
// Make newform 3648.2.a.b in Magma, downloaded from the LMFDB on 30 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3648_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_3648_2_a_b();".
// 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_3648_2_a_b();".
// 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_3648_a();"
function MakeCharacter_3648_a()
    N := 3648;
    order := 1;
    char_gens := [2623, 2053, 1217, 1921];
    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_3648_a_Hecke( : Kf := Kf)
    return MakeCharacter_3648_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], [-1], [-3], [3], [-3], [0], [1], [-1], [0], [8], [-2], [4], [-12], [1], [9], [-6], [6], [1], [10], [-10], [-11], [6], [0], [-6], [-8], [-14], [10], [-18], [10], [-4], [8], [-13], [1], [9], [-5], [18], [-2], [4], [-4], [-18], [-24], [-22], [-13], [-14], [-18], [-5], [-22], [-16], [4], [-21], [15], [-13], [-24], [-13], [28], [-3], [22], [0], [23], [6], [13], [-16], [6], [-7], [-26], [-18], [18], [-12], [23], [-1], [18], [19], [-4], [-16], [-6], [-14], [-23], [-17], [10], [14], [4], [38], [0], [-4], [-20], [15], [-20], [39], [3], [-27], [15], [32], [34], [20], [31], [24], [24], [26], [-14], [5], [-8], [3], [24], [-12], [-32], [-33], [-13], [-42], [42], [-2], [4], [-9], [17], [-28], [-37], [16], [25], [-45], [-21], [10], [40], [34], [-12], [20], [-41], [42], [-46], [-1], [37], [-50], [45], [-26], [28], [-23], [-47], [51], [10], [-44], [-2], [39], [-28], [-13], [11], [-2], [4], [-42], [46], [18], [-7], [-8], [-34], [-3], [29], [30], [-46], [18], [16], [30], [-43], [6], [-52], [26], [-32], [-60], [-14], [52], [52], [-13], [-18], [39], [-32], [14], [60], [-26], [40], [-25], [-28], [-45], [0], [-27], [13], [-48], [10], [-26], [-17], [39], [66], [10], [4], [-52], [-24], [45], [-38], [24], [-52], [12], [-19], [14], [6], [-35], [44], [-20], [6], [40], [12], [47], [13], [-20], [45], [62], [46], [-26], [-49], [8], [8], [4], [40], [60], [-42], [-43], [50], [-28], [-20], [-15], [38], [-23], [-62], [-46], [2], [57], [45], [6], [62], [-30], [-53], [-41], [1], [1], [-52], [22], [42], [-29], [16], [14], [-48], [45], [-35], [54], [-56], [-43], [34], [-49], [68], [-6], [41], [52], [-13], [-24], [32], [-54], [-40], [22], [-59], [-42], [-21], [30], [-54], [71], [-14], [-39], [6], [-64], [-7], [-49], [-50], [-8], [-23], [-66], [-18], [-75], [22], [-32], [61], [58], [-39], [-21], [19], [60], [35], [12], [2], [-3], [4], [10], [-10], [22], [-8], [-83], [70], [-65], [-71], [-46], [-5], [6], [27], [-54], [36], [-52], [-25], [22], [-69], [30], [28], [52], [-24], [10], [-9], [34], [-65], [21], [-4], [13], [42], [-82], [26], [38], [44], [40], [76], [39], [-7], [-66], [-16], [-11], [-75], [-13], [10], [-24], [23], [68], [46], [-3], [-52], [82], [14], [24], [70], [68], [84], [3], [64], [-44], [33], [-42], [66], [40], [-8], [-96], [97], [0], [-89], [6], [72], [-42], [7], [-52], [-46], [-62], [70], [-20], [-16], [1], [12], [15], [79], [32], [0], [11], [-49], [-68], [-90], [-1], [-43], [15], [-52], [60], [28], [10], [-1], [-84], [10], [-46], [52], [-85], [-6], [54], [50], [60], [60], [-10], [-54], [11], [76], [-78], [-54], [77], [27], [36], [0], [15], [22], [17], [62], [-13], [34], [-30], [72], [-38], [10], [-6], [-42], [-16], [-47], [-92], [36], [40], [-8], [-69], [-23], [-59]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3648_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3648_2_a_b();".
// 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_3648_2_a_b(:prec:=1)
    chi := MakeCharacter_3648_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_3648_2_a_b();".
// 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_3648_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3648_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3, 1]>,<7,R![-3, 1]>,<11,R![3, 1]>,<23,R![0, 1]>,<31,R![2, 1]>],Snew);
    return Vf;
end function;