
// Make newform 2790.2.a.k in Magma, downloaded from the LMFDB on 24 December 2025.
// 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_k();".
// 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_k();".
// 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 := 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]];
    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], [0], [-4], [-6], [8], [0], [0], [1], [-4], [6], [8], [12], [6], [6], [2], [2], [6], [8], [8], [-12], [0], [-10], [6], [-10], [0], [2], [18], [-16], [6], [6], [-4], [-18], [8], [14], [2], [24], [6], [0], [2], [18], [14], [-6], [8], [8], [-28], [-12], [14], [-18], [-12], [2], [0], [30], [0], [0], [32], [8], [18], [-22], [-6], [-34], [18], [-16], [-6], [-4], [20], [36], [2], [-30], [-30], [20], [38], [32], [0], [-24], [2], [-36], [14], [-30], [26], [-6], [-16], [8], [36], [-12], [8], [36], [-16], [-24], [6], [-40], [-12], [-4], [-36], [-12], [6], [-40], [2], [2], [-42], [24], [-36], [20], [-22], [12], [-6], [-6], [-10], [-34], [-40], [-6], [-4], [8], [12], [32], [0], [-6], [-6], [-10], [44], [-18], [36], [44], [-30], [-10], [24], [26], [-22], [-52], [-24], [-16], [-4], [-12], [-34], [54], [20], [-6], [-36], [-16], [24], [-16], [12], [-22], [18], [2], [-54], [20], [0], [-22], [0], [44], [-24], [-10], [-24], [-52], [0], [38], [48], [-12], [-18], [-4], [-6], [18], [-24], [-16], [-10], [50], [42], [-36], [14], [30], [-34], [-16], [-12], [-28], [-30], [14], [-10], [38], [24], [14], [-18], [0], [24], [50], [26], [38], [6], [-40], [-48], [-52], [12], [-24], [6], [-58], [-22], [42], [24], [-18], [56], [50], [50], [-30], [18], [-52], [12], [-30], [8], [32], [0], [2], [36], [-24], [-58], [-58], [30], [0], [30], [38], [-4], [-54], [38], [-12], [38], [54], [0], [-16], [-18], [68], [-16], [-4], [48], [44], [-48], [-70], [-66], [12], [-12], [24], [44], [-16], [-46], [42], [6], [20], [-60], [68], [-48], [14], [-18], [48], [-10], [30], [18], [-58], [-64], [42], [-34], [14], [-12], [-34], [-46], [-18], [-28], [6], [-6], [-58], [6], [50], [56], [80], [32], [-58], [-46], [-72], [26], [-34], [-84], [24], [68], [-48], [38], [2], [24], [-52], [30], [68], [72], [42], [36], [-6], [42], [-64], [72], [-40], [-6], [-36], [-40], [-22], [-66], [-16], [12], [-28], [38], [12], [14], [-36], [26], [-72], [-36], [-30], [-16], [84], [74], [-24], [-84], [62], [-12], [-52], [-16], [-90], [38], [-66], [-70], [-52], [62], [48], [-30], [14], [-78], [-76], [12], [-16], [-48], [86], [-90], [86], [-16], [56], [-54], [-48], [-40], [30], [-42], [2], [80], [-12], [42], [44], [-88], [-78], [-64], [86], [-6], [-36], [-36], [-42], [-12], [74], [-48], [96], [-6], [-82], [-16], [-30], [56], [38], [18], [-40], [12], [-90], [56], [32], [-72], [78], [-94], [54], [-28], [60], [30], [68], [90], [20], [60], [-28], [8], [-40], [96], [-10], [54], [30], [2], [-18], [74], [-40], [18], [-76], [72], [14], [-18], [14], [6], [-84], [-52], [-94], [96], [20], [-84], [44], [78], [-12], [-52], [2], [42], [96], [-34], [-66], [-36], [48], [20], [-72], [42], [74], [-30], [-60], [-48], [-52], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2790_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2790_2_a_k();".
// 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_k(: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_k();".
// 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_k( : 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![0, 1]>,<13,R![4, 1]>,<17,R![6, 1]>,<19,R![-8, 1]>],Snew);
    return Vf;
end function;