
// Make newform 7056.2.a.v in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7056_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_7056_2_a_v();".
// 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_7056_2_a_v();".
// 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_7056_a();"
function MakeCharacter_7056_a()
    N := 7056;
    order := 1;
    char_gens := [6175, 1765, 785, 4609];
    v := [1, 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_7056_a_Hecke(Kf)
    return MakeCharacter_7056_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 := [[0], [0], [-1], [0], [5], [-2], [6], [2], [-6], [3], [5], [-2], [8], [4], [-4], [-9], [3], [12], [-2], [-8], [14], [-1], [-17], [18], [-3], [-10], [8], [3], [14], [-8], [-19], [-17], [-6], [10], [6], [15], [-12], [18], [-22], [-6], [20], [20], [-4], [1], [18], [0], [-24], [3], [-7], [20], [22], [2], [17], [15], [2], [24], [25], [-15], [28], [18], [4], [21], [34], [4], [23], [-9], [-8], [-19], [-12], [-6], [-8], [6], [7], [4], [4], [18], [14], [8], [-24], [5], [4], [16], [-32], [-10], [33], [9], [-10], [-29], [-30], [-20], [-28], [24], [25], [9], [34], [-10], [13], [-24], [34], [32], [18], [37], [-5], [-14], [-40], [-7], [21], [-26], [-6], [1], [-25], [-6], [20], [4], [9], [-12], [22], [-28], [29], [-36], [40], [1], [-45], [-21], [-8], [-45], [-18], [10], [-35], [-20], [-34], [2], [29], [-20], [-18], [19], [38], [-40], [-33], [28], [34], [-29], [24], [41], [-26], [30], [22], [12], [32], [-38], [24], [-18], [-28], [-26], [22], [-24], [-44], [-8], [5], [-21], [28], [-60], [-17], [-15], [28], [48], [7], [14], [11], [3], [-33], [-40], [-18], [29], [-41], [0], [56], [53], [-36], [-8], [-25], [33], [26], [42], [2], [-21], [12], [22], [-5], [46], [19], [-5], [38], [-30], [-15], [30], [-34], [24], [44], [-36], [5], [-16], [-6], [-18], [-44], [13], [-48], [-1], [-6], [-18], [-34], [-33], [19], [9], [-66], [-38], [21], [-42], [18], [50], [-4], [-56], [32], [-33], [-31], [60], [-6], [68], [-11], [-56], [-28], [-24], [7], [-46], [34], [-50], [-47], [-11], [-72], [-38], [8], [32], [-25], [-34], [30], [48], [-67], [20], [-40], [-24], [22], [48], [-14], [70], [-27], [-51], [28], [8], [41], [-9], [8], [-27], [-34], [28], [52], [-26], [13], [26], [36], [-23], [-40], [54], [79], [5], [-45], [-16], [-13], [10], [-38], [36], [72], [-25], [2], [52], [-42], [-34], [-49], [-67], [-1], [64], [18], [-27], [-76], [-36], [-50], [7], [-32], [42], [-68], [-28], [6], [-55], [25], [57], [52], [-2], [73], [-34], [-18], [-46], [-66], [-10], [22], [-40], [-54], [74], [-1], [54], [-1], [-8], [20], [10], [-23], [32], [44], [-19], [-32], [-34], [-78], [37], [32], [53], [40], [-20], [-10], [-52], [-30], [-24], [35], [-1], [8], [18], [9], [-55], [11], [-5], [-22], [6], [-62], [49], [90], [15], [-82], [39], [-52], [-80], [-4], [3], [48], [-74], [0], [-88], [-50], [44], [70], [-69], [18], [8], [69], [51], [56], [34], [31], [-16], [38], [-12], [90], [-15], [-42], [26], [-6], [12], [0], [-6], [-84], [-36], [-40], [72], [26], [2], [-31], [26], [-27], [-28], [-74], [15], [11], [-14], [-82], [85], [86], [-90], [-60], [22], [5], [-41], [30], [-72], [-104], [-69], [73], [-51], [3], [-60], [-22], [-43], [16], [76], [42], [12], [33], [-34], [-12], [-1], [-6], [62], [83], [102], [66], [36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7056_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7056_2_a_v();".
// 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_7056_2_a_v(:prec:=1)
    chi := MakeCharacter_7056_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_7056_2_a_v();".
// 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_7056_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7056_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 1]>,<11,R![-5, 1]>,<13,R![2, 1]>,<17,R![-6, 1]>,<23,R![6, 1]>],Snew);
    return Vf;
end function;