
// Make newform 2925.2.a.q in Magma, downloaded from the LMFDB on 08 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2925_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_2925_2_a_q();".
// 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_2925_2_a_q();".
// 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_2925_a();"
function MakeCharacter_2925_a()
    N := 2925;
    order := 1;
    char_gens := [326, 352, 2251];
    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_2925_a_Hecke( : Kf := Kf)
    return MakeCharacter_2925_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], [0], [0], [-3], [1], [1], [-1], [-2], [-3], [2], [-6], [-11], [5], [-4], [-10], [11], [-8], [13], [-12], [5], [-10], [-3], [-12], [15], [-17], [0], [16], [9], [-16], [14], [-10], [6], [18], [-1], [-13], [16], [10], [13], [-12], [-6], [-2], [-7], [8], [13], [0], [4], [-4], [-8], [22], [18], [-27], [13], [-2], [0], [-18], [-8], [4], [22], [18], [-30], [12], [24], [5], [-24], [10], [28], [0], [-4], [-19], [-8], [6], [-24], [-36], [-4], [-14], [-30], [0], [29], [-30], [2], [26], [4], [24], [-4], [-17], [9], [13], [11], [-15], [-27], [-23], [-9], [7], [-28], [-14], [28], [7], [14], [-16], [-30], [32], [-30], [21], [0], [-31], [19], [-18], [4], [-16], [-37], [32], [-13], [-42], [-34], [40], [12], [-15], [47], [22], [-20], [4], [6], [3], [-24], [22], [20], [4], [24], [-38], [49], [10], [-34], [-5], [8], [6], [-40], [36], [-52], [-47], [26], [36], [-27], [-20], [26], [30], [5], [-45], [29], [-29], [34], [-18], [-2], [16], [-21], [-6], [-44], [37], [-1], [-30], [-37], [-24], [-1], [16], [-12], [32], [-12], [-25], [36], [2], [31], [-36], [22], [48], [-45], [19], [36], [-18], [27], [-44], [-2], [-47], [-40], [-22], [58], [-66], [28], [-24], [-49], [8], [19], [20], [-22], [-27], [13], [60], [-29], [-50], [-54], [12], [-59], [-18], [33], [-9], [-59], [15], [-27], [62], [-4], [-57], [-13], [52], [-22], [-48], [14], [-60], [44], [32], [-42], [34], [6], [-29], [-10], [-18], [44], [-11], [53], [45], [-10], [2], [45], [-72], [31], [-16], [16], [34], [30], [-12], [-38], [18], [12], [-48], [15], [-40], [12], [60], [16], [16], [-35], [-40], [-40], [-22], [-2], [18], [-39], [37], [-33], [70], [49], [18], [-17], [26], [-53], [56], [60], [6], [-47], [-5], [-1], [19], [-10], [14], [-13], [-29], [-72], [-42], [19], [-36], [-48], [12], [18], [13], [28], [-32], [18], [48], [48], [-28], [-36], [-46], [24], [-40], [52], [-48], [-12], [-38], [42], [-40], [-39], [36], [73], [66], [-52], [-35], [75], [64], [13], [51], [47], [-48], [60], [21], [-42], [-30], [4], [28], [-59], [-57], [81], [42], [48], [44], [-26], [-62], [-79], [-18], [31], [-34], [-19], [-15], [50], [-18], [53], [-88], [40], [-64], [-2], [-58], [-44], [50], [-6], [83], [13], [3], [36], [86], [-84], [89], [-43], [72], [-71], [6], [-3], [5], [14], [-45], [-54], [-67], [-12], [6], [38], [50], [76], [36], [-20], [-38], [44], [-48], [9], [-19], [-42], [12], [56], [6], [15], [8], [19], [42], [-53], [84], [2], [50], [-18], [-64], [82], [36], [-76], [86], [40], [24], [-3], [-24], [62], [-59], [-68], [85], [10], [26], [-18], [-29], [-54], [-62], [-27], [95], [4], [65], [-89], [46], [-34], [-81], [-21], [-18], [45], [-31], [-21], [94], [34], [95], [74], [96], [55], [-94], [-31], [24], [52], [46], [68], [85], [105], [14], [-14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2925_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2925_2_a_q();".
// 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_2925_2_a_q(:prec:=1)
    chi := MakeCharacter_2925_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_2925_2_a_q();".
// 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_2925_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2925_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 1]>,<7,R![3, 1]>,<11,R![-1, 1]>],Snew);
    return Vf;
end function;