
// Make newform 7056.2.a.br in Magma, downloaded from the LMFDB on 23 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_br();".
// 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_br();".
// 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], [2], [0], [0], [2], [6], [-4], [-4], [-6], [-8], [-10], [-10], [-12], [8], [-6], [-4], [10], [-12], [4], [-2], [-8], [-4], [6], [-10], [10], [0], [16], [14], [14], [-8], [12], [14], [-4], [-6], [8], [10], [4], [-16], [2], [0], [-6], [-4], [18], [-22], [-24], [12], [16], [-12], [-6], [-18], [-12], [22], [-20], [14], [28], [10], [-8], [-10], [30], [-12], [-22], [-4], [0], [6], [2], [20], [-30], [-8], [10], [-10], [-12], [32], [22], [4], [16], [-14], [26], [-10], [14], [36], [6], [12], [-34], [24], [-40], [-34], [10], [18], [-8], [4], [24], [8], [24], [-4], [24], [-6], [-18], [12], [-2], [-28], [18], [-12], [-18], [-4], [-18], [-36], [6], [36], [22], [-32], [-10], [-18], [12], [32], [-26], [-20], [32], [-6], [-24], [-14], [-46], [-38], [-40], [36], [-38], [6], [16], [-16], [-14], [-28], [52], [0], [-10], [6], [14], [-38], [20], [18], [-26], [-4], [42], [-32], [16], [-14], [-48], [-14], [38], [-44], [4], [-2], [-42], [-4], [0], [52], [12], [-8], [30], [-10], [-22], [-24], [38], [48], [-28], [22], [40], [48], [2], [-30], [-6], [-16], [-54], [52], [42], [16], [14], [-20], [18], [0], [-6], [-8], [12], [-26], [14], [-28], [10], [-2], [-60], [-22], [32], [-26], [0], [-4], [34], [-56], [30], [-62], [30], [62], [-32], [-14], [64], [-54], [-10], [68], [18], [48], [24], [-26], [-28], [-14], [2], [-8], [68], [32], [-42], [32], [-10], [12], [26], [-58], [-16], [14], [32], [12], [-26], [-58], [-44], [-48], [24], [-18], [-68], [-48], [-42], [-44], [48], [70], [-54], [-56], [40], [0], [-20], [56], [14], [-18], [-48], [-8], [-12], [-20], [68], [62], [-42], [-60], [30], [34], [72], [-42], [20], [58], [70], [8], [0], [-6], [10], [38], [68], [-46], [30], [-20], [-46], [-54], [-68], [14], [0], [-26], [16], [-16], [62], [-6], [-28], [-32], [-16], [72], [-30], [-52], [-20], [-14], [-6], [-64], [-18], [50], [-68], [30], [28], [-50], [-78], [32], [34], [-36], [-44], [6], [-38], [-40], [24], [-4], [82], [-24], [-14], [-60], [-26], [-48], [-30], [22], [-84], [28], [-2], [-84], [-60], [6], [-10], [-20], [10], [2], [-16], [6], [-58], [20], [4], [-68], [66], [78], [42], [72], [4], [-20], [-52], [-50], [46], [-18], [-8], [-10], [30], [18], [24], [-22], [-24], [58], [20], [0], [90], [28], [-6], [18], [0], [22], [-18], [0], [-28], [-18], [-28], [6], [-90], [12], [16], [52], [58], [58], [32], [-54], [-48], [20], [28], [10], [32], [-34], [28], [20], [-18], [38], [-82], [-30], [6], [8], [78], [-60], [56], [80], [26], [76], [-72], [-62], [66], [-48], [28], [-76], [-86], [32], [30], [-68], [18], [-70], [-66], [-56], [-90], [-94], [88], [78], [-34], [4], [36], [62], [-30], [-8], [-4], [-86], [-6], [84], [80], [54], [-48], [10], [18], [28], [20], [38], [-30], [96], [-18], [60], [-64]];
    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_br();".
// 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_br(: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_br();".
// 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_br( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7056_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![0, 1]>,<13,R![-2, 1]>,<17,R![-6, 1]>,<23,R![4, 1]>],Snew);
    return Vf;
end function;