
// Make newform 5415.2.a.c 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_c();".
// 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_c();".
// 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], [-2], [4], [2], [0], [-4], [-4], [0], [0], [0], [-10], [12], [2], [-4], [2], [16], [0], [-2], [8], [-12], [0], [16], [14], [8], [12], [-10], [-14], [-4], [-14], [6], [8], [-18], [-24], [-18], [-6], [8], [-18], [12], [-10], [-18], [-4], [-2], [12], [12], [-24], [-12], [-14], [-18], [-6], [-26], [22], [-22], [-16], [-4], [12], [22], [-28], [26], [-26], [0], [-18], [-14], [-14], [28], [-16], [-28], [-18], [-2], [-2], [10], [-4], [20], [-8], [-30], [-2], [-12], [26], [-26], [34], [4], [36], [40], [16], [0], [-14], [6], [-22], [16], [38], [-8], [-6], [32], [-8], [24], [12], [28], [-42], [-16], [-38], [-12], [0], [-24], [42], [-32], [-18], [-36], [-30], [16], [-18], [-6], [16], [48], [24], [26], [-24], [-18], [-16], [-18], [-24], [50], [-12], [-16], [-34], [18], [42], [-14], [34], [4], [8], [0], [10], [6], [-46], [-14], [-28], [14], [2], [-4], [6], [-14], [-36], [38], [-24], [10], [-10], [-12], [-40], [-12], [-10], [-46], [24], [-52], [0], [32], [10], [-42], [60], [-52], [-46], [34], [24], [-18], [-16], [-8], [-2], [38], [58], [-24], [10], [46], [-30], [0], [-54], [0], [30], [40], [14], [-2], [0], [-28], [-38], [-28], [-10], [0], [-36], [-14], [58], [20], [-56], [-20], [12], [48], [-46], [14], [10], [2], [36], [68], [-56], [52], [38], [10], [-18], [-16], [-36], [-42], [12], [18], [42], [-22], [52], [-20], [34], [58], [60], [48], [14], [18], [64], [-24], [-50], [28], [-10], [66], [-12], [28], [-62], [-38], [4], [32], [0], [-74], [12], [30], [42], [18], [-4], [20], [-36], [70], [-34], [22], [18], [38], [36], [68], [24], [-14], [66], [-48], [10], [14], [38], [74], [-28], [-38], [-34], [0], [-68], [22], [-44], [18], [4], [44], [-22], [-4], [6], [26], [-80], [14], [60], [-16], [-62], [68], [-58], [2], [-14], [16], [-40], [-12], [-38], [6], [-18], [42], [-38], [-32], [0], [-18], [-4], [54], [-56], [44], [26], [56], [-66], [-36], [-42], [-10], [-30], [-16], [76], [-80], [-16], [20], [-38], [-66], [18], [-20], [18], [52], [0], [-24], [38], [-30], [-32], [18], [70], [68], [-90], [-28], [12], [42], [70], [-92], [52], [32], [-42], [14], [-22], [-44], [-52], [72], [-4], [10], [-66], [-66], [14], [-68], [6], [40], [56], [66], [24], [42], [-8], [-84], [42], [-60], [12], [58], [24], [54], [-82], [-6], [70], [74], [-16], [62], [38], [0], [-72], [-38], [76], [-6], [-24], [-22], [82], [-28], [16], [80], [-16], [-2], [-4], [-94], [74], [-30], [40], [-16], [-6], [-62], [-58], [-12], [24], [28], [-50], [50], [24], [-78], [-38], [30], [90], [-44], [-68], [24], [12], [4], [-30], [-70], [-30], [4], [-78], [-44], [24], [-50], [-88], [24], [46], [0], [-18], [-68], [-40], [-82], [6], [60], [76], [78], [24], [8], [36], [-100], [-36], [52], [102], [-60], [-30], [16], [18]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : 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![2, 1]>,<13,R![-4, 1]>],Snew);
    return Vf;
end function;