
// Make newform 9408.2.a.ca in Magma, downloaded from the LMFDB on 07 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9408_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_9408_2_a_ca();".
// 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_9408_2_a_ca();".
// 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_9408_a();"
function MakeCharacter_9408_a()
    N := 9408;
    order := 1;
    char_gens := [1471, 6469, 3137, 4609];
    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_9408_a_Hecke(Kf)
    return MakeCharacter_9408_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], [-2], [0], [2], [4], [6], [8], [6], [10], [-4], [-6], [-6], [4], [-8], [-2], [-4], [8], [-8], [10], [4], [-4], [12], [-14], [4], [10], [4], [-14], [-10], [-14], [4], [-12], [-10], [12], [-2], [8], [16], [24], [-16], [2], [6], [20], [-6], [-10], [6], [8], [-4], [8], [4], [-20], [14], [-18], [28], [-28], [10], [-26], [-2], [4], [-10], [6], [16], [18], [16], [-16], [24], [-18], [-4], [14], [-18], [0], [6], [-22], [-16], [26], [-36], [0], [18], [-32], [30], [12], [-4], [6], [2], [32], [24], [-18], [-6], [-22], [18], [4], [-20], [0], [-16], [6], [-36], [-16], [-10], [22], [-4], [-34], [16], [6], [4], [6], [0], [-8], [-4], [6], [-30], [8], [-24], [-26], [-2], [4], [-12], [-26], [0], [48], [34], [-14], [-8], [-22], [-30], [22], [-20], [50], [14], [-48], [-44], [-4], [40], [18], [48], [14], [-22], [32], [-6], [4], [-6], [-6], [-52], [-2], [44], [-30], [-4], [24], [16], [42], [48], [-18], [-18], [-34], [-20], [-48], [28], [50], [-24], [-6], [0], [-18], [-46], [42], [-4], [20], [38], [16], [-32], [-8], [-2], [-6], [-54], [48], [-30], [-6], [56], [-2], [32], [-6], [-44], [8], [12], [52], [-6], [6], [-26], [-22], [-38], [-36], [10], [-40], [-28], [58], [20], [42], [-62], [-58], [-38], [2], [2], [48], [22], [-32], [52], [44], [12], [18], [-16], [-6], [34], [32], [30], [54], [-16], [60], [40], [-40], [-64], [-66], [-18], [-38], [14], [-36], [-6], [16], [12], [58], [26], [-10], [36], [30], [54], [-8], [32], [-46], [0], [-32], [52], [38], [46], [16], [-2], [-20], [-8], [46], [-30], [-24], [68], [-60], [16], [-10], [-54], [14], [66], [4], [-30], [50], [10], [16], [42], [56], [52], [-2], [4], [24], [54], [-4], [-18], [6], [-64], [-22], [-24], [-20], [-76], [56], [-4], [-20], [2], [-34], [-50], [36], [-40], [-44], [-64], [-56], [28], [-78], [2], [-18], [44], [-78], [54], [20], [-18], [-4], [18], [-70], [-16], [-50], [12], [-20], [16], [18], [20], [66], [0], [62], [-58], [-44], [-54], [-10], [24], [18], [-18], [68], [-34], [20], [-52], [22], [52], [-26], [-40], [2], [-54], [-64], [50], [36], [0], [-24], [42], [34], [10], [-78], [60], [76], [-16], [4], [14], [-14], [44], [60], [58], [-42], [-82], [-72], [34], [-6], [60], [52], [0], [18], [-48], [-58], [70], [-80], [-86], [14], [-48], [20], [-62], [10], [58], [-6], [-30], [34], [-68], [18], [30], [28], [62], [54], [56], [58], [-26], [-28], [62], [-76], [-10], [-88], [14], [-68], [-54], [90], [-72], [2], [-76], [-96], [72], [-36], [-64], [96], [50], [38], [74], [-8], [-46], [-26], [-24], [-42], [-16], [-30], [16], [-62], [-68], [10], [-90], [-8], [-82], [-102], [-52], [-36], [56], [18], [-58], [-44], [42], [-90], [-18], [-28], [-26], [-72], [-62], [80], [-78], [-84], [-80], [26], [-18], [30], [52], [64], [68], [-54], [-16], [-32], [-40], [-78], [-58], [70], [8], [40], [-44], [-90], [-26], [-38], [-28], [-62], [52], [-56], [-72], [-12], [32], [-96], [82], [-18], [46], [-74], [-62], [68], [-28], [82], [-52], [-4], [54], [-74], [-52], [-98], [88], [-68], [6], [68], [-36], [58], [-24], [-30], [22], [36], [90], [100], [-112], [46], [36], [-90], [-68], [42], [100], [10], [-26], [-4], [64], [84], [104], [24], [-50], [110], [66], [8], [-72], [22], [36], [96], [58]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9408_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9408_2_a_ca();".
// 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_9408_2_a_ca(:prec:=1)
    chi := MakeCharacter_9408_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(3581) - 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_9408_2_a_ca();".
// 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_9408_2_a_ca( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9408_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![-6, 1]>,<19,R![-8, 1]>,<31,R![4, 1]>],Snew);
    return Vf;
end function;