
// Make newform 3675.2.a.m in Magma, downloaded from the LMFDB on 24 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_m();".
// 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_m();".
// 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], [0], [-3], [2], [1], [2], [-8], [-8], [7], [0], [-8], [-10], [-14], [10], [7], [-5], [-12], [-11], [-7], [14], [-6], [9], [-16], [-13], [6], [-15], [-6], [11], [14], [-10], [3], [-4], [-1], [7], [-9], [2], [18], [-12], [22], [-8], [6], [12], [11], [-5], [-15], [20], [13], [24], [-28], [21], [0], [0], [16], [-16], [-12], [5], [6], [-19], [-16], [-12], [-30], [14], [14], [21], [-2], [10], [-22], [24], [30], [-8], [11], [35], [10], [-36], [14], [-12], [5], [24], [-35], [-2], [-2], [-11], [6], [6], [-3], [-2], [33], [-2], [2], [8], [-6], [-19], [40], [-8], [34], [-16], [-3], [-12], [-4], [28], [12], [-3], [-2], [-6], [42], [26], [-45], [-43], [6], [-24], [-12], [45], [24], [-29], [12], [44], [-24], [11], [-1], [-14], [0], [-15], [34], [1], [-4], [43], [-31], [-49], [6], [23], [-43], [-6], [-10], [48], [-17], [30], [-6], [3], [-48], [5], [-24], [-15], [-42], [-6], [18], [-52], [6], [23], [12], [31], [58], [1], [54], [24], [-44], [22], [42], [-22], [-20], [-23], [8], [-40], [-22], [0], [-37], [-6], [-28], [8], [-61], [10], [-35], [21], [44], [-28], [-36], [29], [-38], [-64], [-38], [30], [22], [-22], [-26], [-19], [-4], [-22], [40], [29], [44], [29], [14], [-36], [-12], [-45], [-26], [50], [-34], [-18], [49], [-50], [10], [38], [8], [-9], [-22], [40], [-15], [43], [22], [-35], [64], [-48], [35], [-43], [-18], [-20], [26], [10], [17], [6], [64], [-66], [-37], [-14], [10], [8], [-42], [-31], [-52], [-8], [-20], [-9], [14], [45], [2], [-6], [72], [62], [-36], [-35], [7], [-28], [62], [40], [-6], [57], [-40], [-19], [60], [-24], [-35], [-42], [10], [-42], [49], [-52], [-50], [19], [-52], [49], [-34], [-32], [19], [36], [-34], [-44], [-10], [-21], [44], [-23], [71], [-75], [-23], [10], [-46], [-53], [-34], [-48], [25], [6], [-1], [-17], [20], [45], [18], [-5], [-76], [-48], [18], [-58], [18], [11], [-66], [-56], [-38], [36], [-19], [-22], [-84], [-71], [24], [27], [9], [34], [41], [14], [-57], [-36], [24], [-84], [-75], [38], [-26], [32], [42], [61], [-6], [-25], [70], [54], [-29], [-82], [65], [-36], [-71], [-18], [24], [-83], [38], [-56], [-78], [56], [-72], [-25], [-24], [-70], [31], [50], [72], [-74], [73], [-54], [-26], [1], [67], [28], [12], [43], [-62], [6], [11], [-77], [-66], [-36], [54], [26], [34], [3], [-18], [72], [42], [-1], [-17], [42], [-29], [19], [60], [76], [-6], [-34], [-3], [-10], [-24], [36], [67], [-4], [77], [-90], [-50], [64], [-18], [40], [-24], [-1], [-86], [45], [36], [89], [34], [-18], [43], [74], [-47], [21], [-100], [15], [-64], [26], [-54], [-41], [-16], [-42], [43], [93], [32], [5], [-102], [-89], [32], [18], [7], [38], [68], [34], [1], [10], [-18], [-24], [26], [-28], [-44], [58], [-54], [56], [12], [25], [-48]];
    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_m();".
// 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_m(: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_m();".
// 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_m( : 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![0, 1]>,<13,R![3, 1]>],Snew);
    return Vf;
end function;