
// Make newform 2304.3.e.j in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_e_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_2304_3_e_j();".
// 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_2304_3_e_j();".
// 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, 4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 9, 0, 3], [0, 15, 0, 3], [12, 0, 6, 0]];
    Rf_basisdens := [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_2304_e();"
function MakeCharacter_2304_e()
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    v := [2, 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_2304_e_Hecke();"
function MakeCharacter_2304_e_Hecke(Kf)
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 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, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, -1], [0, 2, 0, 0], [0, 0, 0, -1], [0, -5, 0, 0], [20, 0, 0, 0], [0, 0, -2, 0], [0, 0, 5, 0], [0, 0, 0, -5], [0, 0, 0, -4], [0, 17, 0, 0], [-40, 0, 0, 0], [0, 0, -10, 0], [0, 0, 5, 0], [0, 8, 0, 0], [0, 0, 0, 0], [100, 0, 0, 0], [0, 0, -10, 0], [-20, 0, 0, 0], [0, 0, 0, 5], [0, -30, 0, 0], [0, 3, 0, 0], [40, 0, 0, 0], [0, 0, -25, 0], [0, 0, 0, 9], [0, -40, 0, 0], [0, 0, 0, 5], [0, 15, 0, 0], [0, 0, 0, -1], [0, -12, 0, 0], [0, -25, 0, 0], [172, 0, 0, 0], [0, 0, 15, 0], [0, 0, 0, 15], [0, 0, 0, 16], [-160, 0, 0, 0], [0, 0, 8, 0], [0, 0, 5, 0], [0, -12, 0, 0], [0, 0, 0, 25], [0, 0, 40, 0], [-10, 0, 0, 0], [0, 0, 3, 0], [0, 0, 0, 5], [172, 0, 0, 0], [0, 0, 0, -1], [0, -10, 0, 0], [0, 0, 0, -25], [0, 25, 0, 0], [0, 0, -20, 0], [140, 0, 0, 0], [0, -58, 0, 0], [0, 5, 0, 0], [0, 0, 12, 0], [0, 0, 15, 0], [0, 0, 0, -45], [0, 0, 0, -31], [0, -33, 0, 0], [-80, 0, 0, 0], [0, 0, -43, 0], [-380, 0, 0, 0], [0, 0, 20, 0], [-310, 0, 0, 0], [0, 0, -33, 0], [-500, 0, 0, 0], [100, 0, 0, 0], [0, -10, 0, 0], [0, 0, 0, -20], [0, -145, 0, 0], [0, 0, -70, 0], [0, 0, 0, -21], [0, 0, 0, 64], [92, 0, 0, 0], [0, 0, -20, 0], [0, 0, 35, 0], [0, 0, 0, 4], [0, -53, 0, 0], [-368, 0, 0, 0], [0, -102, 0, 0], [0, 0, 0, 5], [0, 0, -50, 0], [-470, 0, 0, 0], [0, 0, 0, 15], [0, 130, 0, 0], [0, -163, 0, 0], [-680, 0, 0, 0], [0, 0, -45, 0], [0, 0, 0, -31], [0, 110, 0, 0], [0, 0, -30, 0], [0, 0, 0, 61], [0, 52, 0, 0], [640, 0, 0, 0], [0, 0, 10, 0], [0, 0, 85, 0], [0, 203, 0, 0], [20, 0, 0, 0], [0, 0, 0, -35], [-760, 0, 0, 0], [0, 0, 135, 0], [0, 50, 0, 0], [0, -113, 0, 0], [280, 0, 0, 0], [-650, 0, 0, 0], [0, -160, 0, 0], [0, 105, 0, 0], [0, 0, -50, 0], [490, 0, 0, 0], [0, 0, 0, 31], [0, 0, 0, 36], [0, -185, 0, 0], [-1112, 0, 0, 0], [0, 0, 0, -95], [0, 23, 0, 0], [80, 0, 0, 0], [0, 0, -70, 0], [0, 0, -147, 0], [0, 168, 0, 0], [0, 0, 0, -120], [-50, 0, 0, 0], [0, 0, -23, 0], [0, 10, 0, 0], [-400, 0, 0, 0], [0, 0, -135, 0], [0, 0, 0, 115], [0, 0, -90, 0], [0, 0, 0, 9], [0, 0, 0, 59], [-920, 0, 0, 0], [0, 0, -160, 0], [0, 0, 0, -5], [0, 0, 0, 39], [0, -243, 0, 0], [890, 0, 0, 0], [0, 0, -127, 0], [-100, 0, 0, 0], [0, 0, -55, 0], [0, 257, 0, 0], [-772, 0, 0, 0], [0, 0, -65, 0], [0, 0, 0, -109], [0, -180, 0, 0], [0, 0, 0, -35], [0, 0, -110, 0], [0, 0, 0, 96], [0, -135, 0, 0], [-568, 0, 0, 0], [0, 0, 112, 0], [0, 0, 0, 4], [0, -3, 0, 0], [-520, 0, 0, 0], [0, 0, -12, 0], [-40, 0, 0, 0], [0, 0, -160, 0], [0, 0, 0, -165], [0, -333, 0, 0], [470, 0, 0, 0], [0, 0, -85, 0], [0, 80, 0, 0], [0, -125, 0, 0], [0, 0, 0, -99], [0, 378, 0, 0], [0, 325, 0, 0], [0, 0, -60, 0], [0, 0, 0, -45], [0, 0, 0, -24], [1600, 0, 0, 0], [0, 0, 45, 0], [0, 292, 0, 0], [0, 0, 0, -60], [0, 0, -20, 0], [-760, 0, 0, 0], [0, 0, 0, 25], [0, 183, 0, 0], [-248, 0, 0, 0], [0, 0, -145, 0], [0, 0, 0, 139], [0, 0, 0, 40], [0, 0, 0, 1], [0, 22, 0, 0], [0, 0, 0, -141], [0, 215, 0, 0], [0, 0, -180, 0], [0, 0, 285, 0], [0, 0, 0, 39], [1480, 0, 0, 0], [-1540, 0, 0, 0], [0, 0, 20, 0], [1640, 0, 0, 0], [0, -140, 0, 0], [808, 0, 0, 0], [0, 0, -85, 0], [0, 460, 0, 0], [0, 265, 0, 0], [-602, 0, 0, 0], [0, 0, 0, 91], [0, 175, 0, 0], [0, 0, 200, 0], [0, 0, -295, 0], [0, 0, 0, 175], [0, 0, 0, 161], [220, 0, 0, 0], [0, -202, 0, 0], [0, 0, -173, 0], [0, 0, 0, 205], [0, -490, 0, 0], [0, 113, 0, 0], [1160, 0, 0, 0], [-70, 0, 0, 0], [0, 0, -175, 0], [0, 0, 0, -129], [0, -540, 0, 0], [0, 0, -30, 0], [1642, 0, 0, 0], [0, 0, 0, -139], [0, -263, 0, 0], [0, 0, 300, 0], [0, 0, 185, 0], [0, 0, 0, 65], [0, 0, 0, 135], [0, -457, 0, 0], [0, 0, 0, -239], [0, 430, 0, 0], [0, 0, 0, 20], [0, 165, 0, 0], [0, 0, -90, 0], [0, 0, 0, -209], [0, -72, 0, 0], [0, 0, 0, 121], [2432, 0, 0, 0], [0, 0, 0, -235], [0, -7, 0, 0], [2480, 0, 0, 0], [0, 0, -338, 0], [-872, 0, 0, 0], [0, 0, 7, 0], [0, -362, 0, 0], [0, 0, 210, 0], [0, 10, 0, 0], [-740, 0, 0, 0], [0, 0, 0, -219], [0, 0, 0, 240], [0, 265, 0, 0], [0, 0, 70, 0], [0, 0, 0, 41], [0, -428, 0, 0], [1540, 0, 0, 0], [0, 0, 108, 0], [0, 0, 0, 141], [0, -503, 0, 0], [0, 0, -118, 0], [-832, 0, 0, 0], [0, 0, 195, 0], [0, -452, 0, 0], [0, 0, 0, 165], [1120, 0, 0, 0], [0, 0, 125, 0], [-2270, 0, 0, 0], [0, 0, 0, 101], [0, 220, 0, 0], [0, 0, 0, 105], [0, 0, 0, -184], [0, -135, 0, 0], [-140, 0, 0, 0], [0, 0, 145, 0], [0, -487, 0, 0], [2780, 0, 0, 0], [0, 0, 383, 0], [0, 0, 0, -40], [-2360, 0, 0, 0], [-2560, 0, 0, 0], [0, 0, 0, 15], [1160, 0, 0, 0], [0, 0, 0, 231], [0, 90, 0, 0], [0, 0, 0, -140], [-3098, 0, 0, 0], [0, -392, 0, 0], [0, 0, 198, 0], [0, 0, 0, -125], [0, 0, 450, 0], [0, 0, 0, -300], [2420, 0, 0, 0], [0, 0, -360, 0], [3260, 0, 0, 0], [0, 0, 23, 0], [0, 0, 0, -45], [0, 127, 0, 0], [0, 0, 205, 0], [0, -520, 0, 0], [0, 635, 0, 0], [0, 548, 0, 0], [0, 0, 0, 216], [0, 0, -165, 0], [0, 0, 0, -25], [0, 0, -465, 0], [0, 378, 0, 0], [140, 0, 0, 0], [-730, 0, 0, 0], [0, 0, 403, 0], [0, 0, 0, 185]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_e_j();".
// 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_2304_3_e_j(:prec:=4)
    chi := MakeCharacter_2304_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2304_3_e_j();".
// 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_2304_3_e_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![54, 0, 1]>,<7,R![-108, 0, 1]>,<13,R![-108, 0, 1]>,<19,R![-20, 1]>,<31,R![-2700, 0, 1]>],Snew);
    return Vf;
end function;