
// Make newform 9576.2.a.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9576_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_9576_2_a_d();".
// 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_9576_2_a_d();".
// 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_9576_a();"
function MakeCharacter_9576_a()
    N := 9576;
    order := 1;
    char_gens := [7183, 4789, 5321, 4105, 1009];
    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_9576_a_Hecke(Kf)
    return MakeCharacter_9576_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], [0], [-2], [-1], [2], [-4], [4], [-1], [0], [2], [10], [4], [-6], [-12], [-8], [6], [0], [-2], [4], [0], [14], [-2], [6], [-14], [14], [18], [10], [-16], [4], [-14], [14], [-14], [0], [20], [-14], [-2], [-14], [4], [0], [-18], [-20], [-8], [4], [-2], [-2], [20], [-24], [-26], [-24], [-26], [0], [-16], [-10], [-2], [30], [0], [-18], [28], [-2], [-10], [20], [22], [-28], [0], [-22], [30], [0], [-2], [10], [-14], [-24], [-36], [12], [20], [-8], [-16], [-30], [-14], [2], [-18], [6], [-16], [40], [2], [22], [-30], [-26], [-38], [-26], [36], [14], [-24], [-38], [-34], [4], [36], [-30], [10], [-8], [18], [-16], [-14], [12], [-42], [-4], [30], [2], [36], [24], [-2], [22], [-38], [4], [-4], [44], [-30], [-4], [-12], [18], [-24], [32], [-10], [22], [-4], [20], [-26], [-10], [32], [-8], [34], [12], [-16], [-18], [2], [20], [-30], [-6], [4], [-54], [-44], [12], [22], [4], [12], [-32], [48], [-2], [18], [-28], [-32], [4], [-48], [36], [-32], [36], [-8], [-4], [-32], [22], [-6], [30], [38], [-40], [-40], [-46], [-16], [14], [26], [34], [-42], [60], [4], [4], [-30], [-10], [-12], [-28], [50], [-14], [-50], [-4], [12], [-4], [62], [-60], [-66], [28], [36], [50], [28], [-6], [62], [40], [2], [26], [-6], [-2], [-6], [44], [-32], [38], [-10], [-28], [-14], [-10], [-70], [64], [-4], [-36], [8], [-2], [-50], [60], [0], [56], [-62], [24], [2], [24], [-54], [28], [14], [26], [-4], [-12], [62], [62], [72], [-10], [18], [-46], [-40], [50], [-30], [28], [12], [14], [-38], [-66], [-16], [-72], [-28], [32], [0], [30], [20], [12], [-68], [-28], [-24], [42], [60], [36], [42], [-6], [18], [38], [28], [58], [-70], [-14], [4], [-62], [-28], [8], [-36], [6], [20], [-76], [-30], [-52], [32], [10], [0], [10], [36], [-38], [-8], [34], [50], [-24], [-16], [68], [28], [-28], [-32], [46], [-6], [16], [38], [-46], [22], [6], [68], [-40], [-50], [-54], [-54], [60], [20], [-10], [18], [88], [24], [-52], [18], [-16], [-48], [-36], [-34], [80], [-10], [-30], [76], [-28], [34], [42], [-8], [-30], [36], [24], [-62], [-30], [-54], [16], [14], [-88], [-28], [-48], [34], [74], [22], [68], [14], [-92], [22], [12], [-46], [42], [-64], [-84], [56], [46], [-10], [66], [-24], [10], [-12], [-80], [66], [56], [18], [-10], [66], [36], [66], [-36], [-42], [72], [-56], [50], [-48], [8], [28], [-60], [-38], [42], [86], [-58], [34], [56], [-48], [-58], [-84], [-50], [24], [8], [-34], [48], [70], [-98], [-60], [-76], [-40], [-68], [48], [8], [-50], [-4], [56], [78], [-54], [-18], [-68], [48], [34], [34], [-34], [84], [90], [52], [-28], [-6], [-42], [22], [0], [22], [-90], [-48], [42], [46], [66], [12], [-28], [58], [18], [-8], [62], [52], [0], [-74], [-16], [-84], [16], [58], [78], [24], [-56], [-28], [-52], [-18], [-94], [-28], [0], [-26], [32], [10], [8], [32], [-80], [34], [-108], [72], [0], [18], [-74], [-4], [-88], [82], [-104], [-96], [0], [6], [-16], [110], [54], [32], [88], [-18], [6], [84], [-34], [-32], [84], [44], [-46], [-50], [-70], [48], [12], [82], [56], [64], [-50], [-72], [20], [-94], [56], [72], [-6], [70], [22], [-74], [-70], [44], [74], [-22], [-12], [-16], [6], [-88], [72], [-78], [-34], [30], [18], [52], [54], [40], [-16], [-2], [16], [18], [-12], [12], [40], [-72], [62], [-80], [64], [-42], [40], [-46], [62], [20], [30], [42], [70], [72], [-26], [86], [-108], [38], [60], [-10], [50], [-26], [-38], [12], [-14], [-104], [66]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9576_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9576_2_a_d();".
// 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_9576_2_a_d(:prec:=1)
    chi := MakeCharacter_9576_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(3833) - 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_9576_2_a_d();".
// 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_9576_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9576_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<11,R![-2, 1]>,<13,R![4, 1]>,<17,R![-4, 1]>],Snew);
    return Vf;
end function;