
// Make newform 7350.2.a.p in Magma, downloaded from the LMFDB on 16 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_p();".
// 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_p();".
// 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], [4], [-2], [2], [4], [8], [6], [8], [2], [-2], [12], [-8], [-6], [-4], [2], [-12], [8], [-14], [0], [12], [-2], [10], [-6], [8], [20], [-2], [14], [0], [-12], [-10], [-20], [-18], [8], [14], [-12], [-16], [14], [-20], [10], [-16], [14], [-6], [16], [20], [0], [12], [26], [-10], [-16], [-18], [-4], [2], [-24], [-30], [24], [-14], [10], [20], [6], [-4], [-8], [34], [-14], [28], [-2], [-12], [34], [18], [-8], [-32], [-14], [28], [24], [-18], [-2], [18], [-10], [-12], [-26], [0], [-6], [-32], [4], [-14], [38], [34], [16], [-20], [0], [24], [36], [4], [16], [-14], [30], [-28], [14], [-28], [2], [-36], [-6], [-20], [26], [-12], [18], [8], [22], [0], [34], [-10], [44], [24], [-30], [-20], [-32], [34], [28], [10], [-2], [6], [4], [20], [6], [-26], [0], [-24], [30], [36], [-24], [-32], [34], [-2], [14], [-26], [-4], [30], [-38], [-36], [46], [8], [-12], [34], [24], [-26], [-22], [28], [0], [10], [-42], [-28], [16], [12], [48], [-8], [-10], [18], [18], [12], [6], [8], [-36], [-18], [32], [-32], [38], [50], [-42], [4], [-62], [56], [-26], [40], [14], [-52], [-18], [8], [18], [-64], [-28], [50], [-22], [-16], [-38], [26], [-4], [-22], [16], [26], [-44], [4], [50], [12], [-54], [-46], [-22], [-30], [-48], [54], [-24], [-10], [30], [28], [-2], [24], [60], [10], [12], [14], [42], [-40], [-12], [-24], [-58], [-32], [22], [-8], [50], [-58], [-48], [18], [-16], [44], [22], [26], [0], [40], [-12], [26], [-44], [-48], [-54], [-76], [56], [-34], [10], [-44], [40], [-36], [-4], [72], [-34], [2], [-40], [32], [-12], [-68], [-48], [-22], [-42], [24], [-26], [-18], [12], [38], [36], [6], [2], [-16], [44], [26], [-2], [-62], [-12], [54], [30], [-20], [-54], [50], [20], [-78], [0], [42], [8], [-76], [14], [10], [-28], [24], [-40], [-16], [26], [-28], [0], [-2], [58], [80], [6], [22], [-68], [-26], [12], [-54], [-46], [40], [-42], [-4], [28], [2], [2], [-64], [76], [44], [-2], [36], [18], [-24], [34], [-24], [62], [-30], [36], [-72], [-42], [36], [64], [58], [-30], [-28], [6], [-30], [-64], [-74], [14], [52], [36], [48], [-6], [-18], [18], [-56], [28], [-4], [-60], [-2], [18], [22], [-80], [-46], [-58], [10], [-8], [-46], [-68], [-6], [-52], [64], [22], [20], [38], [38], [-32], [6], [74], [48], [-68], [78], [8], [-30], [-2], [48], [20], [12], [-58], [14], [24], [58], [-20], [12], [64], [62], [16], [10], [68], [-80], [-70], [-10], [-78], [-46], [54], [-8], [-98], [-44], [-16], [-64], [22], [-4], [-72], [34], [-58], [68], [-20], [-56], [38], [-24], [46], [92], [-18], [-94], [-2], [32], [90], [-54], [-64], [-6], [34], [60], [-28], [-6], [-22], [-12], [-44], [6], [2], [48], [-56], [18], [64], [-26], [6], [-48], [-36], [50], [-18], [-20], [42], [-52], [72], [22], [-20], [76], [24], [-34], [-90], [106], [70], [60], [-16], [20], [-30], [-58], [96], [30], [-2], [-44], [72], [46], [50], [20], [72], [44], [-2], [46], [30], [14], [-60], [54], [-58], [-52], [24], [76], [38], [-12], [94], [-40], [-44], [-62], [4], [16], [60], [-96], [34]];
    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_p();".
// 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_p(: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_p();".
// 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_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7350_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-4, 1]>,<13,R![2, 1]>,<17,R![-2, 1]>,<19,R![-4, 1]>,<23,R![-8, 1]>,<31,R![-8, 1]>],Snew);
    return Vf;
end function;