
// Make newform 7623.2.a.k 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_k();".
// 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_k();".
// 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 := [[0], [0], [3], [-1], [0], [4], [-3], [-2], [0], [-6], [2], [-10], [6], [-11], [3], [-12], [3], [-8], [5], [12], [-8], [-8], [15], [-15], [14], [3], [-4], [-12], [1], [18], [1], [15], [-6], [-20], [6], [-17], [-4], [-16], [-21], [3], [-24], [20], [18], [-23], [6], [8], [7], [-16], [9], [8], [-18], [6], [10], [12], [21], [0], [18], [16], [-17], [-24], [4], [-9], [-14], [-27], [-16], [-12], [-25], [-14], [-18], [-2], [-18], [24], [14], [-35], [-1], [21], [-12], [2], [0], [-38], [9], [-7], [-12], [38], [22], [-24], [30], [-29], [27], [-4], [36], [-9], [-7], [24], [-1], [-39], [3], [-9], [-38], [-5], [28], [18], [39], [-42], [4], [20], [21], [-15], [-30], [-8], [28], [-41], [18], [38], [-25], [-6], [8], [21], [-6], [-6], [20], [-29], [-45], [-6], [8], [-42], [35], [-48], [26], [-20], [16], [24], [23], [23], [15], [-44], [-27], [22], [-3], [-36], [-2], [-6], [-16], [-24], [-16], [-27], [40], [-9], [26], [-6], [10], [-30], [-13], [-9], [-52], [-6], [-20], [21], [34], [30], [30], [54], [13], [3], [-12], [-24], [32], [46], [22], [-54], [60], [-16], [30], [34], [-4], [-21], [-17], [24], [34], [-8], [17], [48], [-13], [54], [18], [-18], [7], [26], [13], [21], [32], [54], [23], [-30], [30], [-33], [22], [-49], [-42], [12], [-24], [-2], [2], [-20], [-27], [-30], [-10], [66], [66], [-40], [-17], [-42], [8], [39], [-21], [-34], [-59], [21], [0], [42], [49], [58], [72], [-25], [-27], [10], [-6], [18], [-2], [30], [-37], [16], [-23], [-54], [44], [27], [14], [36], [24], [27], [-6], [28], [-22], [-34], [18], [27], [2], [12], [-5], [0], [-41], [-3], [-24], [-28], [18], [12], [23], [22], [-63], [58], [1], [-42], [-32], [64], [30], [26], [6], [-69], [-53], [48], [-10], [68], [80], [1], [-32], [32], [12], [-23], [70], [57], [0], [77], [12], [34], [82], [-66], [5], [12], [-46], [15], [12], [12], [-54], [81], [37], [3], [8], [-21], [48], [28], [-56], [-60], [61], [66], [-52], [-1], [-66], [-22], [-36], [25], [81], [60], [48], [-4], [-72], [70], [24], [-60], [-10], [-18], [-50], [17], [18], [-61], [-72], [-16], [65], [8], [18], [-78], [-62], [12], [10], [-12], [-92], [-33], [86], [-33], [68], [-38], [35], [-72], [-87], [-37], [54], [-12], [50], [47], [-60], [-3], [-68], [17], [-36], [28], [22], [-51], [87], [63], [36], [-54], [22], [-18], [-30], [-66], [-94], [17], [18], [79], [-62], [-84], [-64], [-54], [72], [94], [-43], [63], [42], [64], [-15], [-26], [30], [-36], [43], [54], [-56], [81], [71], [-76], [25], [-96], [-10], [3], [-6], [-40], [84], [22], [-50], [27], [-52], [24], [-50], [24], [-32], [81], [54], [58], [-31], [78], [-64], [12], [82], [-12], [30], [-83], [25], [-69], [90], [-82], [15], [-36], [36], [34], [-102], [12], [-16], [57], [-6], [0], [-82], [-9]];
    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_k();".
// 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_k(: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_k();".
// 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_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7623_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<5,R![-3, 1]>,<13,R![-4, 1]>],Snew);
    return Vf;
end function;