
// Make newform 3450.2.a.z in Magma, downloaded from the LMFDB on 24 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_z();".
// 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_z();".
// 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], [2], [2], [0], [8], [1], [-10], [8], [-8], [-6], [-12], [-8], [-10], [4], [12], [4], [16], [10], [10], [10], [0], [-10], [10], [2], [10], [-8], [-8], [4], [-8], [12], [-20], [-10], [20], [8], [-4], [8], [2], [-16], [16], [-16], [-14], [-26], [-14], [4], [4], [-14], [-16], [6], [-16], [10], [-14], [6], [-4], [18], [28], [6], [-16], [16], [-2], [-28], [16], [10], [-6], [-12], [-22], [24], [-6], [-14], [8], [-18], [24], [8], [-36], [-2], [10], [-16], [-10], [30], [-36], [-28], [-14], [16], [28], [18], [-10], [-30], [-8], [6], [24], [40], [0], [-4], [-36], [6], [-12], [-4], [-38], [20], [2], [-6], [-32], [12], [-2], [0], [6], [32], [10], [-12], [44], [-48], [-8], [-2], [0], [-4], [40], [6], [-30], [-28], [46], [-46], [-24], [-52], [-6], [28], [16], [-14], [52], [44], [28], [26], [8], [30], [-10], [6], [4], [-6], [30], [12], [42], [-24], [6], [10], [-12], [10], [-18], [-4], [0], [22], [0], [-20], [0], [-40], [-36], [26], [-6], [30], [-34], [8], [-8], [56], [30], [20], [-36], [16], [38], [-6], [-18], [-46], [50], [-44], [-14], [-20], [36], [-12], [-26], [-14], [-56], [-16], [-30], [22], [-6], [-4], [-26], [-6], [44], [-42], [24], [-14], [-24], [-20], [-34], [-6], [-44], [26], [12], [20], [24], [-18], [20], [-50], [38], [42], [10], [-34], [52], [-14], [-28], [18], [54], [6], [62], [48], [-2], [-52], [38], [24], [34], [14], [-30], [6], [66], [-44], [6], [56], [-24], [38], [4], [14], [-8], [-14], [-58], [-56], [-12], [-62], [-6], [0], [56], [2], [-52], [-40], [62], [54], [-56], [44], [-18], [-56], [-48], [24], [-4], [68], [22], [18], [-44], [-48], [44], [-58], [-22], [34], [-38], [-30], [28], [-18], [8], [54], [-12], [-56], [6], [-6], [-12], [66], [-6], [70], [8], [28], [-26], [34], [42], [56], [-50], [-40], [-20], [12], [-16], [-26], [42], [64], [-18], [30], [18], [-62], [-10], [-34], [86], [-10], [-58], [44], [-52], [-86], [10], [6], [-52], [-56], [58], [36], [-8], [60], [58], [-72], [-14], [76], [-4], [12], [38], [68], [80], [-30], [-22], [44], [-62], [-18], [28], [-72], [58], [-60], [-28], [24], [66], [-70], [-58], [8], [24], [8], [-64], [-40], [56], [-58], [58], [58], [88], [-74], [46], [78], [12], [-2], [-44], [84], [-10], [-4], [54], [50], [-26], [68], [-54], [52], [18], [90], [-16], [-44], [-90], [32], [-10], [84], [74], [-30], [-74], [62], [56], [28], [72], [-14], [14], [-38], [64], [60], [14], [-60], [2], [-26], [68], [92], [-82], [-56], [96], [-100], [-10], [-12], [-36], [-50], [54], [-36], [-68], [-12], [-82], [74], [4], [-100], [86], [14], [34], [-50], [48], [-10], [52], [40], [-58], [-88], [56], [-82], [74], [-6], [76], [50], [-26], [0], [-24], [60], [64], [-98], [32], [-8], [84], [14], [30], [66], [74], [-20], [32]];
    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_z();".
// 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_z(: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_z();".
// 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_z( : 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![-2, 1]>,<13,R![-2, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;