
// Make newform 5415.2.a.h in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5415_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_5415_2_a_h();".
// 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_5415_2_a_h();".
// 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_5415_a();"
function MakeCharacter_5415_a()
    N := 5415;
    order := 1;
    char_gens := [3611, 2167, 5056];
    v := [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_5415_a_Hecke(Kf)
    return MakeCharacter_5415_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], [-2], [-6], [0], [-6], [0], [-8], [-4], [0], [-4], [0], [-2], [-8], [-2], [-12], [2], [8], [-16], [14], [-8], [0], [0], [12], [-18], [8], [12], [6], [6], [-4], [-2], [-10], [-16], [6], [16], [-2], [-22], [-16], [-6], [20], [-10], [10], [-24], [6], [-20], [20], [-8], [28], [2], [22], [6], [-18], [-6], [22], [-12], [-12], [-20], [-2], [4], [-14], [-30], [8], [34], [10], [6], [-28], [12], [8], [-2], [-34], [10], [-14], [0], [-20], [-8], [-14], [6], [-4], [10], [-30], [-6], [-36], [16], [-24], [-12], [8], [-6], [30], [-14], [12], [18], [-8], [-26], [-24], [-36], [24], [4], [4], [-42], [8], [18], [-36], [8], [-24], [2], [-20], [-10], [20], [-22], [-16], [-2], [34], [8], [-40], [-40], [2], [20], [-18], [-48], [22], [-36], [-18], [36], [32], [-42], [-14], [-18], [26], [-22], [-20], [-24], [40], [-38], [30], [-14], [6], [28], [26], [2], [-20], [6], [-14], [-36], [6], [-8], [26], [-46], [44], [24], [24], [-42], [2], [-48], [52], [0], [-40], [18], [30], [-44], [48], [30], [-14], [8], [42], [-8], [16], [14], [-34], [-14], [-40], [-54], [2], [18], [-56], [-14], [-8], [-2], [-56], [-50], [-10], [-32], [16], [54], [0], [-2], [-12], [20], [18], [46], [-56], [-20], [60], [44], [20], [-10], [-2], [-30], [2], [8], [36], [-16], [-32], [-18], [-42], [30], [48], [-36], [-42], [-4], [10], [66], [50], [20], [36], [-38], [66], [36], [0], [-10], [-22], [-40], [40], [-18], [-44], [-10], [30], [60], [-4], [14], [26], [-36], [72], [32], [-26], [72], [30], [-6], [-18], [68], [-76], [12], [46], [-26], [-6], [-50], [-42], [28], [-36], [36], [34], [-30], [-52], [26], [6], [-30], [-54], [12], [70], [30], [16], [4], [-26], [40], [-14], [36], [12], [18], [-28], [-82], [26], [-72], [38], [36], [28], [18], [-16], [62], [74], [22], [16], [64], [-40], [-22], [14], [2], [74], [-18], [24], [-56], [30], [-72], [10], [56], [80], [2], [8], [86], [-44], [6], [22], [30], [16], [12], [-88], [28], [28], [-14], [50], [-78], [-16], [2], [12], [88], [-12], [30], [-66], [-48], [-6], [-2], [-60], [46], [20], [-36], [2], [14], [36], [-36], [32], [-26], [-34], [-58], [-68], [-16], [-8], [-24], [-54], [-6], [46], [-26], [-64], [-42], [0], [-40], [46], [72], [-22], [-48], [-28], [18], [-60], [16], [-78], [40], [94], [-38], [70], [-70], [66], [-48], [-42], [-18], [-32], [-72], [58], [40], [-14], [-88], [2], [-58], [-28], [44], [24], [80], [14], [-12], [-10], [-22], [-22], [36], [-24], [42], [-70], [-2], [36], [56], [28], [6], [34], [48], [-78], [-58], [90], [26], [-52], [-16], [88], [60], [68], [98], [42], [66], [52], [-82], [60], [-8], [-50], [0], [96], [-70], [36], [-42], [-76], [8], [62], [-18], [68], [12], [-2], [24], [0], [-88], [-40], [36], [96], [50], [-60], [-46], [-40], [-106]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5415_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5415_2_a_h();".
// 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_5415_2_a_h(:prec:=1)
    chi := MakeCharacter_5415_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_5415_2_a_h();".
// 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_5415_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5415_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<7,R![2, 1]>,<11,R![6, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;