
// Make newform 3525.2.a.j in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3525_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_3525_2_a_j();".
// 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_3525_2_a_j();".
// 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_3525_a();"
function MakeCharacter_3525_a()
    N := 3525;
    order := 1;
    char_gens := [2351, 1552, 2026];
    v := [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_3525_a_Hecke(Kf)
    return MakeCharacter_3525_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 := [[1], [-1], [0], [0], [4], [2], [-2], [0], [0], [-6], [-4], [10], [-2], [-8], [1], [2], [-4], [14], [8], [16], [-2], [8], [4], [18], [14], [-10], [16], [4], [-18], [10], [-4], [12], [-14], [16], [6], [4], [18], [-24], [0], [-6], [20], [6], [0], [14], [10], [-4], [-8], [12], [-4], [22], [-14], [0], [2], [-12], [-14], [24], [6], [24], [10], [30], [-12], [-18], [-20], [-32], [6], [-18], [4], [-2], [-12], [22], [6], [0], [-28], [-14], [-4], [24], [-6], [2], [10], [34], [4], [22], [-32], [-18], [24], [36], [-26], [22], [10], [4], [-12], [0], [32], [36], [32], [24], [-30], [2], [28], [-2], [8], [-42], [-36], [-18], [-20], [-18], [-36], [26], [24], [10], [20], [-6], [-42], [-28], [-20], [14], [-4], [0], [-6], [-4], [-10], [46], [-18], [-4], [40], [-22], [-26], [-40], [20], [34], [20], [16], [20], [-6], [18], [-14], [-30], [40], [-50], [-10], [52], [-54], [24], [-12], [-18], [-40], [-38], [-30], [0], [-48], [-38], [-26], [44], [8], [44], [-40], [-4], [10], [-58], [22], [20], [50], [16], [-20], [-34], [-48], [0], [2], [-38], [6], [-4], [14], [24], [30], [20], [38], [-28], [-2], [12], [54], [-40], [60], [-6], [14], [-40], [-42], [34], [44], [-6], [0], [14], [28], [-8], [-2], [28], [54], [-62], [34], [14], [-56], [-26], [-32], [42], [-46], [60], [34], [20], [-52], [-66], [-8], [-18], [-42], [-32], [4], [64], [10], [52], [6], [-24], [-42], [-26], [-16], [-42], [-52], [60], [-66], [-6], [0], [-32], [-36], [-30], [60], [32], [-14], [-16], [-24], [34], [50], [12], [40], [20], [-20], [-4], [-10], [54], [0], [24], [-20], [60], [8], [2], [-38], [24], [-30], [-18], [52], [38], [32], [-34], [-10], [0], [52], [70], [-46], [-30], [-60], [6], [6], [8], [38], [46], [-68], [-42], [68], [38], [76], [-20], [30], [-70], [-20], [24], [-12], [-48], [38], [4], [-8], [54], [6], [68], [-6], [-2], [-76], [-6], [4], [-46], [10], [-64], [-18], [36], [16], [38], [46], [80], [36], [20], [-74], [44], [-50], [-16], [74], [16], [30], [-2], [32], [-72], [26], [4], [40], [6], [34], [-4], [22], [10], [-8], [-86], [-22], [28], [-72], [88], [-38], [-50], [74], [52], [84], [-28], [-52], [-42], [46], [-38], [52], [10], [-30], [-58], [-80], [6], [-12], [38], [24], [-48], [2], [-20], [-26], [58], [4], [-34], [-6], [-48], [44], [-62], [-72], [-62], [54], [24], [-60], [-32], [-26], [86], [-40], [-22], [36], [4], [0], [50], [20], [-94], [84], [8], [14], [98], [-58], [46], [94], [-44], [-42], [-60], [40], [-44], [26], [-28], [8], [26], [-6], [-60], [20], [24], [-26], [-28], [-70], [-40], [66], [86], [6], [60], [-18], [-74], [16], [-46], [-78], [-16], [28], [62], [42], [84], [8], [6], [18], [72], [-68], [-70], [-104], [46], [74], [-24], [60], [70], [-42], [-60], [42], [-64], [-48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3525_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3525_2_a_j();".
// 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_3525_2_a_j(:prec:=1)
    chi := MakeCharacter_3525_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_3525_2_a_j();".
// 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_3525_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3525_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<7,R![0, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;