
// Make newform 115.4.a.c in Magma, downloaded from the LMFDB on 02 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_115_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_115_4_a_c();".
// 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_115_4_a_c();".
// 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 := [-27, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_115_a();"
function MakeCharacter_115_a()
    N := 115;
    order := 1;
    char_gens := [47, 51];
    v := [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_115_a_Hecke( : Kf := Kf)
    return MakeCharacter_115_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 := 4;
    raw_aps := [[-3, 0], [-1, -1], [5, 0], [-2, 5], [-11, -5], [-6, -3], [-43, 7], [-42, 13], [-23, 0], [-220, 10], [-144, -17], [-20, -28], [-291, -11], [320, 8], [228, -74], [-210, -44], [18, -54], [25, -57], [68, -76], [-563, 97], [6, 62], [260, 172], [-658, -144], [-200, 180], [771, -203], [-192, -222], [-203, 91], [696, -130], [296, -303], [-1530, -8], [246, 342], [2022, 8], [1156, 187], [962, 322], [-1340, 3], [1017, 181], [-1468, 190], [-808, -271], [-2112, 344], [-33, 567], [-32, -706], [-2184, -339], [-2502, 118], [334, 354], [-783, 353], [610, -112], [2878, -576], [1880, -784], [1992, 542], [-14, 144], [1558, 798], [-4160, -60], [-4328, -488], [-606, 27], [206, -458], [4183, -519], [-948, 1090], [1016, 635], [-2266, 388], [1052, 1042], [-1078, -1072], [-4282, 680], [-635, 1097], [4996, 48], [4868, -463], [1430, -1571], [4670, 86], [1820, 659], [3439, 1111], [1662, 1112], [728, 640], [-3738, -140], [3436, 1020], [2450, -1850], [3115, -1887], [-11160, 448], [5659, -1135], [-1171, 153], [8478, -594], [-10774, 827], [-3260, -1156], [-1752, 1021], [2088, 566], [-360, 491], [-1228, 907], [-9944, 659], [-9190, 557], [462, 976], [-5410, 138], [-7806, -440], [12366, 722], [5186, -248], [6354, -1618], [-9012, -786], [-7768, 52], [-3362, 1805], [-776, 2402], [-9418, 1252], [17038, 528], [-1504, 840], [-18948, 131], [-2704, 1504], [-10876, -1264], [9006, -528], [-2967, -3757], [11306, -436], [16787, 903], [-9414, 1806], [-17961, 1859], [-9550, 2435], [-3152, -1748], [3698, 3492], [-2375, 1391], [11694, 2195], [-2026, 3102], [7324, -4260], [2264, -1558], [-7734, 1020], [1918, 2277], [9504, 756], [-8739, -1601], [-9414, -712], [4082, -2672], [-8148, -847], [9640, -5030], [-6348, 4911], [12955, -2979], [28968, 893], [-10027, 1599], [8538, -4068], [1164, 912], [-24926, 2321], [2554, 2026], [2848, -5752], [16154, 2303], [-15726, -108], [-6288, -4648], [-2050, -2426], [10814, -5630], [23895, -105], [-2144, -5486], [-3982, -1488], [18386, -2124], [-27514, -670], [-25226, 4664], [9736, 1502], [-370, 8571], [-126, -348], [10980, 946], [-18930, -2502], [-11975, 4669], [3400, -700], [44292, -745], [-37332, 344], [25464, 3758], [-17616, -946], [8992, -5104], [12114, 2844], [22644, -3809], [-12931, -529], [-51508, -37], [10739, -6255], [-20786, -2634], [-24277, 793], [-40236, 2093], [-31755, 1347], [-26717, -4549], [15118, -1436]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_115_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_115_4_a_c();".
// 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_115_4_a_c(:prec:=2)
    chi := MakeCharacter_115_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_115_4_a_c();".
// 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_115_4_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_115_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![3, 1]>],Snew);
    return Vf;
end function;