
// Make newform 2475.2.a.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2475_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_2475_2_a_e();".
// 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_2475_2_a_e();".
// 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_2475_a();"
function MakeCharacter_2475_a()
    N := 2475;
    order := 1;
    char_gens := [551, 2377, 2026];
    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_2475_a_Hecke(Kf)
    return MakeCharacter_2475_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+1)]];
    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], [-1], [1], [-1], [6], [-7], [-6], [6], [-7], [2], [6], [-1], [0], [6], [0], [5], [5], [12], [14], [-4], [6], [-6], [17], [0], [8], [6], [11], [18], [8], [-18], [0], [-4], [6], [-7], [23], [-13], [0], [24], [-18], [5], [12], [-7], [-24], [11], [5], [5], [24], [17], [-12], [30], [-13], [18], [-24], [18], [-18], [-16], [-13], [-12], [29], [-18], [5], [-18], [-13], [30], [-4], [-13], [-24], [-10], [18], [24], [-37], [-31], [-19], [6], [30], [-25], [18], [-13], [36], [-10], [30], [11], [35], [30], [-24], [26], [-6], [-4], [18], [36], [-13], [-18], [-13], [-6], [-36], [-18], [17], [-37], [-28], [12], [-30], [-30], [-31], [-1], [18], [-6], [-12], [17], [-40], [14], [-12], [47], [-19], [-42], [32], [24], [24], [24], [26], [-22], [-30], [-30], [-4], [36], [17], [6], [5], [14], [32], [6], [32], [29], [-30], [29], [18], [-55], [0], [-12], [-7], [0], [-31], [-42], [14], [30], [-19], [-30], [-4], [12], [53], [-30], [-7], [6], [20], [30], [-43], [24], [-7], [24], [6], [-42], [-52], [6], [42], [0], [5], [-10], [-13], [-30], [-6], [-1], [60], [59], [-1], [-42], [17], [36], [-13], [-19], [53], [-60], [29], [36], [-24], [30], [47], [35], [-49], [0], [29], [60], [-31], [-12], [24], [24], [14], [14], [-12], [-6], [-30], [-52], [26], [-34], [42], [-36], [-25], [-30], [42], [17], [29], [30], [-19], [36], [36], [26], [59], [36], [6], [66], [-49], [29], [18], [-40], [12], [-43], [-42], [-48], [-67], [-12], [65], [68], [5], [42], [44], [-30], [62], [-6], [54], [42], [42], [11], [11], [-61], [60], [-18], [47], [12], [-1], [12], [50], [48], [-6], [-55], [-12], [-60], [41], [8], [48], [-10], [-1], [-42], [-10], [-31], [30], [20], [-60], [-12], [-52], [-24], [-25], [-16], [-70], [23], [23], [-79], [-72], [29], [65], [-48], [-6], [-64], [-72], [-19], [-67], [60], [65], [-48], [-16], [-72], [-24], [-6], [-6], [-12], [5], [18], [-19], [60], [30], [-4], [-46], [-60], [11], [-36], [32], [-7], [42], [29], [24], [-22], [-6], [24], [24], [41], [-6], [-61], [0], [-6], [-31], [-12], [20], [-61], [-24], [29], [-72], [74], [-37], [11], [-78], [-78], [-73], [18], [11], [24], [53], [12], [11], [-42], [-67], [-16], [71], [72], [96], [-1], [-12], [-6], [-43], [-1], [30], [42], [-79], [-94], [6], [41], [-67], [66], [-30], [-18], [-72], [36], [-55], [54], [-36], [-18], [-4], [-13], [6], [56], [-7], [-30], [77], [6], [6], [-40], [29], [-18], [78], [17], [24], [-49], [-24], [-36], [-88], [96], [44], [54], [-67], [-22], [20], [12], [-55], [-36], [-78], [-19], [60], [-1], [47], [24], [5], [-72], [-58], [-24], [53], [18], [18], [56], [17], [36], [-49], [-48], [-43], [42], [-78], [53], [-25], [-18], [30], [35], [-24], [-30], [-54], [-43], [90], [-72], [-58], [-90], [-78], [-36], [29], [-108]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2475_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2475_2_a_e();".
// 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_2475_2_a_e(:prec:=1)
    chi := MakeCharacter_2475_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_2475_2_a_e();".
// 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_2475_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2475_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<7,R![1, 1]>,<29,R![-6, 1]>],Snew);
    return Vf;
end function;