
// Make newform 2325.2.a.a 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_a();".
// 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_a();".
// 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_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 := [[-2], [-1], [0], [4], [-1], [6], [-1], [-5], [-5], [-9], [1], [-10], [-6], [-6], [10], [-9], [-14], [10], [-13], [6], [14], [-12], [-5], [-11], [3], [14], [-7], [4], [-2], [6], [-4], [4], [-9], [-6], [-6], [0], [11], [11], [16], [-6], [-4], [-6], [-20], [21], [-22], [10], [-13], [-2], [4], [-6], [-24], [-5], [-4], [5], [0], [12], [18], [8], [-16], [30], [-5], [14], [28], [-30], [-8], [-6], [-34], [34], [12], [13], [19], [26], [-8], [-1], [-20], [9], [-26], [-7], [-23], [12], [20], [18], [-16], [-18], [0], [4], [37], [-14], [3], [-34], [-2], [-10], [2], [-3], [-20], [-12], [29], [30], [40], [25], [-1], [37], [0], [-27], [28], [-2], [3], [42], [-14], [-28], [-19], [10], [-2], [-4], [14], [31], [16], [-7], [16], [30], [-17], [-44], [-2], [2], [-45], [6], [-42], [16], [-36], [-17], [16], [-9], [-35], [28], [18], [37], [9], [4], [-18], [3], [-52], [15], [16], [33], [-4], [-24], [22], [-12], [-32], [-40], [6], [14], [-44], [-30], [-57], [-41], [29], [-5], [9], [-51], [9], [-54], [-24], [-18], [-24], [7], [-56], [-57], [-26], [-6], [-36], [-28], [-28], [31], [16], [54], [-60], [16], [1], [-50], [13], [-23], [-19], [57], [-50], [-41], [7], [48], [-20], [50], [2], [-8], [-18], [6], [54], [15], [24], [-14], [48], [48], [40], [28], [-42], [23], [-18], [-18], [3], [27], [16], [47], [68], [-6], [41], [-18], [-39], [11], [33], [12], [-19], [-22], [-8], [-1], [32], [-25], [54], [-32], [-36], [-33], [20], [-18], [60], [4], [-53], [18], [54], [-45], [47], [-44], [-21], [-5], [-6], [8], [-44], [4], [-7], [36], [70], [-5], [-8], [-6], [53], [76], [-27], [54], [54], [48], [61], [8], [12], [2], [-20], [13], [-28], [41], [57], [-12], [60], [-52], [77], [34], [-75], [-30], [-6], [-22], [-18], [29], [-38], [4], [8], [-65], [-56], [29], [-6], [74], [-11], [-12], [-6], [63], [-19], [-1], [2], [-70], [-1], [-80], [-28], [-3], [70], [-48], [27], [-80], [27], [-45], [26], [12], [50], [-10], [39], [-63], [-15], [-7], [69], [46], [78], [-14], [-76], [-42], [51], [84], [11], [30], [-14], [56], [86], [21], [-36], [-12], [32], [-31], [-64], [-69], [-47], [18], [19], [-81], [5], [20], [86], [-86], [67], [-20], [82], [6], [78], [32], [-38], [-36], [69], [74], [87], [20], [-76], [42], [42], [-34], [39], [-94], [-33], [32], [-87], [-72], [-23], [-69], [40], [-72], [-68], [-34], [-72], [20], [-2], [-40], [20], [12], [0], [-52], [40], [-45], [24], [34], [6], [0], [-9], [-82], [-8], [-93], [-28], [96], [27], [-16], [-82], [8], [-82], [62], [-2], [49], [-66], [-86], [-40], [74], [16], [-65], [-52], [6], [57], [8], [11], [8], [58], [-74], [28], [29], [50], [8], [-12], [-92], [-26], [0], [11], [37], [-36], [20], [-35], [-64], [-8], [32], [-34], [-46], [36], [22], [16], [68]];
    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_a();".
// 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_a(:prec:=1)
    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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2325_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, 1]>,<7,R![-4, 1]>,<11,R![1, 1]>,<13,R![-6, 1]>],Snew);
    return Vf;
end function;