
// Make newform 9520.2.a.k in Magma, downloaded from the LMFDB on 28 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_9520_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_9520_2_a_k();".
// 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_9520_2_a_k();".
// 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_9520_a();"
function MakeCharacter_9520_a()
    N := 9520;
    order := 1;
    char_gens := [5951, 7141, 7617, 1361, 8401];
    v := [1, 1, 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_9520_a_Hecke( : Kf := Kf)
    return MakeCharacter_9520_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], [0], [1], [1], [6], [4], [-1], [-2], [-8], [-4], [0], [-8], [-6], [-12], [0], [2], [-6], [-6], [-8], [-4], [-10], [4], [-14], [-6], [18], [-8], [-12], [14], [12], [10], [-16], [-16], [6], [-4], [6], [8], [24], [-22], [0], [-22], [12], [-2], [0], [6], [20], [8], [-6], [4], [24], [0], [-14], [-24], [-2], [-2], [18], [0], [2], [20], [-8], [30], [-4], [-12], [14], [-16], [26], [-28], [12], [-14], [2], [12], [-14], [-24], [-16], [-6], [-10], [-12], [-6], [38], [-10], [-10], [12], [-6], [0], [-26], [0], [24], [30], [-10], [4], [16], [-34], [0], [0], [-36], [-22], [-8], [40], [6], [-14], [36], [-22], [-6], [30], [-26], [2], [2], [42], [-6], [0], [-22], [16], [-14], [-30], [-32], [40], [-10], [-40], [-8], [36], [48], [24], [6], [-42], [34], [28], [38], [4], [48], [32], [40], [-8], [36], [48], [-38], [30], [-26], [4], [-32], [12], [-18], [20], [-36], [-4], [2], [-28], [-40], [10], [-30], [-34], [8], [-52], [18], [24], [-56], [-54], [12], [-16], [-18], [-30], [10], [-30], [-46], [-32], [-30], [34], [0], [28], [42], [30], [54], [20], [52], [24], [10], [44], [-34], [-62], [-48], [-20], [36], [-64], [-52], [16], [-6], [-8], [36], [-28], [-34], [22], [0], [2], [-58], [36], [-40], [10], [-54], [-30], [0], [2], [-24], [-44], [56], [20], [-46], [14], [-60], [-20], [32], [34], [58], [14], [-36], [-4], [-66], [24], [6], [-40], [6], [0], [-66], [42], [-40], [-34], [-4], [18], [-2], [22], [-24], [24], [-30], [46], [24], [16], [-6], [26], [12], [58], [36], [-66], [52], [-54], [-34], [36], [22], [14], [64], [-48], [64], [-68], [-24], [22], [14], [8], [58], [-36], [-48], [8], [80], [-46], [62], [12], [-24], [50], [-18], [10], [38], [-6], [26], [-2], [42], [50], [32], [-14], [64], [26], [36], [-20], [26], [38], [-6], [20], [-40], [-32], [40], [16], [16], [-18], [12], [-4], [10], [-16], [-84], [-6], [60], [60], [18], [8], [-76], [12], [14], [-58], [26], [-40], [6], [62], [-74], [40], [58], [24], [-50], [-24], [-4], [-66], [16], [40], [66], [-22], [-24], [46], [-66], [-36], [-58], [36], [16], [-66], [-26], [-86], [-12], [-24], [72], [-64], [48], [40], [18], [-2], [-52], [-34], [-2], [70], [80], [2], [70], [-34], [0], [-50], [-46], [78], [44], [-8], [-78], [22], [-58], [30], [-8], [22], [-54], [-48], [44], [-26], [-96], [20], [46], [80], [78], [-98], [-38], [18], [-88], [6], [-80], [-72], [4], [-54], [4], [72], [60], [-20], [34], [6], [86], [-46], [-14], [-68], [2], [-4], [-64], [32], [28], [-46], [-36], [-14], [74], [12], [-78], [-72], [46], [-24], [-34], [58], [-26], [34], [62], [-8], [-74], [72], [96], [62], [-38], [-90], [40], [34], [18], [-56], [22], [-58], [-2], [-12], [-56], [-66], [-44], [94], [62], [36], [-54], [-58], [-44], [-14], [-46], [-42], [48], [54], [-8], [98], [104], [-34], [46], [2], [54], [54], [-20], [84], [-6], [-22], [0], [-62], [-82], [10], [-24], [-34], [76], [92], [-40], [-2], [14], [-98], [106], [-10], [66], [22], [34], [26], [-80], [60], [-20], [2], [-42], [-56], [-6], [-58], [-62], [64], [-96], [-24], [-86], [78], [-10], [-40], [-40], [104], [62], [54], [-110]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9520_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9520_2_a_k();".
// 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_9520_2_a_k(:prec:=1)
    chi := MakeCharacter_9520_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(3449) - 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_9520_2_a_k();".
// 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_9520_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<11,R![-6, 1]>,<13,R![-4, 1]>],Snew);
    return Vf;
end function;