
// Make newform 8280.2.a.p in Magma, downloaded from the LMFDB on 09 January 2026.
// 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_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_8280_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_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 := 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]];
    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], [2], [-2], [0], [1], [6], [0], [-6], [-2], [12], [0], [-10], [12], [-6], [12], [12], [2], [16], [-16], [-14], [6], [-10], [8], [0], [10], [14], [-4], [-20], [14], [12], [22], [16], [2], [20], [-16], [-6], [12], [2], [24], [18], [2], [16], [4], [-20], [0], [-14], [6], [-20], [10], [12], [-2], [-16], [-18], [-8], [2], [18], [20], [-10], [12], [-12], [22], [18], [28], [-10], [4], [14], [-18], [16], [-8], [10], [16], [24], [-10], [2], [10], [-38], [12], [26], [-40], [22], [-24], [-12], [22], [14], [22], [-28], [-16], [16], [-4], [-20], [-4], [32], [-26], [26], [-20], [14], [36], [22], [24], [34], [-32], [-22], [28], [-18], [28], [42], [28], [26], [30], [-32], [-16], [-30], [-20], [32], [-38], [-36], [-14], [-38], [6], [-20], [12], [-18], [-38], [36], [8], [34], [-28], [0], [32], [26], [-10], [-14], [46], [44], [38], [-10], [20], [30], [-4], [48], [-34], [-16], [42], [-42], [-20], [40], [10], [-54], [-20], [-8], [44], [48], [-16], [-42], [22], [14], [-12], [14], [-28], [20], [30], [-24], [8], [-46], [-38], [-6], [12], [-42], [-8], [-10], [0], [-6], [-20], [30], [-32], [-14], [-20], [-60], [-14], [6], [-56], [-10], [50], [-28], [26], [-4], [50], [4], [16], [-50], [0], [-58], [-22], [-30], [-18], [16], [14], [-64], [42], [34], [4], [42], [-32], [-4], [22], [4], [-6], [6], [-8], [-16], [-52], [26], [-28], [-18], [0], [-6], [6], [40], [-66], [-40], [20], [-26], [-42], [12], [8], [-12], [-30], [-16], [16], [-18], [4], [32], [-14], [46], [28], [4], [40], [-20], [-52], [54], [-34], [44], [44], [-68], [32], [72], [18], [-54], [-24], [-46], [-54], [-60], [-30], [44], [42], [-38], [16], [24], [-2], [-30], [-66], [-16], [14], [66], [-4], [26], [22], [-52], [-74], [40], [42], [-44], [-60], [-10], [-78], [-20], [-16], [-48], [56], [-30], [-20], [-4], [-34], [-18], [-8], [70], [14], [-72], [-26], [4], [66], [-66], [56], [42], [4], [-28], [22], [-10], [72], [44], [-40], [-70], [52], [-70], [-64], [82], [8], [-10], [66], [52], [8], [-38], [-60], [-36], [14], [-66], [-40], [-58], [54], [-12], [-42], [26], [-56], [-68], [48], [78], [22], [-38], [16], [44], [0], [36], [-14], [54], [-38], [-88], [-62], [-58], [22], [24], [-42], [12], [-58], [76], [-64], [54], [68], [-70], [86], [-64], [-14], [54], [-72], [4], [-50], [-64], [10], [-50], [24], [20], [52], [34], [42], [-64], [-22], [20], [28], [40], [-10], [-40], [-54], [-4], [0], [38], [-78], [2], [78], [-50], [76], [46], [56], [-32], [-16], [18], [-60], [80], [-70], [42], [-12], [68], [96], [-74], [32], [18], [-56], [102], [30], [46], [32], [30], [-90], [-32], [-30], [30], [-4], [44], [-46], [42], [72], [-64], [-34], [30], [-60], [28], [54], [-96], [46], [26], [-96], [60], [-54], [82], [0], [22], [-4], [-44], [-22], [-76], [-52], [-56], [2], [82], [-70], [-34], [36], [104], [-4], [-78], [-58], [24], [50], [-66], [-20], [56], [98], [74], [60], [-64], [-4], [30], [54], [38], [-34], [-36], [-86], [86], [68], [-16], [-108], [70], [-52], [18], [80], [16], [-30], [-16], [76], [60], [-12], [-30], [44], [2], [78], [96], [-104], [-70], [66], [-46], [110]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8280_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8280_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_8280_2_a_p(: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_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_8280_2_a_p( : 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![-2, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;