
// Make newform 7098.2.a.s in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7098_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_7098_2_a_s();".
// 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_7098_2_a_s();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_7098_a();"
function MakeCharacter_7098_a()
    N := 7098;
    order := 1;
    char_gens := [4733, 5071, 6931];
    v := [1, 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_7098_a_Hecke(Kf)
    return MakeCharacter_7098_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+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 := 2;
    raw_aps := [[1], [-1], [-1], [-1], [1], [0], [-1], [1], [-3], [9], [4], [-9], [-8], [-7], [8], [-10], [-6], [11], [12], [-6], [11], [-12], [-6], [12], [2], [2], [5], [-2], [-7], [-2], [-16], [-3], [-15], [-12], [-6], [1], [-21], [-20], [-11], [-14], [18], [2], [-11], [-12], [2], [-5], [-5], [8], [12], [2], [18], [8], [10], [-11], [-14], [16], [-20], [-4], [-16], [-22], [14], [-26], [-8], [-8], [-14], [12], [8], [-17], [0], [-4], [16], [30], [-8], [-14], [-16], [-35], [-2], [4], [14], [-27], [23], [-22], [18], [-4], [33], [-6], [-15], [-4], [-3], [29], [37], [39], [-32], [-12], [10], [-38], [37], [-31], [-36], [41], [-28], [-2], [-11], [30], [-32], [18], [16], [-28], [39], [-16], [1], [17], [19], [21], [29], [44], [19], [-42], [-13], [-30], [26], [-29], [32], [19], [40], [30], [-14], [8], [-23], [40], [8], [36], [-20], [38], [-42], [-31], [-19], [53], [-2], [-4], [21], [-36], [8], [-3], [-25], [52], [-56], [22], [10], [-48], [22], [59], [-31], [2], [12], [-29], [34], [26], [-20], [30], [-35], [42], [-9], [-4], [39], [-9], [-12], [-14], [-24], [-16], [-27], [-60], [12], [-34], [-45], [-38], [-38], [-4], [40], [-43], [59], [60], [-64], [36], [30], [-56], [-26], [29], [-48], [9], [14], [5], [4], [19], [-2], [-9], [-8], [56], [-2], [28], [28], [17], [2], [10], [16], [-48], [-50], [60], [-37], [52], [69], [42], [-58], [48], [25], [-51], [32], [27], [-64], [64], [32], [56], [21], [56], [-9], [-26], [35], [72], [-11], [23], [34], [52], [-65], [0], [12], [-27], [1], [18], [54], [-44], [36], [50], [13], [-70], [74], [20], [-60], [-50], [0], [27], [-13], [62], [-12], [-48], [30], [-73], [30], [68], [-70], [-57], [22], [50], [14], [-33], [38], [16], [-32], [6], [12], [58], [-46], [32], [31], [6], [-22], [-74], [15], [-37], [-56], [8], [-18], [-37], [-18], [0], [61], [24], [27], [-46], [24], [-31], [23], [-45], [-51], [-30], [-50], [18], [0], [-26], [-21], [39], [-32], [28], [-12], [26], [75], [-32], [-24], [25], [-66], [-44], [-30], [-60], [-60], [-35], [12], [14], [90], [18], [-31], [24], [26], [-62], [20], [-91], [17], [60], [52], [25], [13], [87], [-57], [-5], [40], [28], [8], [-62], [-61], [28], [-94], [-19], [78], [-28], [51], [-28], [78], [-18], [13], [-6], [-72], [-18], [52], [58], [-54], [0], [18], [-33], [55], [66], [-22], [-32], [41], [-65], [-63], [72], [12], [46], [-83], [-65], [-55], [-18], [6], [-10], [-79], [35], [-52], [-24], [-57], [-31], [-9], [-32], [34], [-61], [-1], [-50], [-28], [-83], [-53], [67], [-64], [-58], [-50], [42], [12], [36], [-46], [58], [8], [-29], [-43], [86], [84], [-46], [-87], [-42], [-18], [17], [-10], [-43], [-27], [78], [6], [-3], [64], [-65], [21], [48], [12], [-85], [-66], [106], [59], [-22], [18], [24], [-81], [-78], [18], [90], [25], [-14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7098_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7098_2_a_s();".
// 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_7098_2_a_s(:prec:=1)
    chi := MakeCharacter_7098_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_7098_2_a_s();".
// 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_7098_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7098_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 1]>,<11,R![-1, 1]>,<17,R![1, 1]>],Snew);
    return Vf;
end function;