
// Make newform 112.6.i.g in Magma, downloaded from the LMFDB on 08 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_112_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_112_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_112_6_i_g();".
// 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_112_6_i_g();".
// 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 := [92895579, -25332057, 1863790, 3648839, 657990, 27806, -898, -521, -119, -3, 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], [-284541456761027013555, -115074613295547071477, -59305399941208136095, -12374882267308632540, -527187805285965265, 28536164025490220, 8704766381850349, 1755200034298057, 80250234146853, -17151069986588], [-1167207457164718108487021541, 11482780674704478107745539249, 2130158090435409117164074341, 35044891457936010176916201, -3136904970777860996782615, -1784657554484775854973480, -427334094491622611631563, 390299734521856593718, 3998053266244577812763, -83622367719149452381], [4996227955653678642352316242128, -698926715762252200310386715315, -357587613159464399273479579996, -27141840046274712081346665873, -287867175691381552237971628, -147041569306757069269140145, 60779390018685862527644803, 6899857122784815838184224, -715887898900143812358510, 4360035704821052650099], [-22872299262533116707107668473, 5090763111163946503113669866, 3561651001831299799176109977, 350159017235668660926893700, 7293211844541683227745357, -1141869910255571967018573, -548203343053803274159004, -64109057987136330427586, 2563450462652211073757, 360611666314254475586], [-3044921325816592037954233097037, -1084057526217463647662996949799, -29067451279760453763981702611, 3297686923587929926416425181, 890081637816657484028287537, 239795363362585232795489488, 15500368356152314963501085, -3719842424064503967707398, 395411541118777508399211, -30976930565536023489865], [130091386836880615858202304009, 47140260370309174547656314659, 30272756134278292119981342157, 4970204862730567500865443084, 104675818605670551514032499, -16599162992406692386788596, -2335223515720492139237551, -1110924299273894178932527, 10452245882686456422585, 6029216722011023593244], [4759105204074921941515894824771, -107521626435580352904298649794, -263613804562889518748189229983, -35522474868877232246310481356, -161390835083055286569462491, -29577625740414631952675945, 68079138624492285682720544, 7424627513071807861040474, -365316503545882087209051, -42635134954087600449622], [-354949760412154045590462761058, -113806809052519127485804746964, 1022945265659664888961114319, 3946803287199280616326713649, 210280671758443155327176433, -1222304774170129217593775, 774049127534792461621648, -939968171018909571130001, -8845265752464631950979, 5308531290342553729665], [-58429141728791777745780780627, -9184048700048858000640738060, 1192707820540668364963802168, 956389277630192833103100620, 107285020069158922184064016, -1602216419859621538146012, -302681040743130332694860, -127263552577879154102816, -24153824418477016656728, 2415886865346298664720]];
    Rf_basisdens := [1, 709480988855409244257, 1630237559346934629329140791, 44016414102367234991886801357, 1630237559346934629329140791, 44016414102367234991886801357, 6288059157481033570269543051, 44016414102367234991886801357, 4890712678040803887987422373, 1630237559346934629329140791];
    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_112_i();"
function MakeCharacter_112_i()
    N := 112;
    order := 3;
    char_gens := [15, 85, 17];
    v := [3, 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_112_i_Hecke();"
function MakeCharacter_112_i_Hecke( : Kf := false)
    N := 112;
    order := 3;
    char_gens := [15, 85, 17];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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 := 6;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 3, 0, 0, 1, 0, 0, 0, 0, 0], [0, 6, 0, 0, 0, 0, -1, 0, 0, 0], [16, 13, 0, 0, 0, 0, 0, 0, -1, 0], [-72, -72, 0, 1, -3, -3, -3, 1, 4, -3], [-15, 0, 3, -3, -3, -1, -3, 1, 3, -8], [-34, -34, 0, -4, -21, 1, -6, -4, -5, -6], [0, -206, 15, 6, 0, -5, 12, -11, -5, 5], [0, -743, -48, -1, 0, -8, 58, -7, -8, 8], [-45, 0, -237, -11, 237, -21, -11, 21, 11, 26], [1652, 1652, 0, -11, 288, -6, 46, -11, -5, 46], [0, 2891, 489, 1, 0, -34, 54, -35, -34, 34], [1075, 0, 33, -7, -33, -63, -7, 63, 7, 26], [-372, 0, 546, -30, -546, -10, -30, 10, 30, 66], [0, -1232, 156, -73, 0, -15, 132, 58, -15, 15], [5875, 5875, 0, -10, -519, 1, -311, -10, -11, -311], [17867, 17867, 0, 28, 687, -42, -70, 28, 70, -70], [0, 3923, -273, -161, 0, 14, -130, 175, 14, -14], [-6931, -6931, 0, 102, -1065, 92, 144, 102, 10, 144], [22302, 0, -1146, 164, 1146, 90, 164, -90, -164, 186], [-27793, -27793, 0, -16, 1326, 46, 44, -16, -62, 44], [0, -22267, 3306, 97, 0, 164, 346, 67, 164, -164], [-41666, 0, -1344, 122, 1344, 120, 122, -120, -122, -140], [0, -11373, -414, 626, 0, 204, 56, -422, 204, -204], [28033, 0, -2055, -155, 2055, -635, -155, 635, 155, -220], [-53254, -53254, 0, 28, 5394, 434, -1053, 28, -406, -1053], [0, -34605, -2574, -3, 0, 370, -2294, 373, 370, -370], [0, -47610, 5289, -120, 0, 39, -492, 159, 39, -39], [60900, 60900, 0, -136, -84, -344, 1191, -136, 208, 1191], [-17831, 0, 645, 509, -645, -95, 509, 95, -509, -1356], [105050, 0, -294, 196, 294, 406, 196, -406, -196, -882], [0, -106571, -231, -224, 0, -546, 2440, -322, -546, 546], [124950, 124950, 0, 308, 177, -277, 2108, 308, 585, 2108], [-214022, 0, -1428, -186, 1428, -300, -186, 300, 186, -2732], [0, 193606, -6108, -612, 0, 732, -253, 1344, 732, -732], [86632, 86632, 0, 417, 4290, -20, 704, 417, 437, 704], [-133427, -133427, 0, -1826, -5265, -569, 1137, -1826, -1257, 1137], [0, -38337, -11865, 442, 0, 40, -5384, -402, 40, -40], [-240707, 0, -1416, 1399, 1416, 296, 1399, -296, -1399, 728], [0, -118911, 2631, 771, 0, 1174, -3300, 403, 1174, -1174], [196792, 196792, 0, -267, 8355, -853, -3857, -267, 586, -3857], [-285982, 0, 15042, -2110, -15042, -2178, -2110, 2178, 2110, -2970], [0, 123440, -12072, -1285, 0, 629, 6496, 1914, 629, -629], [223296, 223296, 0, 648, 10299, -2735, -4744, 648, 3383, -4744], [141303, 0, 17499, 893, -17499, 1099, 893, -1099, -893, -4114], [55490, 55490, 0, -961, 2496, 3840, 2208, -961, -4801, 2208], [131434, 0, -1428, -238, 1428, 2604, -238, -2604, 238, -168], [-363158, 0, 6990, -924, -6990, 1498, -924, -1498, 924, 5838], [-155974, -155974, 0, 2311, -16155, 2959, 3651, 2311, -648, 3651], [0, -308888, 26748, 1632, 0, 712, 4929, -920, 712, -712], [0, -153744, -21921, 4035, 0, 3156, -3619, -879, 3156, -3156], [-21209, 0, -25656, 305, 25656, -112, 305, 112, -305, 3516], [616453, 616453, 0, -1872, -10524, 2260, -3416, -1872, -4132, -3416], [-35317, 0, -10200, 6063, 10200, 4212, 6063, -4212, -6063, -7814], [0, -405922, 1329, -4119, 0, 916, -13389, 5035, 916, -916], [603863, 603863, 0, 3886, -26922, -1227, -2833, 3886, 5113, -2833], [993381, 993381, 0, 4330, 46551, -1989, -4815, 4330, 6319, -4815], [0, -301741, -25548, -4449, 0, -4542, -6086, -93, -4542, 4542], [139823, 139823, 0, -3930, 23169, -3135, -3195, -3930, -795, -3195], [804301, 0, -35475, 1401, 35475, 3385, 1401, -3385, -1401, -7860], [-490164, -490164, 0, -1479, 3525, -7059, -14183, -1479, 5580, -14183], [-1189914, 0, -35244, 6732, 35244, 396, 6732, -396, -6732, 21132], [-461732, 0, -66150, -6730, 66150, -7722, -6730, 7722, 6730, -2674], [-926615, -926615, 0, 2368, 42096, 5393, 4139, 2368, -3025, 4139], [0, -367915, -62970, 226, 0, 1144, -19434, 918, 1144, -1144], [0, -641607, -20217, 4011, 0, -7242, 16932, -11253, -7242, 7242], [0, 841729, 53607, -8826, 0, -4254, 10416, 4572, -4254, 4254], [525127, 0, -126183, -8631, 126183, -2911, -8631, 2911, 8631, 5286], [223515, 223515, 0, 12040, -31029, 10790, -12578, 12040, 1250, -12578], [252219, 0, -33453, 12873, 33453, 7763, 12873, -7763, -12873, -24778], [364056, 364056, 0, 3144, -133689, 8125, 8780, 3144, -4981, 8780], [0, 756798, 27228, 15147, 0, 6063, 132, -9084, 6063, -6063], [-788957, -788957, 0, -26, 78612, -8033, -3563, -26, 8007, -3563], [0, -446475, -40263, 6861, 0, 9186, -11228, 2325, 9186, -9186], [352633, 0, 50112, -1167, -50112, -8452, -1167, 8452, 1167, 34802], [0, 497922, 42366, -10067, 0, 1031, 304, 11098, 1031, -1031], [335738, 335738, 0, 12124, 36918, 966, -14777, 12124, 11158, -14777], [0, 1490964, 48774, -18462, 0, 3884, -33527, 22346, 3884, -3884], [0, 780294, 11937, 4617, 0, 11088, -24369, 6471, 11088, -11088], [-876182, -876182, 0, 6072, -109995, 10347, -63092, 6072, -4275, -63092], [-757540, 0, -24330, -12678, 24330, -12262, -12678, 12262, 12678, 11602], [-1027707, 0, 102183, 11173, -102183, 22219, 11173, -22219, -11173, 8188], [-952182, -952182, 0, -2693, 57846, 17542, 44810, -2693, -20235, 44810], [3299677, 0, 27321, 11817, -27321, 21285, 11817, -21285, -11817, -5244], [0, -591990, -121980, 3179, 0, -1997, 42284, -5176, -1997, 1997], [0, -3071680, 67263, 7922, 0, 11729, -21180, 3807, 11729, -11729], [-1301153, 0, 68001, 19637, -68001, 7553, 19637, -7553, -19637, 10746], [0, -192627, -147042, -16822, 0, -7572, 62826, 9250, -7572, 7572], [1863743, 0, 69951, -6263, -69951, -13241, -6263, 13241, 6263, -23990], [-97760, 0, -160692, -31500, 160692, -17388, -31500, 17388, 31500, 15540], [0, -207977, -150381, 3422, 0, 8500, 8292, 5078, 8500, -8500], [-641591, -641591, 0, -21280, -164322, -19061, -8387, -21280, -2219, -8387], [658011, 658011, 0, 13834, 193134, 16697, -7701, 13834, -2863, -7701], [958545, 0, -6600, -9367, 6600, -6572, -9367, 6572, 9367, -56206], [0, -4139017, 1695, -40870, 0, -22584, 10624, 18286, -22584, 22584], [3051298, 0, 70314, 5008, -70314, 7470, 5008, -7470, -5008, -17286], [0, 31070, 58830, 24526, 0, -2404, 40683, -26930, -2404, 2404], [-2921683, -2921683, 0, -2912, 38814, -3318, -14146, -2912, 406, -14146], [0, -690478, 104253, 2412, 0, -169, -4752, -2581, -169, 169], [0, 234622, -60318, -41134, 0, -5032, -34689, 36102, -5032, 5032], [1972523, 0, 210648, 1795, -210648, -9684, 1795, 9684, -1795, 28758], [-405780, -405780, 0, -45252, 75660, 2884, 19111, -45252, -48136, 19111], [5651685, 5651685, 0, -32722, 116559, -14748, 66568, -32722, -17974, 66568], [0, 253365, -98142, 36166, 0, 10120, 26454, -26046, 10120, -10120], [-815524, -815524, 0, 24133, 245073, 32781, 5613, 24133, -8648, 5613], [-3910479, -3910479, 0, -13564, -78072, -33772, -27296, -13564, 20208, -27296], [1421776, 0, -37086, 15538, 37086, -6002, 15538, 6002, -15538, 73766], [0, 317896, -250455, -37051, 0, 8316, 44573, 45367, 8316, -8316], [-3372639, -3372639, 0, 2456, 324384, -22231, 90755, 2456, 24687, 90755], [7445265, 0, 101409, -3675, -101409, -26691, -3675, 26691, 3675, 45404], [0, 917350, -63510, -2503, 0, 25775, 15980, 28278, 25775, -25775], [-3804340, -3804340, 0, -12316, 322578, 40878, 40449, -12316, -53194, 40449], [-17941, 0, -141699, 46745, 141699, 54397, 46745, -54397, -46745, 97954], [5384617, 5384617, 0, 31510, 148449, 9402, -120302, 31510, 22108, -120302], [-2607698, 0, 576126, 50684, -576126, 78786, 50684, -78786, -50684, -25806], [-1762043, -1762043, 0, 13400, -182172, 34388, -24512, 13400, -20988, -24512], [-3778172, 0, -51414, -5014, 51414, 52686, -5014, -52686, 5014, -59110], [-7543551, -7543551, 0, 15676, -3288, 34021, -55909, 15676, -18345, -55909], [0, -563683, 438207, -23229, 0, 17254, 75512, 40483, 17254, -17254], [-2628450, 0, 343824, -57370, -343824, -42576, -57370, 42576, 57370, -51584], [361873, 361873, 0, 6246, -92463, 14997, 5139, 6246, -8751, 5139], [-3172889, 0, 613605, -30571, -613605, -56219, -30571, 56219, 30571, -44782], [0, -3088012, -380136, -10604, 0, 1300, 22913, 11904, 1300, -1300], [-8130046, -8130046, 0, -2373, -560157, 18325, 21205, -2373, -20698, 21205], [0, -4197447, -679659, 12884, 0, 35566, -94676, 22682, 35566, -35566], [-526950, 0, -225360, -70096, 225360, -55176, -70096, 55176, 70096, 181516], [0, -8147669, 531873, 63337, 0, -6362, -4146, -69699, -6362, 6362], [0, 2728299, -345444, 31495, 0, -30510, 208322, -62005, -30510, 30510], [5758018, 0, -385878, 35564, 385878, 20726, 35564, -20726, -35564, -62570], [0, 4613069, -752589, -37105, 0, 7458, 17996, 44563, 7458, -7458], [6437718, 6437718, 0, 33211, 378429, -18693, 85127, 33211, 51904, 85127], [-13997646, 0, -293706, -66324, 293706, -74286, -66324, 74286, 66324, 94054], [0, 16942780, 141504, 15283, 0, -5979, 85628, -21262, -5979, 5979], [4305663, 0, -396105, 15229, 396105, 18055, 15229, -18055, -15229, -120074], [0, -2177330, 352731, -26877, 0, 16948, -282109, 43825, 16948, -16948], [4667963, 0, -635751, -86247, 635751, -106855, -86247, 106855, 86247, -214486], [6755654, 6755654, 0, -87284, -62418, -82010, 19635, -87284, -5274, 19635], [1356903, 1356903, 0, -16672, -820551, -66056, 23884, -16672, 49384, 23884], [-13422481, 0, 31107, 3829, -31107, -50813, 3829, 50813, -3829, 88006], [3076183, 3076183, 0, -68420, 187794, 7638, 152224, -68420, -76058, 152224], [-19274320, 0, 384438, -53134, -384438, -79918, -53134, 79918, 53134, -90474], [0, 327196, 470304, -21468, 0, -91764, -72575, -70296, -91764, 91764], [20285665, 20285665, 0, -7704, 202632, 33105, -80965, -7704, -40809, -80965], [970168, 0, -351342, 46934, 351342, 89670, 46934, -89670, -46934, 48898], [5087592, 5087592, 0, 51688, -90516, -52248, -207253, 51688, 103936, -207253], [11644872, 0, 249444, 23348, -249444, 2668, 23348, -2668, -23348, 195860], [8857697, 0, -230589, 76733, 230589, 65487, 76733, -65487, -76733, -41888], [9598695, 9598695, 0, 14900, -98838, 5326, -318064, 14900, 9574, -318064], [0, -8247280, 663939, 59910, 0, 10541, 160808, -49369, 10541, -10541], [0, 14432477, -556038, 33977, 0, 54028, -29350, 20051, 54028, -54028], [0, -7673142, 596250, 188618, 0, 67056, 286259, -121562, 67056, -67056], [-17862314, 0, -20742, -58674, 20742, 24654, -58674, -24654, 58674, 106970], [12846390, 0, 349164, -28386, -349164, 36412, -28386, -36412, 28386, -378720], [0, 1349476, -516138, -167159, 0, -27743, -459272, 139416, -27743, 27743], [8825439, 8825439, 0, -43060, 323679, -84426, 506734, -43060, 41366, 506734], [-11882960, 0, 1073388, 41352, -1073388, 106988, 41352, -106988, -41352, -52712], [0, 13673418, -1284372, 114835, 0, -14461, -32548, -129296, -14461, 14461], [0, -22821987, -151476, -9088, 0, 23524, -208974, 32612, 23524, -23524], [12489278, 0, 86340, 42148, -86340, 116996, 42148, -116996, -42148, -165768], [2371959, 2371959, 0, 7782, -471573, 31959, 19467, 7782, -24177, 19467], [0, -15472127, -1204587, -65276, 0, 6222, -235660, 71498, 6222, -6222], [-403983, 0, 478377, -150111, -478377, 26189, -150111, -26189, 150111, -104968], [4223602, 0, 1282950, -152872, -1282950, -181110, -152872, 181110, 152872, -370190], [0, -19000623, 533367, 96502, 0, 62706, 388720, -33796, 62706, -62706], [18206120, 18206120, 0, 80232, -73161, 193317, 101732, 80232, -113085, 101732], [-349284, -349284, 0, 8169, -270078, -40404, -325668, 8169, 48573, -325668], [27180501, 27180501, 0, 19908, 882366, -88337, -93799, 19908, 108245, -93799], [-19417807, -19417807, 0, 107226, -1183827, 134421, 30639, 107226, -27195, 30639]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_112_i_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_112_6_i_g();".
// 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_112_6_i_g(:prec:=10)
    chi := MakeCharacter_112_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_112_6_i_g();".
// 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_112_6_i_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_112_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![386099434161, -56279254437, 11756456271, -274970430, 82403301, -1914555, 390805, -3142, 807, -13, 1]>],Snew);
    return Vf;
end function;