
// Make newform 2310.2.a.p in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2310_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_2310_2_a_p();".
// 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_2310_2_a_p();".
// 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_2310_a();"
function MakeCharacter_2310_a()
    N := 2310;
    order := 1;
    char_gens := [1541, 1387, 661, 211];
    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_2310_a_Hecke(Kf)
    return MakeCharacter_2310_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 := [[1], [-1], [1], [1], [-1], [4], [-4], [8], [2], [-6], [-4], [4], [2], [4], [-4], [2], [0], [10], [14], [8], [-10], [4], [-4], [16], [-14], [-6], [-16], [12], [-4], [4], [0], [-12], [4], [12], [-10], [12], [6], [14], [18], [-18], [-4], [-8], [20], [-14], [10], [12], [6], [-24], [-4], [-12], [-14], [-6], [2], [24], [-14], [-20], [-14], [16], [10], [-4], [-14], [-14], [-14], [-30], [6], [-6], [-20], [-34], [-12], [-14], [-6], [14], [-24], [6], [-20], [-12], [-14], [-30], [-6], [14], [-28], [-2], [-14], [-14], [40], [-12], [-6], [10], [14], [16], [36], [24], [0], [-28], [-8], [-30], [-30], [-32], [-42], [16], [-32], [10], [24], [40], [18], [6], [-36], [-32], [12], [14], [32], [-6], [-4], [46], [8], [18], [12], [36], [-10], [36], [40], [-50], [-26], [-36], [-26], [2], [30], [-42], [48], [-20], [34], [16], [32], [-28], [-6], [-2], [26], [-30], [34], [48], [56], [-6], [4], [52], [16], [-54], [4], [0], [50], [34], [2], [40], [-2], [6], [-38], [-16], [-48], [-12], [-22], [10], [16], [30], [8], [-36], [32], [-12], [8], [-16], [-22], [-46], [-8], [-24], [30], [10], [-40], [-24], [-30], [-54], [28], [-14], [-8], [-60], [-52], [-36], [-46], [30], [6], [20], [26], [24], [-2], [-12], [32], [6], [-12], [-10], [50], [36], [48], [14], [-50], [-56], [-50], [30], [40], [2], [20], [-4], [44], [-62], [50], [-2], [-44], [28], [20], [30], [32], [42], [58], [62], [-52], [24], [50], [4], [-24], [4], [-18], [-48], [8], [12], [-32], [-20], [44], [60], [64], [-14], [-30], [6], [-36], [-6], [-60], [0], [40], [-74], [16], [-8], [-40], [68], [38], [36], [2], [50], [62], [-18], [-70], [-48], [14], [-58], [58], [-26], [32], [36], [10], [12], [38], [26], [18], [-12], [56], [6], [64], [-58], [-14], [-64], [-50], [-32], [0], [-16], [34], [-36], [26], [-16], [-66], [58], [74], [0], [-46], [-58], [-80], [38], [-50], [12], [26], [76], [66], [30], [-8], [-50], [84], [70], [38], [50], [-52], [-68], [24], [-62], [16], [28], [16], [-42], [-10], [30], [-44], [-30], [68], [34], [-36], [18], [-86], [-12], [16], [22], [-42], [-64], [6], [-14], [-40], [20], [-16], [70], [-48], [-18], [-8], [-24], [-26], [84], [-58], [-64], [6], [52], [-64], [-36], [26], [-48], [42], [-72], [-44], [74], [-36], [-6], [48], [50], [18], [4], [-24], [56], [-26], [-84], [62], [-18], [-14], [-34], [22], [-72], [20], [42], [74], [-56], [-42], [-44], [-6], [-72], [-46], [-88], [32], [-76], [-30], [-38], [6], [-14], [70], [68], [72], [82], [-20], [-76], [96], [-82], [76], [-56], [62], [38], [-36], [-32], [-16], [58], [24], [-40], [-92], [78], [78], [-8], [88], [42], [-18], [32], [-104], [88], [-4], [-8], [74], [54], [52], [2], [70], [-70], [42], [-56], [-50], [26], [-30], [92], [-82], [36], [-10], [-14], [-36], [-40], [-50], [-84]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2310_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2310_2_a_p();".
// 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_2310_2_a_p(:prec:=1)
    chi := MakeCharacter_2310_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_2310_2_a_p();".
// 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_2310_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2310_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<13,R![-4, 1]>,<17,R![4, 1]>,<19,R![-8, 1]>,<23,R![-2, 1]>,<29,R![6, 1]>,<31,R![4, 1]>],Snew);
    return Vf;
end function;