
// Make newform 3450.2.a.o in Magma, downloaded from the LMFDB on 28 July 2026.
// 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_o();".
// 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_o();".
// 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 := 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]];
    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], [0], [-2], [0], [2], [1], [-6], [-4], [10], [-6], [-2], [0], [-12], [12], [-10], [-14], [0], [-2], [-10], [0], [12], [10], [-18], [-14], [0], [2], [-12], [4], [-12], [-12], [20], [0], [8], [-14], [-8], [24], [6], [12], [2], [-12], [-14], [-18], [-10], [8], [16], [-12], [26], [6], [-24], [14], [-12], [6], [-12], [-6], [20], [-14], [-12], [10], [-12], [16], [-24], [22], [6], [32], [10], [-12], [2], [-30], [36], [10], [10], [26], [-12], [12], [-14], [24], [-10], [0], [2], [24], [-26], [8], [-36], [18], [34], [-30], [-8], [12], [0], [-8], [12], [-40], [24], [-30], [-12], [34], [2], [-8], [36], [36], [0], [38], [34], [36], [6], [0], [38], [-8], [-38], [12], [-10], [14], [-48], [22], [-48], [-6], [36], [14], [-38], [48], [-36], [32], [-48], [50], [-24], [-26], [46], [-4], [12], [-22], [-2], [6], [2], [-24], [22], [-12], [6], [8], [6], [4], [36], [2], [-12], [10], [-18], [-28], [-24], [-14], [12], [16], [0], [34], [48], [-34], [-30], [46], [24], [-12], [-24], [4], [-36], [-36], [24], [20], [-26], [38], [-6], [-60], [14], [-12], [10], [-28], [-48], [-4], [-18], [-62], [14], [4], [-36], [-14], [18], [48], [12], [34], [46], [-46], [24], [46], [60], [-46], [-66], [0], [-24], [-46], [22], [-48], [48], [-12], [-28], [34], [-10], [-24], [-42], [-46], [-36], [-54], [8], [-2], [54], [-26], [-48], [0], [26], [-56], [30], [-24], [-30], [-22], [-22], [-42], [-38], [-36], [-46], [12], [24], [-50], [36], [46], [14], [50], [6], [10], [-48], [62], [36], [-36], [-48], [-36], [56], [4], [-70], [-66], [48], [28], [0], [-10], [72], [-50], [-72], [-12], [-10], [66], [12], [-70], [-38], [-18], [58], [58], [-72], [14], [-14], [-18], [62], [60], [24], [-38], [54], [-34], [22], [10], [50], [22], [-56], [60], [-34], [2], [0], [-48], [38], [-12], [-10], [-32], [0], [-14], [12], [8], [-66], [0], [48], [-54], [36], [46], [24], [-22], [-30], [60], [28], [-50], [-60], [50], [36], [50], [-62], [-84], [50], [-60], [-50], [-24], [-12], [60], [-44], [24], [-34], [60], [-48], [-14], [66], [-46], [-74], [-18], [16], [72], [-34], [2], [-44], [24], [-84], [-10], [18], [-88], [84], [-34], [36], [38], [36], [38], [34], [-50], [-72], [6], [-70], [48], [12], [-70], [-8], [0], [12], [-4], [10], [-6], [-62], [-58], [-78], [36], [60], [42], [-48], [58], [78], [-96], [60], [28], [-26], [42], [34], [98], [-60], [56], [72], [0], [-10], [34], [60], [-72], [94], [60], [10], [60], [12], [4], [54], [-46], [72], [44], [34], [-62], [-72], [-34], [6], [60], [16], [-48], [-14], [-46], [-12], [74], [-18], [-34], [42], [58], [48], [-42], [56], [-98], [-66], [58], [-36], [-38], [66], [-48], [62], [58], [90], [0], [52], [-12], [12], [12], [-38], [-72], [12], [-26], [42], [-84], [-54], [-16], [-72]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3450_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3450_2_a_o();".
// 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_o(: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_o();".
// 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_o( : 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![0, 1]>,<13,R![2, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;