
// Make newform 8400.2.a.bm in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8400_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_8400_2_a_bm();".
// 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_8400_2_a_bm();".
// 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_8400_a();"
function MakeCharacter_8400_a()
    N := 8400;
    order := 1;
    char_gens := [3151, 2101, 2801, 5377, 3601];
    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_8400_a_Hecke(Kf)
    return MakeCharacter_8400_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], [0], [-1], [-4], [2], [2], [2], [6], [6], [-6], [-4], [0], [4], [-4], [-2], [-4], [-2], [12], [8], [14], [-16], [-16], [16], [-14], [8], [0], [-18], [18], [-6], [0], [-4], [10], [14], [6], [16], [2], [24], [8], [14], [20], [22], [-8], [24], [18], [-10], [8], [16], [-4], [2], [-18], [-8], [26], [12], [22], [-30], [-4], [22], [4], [10], [20], [-22], [-4], [20], [22], [-26], [24], [8], [-10], [6], [30], [-8], [-16], [-24], [28], [-12], [-30], [-34], [6], [-26], [24], [-2], [24], [-34], [6], [6], [18], [-28], [-24], [-24], [-28], [4], [12], [12], [4], [12], [-28], [12], [-12], [10], [28], [30], [-24], [6], [8], [-18], [-16], [-14], [-8], [30], [-8], [-4], [6], [30], [-24], [18], [-4], [-40], [-26], [36], [-2], [12], [-42], [14], [-22], [6], [-38], [-28], [-16], [42], [16], [-34], [-4], [8], [0], [30], [10], [-28], [-18], [-18], [22], [-6], [-8], [30], [38], [16], [-26], [-10], [6], [54], [36], [-36], [56], [-12], [8], [-48], [-20], [48], [10], [-60], [18], [-6], [36], [24], [-42], [24], [-36], [22], [30], [-6], [-4], [18], [-24], [8], [2], [-52], [-36], [-14], [32], [58], [4], [-16], [-8], [-42], [-46], [-64], [52], [-4], [-46], [-60], [-30], [38], [60], [24], [2], [-6], [30], [-64], [-58], [-16], [18], [-34], [22], [26], [64], [-18], [34], [-10], [6], [-18], [-4], [52], [16], [-8], [0], [-26], [-20], [-28], [26], [6], [-22], [-62], [46], [40], [20], [-34], [-66], [24], [-32], [12], [40], [26], [64], [-18], [-68], [0], [-18], [18], [-76], [36], [-30], [22], [-8], [2], [14], [-56], [64], [40], [28], [46], [52], [-4], [50], [-10], [-30], [-36], [54], [12], [-54], [14], [-24], [-10], [66], [26], [18], [66], [-54], [-68], [-44], [-2], [-22], [-12], [26], [-4], [10], [56], [-18], [-66], [50], [60], [-56], [-72], [-48], [-54], [44], [-48], [-68], [14], [-30], [-72], [-30], [48], [-42], [-40], [64], [0], [-70], [14], [-20], [-28], [-46], [46], [-44], [-22], [-16], [16], [18], [-14], [80], [-56], [8], [-18], [-58], [32], [30], [-14], [84], [72], [26], [70], [22], [12], [8], [32], [-42], [-22], [16], [20], [-46], [74], [6], [78], [22], [48], [-28], [-72], [-74], [-86], [-14], [-40], [-8], [70], [-36], [16], [30], [44], [34], [-8], [36], [-54], [42], [-92], [-38], [0], [54], [54], [-84], [-60], [-94], [-42], [-88], [-56], [-18], [4], [-28], [28], [-6], [92], [-62], [52], [-2], [-66], [30], [22], [-20], [36], [0], [-82], [-20], [-10], [-48], [90], [36], [86], [-50], [4], [4], [42], [-84], [-92], [26], [102], [-12], [-68], [-24], [16], [-54], [76], [-24], [54], [-22], [66], [-52], [-38], [-4], [-14], [44], [90], [4], [-20], [-22], [74], [14], [80], [64], [-28], [-80], [48], [-18], [-44], [-54], [-98], [-98], [40], [-82], [66], [-46], [6], [-78], [24], [-2], [36], [20], [-60], [-22], [80], [110], [86], [-28], [-62], [16], [74], [-38], [40], [22], [-42], [4], [8], [30], [58], [-16], [72], [-30], [44], [52], [50], [-14], [-84], [-2], [-30], [-4], [68], [-44], [6], [100], [-40], [-100], [76], [-6], [-94], [44], [-114], [-48], [46], [92], [-26], [30], [-10], [-36], [-66], [46], [44], [-42], [-104], [-16], [-54], [-2], [20], [-86], [92], [-22], [72], [2], [-46], [-60], [94], [100], [90], [-10], [-40], [2], [-72], [94], [28], [48], [18], [6], [58], [-40], [-4], [64], [80], [-98], [42], [4], [40], [36], [-102], [-48], [8], [32], [100], [-42], [62], [-118], [12], [118], [66], [14], [72], [-52], [-74]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8400_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8400_2_a_bm();".
// 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_8400_2_a_bm(:prec:=1)
    chi := MakeCharacter_8400_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(3833) - 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_8400_2_a_bm();".
// 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_8400_2_a_bm( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![4, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>,<19,R![-2, 1]>,<23,R![-6, 1]>],Snew);
    return Vf;
end function;