
// Make newform 7448.2.a.h in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7448_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_7448_2_a_h();".
// 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_7448_2_a_h();".
// 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_7448_a();"
function MakeCharacter_7448_a()
    N := 7448;
    order := 1;
    char_gens := [1863, 3725, 3041, 3137];
    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_7448_a_Hecke(Kf)
    return MakeCharacter_7448_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 := [[0], [0], [-3], [0], [-1], [0], [6], [1], [-3], [6], [-2], [0], [-6], [9], [3], [-2], [12], [-11], [-8], [6], [15], [-12], [-7], [8], [6], [-9], [14], [-4], [4], [-18], [-16], [8], [-3], [17], [-3], [6], [7], [-19], [14], [-8], [-10], [-22], [5], [-20], [3], [-5], [-20], [2], [-12], [-14], [14], [-24], [-4], [-3], [22], [-24], [-24], [-15], [17], [6], [-1], [-14], [-22], [0], [13], [-2], [-22], [6], [-5], [-22], [30], [-5], [8], [32], [-28], [8], [18], [-34], [0], [-2], [5], [2], [-6], [24], [-20], [28], [36], [-25], [-1], [13], [-33], [-3], [26], [-33], [-13], [-1], [-6], [12], [-18], [11], [-2], [-15], [-24], [-18], [23], [-7], [-12], [-3], [14], [16], [-32], [-6], [45], [1], [-23], [44], [32], [-25], [14], [50], [-32], [28], [48], [-42], [4], [-19], [13], [-32], [33], [-26], [0], [-32], [-40], [-53], [-25], [-19], [36], [-52], [-44], [-29], [30], [25], [-45], [16], [24], [-4], [9], [24], [-13], [18], [-32], [54], [-44], [58], [26], [50], [-37], [25], [43], [10], [12], [56], [-40], [8], [12], [-6], [52], [-42], [-8], [-29], [-6], [-46], [-36], [47], [-38], [17], [13], [41], [-2], [1], [43], [46], [-2], [22], [-32], [-10], [-30], [28], [32], [-43], [-40], [0], [14], [-2], [-32], [-6], [-7], [-1], [-19], [-27], [-40], [-26], [-38], [-40], [24], [39], [-9], [-54], [-26], [-22], [27], [38], [7], [-46], [-66], [4], [-57], [-38], [18], [39], [-62], [32], [-62], [-8], [24], [11], [-38], [-50], [14], [65], [63], [-50], [64], [48], [-44], [-56], [51], [50], [-75], [-30], [-56], [41], [11], [20], [48], [31], [44], [-58], [-4], [37], [15], [50], [-72], [68], [5], [45], [41], [-42], [12], [10], [24], [-26], [14], [-74], [-67], [66], [14], [-43], [8], [19], [-72], [18], [-17], [-65], [14], [52], [39], [-66], [-74], [-37], [32], [-64], [-33], [76], [12], [73], [-34], [-36], [63], [-24], [-3], [3], [64], [78], [4], [62], [2], [-42], [-24], [59], [67], [-14], [41], [-54], [-79], [72], [76], [46], [-16], [-57], [-85], [-21], [32], [-14], [-8], [-46], [45], [76], [62], [69], [2], [-42], [42], [30], [-39], [-50], [2], [-52], [24], [-26], [-73], [-12], [32], [-91], [20], [9], [46], [60], [69], [7], [32], [-43], [-4], [-14], [-56], [-6], [10], [82], [-46], [11], [10], [-2], [51], [24], [-58], [26], [-9], [45], [-19], [-26], [-18], [38], [22], [84], [-5], [6], [42], [-28], [-22], [-36], [10], [-9], [-6], [-8], [-34], [98], [23], [69], [-51], [90], [30], [71], [24], [59], [-26], [44], [65], [-33], [-56], [-70], [-48], [-22], [-87], [8], [-62], [-66], [-64], [-96], [-24], [-17], [-10], [3], [-36], [-64], [24], [27], [-98], [-20], [-64], [-84], [4], [-93], [-72], [52], [13], [-65], [-54], [-80], [-41], [74], [24], [92], [-57], [-54], [26], [10], [-6], [71], [-13], [-53]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7448_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7448_2_a_h();".
// 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_7448_2_a_h(:prec:=1)
    chi := MakeCharacter_7448_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_7448_2_a_h();".
// 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_7448_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7448_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![3, 1]>,<11,R![1, 1]>,<13,R![0, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;