
// Make newform 9025.2.a.j in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9025_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_9025_2_a_j();".
// 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_9025_2_a_j();".
// 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_9025_a();"
function MakeCharacter_9025_a()
    N := 9025;
    order := 1;
    char_gens := [5777, 3251];
    v := [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_9025_a_Hecke(Kf)
    return MakeCharacter_9025_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+1)]];
    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], [4], [-1], [-2], [2], [0], [-6], [-9], [7], [2], [-2], [-2], [-6], [4], [-9], [-7], [-10], [-1], [-10], [-1], [6], [11], [-6], [15], [16], [-10], [-15], [-12], [-6], [12], [-12], [20], [-1], [-9], [-4], [-4], [-12], [-24], [-15], [6], [-3], [16], [18], [13], [-5], [-2], [14], [17], [8], [-19], [-1], [13], [-6], [0], [-3], [3], [28], [-10], [-14], [4], [16], [0], [-30], [-2], [-20], [34], [12], [14], [8], [-20], [16], [-12], [29], [6], [-33], [8], [-3], [5], [9], [15], [-21], [-4], [13], [-4], [-31], [-34], [-25], [4], [-10], [15], [-38], [13], [28], [26], [-18], [-15], [16], [23], [22], [-14], [-24], [-15], [13], [-6], [24], [32], [12], [-17], [-14], [-24], [-18], [4], [1], [-21], [46], [6], [-10], [-20], [15], [-20], [0], [30], [3], [6], [25], [-21], [-8], [6], [-29], [50], [-41], [-26], [-18], [5], [34], [22], [24], [-43], [-9], [21], [30], [6], [-2], [-48], [16], [-18], [1], [20], [-58], [37], [8], [-16], [12], [-41], [16], [51], [56], [-35], [42], [-24], [-22], [-36], [-44], [36], [-44], [-26], [-45], [-18], [45], [22], [21], [14], [20], [-6], [11], [-6], [58], [34], [46], [-39], [-22], [48], [54], [21], [60], [4], [-14], [-57], [52], [-8], [11], [23], [-4], [-46], [-30], [-2], [48], [-52], [-38], [33], [28], [-14], [36], [42], [43], [52], [-15], [-44], [40], [25], [-10], [38], [-32], [-10], [48], [42], [-44], [16], [-17], [-56], [11], [20], [-40], [58], [-68], [-16], [-54], [-17], [-58], [17], [49], [-26], [0], [-48], [31], [2], [-21], [61], [24], [24], [-52], [-70], [-20], [52], [-12], [28], [1], [-6], [-28], [15], [48], [54], [-22], [-27], [19], [-44], [8], [-44], [44], [0], [-11], [26], [14], [-29], [59], [-33], [-64], [30], [-26], [-72], [36], [-57], [-26], [-76], [12], [46], [38], [-39], [36], [51], [-48], [62], [36], [-32], [26], [24], [-17], [5], [-35], [-64], [-42], [-47], [-14], [-65], [-64], [-40], [-60], [-80], [24], [42], [-67], [78], [-29], [-50], [-18], [-53], [-24], [-16], [-14], [-15], [53], [62], [-18], [75], [47], [-37], [-40], [2], [-32], [2], [7], [38], [6], [47], [-68], [-40], [64], [36], [-23], [-74], [21], [-56], [60], [54], [34], [-54], [-71], [68], [-24], [12], [49], [80], [8], [-85], [-27], [12], [-81], [54], [17], [-68], [73], [28], [-13], [-78], [48], [-25], [-34], [28], [-72], [15], [-42], [16], [-90], [78], [30], [-44], [71], [32], [5], [-36], [-27], [-8], [40], [11], [-21], [-34], [7], [50], [69], [-56], [-6], [4], [-5], [16], [-71], [18], [-2], [12], [50], [-54], [-87], [-62], [-53], [78], [-11], [39], [-64], [29], [70], [-14], [54], [16], [-38], [-16], [-68], [-7], [-58], [-12], [-34], [-36], [66], [5], [-72], [15], [-99], [0], [-14], [-4], [-39], [-80], [-8], [-1], [44], [-12], [-36], [15], [-83], [-75]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9025_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9025_2_a_j();".
// 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_9025_2_a_j(:prec:=1)
    chi := MakeCharacter_9025_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_9025_2_a_j();".
// 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_9025_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 1]>,<3,R![0, 1]>,<7,R![-4, 1]>,<11,R![1, 1]>,<29,R![9, 1]>],Snew);
    return Vf;
end function;