
// Make newform 2790.2.a.p 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_p();".
// 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_p();".
// 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], [-2], [4], [-4], [-2], [-8], [8], [-4], [-1], [-12], [-10], [8], [4], [-6], [-2], [10], [-6], [-6], [-4], [-8], [-4], [0], [-18], [18], [-14], [-8], [18], [6], [4], [-10], [-6], [4], [18], [-16], [-22], [-6], [8], [-22], [20], [18], [-18], [-2], [-10], [16], [24], [-8], [-4], [22], [10], [4], [10], [-20], [18], [0], [12], [8], [8], [-6], [2], [6], [-2], [30], [20], [22], [-12], [-32], [4], [-30], [-34], [-10], [-8], [34], [16], [16], [28], [-34], [-4], [6], [10], [2], [30], [4], [-40], [-4], [-28], [36], [40], [-36], [0], [10], [-12], [8], [-36], [-12], [-16], [-18], [0], [2], [2], [-6], [0], [28], [-12], [-6], [-28], [30], [30], [-26], [-14], [8], [-42], [-20], [8], [4], [-8], [-8], [6], [2], [46], [8], [-6], [-4], [28], [-2], [-42], [24], [-2], [6], [20], [16], [16], [36], [20], [-34], [-14], [28], [30], [-28], [-16], [4], [28], [-36], [-46], [-42], [14], [-10], [-20], [-16], [-2], [0], [20], [56], [-10], [-24], [4], [-32], [-2], [-12], [-28], [-22], [0], [2], [46], [0], [-56], [18], [-38], [30], [-24], [-2], [42], [-34], [-32], [20], [-12], [6], [2], [-10], [-22], [-12], [-30], [18], [-16], [12], [-26], [-30], [-26], [42], [-20], [24], [-52], [0], [-48], [26], [-50], [-18], [46], [-56], [-14], [-8], [22], [-14], [-30], [-34], [20], [-20], [58], [8], [-28], [-28], [-34], [-52], [0], [-26], [-62], [-26], [-48], [-30], [-2], [20], [-54], [-70], [-68], [-26], [10], [-48], [28], [38], [-28], [64], [4], [-16], [52], [-48], [-22], [66], [-24], [-12], [-32], [-36], [28], [66], [38], [-70], [56], [-48], [4], [-48], [34], [30], [-48], [38], [66], [74], [30], [-8], [54], [-2], [26], [52], [22], [-34], [-54], [52], [42], [-54], [-2], [-6], [10], [-40], [-44], [-16], [6], [-34], [48], [-14], [22], [0], [56], [-36], [-32], [-58], [66], [32], [64], [74], [-4], [16], [-2], [44], [-74], [-6], [0], [-52], [40], [-42], [-8], [-8], [34], [-38], [-72], [-12], [-4], [-2], [-60], [-2], [-68], [-18], [0], [24], [-22], [-16], [12], [-46], [12], [60], [-34], [-36], [52], [20], [18], [-54], [-6], [34], [-36], [46], [-24], [-90], [78], [30], [-4], [84], [48], [56], [6], [-30], [6], [68], [16], [-34], [-52], [-48], [2], [78], [-86], [48], [-12], [70], [44], [68], [30], [-60], [-34], [-42], [20], [-24], [98], [60], [14], [8], [96], [-54], [38], [-68], [30], [28], [38], [-30], [56], [36], [50], [-24], [-64], [-4], [-6], [18], [-34], [-8], [32], [58], [56], [-66], [60], [-92], [-68], [72], [-56], [40], [22], [-86], [54], [-22], [-38], [-38], [80], [18], [12], [44], [62], [66], [42], [42], [-72], [20], [-50], [80], [-36], [96], [-64], [18], [12], [-44], [42], [22], [0], [58], [50], [84], [-76], [20], [72], [2], [34], [-66], [-36], [-24], [-68], [-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_p();".
// 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_p(: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_p();".
// 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_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2790_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![-4, 1]>,<13,R![4, 1]>,<17,R![2, 1]>,<19,R![8, 1]>],Snew);
    return Vf;
end function;