
// Make newform 7623.2.a.p in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7623_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_7623_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_7623_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_7623_a();"
function MakeCharacter_7623_a()
    N := 7623;
    order := 1;
    char_gens := [848, 4357, 4600];
    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_7623_a_Hecke(Kf)
    return MakeCharacter_7623_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 := [[2], [0], [-1], [-1], [0], [4], [-1], [0], [4], [0], [-2], [6], [2], [3], [7], [12], [-5], [12], [5], [-6], [-2], [8], [-15], [9], [2], [13], [8], [12], [7], [-6], [11], [-15], [-8], [-2], [-2], [5], [2], [-20], [9], [9], [0], [16], [-2], [-5], [-22], [0], [-3], [10], [27], [30], [-6], [6], [10], [20], [-19], [-8], [2], [2], [25], [-24], [22], [9], [-22], [21], [20], [-28], [-5], [14], [-10], [-6], [30], [26], [-32], [19], [19], [1], [-6], [-2], [-14], [24], [-23], [25], [30], [-32], [-32], [-28], [34], [-27], [-3], [-32], [12], [37], [-31], [-34], [-1], [-13], [3], [-45], [38], [1], [32], [-22], [-31], [-18], [20], [30], [-31], [-17], [-16], [-4], [-34], [-7], [-36], [-16], [-5], [-18], [14], [-11], [-40], [2], [-8], [-19], [-15], [-44], [32], [-50], [-13], [-24], [18], [42], [32], [-16], [-41], [35], [-15], [14], [9], [-14], [-23], [48], [4], [32], [0], [16], [34], [-43], [56], [49], [-12], [6], [18], [-26], [-41], [37], [-12], [-12], [16], [-3], [2], [42], [12], [2], [31], [55], [42], [-48], [40], [20], [14], [-42], [-26], [46], [-14], [62], [-34], [7], [5], [4], [-22], [-2], [-19], [52], [-9], [6], [-42], [-54], [45], [48], [-25], [55], [-4], [-4], [27], [-42], [-36], [-21], [26], [-5], [-66], [-56], [-30], [18], [-6], [-20], [25], [62], [-54], [30], [36], [-2], [45], [18], [36], [-13], [-15], [8], [-13], [3], [60], [42], [-17], [36], [60], [-13], [-33], [38], [-6], [48], [-8], [44], [-29], [-68], [-17], [10], [-20], [17], [-18], [6], [-4], [11], [-52], [-52], [18], [66], [-34], [45], [56], [-20], [-15], [-34], [1], [-5], [48], [50], [54], [38], [-17], [-42], [57], [-2], [-65], [-60], [38], [30], [6], [-14], [-6], [51], [41], [-66], [-16], [48], [-68], [-61], [-16], [38], [-18], [-53], [46], [-33], [80], [65], [-8], [-28], [-2], [78], [45], [-48], [58], [-7], [-14], [28], [24], [-9], [27], [-59], [-86], [11], [-12], [-40], [72], [42], [-65], [-40], [-4], [-53], [-66], [-36], [42], [27], [-81], [44], [30], [-52], [18], [50], [40], [20], [4], [-14], [28], [-11], [-26], [-85], [62], [-76], [-3], [-34], [-66], [-52], [-50], [72], [8], [20], [60], [39], [-62], [41], [-26], [80], [11], [-42], [15], [35], [-2], [-34], [40], [23], [12], [93], [-76], [-15], [36], [66], [-74], [-33], [-69], [37], [82], [-8], [26], [-78], [36], [-10], [-12], [25], [-6], [77], [30], [6], [-36], [-10], [-40], [-56], [-71], [-29], [-2], [68], [-45], [60], [38], [-24], [-7], [58], [-68], [-3], [15], [98], [-1], [88], [-34], [-49], [22], [82], [-18], [-2], [-34], [15], [-80], [4], [14], [-58], [48], [93], [86], [42], [33], [-88], [-10], [76], [-48], [-10], [50], [23], [11], [15], [-68], [-48], [-41], [24], [-14], [88], [12], [-52], [-104], [21], [72], [74], [88], [-55]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7623_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7623_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_7623_2_a_p(:prec:=1)
    chi := MakeCharacter_7623_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_7623_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_7623_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7623_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 1]>,<5,R![1, 1]>,<13,R![-4, 1]>],Snew);
    return Vf;
end function;