
// Make newform 7448.2.a.a in Magma, downloaded from the LMFDB on 24 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_a();".
// 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_a();".
// 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], [-2], [-1], [0], [-3], [-4], [-2], [-1], [-7], [2], [-6], [-10], [-8], [7], [-9], [6], [-14], [-5], [14], [8], [-1], [10], [-17], [0], [-12], [-11], [0], [-18], [-16], [2], [8], [12], [-7], [-9], [3], [-12], [-7], [11], [12], [12], [-18], [16], [13], [-24], [-3], [-1], [22], [-20], [8], [-6], [-6], [-24], [-10], [15], [-10], [16], [-24], [-15], [-5], [-10], [13], [24], [16], [24], [-7], [-4], [6], [28], [13], [2], [-26], [-25], [-12], [4], [18], [-12], [6], [6], [-18], [34], [23], [-8], [0], [16], [-26], [12], [-32], [-1], [-7], [33], [29], [-7], [-2], [-27], [13], [-9], [6], [-10], [40], [33], [-34], [35], [0], [-6], [21], [5], [-36], [-11], [8], [6], [-14], [18], [5], [43], [25], [-10], [-36], [7], [-14], [6], [-16], [-50], [32], [-14], [-20], [-33], [-13], [8], [49], [14], [28], [6], [30], [-23], [-21], [-15], [-22], [28], [42], [-41], [-42], [31], [23], [8], [-4], [-54], [-37], [32], [17], [-30], [-8], [2], [-36], [-32], [-38], [-58], [39], [-51], [7], [0], [-36], [46], [-48], [6], [-18], [-8], [-38], [62], [-26], [-27], [-16], [-50], [-44], [-33], [30], [37], [-45], [-49], [-44], [-33], [47], [-18], [18], [10], [36], [30], [18], [-18], [-10], [-3], [-42], [-20], [-4], [-2], [-36], [54], [17], [-67], [-3], [-51], [-18], [-28], [-28], [-22], [-52], [-31], [-45], [-52], [-30], [-16], [43], [34], [47], [-42], [0], [42], [39], [-2], [-16], [13], [72], [-56], [-30], [-20], [28], [-59], [4], [18], [-8], [-45], [5], [-68], [28], [-42], [16], [20], [-5], [-18], [51], [-30], [-68], [47], [51], [10], [18], [-45], [-16], [28], [40], [37], [1], [-62], [-24], [20], [-21], [3], [51], [-46], [-24], [22], [38], [-48], [22], [-66], [-27], [-8], [6], [-23], [-10], [29], [40], [-20], [31], [-61], [2], [32], [53], [-80], [-60], [-15], [-38], [32], [51], [54], [-36], [-75], [54], [-46], [83], [-20], [-37], [33], [-16], [-16], [-24], [14], [64], [-10], [-66], [-63], [51], [50], [13], [-18], [27], [-64], [-58], [-18], [-48], [37], [3], [45], [-18], [68], [-36], [-50], [87], [12], [14], [-39], [-36], [2], [8], [-68], [-19], [2], [-38], [24], [-54], [34], [-11], [-56], [44], [-45], [-52], [59], [58], [56], [-7], [43], [38], [-15], [-70], [50], [-4], [72], [-38], [54], [26], [-3], [26], [-50], [89], [8], [80], [-36], [-57], [47], [21], [-16], [82], [78], [-54], [28], [49], [78], [-34], [-24], [-70], [-56], [-96], [-9], [32], [40], [74], [30], [-5], [29], [21], [14], [-42], [-49], [0], [-37], [-70], [-16], [37], [-59], [44], [42], [-52], [84], [43], [-16], [30], [-42], [10], [-48], [-80], [-87], [76], [95], [-12], [62], [-6], [-13], [-14], [-54], [-14], [-4], [-92], [45], [-66], [-44], [65], [-67], [-18], [38], [51], [6], [0], [2], [-57], [-20], [20], [8], [-78], [99], [-23], [-65]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7448_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![1, 1]>,<11,R![3, 1]>,<13,R![4, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;