
// Make newform 3450.2.a.u 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_u();".
// 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_u();".
// 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], [-4], [2], [-4], [6], [-4], [1], [8], [8], [10], [6], [-6], [4], [14], [4], [6], [-14], [10], [-14], [-8], [4], [0], [8], [-8], [-4], [-8], [-14], [10], [-2], [16], [-6], [-20], [-10], [20], [2], [8], [8], [-10], [-4], [-2], [-16], [22], [22], [16], [-20], [-14], [-8], [-22], [6], [26], [10], [-2], [-18], [-16], [12], [-8], [12], [8], [22], [-26], [-4], [-14], [4], [18], [12], [20], [-12], [-30], [-14], [-16], [12], [18], [-16], [24], [-14], [28], [20], [-10], [-30], [6], [8], [-32], [-20], [28], [30], [-16], [0], [10], [36], [24], [10], [-24], [20], [0], [24], [0], [14], [10], [-28], [2], [12], [-44], [-24], [34], [-12], [-18], [14], [-26], [-30], [14], [-18], [-20], [-8], [-24], [-34], [28], [42], [-6], [-22], [34], [26], [-12], [-4], [-6], [34], [46], [-32], [-14], [-4], [-32], [-16], [-34], [-6], [14], [-42], [-50], [-54], [-6], [-12], [12], [-42], [48], [-38], [-12], [4], [-6], [-4], [36], [-20], [-48], [-8], [-12], [-46], [-36], [-16], [-18], [0], [38], [20], [-38], [-22], [6], [-10], [0], [4], [-4], [18], [6], [-10], [-10], [16], [-32], [-20], [60], [36], [28], [-8], [58], [50], [-42], [28], [-18], [-16], [10], [36], [-46], [6], [-42], [-50], [36], [16], [8], [60], [-14], [14], [42], [26], [-36], [-30], [-28], [-32], [2], [6], [-26], [56], [-20], [-14], [-4], [-30], [48], [-36], [-28], [18], [22], [-10], [50], [0], [-62], [-58], [48], [-30], [-24], [-20], [54], [-70], [6], [-4], [-44], [-40], [-56], [-32], [14], [-26], [0], [-26], [54], [36], [38], [44], [20], [-34], [-70], [30], [22], [38], [42], [-20], [48], [78], [-40], [56], [46], [42], [-8], [42], [38], [-46], [14], [-68], [16], [-66], [-74], [18], [-64], [-6], [0], [-14], [18], [6], [30], [24], [-24], [-62], [14], [-68], [10], [-50], [6], [32], [-32], [-40], [22], [48], [50], [4], [6], [-44], [30], [30], [-72], [58], [-10], [-16], [-34], [-40], [86], [20], [-76], [52], [34], [-72], [44], [52], [-38], [-84], [82], [84], [88], [12], [70], [-8], [-76], [24], [74], [-40], [-58], [48], [74], [20], [4], [-12], [-74], [54], [58], [36], [44], [0], [66], [-82], [2], [68], [12], [-4], [-28], [62], [14], [62], [-32], [16], [22], [-8], [-80], [-78], [36], [-2], [-32], [60], [-70], [-4], [78], [-40], [-8], [-34], [-6], [-44], [-30], [78], [-40], [-26], [6], [80], [62], [-36], [50], [42], [-44], [-58], [-40], [76], [0], [82], [-88], [-56], [-56], [36], [44], [0], [-22], [34], [2], [2], [-22], [-56], [84], [44], [-52], [-54], [-48], [-2], [-102], [24], [-92], [24], [-28], [80], [-44], [32], [-88], [-34], [94], [-32], [18], [8], [52], [-50], [-22], [86], [-52], [86], [-34], [-12], [28], [-52], [76], [90], [-90], [42], [16], [-14], [-22], [-80], [-36], [14], [-42], [48], [-70], [-92], [-34]];
    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_u();".
// 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_u(: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_u();".
// 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_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![-2, 1]>,<13,R![4, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;