
// Make newform 1530.2.a.d in Magma, downloaded from the LMFDB on 22 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1530_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_1530_2_a_d();".
// 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_1530_2_a_d();".
// 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_1530_a();"
function MakeCharacter_1530_a()
    N := 1530;
    order := 1;
    char_gens := [1361, 307, 1261];
    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_1530_a_Hecke( : Kf := Kf)
    return MakeCharacter_1530_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], [0], [1], [-4], [4], [-2], [-1], [-4], [4], [-2], [4], [-6], [-2], [-12], [-8], [2], [-12], [2], [4], [4], [-14], [-12], [4], [-10], [18], [10], [0], [-12], [10], [6], [-16], [-20], [6], [-20], [2], [-8], [-10], [-20], [-12], [14], [-12], [-22], [8], [18], [14], [4], [4], [-8], [20], [6], [-18], [24], [10], [28], [-18], [16], [30], [0], [2], [-10], [-20], [26], [-4], [-20], [10], [-10], [28], [-22], [-20], [22], [-2], [24], [28], [38], [20], [-8], [18], [-30], [6], [10], [-12], [-18], [4], [-14], [-12], [36], [6], [10], [-14], [40], [-36], [-20], [20], [12], [4], [-44], [-38], [-18], [28], [18], [20], [26], [36], [6], [-44], [18], [-44], [-18], [-24], [-6], [28], [-2], [30], [-28], [-16], [6], [28], [16], [30], [28], [14], [34], [-18], [-36], [-44], [18], [-14], [44], [8], [-18], [12], [-28], [12], [-34], [6], [-46], [-6], [36], [26], [-18], [-28], [30], [-28], [-44], [-34], [36], [-46], [22], [28], [-32], [-22], [30], [-28], [12], [28], [-20], [-56], [6], [-22], [-10], [28], [6], [-16], [-28], [30], [28], [-4], [-38], [-6], [-18], [-60], [-10], [-12], [10], [48], [-42], [-28], [-18], [-16], [-18], [-8], [-20], [-54], [22], [-24], [-6], [34], [44], [66], [-12], [26], [44], [-4], [10], [-44], [-18], [50], [-46], [54], [40], [-42], [-12], [14], [34], [-28], [-14], [-24], [-52], [-42], [-36], [2], [66], [-36], [-12], [4], [66], [40], [30], [28], [-54], [-26], [-52], [46], [-4], [28], [-2], [-34], [-12], [72], [28], [-34], [-20], [-8], [6], [12], [48], [-6], [-18], [20], [24], [-76], [-4], [56], [-2], [6], [-36], [-4], [-60], [28], [32], [70], [46], [16], [-70], [-30], [36], [58], [-76], [14], [-22], [-52], [68], [26], [58], [-2], [-52], [18], [-42], [20], [42], [-6], [-4], [10], [8], [18], [64], [60], [-10], [26], [-52], [40], [-20], [-36], [-50], [-84], [32], [-14], [-34], [-56], [-50], [-34], [-12], [54], [68], [-62], [14], [8], [2], [60], [-20], [-54], [-46], [-20], [4], [-76], [2], [-20], [2], [56], [46], [12], [22], [22], [4], [24], [-70], [-12], [-44], [-86], [30], [44], [26], [-70], [56], [6], [2], [-76], [76], [20], [-66], [58], [38], [44], [20], [4], [20], [-90], [6], [26], [72], [-26], [22], [30], [0], [26], [-12], [58], [44], [-28], [-82], [36], [-70], [-6], [-20], [70], [-58], [0], [-36], [-2], [32], [10], [-26], [-40], [60], [20], [-54], [14], [24], [-30], [60], [-52], [92], [-38], [-24], [-22], [-28], [-84], [-62], [-34], [-86], [-18], [22], [-20], [54], [-52], [92], [64], [46], [-60], [16], [50], [14], [36], [-20], [48], [-22], [-32], [-58], [28], [-54], [34], [-50], [72], [-26], [18], [-84], [-26], [46], [-36], [36], [-54], [-38], [-84], [68], [-38], [-58], [36], [4], [-82], [0], [-46], [-62], [36], [-36], [-6], [2], [-36], [-74], [-60], [20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1530_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1530_2_a_d();".
// 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_1530_2_a_d(:prec:=1)
    chi := MakeCharacter_1530_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_1530_2_a_d();".
// 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_1530_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1530_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![-4, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;