
// Make newform 2304.3.h.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_h_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_h_a();".
// 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_h_a();".
// 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];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 2, 0], [0, 1, 0, 1], [0, 1, 0, -1]];
    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_h();"
function MakeCharacter_2304_h()
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_h_Hecke();"
function MakeCharacter_2304_h_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, 0, -5], [-12, 0, 0, 0], [0, 0, 0, 4], [0, -4, 0, 0], [0, 0, -7, 0], [0, -8, 0, 0], [0, 0, -28, 0], [0, 0, 0, -21], [-4, 0, 0, 0], [0, -15, 0, 0], [0, 0, -15, 0], [0, 4, 0, 0], [0, 0, 12, 0], [0, 0, 0, -35], [0, 0, 0, -56], [0, -7, 0, 0], [0, -44, 0, 0], [0, 0, -20, 0], [80, 0, 0, 0], [100, 0, 0, 0], [0, 0, 0, -92], [0, 0, 105, 0], [-112, 0, 0, 0], [0, 0, 0, -45], [44, 0, 0, 0], [0, 0, 0, 0], [0, 68, 0, 0], [0, 0, 15, 0], [-20, 0, 0, 0], [0, 0, 0, 40], [0, 0, 63, 0], [0, -60, 0, 0], [0, 0, 0, -91], [196, 0, 0, 0], [0, 119, 0, 0], [0, -84, 0, 0], [0, 0, 120, 0], [0, 0, 0, -147], [0, 0, 0, -144], [0, 108, 0, 0], [0, 0, -264, 0], [-178, 0, 0, 0], [0, 0, 0, 147], [-196, 0, 0, 0], [0, -28, 0, 0], [28, 0, 0, 0], [0, 0, 0, 60], [0, -92, 0, 0], [0, 0, 215, 0], [0, 0, -160, 0], [384, 0, 0, 0], [0, 0, 0, 140], [0, 0, 65, 0], [0, 0, -280, 0], [0, 0, 0, -59], [140, 0, 0, 0], [0, 124, 0, 0], [0, 0, -119, 0], [0, -184, 0, 0], [0, 0, 0, -147], [0, -276, 0, 0], [0, 0, 24, 0], [222, 0, 0, 0], [0, 0, 0, -315], [0, 260, 0, 0], [48, 0, 0, 0], [0, 0, 0, -252], [0, 105, 0, 0], [0, 0, -193, 0], [0, 0, 420, 0], [172, 0, 0, 0], [0, -1, 0, 0], [0, 368, 0, 0], [0, 0, 0, 0], [0, 0, 0, 325], [0, 169, 0, 0], [0, 0, 105, 0], [-144, 0, 0, 0], [0, 0, 0, -540], [0, -340, 0, 0], [0, 0, -4, 0], [-482, 0, 0, 0], [756, 0, 0, 0], [0, 0, 0, 140], [0, 0, 55, 0], [-464, 0, 0, 0], [0, 0, 0, 35], [20, 0, 0, 0], [0, 0, 0, 492], [0, 0, 476, 0], [-388, 0, 0, 0], [0, 0, 0, 0], [0, 48, 0, 0], [0, 0, -300, 0], [0, 0, 0, 219], [0, 0, 511, 0], [0, -176, 0, 0], [0, 468, 0, 0], [0, 28, 0, 0], [0, 0, 0, 227], [0, 0, 0, 332], [0, 0, -289, 0], [0, -408, 0, 0], [242, 0, 0, 0], [0, 0, 0, 280], [0, 0, 145, 0], [0, 0, 420, 0], [-286, 0, 0, 0], [-100, 0, 0, 0], [0, -385, 0, 0], [0, 0, -615, 0], [0, -128, 0, 0], [-196, 0, 0, 0], [0, 0, -455, 0], [0, 60, 0, 0], [0, 0, -500, 0], [0, 0, 0, 93], [0, 0, 0, -840], [0, 65, 0, 0], [610, 0, 0, 0], [0, 0, 0, 427], [0, 0, 0, 700], [0, 316, 0, 0], [0, 0, 0, -21], [0, -28, 0, 0], [0, 0, 436, 0], [-372, 0, 0, 0], [0, -260, 0, 0], [0, 520, 0, 0], [0, 0, 440, 0], [-180, 0, 0, 0], [0, -76, 0, 0], [0, 0, 385, 0], [-526, 0, 0, 0], [0, 0, 0, 1005], [0, 536, 0, 0], [0, 0, 0, 1085], [0, 0, -95, 0], [0, -272, 0, 0], [0, 0, 0, -621], [868, 0, 0, 0], [0, 0, 0, -1000], [0, -604, 0, 0], [0, 0, 756, 0], [0, 641, 0, 0], [0, 0, -735, 0], [0, 516, 0, 0], [0, 0, 40, 0], [0, -231, 0, 0], [0, 0, -175, 0], [0, 4, 0, 0], [0, 0, -632, 0], [0, 100, 0, 0], [0, 0, -56, 0], [884, 0, 0, 0], [0, 0, 761, 0], [-78, 0, 0, 0], [0, 0, 0, -189], [0, 0, 0, 408], [0, 0, -945, 0], [-20, 0, 0, 0], [0, 0, 0, -84], [0, 0, -297, 0], [0, 0, -840, 0], [-116, 0, 0, 0], [0, -65, 0, 0], [1616, 0, 0, 0], [0, 0, 0, 427], [0, 0, 0, -336], [0, -57, 0, 0], [0, 0, -896, 0], [32, 0, 0, 0], [116, 0, 0, 0], [0, 0, 1225, 0], [0, -928, 0, 0], [0, 0, 0, -459], [308, 0, 0, 0], [0, 727, 0, 0], [212, 0, 0, 0], [0, 0, 0, 700], [0, -444, 0, 0], [0, 0, 1113, 0], [0, 0, -528, 0], [0, 0, 0, -21], [0, 476, 0, 0], [0, -936, 0, 0], [-1696, 0, 0, 0], [0, 0, 984, 0], [-1472, 0, 0, 0], [0, 0, 0, 720], [0, 152, 0, 0], [0, 0, 0, 1379], [0, 0, 0, 168], [0, 0, -1337, 0], [850, 0, 0, 0], [0, 140, 0, 0], [0, 0, 1583, 0], [0, 0, 1008, 0], [0, 0, 0, 445], [188, 0, 0, 0], [0, 812, 0, 0], [-560, 0, 0, 0], [0, 0, 0, 540], [0, 0, 0, -843], [644, 0, 0, 0], [0, 0, 0, -828], [0, 0, 1015, 0], [0, 868, 0, 0], [978, 0, 0, 0], [0, 0, 0, -1029], [1412, 0, 0, 0], [0, 0, 0, -1120], [0, 0, 1204, 0], [-110, 0, 0, 0], [1988, 0, 0, 0], [0, 0, -329, 0], [0, 0, 0, 0], [0, 0, 0, -1203], [0, 428, 0, 0], [1476, 0, 0, 0], [0, 0, 201, 0], [-2380, 0, 0, 0], [0, 0, 0, 540], [0, -847, 0, 0], [0, 0, -777, 0], [0, 0, -1116, 0], [-492, 0, 0, 0], [0, 0, 0, -1304], [0, -1164, 0, 0], [0, 140, 0, 0], [-316, 0, 0, 0], [0, 0, -409, 0], [0, 780, 0, 0], [0, 0, 1460, 0], [-1360, 0, 0, 0], [0, 0, 0, -635], [0, 0, 0, 1316], [0, 0, -820, 0], [0, 0, 0, -228], [0, -748, 0, 0], [-52, 0, 0, 0], [0, -313, 0, 0], [0, 0, -63, 0], [0, 0, 1036, 0], [-1772, 0, 0, 0], [0, 0, 0, -696], [0, -332, 0, 0], [0, 0, 392, 0], [0, -780, 0, 0], [0, 0, 609, 0], [0, 0, 1180, 0], [1184, 0, 0, 0], [0, 0, 0, 693], [0, 0, 0, 840], [0, 908, 0, 0], [0, 1360, 0, 0], [0, 0, 0, -195], [2450, 0, 0, 0], [2692, 0, 0, 0], [0, 0, 0, -1008], [0, -1220, 0, 0], [0, 759, 0, 0], [0, 0, -137, 0], [0, -524, 0, 0], [0, 0, 0, -1331], [0, 0, 1735, 0], [0, -280, 0, 0], [0, 0, 0, 53], [0, 617, 0, 0], [0, -728, 0, 0], [-640, 0, 0, 0], [-2084, 0, 0, 0], [-1568, 0, 0, 0], [-1628, 0, 0, 0], [0, 0, 0, -2028], [0, 793, 0, 0], [446, 0, 0, 0], [0, 0, 0, 560], [0, 0, -140, 0], [-1180, 0, 0, 0], [0, 0, 92, 0], [0, 767, 0, 0], [0, 56, 0, 0], [0, 0, 1024, 0], [-2272, 0, 0, 0], [0, 0, 0, 405], [-1340, 0, 0, 0], [0, 0, 2015, 0], [0, 0, 0, -259], [0, 0, 0, 2520], [0, 0, 905, 0], [0, 0, 0, -1680], [0, -969, 0, 0], [0, 0, 0, -1165], [3404, 0, 0, 0], [0, 0, 0, 1155], [0, 0, 0, -684], [0, -200, 0, 0], [-3710, 0, 0, 0], [0, 0, 0, 2515], [-2764, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_h_a();".
// 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_h_a(:prec:=4)
    chi := MakeCharacter_2304_h();
    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_h_a();".
// 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_h_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-50, 0, 1]>,<7,R![12, 1]>],Snew);
    return Vf;
end function;