
// Make newform 2496.2.a.v in Magma, downloaded from the LMFDB on 09 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2496_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_2496_2_a_v();".
// 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_2496_2_a_v();".
// 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_2496_a();"
function MakeCharacter_2496_a()
    N := 2496;
    order := 1;
    char_gens := [703, 1093, 833, 769];
    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_2496_a_Hecke( : Kf := Kf)
    return MakeCharacter_2496_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]];
    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], [0], [-2], [0], [-1], [-6], [2], [0], [6], [-2], [-2], [-12], [-4], [0], [-6], [12], [-2], [-10], [-12], [14], [-8], [12], [0], [-10], [-18], [16], [-12], [-14], [6], [4], [12], [-12], [-16], [-12], [10], [-2], [14], [24], [6], [12], [-2], [0], [2], [24], [-20], [-4], [-26], [24], [-14], [6], [12], [-10], [12], [6], [24], [-18], [-26], [10], [-12], [-16], [12], [2], [-24], [-22], [24], [-10], [2], [-12], [22], [12], [24], [-20], [-14], [26], [-24], [-30], [22], [24], [-10], [-12], [34], [-24], [14], [-8], [-12], [0], [38], [0], [22], [36], [12], [-2], [12], [14], [24], [0], [30], [-16], [10], [-16], [12], [12], [-18], [-40], [2], [-12], [0], [48], [26], [4], [22], [-36], [-22], [10], [18], [14], [24], [30], [-36], [-2], [-10], [-6], [-24], [14], [42], [10], [0], [-44], [-38], [2], [-12], [-32], [-2], [-12], [-22], [-12], [26], [18], [6], [-22], [12], [40], [48], [-2], [-36], [22], [-42], [44], [0], [-50], [-42], [-16], [-24], [8], [-48], [16], [-36], [-34], [12], [-36], [-18], [-50], [12], [24], [-60], [52], [10], [-10], [30], [48], [-50], [24], [14], [40], [-6], [50], [12], [16], [58], [-26], [-12], [-50], [24], [-48], [-6], [-26], [-46], [14], [-48], [-34], [-36], [32], [-60], [-12], [-42], [50], [-62], [24], [24], [-24], [-44], [58], [-34], [-60], [18], [-2], [-60], [24], [-40], [14], [-54], [-56], [-24], [24], [-46], [-8], [-30], [-12], [36], [58], [-50], [0], [46], [-60], [-62], [-6], [0], [-20], [-36], [-74], [32], [58], [-54], [-16], [12], [14], [-36], [36], [-48], [60], [56], [40], [-26], [-24], [-24], [-2], [-60], [26], [24], [58], [60], [-60], [-70], [6], [-48], [-38], [26], [-54], [2], [-44], [36], [10], [-14], [48], [20], [-24], [-60], [50], [-42], [46], [-10], [-10], [40], [-70], [-38], [-24], [46], [-22], [12], [-24], [-50], [72], [-50], [-22], [48], [-34], [36], [34], [54], [42], [-36], [48], [12], [46], [-54], [28], [-30], [12], [-46], [-22], [48], [-56], [-36], [80], [62], [36], [-14], [36], [-38], [-72], [-24], [30], [-4], [-48], [50], [-24], [72], [38], [-6], [68], [50], [-6], [10], [-24], [62], [38], [-10], [24], [30], [-62], [30], [-56], [-48], [74], [12], [-50], [36], [-58], [16], [-26], [-42], [12], [-32], [-24], [36], [-38], [-82], [0], [18], [38], [38], [84], [-44], [22], [-42], [48], [-72], [6], [0], [-26], [-6], [72], [-24], [-52], [38], [-48], [34], [-34], [-36], [44], [-24], [-54], [40], [34], [84], [48], [2], [42], [86], [60], [-36], [-2], [96], [86], [-12], [82], [70], [2], [-24], [-10], [84], [0], [-28], [-72], [-10], [34], [18], [68], [-24], [-26], [90], [-26], [96], [-72], [40], [-62], [-24], [74], [-48], [2], [-6], [84], [-100], [2], [-30], [-84], [-8], [-96], [96], [-18], [-26], [36], [84], [-94], [-60], [36], [-48], [38], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2496_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2496_2_a_v();".
// 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_2496_2_a_v(:prec:=1)
    chi := MakeCharacter_2496_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_2496_2_a_v();".
// 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_2496_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2496_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![2, 1]>,<11,R![0, 1]>,<17,R![6, 1]>,<19,R![-2, 1]>],Snew);
    return Vf;
end function;