
// Make newform 3450.2.a.h 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_h();".
// 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_h();".
// 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], [-3], [-3], [3], [4], [-3], [1], [3], [-10], [8], [-9], [-1], [-2], [6], [8], [12], [-8], [-14], [-13], [-17], [-1], [-6], [-6], [-6], [-13], [12], [0], [-6], [-16], [10], [18], [8], [-6], [-22], [-6], [-6], [-2], [-15], [-10], [-16], [-15], [6], [17], [11], [22], [6], [12], [6], [1], [-10], [-10], [-16], [18], [8], [-5], [22], [13], [-10], [0], [0], [-8], [24], [-24], [-31], [-26], [-14], [0], [-5], [-9], [31], [-27], [16], [12], [29], [38], [-30], [-28], [-23], [19], [-10], [16], [-4], [-4], [-26], [-26], [-22], [-9], [4], [-11], [33], [2], [12], [34], [35], [34], [4], [-11], [-13], [-22], [32], [-5], [-20], [-4], [7], [-24], [21], [6], [-14], [-2], [-20], [16], [20], [5], [-16], [31], [18], [3], [-33], [-10], [-3], [0], [-38], [-48], [-36], [26], [16], [16], [-14], [50], [29], [45], [20], [-27], [36], [40], [5], [-24], [21], [16], [-55], [16], [9], [11], [33], [23], [-22], [20], [-54], [22], [-40], [48], [8], [1], [55], [32], [-41], [22], [-6], [-2], [24], [44], [43], [-38], [39], [-26], [49], [12], [-26], [-45], [31], [-31], [46], [-62], [-12], [62], [45], [-15], [4], [-42], [-5], [49], [-21], [39], [12], [34], [-31], [1], [-12], [-59], [4], [-53], [-34], [-29], [50], [-24], [-10], [60], [-14], [-18], [22], [41], [-40], [-59], [-39], [-56], [-16], [51], [8], [-31], [-11], [-1], [51], [-6], [-14], [-64], [-70], [45], [-6], [9], [13], [-50], [-29], [-56], [73], [-30], [-24], [-60], [20], [30], [-16], [27], [33], [36], [35], [72], [-8], [52], [26], [-57], [-44], [10], [50], [58], [-48], [10], [-28], [-21], [27], [54], [42], [-15], [-28], [59], [50], [-18], [37], [-7], [-2], [40], [-57], [-78], [-34], [-46], [11], [-60], [-30], [39], [-63], [71], [52], [16], [-45], [55], [-56], [-48], [-55], [64], [-44], [-68], [-12], [-33], [26], [24], [54], [-52], [74], [-60], [13], [-78], [-3], [-21], [-36], [-46], [48], [7], [-29], [6], [-2], [-76], [16], [19], [-10], [-68], [-23], [-10], [-70], [-72], [-33], [22], [-64], [2], [-40], [32], [34], [-24], [-8], [10], [54], [88], [58], [53], [62], [-66], [-18], [-60], [54], [-67], [30], [-50], [-9], [-26], [-66], [-52], [74], [10], [4], [-73], [28], [86], [-32], [61], [-48], [64], [0], [78], [42], [-19], [76], [32], [22], [3], [32], [-14], [15], [87], [-7], [-93], [6], [46], [9], [12], [-21], [10], [22], [50], [80], [-12], [6], [10], [18], [22], [-9], [-45], [30], [47], [62], [-90], [-7], [-30], [20], [-14], [-38], [-9], [-96], [-54], [38], [-28], [40], [64], [93], [44], [-68], [-8], [64], [9], [-6], [-19], [14], [-59], [54], [-83], [14], [54], [-40], [-60], [42], [65], [-42], [-6], [-41], [99], [-95], [-26], [63], [60], [-52], [40], [39], [-40], [6], [-8], [44], [-101], [-45], [-71], [19], [14], [-60]];
    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_h();".
// 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_h(: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_h();".
// 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_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3450_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![3, 1]>,<11,R![3, 1]>,<13,R![-3, 1]>,<17,R![-4, 1]>],Snew);
    return Vf;
end function;