
// Make newform 2790.2.a.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2790_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_2790_2_a_c();".
// 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_2790_2_a_c();".
// 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_2790_a();"
function MakeCharacter_2790_a()
    N := 2790;
    order := 1;
    char_gens := [2171, 1117, 1801];
    v := [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_2790_a_Hecke(Kf)
    return MakeCharacter_2790_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 := [[-1], [0], [-1], [0], [4], [6], [-2], [4], [8], [-6], [-1], [-2], [-10], [-4], [0], [10], [12], [-2], [-4], [0], [2], [0], [-4], [14], [18], [-6], [16], [4], [-18], [14], [0], [4], [22], [-4], [10], [8], [14], [-4], [-24], [-14], [-4], [-10], [24], [-14], [10], [-8], [4], [-16], [-4], [-10], [-10], [0], [18], [-12], [-2], [8], [10], [-16], [-2], [-10], [20], [26], [28], [0], [18], [-30], [-20], [26], [-28], [30], [14], [16], [32], [38], [12], [32], [18], [-18], [-26], [10], [20], [38], [-24], [-22], [-8], [-28], [-42], [2], [-6], [-16], [-36], [-24], [8], [20], [4], [24], [10], [-42], [-4], [-34], [28], [18], [-36], [-34], [-20], [2], [-12], [-18], [32], [-22], [8], [14], [22], [12], [-40], [-10], [20], [-24], [18], [4], [-10], [10], [26], [-44], [20], [50], [-10], [-16], [-48], [-2], [4], [-24], [-16], [14], [-2], [34], [-6], [-12], [18], [-18], [-4], [-30], [-40], [-12], [-2], [48], [22], [6], [-52], [16], [46], [22], [-44], [-24], [-12], [16], [-56], [38], [-38], [-22], [-36], [6], [-8], [12], [-18], [24], [32], [-42], [-14], [26], [-60], [14], [0], [-6], [16], [30], [-4], [10], [16], [-34], [-24], [12], [22], [54], [16], [2], [-2], [28], [-38], [-24], [-38], [36], [52], [10], [-20], [-42], [-30], [62], [-2], [-8], [18], [16], [54], [-30], [12], [18], [48], [12], [70], [-36], [-54], [-30], [-16], [-60], [-40], [42], [-40], [-50], [8], [-30], [22], [40], [14], [56], [-52], [70], [-26], [0], [-24], [-68], [-58], [-60], [16], [30], [44], [-32], [-14], [10], [36], [72], [-4], [12], [40], [-18], [30], [-48], [48], [28], [-36], [-48], [-50], [-2], [-24], [-22], [-14], [36], [-74], [12], [-70], [-22], [8], [12], [54], [-2], [-66], [68], [-78], [22], [-28], [58], [14], [-28], [-14], [-64], [2], [-48], [-44], [30], [-38], [-20], [0], [-40], [-56], [70], [-60], [-48], [-6], [42], [-24], [6], [34], [-20], [-42], [-84], [-74], [26], [-16], [42], [36], [52], [-54], [-46], [-48], [-20], [-68], [-14], [36], [-82], [-8], [-26], [80], [50], [14], [52], [40], [74], [-36], [80], [-14], [6], [52], [-14], [66], [40], [54], [-14], [4], [-28], [-16], [-22], [46], [-30], [16], [-84], [-68], [-44], [46], [-66], [42], [-16], [-34], [22], [-78], [-88], [-30], [-60], [70], [60], [-32], [-22], [68], [66], [-30], [-80], [-10], [-42], [16], [52], [-82], [24], [70], [30], [-48], [-36], [-52], [66], [50], [-24], [90], [20], [60], [64], [-86], [-8], [38], [-36], [-24], [-94], [-18], [34], [42], [-42], [-56], [-82], [52], [-24], [-64], [30], [76], [0], [-78], [26], [-4], [-84], [-48], [-6], [0], [6], [12], [-30], [-50], [62], [-8], [38], [-30], [24], [-82], [-42], [84], [60], [-6], [10], [68], [4], [-22], [66], [0], [-80], [-82], [24], [-6], [-98], [16], [92], [-54], [66], [-84], [78], [12], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2790_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2790_2_a_c();".
// 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_2790_2_a_c(:prec:=1)
    chi := MakeCharacter_2790_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_2790_2_a_c();".
// 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_2790_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2790_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![-4, 1]>,<13,R![-6, 1]>,<17,R![2, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;