
// Make newform 3520.2.a.t in Magma, downloaded from the LMFDB on 07 December 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_t();".
// 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_t();".
// 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], [0], [1], [-2], [1], [0], [0], [8], [-8], [-10], [8], [10], [-2], [6], [-8], [-14], [4], [-10], [-4], [0], [-8], [-4], [-10], [6], [-10], [-14], [-4], [-2], [2], [-10], [10], [-16], [-18], [4], [-6], [-12], [14], [-8], [10], [12], [20], [14], [0], [-8], [12], [-16], [16], [-16], [18], [-6], [-4], [28], [-18], [4], [2], [-10], [-14], [12], [8], [-18], [-10], [-24], [-26], [0], [-6], [6], [20], [32], [-10], [14], [-6], [-12], [-24], [4], [20], [4], [-22], [-34], [-18], [34], [-20], [6], [-8], [-26], [16], [24], [6], [12], [10], [0], [-36], [20], [-24], [4], [-4], [-6], [-14], [22], [-14], [42], [-30], [12], [42], [-6], [28], [-2], [-12], [8], [-24], [14], [42], [4], [30], [-36], [0], [30], [-40], [-12], [-14], [20], [34], [8], [24], [4], [-4], [46], [-10], [16], [28], [-4], [20], [6], [32], [14], [-30], [-26], [42], [-18], [38], [-18], [20], [-26], [-28], [22], [14], [0], [-4], [48], [-20], [-52], [-20], [-22], [-56], [26], [-28], [48], [-32], [2], [-20], [50], [52], [-52], [-14], [-20], [-2], [-44], [16], [4], [30], [-6], [12], [-50], [28], [-44], [48], [-14], [4], [26], [34], [-14], [-44], [36], [-18], [44], [56], [54], [-20], [-28], [-30], [0], [22], [-50], [28], [-38], [-30], [34], [54], [34], [-36], [-10], [14], [-40], [34], [14], [12], [-18], [-8], [42], [6], [-52], [-68], [-42], [-44], [28], [60], [-54], [-58], [-46], [12], [-22], [-26], [-4], [-54], [-4], [14], [-6], [6], [24], [-54], [24], [30], [-44], [-64], [-10], [-44], [46], [-2], [-48], [-36], [40], [24], [0], [-4], [14], [-44], [4], [48], [36], [12], [38], [16], [66], [36], [10], [28], [36], [6], [18], [-18], [-44], [54], [-38], [-58], [16], [-54], [52], [-50], [-38], [-2], [-72], [18], [28], [-10], [-40], [28], [16], [-36], [54], [-58], [0], [-26], [-80], [78], [10], [82], [-24], [-34], [-8], [16], [-70], [-6], [-48], [4], [-72], [0], [-70], [-64], [18], [-4], [-34], [68], [-4], [64], [12], [76], [-86], [52], [-70], [-64], [44], [-62], [-30], [-78], [84], [42], [-2], [28], [-12], [-58], [10], [88], [-10], [14], [8], [60], [-2], [-36], [40], [54], [12], [14], [-42], [4], [58], [48], [0], [50], [-12], [-10], [-82], [6], [-30], [-46], [-8], [2], [48], [-38], [-56], [24], [-22], [-68], [-22], [-50], [66], [-74], [24], [24], [-48], [-12], [-60], [-88], [18], [-48], [-72], [-64], [78], [44], [38], [38], [-12], [-20], [-10], [-6], [-72], [-74], [60], [12], [-28], [10], [-28], [18], [98], [34], [64], [56], [60], [-24], [86], [18], [-36], [14], [34], [12], [-28], [32], [-4], [40], [-54], [-4], [-50], [58], [86], [22], [-6], [-54], [12], [46], [34], [-12], [92], [36], [-24], [24], [52], [96], [-18], [48], [52], [50], [-6], [-10], [64], [-20], [-16], [34], [10], [16], [-62], [-100], [4]];
    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_t();".
// 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_t(: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_t();".
// 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_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<7,R![2, 1]>,<13,R![0, 1]>,<17,R![0, 1]>,<19,R![-8, 1]>],Snew);
    return Vf;
end function;