
// Make newform 3450.2.a.j in Magma, downloaded from the LMFDB on 25 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_j();".
// 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_j();".
// 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], [0], [-6], [-2], [0], [1], [6], [8], [-10], [-6], [8], [-8], [6], [-4], [-6], [-8], [-8], [-10], [-8], [8], [-6], [-18], [6], [8], [0], [-6], [6], [8], [4], [6], [-4], [-6], [-16], [-18], [-12], [16], [18], [-4], [2], [-24], [-18], [-22], [-16], [4], [0], [0], [18], [22], [8], [26], [8], [-18], [8], [22], [16], [10], [-22], [-24], [6], [4], [0], [6], [2], [-20], [22], [-36], [-26], [-18], [16], [24], [-26], [-24], [32], [2], [-6], [26], [10], [16], [-22], [-8], [14], [8], [-20], [-14], [38], [6], [40], [-8], [0], [-16], [36], [-20], [-16], [-26], [-14], [16], [-10], [-4], [-10], [-32], [-14], [8], [-2], [-12], [30], [0], [10], [16], [46], [-2], [-16], [-40], [-22], [16], [24], [42], [-48], [2], [30], [6], [4], [-12], [42], [26], [40], [16], [-2], [-4], [8], [0], [-34], [42], [-6], [-2], [32], [54], [-6], [28], [-10], [-56], [48], [-10], [24], [50], [-10], [20], [-48], [-14], [50], [-36], [-16], [-8], [40], [-16], [-14], [-2], [18], [-44], [6], [56], [-32], [-50], [48], [40], [-38], [-6], [34], [-24], [22], [56], [-2], [16], [42], [20], [-42], [-24], [-14], [48], [-32], [-38], [6], [-24], [-30], [-38], [-16], [10], [0], [46], [-44], [40], [-34], [-56], [-34], [18], [-34], [54], [-56], [42], [-32], [26], [50], [64], [18], [40], [-52], [-6], [20], [14], [-26], [8], [24], [24], [-62], [32], [2], [-24], [-54], [6], [40], [-14], [-32], [-44], [-26], [-18], [0], [24], [-76], [66], [56], [-48], [-6], [-24], [8], [-6], [46], [-20], [8], [-48], [-44], [-8], [14], [-50], [-48], [-56], [32], [48], [48], [6], [-30], [24], [-22], [74], [44], [18], [-32], [-22], [22], [-80], [-72], [30], [38], [14], [-40], [-54], [42], [-24], [-6], [-58], [-80], [-50], [-48], [-2], [-32], [36], [62], [-38], [-8], [-80], [72], [0], [50], [12], [-72], [14], [62], [-72], [82], [18], [72], [6], [0], [50], [18], [-56], [-14], [36], [28], [38], [-74], [40], [-4], [40], [46], [-52], [26], [24], [-30], [-80], [-22], [74], [44], [80], [-38], [-36], [-80], [-74], [-30], [-56], [22], [62], [32], [6], [90], [-72], [36], [-64], [-66], [-26], [90], [-80], [12], [32], [-52], [34], [70], [-22], [-80], [2], [22], [-26], [48], [-74], [60], [-42], [-36], [-88], [22], [44], [-74], [-42], [56], [-14], [-42], [96], [-40], [-66], [48], [46], [-6], [72], [-96], [-68], [22], [-70], [-64], [-78], [-36], [-92], [24], [34], [24], [-6], [-60], [8], [86], [42], [-10], [42], [38], [64], [46], [0], [-72], [96], [26], [-64], [40], [82], [-54], [-100], [-20], [64], [70], [-48], [66], [80], [86], [-50], [-66], [-56], [-58], [-42], [-64], [6], [18], [-64], [12], [-66], [82], [48], [64], [46], [-90], [-48], [8], [-98], [96], [26], [-42], [-32], [-52], [70], [42], [64], [-38], [92], [24]];
    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_j();".
// 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_j(: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_j();".
// 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_j( : 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![0, 1]>,<13,R![6, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;