
// Make newform 4275.2.a.n in Magma, downloaded from the LMFDB on 18 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_4275_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_4275_2_a_n();".
// 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_4275_2_a_n();".
// 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_4275_a();"
function MakeCharacter_4275_a()
    N := 4275;
    order := 1;
    char_gens := [1901, 1027, 1351];
    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_4275_a_Hecke( : Kf := Kf)
    return MakeCharacter_4275_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 := [[1], [0], [0], [2], [-2], [-2], [2], [1], [0], [6], [-4], [2], [-2], [-10], [0], [-10], [0], [-10], [-4], [8], [4], [-8], [-12], [-10], [-18], [0], [16], [-8], [-6], [6], [12], [18], [-6], [-4], [-8], [-12], [16], [-14], [0], [-6], [-24], [-6], [18], [10], [-6], [-8], [-4], [-8], [4], [-22], [14], [-2], [-14], [14], [30], [24], [6], [0], [8], [-22], [-22], [2], [-20], [-6], [24], [14], [-16], [22], [36], [2], [-34], [-22], [-34], [-18], [16], [36], [36], [-8], [-30], [22], [-18], [-18], [-12], [26], [-16], [-4], [-10], [40], [-32], [14], [-4], [18], [16], [26], [-20], [-16], [42], [42], [16], [2], [28], [34], [12], [6], [28], [-24], [-12], [26], [36], [-10], [16], [0], [42], [-20], [-8], [-18], [26], [-24], [-34], [-20], [18], [-26], [-18], [48], [36], [24], [10], [-42], [14], [-36], [28], [-20], [44], [-4], [12], [-34], [14], [16], [18], [40], [12], [-24], [-18], [-12], [26], [0], [20], [-30], [-44], [4], [14], [-4], [-38], [52], [-48], [16], [40], [-48], [4], [-50], [-36], [-18], [22], [12], [-22], [-24], [32], [8], [-6], [6], [12], [-10], [-6], [16], [-8], [0], [44], [-24], [-32], [-50], [26], [12], [-58], [-66], [-24], [48], [-30], [40], [-58], [-42], [-14], [-44], [28], [38], [44], [-50], [-30], [-4], [-66], [32], [22], [0], [-2], [2], [42], [18], [16], [-28], [-36], [20], [-40], [-52], [22], [-20], [4], [46], [-18], [42], [8], [-2], [-34], [68], [-10], [58], [-68], [-74], [62], [-28], [36], [-46], [28], [64], [32], [-2], [6], [48], [34], [14], [-6], [-44], [24], [-12], [-38], [58], [2], [-58], [18], [-32], [-64], [72], [-68], [48], [48], [54], [14], [6], [-26], [-56], [-2], [20], [32], [-24], [-26], [14], [62], [-16], [-58], [72], [80], [6], [-10], [12], [36], [-48], [70], [-50], [-36], [-70], [-2], [18], [36], [-48], [-24], [-2], [-42], [26], [-32], [-66], [16], [-58], [8], [-52], [18], [-20], [-26], [-4], [-8], [-22], [-48], [14], [36], [-66], [-64], [60], [4], [70], [24], [-34], [-6], [-68], [16], [36], [74], [44], [48], [90], [-86], [80], [-32], [4], [-40], [-20], [-26], [44], [42], [-86], [-76], [-40], [-40], [42], [-50], [62], [0], [4], [12], [84], [-82], [-6], [-50], [58], [6], [-42], [-14], [28], [46], [-76], [18], [12], [-84], [-42], [40], [34], [-76], [-8], [-14], [-6], [-18], [-26], [-82], [-4], [-48], [84], [36], [76], [-6], [-6], [18], [64], [-50], [-38], [76], [-96], [-6], [-8], [-64], [8], [-18], [-56], [8], [86], [-78], [70], [30], [22], [56], [-40], [88], [48], [74], [72], [70], [-66], [30], [-22], [-68], [-30], [44], [-54], [40], [-68], [-34], [-2], [-12], [38], [-2], [32], [8], [62], [-76], [78], [-34], [42], [-48], [-52], [-60], [-24], [-36], [36], [-62], [4], [-6], [-98], [-8], [80], [2], [-78], [-72], [-72], [100], [78]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4275_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4275_2_a_n();".
// 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_4275_2_a_n(:prec:=1)
    chi := MakeCharacter_4275_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_4275_2_a_n();".
// 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_4275_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4275_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<7,R![-2, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;