
// Make newform 2880.2.a.c in Magma, downloaded from the LMFDB on 04 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_2880_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_2880_2_a_c();".
// 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_2880_2_a_c();".
// 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_2880_a();"
function MakeCharacter_2880_a()
    N := 2880;
    order := 1;
    char_gens := [2431, 901, 641, 577];
    v := [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_2880_a_Hecke( : Kf := Kf)
    return MakeCharacter_2880_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], [-4], [4], [-6], [-2], [-4], [0], [10], [-4], [10], [-2], [4], [-8], [2], [12], [10], [-12], [0], [10], [-4], [4], [6], [-14], [2], [4], [-12], [2], [6], [12], [20], [22], [-4], [10], [4], [2], [12], [8], [18], [-12], [-6], [-16], [2], [2], [-20], [4], [28], [-12], [10], [-2], [16], [18], [-28], [-26], [8], [-6], [-28], [-14], [22], [-20], [18], [20], [-8], [-6], [-22], [28], [-14], [12], [-22], [6], [16], [36], [26], [-12], [16], [26], [-14], [6], [-22], [12], [26], [-8], [2], [-20], [12], [38], [-6], [18], [-4], [-12], [24], [-20], [-4], [20], [16], [-6], [-18], [4], [-14], [-20], [2], [-12], [30], [-20], [-14], [20], [6], [-48], [10], [-4], [-38], [-18], [-4], [-12], [-42], [-36], [0], [-30], [-44], [2], [-14], [-22], [28], [12], [-6], [10], [8], [-4], [-6], [52], [-24], [-4], [2], [14], [18], [34], [28], [18], [-18], [52], [34], [12], [52], [-14], [0], [26], [6], [20], [32], [2], [-2], [28], [-8], [20], [32], [4], [54], [10], [-46], [-52], [-10], [-28], [36], [-18], [8], [-4], [42], [2], [26], [-36], [-22], [24], [26], [12], [-42], [-52], [-38], [20], [26], [-12], [28], [50], [30], [0], [58], [42], [-12], [10], [48], [-14], [52], [-28], [18], [12], [-34], [2], [-22], [6], [40], [18], [20], [2], [50], [-44], [10], [20], [-28], [22], [-12], [18], [58], [-44], [68], [16], [10], [52], [-10], [-24], [-30], [26], [-20], [-26], [60], [36], [18], [-66], [-56], [-60], [12], [-22], [12], [20], [38], [4], [24], [34], [-54], [36], [0], [20], [-52], [-20], [-54], [38], [64], [-44], [-12], [28], [0], [42], [-18], [-16], [42], [-54], [20], [26], [60], [-30], [-54], [44], [-4], [26], [-14], [14], [-12], [66], [-10], [68], [-6], [-22], [4], [10], [12], [2], [-44], [60], [74], [58], [20], [72], [-36], [32], [-78], [36], [-80], [18], [-6], [4], [-50], [-30], [60], [-18], [-36], [34], [-30], [-76], [34], [36], [-68], [-22], [-30], [-68], [-36], [52], [34], [12], [34], [-64], [-22], [-48], [-46], [70], [76], [-24], [-22], [68], [-56], [-62], [62], [20], [-6], [74], [-28], [54], [34], [-4], [20], [-48], [58], [-46], [34], [36], [-36], [28], [-84], [-22], [-26], [26], [-36], [10], [6], [-30], [44], [-86], [-20], [10], [28], [-72], [-14], [76], [58], [42], [-4], [-30], [-66], [-80], [44], [-82], [-24], [26], [-58], [0], [-60], [4], [-70], [42], [44], [90], [60], [-92], [0], [-78], [4], [-38], [60], [96], [66], [6], [-70], [66], [54], [76], [54], [-68], [-24], [-100], [-46], [-68], [40], [34], [-54], [-4], [12], [72], [10], [36], [62], [20], [-54], [-102], [78], [-68], [86], [-94], [-12], [26], [-26], [12], [-36], [34], [-6], [-36], [60], [74], [-38], [0], [-52], [62], [48], [2], [58], [48], [20], [-54], [-54], [52], [22], [68], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2880_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2880_2_a_c();".
// 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_2880_2_a_c(:prec:=1)
    chi := MakeCharacter_2880_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_2880_2_a_c();".
// 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_2880_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2880_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![-4, 1]>,<13,R![6, 1]>,<17,R![2, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;