
// Make newform 7128.2.a.d in Magma, downloaded from the LMFDB on 16 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_7128_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_7128_2_a_d();".
// 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_7128_2_a_d();".
// 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_7128_a();"
function MakeCharacter_7128_a()
    N := 7128;
    order := 1;
    char_gens := [1783, 3565, 5105, 6481];
    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_7128_a_Hecke( : Kf := Kf)
    return MakeCharacter_7128_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], [-1], [0], [1], [2], [-6], [6], [7], [-10], [0], [2], [-12], [-8], [-1], [9], [0], [8], [13], [0], [-2], [-14], [-6], [13], [10], [6], [-1], [-8], [2], [6], [12], [-6], [-18], [-2], [-24], [-12], [13], [-15], [-10], [-6], [-9], [-21], [-25], [16], [6], [-7], [0], [-15], [-14], [-13], [-14], [8], [26], [9], [-6], [-12], [-23], [2], [6], [4], [6], [-30], [-4], [33], [-13], [-17], [35], [-16], [0], [-28], [1], [-26], [-1], [4], [-19], [-23], [18], [-11], [11], [-8], [19], [-11], [-20], [1], [-10], [20], [-6], [34], [-6], [-41], [-1], [26], [16], [-10], [-1], [-6], [-27], [-26], [-38], [26], [36], [14], [-24], [14], [-12], [25], [-3], [-4], [24], [-16], [32], [-8], [-17], [-36], [35], [-23], [-25], [9], [-10], [20], [46], [50], [32], [-1], [33], [20], [11], [1], [-12], [-32], [10], [-34], [48], [-31], [20], [-16], [-42], [-48], [18], [30], [16], [-16], [-51], [-6], [21], [-21], [-16], [-24], [29], [52], [-32], [31], [-29], [2], [-44], [29], [40], [51], [0], [-38], [11], [30], [4], [-21], [-45], [-32], [-56], [-22], [0], [-3], [48], [13], [0], [20], [-35], [39], [26], [-3], [-14], [-38], [8], [38], [-3], [-24], [3], [-15], [-44], [29], [-20], [52], [53], [-24], [64], [42], [18], [45], [8], [-46], [0], [62], [38], [-64], [23], [46], [-60], [26], [43], [2], [8], [-12], [22], [53], [-61], [12], [-32], [-69], [-16], [20], [27], [46], [-50], [-40], [-53], [-19], [20], [-44], [22], [35], [28], [10], [-47], [-56], [-16], [-30], [51], [-58], [30], [-40], [67], [-28], [-31], [-48], [25], [38], [-28], [28], [-40], [40], [-20], [40], [50], [18], [63], [19], [18], [-4], [9], [-38], [23], [-6], [70], [-46], [22], [-38], [-31], [-65], [41], [27], [30], [68], [-55], [31], [41], [-50], [44], [29], [-69], [22], [34], [-18], [-36], [-55], [82], [60], [22], [-33], [21], [18], [28], [22], [67], [-23], [68], [-72], [84], [20], [-16], [21], [-60], [22], [0], [40], [68], [57], [-31], [-25], [-39], [-71], [15], [-42], [72], [26], [-8], [-4], [-54], [54], [81], [60], [9], [28], [4], [-23], [66], [76], [-36], [-45], [27], [1], [-84], [56], [2], [58], [-42], [66], [-24], [-52], [-35], [-52], [15], [34], [-11], [57], [-50], [1], [37], [-48], [-86], [-19], [-6], [-22], [-32], [7], [69], [12], [-50], [-22], [-8], [-50], [12], [83], [60], [66], [-45], [50], [-51], [-26], [-78], [46], [72], [-20], [67], [-48], [14], [-10], [-79], [59], [-18], [-10], [64], [-26], [3], [-27], [-48], [-24], [-42], [-37], [-9], [-38], [68], [-7], [17], [-50], [-37], [73], [-45], [-68], [-88], [-22], [-9], [8], [-14], [43], [28], [41], [-32], [80], [-15], [62], [3], [-25], [4], [-60], [19], [-52], [76], [35], [-36], [15], [58], [-2], [21], [-54], [-96], [-54], [7], [1], [41], [-4], [-44], [36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7128_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7128_2_a_d();".
// 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_7128_2_a_d(:prec:=1)
    chi := MakeCharacter_7128_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_7128_2_a_d();".
// 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_7128_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7128_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 1]>,<7,R![0, 1]>,<13,R![-2, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;