
// Make newform 8400.2.a.ct in Magma, downloaded from the LMFDB on 11 September 2025.
// 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_ct();".
// 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_ct();".
// 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 := 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]];
    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], [6], [-2], [0], [4], [-6], [6], [-8], [-2], [12], [-4], [12], [6], [0], [-10], [8], [-6], [10], [4], [-12], [12], [10], [-12], [8], [-6], [14], [6], [-4], [-12], [-6], [4], [-6], [-8], [-14], [-16], [-12], [12], [-6], [2], [-6], [10], [6], [16], [4], [8], [24], [2], [-6], [-18], [-10], [0], [12], [-6], [24], [16], [22], [-18], [20], [12], [20], [12], [-26], [30], [-20], [22], [-18], [-10], [0], [-6], [32], [10], [-20], [-24], [30], [-14], [6], [14], [24], [26], [-6], [-26], [-8], [30], [-30], [22], [12], [-28], [0], [12], [32], [18], [4], [-24], [24], [0], [-4], [2], [8], [-42], [0], [6], [40], [22], [-24], [-48], [-30], [2], [32], [-38], [18], [-20], [-20], [-18], [-4], [-36], [18], [6], [38], [34], [-12], [-42], [4], [6], [-10], [0], [8], [46], [-32], [6], [-32], [-38], [-36], [26], [-12], [-4], [-36], [-54], [-20], [18], [-4], [-6], [-22], [-12], [-14], [-12], [4], [-6], [-14], [24], [-4], [36], [-4], [-54], [16], [12], [-2], [-24], [-30], [-42], [-28], [-12], [-6], [-48], [40], [10], [14], [-36], [30], [14], [18], [10], [-56], [-24], [16], [30], [-16], [14], [-28], [-60], [22], [0], [-6], [12], [-38], [-28], [38], [-24], [58], [-6], [-20], [12], [42], [24], [62], [46], [-12], [12], [-30], [40], [22], [-10], [12], [60], [-56], [66], [-54], [4], [-2], [-24], [-64], [-36], [60], [2], [-40], [24], [-54], [-54], [38], [40], [42], [-16], [-36], [-46], [36], [-18], [8], [-6], [-14], [-20], [-56], [42], [44], [0], [-10], [-66], [-6], [-72], [30], [28], [-16], [2], [36], [60], [8], [-24], [40], [-6], [58], [0], [-18], [38], [-36], [6], [-22], [-52], [-12], [-2], [-52], [-66], [-46], [-38], [-72], [52], [-54], [0], [32], [18], [-58], [20], [-38], [-8], [-14], [56], [-30], [2], [-82], [24], [-24], [28], [48], [38], [-4], [-6], [46], [78], [-56], [-60], [-78], [-36], [-30], [-72], [-62], [-60], [40], [72], [0], [-76], [22], [-6], [-20], [-6], [40], [22], [-42], [-70], [-6], [22], [72], [6], [-90], [-4], [42], [-10], [36], [-18], [-14], [-18], [4], [10], [12], [8], [-66], [-34], [-56], [-28], [78], [66], [38], [-30], [-80], [-24], [16], [48], [-34], [-84], [-34], [-16], [-86], [-78], [72], [64], [-6], [-90], [74], [44], [24], [12], [-68], [-38], [18], [8], [-10], [24], [48], [36], [54], [-18], [10], [36], [-42], [-66], [44], [10], [-48], [-28], [-34], [-30], [52], [-66], [-30], [-8], [-98], [84], [30], [-74], [96], [-86], [12], [-90], [-16], [54], [-68], [-96], [-80], [22], [32], [-12], [14], [96], [54], [-28], [90], [-50], [-56], [72], [-80], [54], [-58], [-42], [-4], [36], [96], [-8], [46], [-30], [-28], [-24], [46], [18], [54], [28], [-86], [48], [6], [-16], [-72], [36], [54], [10], [-42], [-36], [70], [12], [-66], [-66], [-92], [-36], [38], [-42], [88], [-24], [-62], [12], [-94], [-10], [32], [16], [-48], [-18], [38], [-30], [-82], [-18], [68], [-24], [74], [50], [8], [84], [18], [48], [34], [42], [50], [-24], [94], [-18], [-20], [-20], [-6], [-70], [-76], [-74], [40], [60], [6], [100], [68], [-78], [36], [50], [-30], [-26], [-66], [40], [0], [-42], [58], [-96], [-14], [60], [-64], [42], [110], [60], [100], [52], [82], [-36], [62], [24], [6], [-34], [20], [-18], [112], [-8], [-42], [-40], [6], [-64], [-50], [72], [18], [-104], [-74], [44], [36], [-108], [-26], [-18], [88], [58], [-72], [-58], [-90], [56], [22], [76], [18], [54], [-94], [-12], [34], [96], [66], [84], [-64], [-78]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8400_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8400_2_a_ct();".
// 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_ct(: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_ct();".
// 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_ct( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-6, 1]>,<13,R![2, 1]>,<17,R![0, 1]>,<19,R![-4, 1]>,<23,R![6, 1]>],Snew);
    return Vf;
end function;