
// Make newform 3450.2.a.b in Magma, downloaded from the LMFDB on 06 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3450_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_3450_2_a_b();".
// 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_3450_2_a_b();".
// 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_3450_a();"
function MakeCharacter_3450_a()
    N := 3450;
    order := 1;
    char_gens := [1151, 277, 1201];
    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_3450_a_Hecke(Kf)
    return MakeCharacter_3450_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 := [[-1], [-1], [0], [0], [-2], [-4], [-6], [-8], [1], [4], [0], [2], [-2], [-2], [12], [6], [8], [2], [6], [10], [2], [-8], [-8], [-12], [16], [12], [0], [12], [-10], [-18], [-2], [-4], [-2], [4], [14], [-4], [10], [16], [16], [10], [16], [10], [24], [-26], [18], [-16], [4], [2], [-28], [-10], [-6], [-14], [-6], [2], [-22], [0], [8], [24], [28], [12], [-14], [14], [-28], [26], [-28], [-18], [28], [28], [12], [26], [-10], [-16], [-16], [-6], [12], [32], [26], [-28], [-8], [-10], [30], [-14], [-8], [-16], [28], [4], [30], [8], [-12], [26], [24], [-16], [26], [20], [-4], [16], [20], [-20], [-30], [-22], [-4], [-38], [24], [-24], [20], [-38], [-4], [-6], [-34], [-26], [10], [30], [18], [32], [-32], [-4], [10], [28], [-2], [-26], [-50], [10], [10], [12], [20], [-14], [38], [-2], [-28], [42], [-36], [24], [32], [6], [-22], [-2], [6], [-22], [-14], [-46], [4], [0], [22], [-12], [-38], [-12], [-36], [-18], [-28], [20], [20], [4], [-32], [-36], [22], [-4], [40], [30], [-32], [38], [-4], [-26], [42], [-14], [42], [-48], [12], [4], [50], [-14], [50], [-26], [16], [40], [28], [-24], [44], [-40], [44], [-18], [-54], [42], [44], [58], [-16], [58], [4], [-34], [54], [-42], [14], [20], [-28], [36], [-32], [-34], [46], [26], [-26], [44], [-14], [52], [48], [-6], [58], [-30], [8], [60], [-30], [20], [58], [60], [16], [-48], [-30], [46], [-18], [-30], [24], [30], [-26], [-40], [-54], [44], [60], [-66], [-34], [30], [24], [-24], [-40], [-36], [-56], [38], [-62], [24], [-2], [-58], [-72], [-26], [-24], [20], [14], [18], [18], [-50], [-10], [62], [32], [64], [-66], [20], [8], [46], [-42], [36], [62], [42], [22], [-18], [32], [-60], [46], [6], [14], [-28], [-38], [-36], [14], [-34], [14], [2], [-16], [32], [58], [-18], [-4], [-14], [30], [-46], [-64], [80], [-8], [42], [32], [-54], [76], [54], [20], [-34], [-2], [12], [46], [42], [-56], [38], [16], [-30], [24], [-68], [52], [-6], [-56], [-76], [-48], [-78], [-20], [14], [44], [80], [76], [-26], [-60], [12], [72], [-46], [60], [54], [-40], [50], [-72], [52], [-24], [54], [-46], [42], [24], [84], [-8], [10], [-50], [-82], [-28], [-12], [-32], [52], [74], [-14], [-18], [-52], [56], [-6], [-36], [-8], [-22], [40], [-18], [80], [52], [-54], [68], [-2], [36], [-44], [-22], [30], [84], [-42], [-54], [-16], [22], [-10], [40], [50], [-36], [74], [54], [-56], [38], [-52], [-92], [-72], [-86], [-80], [-40], [-20], [84], [-100], [12], [66], [-30], [78], [26], [-50], [20], [-12], [-84], [92], [6], [-24], [-74], [-34], [-68], [-28], [24], [-52], [-88], [48], [36], [-60], [62], [26], [12], [62], [92], [-20], [-50], [-14], [34], [-40], [14], [34], [-104], [-8], [20], [40], [50], [-66], [54], [56], [-30], [74], [-48], [88], [-82], [74], [28], [-22], [-44], [-58]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3450_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3450_2_a_b();".
// 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_3450_2_a_b(:prec:=1)
    chi := MakeCharacter_3450_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_3450_2_a_b();".
// 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_3450_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![2, 1]>,<13,R![4, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;