
// Make newform 315.3.bi.a in Magma, downloaded from the LMFDB on 12 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_315_bi();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_315_bi_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_315_3_bi_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.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_315_3_bi_a();".
// 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
        poly := [49, 0, 7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 1, 0], [0, 14, 0, -1], [0, 21, 0, 2]];
    Rf_basisdens := [1, 7, 7, 7];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_315_bi();"
function MakeCharacter_315_bi()
    N := 315;
    order := 6;
    char_gens := [281, 127, 136];
    v := [6, 3, 5];
    // 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_315_bi_Hecke();"
function MakeCharacter_315_bi_Hecke(Kf)
    N := 315;
    order := 6;
    char_gens := [281, 127, 136];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [1, 1, 0, 0]];
    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]];
    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 := 3;
    raw_aps := [[-1, 1, 0, 0], [0, 0, 0, 0], [5, 5, 0, 0], [0, 0, 1, 0], [0, 0, -1, 3], [0, 0, 3, -2], [0, 2, 0, 0], [13, -13, 0, 0], [3, -3, 0, 0], [0, 0, -6, 4], [-44, -22, 0, 0], [0, 0, -4, 5], [0, 0, 1, 4], [0, 0, 2, 8], [-41, -41, 0, 0], [26, 13, 0, 0], [0, 0, -10, 2], [32, -32, 0, 0], [0, 0, -10, 2], [0, 0, 12, -8], [0, 0, 2, -6], [-104, -104, 0, 0], [32, 0, 0, 0], [0, 0, 8, -10], [0, 0, 12, -8], [0, 0, 30, -6], [0, 0, 16, 8], [64, -64, 0, 0], [0, 56, 0, 0], [-94, -188, 0, 0], [0, 0, -3, -12], [0, 0, -12, 15], [192, 96, 0, 0], [68, 136, 0, 0], [0, 0, 4, 2], [0, 170, 0, 0], [0, 0, -1, 3], [0, 0, 8, -10], [89, 0, 0, 0], [97, 97, 0, 0], [0, 0, -7, 21], [-60, -120, 0, 0], [0, 0, 4, 2], [0, 0, 20, -4], [-45, -90, 0, 0], [-220, -110, 0, 0], [287, 0, 0, 0], [0, 0, 12, -8], [0, 134, 0, 0], [-18, 18, 0, 0], [214, -214, 0, 0], [0, 0, -42, 28], [-18, -9, 0, 0], [0, 0, -9, -36], [-56, -56, 0, 0], [-360, -180, 0, 0], [0, 0, 70, -14], [-146, 146, 0, 0], [0, 0, 30, -6], [0, 0, 39, -26], [0, 0, 8, -24], [455, 0, 0, 0], [0, 0, 12, -8], [0, 0, -70, 14], [0, 0, -32, -16], [116, -116, 0, 0], [175, 175, 0, 0], [0, 0, -2, -8], [100, 50, 0, 0], [4, 8, 0, 0], [0, 518, 0, 0], [0, 0, -20, -10], [0, 0, -1, 3], [0, 0, -72, 90], [-67, 0, 0, 0], [547, 547, 0, 0], [0, 0, 26, -78], [0, 0, 16, 8], [0, 0, 34, 17], [592, 296, 0, 0], [0, 0, -11, -44], [752, 0, 0, 0], [0, 0, 38, -114], [0, 0, -114, 76], [-88, 88, 0, 0], [260, -260, 0, 0], [0, 0, -33, 22], [0, 0, 40, -50], [0, 0, -4, -16], [0, 0, 7, 28], [340, 340, 0, 0], [0, 0, -120, 24], [0, 0, 70, -14], [0, 0, -24, 16], [-218, -218, 0, 0], [290, 0, 0, 0], [0, 0, 0, 0], [0, 0, 55, -11], [0, 0, 40, 20], [-470, -470, 0, 0], [0, 0, -4, -16], [-298, -149, 0, 0], [0, -226, 0, 0], [0, 0, 10, 5], [0, -406, 0, 0], [0, 0, 56, -168], [-556, 0, 0, 0], [892, 892, 0, 0], [0, 0, 8, -24], [-380, -760, 0, 0], [0, 0, -74, -37], [0, 0, -125, 25], [152, 304, 0, 0], [-774, -387, 0, 0], [662, 0, 0, 0], [0, 0, -49, 147], [0, 0, -78, 52], [0, -1009, 0, 0], [-141, 141, 0, 0], [0, 0, -96, 64], [688, 344, 0, 0], [0, 0, -16, -64], [679, 679, 0, 0], [-1468, -734, 0, 0], [8, -8, 0, 0], [0, 0, 120, -80], [-146, -146, 0, 0], [0, 0, 80, -100], [0, 0, 39, -26], [0, 0, -74, -37], [0, -253, 0, 0], [-415, -830, 0, 0], [-620, -620, 0, 0], [0, 0, -30, -120], [0, 0, -60, 75], [173, 346, 0, 0], [0, -439, 0, 0], [0, 0, 44, -132], [-46, 0, 0, 0], [0, 0, -43, 129], [-663, -1326, 0, 0], [0, 0, -44, -22], [0, 0, 170, -34], [822, 1644, 0, 0], [-268, -134, 0, 0], [0, 0, 20, 80], [0, 0, 237, -158], [0, -730, 0, 0], [396, -396, 0, 0], [139, -139, 0, 0], [0, 0, 132, -165], [0, 0, -27, -108], [0, 0, 30, 120], [-1070, -1070, 0, 0], [0, 0, 210, -42], [0, 0, -258, 172], [-350, -350, 0, 0], [0, 0, -188, 235], [0, 0, -258, 172], [0, 0, 110, -22], [-34, 34, 0, 0], [402, 804, 0, 0], [0, 0, 10, 40], [0, 0, -28, 35], [-1052, -526, 0, 0], [0, 287, 0, 0], [0, 1100, 0, 0], [0, 0, -40, 120]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_315_bi_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_315_3_bi_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_315_3_bi_a(:prec:=4)
    chi := MakeCharacter_315_bi();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_315_3_bi_a();".
// 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_315_3_bi_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_315_bi();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![3, 3, 1]>],Snew);
    return Vf;
end function;