
// Make newform 6975.2.a.k in Magma, downloaded from the LMFDB on 04 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_6975_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_6975_2_a_k();".
// 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_6975_2_a_k();".
// 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_6975_a();"
function MakeCharacter_6975_a()
    N := 6975;
    order := 1;
    char_gens := [776, 5302, 1801];
    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_6975_a_Hecke( : Kf := Kf)
    return MakeCharacter_6975_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]];
    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], [0], [0], [3], [4], [7], [3], [-5], [-1], [1], [0], [8], [10], [2], [-9], [12], [-2], [11], [2], [-10], [-10], [3], [-7], [-1], [10], [-11], [6], [2], [6], [-16], [6], [-17], [-20], [-4], [-24], [7], [11], [16], [16], [20], [8], [-10], [21], [6], [-20], [11], [-6], [12], [4], [-24], [15], [18], [21], [26], [24], [-26], [10], [0], [2], [15], [24], [-8], [-24], [18], [8], [-24], [28], [-4], [9], [3], [-30], [8], [3], [-12], [-3], [14], [29], [-31], [20], [-24], [2], [-16], [2], [-36], [6], [25], [-8], [27], [-14], [6], [-6], [-28], [1], [10], [28], [-15], [12], [-28], [-35], [19], [25], [34], [17], [-20], [-2], [-5], [-4], [-24], [42], [29], [-10], [-12], [-20], [-40], [-41], [34], [-3], [-10], [10], [19], [-22], [-18], [-14], [23], [0], [-40], [28], [0], [-45], [-38], [-29], [49], [-34], [-6], [21], [37], [-8], [-26], [19], [-36], [23], [12], [-23], [10], [48], [10], [54], [44], [-4], [22], [10], [2], [-8], [23], [-29], [-11], [23], [-7], [-19], [-51], [22], [-42], [-36], [-38], [-25], [2], [-57], [-22], [22], [60], [8], [-4], [3], [48], [-6], [4], [-22], [49], [42], [-27], [5], [41], [17], [-16], [3], [-37], [-28], [6], [12], [-28], [-6], [28], [66], [34], [63], [-14], [26], [-48], [-6], [16], [-10], [26], [-17], [66], [70], [31], [27], [-64], [-21], [18], [-6], [13], [20], [49], [-29], [-47], [-22], [-35], [-14], [-36], [-49], [44], [-13], [-14], [-52], [62], [47], [0], [-68], [58], [-20], [-45], [-54], [-16], [-65], [27], [-42], [71], [-33], [-36], [-34], [74], [40], [-67], [-30], [-52], [-37], [50], [-34], [73], [30], [49], [46], [18], [12], [77], [2], [-28], [18], [0], [-23], [64], [65], [1], [-28], [42], [-26], [25], [52], [1], [-46], [-28], [-6], [14], [-79], [-40], [-44], [-48], [-33], [-32], [69], [-44], [74], [49], [36], [26], [63], [-43], [-45], [-10], [-58], [-81], [12], [-4], [-83], [0], [68], [55], [-78], [-1], [-73], [-20], [30], [-30], [38], [79], [85], [81], [21], [-43], [-2], [-88], [24], [0], [32], [-81], [24], [-9], [-54], [-34], [48], [28], [77], [48], [-8], [68], [1], [0], [31], [73], [-26], [47], [-21], [41], [-12], [-26], [78], [7], [44], [-58], [74], [-90], [-58], [42], [0], [69], [38], [-33], [-90], [-58], [-4], [6], [44], [-45], [-74], [47], [-12], [-75], [-86], [17], [-65], [-56], [-48], [-72], [-66], [68], [58], [10], [18], [-20], [-20], [4], [-24], [-8], [-5], [-36], [-50], [-66], [54], [51], [94], [66], [-17], [-48], [-14], [75], [-54], [-4], [0], [50], [-34], [-22], [69], [-86], [2], [-12], [70], [-86], [-5], [62], [-102], [-67], [34], [51], [16], [-62], [-42], [62], [49], [-26], [0], [8], [-56], [-70], [64], [-81], [-31], [-42], [12], [-87], [-2], [-16], [-100], [-70], [58], [30], [90], [-12], [84]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6975_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6975_2_a_k();".
// 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_6975_2_a_k(:prec:=1)
    chi := MakeCharacter_6975_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_6975_2_a_k();".
// 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_6975_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6975_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<7,R![0, 1]>,<11,R![-3, 1]>,<13,R![-4, 1]>,<17,R![-7, 1]>,<29,R![1, 1]>],Snew);
    return Vf;
end function;