
// Make newform 7600.2.a.r in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7600_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_7600_2_a_r();".
// 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_7600_2_a_r();".
// 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_7600_a();"
function MakeCharacter_7600_a()
    N := 7600;
    order := 1;
    char_gens := [4751, 5701, 5777, 401];
    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_7600_a_Hecke(Kf)
    return MakeCharacter_7600_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], [0], [2], [0], [-6], [-2], [1], [-2], [-2], [-4], [10], [-10], [6], [-6], [-6], [4], [2], [-2], [-12], [6], [-8], [-2], [2], [18], [-10], [-6], [-18], [10], [10], [6], [-16], [6], [-12], [10], [-12], [-18], [6], [14], [-6], [12], [-10], [-20], [-22], [-2], [-8], [-8], [2], [14], [-18], [22], [24], [14], [-16], [18], [-26], [26], [28], [14], [22], [22], [2], [22], [-12], [-10], [-30], [-16], [2], [-6], [-26], [6], [24], [26], [-6], [-20], [-6], [-6], [-10], [18], [-10], [-4], [10], [36], [18], [8], [30], [-34], [38], [-34], [-26], [10], [-24], [-26], [-32], [36], [-26], [30], [42], [26], [-34], [-2], [-26], [-38], [22], [-40], [38], [18], [-2], [-48], [-10], [22], [22], [-34], [-4], [12], [30], [-26], [18], [-18], [-36], [26], [10], [2], [-30], [8], [-30], [14], [-8], [2], [6], [28], [10], [-4], [-10], [-6], [-14], [42], [14], [-38], [-6], [40], [10], [38], [-18], [42], [0], [-2], [42], [4], [-54], [26], [14], [-34], [22], [-10], [-52], [-32], [54], [46], [14], [-22], [-14], [18], [0], [-6], [18], [28], [-2], [-6], [-58], [12], [30], [-60], [14], [32], [-6], [40], [6], [-6], [50], [-62], [8], [2], [-22], [-18], [-50], [2], [42], [54], [-60], [34], [-18], [-32], [-62], [-54], [10], [-50], [-10], [-2], [30], [-2], [-28], [-62], [58], [-60], [-18], [0], [-62], [10], [0], [-42], [42], [-26], [38], [-32], [6], [-14], [50], [-58], [30], [-22], [32], [42], [70], [-2], [-22], [10], [32], [-2], [40], [14], [20], [20], [54], [22], [-30], [22], [22], [60], [60], [-78], [32], [-74], [-34], [-30], [24], [18], [-64], [-44], [14], [-34], [18], [-46], [-58], [22], [4], [70], [-2], [42], [62], [34], [6], [30], [-46], [22], [-44], [-50], [58], [-22], [-66], [-50], [46], [78], [32], [-54], [70], [18], [30], [6], [0], [10], [-12], [66], [70], [2], [-12], [-26], [-22], [-32], [-50], [-14], [10], [66], [56], [26], [-46], [-36], [62], [-4], [-30], [-58], [58], [72], [2], [48], [50], [14], [-2], [24], [30], [70], [-18], [18], [26], [74], [-62], [-12], [60], [-58], [6], [-16], [-26], [-78], [-46], [-26], [-66], [52], [-30], [-82], [78], [18], [-6], [-56], [-42], [-80], [-86], [26], [66], [-42], [10], [-30], [30], [2], [36], [-14], [-60], [58], [-34], [60], [70], [56], [2], [-14], [-14], [-74], [10], [24], [56], [-74], [26], [78], [-6], [22], [-60], [-94], [-54], [22], [-30], [54], [-32], [76], [-34], [-14], [-20], [38], [-60], [-76], [-34], [-90], [34], [30], [-82], [26], [30], [-60], [66], [60], [70], [-74], [-42], [-2], [10], [-12], [10], [-76], [42], [24], [2], [-40], [42], [10], [6], [38], [42], [66], [-20], [-50], [2], [-94], [-36], [-38], [14], [50], [-64], [54], [-2], [-80], [-42], [22], [42], [-38], [-30], [58], [52], [-38], [-54], [-6], [-2], [40], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7600_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7600_2_a_r();".
// 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_7600_2_a_r(:prec:=1)
    chi := MakeCharacter_7600_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_7600_2_a_r();".
// 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_7600_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<7,R![-2, 1]>,<11,R![0, 1]>,<13,R![6, 1]>],Snew);
    return Vf;
end function;