
// Make newform 3675.2.a.l in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3675_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_3675_2_a_l();".
// 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_3675_2_a_l();".
// 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_3675_a();"
function MakeCharacter_3675_a()
    N := 3675;
    order := 1;
    char_gens := [1226, 1177, 2551];
    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_3675_a_Hecke(Kf)
    return MakeCharacter_3675_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], [-6], [-2], [4], [6], [0], [-2], [10], [4], [-2], [4], [0], [-6], [8], [2], [16], [10], [-6], [4], [8], [-6], [-2], [6], [8], [-4], [2], [-6], [-20], [-4], [6], [-2], [-14], [8], [18], [-4], [12], [0], [-14], [6], [-18], [8], [-2], [-14], [-16], [-24], [8], [10], [26], [-6], [-22], [0], [16], [24], [14], [14], [28], [-2], [20], [24], [4], [24], [6], [18], [-24], [24], [-12], [2], [20], [22], [0], [0], [-28], [-20], [26], [22], [-30], [22], [12], [18], [-14], [-34], [-6], [4], [10], [20], [-30], [-36], [-24], [-24], [-12], [10], [4], [-36], [-30], [-38], [20], [6], [16], [-38], [-36], [-30], [36], [14], [-12], [44], [-2], [26], [24], [-4], [26], [-10], [-20], [-2], [-20], [-20], [-2], [-18], [18], [36], [32], [-28], [50], [-10], [26], [-12], [-40], [22], [-44], [-40], [-12], [40], [22], [18], [24], [-4], [-16], [-18], [-34], [-2], [-44], [12], [6], [-12], [30], [-24], [26], [24], [4], [-14], [48], [8], [-16], [6], [28], [14], [2], [42], [36], [-6], [-40], [8], [-30], [24], [4], [14], [-14], [-24], [18], [30], [50], [-8], [38], [-42], [8], [-6], [16], [14], [8], [0], [32], [-36], [16], [-6], [-28], [44], [-50], [-28], [-34], [20], [-28], [14], [12], [-12], [-30], [-40], [24], [-36], [6], [-22], [-14], [-46], [-64], [-48], [38], [36], [62], [2], [-28], [-22], [16], [48], [-20], [10], [-36], [-2], [32], [-10], [-22], [-22], [-58], [48], [-60], [10], [56], [38], [-72], [18], [24], [6], [64], [30], [4], [8], [-58], [-18], [-54], [52], [60], [-34], [48], [-34], [-24], [-60], [-32], [-60], [40], [16], [4], [78], [16], [62], [-48], [-46], [10], [52], [72], [6], [-12], [-68], [50], [26], [-12], [-22], [-66], [42], [-48], [-10], [30], [52], [-10], [44], [-42], [16], [-12], [50], [-10], [-48], [16], [-4], [0], [-10], [44], [6], [28], [74], [-86], [-34], [78], [72], [-46], [-20], [-40], [-14], [-50], [44], [32], [68], [26], [-42], [0], [20], [-20], [64], [12], [-30], [-30], [56], [-48], [30], [18], [-8], [-24], [-86], [52], [86], [-74], [66], [-62], [-12], [14], [16], [-42], [-14], [44], [4], [24], [34], [-50], [30], [78], [-92], [-80], [-60], [-74], [64], [-70], [-64], [64], [-78], [-90], [-48], [54], [38], [34], [-48], [-28], [-24], [38], [4], [-42], [-24], [-58], [92], [20], [-52], [-14], [-72], [-16], [82], [-16], [-66], [-68], [-44], [12], [40], [-10], [42], [-14], [-72], [-70], [94], [52], [28], [-50], [-58], [-30], [26], [62], [-6], [-20], [-82], [-50], [-52], [-28], [70], [24], [72], [-26], [4], [6], [-28], [90], [32], [94], [42], [52], [-42], [54], [-46], [0], [72], [-30], [-2], [-92], [-30], [4], [16], [90], [-6], [-92], [32], [64], [14], [26], [-48], [-60], [-8], [-6], [-10], [-48], [0], [-66], [-72], [12], [-54], [66], [84]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3675_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3675_2_a_l();".
// 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_3675_2_a_l(:prec:=1)
    chi := MakeCharacter_3675_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_3675_2_a_l();".
// 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_3675_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3675_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<11,R![6, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;