
// Make newform 7800.2.a.e in Magma, downloaded from the LMFDB on 14 May 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_e();".
// 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_e();".
// 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], [0], [0], [1], [-2], [0], [-4], [6], [8], [-10], [-6], [-4], [8], [-2], [0], [6], [-12], [8], [-6], [8], [12], [-14], [10], [6], [-4], [-4], [-6], [-2], [-12], [12], [2], [-4], [-14], [8], [-2], [-20], [0], [6], [-20], [-10], [-8], [10], [6], [-8], [28], [16], [-4], [-22], [6], [-8], [2], [12], [14], [-12], [14], [16], [-2], [-30], [4], [-26], [-4], [-16], [-10], [14], [-8], [-2], [28], [-22], [-14], [-16], [-12], [-26], [-8], [0], [-18], [-34], [-6], [-30], [-36], [-30], [16], [-18], [-16], [-4], [26], [18], [-14], [40], [-12], [-24], [-24], [4], [-24], [12], [10], [-6], [-4], [34], [-36], [14], [-12], [10], [4], [2], [44], [18], [40], [10], [28], [6], [-6], [8], [-8], [18], [-28], [28], [-50], [-4], [-38], [-10], [-42], [-12], [16], [30], [10], [8], [-20], [-34], [8], [8], [-32], [-34], [34], [-30], [-26], [-36], [-42], [-6], [16], [-6], [36], [-4], [-26], [24], [-10], [30], [-12], [-24], [14], [-30], [20], [-36], [12], [24], [24], [42], [22], [10], [-44], [-34], [32], [-20], [42], [-8], [-16], [-26], [50], [46], [-56], [18], [32], [34], [-24], [26], [-16], [-30], [44], [-58], [-16], [12], [46], [58], [48], [54], [-50], [4], [-46], [24], [-10], [-12], [60], [-6], [-4], [38], [-22], [46], [34], [-36], [18], [-32], [22], [-30], [-48], [-18], [8], [36], [34], [-52], [-34], [-26], [-12], [12], [16], [-38], [-4], [18], [-72], [62], [14], [16], [66], [16], [12], [-2], [46], [-8], [12], [0], [62], [60], [-32], [-54], [36], [-8], [26], [70], [4], [24], [-44], [28], [44], [34], [26], [48], [64], [-16], [40], [-20], [14], [-30], [24], [-6], [54], [56], [14], [52], [-34], [-22], [44], [60], [-54], [-74], [2], [44], [-30], [10], [44], [22], [-42], [-28], [2], [-32], [-10], [-16], [-76], [2], [18], [-12], [76], [-16], [-36], [26], [-28], [0], [-26], [-10], [48], [34], [-34], [84], [50], [72], [-26], [-10], [-24], [6], [-52], [-28], [-26], [78], [-80], [-52], [-84], [82], [76], [86], [80], [6], [-12], [-30], [-62], [-4], [0], [90], [-24], [0], [-30], [34], [-52], [26], [-50], [40], [2], [-46], [32], [-28], [-28], [30], [10], [30], [-48], [20], [80], [60], [-38], [-14], [82], [-76], [54], [78], [-54], [-40], [-18], [-60], [-10], [20], [-48], [22], [-16], [-54], [50], [44], [70], [38], [-96], [-96], [-58], [-88], [6], [10], [-44], [88], [-52], [-18], [-98], [-40], [90], [20], [-68], [16], [-2], [24], [-42], [0], [16], [-46], [-14], [-34], [-102], [-46], [88], [-6], [-32], [-24], [64], [-50], [-92], [76], [34], [-90], [0], [-44], [-80], [-50], [64], [42], [-68], [-6], [-102], [46], [0], [26], [66], [-24], [46], [58], [-20], [-56], [54], [-66], [36], [-4], [46], [6], [80], [92], [-62], [-76], [-2], [38], [72], [-84], [-70], [14], [36], [58], [-80], [-40], [18], [64], [-100], [-24], [-98], [-78], [-30], [-46], [68], [0], [4], [10], [2], [-72], [82], [-74], [-12], [-16], [98], [-2], [-4], [-56], [4], [50], [98], [102], [58], [-28], [86], [74], [20], [-8], [-68], [62], [92], [-78], [-80], [36], [2], [-12], [56], [68], [-56], [26]];
    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_e();".
// 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_e(: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_e();".
// 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_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![0, 1]>,<17,R![2, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;