
// Make newform 8280.2.a.m in Magma, downloaded from the LMFDB on 23 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_m();".
// 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_m();".
// 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], [-2], [0], [-2], [-2], [-4], [-1], [0], [8], [4], [-4], [6], [0], [-2], [-6], [-6], [10], [6], [-14], [4], [4], [14], [16], [-8], [-10], [20], [2], [18], [16], [18], [6], [4], [6], [-16], [-8], [8], [0], [-2], [-6], [18], [4], [6], [-2], [4], [4], [4], [-4], [-26], [6], [30], [-14], [-24], [-6], [8], [-4], [8], [18], [6], [14], [10], [8], [30], [-16], [34], [-28], [8], [20], [30], [6], [-16], [-18], [0], [-4], [-32], [18], [2], [34], [-10], [12], [-18], [24], [-28], [32], [28], [0], [-16], [-12], [16], [12], [4], [16], [-30], [28], [-24], [16], [22], [-26], [2], [8], [-46], [12], [-22], [-24], [38], [-8], [10], [30], [-30], [32], [8], [34], [-32], [-4], [42], [-26], [-24], [50], [12], [-14], [2], [-26], [-24], [20], [-18], [-2], [6], [22], [-4], [-44], [-16], [8], [32], [-20], [-10], [42], [38], [18], [-32], [36], [20], [12], [-4], [2], [12], [-22], [22], [12], [28], [-10], [-26], [16], [36], [14], [28], [16], [0], [-24], [54], [24], [-54], [-4], [-8], [30], [0], [8], [-2], [-30], [-42], [-20], [58], [-32], [-20], [-32], [-6], [-12], [8], [-2], [-26], [40], [0], [-22], [2], [-8], [34], [-18], [-6], [-54], [-14], [-50], [8], [36], [-4], [-52], [30], [-26], [-48], [-58], [-24], [42], [32], [42], [26], [20], [-18], [4], [24], [-12], [52], [-38], [-28], [6], [-28], [30], [22], [56], [-36], [48], [-6], [-26], [-60], [-48], [-34], [60], [10], [42], [-18], [-22], [-22], [2], [-20], [-20], [-32], [-26], [-32], [-50], [-38], [-38], [10], [-36], [4], [-68], [-46], [54], [2], [-8], [0], [28], [-24], [48], [-6], [-48], [-26], [-54], [58], [18], [-10], [74], [14], [-10], [-36], [-6], [64], [-6], [-72], [-22], [-14], [34], [2], [78], [-6], [60], [-60], [2], [-16], [-80], [2], [38], [16], [0], [-8], [-24], [46], [36], [6], [-44], [-22], [-72], [-84], [6], [-76], [-50], [36], [46], [-66], [-16], [38], [42], [80], [-64], [22], [36], [20], [28], [14], [40], [46], [56], [50], [-52], [-30], [30], [24], [-32], [-38], [-2], [2], [-44], [-16], [-44], [32], [-36], [76], [-6], [-62], [-24], [0], [48], [-6], [2], [66], [40], [60], [-64], [-72], [-82], [90], [38], [-22], [26], [54], [-40], [-40], [50], [-22], [38], [4], [0], [-6], [4], [66], [-32], [2], [34], [-10], [68], [-80], [-38], [-40], [68], [4], [52], [76], [84], [46], [42], [-10], [-58], [54], [-20], [-84], [30], [-20], [46], [30], [28], [76], [-98], [46], [-70], [74], [12], [-2], [20], [-36], [-96], [-46], [-34], [40], [70], [50], [70], [44], [0], [-4], [-68], [86], [40], [92], [62], [34], [-22], [10], [0], [24], [16], [84], [-26], [14], [50], [54], [84], [-28], [-56], [12], [-46], [-72], [2], [-64], [26], [16], [-12], [54], [14], [-2], [-28], [-72], [28], [-90], [-82], [-28], [-4], [-48], [2], [10], [-66], [74], [-72], [-16], [-36], [-94], [26], [48], [-46], [-42], [-92], [4], [14], [78], [-92], [84], [-68], [-24], [-8], [-96], [54], [-46], [28], [14], [20], [88], [84], [-22], [68], [-34], [40], [-76], [2], [-12], [-48], [28], [66], [78], [86], [-8], [52], [-92], [48], [34], [-50], [-54], [-84]];
    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_m();".
// 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_m(: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_m();".
// 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_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8280_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![0, 1]>,<13,R![2, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;