
// Make newform 2535.2.a.i in Magma, downloaded from the LMFDB on 05 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2535_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_2535_2_a_i();".
// 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_2535_2_a_i();".
// 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_2535_a();"
function MakeCharacter_2535_a()
    N := 2535;
    order := 1;
    char_gens := [1691, 1522, 1861];
    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_2535_a_Hecke( : Kf := Kf)
    return MakeCharacter_2535_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], [1], [1], [1], [-6], [0], [0], [4], [-6], [-6], [-5], [-2], [0], [11], [-6], [0], [-6], [-1], [-11], [6], [-5], [11], [-12], [-12], [-17], [6], [-13], [18], [7], [-6], [-1], [18], [-12], [-19], [-24], [16], [-1], [-11], [12], [-6], [-12], [-10], [6], [13], [-12], [17], [-13], [-8], [0], [-14], [-6], [0], [22], [0], [-12], [12], [-24], [7], [-10], [6], [23], [-6], [25], [12], [23], [12], [7], [-7], [12], [-11], [18], [-12], [5], [23], [1], [-24], [24], [13], [12], [-5], [-24], [-17], [-24], [11], [11], [-36], [-6], [31], [0], [19], [42], [0], [28], [30], [40], [42], [18], [-36], [-4], [7], [-7], [12], [42], [30], [-28], [-14], [-42], [30], [18], [26], [-40], [-23], [-6], [19], [37], [-30], [-11], [-6], [48], [-36], [49], [11], [0], [-18], [-5], [-12], [19], [6], [-25], [-29], [-20], [-6], [-40], [14], [-48], [-38], [18], [13], [12], [-24], [-5], [6], [-16], [0], [-31], [18], [-35], [42], [-13], [54], [46], [-48], [-7], [-42], [-28], [-36], [-16], [-12], [14], [18], [48], [-42], [4], [12], [-48], [42], [32], [-49], [-29], [18], [24], [-50], [-30], [19], [-16], [-54], [1], [-30], [-7], [11], [-32], [-30], [41], [18], [-66], [18], [-7], [4], [-5], [24], [65], [-12], [47], [-42], [30], [-24], [13], [17], [-18], [42], [-60], [-19], [-5], [-7], [-6], [12], [-11], [-60], [-12], [20], [11], [60], [-49], [42], [0], [-26], [-31], [6], [-66], [-60], [5], [-59], [12], [1], [12], [-13], [-6], [12], [-37], [36], [-31], [35], [40], [-30], [-1], [18], [-41], [42], [-24], [-18], [-24], [8], [-64], [61], [-18], [-36], [-29], [-42], [1], [72], [-65], [-6], [-6], [5], [-30], [-18], [14], [-8], [-24], [-74], [20], [36], [55], [47], [12], [23], [-54], [-42], [4], [72], [-13], [67], [-26], [-52], [5], [40], [-18], [-26], [13], [12], [-42], [40], [-18], [37], [52], [6], [-43], [78], [-29], [-54], [-60], [-54], [-60], [18], [-37], [-12], [56], [66], [-18], [61], [-49], [-30], [56], [-42], [23], [-5], [18], [-10], [42], [2], [-12], [42], [12], [-31], [18], [29], [48], [54], [79], [18], [-19], [-59], [0], [43], [-54], [-31], [-5], [-44], [78], [-42], [-50], [66], [53], [0], [31], [-18], [-47], [-6], [-50], [17], [70], [36], [-30], [71], [-42], [30], [-31], [-8], [78], [-12], [13], [-38], [72], [-37], [-70], [24], [-84], [-24], [84], [36], [-38], [0], [-6], [6], [-31], [-73], [12], [67], [-10], [60], [65], [18], [0], [-40], [-58], [-60], [-60], [37], [-66], [-19], [24], [-90], [-56], [-30], [67], [-24], [-8], [17], [-11], [78], [-98], [42], [0], [32], [-12], [41], [-80], [18], [-76], [0], [25], [42], [-53], [42], [-60], [-91], [55], [36], [-23], [84], [83], [48], [-78], [32], [77], [48], [72], [53], [24], [60], [6], [-23], [-12], [-12], [-71], [78], [-30], [-24], [79], [66]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2535_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2535_2_a_i();".
// 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_2535_2_a_i(:prec:=1)
    chi := MakeCharacter_2535_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_2535_2_a_i();".
// 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_2535_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2535_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<7,R![-1, 1]>,<11,R![6, 1]>],Snew);
    return Vf;
end function;