
// Make newform 7360.2.a.t in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7360_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_7360_2_a_t();".
// 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_7360_2_a_t();".
// 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_7360_a();"
function MakeCharacter_7360_a()
    N := 7360;
    order := 1;
    char_gens := [1151, 5061, 4417, 6721];
    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_7360_a_Hecke(Kf)
    return MakeCharacter_7360_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 := [[0], [1], [-1], [0], [-2], [5], [-4], [2], [-1], [3], [7], [2], [-9], [4], [-9], [6], [0], [-2], [2], [-1], [1], [-14], [0], [16], [-4], [-10], [0], [-10], [4], [-16], [-7], [-15], [-12], [-9], [-14], [15], [4], [23], [16], [-14], [-9], [2], [14], [5], [-25], [-10], [-28], [-8], [4], [0], [-25], [-9], [-10], [-8], [19], [14], [7], [0], [-13], [-32], [22], [-16], [-16], [-9], [8], [-2], [19], [8], [28], [-17], [-27], [30], [-24], [-14], [16], [16], [-8], [-3], [-30], [-35], [12], [8], [28], [-6], [-15], [25], [6], [32], [7], [-24], [6], [16], [-23], [-1], [-3], [22], [15], [36], [-28], [35], [7], [24], [-36], [2], [-4], [-11], [9], [-2], [-32], [5], [-20], [-12], [-26], [48], [-40], [-24], [34], [7], [-19], [0], [10], [-9], [30], [-5], [-20], [16], [12], [32], [46], [-52], [23], [24], [4], [-26], [13], [-34], [-24], [4], [-18], [10], [29], [6], [3], [8], [46], [-12], [-22], [-41], [-43], [57], [50], [-28], [-52], [-33], [30], [50], [60], [23], [-26], [48], [-15], [-26], [-3], [-22], [-26], [50], [-32], [-14], [46], [-3], [-18], [-49], [8], [-8], [-31], [-36], [31], [33], [58], [64], [-47], [-8], [19], [-54], [-54], [12], [-30], [-18], [5], [-48], [39], [20], [-10], [30], [-26], [40], [-50], [40], [-16], [9], [60], [-8], [23], [-18], [-28], [-33], [-42], [-3], [55], [43], [-9], [5], [-36], [-22], [48], [44], [-21], [-51], [18], [57], [13], [-26], [53], [-72], [53], [45], [42], [-3], [70], [20], [-66], [-8], [-10], [18], [-28], [-12], [48], [-58], [41], [-39], [8], [25], [-21], [29], [6], [39], [39], [-30], [66], [-2], [36], [30], [26], [44], [-18], [-1], [26], [-62], [-5], [13], [36], [-24], [-10], [-56], [-50], [-30], [42], [-32], [-38], [15], [55], [-58], [28], [32], [13], [-1], [20], [-18], [-70], [0], [-75], [46], [-42], [-70], [-32], [48], [26], [54], [40], [78], [-74], [76], [2], [-6], [39], [46], [-60], [69], [-85], [11], [28], [-46], [44], [-9], [44], [51], [52], [36], [70], [30], [-53], [-26], [64], [23], [6], [32], [63], [39], [-16], [-1], [-44], [28], [-39], [40], [-66], [80], [-70], [79], [46], [-56], [-17], [17], [25], [-17], [74], [-33], [70], [-40], [63], [40], [1], [12], [10], [-30], [-54], [-44], [-31], [-52], [34], [38], [-79], [17], [-70], [-18], [80], [42], [-76], [-2], [57], [-36], [66], [-51], [1], [70], [47], [-11], [33], [-70], [-22], [75], [-52], [-29], [-20], [-72], [37], [75], [82], [-74], [36], [-29], [-42], [-78], [-52], [-53], [24], [21], [49], [-22], [-14], [-64], [-36], [-54], [-87], [52], [44], [-66], [40], [-24], [84], [42], [-31], [-79], [68], [-60], [-83], [11], [-74], [-27], [64], [-36], [91], [21], [-70], [86], [-50], [27], [-96], [68], [-12], [0], [-28], [4], [16], [76], [-31], [-63], [-74], [-90], [-67], [-101]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7360_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7360_2_a_t();".
// 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_7360_2_a_t(:prec:=1)
    chi := MakeCharacter_7360_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_7360_2_a_t();".
// 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_7360_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7360_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<7,R![0, 1]>,<11,R![2, 1]>,<13,R![-5, 1]>,<17,R![4, 1]>],Snew);
    return Vf;
end function;