
// Make newform 3120.2.a.r in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3120_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_3120_2_a_r();".
// 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_3120_2_a_r();".
// 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_3120_a();"
function MakeCharacter_3120_a()
    N := 3120;
    order := 1;
    char_gens := [1951, 2341, 2081, 2497, 2641];
    v := [1, 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_3120_a_Hecke(Kf)
    return MakeCharacter_3120_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], [0], [1], [-6], [-4], [0], [-2], [0], [-2], [-2], [4], [-4], [-10], [-8], [-2], [4], [12], [-6], [0], [16], [-10], [2], [-10], [8], [-12], [6], [-14], [-8], [-4], [-18], [-4], [2], [-8], [14], [-4], [20], [6], [-12], [6], [0], [-22], [-6], [-8], [-4], [16], [-8], [14], [-6], [-12], [10], [4], [-14], [0], [14], [8], [22], [-2], [12], [18], [-12], [-32], [26], [-14], [12], [-14], [12], [-18], [6], [-4], [-32], [6], [4], [20], [-2], [22], [-18], [34], [-36], [-2], [12], [18], [16], [20], [14], [10], [2], [40], [-20], [-4], [-24], [20], [4], [-16], [2], [18], [20], [-10], [12], [10], [20], [-30], [20], [-38], [40], [46], [-16], [-22], [16], [30], [-18], [28], [-40], [-30], [20], [24], [6], [-44], [-10], [-30], [22], [-24], [4], [6], [30], [-8], [-16], [-34], [36], [36], [8], [22], [-18], [18], [18], [4], [14], [18], [52], [-30], [-24], [8], [-34], [-12], [14], [10], [52], [36], [22], [-38], [20], [-24], [-4], [-16], [8], [6], [58], [42], [-24], [10], [-8], [-4], [-18], [-52], [-56], [-26], [-6], [14], [-24], [-50], [-48], [-38], [40], [26], [12], [18], [-8], [-50], [24], [-12], [6], [-10], [44], [-50], [62], [28], [-46], [4], [18], [8], [-12], [-38], [4], [58], [-14], [-2], [-50], [-8], [-30], [-56], [-10], [18], [-8], [30], [-64], [-20], [22], [-60], [-14], [62], [-16], [-56], [36], [10], [-24], [66], [68], [42], [-42], [48], [-18], [-32], [-52], [-10], [66], [-24], [-32], [-40], [-50], [-44], [8], [-6], [-28], [44], [10], [18], [-36], [8], [-32], [-12], [-64], [-2], [-66], [-48], [-72], [-16], [4], [8], [62], [54], [-52], [-38], [54], [-24], [-10], [-12], [38], [18], [-56], [-44], [-66], [-66], [-18], [44], [18], [-66], [-36], [-18], [-18], [-12], [-6], [72], [-78], [-64], [48], [-34], [26], [60], [48], [0], [24], [-26], [12], [32], [50], [-14], [-72], [-70], [-10], [-28], [54], [56], [-66], [-74], [40], [-42], [-4], [-12], [26], [82], [-8], [52], [-4], [-62], [-4], [-2], [84], [-58], [-72], [-54], [10], [4], [68], [74], [0], [12], [2], [50], [4], [-62], [-26], [-16], [6], [-30], [-4], [-20], [32], [14], [-42], [-58], [64], [-48], [20], [72], [-50], [-42], [-46], [-32], [-34], [-38], [-30], [64], [18], [20], [38], [68], [20], [6], [-92], [-38], [-22], [24], [-42], [42], [36], [88], [50], [-4], [46], [42], [-24], [-24], [-84], [-6], [-94], [-72], [90], [12], [-52], [60], [-66], [-8], [-98], [-88], [48], [-38], [50], [10], [42], [-66], [32], [-2], [76], [28], [56], [22], [-28], [56], [58], [-30], [-80], [-60], [44], [-70], [-24], [-70], [100], [10], [14], [10], [-40], [6], [-54], [8], [-18], [-18], [4], [72], [34], [-10], [-100], [36], [26], [38], [84], [48], [-2], [16], [-66], [62], [60], [52], [50], [90], [-84], [54], [-60], [-104]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3120_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3120_2_a_r();".
// 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_3120_2_a_r(:prec:=1)
    chi := MakeCharacter_3120_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_3120_2_a_r();".
// 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_3120_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3120_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![0, 1]>,<17,R![6, 1]>,<19,R![4, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;