
// Make newform 8624.2.a.ca in Magma, downloaded from the LMFDB on 21 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_8624_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_8624_2_a_ca();".
// 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_8624_2_a_ca();".
// 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
        poly := [-7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_8624_a();"
function MakeCharacter_8624_a()
    N := 8624;
    order := 1;
    char_gens := [5391, 6469, 7745, 3137];
    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_8624_a_Hecke( : Kf := Kf)
    return MakeCharacter_8624_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], [0, 1], [1, 1], [0, 0], [-1, 0], [5, 0], [6, 0], [3, -1], [1, 1], [1, -2], [4, 0], [1, -1], [3, -3], [4, 0], [-8, -2], [-1, -1], [-2, 1], [9, -2], [4, -3], [-7, -1], [3, 1], [0, -1], [-8, -2], [-4, -4], [-11, 2], [3, 0], [-5, -3], [3, -3], [0, 4], [-13, 2], [-8, 3], [-13, -1], [3, 6], [4, 0], [-10, 2], [6, -1], [0, -8], [2, 1], [-2, -5], [5, 2], [12, -3], [-10, 0], [8, 2], [7, -7], [3, 6], [9, 5], [7, 3], [15, -1], [8, 2], [-16, 0], [9, 3], [4, -5], [-1, 9], [-2, -2], [-11, 4], [-12, 3], [16, 4], [10, 3], [-17, 2], [-15, 3], [5, 1], [12, 0], [-9, 5], [-9, -3], [13, -4], [12, 0], [18, -1], [17, 3], [-3, -9], [14, 6], [-14, -8], [18, -3], [23, -5], [5, -6], [-14, -3], [5, 11], [3, -9], [10, 8], [11, -4], [11, 3], [-6, -6], [-12, -8], [-16, 5], [-16, 0], [4, 3], [-8, 10], [-6, 6], [-18, -8], [-11, -8], [-4, 10], [-16, 2], [18, 3], [20, 4], [-16, 2], [-2, -6], [12, 3], [10, 4], [-18, 6], [3, -7], [-3, -2], [-9, -7], [0, 12], [-21, 3], [-20, -8], [37, 3], [1, -10], [0, 3], [15, 3], [-12, 12], [-14, -4], [-16, -2], [2, 0], [-11, -2], [-24, -4], [-11, -9], [-3, -6], [12, -7], [-15, -9], [5, -1], [-4, -4], [18, 4], [-7, -3], [-18, 6], [6, 3], [32, -5], [9, -6], [17, -9], [-12, 6], [12, 2], [15, 10], [-8, 6], [40, -2], [16, 0], [-2, 8], [-6, 0], [6, -8], [-8, 4], [-23, 9], [-14, -8], [40, 4], [4, -12], [-13, 2], [28, 6], [1, 13], [9, 13], [-12, -6], [18, -8], [36, 0], [-14, -9], [31, 7], [-7, 6], [-9, 6], [-14, 3], [-16, 11], [-16, 10], [-4, 2], [30, 2], [25, 4], [-7, -15], [3, -18], [18, 6], [-7, -7], [-16, 16], [18, 3], [10, 10], [32, 2], [-17, -15], [-32, -4], [-52, 0], [6, -12], [16, 10], [17, 3], [-48, -3], [34, 8], [-45, -1], [7, -2], [11, 1], [2, -16], [18, 14], [-10, 0], [3, 5], [32, -10], [24, -8], [-12, 0], [21, 15], [-5, -11], [-51, -2], [-50, 3], [-3, -11], [-26, -5], [14, 6], [-27, -9], [6, 11], [-15, -15], [-24, 0], [15, 0], [24, 4], [-53, 5], [-29, 1], [-10, 5], [13, 10], [26, 7], [35, 3], [-37, 9], [-16, 11], [4, 16], [16, 0], [-13, -13], [15, 15], [14, 7], [-23, 11], [-18, -12], [-32, 12], [-6, 18], [24, -3], [-3, 4], [-24, -7], [-15, 9], [-28, -4], [-30, 12], [-21, -8], [42, -7], [3, 12], [-5, -21], [0, 6], [9, 16], [-45, 0], [27, -15], [4, 15], [-24, 6], [-12, 4], [24, 2], [40, -9], [-10, -4], [-12, -13], [54, 3], [30, -14], [-23, 10], [46, -5], [31, -11], [10, -2], [-4, 16], [36, 8], [17, 15], [11, 17], [0, 6], [37, -15], [-10, 11], [29, 13], [-28, 5], [33, 4], [-28, -16], [-17, 7], [0, -8], [-13, -4], [16, -2], [6, -8], [-32, 0], [19, -11], [-45, -5], [-4, 4], [12, 24], [60, -8], [19, 8], [-7, 2], [56, 1], [-20, 16], [21, 18], [-3, 5], [16, -20], [22, 8], [-8, 12], [-3, -8], [-14, 18], [15, 7], [11, 1], [-6, -15], [29, 6], [-19, -21], [5, 17], [-18, 15], [12, -22], [18, 3], [63, -2], [-28, 10], [-24, 6], [33, -2], [22, 16], [-9, -1], [-48, 0], [-50, -8], [-28, -19], [39, -9], [0, 6], [27, -8], [37, 4], [15, -13], [12, 0], [-28, -22], [13, -27], [21, 13], [57, 0], [10, -15], [-20, 13], [48, -7], [21, -2], [-12, 3], [8, -24], [-20, -26], [-40, 12], [-6, -15], [9, 3], [-48, 12], [54, -10], [30, -9], [-17, -1], [46, -17], [-42, 6], [-68, -4], [45, -15], [-11, 3], [-49, -6], [5, 2], [-15, -1], [-31, -7], [-1, 0], [-38, 6], [-34, 10], [-6, -27], [-3, -24], [18, 10], [-27, 3], [-32, -9], [-13, -13], [-45, 11], [-36, 3], [-11, 17], [-80, 4], [-21, -14], [30, 2], [21, -17], [42, 18], [21, -12], [-11, -15], [13, -5], [-3, 9], [-13, 3], [72, -7], [36, 12], [-45, 3], [4, 0], [44, -12], [41, -13], [44, -5], [14, -6], [-33, 21], [8, -4], [-32, -14], [-15, 3], [-48, 0], [27, -20], [-53, 13], [60, -12], [-36, 12], [8, 1], [17, 18], [6, -6], [-18, 14], [18, 4], [26, -19], [-22, -2], [14, 5], [-12, 0], [-10, -23], [32, 24], [24, 27], [78, -6], [-14, -4], [2, -16], [24, 4], [31, 19], [14, -16], [-46, -11], [-66, -12], [-12, -4], [-32, -14], [1, -3], [23, 9], [20, 4], [-12, -30], [-39, 4], [1, -23], [-10, 17], [-42, -1], [37, -5], [12, 4], [54, 5], [-27, 12], [-26, 15], [-23, -8], [79, -4], [57, -12], [-68, -9], [33, -12], [-31, -16], [42, -10], [-21, 13], [0, 0], [88, -6], [-42, -3], [-41, -1], [-26, -2], [26, -7], [-16, -2], [65, -3], [34, -8], [-26, -5], [60, 8], [57, 12], [-70, 11], [61, -5], [7, 32], [44, 2], [-55, -1], [69, 10], [4, 4], [-50, -17], [-18, -12], [4, 15], [6, -21]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8624_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8624_2_a_ca();".
// 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_8624_2_a_ca(:prec:=2)
    chi := MakeCharacter_8624_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_8624_2_a_ca();".
// 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_8624_2_a_ca( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8624_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-7, 0, 1]>,<5,R![-6, -2, 1]>,<13,R![-5, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;