
// Make newform 2310.2.a.m in Magma, downloaded from the LMFDB on 24 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_m();".
// 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_m();".
// 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], [2], [2], [4], [-4], [-6], [0], [2], [6], [12], [8], [-6], [8], [-14], [12], [8], [10], [4], [16], [10], [6], [-6], [8], [-12], [2], [-6], [-8], [12], [2], [-4], [-6], [-12], [2], [4], [-8], [-6], [-20], [26], [-24], [2], [6], [0], [12], [24], [-8], [10], [26], [12], [-18], [0], [-18], [24], [-6], [-8], [-18], [-22], [24], [-22], [8], [-24], [-18], [-22], [-4], [2], [-20], [2], [30], [-4], [-8], [6], [20], [32], [-10], [-14], [2], [-10], [-40], [22], [36], [-2], [0], [-36], [18], [10], [18], [36], [-20], [-24], [4], [12], [-36], [16], [10], [-30], [-16], [-46], [28], [-18], [-16], [2], [20], [-2], [12], [26], [24], [-26], [8], [-2], [-6], [0], [0], [2], [44], [-24], [-22], [-44], [2], [50], [18], [-28], [32], [-30], [-10], [-16], [16], [10], [36], [16], [40], [10], [22], [-10], [-38], [-48], [18], [26], [-44], [-14], [-4], [-36], [10], [0], [-6], [-38], [40], [12], [22], [42], [20], [24], [-4], [24], [-36], [-6], [50], [-14], [-28], [42], [48], [-40], [10], [-8], [24], [-14], [42], [42], [20], [-46], [-36], [-38], [16], [-30], [20], [30], [16], [2], [-28], [-12], [-38], [58], [-36], [-14], [-50], [-44], [10], [40], [-42], [36], [-60], [58], [60], [30], [-6], [-14], [-46], [56], [-6], [56], [10], [-10], [16], [-30], [8], [36], [18], [-32], [34], [-6], [-28], [36], [40], [10], [56], [-50], [-12], [-6], [-6], [-16], [-46], [36], [-24], [-62], [14], [56], [-48], [12], [10], [-12], [-44], [18], [-60], [-32], [-46], [46], [44], [-32], [-12], [-28], [56], [-34], [66], [24], [-56], [-64], [-52], [-16], [70], [46], [12], [42], [-30], [36], [6], [40], [26], [-22], [-40], [-4], [74], [-54], [22], [48], [-58], [-54], [-4], [-58], [-14], [-76], [46], [-76], [-38], [8], [-44], [74], [-38], [-60], [8], [24], [-16], [10], [24], [0], [-38], [-18], [-80], [54], [14], [-28], [-6], [-20], [-58], [66], [-72], [-6], [-12], [16], [-22], [38], [44], [36], [20], [-62], [-4], [26], [0], [-26], [48], [-18], [74], [-52], [-64], [-34], [-48], [52], [6], [18], [-4], [-38], [-62], [20], [-22], [-30], [-76], [-76], [-56], [46], [-46], [2], [16], [8], [-20], [-52], [62], [34], [2], [-64], [-14], [-30], [-54], [-32], [-30], [-20], [-38], [-4], [88], [-54], [-20], [-62], [6], [56], [-14], [50], [-8], [-12], [66], [-28], [14], [-18], [-52], [60], [-92], [-94], [-70], [24], [34], [-28], [96], [-72], [-54], [48], [18], [72], [20], [-38], [38], [-38], [10], [-70], [-48], [-30], [76], [56], [8], [10], [20], [0], [-46], [-38], [100], [-44], [8], [-102], [32], [-30], [68], [26], [34], [34], [80], [6], [42], [-8], [82], [-30], [28], [-72], [58], [54], [-44], [84], [-22], [74], [60], [8], [102], [8], [102], [82], [-84], [-4], [78], [18], [-36], [-22], [24], [-16]];
    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_m();".
// 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_m(: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_m();".
// 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_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2310_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<13,R![-2, 1]>,<17,R![-2, 1]>,<19,R![-4, 1]>,<23,R![4, 1]>,<29,R![6, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;