
// Make newform 3380.2.a.c in Magma, downloaded from the LMFDB on 05 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3380_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_3380_2_a_c();".
// 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_3380_2_a_c();".
// 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_3380_a();"
function MakeCharacter_3380_a()
    N := 3380;
    order := 1;
    char_gens := [1691, 677, 1861];
    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_3380_a_Hecke( : Kf := Kf)
    return MakeCharacter_3380_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 := [[0], [-2], [1], [-2], [0], [0], [-6], [4], [6], [6], [4], [-2], [-6], [-10], [6], [-6], [-12], [2], [-2], [12], [-2], [8], [-6], [6], [-2], [6], [14], [-6], [-2], [-6], [2], [0], [-18], [-4], [6], [-20], [-22], [10], [-18], [-6], [-12], [-10], [-12], [-26], [-18], [8], [-16], [10], [6], [-14], [-6], [24], [-14], [0], [-6], [-18], [18], [-20], [26], [-6], [14], [30], [-2], [12], [-22], [6], [-8], [2], [-30], [10], [-18], [-24], [-22], [26], [28], [-6], [-6], [-2], [30], [34], [36], [-26], [-36], [2], [8], [6], [-6], [-26], [-30], [-14], [-30], [24], [-26], [0], [4], [-18], [-6], [-6], [14], [-14], [26], [30], [-18], [30], [8], [22], [6], [-18], [0], [-10], [-22], [-2], [6], [-20], [28], [-18], [-14], [42], [42], [36], [22], [-46], [18], [42], [-8], [-30], [34], [-24], [-46], [22], [-20], [-6], [-4], [-22], [-42], [-2], [30], [-26], [42], [-6], [16], [54], [38], [30], [2], [48], [-50], [18], [-4], [-6], [-26], [-18], [14], [18], [-46], [12], [-16], [42], [-22], [18], [-18], [-6], [22], [-24], [-18], [18], [-4], [26], [-50], [18], [-36], [-14], [12], [-2], [-16], [42], [-8], [-54], [-10], [-46], [22], [48], [-22], [30], [42], [30], [-46], [-14], [-62], [-12], [-22], [-54], [-40], [30], [-6], [-6], [-14], [26], [30], [30], [-6], [-28], [22], [-46], [-60], [-6], [-32], [30], [6], [56], [26], [42], [14], [6], [48], [10], [50], [18], [-18], [-18], [26], [-32], [-18], [34], [-54], [-22], [-30], [0], [-22], [24], [-46], [-52], [-68], [54], [62], [30], [58], [-42], [-60], [60], [-54], [-64], [-10], [10], [30], [24], [-50], [72], [4], [78], [-50], [-18], [6], [14], [-6], [-12], [-58], [46], [42], [70], [38], [42], [-14], [50], [-42], [44], [-6], [-42], [58], [-78], [62], [-2], [-26], [-16], [-22], [58], [-18], [10], [-38], [-48], [-42], [4], [42], [58], [-74], [-36], [26], [30], [-80], [-42], [18], [42], [-18], [-48], [26], [18], [-52], [-78], [-60], [46], [50], [54], [-88], [-18], [32], [-2], [66], [-10], [-24], [2], [54], [-30], [-30], [-82], [30], [74], [36], [-12], [-26], [-42], [8], [-2], [-30], [34], [30], [-34], [52], [10], [-54], [-54], [-50], [42], [-40], [42], [16], [6], [-50], [-90], [-86], [2], [-2], [42], [54], [20], [-18], [12], [-70], [46], [36], [18], [-80], [-74], [78], [62], [-34], [-78], [-72], [72], [-78], [-54], [-74], [-54], [-54], [36], [-70], [74], [30], [34], [-10], [48], [68], [-78], [90], [-76], [50], [-12], [12], [46], [6], [2], [18], [78], [-74], [-18], [4], [-6], [28], [50], [-38], [42], [10], [-42], [36], [50], [-36], [-46], [-8], [-6], [32], [54], [-50], [90], [22], [-18], [-90], [20], [70], [-18], [-38], [60], [-46], [18], [-18], [32], [-46], [-66], [-96], [2], [30], [6], [-30], [-2], [6], [60], [-74], [102], [-18], [66], [88], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3380_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3380_2_a_c();".
// 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_3380_2_a_c(:prec:=1)
    chi := MakeCharacter_3380_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_3380_2_a_c();".
// 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_3380_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3380_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<7,R![2, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;