], [38], [20], [10], [-2], [-6], [48], [-44], [-22], [22], [48], [-8], [18], [-40], [22], [48], [-50], [-26], [38], [6], [-24], [-32], [-20], [-14], [-16], [14], [-18], [-52], [4], [-10], [14], [32], [-48], [56], [56], [-56], [-16], [-34], [-8], [8], [-42], [22], [28], [12], [24], [36], [22], [2], [26], [-12], [-42], [-40], [-54], [32], [-6], [-58], [-8], [24], [-26], [-26], [-20], [2], [60], [12], [46], [-38], [-50], [-22], [60], [-2], [-48], [16], [64], [-52], [30], [-18], [30], [12], [-32], [12], [-12], [-22], [-34], [-32], [14], [-10], [52], [12], [-8], [46], [-42], [-16], [-12], [44], [-18], [-48], [18], [16], [16], [6], [-50], [-28], [-10], [-36], [-50], [58], [72], [-4], [-24], [-54], [0], [-70], [18], [-16], [24], [-22], [-24], [44], [24], [-16], [56], [48], [62], [-20], [24], [30], [-48], [54], [-24], [-38], [16], [-56], [26], [-54], [60], [6], [-10], [-66], [-2], [-28], [-12], [-70], [14], [-60], [36], [60], [-24], [-10], [-30], [-14], [-30], [-46], [0], [10], [-54], [12], [38], [70], [36], [0], [-34], [-64], [-58], [14], [-48], [30], [-64], [-70], [30], [22], [-52], [20], [-72], [50], [-18], [-4], [6], [60], [86], [-22], [-60], [16], [60], [64], [58], [-84], [14], [24], [86], [-32], [28], [30], [28], [-36], [34], [-84], [12], [-62], [34], [20], [-18], [-42], [-46], [60], [-2], [10], [-46], [-8], [-78], [-30], [-14], [32], [-68], [70], [-16], [22], [-56], [-46], [56], [14], [6], [24], [-8], [-36], [-20], [-58], [-14], [-68], [38], [66], [-46], [48], [-44], [-54], [-42], [-36], [60], [-18], [84], [14], [-46], [-48], [-20], [-52], [-26], [-12], [-22], [-2], [92], [-36], [-92], [-26], [16], [62], [24], [-48], [14], [90], [70], [24], [32], [-98], [20], [-6], [88], [18], [-54], [-66], [-64], [50], [48], [28], [20], [52], [86], [50], [74], [100], [76], [78], [34], [6], [68], [60], [48], [-30], [60], [14], [76], [66], [6], [52], [12], [-78], [-2], [-8], [0], [20], [48], [106], [46], [32], [4], [-66], [-32], [-68], [36], [-22], [-96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8112_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8112_2_a_bi();".
// 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_8112_2_a_bi(:prec:=1)
    chi := MakeCharacter_8112_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_8112_2_a_bi();".
// 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_8112_2_a_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8112_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 1]>,<7,R![2, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;