
// Make newform 7920.2.a.e in Magma, downloaded from the LMFDB on 21 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_7920_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_7920_2_a_e();".
// 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_7920_2_a_e();".
// 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_7920_a();"
function MakeCharacter_7920_a()
    N := 7920;
    order := 1;
    char_gens := [991, 5941, 3521, 6337, 6481];
    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_7920_a_Hecke( : Kf := Kf)
    return MakeCharacter_7920_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], [-1], [-6], [-3], [5], [-2], [5], [-5], [-1], [2], [-12], [-2], [13], [2], [1], [-16], [15], [10], [-2], [-14], [-9], [-16], [10], [16], [-4], [10], [16], [8], [7], [-12], [4], [15], [-18], [7], [1], [1], [6], [4], [22], [12], [7], [24], [1], [1], [-26], [18], [6], [-11], [-2], [18], [-10], [22], [-13], [-20], [-24], [16], [-18], [-10], [-6], [22], [21], [-30], [3], [20], [29], [-2], [10], [18], [24], [12], [14], [2], [-14], [2], [22], [21], [-32], [-4], [-36], [-8], [4], [-16], [12], [6], [33], [-5], [-6], [27], [20], [12], [-13], [8], [0], [-8], [26], [4], [15], [36], [30], [-18], [-36], [19], [-14], [-39], [22], [-27], [14], [21], [-10], [-12], [0], [-15], [9], [-7], [-6], [19], [49], [-22], [-1], [0], [-19], [-34], [31], [46], [1], [26], [-32], [8], [33], [7], [-2], [30], [10], [9], [48], [-34], [6], [-1], [-50], [-44], [-52], [4], [0], [22], [21], [22], [-46], [2], [-50], [-5], [-40], [29], [3], [32], [-59], [10], [41], [41], [-29], [-55], [44], [8], [-14], [-16], [8], [-6], [9], [-21], [-22], [10], [55], [24], [53], [4], [-16], [-43], [5], [-10], [-33], [-15], [-47], [38], [-30], [20], [44], [-18], [-36], [-56], [20], [46], [52], [24], [-34], [24], [2], [-30], [35], [5], [-2], [-7], [-4], [24], [33], [-40], [0], [-18], [-4], [-47], [60], [-64], [17], [-60], [27], [37], [58], [-24], [47], [50], [-14], [3], [4], [-56], [-63], [-66], [27], [0], [36], [-6], [31], [70], [10], [-44], [43], [19], [-30], [48], [-23], [57], [-12], [-32], [-26], [2], [-62], [48], [-30], [-15], [11], [-34], [0], [-42], [19], [4], [39], [72], [-54], [18], [37], [-24], [-28], [-35], [-46], [60], [-20], [-20], [-25], [-44], [-18], [-6], [-29], [-4], [-14], [-77], [26], [24], [-42], [14], [-27], [-47], [-4], [-24], [-55], [-68], [-3], [14], [58], [29], [10], [-48], [-12], [41], [-63], [30], [-22], [16], [87], [-20], [-34], [65], [32], [-40], [36], [22], [40], [-28], [16], [-79], [-8], [-20], [-30], [-36], [-15], [33], [10], [-10], [12], [-22], [44], [7], [-58], [-70], [-50], [-45], [19], [-30], [-44], [69], [-72], [79], [-42], [44], [38], [69], [3], [58], [93], [-19], [43], [-78], [42], [-7], [65], [29], [45], [80], [-17], [60], [-61], [-23], [2], [-20], [45], [-31], [75], [-15], [-39], [54], [60], [82], [60], [-12], [81], [17], [24], [98], [-29], [44], [24], [-80], [56], [-9], [-42], [-10], [-60], [78], [-13], [20], [-31], [-36], [-62], [44], [59], [-16], [0], [-96], [49], [24], [18], [29], [-29], [-24], [-64], [8], [86], [62], [9], [62], [55], [95], [-86], [68], [-12], [6], [6], [29], [2], [-21], [10], [-42], [48], [-12], [-20], [-78], [12], [-12], [-16], [58], [-51], [68], [-4], [76], [91], [-38], [-29], [-31], [20], [-40], [-2], [29], [39], [-78], [-54], [45], [-46], [98], [14], [-60], [100], [57], [53], [-23], [68], [-80], [-27], [38], [-45], [-35], [85], [26], [48], [48], [8], [-72], [0], [-77], [48], [54], [-88], [-60], [-99], [0], [24], [2], [77], [-8], [28], [36], [56], [-16], [101], [16], [-54], [26], [-20], [-36], [-40], [-93], [6], [-86], [-36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7920_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7920_2_a_e();".
// 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_7920_2_a_e(:prec:=1)
    chi := MakeCharacter_7920_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_7920_2_a_e();".
// 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_7920_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7920_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 1]>,<13,R![6, 1]>,<17,R![3, 1]>,<19,R![-5, 1]>,<23,R![2, 1]>,<29,R![-5, 1]>],Snew);
    return Vf;
end function;