
// Make newform 8280.2.a.r in Magma, downloaded from the LMFDB on 13 May 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_r();".
// 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_r();".
// 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], [0], [-2], [6], [0], [1], [-6], [0], [-6], [6], [0], [8], [6], [12], [10], [0], [0], [-6], [8], [0], [-10], [-6], [10], [-8], [8], [10], [-2], [8], [20], [-18], [-12], [6], [-8], [2], [4], [0], [10], [-4], [18], [8], [18], [18], [16], [-4], [16], [24], [2], [22], [-16], [10], [24], [-18], [24], [-6], [-8], [14], [-26], [-16], [-26], [-12], [24], [2], [-6], [-12], [2], [20], [22], [-2], [-32], [8], [10], [8], [0], [-34], [14], [-26], [10], [0], [-38], [24], [10], [0], [-12], [-18], [2], [10], [8], [32], [16], [32], [-12], [36], [-16], [-6], [-18], [-8], [22], [12], [6], [8], [14], [8], [-14], [12], [30], [-24], [42], [32], [-14], [6], [32], [-40], [6], [-8], [24], [34], [0], [18], [-14], [6], [12], [12], [-10], [10], [32], [32], [-14], [20], [-40], [16], [34], [-10], [-22], [46], [8], [6], [-26], [-28], [-22], [8], [8], [-10], [-24], [-10], [6], [-20], [16], [-10], [-2], [-36], [0], [-32], [8], [0], [-18], [26], [-18], [-36], [-2], [-40], [32], [38], [-16], [-16], [14], [-54], [10], [-24], [-42], [-56], [-22], [24], [-58], [28], [58], [8], [2], [-32], [48], [-34], [-42], [-24], [-34], [-34], [-8], [10], [24], [-46], [60], [-56], [2], [16], [-26], [2], [-46], [-2], [-56], [-42], [56], [-50], [-62], [-16], [58], [-8], [-60], [-58], [-52], [-30], [-6], [-56], [48], [64], [18], [-16], [30], [8], [34], [22], [-8], [46], [64], [28], [-26], [70], [40], [56], [36], [6], [-40], [16], [-26], [32], [8], [-70], [-2], [-36], [-48], [-40], [-20], [-40], [-2], [-18], [-72], [-8], [64], [-32], [0], [-6], [-66], [40], [-22], [18], [12], [-30], [-8], [-30], [74], [-32], [-16], [-50], [-6], [30], [8], [54], [-10], [-48], [-30], [-58], [8], [26], [-16], [18], [-16], [12], [-50], [42], [-72], [16], [24], [0], [-30], [-68], [80], [42], [-50], [-64], [46], [-18], [32], [-74], [-80], [6], [-18], [8], [-54], [-60], [-68], [2], [6], [-72], [4], [-40], [34], [84], [74], [40], [22], [-48], [54], [22], [44], [48], [10], [-36], [-40], [-14], [-34], [-40], [-46], [2], [-16], [-18], [10], [-8], [4], [-32], [14], [-10], [18], [-40], [-92], [-16], [68], [50], [-18], [-22], [16], [-10], [62], [-54], [-48], [38], [28], [38], [-4], [-72], [54], [20], [42], [42], [-24], [66], [-58], [32], [72], [-2], [-80], [82], [70], [8], [96], [-84], [58], [66], [16], [34], [-36], [-20], [56], [30], [8], [78], [4], [72], [18], [-10], [-70], [-74], [46], [-64], [-50], [16], [24], [8], [-66], [-24], [-88], [-14], [-14], [28], [-4], [48], [82], [16], [14], [-16], [-86], [-18], [-66], [24], [-66], [58], [8], [26], [-50], [56], [-36], [34], [58], [-104], [-80], [74], [42], [-104], [104], [-58], [48], [6], [26], [-32], [12], [-38], [-30], [-8], [-26], [52], [32], [-94], [-48], [-76], [56], [46], [-10], [-38], [-10], [-12], [56], [-84], [-50], [38], [-80], [2], [-2], [-4], [-48], [-46], [-70], [-12], [80], [-36], [54], [18], [-78], [-74], [20], [66], [102], [-20], [-80], [-32], [-74], [-52], [66], [-64], [0], [102], [112], [-40], [-84], [104], [114], [12], [10], [66], [-8], [-64], [-94], [42], [-10], [10]];
    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_r();".
// 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_r(: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_r();".
// 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_r( : 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![0, 1]>,<13,R![2, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;