
// Make newform 475.3.c.f in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_475_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_475_c_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_475_3_c_f();".
// 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_475_3_c_f();".
// 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 := [28561, 0, -7098, 0, 771, 0, -42, 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, 4901, 0, -771, 0, 42, 0, -1], [0, 9295, 0, -771, 0, 42, 0, -1], [0, 10985, 0, -2809, 0, 251, 0, -10], [3549, 0, -602, 0, 42, 0, -1, 0], [-7098, 0, 940, 0, -42, 0, 1, 0], [0, 93457, 0, -16919, 0, 1259, 0, -34], [-32955, 0, 7424, 0, -628, 0, 23, 0]];
    Rf_basisdens := [1, 2197, 2197, 4394, 338, 169, 4394, 338];
    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_475_c();"
function MakeCharacter_475_c()
    N := 475;
    order := 2;
    char_gens := [77, 401];
    v := [2, 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_475_c_Hecke();"
function MakeCharacter_475_c_Hecke(Kf)
    N := 475;
    order := 2;
    char_gens := [77, 401];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 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, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 3, 0, 0, 0], [0, -5, 0, 4, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, -1, 0], [-6, 0, 0, 0, -7, 1, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, -1], [0, 0, 0, 0, 0, -1, 0, 1], [0, 12, 0, -12, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 3], [0, 0, -2, 0, 0, 0, -3, 0], [0, 0, -2, 0, 0, 0, -2, 0], [0, -19, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 0, 4], [22, 0, 0, 0, 1, 0, 0, 0], [0, -8, 0, -19, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, 0, -1], [0, 0, -7, 0, 0, 0, -5, 0], [0, 0, 0, 0, 0, 4, 0, -6], [0, 0, 12, 0, 0, 0, 4, 0], [0, 0, 0, 0, 0, 1, 0, 3], [0, 0, 0, -10, 0, 0, 0, 0], [52, 0, 0, 0, 105, 0, 0, 0], [0, 45, 0, -5, 0, 0, 0, 0], [0, 4, 0, 47, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 3], [0, 35, 0, 17, 0, 0, 0, 0], [0, 38, 0, 30, 0, 0, 0, 0], [-16, 0, 0, 0, 98, 0, 0, 0], [0, 0, 9, 0, 0, 0, -13, 0], [-2, 0, 0, 0, 126, 0, 0, 0], [74, 0, 0, 0, -46, 0, 0, 0], [0, 0, 0, 0, 0, 6, 0, 8], [0, 0, 14, 0, 0, 0, 12, 0], [0, 0, -16, 0, 0, 0, 17, 0], [0, -19, 0, 33, 0, 0, 0, 0], [0, -52, 0, -8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 16, 0, -4], [0, 0, 0, 0, 0, 2, 0, -6], [-131, 0, 0, 0, -29, 0, 0, 0], [0, -118, 0, -16, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, 3, 0], [139, 0, 0, 0, -109, 0, 0, 0], [0, 0, 0, 0, 0, 3, 0, 12], [0, 24, 0, -48, 0, 0, 0, 0], [0, -2, 0, 89, 0, 0, 0, 0], [-244, 0, 0, 0, 16, 0, 0, 0], [0, 0, -12, 0, 0, 0, -26, 0], [63, 0, 0, 0, 241, 0, 0, 0], [0, 0, 0, 0, 0, -8, 0, -6], [222, 0, 0, 0, -27, 0, 0, 0], [0, 95, 0, -93, 0, 0, 0, 0], [0, 0, -8, 0, 0, 0, 18, 0], [0, 0, 0, 0, 0, -10, 0, 4], [65, 0, 0, 0, 41, 0, 0, 0], [0, 0, -26, 0, 0, 0, 8, 0], [0, 0, 0, 0, 0, -11, 0, 19], [0, 0, -8, 0, 0, 0, 18, 0], [0, 129, 0, -62, 0, 0, 0, 0], [0, -78, 0, -28, 0, 0, 0, 0], [-55, 0, 0, 0, 165, 0, 0, 0], [0, 0, -69, 0, 0, 0, 11, 0], [0, 39, 0, 110, 0, 0, 0, 0], [0, 0, 0, 0, 0, -23, 0, -14], [0, -29, 0, -133, 0, 0, 0, 0], [0, 0, -36, 0, 0, 0, 14, 0], [-2, 0, 0, 0, 28, 0, 0, 0], [0, 0, 45, 0, 0, 0, -11, 0], [-371, 0, 0, 0, 11, 0, 0, 0], [0, 0, 10, 0, 0, 0, -36, 0], [0, -41, 0, -118, 0, 0, 0, 0], [0, 0, 0, 0, 0, 11, 0, -12], [0, -131, 0, -65, 0, 0, 0, 0], [-288, 0, 0, 0, 3, 0, 0, 0], [0, 0, -58, 0, 0, 0, -16, 0], [0, 0, 0, 0, 0, 5, 0, -21], [0, 0, 0, 0, 0, -39, 0, -7], [376, 0, 0, 0, -105, 0, 0, 0], [0, 0, 0, 0, 0, -26, 0, 15], [0, 0, 0, 0, 0, 1, 0, 9], [0, -239, 0, -15, 0, 0, 0, 0], [0, 0, 0, 0, 0, 23, 0, 7], [0, 0, -8, 0, 0, 0, 43, 0], [0, 0, 0, 0, 0, -10, 0, 16], [0, 0, 3, 0, 0, 0, 9, 0], [182, 0, 0, 0, 361, 0, 0, 0], [0, 0, 14, 0, 0, 0, -56, 0], [0, 0, 64, 0, 0, 0, 13, 0], [302, 0, 0, 0, -8, 0, 0, 0], [0, -46, 0, -94, 0, 0, 0, 0], [-168, 0, 0, 0, -283, 0, 0, 0], [-144, 0, 0, 0, 481, 0, 0, 0], [0, 0, 13, 0, 0, 0, -26, 0], [0, 0, 0, 0, 0, -26, 0, -14], [0, 0, 0, 0, 0, -24, 0, 42], [0, 98, 0, 39, 0, 0, 0, 0], [-182, 0, 0, 0, -81, 0, 0, 0], [0, 204, 0, -188, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -72, 0], [0, -246, 0, 116, 0, 0, 0, 0], [0, 0, 0, 0, 0, -14, 0, 20], [-770, 0, 0, 0, 237, 0, 0, 0], [0, 0, 13, 0, 0, 0, 91, 0], [0, 0, -50, 0, 0, 0, -5, 0], [0, 0, -54, 0, 0, 0, -26, 0], [0, 0, 0, 0, 0, 49, 0, 9], [0, 0, 0, 0, 0, -51, 0, 13], [0, -86, 0, -12, 0, 0, 0, 0], [0, 0, 40, 0, 0, 0, -15, 0], [0, 0, -24, 0, 0, 0, 6, 0], [840, 0, 0, 0, 1, 0, 0, 0], [216, 0, 0, 0, 342, 0, 0, 0], [0, 0, 0, 0, 0, -37, 0, -31], [0, 0, -30, 0, 0, 0, 18, 0], [0, 0, -5, 0, 0, 0, 36, 0], [0, 0, 2, 0, 0, 0, -26, 0], [0, 0, 0, 0, 0, -15, 0, -32], [0, 0, 0, 0, 0, 48, 0, -13], [0, -112, 0, 290, 0, 0, 0, 0], [0, -349, 0, 210, 0, 0, 0, 0], [0, 30, 0, -12, 0, 0, 0, 0], [-420, 0, 0, 0, -116, 0, 0, 0], [-576, 0, 0, 0, -202, 0, 0, 0], [-280, 0, 0, 0, -374, 0, 0, 0], [-799, 0, 0, 0, 307, 0, 0, 0], [0, 0, -69, 0, 0, 0, -30, 0], [0, 0, 52, 0, 0, 0, -84, 0], [32, 0, 0, 0, -371, 0, 0, 0], [0, 5, 0, 121, 0, 0, 0, 0], [0, 0, 0, 0, 0, -35, 0, -13], [0, 0, 100, 0, 0, 0, -32, 0], [-175, 0, 0, 0, -606, 0, 0, 0], [-421, 0, 0, 0, -100, 0, 0, 0], [0, 211, 0, 54, 0, 0, 0, 0], [0, -88, 0, 445, 0, 0, 0, 0], [0, 227, 0, -306, 0, 0, 0, 0], [603, 0, 0, 0, -386, 0, 0, 0], [0, 0, 0, 0, 0, 43, 0, 18], [262, 0, 0, 0, -556, 0, 0, 0], [0, 0, 89, 0, 0, 0, 48, 0], [0, 42, 0, 137, 0, 0, 0, 0], [0, 0, 0, 0, 0, -24, 0, -17], [0, 0, 0, 0, 0, -68, 0, 36], [0, 0, 32, 0, 0, 0, 90, 0], [0, 274, 0, 346, 0, 0, 0, 0], [-408, 0, 0, 0, 519, 0, 0, 0], [0, -213, 0, 181, 0, 0, 0, 0], [0, -311, 0, 12, 0, 0, 0, 0], [-1380, 0, 0, 0, -22, 0, 0, 0], [0, 0, 132, 0, 0, 0, -12, 0], [0, -317, 0, -19, 0, 0, 0, 0], [0, -248, 0, 281, 0, 0, 0, 0], [0, 0, 0, 0, 0, 36, 0, -52], [-285, 0, 0, 0, 623, 0, 0, 0], [-385, 0, 0, 0, -402, 0, 0, 0], [0, 0, -3, 0, 0, 0, 27, 0], [0, 0, 0, 0, 0, -4, 0, 31], [0, 0, -28, 0, 0, 0, -6, 0], [0, -83, 0, 105, 0, 0, 0, 0], [0, 0, -78, 0, 0, 0, -96, 0], [0, 0, 0, 0, 0, -24, 0, -18], [0, -129, 0, -83, 0, 0, 0, 0], [0, 16, 0, -292, 0, 0, 0, 0], [0, 0, 0, 0, 0, 79, 0, -29], [0, 0, -150, 0, 0, 0, -3, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_475_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_475_3_c_f();".
// 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_475_3_c_f(:prec:=8)
    chi := MakeCharacter_475_c();
    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_475_3_c_f();".
// 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_475_3_c_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_475_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![17, 0, 10, 0, 1]>,<7,R![2201, 0, -94, 0, 1]>],Snew);
    return Vf;
end function;