
// Make newform 315.10.a.o in Magma, downloaded from the LMFDB on 07 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_315_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_315_10_a_o();".
// 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_10_a_o();".
// 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 := [51239796804, 1793109513, -759677093, -7013491, 2674143, 5227, -2943, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-736, 1, 1, 0, 0, 0, 0, 0], [3565684744244, 300084677529, -43345120552, -1392176191, 108855364, 831075, -59216, 67], [5986461701612, -104272438449, -37232226392, -907999177, 36125308, 2341909, -7440, -1179], [-5612577959916, -317612425551, 37232226392, 1283007945, -36125308, -2341909, 7440, 1179], [626401057148, -1637144624605, -16965363864, 9939069163, 41835468, -14004175, -24496, 5569], [-19198705417580, -1445982767727, 124214580680, 7143843033, -152876380, -9004117, 50688, 3307]];
    Rf_basisdens := [1, 1, 1, 187504384, 375008768, 375008768, 375008768, 750017536];
    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_a();"
function MakeCharacter_315_a()
    N := 315;
    order := 1;
    char_gens := [281, 127, 136];
    v := [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_315_a_Hecke( : Kf := Kf)
    return MakeCharacter_315_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 := 10;
    raw_aps := [[3, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-625, 0, 0, 0, 0, 0, 0, 0], [2401, 0, 0, 0, 0, 0, 0, 0], [-943, -36, 12, -1, 2, 1, 0, -5], [10253, 912, -77, -5, 1, 1, 2, 2], [23124, -1228, 19, -9, 12, 2, 19, -3], [-45303, 3858, 375, 3, 44, -54, 0, 15], [203980, -4074, -276, -65, 56, -33, -69, 133], [1208314, -34004, 43, 164, 81, -262, -42, 239], [-522069, -286, 1252, -112, 362, 114, -189, 126], [-3299868, 16396, -4783, 473, 1268, 130, -117, 673], [3565148, 100422, 12785, -139, 1101, -1627, -409, 768], [-4119490, 90262, 9196, -1269, -284, -745, 391, 821], [12306742, 311134, 22962, 445, -1712, -197, 1695, -1015], [19943865, 615568, 20566, -359, -1090, -4911, -1222, 4759], [21718858, -15384, 31528, -4780, -1938, 5062, -20, -560], [-18751220, -387356, 8433, -2198, -5523, -2010, -1768, -7383], [-20137430, 163468, 96691, -1324, -4911, 5118, 5002, -22593], [25118045, 2108912, 172425, -3946, 4734, -24015, 5407, 25646], [-10803785, 484664, -45685, 213, 14217, -21581, 3146, 2236], [23169074, 4150838, -161981, -6501, -9791, 30865, 21429, -16926], [-3761558, 5451534, 356889, 4413, 4429, 16997, -9595, -23770], [163817820, 948812, -308280, 16726, 14766, 40104, 9222, -21674], [269593857, 19730576, 658055, 8559, 12873, -10951, -21598, 27722], [220832234, 6336370, -546291, -16611, -56255, 75901, 12227, -14594], [-129235040, 29524248, -671954, 10636, -9746, 34748, -72628, -19398], [153004622, -6737330, -1487446, -64161, -14660, -97235, -38359, 5433], [52742072, 20334162, -1010051, 55271, 55321, -137905, 25811, 17316], [447786557, 24597208, -2188898, 43217, 80790, -6299, 33914, 57287], [-590782576, 10849966, -472245, -132947, -159467, -206185, -88117, -102002], [269711884, 38754628, -1986846, -125238, -17502, 35850, -97360, 77730], [-101659329, -103973772, -1643106, 168863, -265116, -116187, -40254, -233557], [-1225398535, 92408002, -1953477, 77265, -56854, -280474, -93178, 302147], [2413865996, 32516780, -2183307, 8254, 64571, 106308, 111898, 284459], [-910376978, 126685098, -506967, -255021, 2529, 292055, -203511, 45034], [-167866193, 69554496, -1842391, 222767, -691247, 138571, 277264, -562326], [-1012738818, 105943322, 6498362, -62871, 72486, -18823, -28507, 838313], [3717935540, -22835864, -1720025, 63580, -374035, -1322114, -97954, 318653], [7146603000, 34360268, -1344298, -27052, 175626, -240568, 46098, 233854], [7097457105, 36434688, 8855458, 771843, 1122630, 959591, -247392, -71741], [2382384386, -46198152, 1003003, 888074, 1422179, 573810, 282494, 524639], [5525668403, -197158324, 8323837, -505610, 338058, 666273, -151175, -1200250], [-5041652236, -239592484, 12564064, 308736, -742472, -1235848, 355050, -563220], [9795308579, -132481644, -3090362, -542275, -1394438, 1076505, -242392, -791081], [-4128379553, -427448898, -25963098, -362508, 850106, -984016, -1336245, -133810], [51835320, -741722778, -6506501, -478129, -3685949, -1526525, 82571, -1597924], [1781506602, -293086432, 43483938, 264154, 644014, 1896590, -1272716, 825004], [13721730518, -184759798, -25500637, 1128791, 699523, -1443085, 121631, 4664738], [-12594319248, -631290728, -21781185, -1713948, 856771, 273064, 1217750, 1073585], [2246411837, -356493640, -62650128, -24659, -1042876, 2608301, 145866, -1959507], [5018849549, -188683492, -36204565, -114826, 729374, -3982597, 750855, 1299210], [13312348844, -844108488, -9824064, -950214, -640984, -547550, -734064, 5186866], [10703301254, 511723528, -113017880, 1041442, 1279152, -2493006, -191750, 1924160], [17577369486, 571981808, -27792681, -258571, 934790, 6268930, -1838909, -2364667], [3039785422, -127970386, 7423908, -3111277, -1700170, 237853, 1373725, -6304385], [17699794222, -326860290, -70813487, -2665781, 2071321, -4417609, -920019, 5323728], [6868021037, -395115886, 45591418, 647498, -1719628, 13044064, 196071, -8792714], [-25872783406, -904974436, 76533307, -1999725, 4887682, 6050272, -3141321, -459171], [-9628381562, 1126958908, -72935546, -628372, 4269674, -2638056, -4138098, 5806026], [42432670556, -834007168, 47145784, 1683656, -1335144, -9221880, -531504, 3877776], [31172028722, -1459620380, -68901498, 1845626, -2436446, 748806, 6322506, -5614012], [2332414190, 689289572, 62793092, 4282442, -3245834, -7003656, 2162600, -6612440], [91837666758, 2064793628, 209840084, 4202834, -11759310, 4987916, -2615648, -10595844], [104162595737, -297656392, 133781043, -87053, -5283311, -3798303, -2151674, 10321586], [-18407332735, 1271210316, -202881102, 2870783, 15393310, 1182299, -5281664, 2681413], [30036127152, -370349982, 70056765, -3575659, 3951629, -13230407, 6265617, -6243328], [-10464078388, -405234536, 808594, 2247090, -16215698, -3174898, 5922304, 5451404], [117892592464, 964606010, 317632336, 7173223, -9557658, 5123417, -8834713, -11433169], [-61981889512, -1149651472, -352757919, -160504, -9717331, 720028, -6175058, -2453053], [30173696882, -2070278548, 423405495, 4120077, 10281692, -25944428, 1282243, 9504559], [-94654365757, 2289425452, -346795309, 6228396, -2132630, -13460483, 9349031, -4449012], [104686067534, 3732438528, -231680946, -11363354, 14712874, 8354658, -8458248, 9430632], [140085012280, 9476340, 361461355, 16539115, -10105080, 4396130, -1738975, -18884525], [49052615318, 5412654950, -13969307, -6705825, 18992955, 38202805, 6839685, -6233240], [112594023716, -10251493456, 397412267, -2387488, 29725, 19926294, -13513558, -1775659], [-9904879900, 2457605856, 223365093, -19605340, -27036153, 2026798, -7753104, 2952753], [75513950605, -2402146408, -233873327, -19434067, 3031021, -76154467, 9391164, 65117004], [-222500046040, 6181108280, -603540350, 5377772, 18142002, -12562276, 14980200, 256210], [-13104108902, 868283916, -179300668, 38625310, 14997148, -13542602, -16063154, 29088446], [-78433998818, -4986955964, -264586410, 10843356, 47871826, 20056064, -14999508, 49634536], [-74473708766, -3051970432, -77343276, 8157498, 26281114, 45071196, -7244630, 30147260], [-94257686715, 11110314952, 199678565, 11878, 45402478, 6533693, 33559147, -38776214], [1462014297, -11277402448, -15250743, -4241749, -64113329, 6489865, -29023694, -3795664], [-74836421029, -21787847262, 161785798, -4830872, -7795436, -7103734, 12301115, -79432720], [-368555941572, -10990950658, -742260322, -13718883, 57160524, 72711207, -23889579, -22140637], [-354605970188, 5077424452, -675225344, 10260678, 68470612, -22751822, 4600678, 39800494], [-148254136894, -24564397188, -381709692, 4484540, -97365974, 21525606, 12332532, -17418218], [-304069843422, 10317990744, -411956656, -16578336, -62387480, 14304312, -27262032, -82443956], [-255500421520, 678545022, 302732201, 10122793, 90442891, -48639957, 20089219, 67203184], [-170784774410, 15264501722, 82444417, 26610039, -63088043, 29244463, -15601873, -72667344], [233903811822, 943236464, 1079187214, 9945796, -47011468, -1452162, -22902390, 98457060], [243543131944, -36680722978, 200027111, 37480901, 72340321, 39955615, 15329831, -37792714], [6907270665, 38017565896, 865867822, -21624161, -104048758, -11121413, -2651164, -88711157], [559770641082, -14937400778, 10308025, -68985677, 60987431, 33414353, 13039669, 65404384], [-458671859656, -24567003516, -74730309, -52399738, 29958801, -36210576, 48405216, 13057923], [165950935998, 24579970410, 2572408125, 10965765, -8264195, -5897671, 456879, 68571422], [-482267958804, 32212996958, 799470047, 79276551, -12223191, 60585041, -35322039, 482978], [14508010582, -42951938384, -1367946346, -40002240, 83663668, 23008618, -43963914, 50097236], [-391592187382, -36678974040, 620511866, -18381404, 112943650, 8223804, 9837724, 19490254], [313779457622, -47336309188, -1209591641, -63500176, -123834499, -50802878, -87929322, 183385123], [-201513196211, 18460740308, -626361226, -14866821, -52572450, -113971021, -111754800, 233769745], [598752703596, -50156968896, 3369516360, -9369284, 47955916, 139542144, -6910528, -60750524], [-368633676200, 11033183768, -128388046, 113397280, -158602234, 99589844, -9839712, -97219866], [-298627653264, -65498156250, 594051845, -25897233, 126217745, -20673297, 8447563, 114859198], [-871515913911, 6199922592, -1584234233, 90658759, -46471179, -114947643, 88017758, 5174678], [1226959977004, 28490102576, -1181212124, 4756448, 68487212, 124458208, 138328688, 4146724], [-790644859506, -83018573632, -955351101, -24574731, -102906010, -98449666, -36456525, -93041855], [1165239437231, 3877156996, -1042048051, 20183286, -257377398, -230146783, -6152251, 41266462], [-1652760520396, 9142120724, -744725260, -5205532, 134462872, -244582056, -77200174, 56977816], [-552808537370, -34348409900, -430620404, 68110440, -39466240, 284720420, 56203630, -220182520], [-1091156102882, -79803759268, -3347141463, 19883593, -154696794, 104275608, 96498349, -69332861], [-151550668403, 79522182156, 167979854, -70573091, 335856520, 64588911, -31140058, 156715301], [-115328127999, -649939410, 2027396015, 94366809, -91348540, -253164024, -175589234, 245639161], [-407502488586, 39649410598, -2982197381, 146974909, -328596361, -265595115, -19827521, 116696674], [878999376838, 3719837624, -2835686134, -116095914, -62493170, 19328794, 159127568, 26765104], [278540093986, 87511896464, -3974822456, 58794164, 224388742, -77342882, -78177510, 130806510], [306989694100, 26981868268, -3067874321, 113704186, 306666237, -35014084, 49115108, -389179501], [1463634065171, -76118696860, 298253474, 186146545, 132327286, -89650931, -34564672, 101686879], [2254531978991, -11707503536, -2694710860, -204189657, -137731882, -101896067, -116240818, 81185591], [-2766090665900, 80913127048, 1473504099, -172286604, -351922841, 294061368, 3863222, -305507723], [202308053194, 17205454052, -4785432374, -45141754, -46975414, -8913490, -106573434, 90705296], [1785509105070, 10991703468, -845134872, -150297524, -49352062, 33388242, 117180868, -177097646], [773211049552, -113281853030, -4144281092, -221682389, 291948798, 119706425, -63513229, 131921483], [458152393491, -44684832422, 2364545127, 124531657, -19135136, 21319012, 273565616, 317590101], [1422460351090, 63623180904, -4242388731, 164876834, -96811925, -7640272, 91535664, 90001047], [925271866282, 4169014272, -3074495088, 26467806, 448862206, -175313480, 93808446, 243508176], [2946324749036, 17423344012, -2699830260, 154739142, 892495132, 389190814, -89907410, 163924430], [3194993996502, 100934450684, 277501046, 43218452, -322260598, 439153640, 387378182, -916681558], [1663994922053, -838394224, -4986656009, -538416631, 178497483, -50395551, 9528012, -87709518], [2145161555796, -103523163318, 2062460205, -106026287, -241409839, 56989793, -129490611, 208160404], [3831363254306, -93050233750, 8990671974, 98638489, 205538640, -452898013, 227073323, -268756341], [-2414486468100, -26537583378, 5863423349, 291627997, -393920589, 300055659, -269697515, -938681330], [927202644950, 122329189932, -302459645, 510742407, -467800874, -208051336, 294628055, -456126335], [5334317495240, 99718771548, -3346424234, -82752136, -238613522, -974458080, -69188032, 78336244], [2964784441276, 114533860128, -8867137132, -274314634, -132156536, -1161512422, -76798088, 307552542], [4099048337120, -58070953116, -4847413076, 163838562, -241439682, 737999104, 227238444, 340620452], [1770310285126, 136298794120, -3143487128, -109204760, 194309030, -1255215310, -559307930, 845376570], [7671257011460, -103831209328, 2477412220, 163137586, -593021424, -522849746, -404480648, 937771466], [3315225691118, -68808544400, -2468084634, -104416602, -427422318, 371262474, 576801564, -1801521308], [-1729480513583, 283172965710, 9089559813, -242202809, -834324780, 300090004, 417590844, -229253401], [8702741845334, -30058902528, 12812512903, 193706162, -371612251, -63460440, 77771244, -857414023], [4734033378872, -352823538014, 2506619631, -175262061, 797841113, 569371909, 288653177, -240010528], [2363270608218, 126316522490, 7207862784, 123276353, -1602060086, 770157151, 8861179, -576366063], [-3809382326974, 253070637136, 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96249740576, 361773625072, -46432425800, 52166409412, 84621061096], [2205069388707754, -26290453535354, 3216247613864, -7327282773, -229959183242, -169059712067, -22283648987, 343613782019], [3845227096587632, -34466635849644, -852612911538, -58725556518, 190740122250, -109028274946, -71620325718, 100643202436], [1946601918220187, -11967107192338, -1660401164137, -29156414707, -45006384194, 485645678458, 227942986124, -275078519261], [735756677253438, -131399829779904, -6229716107224, -102216452888, 69789057406, 33888938258, -65025036410, -234814480970]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_315_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_315_10_a_o();".
// 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_10_a_o(:prec:=8)
    chi := MakeCharacter_315_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 10));
    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_315_10_a_o();".
// 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_10_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_315_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,10,sign)));
    Vf := Kernel([<2,R![39425937408, 5879385600, -557113472, -37060000, 2205048, 56500, -2670, -25, 1]>],Snew);
    return Vf;
end function;