
// Make newform 1230.2.a.c in Magma, downloaded from the LMFDB on 14 July 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1230_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_1230_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_1230_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_1230_a();"
function MakeCharacter_1230_a()
    N := 1230;
    order := 1;
    char_gens := [821, 247, 211];
    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_1230_a_Hecke( : Kf := Kf)
    return MakeCharacter_1230_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], [-1], [0], [-4], [2], [-6], [0], [0], [-2], [-8], [-6], [1], [4], [4], [-6], [0], [-2], [-4], [-8], [2], [0], [-4], [-6], [6], [-10], [4], [12], [10], [-2], [12], [-16], [2], [-12], [-18], [-16], [2], [-12], [4], [-14], [12], [10], [0], [-26], [26], [16], [0], [12], [20], [26], [18], [-8], [-14], [0], [18], [12], [10], [-8], [-6], [-6], [4], [2], [-12], [32], [14], [2], [32], [-6], [28], [-10], [-10], [16], [4], [34], [28], [-4], [2], [-14], [-6], [-38], [16], [10], [-8], [-6], [-16], [-4], [-22], [-26], [-22], [16], [28], [0], [20], [0], [-16], [-4], [-2], [-38], [-4], [22], [-12], [-38], [12], [-14], [-40], [-26], [12], [-6], [-8], [-14], [-12], [-38], [-18], [-20], [8], [-30], [-28], [16], [-6], [12], [30], [-10], [-6], [-36], [0], [50], [10], [40], [-16], [-14], [-28], [8], [8], [-22], [-30], [18], [42], [-28], [2], [26], [12], [-6], [-32], [-12], [38], [8], [10], [-50], [20], [56], [-38], [42], [-28], [-12], [-52], [0], [40], [-30], [30], [-38], [36], [-42], [-40], [-36], [-38], [-48], [-48], [-38], [2], [-54], [60], [-26], [48], [-22], [-8], [42], [40], [-14], [32], [50], [44], [-32], [-6], [46], [0], [50], [10], [36], [-14], [-48], [-14], [-36], [12], [-62], [28], [6], [26], [50], [-14], [-12], [-22], [-32], [-6], [10], [12], [42], [16], [60], [-46], [-60], [-58], [6], [-44], [12], [0], [42], [-56], [-38], [4], [-30], [-54], [-64], [18], [-16], [-28], [50], [-34], [-40], [-72], [-32], [58], [64], [8], [-30], [36], [36], [-22], [-6], [40], [-16], [-36], [8], [32], [46], [30], [-48], [60], [-28], [60], [64], [-62], [66], [-72], [-6], [-30], [-40], [-14], [4], [58], [54], [-4], [-4], [-30], [34], [6], [-44], [-66], [-54], [-68], [50], [2], [28], [26], [-24], [-10], [4], [52], [26], [2], [-84], [84], [-40], [-56], [62], [-12], [-32], [-18], [10], [56], [50], [70], [84], [-30], [-40], [18], [-50], [-48], [42], [4], [28], [74], [42], [-16], [20], [4], [-54], [20], [-2], [-16], [-38], [-4], [6], [42], [-52], [-32], [26], [-56], [-48], [6], [-30], [84], [10], [-30], [-32], [-18], [-46], [8], [4], [-52], [-38], [34], [18], [40], [12], [-60], [20], [42], [-34], [34], [52], [10], [-58], [62], [72], [2], [48], [-2], [-52], [-72], [26], [72], [2], [-66], [-44], [10], [-6], [24], [-24], [-90], [24], [98], [10], [44], [-48], [52], [86], [66], [-44], [10], [60], [40], [72], [-18], [24], [-94], [8], [-64], [82], [-62], [62], [70], [6], [-20], [-50], [-72], [-24], [48], [-22], [-52], [-60], [34], [2], [4], [-4], [-48], [30], [64], [26], [28], [38], [14], [-46], [76], [-78], [2], [-80], [34], [66], [-4], [56], [50], [42], [20], [-48], [6], [50], [16], [80], [-54], [-96], [-54], [42], [48], [-84], [-38], [82], [60], [-86], [-96], [-96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1230_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1230_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_1230_2_a_c(:prec:=1)
    chi := MakeCharacter_1230_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_1230_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_1230_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1230_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![4, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;