
// Make newform 8280.2.a.q in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8280_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_8280_2_a_q();".
// 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_8280_2_a_q();".
// 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_8280_a();"
function MakeCharacter_8280_a()
    N := 8280;
    order := 1;
    char_gens := [2071, 4141, 4601, 1657, 3961];
    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_8280_a_Hecke(Kf)
    return MakeCharacter_8280_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], [0], [1], [0], [-4], [4], [-4], [4], [-1], [-6], [4], [-2], [0], [-4], [0], [-6], [-6], [-4], [4], [-4], [-2], [10], [18], [-2], [2], [10], [0], [10], [-16], [-16], [-22], [-6], [-12], [-12], [-10], [12], [-22], [16], [-8], [-22], [6], [16], [8], [-10], [10], [14], [-20], [10], [-10], [-12], [26], [-24], [2], [-16], [22], [-4], [-22], [-12], [28], [22], [-4], [-14], [-20], [-12], [-10], [-10], [20], [14], [-4], [-10], [-18], [-24], [-12], [26], [32], [-8], [26], [-24], [6], [-2], [12], [-8], [-32], [18], [24], [-40], [12], [-34], [-10], [-6], [6], [16], [-14], [-14], [12], [16], [-30], [18], [4], [-26], [-16], [30], [6], [-10], [-8], [-38], [-24], [22], [4], [-34], [-2], [-30], [-20], [48], [38], [-30], [-28], [-8], [10], [48], [-28], [-42], [6], [-16], [20], [2], [48], [-16], [-20], [34], [4], [24], [10], [-34], [-8], [-34], [18], [20], [-18], [4], [4], [-30], [-6], [-14], [14], [-24], [16], [10], [52], [24], [-8], [14], [28], [0], [-12], [56], [-6], [20], [-26], [-6], [12], [-8], [-42], [-44], [-8], [-40], [40], [8], [10], [6], [60], [-14], [8], [-26], [-16], [42], [-4], [54], [4], [44], [-14], [-40], [44], [-30], [-24], [18], [12], [-12], [26], [48], [6], [-44], [-20], [6], [26], [4], [-14], [38], [-52], [48], [-6], [40], [-8], [30], [-20], [30], [6], [-24], [-44], [44], [-66], [22], [36], [-14], [12], [14], [42], [24], [36], [42], [38], [22], [-4], [-36], [-44], [30], [0], [-12], [-56], [58], [24], [-16], [38], [24], [-44], [-36], [18], [46], [-26], [0], [-2], [44], [18], [10], [42], [-24], [-30], [24], [36], [-8], [6], [6], [20], [62], [18], [46], [40], [-4], [62], [-50], [48], [30], [-14], [-70], [22], [-60], [-6], [70], [44], [-6], [-46], [-44], [-62], [50], [-74], [62], [16], [18], [-22], [72], [48], [10], [-12], [16], [-76], [-60], [14], [-42], [-32], [-36], [-46], [50], [-34], [-24], [-36], [-70], [54], [78], [82], [4], [34], [-2], [2], [-60], [28], [18], [44], [84], [40], [60], [-8], [74], [10], [52], [48], [-34], [-54], [-24], [62], [8], [8], [-14], [-54], [-74], [60], [-22], [76], [-28], [52], [-10], [74], [-2], [-44], [32], [16], [72], [92], [-4], [-42], [-8], [44], [-36], [-70], [-6], [-54], [74], [-74], [-68], [16], [10], [28], [26], [58], [-12], [52], [54], [-16], [44], [42], [-24], [6], [36], [-8], [32], [-80], [58], [66], [44], [86], [-18], [-44], [-32], [-10], [-30], [-64], [-42], [16], [-14], [-30], [58], [30], [12], [-70], [10], [0], [24], [-56], [4], [-4], [-52], [-90], [-42], [66], [-28], [-64], [-54], [-50], [30], [-96], [-18], [94], [26], [-28], [24], [-46], [-88], [62], [-72], [20], [-30], [10], [102], [-62], [16], [-38], [90], [-12], [14], [-84], [-96], [34], [74], [40], [38], [-106], [18], [26], [-48], [76], [76], [-62], [-60], [4], [-44], [-80], [50], [86], [-70], [-20], [54], [-20], [-62], [-66], [-40], [-82], [-30], [60], [8], [-14], [52], [68], [-56], [28], [-60], [-82], [-98], [30], [74], [58], [-100], [36], [-78], [-96], [54], [-48], [14], [-98], [-10], [-54], [76], [-106], [-72], [40], [66], [-54], [38], [46], [-22], [0], [-6], [-30], [-58], [-30]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8280_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8280_2_a_q();".
// 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_8280_2_a_q(:prec:=1)
    chi := MakeCharacter_8280_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(3457) - 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_8280_2_a_q();".
// 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_8280_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8280_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![4, 1]>,<13,R![-4, 1]>,<17,R![4, 1]>],Snew);
    return Vf;
end function;