
// Make newform 7800.2.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7800_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_7800_2_a_a();".
// 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_7800_2_a_a();".
// 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_7800_a();"
function MakeCharacter_7800_a()
    N := 7800;
    order := 1;
    char_gens := [1951, 3901, 5201, 7177, 4201];
    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_7800_a_Hecke(Kf)
    return MakeCharacter_7800_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], [-4], [0], [-1], [-2], [0], [0], [-2], [-4], [-6], [-6], [-4], [4], [10], [0], [-2], [-8], [4], [6], [-8], [-8], [-6], [14], [6], [-16], [-12], [14], [-2], [-8], [-20], [6], [20], [6], [-4], [2], [0], [12], [18], [4], [-10], [8], [14], [10], [24], [-20], [-4], [24], [22], [22], [12], [2], [-20], [-2], [8], [-18], [4], [10], [-6], [-20], [10], [-16], [-8], [22], [-30], [8], [-2], [20], [14], [14], [-4], [16], [10], [8], [-4], [6], [34], [18], [-6], [12], [-10], [12], [14], [8], [36], [-30], [22], [30], [20], [-12], [4], [4], [-36], [16], [8], [-2], [-6], [20], [30], [12], [-30], [20], [26], [-4], [-34], [0], [-34], [0], [10], [48], [-38], [-42], [0], [28], [-46], [8], [-48], [-14], [44], [38], [14], [-38], [8], [0], [30], [22], [0], [32], [18], [8], [-12], [0], [10], [-22], [50], [-6], [32], [-30], [-38], [24], [54], [-32], [-24], [-2], [-20], [26], [-26], [20], [-52], [34], [18], [-44], [8], [-52], [-56], [32], [-30], [-10], [-18], [-32], [6], [20], [36], [-18], [-4], [-40], [26], [34], [10], [24], [-2], [24], [6], [-32], [26], [32], [6], [-32], [14], [20], [-12], [58], [22], [-12], [54], [2], [-40], [-22], [36], [46], [32], [44], [14], [36], [-42], [18], [-30], [14], [-8], [-18], [24], [42], [-46], [-32], [-30], [-20], [-68], [10], [-52], [-18], [-10], [8], [24], [-60], [-22], [64], [18], [-12], [-46], [-58], [-36], [-62], [-36], [-60], [-42], [38], [24], [-64], [48], [50], [-20], [-36], [58], [68], [28], [-30], [-22], [-36], [16], [72], [12], [8], [62], [-2], [56], [-76], [40], [-40], [-56], [-46], [18], [-68], [58], [-14], [-24], [-10], [80], [-70], [-58], [24], [-52], [6], [-62], [62], [44], [-66], [-38], [-8], [58], [-50], [40], [22], [0], [-2], [-76], [32], [-34], [-6], [36], [32], [-20], [-24], [-58], [24], [16], [14], [58], [60], [-62], [14], [-36], [-58], [-48], [-78], [-34], [-24], [42], [-60], [-32], [54], [-46], [-16], [12], [-28], [-50], [12], [14], [28], [26], [0], [-26], [82], [-52], [-4], [26], [32], [28], [-66], [-30], [20], [-58], [30], [20], [-26], [50], [40], [64], [-24], [-6], [30], [66], [-72], [48], [88], [80], [14], [-50], [90], [-8], [-6], [-42], [86], [72], [-30], [36], [70], [-24], [-12], [42], [48], [-10], [30], [-8], [54], [-74], [36], [-24], [-50], [52], [-54], [58], [-64], [88], [4], [-26], [2], [92], [10], [-52], [-20], [-76], [54], [16], [66], [-48], [96], [-66], [-14], [54], [46], [6], [-36], [-18], [72], [44], [-68], [74], [48], [96], [-62], [10], [16], [-68], [60], [38], [44], [-38], [84], [-90], [62], [78], [-60], [-74], [6], [-32], [98], [-46], [64], [-40], [-18], [10], [36], [84], [-42], [-18], [12], [56], [-82], [64], [-66], [58], [76], [4], [-10], [2], [-28], [90], [0], [64], [-70], [-24], [-60], [84], [2], [-30], [90], [102], [-20], [-4], [96], [34], [22], [-32], [34], [-18], [-68], [84], [-78], [-18], [-8], [-12], [-8], [-102], [-82], [86], [62], [-84], [74], [-26], [-44], [76], [-12], [-58], [16], [46], [24], [72], [66], [12], [-36], [-72], [-84], [34]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7800_2_a_a();".
// 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_7800_2_a_a(:prec:=1)
    chi := MakeCharacter_7800_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(3361) - 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_7800_2_a_a();".
// 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_7800_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![0, 1]>,<17,R![2, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;