
// Make newform 3520.2.a.h in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3520_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_3520_2_a_h();".
// 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_3520_2_a_h();".
// 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_3520_a();"
function MakeCharacter_3520_a()
    N := 3520;
    order := 1;
    char_gens := [2751, 1541, 2817, 321];
    v := [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_3520_a_Hecke(Kf)
    return MakeCharacter_3520_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+1)]];
    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], [-1], [-1], [-3], [1], [6], [-7], [5], [6], [-5], [3], [-3], [2], [4], [2], [1], [-10], [-7], [8], [-7], [14], [-10], [-6], [-15], [-12], [-2], [-4], [8], [10], [-16], [-8], [17], [-12], [-20], [-15], [-2], [-3], [19], [-3], [-14], [0], [-2], [-12], [-11], [12], [25], [-23], [6], [-2], [-10], [9], [-10], [-18], [2], [-2], [-9], [20], [8], [12], [-18], [-6], [6], [-2], [3], [-6], [7], [-28], [-17], [18], [-30], [34], [20], [-28], [6], [-30], [-34], [30], [2], [-13], [-20], [0], [-32], [8], [-16], [-20], [4], [-30], [3], [-27], [-34], [23], [0], [12], [-3], [20], [-24], [-20], [22], [-16], [23], [8], [-18], [-6], [0], [27], [38], [-27], [14], [-45], [42], [47], [26], [8], [20], [33], [-33], [19], [42], [31], [15], [-2], [29], [-28], [-31], [-38], [-7], [10], [-25], [22], [-24], [40], [21], [-17], [-38], [-18], [-10], [-19], [28], [-18], [-30], [7], [18], [16], [8], [20], [0], [26], [-7], [30], [6], [-38], [2], [9], [-8], [-57], [-27], [-40], [35], [38], [-17], [-27], [39], [27], [-48], [-12], [-54], [8], [32], [50], [21], [-35], [18], [18], [-11], [0], [-5], [-28], [-12], [-9], [45], [-38], [57], [-29], [-27], [-34], [-10], [52], [4], [-50], [48], [24], [4], [-38], [28], [-12], [-6], [32], [-34], [-22], [-49], [-5], [18], [27], [-40], [20], [-43], [40], [24], [-30], [32], [-37], [-52], [-4], [-27], [-40], [-13], [-33], [-18], [-48], [51], [18], [-10], [-35], [-4], [-32], [-55], [-46], [-35], [-8], [52], [46], [55], [-22], [-18], [4], [47], [-5], [-54], [-20], [-57], [29], [12], [-4], [-70], [-6], [10], [52], [42], [-55], [71], [-38], [52], [2], [-5], [-24], [25], [28], [18], [2], [-17], [16], [-52], [-35], [26], [-12], [80], [-80], [57], [-16], [6], [-22], [33], [-56], [50], [-7], [-54], [-32], [-10], [22], [27], [61], [-52], [32], [33], [68], [43], [-26], [2], [5], [-30], [-52], [-12], [-11], [7], [-34], [30], [48], [-69], [60], [-2], [-21], [-28], [20], [-36], [-38], [88], [68], [40], [15], [56], [-44], [70], [52], [19], [-43], [-50], [-10], [8], [-66], [20], [-73], [38], [38], [46], [39], [-13], [-10], [-36], [-63], [-24], [-17], [22], [40], [34], [37], [-57], [-90], [-31], [-43], [-23], [-34], [58], [55], [-87], [-39], [35], [-52], [53], [28], [87], [-23], [38], [68], [31], [-55], [39], [-65], [47], [-82], [-44], [2], [72], [-28], [-25], [3], [-56], [-18], [1], [-8], [32], [-20], [96], [65], [-62], [-78], [0], [-42], [49], [0], [43], [48], [-26], [92], [-97], [0], [-64], [88], [67], [-76], [82], [5], [-9], [0], [8], [8], [74], [-10], [15], [2], [-87], [95], [-6], [-28], [48], [-30], [-42], [7], [-58], [9], [90], [-86], [32], [84], [52], [38], [8], [60], [-8], [38], [-39], [60], [32], [52], [45], [-26], [87], [29], [80], [52], [10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3520_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3520_2_a_h();".
// 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_3520_2_a_h(:prec:=1)
    chi := MakeCharacter_3520_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_3520_2_a_h();".
// 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_3520_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<7,R![3, 1]>,<13,R![-6, 1]>,<17,R![7, 1]>,<19,R![-5, 1]>],Snew);
    return Vf;
end function;