
// Make newform 1400.2.a.c in Magma, downloaded from the LMFDB on 13 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1400_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_1400_2_a_c();".
// 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_1400_2_a_c();".
// 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_1400_a();"
function MakeCharacter_1400_a()
    N := 1400;
    order := 1;
    char_gens := [351, 701, 1177, 801];
    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_1400_a_Hecke(Kf)
    return MakeCharacter_1400_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 := [[0], [-2], [0], [1], [1], [-4], [0], [6], [-3], [-3], [0], [-9], [2], [-9], [6], [-6], [8], [-10], [1], [-7], [2], [-9], [-12], [-4], [-16], [-14], [6], [-20], [-11], [9], [1], [-12], [22], [-6], [-17], [17], [14], [-12], [12], [2], [16], [22], [20], [1], [-3], [-18], [4], [-26], [22], [-26], [19], [-16], [18], [20], [0], [15], [4], [4], [22], [-7], [-14], [-8], [12], [4], [22], [33], [-17], [-34], [-17], [2], [0], [-3], [-10], [-15], [-5], [24], [-35], [-34], [-39], [14], [12], [23], [-4], [0], [-4], [36], [-9], [-11], [12], [16], [18], [12], [-17], [-39], [-4], [-26], [-36], [30], [20], [3], [-11], [33], [2], [37], [15], [-2], [22], [-44], [15], [8], [-20], [-5], [23], [-16], [-45], [49], [-8], [-42], [-6], [44], [14], [-14], [-18], [23], [26], [-18], [22], [26], [-8], [4], [-43], [0], [4], [-17], [32], [38], [-40], [-38], [28], [43], [14], [2], [-23], [7], [-20], [34], [-28], [38], [26], [7], [46], [-40], [25], [2], [-12], [23], [-29], [-42], [-10], [-56], [-20], [-11], [-40], [-26], [3], [24], [23], [-38], [19], [36], [-45], [10], [-9], [30], [-26], [-36], [-33], [-18], [-24], [44], [23], [48], [3], [6], [-11], [-10], [-11], [34], [-47], [30], [18], [9], [29], [-26], [43], [-18], [5], [-1], [-26], [40], [18], [26], [-52], [-2], [54], [-6], [38], [-40], [-65], [-42], [-13], [-28], [39], [48], [30], [48], [47], [10], [1], [-45], [61], [-20], [-45], [21], [32], [30], [-34], [-9], [56], [65], [51], [60], [60], [27], [64], [28], [50], [-35], [25], [-48], [33], [4], [4], [-21], [-72], [-56], [-62], [18], [-40], [-24], [38], [-50], [-40], [-52], [18], [60], [-35], [20], [-54], [6], [-36], [-57], [12], [8], [42], [32], [-5], [-50], [-9], [31], [36], [52], [-52], [-41], [-52], [44], [-33], [-30], [-49], [-8], [70], [-80], [-68], [-76], [-30], [57], [-47], [9], [60], [-30], [-21], [-28], [-81], [-46], [-23], [72], [22], [68], [-6], [42], [14], [-14], [-61], [-25], [-57], [2], [-23], [36], [-69], [-58], [-54], [6], [29], [28], [-51], [-56], [-26], [32], [70], [21], [-16], [22], [26], [-23], [13], [-52], [-45], [50], [85], [-49], [34], [-77], [74], [4], [-36], [-54], [85], [-16], [-60], [-10], [-58], [75], [-2], [75], [15], [-3], [38], [19], [-62], [42], [-14], [17], [10], [-38], [19], [-62], [-36], [6], [34], [76], [29], [72], [-24], [73], [66], [46], [-14], [79], [75], [13], [80], [9], [54], [-92], [49], [-24], [-71], [72], [42], [-18], [30], [45], [-7], [13], [-62], [-28], [-7], [-20], [-61], [34], [-55], [16], [9], [-94], [5], [-17], [-52], [-32], [-76], [5], [36], [-55], [13], [-76], [6], [52], [17], [54], [46], [-104], [20], [-73], [-84], [-45], [59], [-30], [24], [-30], [4], [-48], [-106], [18], [32], [-12], [36], [-46], [91], [-42], [28], [-42]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1400_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1400_2_a_c();".
// 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_1400_2_a_c(:prec:=1)
    chi := MakeCharacter_1400_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_1400_2_a_c();".
// 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_1400_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<11,R![-1, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;