
// Make newform 7623.2.a.e 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_e();".
// 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_e();".
// 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 := [[-1], [0], [-1], [-1], [0], [-5], [-7], [6], [4], [9], [-2], [9], [-7], [6], [-2], [-3], [-2], [-6], [8], [-6], [10], [14], [0], [9], [17], [-14], [-16], [-18], [1], [15], [8], [-12], [-2], [-20], [13], [2], [2], [10], [-12], [-6], [-6], [-5], [10], [-5], [-7], [6], [24], [-8], [6], [15], [-9], [-12], [-2], [-4], [17], [-8], [11], [8], [-11], [-18], [-14], [-9], [-28], [18], [-19], [14], [34], [-13], [-10], [15], [33], [-28], [-14], [-26], [-32], [16], [9], [-29], [-17], [9], [-32], [1], [30], [-11], [4], [-4], [19], [-33], [-21], [-26], [-24], [-14], [-22], [-28], [-16], [-16], [-6], [-30], [20], [-2], [2], [38], [-22], [-42], [32], [-15], [2], [37], [-10], [5], [-28], [-31], [-45], [14], [46], [-9], [-4], [-32], [-46], [-22], [-17], [14], [51], [-44], [-34], [-35], [26], [-6], [-6], [3], [2], [-10], [46], [17], [45], [-19], [-30], [40], [-2], [-18], [-2], [-10], [12], [34], [-29], [26], [-25], [-38], [0], [30], [-27], [49], [22], [-8], [-30], [36], [-14], [33], [29], [3], [-42], [-37], [-50], [28], [-57], [30], [-14], [47], [-1], [-6], [-20], [58], [-44], [-1], [44], [1], [20], [-47], [50], [7], [44], [-62], [30], [9], [-54], [-6], [-27], [12], [-49], [40], [-34], [-10], [-6], [27], [42], [-39], [35], [-62], [33], [-14], [-30], [18], [-21], [1], [-14], [-22], [-12], [36], [39], [-14], [-33], [-21], [30], [-28], [36], [41], [-58], [18], [36], [-51], [49], [36], [39], [44], [66], [-43], [69], [-18], [-8], [-10], [-59], [52], [-38], [-5], [-38], [-22], [30], [21], [68], [50], [-16], [8], [42], [-39], [26], [-6], [-4], [46], [-24], [-34], [-14], [-23], [0], [-19], [63], [62], [25], [42], [-57], [-62], [-32], [54], [71], [-54], [21], [34], [-42], [-30], [-40], [6], [-13], [36], [37], [-4], [-46], [14], [60], [37], [-74], [-18], [-46], [-58], [-44], [35], [-20], [-54], [30], [78], [-56], [17], [-35], [-38], [-9], [-36], [78], [34], [-86], [11], [30], [68], [54], [-15], [64], [50], [-22], [-50], [42], [-9], [18], [45], [-6], [-43], [-9], [56], [-18], [5], [-2], [68], [10], [-5], [-68], [7], [55], [-40], [71], [2], [18], [8], [-36], [-79], [25], [-66], [-10], [14], [-36], [-42], [49], [-22], [25], [-82], [-31], [54], [51], [20], [70], [2], [7], [-52], [6], [39], [14], [54], [-39], [54], [1], [-51], [-48], [-2], [-29], [-56], [-10], [54], [-12], [-4], [-12], [-53], [-57], [-4], [-33], [-18], [12], [38], [5], [-38], [1], [-32], [-2], [-55], [-87], [-54], [-46], [-69], [8], [-2], [-86], [12], [-18], [-37], [-64], [-38], [38], [-10], [4], [-8], [-12], [-2], [2], [-15], [-56], [25], [23], [83], [-48], [42], [74], [90], [-27], [-13], [-16], [-14], [-39], [-31], [-16], [-10], [-13], [87], [40], [0], [58], [90], [-23], [58], [-6], [-70], [13], [87], [-42], [26], [28], [-34]];
    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_e();".
// 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_e(: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_e();".
// 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_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7623_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<5,R![1, 1]>,<13,R![5, 1]>],Snew);
    return Vf;
end function;