
// Make newform 3920.2.a.s in Magma, downloaded from the LMFDB on 17 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_3920_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_3920_2_a_s();".
// 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_3920_2_a_s();".
// 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_3920_a();"
function MakeCharacter_3920_a()
    N := 3920;
    order := 1;
    char_gens := [1471, 981, 3137, 3041];
    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_3920_a_Hecke( : Kf := Kf)
    return MakeCharacter_3920_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], [4], [-4], [-2], [-8], [6], [6], [8], [4], [6], [-4], [2], [-8], [0], [6], [0], [-16], [6], [14], [-6], [4], [0], [14], [18], [12], [12], [10], [12], [-10], [16], [2], [-16], [12], [-14], [-20], [10], [-8], [-14], [22], [8], [4], [-4], [-24], [26], [-6], [0], [-2], [-12], [30], [12], [-14], [24], [-10], [10], [8], [26], [-8], [32], [-26], [-18], [12], [-14], [16], [-30], [-2], [24], [20], [22], [20], [-36], [6], [2], [18], [-10], [36], [6], [40], [-2], [-8], [-24], [18], [10], [18], [-12], [8], [16], [20], [-36], [28], [-12], [2], [-10], [-8], [-2], [-8], [14], [16], [-22], [-4], [-2], [-48], [-18], [24], [-10], [-12], [-42], [-6], [-4], [-16], [18], [-48], [12], [-34], [12], [42], [18], [-22], [-24], [44], [-34], [38], [-32], [44], [-30], [-36], [-36], [24], [-10], [6], [-34], [42], [8], [18], [42], [-28], [22], [-4], [-48], [2], [-40], [-38], [6], [-4], [-12], [46], [30], [16], [-20], [-48], [-40], [-8], [14], [38], [-46], [8], [-6], [36], [20], [2], [36], [40], [42], [-46], [-38], [-44], [2], [16], [26], [-16], [6], [-4], [38], [-44], [-46], [28], [12], [-10], [-58], [4], [10], [62], [-16], [26], [-8], [-2], [24], [52], [34], [8], [-26], [-30], [-18], [-2], [-28], [-18], [-8], [58], [30], [12], [-62], [64], [-16], [42], [20], [34], [10], [28], [0], [-8], [6], [-4], [46], [36], [46], [38], [-56], [34], [36], [-24], [22], [22], [64], [-4], [-20], [-2], [36], [-8], [58], [72], [36], [30], [6], [20], [48], [-16], [52], [4], [-50], [-50], [56], [52], [-4], [20], [52], [-18], [62], [-12], [70], [-14], [-36], [-10], [48], [-6], [6], [20], [-8], [-6], [34], [-50], [-44], [46], [6], [-24], [-26], [-14], [56], [-42], [-32], [-50], [20], [-64], [-34], [-70], [-36], [12], [48], [44], [-70], [16], [-40], [2], [6], [-24], [78], [-50], [-24], [-54], [36], [-50], [66], [56], [-54], [-20], [-72], [54], [-18], [-32], [64], [12], [82], [48], [50], [-24], [38], [76], [-42], [-30], [32], [52], [-10], [4], [-24], [-34], [34], [44], [26], [-30], [68], [10], [-66], [44], [-40], [76], [-42], [14], [-2], [-80], [-32], [-4], [48], [62], [30], [38], [-12], [-10], [-6], [26], [64], [-2], [44], [26], [-64], [-24], [-6], [-20], [42], [-50], [-4], [86], [-10], [48], [52], [34], [-20], [-10], [-10], [-20], [36], [24], [-38], [66], [-20], [-22], [-12], [-4], [-44], [-42], [-48], [-82], [-60], [40], [94], [-66], [-26], [-62], [-86], [84], [-30], [84], [52], [-88], [42], [72], [52], [82], [90], [84], [-56], [-80], [26], [-16], [-42], [-68], [54], [-30], [18], [-52], [-90], [90], [96], [-2], [-78], [16], [36], [14], [54], [-8], [20], [-38], [-78], [-64], [76], [30], [84], [30], [-22], [92], [28], [54], [18], [0], [-54], [-28], [56]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3920_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3920_2_a_s();".
// 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_3920_2_a_s(:prec:=1)
    chi := MakeCharacter_3920_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_3920_2_a_s();".
// 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_3920_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3920_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<11,R![4, 1]>,<13,R![-2, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;