
// Make newform 525.3.s.g in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_525_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_525_s_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_525_3_s_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_525_3_s_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 := [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, 2, 0, 1, 0, -1, 0, 1], [0, -1, 0, 2, 0, 2, 0, -1], [0, 1, 0, 1, 0, 1, 0, -2], [0, 1, 0, -1, 0, 1, 0, 2]];
    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_525_s();"
function MakeCharacter_525_s()
    N := 525;
    order := 6;
    char_gens := [176, 127, 451];
    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_525_s_Hecke();"
function MakeCharacter_525_s_Hecke(Kf)
    N := 525;
    order := 6;
    char_gens := [176, 127, 451];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 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, 1, 0, 0, -1, 0, 0, 0], [0, 1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 5, 0, 0, 0, 0], [0, 0, 2, 0, 0, 2, 0, 0], [0, -10, 0, 5, 4, 0, 0, -2], [0, -6, 0, -6, -2, 0, 0, -2], [-5, 0, -5, 0, 0, -6, -6, 0], [0, 20, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 0, -12, 0], [0, 0, 0, 0, 0, 8, -16, 0], [0, -25, 0, 0, 10, 0, 0, 0], [10, 0, -20, 0, 0, 28, -14, 0], [0, 0, 0, 50, 0, 0, 0, -4], [0, -24, 0, 48, 6, 0, 0, -12], [0, 22, 0, -22, -30, 0, 0, 30], [8, 0, -4, 0, 0, -18, 36, 0], [-39, 0, -39, 0, 0, -6, -6, 0], [0, 1, 0, -1, -38, 0, 0, 38], [-26, 0, 0, 0, 0, 0, -20, 0], [0, -15, 0, -15, 4, 0, 0, 4], [-19, 0, 19, 0, 0, 16, -16, 0], [0, 48, 0, -24, 20, 0, 0, -10], [12, 0, 12, 0, 0, 10, 10, 0], [0, -34, 0, 17, -48, 0, 0, 24], [36, 0, -18, 0, 0, -2, 4, 0], [0, 41, 0, -82, -4, 0, 0, 8], [0, 166, 0, 0, -6, 0, 0, 0], [0, 0, -49, 0, 0, -6, 0, 0], [0, 0, 0, 70, 0, 0, 0, 32], [0, 0, 0, 151, 0, 0, 0, -4], [20, 0, 20, 0, 0, -4, -4, 0], [0, -244, 0, 244, 12, 0, 0, -12], [47, 0, -94, 0, 0, 68, -34, 0], [10, 0, -10, 0, 0, -86, 86, 0], [0, 0, -107, 0, 0, 52, 0, 0], [0, -29, 0, -29, -2, 0, 0, -2], [0, -119, 0, 0, -66, 0, 0, 0], [0, -20, 0, 10, 68, 0, 0, -34], [0, 110, 0, -220, 12, 0, 0, -24], [0, 0, -134, 0, 0, -24, 0, 0], [-44, 0, 88, 0, 0, 56, -28, 0], [146, 0, -146, 0, 0, -54, 54, 0], [0, 26, 0, -26, 36, 0, 0, -36], [0, 0, 0, 62, 0, 0, 0, 32], [-46, 0, 23, 0, 0, -56, 112, 0], [137, 0, 0, 0, 0, 0, 30, 0], [0, 6, 0, -3, 144, 0, 0, -72], [0, 98, 0, 98, 24, 0, 0, 24], [-21, 0, -21, 0, 0, 74, 74, 0], [0, 98, 0, 0, 78, 0, 0, 0], [-64, 0, 0, 0, 0, 0, -72, 0], [-202, 0, 101, 0, 0, -52, 104, 0], [-8, 0, 16, 0, 0, 60, -30, 0], [0, -104, 0, 208, 2, 0, 0, -4], [0, -176, 0, 176, -80, 0, 0, 80], [64, 0, -32, 0, 0, 78, -156, 0], [-140, 0, -140, 0, 0, -48, -48, 0], [0, -101, 0, 101, 174, 0, 0, -174], [82, 0, 0, 0, 0, 0, -98, 0], [0, -21, 0, -21, 102, 0, 0, 102], [0, -52, 0, 26, -112, 0, 0, 56], [0, -96, 0, 48, -56, 0, 0, 28], [20, 0, -10, 0, 0, -52, 104, 0], [0, 44, 0, -88, 32, 0, 0, -64], [0, -218, 0, 0, 84, 0, 0, 0], [475, 0, -475, 0, 0, -30, 30, 0], [0, 0, 0, -122, 0, 0, 0, 132], [0, -82, 0, 82, 98, 0, 0, -98], [-124, 0, 248, 0, 0, 80, -40, 0], [0, 234, 0, 234, -58, 0, 0, -58], [-296, 0, 296, 0, 0, 118, -118, 0], [0, 196, 0, 196, -40, 0, 0, -40], [0, 199, 0, 0, 154, 0, 0, 0], [211, 0, 0, 0, 0, 0, -2, 0], [0, 10, 0, -20, 44, 0, 0, -88], [0, 0, 244, 0, 0, 78, 0, 0], [0, 92, 0, -184, -56, 0, 0, 112], [80, 0, -80, 0, 0, -202, 202, 0], [-274, 0, 137, 0, 0, -64, 128, 0], [124, 0, -248, 0, 0, -212, 106, 0], [-631, 0, 0, 0, 0, 0, -38, 0], [0, 0, 290, 0, 0, 132, 0, 0], [0, 512, 0, -256, 0, 0, 0, 0], [165, 0, 165, 0, 0, -20, -20, 0], [0, -610, 0, 0, 72, 0, 0, 0], [-304, 0, 0, 0, 0, 0, -86, 0], [0, 401, 0, 0, -52, 0, 0, 0], [-216, 0, 432, 0, 0, -168, 84, 0], [0, 0, 0, -223, 0, 0, 0, -180], [0, 2, 0, -4, 48, 0, 0, -96], [-392, 0, 196, 0, 0, 18, -36, 0], [0, -134, 0, 134, 68, 0, 0, -68], [562, 0, 0, 0, 0, 0, 120, 0], [-103, 0, 103, 0, 0, 102, -102, 0], [0, -452, 0, 226, -136, 0, 0, 68], [450, 0, 450, 0, 0, -28, -28, 0], [592, 0, -296, 0, 0, -8, 16, 0], [0, -196, 0, 392, 68, 0, 0, -136], [-377, 0, 377, 0, 0, 118, -118, 0], [0, 0, 0, -146, 0, 0, 0, 368], [0, -838, 0, 838, -56, 0, 0, 56], [0, -444, 0, -444, -70, 0, 0, -70], [-530, 0, 530, 0, 0, 142, -142, 0], [0, 0, -275, 0, 0, 150, 0, 0], [0, -100, 0, -100, 136, 0, 0, 136], [0, 832, 0, -416, -120, 0, 0, 60], [0, 254, 0, -508, 56, 0, 0, -112], [0, 0, -74, 0, 0, 290, 0, 0], [11, 0, -22, 0, 0, -128, 64, 0], [0, -25, 0, 50, -60, 0, 0, 120], [0, 326, 0, -326, -136, 0, 0, 136], [0, 0, 0, 50, 0, 0, 0, 84], [136, 0, -68, 0, 0, -176, 352, 0], [197, 0, 0, 0, 0, 0, 132, 0], [0, 0, 296, 0, 0, 172, 0, 0], [0, -210, 0, 105, -508, 0, 0, 254], [0, 136, 0, 136, -170, 0, 0, -170], [0, 314, 0, 0, -366, 0, 0, 0], [668, 0, 0, 0, 0, 0, -30, 0], [258, 0, -129, 0, 0, 34, -68, 0], [0, 0, 0, -211, 0, 0, 0, -168], [0, 284, 0, -568, -140, 0, 0, 280], [0, 220, 0, -220, 106, 0, 0, -106], [-393, 0, -393, 0, 0, -150, -150, 0], [160, 0, 0, 0, 0, 0, -102, 0], [-589, 0, 589, 0, 0, 54, -54, 0], [-74, 0, -74, 0, 0, -46, -46, 0], [0, 834, 0, -417, 136, 0, 0, -68], [0, -137, 0, 274, 14, 0, 0, -28], [0, 0, 467, 0, 0, -274, 0, 0], [0, 0, 0, 508, 0, 0, 0, 72], [229, 0, -229, 0, 0, 300, -300, 0], [0, 0, 0, 121, 0, 0, 0, -114], [390, 0, 390, 0, 0, 98, 98, 0], [108, 0, -216, 0, 0, -32, 16, 0], [0, 268, 0, 268, 208, 0, 0, 208], [0, -391, 0, -391, -122, 0, 0, -122], [0, 920, 0, -460, 88, 0, 0, -44], [0, 0, 580, 0, 0, 208, 0, 0], [-619, 0, 1238, 0, 0, 196, -98, 0], [302, 0, -302, 0, 0, 258, -258, 0], [0, -331, 0, 331, -60, 0, 0, 60], [0, 0, 0, -712, 0, 0, 0, 44], [-474, 0, 237, 0, 0, 18, -36, 0], [812, 0, -1624, 0, 0, -104, 52, 0], [0, 1440, 0, -720, 72, 0, 0, -36], [0, 282, 0, 282, -80, 0, 0, -80], [132, 0, 132, 0, 0, -40, -40, 0], [0, -418, 0, 0, 62, 0, 0, 0], [0, -787, 0, 0, -126, 0, 0, 0], [-246, 0, 492, 0, 0, 376, -188, 0], [0, 0, 0, 857, 0, 0, 0, 94], [0, -542, 0, 1084, -64, 0, 0, 128], [0, 523, 0, -523, 438, 0, 0, -438], [-644, 0, 0, 0, 0, 0, -444, 0], [-274, 0, 274, 0, 0, -244, 244, 0], [-32, 0, -32, 0, 0, -310, -310, 0], [0, -760, 0, 380, -184, 0, 0, 92], [244, 0, -122, 0, 0, -120, 240, 0], [0, -614, 0, 0, 468, 0, 0, 0], [0, 0, 0, 1012, 0, 0, 0, 258], [0, 0, 0, 127, 0, 0, 0, -192], [576, 0, 576, 0, 0, -94, -94, 0], [0, -904, 0, 904, 354, 0, 0, -354], [0, 194, 0, 194, -186, 0, 0, -186], [0, 0, -542, 0, 0, -424, 0, 0], [0, 37, 0, 37, -102, 0, 0, -102]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_525_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_525_3_s_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_525_3_s_g(:prec:=8)
    chi := MakeCharacter_525_s();
    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_525_3_s_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_525_3_s_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_525_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![625, 0, -350, 0, 171, 0, -14, 0, 1]>,<11,R![400, 80, 36, -4, 1]>,<13,R![9, 0, -294, 0, 1]>],Snew);
    return Vf;
end function;