
// Make newform 7350.2.a.m in Magma, downloaded from the LMFDB on 23 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_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_7350_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_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], [0], [2], [2], [-2], [8], [-8], [-4], [-6], [-10], [2], [6], [6], [-12], [-2], [-14], [6], [10], [4], [-12], [14], [-14], [6], [4], [0], [-2], [10], [8], [-8], [-2], [-22], [20], [20], [-2], [2], [2], [-12], [-12], [-2], [22], [-20], [-6], [-16], [-4], [-20], [16], [10], [-14], [18], [-24], [-24], [30], [0], [-30], [20], [-2], [18], [20], [-24], [12], [-8], [6], [34], [28], [-12], [28], [-30], [6], [-14], [16], [-22], [12], [-26], [-4], [-38], [10], [-24], [-12], [-18], [-10], [-14], [36], [-4], [-26], [-32], [14], [-4], [-36], [-24], [-32], [-4], [-12], [26], [-14], [-18], [-4], [-30], [26], [18], [24], [10], [4], [-26], [0], [-14], [-34], [-8], [-12], [10], [-6], [22], [-24], [-42], [-36], [14], [14], [16], [22], [-20], [-12], [24], [-34], [-8], [26], [-8], [36], [2], [-4], [-32], [-24], [34], [34], [20], [-24], [-28], [-20], [18], [-22], [0], [-4], [-12], [-22], [-24], [10], [-10], [-42], [32], [-42], [18], [10], [-30], [-38], [22], [-16], [-22], [-42], [-14], [24], [34], [32], [0], [34], [-18], [16], [-6], [-26], [-28], [-40], [-42], [10], [-36], [-4], [-30], [20], [4], [-44], [50], [-28], [52], [58], [-14], [0], [46], [-38], [44], [-50], [24], [34], [24], [36], [-30], [32], [26], [-10], [-58], [22], [30], [-28], [40], [-2], [68], [40], [-8], [-68], [-12], [-50], [-18], [12], [22], [56], [-28], [64], [-52], [32], [30], [32], [-34], [-22], [24], [-46], [-28], [24], [26], [62], [-42], [24], [-68], [30], [22], [8], [46], [44], [66], [-8], [-62], [-60], [-8], [-48], [62], [-16], [26], [-22], [-24], [76], [0], [-52], [24], [54], [14], [-32], [20], [0], [60], [46], [76], [-48], [26], [-4], [4], [62], [6], [-66], [38], [12], [-18], [-50], [-46], [22], [-2], [-74], [-4], [-34], [32], [64], [-22], [50], [-12], [70], [20], [18], [50], [68], [-62], [-16], [30], [28], [26], [36], [48], [26], [0], [-18], [42], [-20], [-40], [48], [-4], [70], [-82], [88], [-24], [-12], [8], [-16], [14], [14], [26], [22], [-48], [82], [10], [-8], [-68], [28], [-62], [-14], [-66], [-50], [-68], [-18], [-56], [62], [48], [36], [-12], [-24], [-34], [-10], [-66], [-72], [-44], [-12], [68], [50], [2], [28], [-56], [82], [2], [70], [72], [66], [-4], [-30], [-10], [-72], [56], [66], [16], [-48], [-64], [-86], [-42], [72], [44], [-62], [24], [34], [-78], [72], [-16], [-28], [8], [56], [-52], [-26], [-72], [-66], [24], [16], [44], [62], [-24], [-30], [34], [-26], [-26], [42], [-62], [0], [22], [50], [54], [-12], [-78], [-14], [-38], [82], [-36], [0], [44], [-90], [-64], [0], [-30], [-44], [84], [10], [-78], [92], [18], [-38], [-8], [22], [-34], [28], [60], [62], [42], [84], [92], [-60], [46], [-78], [-8], [-2], [66], [-12], [-14], [88], [-40], [70], [-80], [-60], [-26], [2], [-64], [64], [-96], [-20], [22], [-58], [22], [-26], [-34], [22], [-76], [-24], [-6], [106], [26], [-52], [-90], [68], [-50], [84], [-66], [110], [-48], [60], [-90], [4], [-36], [38], [-16], [22], [10], [28], [4], [-24], [-10], [-68], [4], [-76], [-44], [-50], [10], [-12], [12], [-48], [66]];
    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_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_7350_2_a_m(: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_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_7350_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7350_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![0, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>,<19,R![2, 1]>,<23,R![-8, 1]>,<31,R![4, 1]>],Snew);
    return Vf;
end function;