
// Make newform 3716.1.t.a in Magma, downloaded from the LMFDB on 18 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3716_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3716_t_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_3716_1_t_a();".
// 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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := CyclotomicField(232);
    end if;
    return [ #coeff eq Kf!0 select 0 else &+[ elt[1]*Kf.1^elt[2] : elt in coeff]  : coeff in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_3716_t();"
function MakeCharacter_3716_t()
    N := 3716;
    order := 232;
    char_gens := [1859, 1861];
    v := [116, 169];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3716_t_Hecke();"
function MakeCharacter_3716_t_Hecke(Kf)
    N := 3716;
    order := 232;
    char_gens := [1859, 1861];
    char_values := [[[-1, 0]], [[1, 53]]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 1;
    raw_aps := [[[1, 5]], [], [[-1, 1], [-1, 27]], [], [], [[-1, 18], [1, 47]], [[1, 77], [1, 88]], [], [], [[-1, 5], [1, 61]], [], [[1, 3], [-1, 77]], [[1, 65], [-1, 98]], [], [], [[1, 16], [-1, 40]], [], [[1, 34], [-1, 96]], [], [], [[-1, 102], [1, 113]], [], [], [[1, 86], [1, 110]], [[-1, 41], [1, 84]], [[-1, 12], [1, 46]], [], [], [[-1, 39], [-1, 56]], [[-1, 68], [1, 74]], [], [], [[1, 22], [-1, 100]], [], [[-1, 70], [1, 96]], [], [[1, 9], [1, 109]], [], [], [[1, 1], [1, 31]], [], [[1, 18], [1, 37]], [], [[-1, 31], [-1, 67]], [[-1, 24], [-1, 63]], [], [], [], [], [[-1, 51], [-1, 110]], [[1, 44], [-1, 105]], [], [[-1, 36], [-1, 57]], [], [[-1, 91], [-1, 97]], [], [[1, 43], [-1, 44]], [], [[1, 51], [1, 76]], [[1, 27], [-1, 99]], [], [[1, 60], [-1, 69]], [], [], [[-1, 108], [-1, 109]], [[-1, 64], [1, 111]], [], [[1, 26], [-1, 94]], [], [[-1, 34], [1, 80]], [[-1, 76], [-1, 90]], [], [], [[-1, 84], [1, 89]], [], [], [[1, 92], [1, 115]], [[1, 11], [-1, 50]], [[1, 83], [1, 112]], [[1, 48], [-1, 75]], [], [[-1, 45], [1, 64]], [], [[1, 12], [-1, 53]], [], [], [[1, 40], [1, 91]], [[1, 29], [1, 42]], [[1, 4], [1, 71]], [], [], [], [], [], [], [], [[-1, 73], [-1, 113]], [[-1, 59], [1, 73]], [], [[1, 62], [-1, 83]], [], [[-1, 28], [1, 70]], [], [[-1, 10], [1, 38]], [], [[-1, 13], [1, 23]], [], [[1, 73], [1, 105]], [], [[1, 79], [-1, 114]], [], [[-1, 42], [1, 93]], [[1, 7], [1, 25]], [], [], [[-1, 72], [-1, 84]], [], [], [[1, 59], [1, 62]], [], [[1, 14], [1, 17]], [[1, 30], [1, 50]], [[-1, 35], [1, 60]], [], [], [[-1, 69], [-1, 76]], [[1, 52], [1, 98]], [], [], [[-1, 91], [-1, 99]], [], [], [], [[-1, 20], [-1, 53]], [[1, 20], [-1, 32]], [[1, 87], [-1, 107]], [[-1, 6], [-1, 16]], [], [[-1, 85], [-1, 90]], [[-1, 51], [-1, 60]], [], [[1, 21], [-1, 42]], [], [], [[-1, 15], [-1, 61]], [], [[1, 10], [1, 82]], [[-1, 24], [-1, 112]], [], [], [[-1, 9], [1, 76]], [[-1, 11], [-1, 82]], [], [], [], [], [], [[1, 58]], [[1, 5], [1, 25]], [[-1, 86], [1, 103]], [], [[1, 8], [1, 75]], [], [], [[-1, 31], [-1, 62]], [], [], [[-1, 14], [1, 55]], [[1, 33], [1, 35]], [[1, 10], [-1, 32]], [], [[1, 24], [1, 44]], [], [[1, 102], [1, 109]], [], [[-1, 30], [-1, 81]], [], [[-1, 67], [-1, 108]], [], [[1, 43], [1, 90]], [], [], [[-1, 12], [-1, 55]], [[1, 24], [-1, 28]], [], [[1, 15], [-1, 23]], [[-1, 32], [-1, 42]], [], [[1, 20], [1, 66]], [], [[-1, 29], [-1, 106]], [], [], [[1, 6], [-1, 89]], [], [[-1, 79], [-1, 106]], [[-1, 37], [1, 52]], [[1, 7], [-1, 11]], [[-1, 47], [-1, 109]], [], [[1, 19], [1, 110]], [], [[1, 100], [1, 107]], [[1, 21], [1, 67]], [], [[1, 51], [-1, 88]], [], [], [[1, 78], [1, 86]], [], [[-1, 40], [-1, 48]], [[-1, 16], [1, 82]], [], [], [], [[1, 16], [-1, 68]], [], [[1, 30], [-1, 53]], [], [[1, 3], [1, 34]], [[1, 55], [-1, 107]], [], [[1, 33], [1, 98]], [], [], [[1, 38], [1, 66]], [[1, 9], [-1, 96]], [], [], [], [[1, 28], [-1, 96]], [], [], [[-1, 26], [-1, 105]], [], [], [[1, 38], [1, 115]], [[-1, 29], [-1, 98]], [], [], [], [], [], [[1, 13], [-1, 60]], [[1, 73], [-1, 85]], [], [], [], [], [], [[1, 17], [1, 39]], [[-1, 18], [1, 54]], [], [[1, 53], [-1, 54]], [[1, 27], [-1, 55]], [], [[-1, 4], [-1, 48]], [], [[1, 50], [1, 64]], [[-1, 83], [1, 97]], [], [], [[-1, 34], [1, 94]], [[1, 69], [1, 114]], [[-1, 1], [1, 16]], [], [[1, 79], [-1, 87]], [[-1, 9], [1, 113]], [], [[-1, 95], [1, 105]], [[-1, 79], [1, 92]], [], [[1, 24], [-1, 26]], [], [[1, 104], [1, 108]], [], [], [[-1, 27], [-1, 74]], [[-1, 52], [-1, 73]], [], [], [], [], [[1, 75], [-1, 94]], [], [], [[1, 89], [-1, 108]], [[-1, 3], [-1, 69]], [], [[-1, 66], [-1, 75]], [[-1, 64], [1, 105]], [], [[1, 28], [1, 102]], [], [[-1, 12], [-1, 97]], [[-1, 65], [1, 85]], [], [[-1, 0], [-1, 76]], [], [], [[1, 81], [1, 106]], [[-1, 17], [-1, 20]], []];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3716_t_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3716_1_t_a();".
// 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_3716_1_t_a(:prec:=112)
    chi := MakeCharacter_3716_t();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 1));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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;

