
// Make newform 7350.2.a.cx in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7350_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_7350_2_a_cx();".
// 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_7350_2_a_cx();".
// 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_7350_a();"
function MakeCharacter_7350_a()
    N := 7350;
    order := 1;
    char_gens := [4901, 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_7350_a_Hecke(Kf)
    return MakeCharacter_7350_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], [3], [5], [0], [-5], [9], [0], [10], [1], [-9], [-8], [3], [3], [-12], [-8], [-8], [-6], [2], [8], [0], [-6], [8], [0], [8], [-6], [14], [18], [13], [9], [18], [-20], [0], [-10], [5], [16], [-3], [9], [-15], [-8], [18], [10], [-15], [16], [5], [-28], [0], [-14], [6], [30], [1], [-9], [-12], [24], [-12], [16], [-26], [-21], [-22], [-21], [-28], [12], [-4], [18], [11], [-20], [-30], [28], [24], [12], [-19], [10], [-19], [27], [-6], [14], [15], [-2], [9], [14], [-24], [-16], [-8], [24], [-33], [10], [-12], [1], [0], [18], [40], [12], [-16], [-12], [-6], [-27], [-22], [-10], [-8], [-39], [-6], [27], [-16], [-22], [-42], [24], [18], [-38], [-25], [-11], [-12], [-29], [-34], [33], [-34], [-9], [-39], [12], [40], [-20], [3], [12], [28], [-42], [-28], [-18], [23], [-1], [5], [39], [2], [-2], [33], [-35], [33], [14], [30], [-27], [7], [-30], [-32], [-6], [40], [24], [-37], [-6], [4], [21], [-5], [33], [-14], [12], [10], [6], [-28], [-39], [44], [18], [18], [-12], [28], [-15], [-6], [9], [-58], [14], [-13], [21], [36], [58], [-42], [4], [4], [18], [-7], [42], [-25], [10], [-5], [45], [-11], [-18], [-63], [24], [-29], [32], [-31], [42], [-22], [-66], [-61], [-30], [60], [0], [-19], [-62], [-42], [-12], [-48], [10], [14], [-53], [24], [-21], [-20], [-36], [-21], [-23], [-8], [-18], [-8], [42], [0], [25], [13], [-63], [27], [-39], [-4], [-38], [-3], [-11], [48], [62], [6], [0], [-67], [36], [-65], [43], [50], [-21], [14], [24], [-26], [39], [-39], [-18], [24], [25], [32], [74], [30], [6], [11], [21], [-4], [-51], [-2], [63], [-75], [-47], [-30], [69], [14], [38], [-3], [56], [61], [18], [-62], [26], [-24], [-47], [0], [3], [58], [33], [-56], [34], [-34], [8], [-58], [29], [48], [-52], [-79], [15], [81], [-28], [39], [16], [-22], [-60], [16], [-27], [-20], [-33], [12], [54], [-36], [-39], [-59], [0], [-62], [42], [27], [-10], [-4], [-3], [2], [18], [20], [58], [-12], [58], [6], [-5], [51], [30], [-30], [-14], [15], [-29], [21], [12], [-64], [-6], [-44], [46], [30], [79], [24], [-35], [-55], [14], [0], [42], [-34], [54], [-86], [12], [35], [42], [38], [-36], [-5], [35], [-62], [90], [-66], [14], [-54], [36], [28], [10], [60], [39], [-23], [-38], [-6], [53], [-46], [42], [84], [0], [-48], [12], [-17], [-42], [48], [-69], [38], [-62], [-51], [-65], [11], [75], [1], [12], [54], [10], [34], [75], [-60], [56], [-66], [62], [42], [-12], [43], [84], [-23], [-39], [68], [-85], [40], [24], [-10], [-42], [-36], [-4], [42], [-14], [-56], [39], [-55], [-42], [-32], [42], [13], [-60], [-66], [-20], [37], [-18], [44], [-27], [-64], [-78], [-36], [-73], [-20], [102], [-48], [11], [-36], [-15], [-30], [-7], [24], [-48], [-16], [51], [72], [57], [-11], [-42], [22], [105], [23], [-75], [62], [30], [-46], [38], [-26], [16], [42], [-21], [-40], [18], [-35], [60], [32], [-57], [61], [70], [-2], [30], [36], [81], [4], [-60], [-10], [9], [-31], [108], [-25], [104], [24], [74], [-94], [-2], [-10], [-54], [30], [-56], [43], [-78], [-24], [101]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7350_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7350_2_a_cx();".
// 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_7350_2_a_cx(:prec:=1)
    chi := MakeCharacter_7350_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(3361) - 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_7350_2_a_cx();".
// 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_7350_2_a_cx( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7350_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-3, 1]>,<13,R![-5, 1]>,<17,R![0, 1]>,<19,R![5, 1]>,<23,R![-9, 1]>,<31,R![-10, 1]>],Snew);
    return Vf;
end function;