
// Make newform 2700.2.a.h in Magma, downloaded from the LMFDB on 20 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_2700_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_2700_2_a_h();".
// 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_2700_2_a_h();".
// 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_2700_a();"
function MakeCharacter_2700_a()
    N := 2700;
    order := 1;
    char_gens := [1351, 1001, 2377];
    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_2700_a_Hecke( : Kf := Kf)
    return MakeCharacter_2700_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], [-2], [2], [6], [1], [-3], [7], [-6], [1], [-8], [-10], [-2], [8], [9], [10], [5], [8], [6], [-2], [15], [13], [2], [8], [16], [-16], [12], [-11], [18], [16], [0], [3], [4], [10], [4], [-10], [-4], [15], [-1], [-20], [5], [-24], [-2], [-13], [-24], [3], [4], [1], [-13], [2], [-28], [7], [12], [-9], [16], [24], [-11], [-26], [6], [26], [-33], [-12], [18], [-20], [-11], [28], [22], [24], [25], [2], [12], [-2], [12], [-25], [-25], [-10], [22], [28], [5], [2], [-5], [-22], [-38], [33], [-15], [-6], [28], [-24], [-24], [21], [2], [-18], [-12], [9], [-21], [-24], [-2], [-22], [22], [22], [-18], [-4], [14], [37], [22], [15], [-9], [6], [5], [14], [16], [1], [-4], [-37], [32], [-6], [-21], [-23], [-38], [-10], [6], [-38], [-23], [-17], [-28], [-6], [-16], [-12], [8], [-29], [-12], [11], [-20], [-18], [-15], [-39], [10], [-41], [6], [-8], [4], [54], [7], [6], [4], [-4], [-3], [23], [39], [-46], [30], [-42], [27], [-24], [54], [0], [6], [-28], [20], [-3], [38], [20], [24], [-6], [-37], [-17], [-30], [11], [39], [-24], [-57], [2], [32], [31], [-42], [-43], [30], [-2], [-22], [34], [28], [-40], [42], [29], [0], [34], [-52], [-31], [-38], [0], [43], [-65], [-48], [33], [15], [9], [36], [30], [56], [-18], [-24], [48], [-17], [-34], [-30], [48], [-11], [16], [4], [-58], [18], [8], [-12], [-32], [-35], [-10], [-32], [11], [-35], [6], [-59], [-18], [-22], [-63], [-21], [67], [70], [-16], [-40], [2], [1], [-16], [60], [34], [13], [45], [-61], [0], [-18], [12], [41], [64], [1], [58], [12], [-12], [6], [20], [1], [12], [32], [17], [-71], [29], [-48], [61], [-70], [5], [-22], [18], [7], [7], [-16], [15], [-28], [0], [-52], [52], [-18], [-73], [-40], [-10], [-11], [46], [-22], [35], [-43], [41], [-30], [-24], [-31], [-61], [-65], [32], [-46], [12], [-58], [-25], [-16], [-16], [-81], [66], [12], [-62], [-36], [-3], [-54], [-40], [42], [14], [45], [32], [-3], [83], [34], [20], [35], [-50], [-54], [33], [36], [60], [-66], [-83], [-1], [40], [-2], [-86], [60], [-13], [-4], [36], [-68], [45], [62], [27], [60], [-20], [19], [46], [46], [-4], [-76], [-60], [36], [55], [21], [-74], [70], [-36], [53], [74], [-61], [-62], [78], [39], [92], [28], [27], [-88], [36], [-30], [60], [14], [-51], [90], [-42], [46], [-65], [-62], [-96], [24], [44], [-12], [94], [-99], [-36], [6], [52], [-43], [1], [-66], [69], [-12], [-40], [40], [66], [48], [-28], [60], [-1], [-46], [65], [84], [24], [75], [62], [30], [52], [-22], [-99], [-48], [-28], [-44], [58], [-1], [-62], [11], [-12], [-98], [-2], [74], [6], [-66], [-7], [24], [96], [-4], [30], [-40], [87], [-21], [-83], [-44], [-46], [-76], [0], [7], [-39], [60], [-28], [-23], [12], [-76], [-45], [71], [52], [-21], [-70]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2700_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2700_2_a_h();".
// 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_2700_2_a_h(:prec:=1)
    chi := MakeCharacter_2700_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_2700_2_a_h();".
// 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_2700_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2700_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![-2, 1]>,<13,R![-6, 1]>,<17,R![-1, 1]>],Snew);
    return Vf;
end function;