
// Make newform 3520.2.a.e in Magma, downloaded from the LMFDB on 06 September 2025.
// 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_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_3520_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_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 := 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]];
    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], [-2], [-1], [4], [1], [-4], [4], [-4], [-6], [10], [-4], [-2], [-10], [-8], [6], [-2], [-4], [10], [-2], [0], [12], [-8], [0], [-10], [-2], [14], [6], [-12], [14], [6], [8], [-4], [-10], [-20], [-18], [-24], [18], [-10], [-8], [-16], [-24], [-10], [-20], [-20], [8], [12], [12], [10], [-16], [-22], [-24], [-24], [-10], [-8], [10], [0], [-18], [16], [12], [-26], [24], [-20], [12], [-8], [-14], [18], [36], [12], [-20], [10], [34], [-32], [14], [8], [-4], [14], [6], [-22], [-2], [-6], [8], [26], [0], [6], [8], [30], [2], [0], [-18], [10], [6], [24], [-34], [-4], [-28], [-36], [18], [6], [-4], [30], [-8], [32], [-4], [-30], [-20], [-10], [-2], [36], [16], [30], [16], [16], [30], [-48], [32], [2], [46], [-14], [-10], [-44], [14], [-36], [-28], [2], [-40], [-38], [6], [-16], [34], [0], [-20], [0], [-40], [-38], [-14], [-10], [38], [-4], [-14], [-18], [28], [18], [54], [-12], [-14], [-36], [-8], [12], [-8], [30], [-32], [-2], [-26], [-40], [-26], [-44], [16], [46], [-8], [-10], [38], [-16], [56], [-20], [-10], [6], [-12], [16], [-34], [-34], [-12], [-18], [48], [48], [-8], [50], [-28], [34], [-44], [30], [42], [44], [-38], [-8], [10], [54], [-48], [-46], [50], [48], [38], [8], [32], [22], [56], [-30], [-10], [-18], [16], [32], [66], [8], [14], [46], [8], [-30], [-44], [44], [62], [52], [24], [-54], [-66], [30], [40], [22], [12], [2], [-22], [-34], [50], [8], [42], [46], [-60], [-10], [54], [36], [-16], [-60], [-30], [44], [40], [-26], [-30], [12], [-62], [-36], [76], [-28], [-2], [-52], [62], [54], [72], [-24], [-18], [-24], [4], [8], [-20], [42], [-30], [-2], [-32], [60], [6], [24], [-30], [-8], [-72], [24], [-30], [-44], [18], [-20], [-2], [-54], [-16], [60], [-30], [-2], [46], [-40], [-32], [74], [-42], [26], [-10], [-12], [56], [28], [28], [38], [80], [-36], [-34], [-60], [8], [42], [-58], [18], [-80], [-76], [68], [-18], [-32], [-66], [-44], [44], [40], [48], [-48], [-54], [52], [50], [-10], [-46], [0], [64], [-84], [42], [10], [-30], [-64], [-34], [-16], [-48], [-18], [-30], [84], [-50], [34], [50], [-64], [-30], [16], [-26], [-68], [56], [34], [-70], [16], [20], [76], [10], [-50], [-24], [-70], [68], [66], [-78], [30], [20], [-18], [-60], [-74], [-38], [48], [-50], [76], [-14], [-30], [-28], [34], [-36], [-20], [-84], [0], [6], [-44], [26], [14], [76], [74], [-18], [8], [80], [-18], [16], [80], [0], [-42], [-16], [2], [-48], [80], [56], [34], [88], [50], [-14], [-16], [92], [-20], [14], [88], [-14], [-52], [66], [34], [86], [-60], [78], [-24], [48], [24], [58], [-8], [-78], [30], [70], [28], [-6], [-66], [-88], [-94], [50], [-46], [76], [-88], [28], [-66], [28], [-76], [-50], [-40], [74], [-78], [-20], [-6], [-20], [18], [-36], [-70], [-18], [-34], [-46], [88], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3520_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3520_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_3520_2_a_e(: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_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_3520_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<7,R![-4, 1]>,<13,R![4, 1]>,<17,R![-4, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;