
// Make newform 7350.2.a.ce in Magma, downloaded from the LMFDB on 11 June 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_ce();".
// 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_ce();".
// 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]];
    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], [4], [0], [-4], [0], [9], [-1], [-8], [0], [10], [6], [3], [3], [-10], [10], [-6], [-2], [-1], [9], [6], [1], [-18], [-8], [3], [14], [0], [-5], [-9], [-18], [2], [18], [-1], [4], [16], [-6], [-18], [12], [8], [0], [19], [-6], [20], [14], [19], [27], [-4], [24], [-24], [-1], [27], [-6], [6], [21], [11], [-8], [6], [-14], [-33], [-8], [24], [31], [-9], [20], [7], [-12], [26], [-24], [30], [19], [-8], [8], [-18], [-6], [4], [24], [-25], [0], [-22], [12], [34], [35], [33], [12], [1], [30], [-8], [-36], [-18], [-41], [-33], [2], [12], [-3], [-18], [4], [26], [-8], [-3], [-39], [-36], [-34], [-23], [-21], [-24], [-18], [11], [7], [16], [6], [-34], [-7], [-30], [34], [18], [-3], [-24], [14], [-29], [33], [-33], [8], [-15], [-10], [-18], [13], [10], [50], [-42], [-7], [-38], [12], [-19], [-6], [-50], [33], [0], [2], [-3], [40], [-15], [-4], [-24], [10], [42], [50], [-6], [-32], [-6], [-32], [24], [-8], [6], [8], [-6], [-35], [-9], [0], [6], [1], [-39], [-42], [-36], [-13], [-14], [-49], [33], [-27], [-58], [-60], [-5], [41], [-36], [2], [15], [16], [26], [49], [45], [34], [18], [54], [3], [34], [22], [-13], [12], [31], [51], [20], [-6], [33], [54], [-10], [-62], [-30], [-42], [-21], [8], [-14], [26], [48], [-33], [-16], [27], [24], [-58], [46], [-9], [1], [-69], [-36], [2], [-5], [0], [-18], [-30], [-40], [56], [-30], [-11], [15], [-64], [-42], [-36], [13], [-36], [-74], [-16], [-49], [6], [58], [-60], [35], [-15], [60], [-72], [-12], [56], [-5], [-70], [-12], [-60], [25], [60], [32], [-24], [-2], [-54], [-30], [2], [-69], [51], [-22], [52], [3], [-29], [16], [9], [62], [-44], [6], [-70], [-81], [-12], [4], [-39], [-16], [34], [43], [17], [-5], [16], [3], [-16], [38], [48], [54], [-37], [-12], [20], [-50], [-24], [25], [9], [-7], [-12], [-6], [-45], [36], [-60], [4], [-9], [80], [-42], [36], [46], [58], [-39], [-7], [-63], [2], [-50], [-39], [-58], [6], [-14], [-6], [30], [60], [-32], [-48], [38], [-21], [48], [-53], [-24], [-64], [46], [33], [16], [-48], [-37], [62], [22], [-18], [-75], [2], [81], [32], [24], [-10], [12], [20], [-54], [41], [19], [82], [36], [-87], [77], [-9], [9], [-10], [46], [-6], [15], [-4], [-11], [-42], [55], [-10], [-42], [-30], [-45], [-12], [-6], [46], [24], [30], [12], [-56], [-35], [42], [-56], [-7], [-51], [-64], [66], [9], [80], [16], [24], [12], [-11], [66], [-62], [-6], [-30], [16], [84], [50], [-42], [32], [4], [4], [12], [-55], [6], [-9], [58], [78], [-23], [-79], [42], [80], [-33], [-76], [-48], [4], [6], [21], [29], [-62], [90], [-26], [-81], [1], [21], [-9], [26], [70], [-75], [-84], [-56], [-72], [6], [51], [106], [78], [-33], [70], [-78], [27], [48], [38], [-12], [86], [87], [104], [66], [10], [-48], [-19], [20], [-26], [-25], [-51], [-102], [68], [-18], [53], [6], [-104], [12], [74], [-34], [-2], [42], [45], [-36], [-77], [-33], [-28], [-27], [-104], [-72], [-70], [32], [87], [-16], [22], [43], [44], [9], [-24], [-70], [61], [12], [42], [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_ce();".
// 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_ce(: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_ce();".
// 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_ce( : 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![-4, 1]>,<17,R![0, 1]>,<19,R![4, 1]>,<23,R![0, 1]>,<31,R![1, 1]>],Snew);
    return Vf;
end function;