
// Make newform 7245.2.a.l in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7245_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_7245_2_a_l();".
// 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_7245_2_a_l();".
// 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_7245_a();"
function MakeCharacter_7245_a()
    N := 7245;
    order := 1;
    char_gens := [5636, 5797, 5176, 1891];
    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_7245_a_Hecke(Kf)
    return MakeCharacter_7245_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], [0], [-1], [-1], [0], [2], [6], [-4], [1], [2], [-4], [-6], [-2], [4], [0], [-2], [8], [-2], [4], [-8], [-2], [4], [4], [6], [-14], [18], [-16], [4], [6], [-6], [-16], [-16], [-6], [0], [18], [0], [-10], [-12], [-8], [-18], [-4], [-10], [4], [-6], [-14], [-16], [-20], [16], [-20], [6], [-2], [0], [-30], [-4], [-6], [0], [10], [12], [-2], [-2], [-12], [10], [20], [-12], [-14], [10], [-20], [6], [4], [-18], [-14], [-28], [-16], [26], [-8], [24], [18], [26], [30], [10], [-20], [-10], [-20], [18], [12], [-20], [-18], [22], [42], [24], [12], [24], [16], [-28], [-4], [-32], [18], [6], [28], [14], [-4], [6], [28], [6], [-40], [38], [-28], [-30], [16], [2], [8], [2], [18], [4], [-20], [-26], [12], [-16], [26], [-48], [22], [10], [-14], [-12], [-8], [-30], [-26], [20], [32], [14], [-20], [-24], [36], [2], [22], [50], [-14], [-4], [-46], [-26], [-32], [-22], [-16], [-12], [-10], [32], [-14], [-6], [0], [-32], [-18], [-18], [28], [0], [-28], [-36], [4], [30], [-22], [18], [-12], [-6], [-8], [12], [42], [48], [56], [-54], [-38], [46], [0], [-42], [-20], [22], [60], [22], [-60], [-54], [-24], [-18], [-8], [-12], [6], [-30], [0], [42], [-2], [-44], [-22], [60], [6], [-36], [32], [2], [60], [46], [58], [-38], [30], [40], [-30], [-44], [10], [-30], [60], [-66], [-56], [-20], [-10], [-40], [-54], [-6], [24], [4], [-12], [42], [40], [-18], [-16], [42], [6], [24], [30], [-40], [-12], [38], [62], [-48], [-16], [-68], [-34], [4], [20], [38], [-20], [0], [-14], [-74], [12], [-44], [4], [24], [-8], [-50], [66], [60], [16], [-60], [56], [-40], [2], [-50], [48], [-70], [-34], [4], [-2], [-12], [-18], [62], [48], [12], [70], [54], [-14], [-76], [-30], [22], [-68], [-38], [6], [-12], [34], [28], [-18], [80], [36], [-34], [2], [-36], [-48], [68], [-40], [-42], [4], [-64], [54], [62], [44], [30], [-46], [60], [86], [-4], [-74], [18], [-16], [-74], [-48], [28], [-54], [-18], [-12], [12], [-8], [-38], [44], [-50], [-20], [38], [-24], [26], [-34], [4], [40], [58], [-56], [-24], [26], [30], [52], [-2], [18], [56], [2], [2], [-8], [44], [8], [62], [14], [58], [-44], [4], [-80], [76], [54], [38], [-30], [-88], [-26], [58], [50], [-44], [-26], [12], [-42], [28], [24], [74], [48], [-46], [-78], [80], [6], [74], [88], [20], [6], [-72], [-14], [-42], [-48], [24], [-60], [26], [-26], [64], [-30], [-60], [-8], [-80], [50], [72], [78], [40], [-20], [-14], [62], [-10], [34], [-46], [48], [62], [28], [0], [48], [50], [68], [48], [18], [86], [-44], [-20], [-60], [6], [-72], [54], [-16], [42], [-58], [54], [-96], [54], [-62], [12], [98], [62], [-4], [-8], [22], [-78], [-44], [-64], [-2], [42], [-16], [-56], [-18], [56], [10], [22], [48], [0], [-18], [-26], [-44], [-106], [40], [-36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7245_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7245_2_a_l();".
// 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_7245_2_a_l(:prec:=1)
    chi := MakeCharacter_7245_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_7245_2_a_l();".
// 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_7245_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7245_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<11,R![0, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;