
// Make newform 2325.2.a.v in Magma, downloaded from the LMFDB on 08 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2325_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_2325_2_a_v();".
// 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_2325_2_a_v();".
// 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 := [4, 3, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-1, -3, 0, 1], [-3, -1, 1, 0], [2, -4, -1, 1]];
    Rf_basisdens := [1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_2325_a();"
function MakeCharacter_2325_a()
    N := 2325;
    order := 1;
    char_gens := [776, 652, 1801];
    v := [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_2325_a_Hecke( : Kf := Kf)
    return MakeCharacter_2325_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], [1, 0, 0, 0], [0, 0, 0, 0], [-1, 1, 0, -1], [2, 1, 1, -1], [-1, 0, -1, -2], [3, 0, 1, 1], [4, 0, 0, 0], [-1, -2, 1, -1], [0, 0, 0, -1], [-1, 0, 0, 0], [-3, -1, -2, 1], [0, -1, 3, 1], [0, -1, 1, 3], [-1, 2, -3, -1], [1, 0, -1, 1], [2, 2, 2, 1], [0, 2, 0, -4], [-1, 0, -3, 0], [6, -1, 1, -2], [7, 1, -2, -1], [3, 1, -6, 0], [1, -2, -1, 1], [-4, -3, -1, 0], [4, 1, 1, -1], [-6, 1, -1, 1], [-5, 2, -1, 4], [-9, 2, -3, -3], [-1, 1, -4, 0], [2, 2, 6, 0], [-14, 0, -2, 4], [4, 0, -2, 1], [1, 0, -1, -1], [10, -6, 0, 2], [0, 3, 3, 1], [9, 1, -4, 0], [-2, -5, 5, 3], [9, 2, 5, -4], [-8, -2, -2, 0], [2, -4, 0, -2], [-6, -2, 0, 0], [0, -2, 4, -4], [2, -4, 0, -3], [2, 3, -3, -1], [7, 0, -7, 1], [1, 3, -6, -6], [8, 4, 4, -2], [6, -3, 1, 1], [1, -4, -3, 5], [2, 4, -4, -2], [-11, 6, -3, -3], [6, -5, 3, 7], [4, 2, 4, 2], [-10, 0, -6, 6], [3, -4, 5, 5], [6, -6, 0, -2], [-8, 0, -8, -1], [-6, -4, 2, 8], [3, 1, 6, -1], [8, 4, -2, -4], [1, -3, -6, 5], [11, -6, 3, 3], [9, 1, 2, 3], [4, -7, -7, 4], [9, 6, 3, 0], [17, 4, 7, 3], [-7, -5, -6, 4], [1, 3, -6, -3], [4, 0, 4, -6], [5, -5, 0, -4], [-5, 2, -9, -9], [8, -8, 2, 3], [16, 0, 8, 0], [2, 2, 2, 0], [8, 2, 2, 6], [5, 2, 3, -7], [-2, 1, 1, 6], [-12, 0, -6, 2], [-16, -3, -9, 6], [8, 4, 2, -6], [-16, 1, 1, -4], [-5, 9, 4, -2], [0, 3, -5, -6], [3, 9, 10, -5], [16, 4, 4, 4], [15, -4, 3, -1], [4, -3, 3, -2], [-3, -9, -2, 3], [-28, -2, 2, -1], [-6, 7, -9, -3], [13, 0, 9, -5], [-8, 1, -7, -6], [0, 2, 10, 0], [-12, -6, 10, 0], [8, 8, 4, 0], [3, 6, -3, -5], [-18, -5, -5, -2], [4, -5, 11, 1], [12, -2, -2, 4], [-1, -9, 2, 12], [-7, -4, -5, 2], [-15, 2, -7, 5], [-3, -8, 5, 9], [-6, 4, -10, -9], [-18, 4, -6, -4], [-18, -3, -5, -5], [-6, 10, 0, -6], [12, -6, 0, 2], [6, -3, 7, 6], [0, -2, -4, 2], [3, 0, 9, 2], [1, 0, -11, -2], [-18, -2, -6, -6], [-17, -1, -12, -10], [4, 0, 12, 4], [6, 4, -6, 3], [-4, -2, 6, -4], [2, -8, -6, 6], [-15, 2, 1, 9], [6, -3, 3, 6], [6, -4, 8, 10], [-7, -10, -1, -2], [1, 4, -5, -11], [11, 8, -5, -5], [-4, 4, 8, 2], [-26, 1, -1, 7], [10, -4, 0, 4], [-18, 3, -1, 9], [7, -6, 7, 6], [-10, -6, 10, 10], [-11, -1, -2, 2], [-26, -6, 0, -2], [-2, 6, -12, -10], [-3, -2, 7, -6], [-20, -8, -12, -7], [11, -11, 8, 16], [7, -8, 1, 9], [0, -9, 13, -3], [1, 0, 11, 7], [-6, 5, -15, -4], [2, 4, 6, 4], [-26, -1, 3, 4], [-22, 1, -11, 3], [-11, 12, -3, -1], [26, 0, 8, -2], [12, -6, 4, 5], [4, 3, -1, 3], [-6, -2, 10, 4], [8, 0, 4, 0], [-12, -2, -2, -2], [10, -11, 3, 7], [-18, 3, 3, 8], [0, -3, -9, -1], [-15, 16, 1, -5], [11, 3, 10, -1], [42, 3, 3, 1], [2, -8, 2, -10], [12, 4, -4, 7], [-4, -1, 3, 9], [-12, -5, -19, 2], [7, -8, 3, 13], [15, 2, 11, -9], [30, -3, 1, 9], [6, 9, 7, -10], [-10, -12, 12, 0], [-2, 10, 12, -4], [2, -8, 6, 0], [-6, 6, 14, 0], [14, 2, -2, -8], [15, -4, 5, 5], [4, -15, -1, 15], [16, 10, 12, 2], [38, 1, 7, 0], [6, -3, 7, 5], [-32, 8, -4, 6], [-14, 6, 4, -7], [-12, -10, 6, 8], [16, 2, -4, -6], [-9, 3, -6, 3], [40, -2, 0, 6], [1, -7, -2, 7], [0, 12, 8, -12], [-8, 7, 3, -7], [-19, 10, 5, -1], [37, 0, 1, 5], [-22, -2, 12, 3], [-6, 3, -11, 1], [3, 2, -9, -2], [12, -4, 10, 8], [18, 0, 8, -5], [-13, 5, -10, -5], [-7, -4, 17, 3], [9, -13, -2, -6], [12, -10, -6, 3], [-9, 10, -15, -5], [-5, 8, -19, -9], [10, 0, -12, -6], [12, -6, -16, 4], [-10, 4, 4, 4], [15, -2, 5, 5], [14, -6, 6, 14], [-2, -8, 2, 12], [18, 3, -3, 5], [2, 4, -12, 0], [-4, 14, 4, 1], [-5, 8, 1, -5], [-14, -4, 18, 4], [5, 10, -9, 9], [10, -3, 3, -1], [-4, 4, -20, -4], [11, 2, 9, 14], [-8, 5, 15, -4], [-1, 5, 4, 7], [-3, -14, -9, 5], [34, 9, 21, -1], [-3, -13, 12, 12], [1, 3, -12, -3], [8, -9, 3, 13], [5, -14, 11, 9], [-12, -2, -8, -2], [24, 0, 10, 10], [-2, -10, 4, 2], [-28, -3, 5, 5], [-7, -12, 3, -12], [7, 6, -11, -17], [-32, 0, 2, 2], [3, -4, 13, -11], [-12, -5, 13, 11], [32, 0, 4, -2], [16, 3, 11, 8], [-3, 8, -19, -12], [-40, -8, -16, -6], [6, 8, 6, -10], [-20, -3, -1, 0], [34, 3, 11, -1], [4, -10, -2, 2], [6, 0, 8, 2], [-16, 8, -2, -6], [-2, 5, -11, 1], [24, -7, 7, 3], [7, 0, 7, -1], [28, 10, -2, -6], [-16, 9, -13, 5], [8, -6, -4, 14], [-13, 16, -3, -13], [-18, -8, -20, -5], [10, 1, 9, -3], [-18, 7, -5, -9], [10, -4, -2, 8], [27, -6, 1, 3], [-22, 18, 2, 0], [60, 1, 5, 3], [16, 0, -4, 10], [-33, 1, -12, -14], [-4, -6, 8, 16], [24, -6, -4, -7], [-6, -2, 14, 2], [36, -13, 9, 15], [41, 4, 7, 3], [14, -2, 14, 16], [-35, -10, -11, 6], [7, 2, 1, -17], [-12, 4, -22, 6], [0, -1, 15, 15], [-27, -6, -11, 7], [2, 6, -8, 4], [18, 6, -2, 14], [-14, 1, 7, -5], [35, -7, 12, 9], [22, -2, 22, 2], [-1, 7, 2, -20], [20, -15, 3, 9], [1, -9, -6, 11], [28, -14, 10, 18], [18, 8, 8, 14], [-11, 13, -14, 1], [41, -10, 7, 1], [-20, 0, -10, -10], [-20, 2, 8, 8], [-4, 0, 4, -14], [29, -4, -7, -5], [-10, -18, -4, 6], [11, -12, -9, 5], [2, -10, -6, -4], [3, 8, -15, 4], [-10, -10, 0, 4], [-51, -3, 12, -1], [-11, 2, -3, -3], [12, 2, -10, 8], [-18, 10, -8, -17], [4, 9, 21, 1], [5, 6, 11, -3], [11, -6, 3, 9], [-4, -3, 9, 20], [-25, -14, -7, -6], [-2, -5, -9, -8], [51, -1, 4, -2], [-9, -2, 3, 15], [-2, 16, -10, -12], [12, 6, 22, -8], [24, -5, 7, 15], [-7, 6, -3, 9], [-44, 4, -8, 4], [13, 12, 1, -11], [-27, 5, -8, -16], [-22, 6, -6, -4], [-41, 12, -1, -3], [10, 10, 14, 4], [-10, 12, 6, 4], [-54, -10, -6, -6], [-19, 2, 3, -3], [0, 13, -5, -10], [-20, -8, 14, 8], [-8, -14, 6, 0], [-11, -8, -31, 5], [-26, 0, -4, 4], [18, -2, 8, 0], [22, -23, 5, 17], [4, 4, 14, -8], [-30, 9, -11, 4], [27, 7, -12, -16], [31, 4, -13, 0], [-32, 1, -19, -21], [-7, 12, -5, 12], [-19, 0, -21, 11], [0, 12, -2, 0], [-6, 14, -12, -18], [25, 12, 15, -4], [-6, 4, 14, 2], [-7, -4, -1, 19], [70, 2, 6, 6], [25, -12, -1, 11], [0, 2, 6, -6], [5, -6, 15, 5], [8, 2, 2, -10], [-3, 8, 21, -3], [44, -2, 12, 6], [-15, -6, -11, 9], [-7, -5, -24, 4], [-12, 1, -9, 1], [1, 10, 11, -12], [-17, -8, 9, -1], [-50, 4, 2, -3], [-12, 8, -28, -4], [-40, 16, -2, -10], [2, 8, 4, 19], [33, 11, -12, 0], [-12, -14, -22, 20], [12, -2, -6, 8], [-13, -8, -3, 21], [17, -5, 6, -2], [3, 12, -9, -8], [-26, -2, -10, -12], [-22, -17, -17, 13], [22, -4, -8, -6], [33, -10, 5, -5], [-42, -8, 2, -8], [14, 1, -7, 13], [-17, -2, -13, -3], [-5, 22, 13, -17], [-4, -13, -9, -5], [-58, -7, -3, 2], [17, 10, -1, 9], [54, 2, 2, 7], [-17, 15, -14, -21], [1, 11, -18, -5], [25, 2, 1, -27], [-28, 19, 13, -9], [-15, 13, -2, -14], [24, -4, -22, 7], [44, 0, 4, 6], [17, -10, 7, 19], [-4, -5, -17, -7], [-10, -4, -22, -6], [-47, 4, -15, -10], [-48, 1, 7, -7], [20, -4, 14, 21], [26, 11, 9, -15], [10, -5, -23, 11], [-5, 7, 8, -9], [-4, -11, -1, 14], [33, 6, 9, -23], [48, 6, 10, -6], [-1, 20, -11, -13], [48, -4, 16, 0], [35, -6, 9, -17], [30, -14, 12, 8], [-5, 0, -9, 0], [-26, -4, -22, -4], [45, 4, 9, 15], [4, -18, 4, -8], [-3, 10, 5, -15], [12, 7, -5, 4], [-11, -13, 8, 23], [-6, 5, 23, 0], [40, -8, -6, 6], [6, -8, -14, 0], [22, -1, 13, 9], [-27, -11, -12, 18], [-22, -16, 18, 13], [52, 0, 10, 8], [-41, -10, -9, 19], [-1, 26, 3, 0], [-13, 4, -3, -7], [16, -14, 2, -9], [-6, 22, 0, -10], [-32, 23, -1, -5], [-6, -5, 7, -4], [-20, -17, -7, 5], [16, -17, -11, 13], [13, 0, -23, 0], [38, 2, -10, 12], [24, -6, -22, 6], [5, 3, 2, 10], [0, 10, -24, -8], [-36, -4, -2, 8], [-18, 5, -19, -1], [-7, -19, -22, 19], [-13, 8, 1, 21], [21, 12, 17, -1], [32, 1, 19, -16], [9, -7, 8, 7], [15, 4, -13, -7], [30, -18, 6, 15], [-50, 7, -7, 1], [-39, -8, -1, 21], [17, 16, -23, -23], [-6, -6, 12, 3], [-24, 4, 0, -12], [30, 12, 10, 14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2325_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2325_2_a_v();".
// 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_2325_2_a_v(:prec:=4)
    chi := MakeCharacter_2325_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_2325_2_a_v();".
// 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_2325_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2325_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -12, -6, 2, 1]>,<7,R![-32, -46, -10, 4, 1]>,<11,R![32, 56, -12, -6, 1]>,<13,R![4, -94, -50, 2, 1]>],Snew);
    return Vf;
end function;