
// Make newform 3450.2.a.ba in Magma, downloaded from the LMFDB on 19 April 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_ba();".
// 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_ba();".
// 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], [2], [6], [2], [0], [-4], [1], [-2], [-8], [4], [2], [8], [0], [2], [-4], [0], [0], [-8], [-6], [-14], [6], [-16], [-2], [2], [10], [6], [12], [-16], [12], [8], [-20], [-4], [10], [20], [4], [-12], [0], [-6], [0], [4], [8], [18], [22], [-22], [28], [28], [-10], [4], [-2], [0], [18], [-18], [6], [-4], [-30], [-4], [-10], [-16], [-28], [-22], [12], [-32], [18], [-30], [-20], [-22], [32], [10], [-6], [32], [-2], [4], [-36], [12], [10], [-14], [8], [-10], [26], [24], [-36], [26], [0], [-4], [10], [-18], [-14], [-32], [-30], [-40], [32], [-32], [36], [-28], [30], [-4], [-40], [-6], [12], [-10], [-18], [24], [-32], [14], [-24], [6], [16], [-38], [-4], [16], [40], [4], [-34], [-24], [16], [-24], [-34], [30], [40], [30], [-18], [-16], [20], [6], [24], [32], [2], [48], [12], [-52], [26], [-20], [30], [-2], [42], [-8], [30], [-50], [36], [18], [32], [-30], [26], [-44], [18], [-10], [-36], [48], [6], [0], [20], [40], [44], [-12], [-30], [2], [-2], [-30], [-16], [-24], [-32], [50], [-28], [-28], [16], [-34], [34], [54], [30], [-54], [20], [-30], [44], [20], [-28], [6], [58], [20], [24], [-18], [38], [34], [-60], [34], [50], [-16], [-10], [-48], [34], [-48], [-40], [-18], [-10], [-36], [42], [56], [-52], [24], [-22], [20], [-2], [38], [-50], [2], [6], [36], [-6], [-52], [50], [-58], [14], [-30], [8], [-10], [52], [38], [8], [18], [-10], [34], [38], [-22], [-12], [-18], [-16], [16], [30], [60], [-26], [36], [18], [30], [-52], [28], [-22], [-18], [-16], [-32], [22], [4], [-64], [54], [62], [56], [52], [58], [-44], [-48], [12], [-52], [-60], [-58], [18], [-4], [-12], [-24], [22], [42], [42], [54], [26], [16], [38], [-76], [18], [-12], [-20], [70], [2], [72], [2], [-22], [22], [-32], [12], [-10], [2], [-50], [72], [-34], [16], [48], [-44], [-64], [-34], [46], [-32], [54], [18], [30], [34], [-70], [-26], [10], [46], [78], [-60], [-20], [82], [-2], [-2], [44], [-52], [26], [52], [28], [-36], [-6], [72], [-74], [-60], [-68], [52], [70], [12], [8], [66], [2], [8], [10], [30], [20], [-56], [34], [64], [-28], [8], [38], [34], [30], [8], [-32], [20], [56], [20], [0], [70], [-62], [-6], [72], [38], [-58], [2], [52], [38], [44], [-36], [18], [-36], [22], [50], [-42], [-24], [2], [36], [-18], [-30], [24], [24], [-2], [-8], [-54], [-92], [90], [-78], [-74], [-26], [-88], [84], [-16], [30], [-66], [34], [-64], [-12], [-98], [12], [34], [18], [4], [52], [-82], [-36], [32], [-4], [38], [-16], [-28], [-2], [-26], [68], [-92], [44], [22], [-38], [92], [-88], [-10], [-34], [18], [-50], [64], [6], [20], [-52], [14], [28], [24], [-34], [-22], [-42], [-80], [-78], [46], [40], [32], [-84], [-56], [-86], [-52], [72], [-4], [46], [-42], [6], [74], [-28], [-40]];
    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_ba();".
// 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_ba(: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_ba();".
// 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_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, 1]>,<11,R![-6, 1]>,<13,R![-2, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;