
// Make newform 350.3.i.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_350_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_350_i_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_350_3_i_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_350_3_i_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 := [1, 0, 0, 0, -1, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 1, 0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0, 0, -1], [0, 0, 0, 1, 0, 1, 0, -1], [0, 1, 0, 1, 0, -1, 0, 0]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1];
    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_350_i();"
function MakeCharacter_350_i()
    N := 350;
    order := 6;
    char_gens := [127, 101];
    v := [3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_350_i_Hecke();"
function MakeCharacter_350_i_Hecke(Kf)
    N := 350;
    order := 6;
    char_gens := [127, 101];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 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+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 := 3;
    raw_aps := [[0, 0, 0, 0, 0, -1, 0, 0], [0, 1, 0, -2, 0, -2, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, -3, 0, 1, -5, 0], [9, 0, -9, 0, 3, 0, 0, -3], [0, -12, 0, 6, 0, -2, -2, 0], [0, -5, 0, 10, 0, 4, -2, 0], [-2, 0, 1, 0, 1, 0, 0, -2], [0, -15, 0, 15, 0, -9, 0, 0], [-12, 0, 0, 0, 0, 0, 0, 6], [-7, 0, -7, 0, -15, 0, 0, -15], [0, -31, 0, 31, 0, 24, 0, 0], [2, 0, -4, 0, -20, 0, 0, 10], [0, 0, 0, -2, 0, 6, -6, 0], [0, -29, 0, -29, 0, 1, -2, 0], [0, -39, 0, 0, 0, 0, -12, 0], [13, 0, 13, 0, -25, 0, 0, -25], [-14, 0, 7, 0, 32, 0, 0, -64], [0, 29, 0, 0, 0, 0, -45, 0], [-6, 0, 0, 0, 0, 0, 0, 30], [0, -53, 0, 106, 0, 32, -16, 0], [0, 0, -55, 0, 15, 0, 0, 0], [0, 136, 0, -68, 0, 4, 4, 0], [126, 0, -63, 0, 24, 0, 0, -48], [0, 44, 0, -22, 0, 26, 26, 0], [-9, 0, -9, 0, -18, 0, 0, -18], [0, 51, 0, 51, 0, -13, 26, 0], [0, 33, 0, -33, 0, 57, 0, 0], [-35, 0, 35, 0, 78, 0, 0, -78], [0, 0, 0, 60, 0, 18, -18, 0], [0, 0, 0, 154, 0, -66, 66, 0], [-162, 0, 81, 0, 3, 0, 0, -6], [0, -75, 0, 0, 0, 0, 30, 0], [0, 0, 0, 0, -56, 0, 0, 28], [0, 0, 33, 0, 42, 0, 0, 0], [113, 0, -113, 0, 15, 0, 0, -15], [0, -63, 0, 126, 0, 96, -48, 0], [0, 65, 0, -65, 0, 111, 0, 0], [0, -20, 0, 10, 0, 76, 76, 0], [0, -27, 0, -27, 0, 48, -96, 0], [147, 0, -147, 0, -27, 0, 0, 27], [16, 0, -32, 0, -104, 0, 0, 52], [0, 0, -159, 0, -63, 0, 0, 0], [0, -61, 0, 0, 0, 0, -66, 0], [0, 0, 0, -48, 0, -162, 162, 0], [-149, 0, -149, 0, -37, 0, 0, -37], [118, 0, 0, 0, 0, 0, 0, 174], [0, -312, 0, 156, 0, 60, 60, 0], [0, 45, 0, -90, 0, -126, 63, 0], [-158, 0, 79, 0, 90, 0, 0, -180], [0, -27, 0, 27, 0, -174, 0, 0], [36, 0, 0, 0, 0, 0, 0, 162], [-11, 0, -11, 0, -28, 0, 0, -28], [190, 0, -380, 0, -136, 0, 0, 68], [0, -87, 0, -87, 0, 102, -204, 0], [0, -201, 0, 0, 0, 0, 165, 0], [115, 0, 115, 0, 44, 0, 0, 44], [-34, 0, 17, 0, 19, 0, 0, -38], [0, -61, 0, 0, 0, 0, 240, 0], [-72, 0, 0, 0, 0, 0, 0, 330], [0, 47, 0, -94, 0, -186, 93, 0], [0, 16, 0, -8, 0, -128, -128, 0], [0, -160, 0, 80, 0, 48, 48, 0], [115, 0, 115, 0, -7, 0, 0, -7], [0, 17, 0, 17, 0, 52, -104, 0], [0, 465, 0, -465, 0, 12, 0, 0], [0, 0, -85, 0, -99, 0, 0, 0], [0, 0, 0, 188, 0, -54, 54, 0], [0, 93, 0, 0, 0, 0, -201, 0], [98, 0, -196, 0, 28, 0, 0, -14], [0, 221, 0, -442, 0, -28, 14, 0], [0, 0, 63, 0, 171, 0, 0, 0], [0, -209, 0, 418, 0, 218, -109, 0], [0, -1, 0, 1, 0, 180, 0, 0], [-364, 0, 0, 0, 0, 0, 0, -102], [0, 133, 0, 133, 0, -101, 202, 0], [9, 0, -9, 0, -114, 0, 0, 114], [0, 37, 0, 37, 0, 126, -252, 0], [0, 0, -117, 0, 378, 0, 0, 0], [-243, 0, -243, 0, 46, 0, 0, 46], [-184, 0, 368, 0, -176, 0, 0, 88], [-4, 0, 0, 0, 0, 0, 0, -114], [-183, 0, 183, 0, 291, 0, 0, -291], [0, -4, 0, 2, 0, 18, 18, 0], [-42, 0, 21, 0, 45, 0, 0, -90], [0, 171, 0, -171, 0, -45, 0, 0], [60, 0, 0, 0, 0, 0, 0, -138], [0, -7, 0, 7, 0, -108, 0, 0], [-346, 0, 692, 0, -236, 0, 0, 118], [0, 0, 0, -46, 0, 198, -198, 0], [0, -399, 0, -399, 0, -39, 78, 0], [-359, 0, -359, 0, 47, 0, 0, 47], [0, 241, 0, 0, 0, 0, -21, 0], [-12, 0, 0, 0, 0, 0, 0, -546], [0, 0, -23, 0, -105, 0, 0, 0], [0, -56, 0, 28, 0, 28, 28, 0], [658, 0, -329, 0, 38, 0, 0, -76], [233, 0, 233, 0, 184, 0, 0, 184], [0, -27, 0, -27, 0, 223, -446, 0], [0, 0, -677, 0, -126, 0, 0, 0], [0, 0, 0, -68, 0, 246, -246, 0], [0, -423, 0, 0, 0, 0, -114, 0], [0, -123, 0, 246, 0, -570, 285, 0], [0, 0, 123, 0, 408, 0, 0, 0], [397, 0, -397, 0, -117, 0, 0, 117], [0, -3, 0, 6, 0, -536, 268, 0], [0, 0, 0, 0, 0, -336, -336, 0], [0, -413, 0, -413, 0, -38, 76, 0], [9, 0, -9, 0, 369, 0, 0, -369], [74, 0, -148, 0, 660, 0, 0, -330], [0, -185, 0, -185, 0, -115, 230, 0], [0, -749, 0, 0, 0, 0, 246, 0], [0, 0, 0, 696, 0, -186, 186, 0], [225, 0, 225, 0, 251, 0, 0, 251], [182, 0, 0, 0, 0, 0, 0, -402], [1077, 0, -1077, 0, 60, 0, 0, -60], [0, -616, 0, 308, 0, -168, -168, 0], [0, -45, 0, 90, 0, 66, -33, 0], [0, -357, 0, 357, 0, 24, 0, 0], [18, 0, 0, 0, 0, 0, 0, -222], [193, 0, 193, 0, -128, 0, 0, -128], [0, 0, 0, 364, 0, -186, 186, 0], [0, -183, 0, -183, 0, -174, 348, 0], [0, -399, 0, 0, 0, 0, -279, 0], [-458, 0, 229, 0, -237, 0, 0, 474], [-942, 0, 0, 0, 0, 0, 0, -72], [0, 0, -149, 0, -588, 0, 0, 0], [370, 0, -185, 0, -187, 0, 0, 374], [0, -272, 0, 136, 0, -204, -204, 0], [0, 29, 0, 29, 0, 216, -432, 0], [-109, 0, 109, 0, -63, 0, 0, 63], [0, 0, 0, 510, 0, 330, -330, 0], [0, 0, 305, 0, -375, 0, 0, 0], [0, 0, 0, -520, 0, 294, -294, 0], [494, 0, -247, 0, -398, 0, 0, 796], [-98, 0, 196, 0, -140, 0, 0, 70], [0, 77, 0, -154, 0, -292, 146, 0], [0, 107, 0, -214, 0, -94, 47, 0], [0, 540, 0, -270, 0, 342, 342, 0], [627, 0, -627, 0, -96, 0, 0, 96], [-396, 0, 792, 0, 432, 0, 0, -216], [0, 0, 507, 0, 54, 0, 0, 0], [0, -161, 0, 0, 0, 0, -615, 0], [0, 0, 0, 654, 0, 570, -570, 0], [-103, 0, -103, 0, -178, 0, 0, -178], [-508, 0, 1016, 0, 544, 0, 0, -272], [0, 1364, 0, -682, 0, 22, 22, 0], [0, 49, 0, -98, 0, -968, 484, 0], [-194, 0, 97, 0, 173, 0, 0, -346], [0, 525, 0, -525, 0, -417, 0, 0], [0, 703, 0, -703, 0, -282, 0, 0], [-312, 0, 624, 0, 264, 0, 0, -132], [0, 0, 0, -542, 0, 18, -18, 0], [0, 583, 0, 583, 0, 49, -98, 0], [0, 217, 0, 0, 0, 0, 111, 0], [1278, 0, 0, 0, 0, 0, 0, -102], [0, 0, 1063, 0, -429, 0, 0, 0], [-10, 0, 5, 0, -392, 0, 0, 784], [0, -48, 0, 24, 0, -512, -512, 0], [-823, 0, -823, 0, 28, 0, 0, 28], [0, -33, 0, 33, 0, 525, 0, 0], [0, 0, 0, 996, 0, 42, -42, 0], [0, 0, 0, -1298, 0, 54, -54, 0], [-322, 0, 161, 0, -101, 0, 0, 202], [0, -771, 0, 0, 0, 0, 162, 0], [0, -317, 0, 634, 0, 430, -215, 0], [599, 0, -599, 0, 903, 0, 0, -903], [0, 421, 0, -842, 0, 120, -60, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_350_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_350_3_i_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_350_3_i_a(:prec:=8)
    chi := MakeCharacter_350_i();
    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_350_3_i_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_350_3_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_350_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![81, 0, 162, 0, 315, 0, 18, 0, 1]>],Snew);
    return Vf;
end function;