
// Make newform 7600.2.a.d 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_d();".
// 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_d();".
// 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], [0], [4], [-4], [2], [-1], [-4], [-6], [8], [4], [-2], [4], [-8], [0], [8], [2], [-14], [8], [-6], [4], [16], [-18], [4], [10], [6], [6], [-2], [-12], [18], [-12], [2], [-12], [10], [-4], [-2], [-4], [-2], [8], [0], [-2], [-8], [-12], [26], [0], [0], [-2], [-18], [26], [22], [0], [-18], [-12], [16], [-4], [-26], [16], [-18], [-18], [0], [-16], [-18], [-24], [14], [12], [-32], [-36], [8], [22], [-14], [-32], [16], [-32], [-12], [-30], [-26], [22], [-22], [-38], [36], [10], [-12], [4], [-20], [24], [34], [-10], [-34], [4], [-24], [0], [-34], [-12], [20], [12], [10], [-30], [-14], [-22], [-2], [6], [10], [18], [28], [-34], [-8], [22], [-44], [-34], [-42], [-26], [-30], [20], [32], [30], [-16], [-12], [-18], [-16], [10], [-48], [0], [-6], [-12], [2], [46], [0], [40], [-54], [44], [26], [20], [34], [-10], [14], [28], [38], [24], [-6], [12], [14], [-32], [-54], [54], [-4], [42], [-40], [28], [-54], [-32], [14], [-20], [-30], [-26], [20], [-56], [-14], [50], [-18], [-32], [-24], [4], [-12], [-24], [10], [28], [-2], [-34], [-18], [-4], [-50], [-32], [22], [48], [-58], [-4], [42], [-14], [26], [52], [60], [20], [-16], [4], [-50], [-8], [-14], [-38], [-56], [16], [24], [68], [18], [-4], [60], [-22], [-30], [6], [0], [-30], [28], [-4], [62], [36], [-66], [8], [54], [-30], [-20], [-10], [42], [12], [-22], [-32], [62], [-40], [2], [-22], [10], [18], [40], [-50], [-4], [14], [-42], [-64], [-8], [2], [-52], [18], [44], [60], [62], [8], [72], [14], [66], [-36], [60], [-62], [44], [32], [18], [60], [48], [56], [24], [52], [72], [26], [66], [-52], [10], [74], [-36], [-42], [26], [-24], [-14], [46], [-54], [42], [-24], [-18], [-16], [-30], [62], [-2], [-54], [14], [34], [18], [56], [44], [48], [44], [10], [-50], [12], [18], [-24], [28], [-26], [40], [32], [54], [-28], [32], [18], [-58], [-28], [-36], [-48], [-56], [-54], [-16], [-78], [-12], [-84], [-22], [-24], [-64], [6], [-60], [-44], [-78], [10], [16], [66], [-16], [26], [-46], [18], [-12], [-66], [68], [-60], [46], [-30], [68], [-22], [58], [-14], [26], [-42], [-32], [-86], [-54], [14], [2], [8], [80], [84], [12], [4], [-18], [-24], [-38], [-68], [-64], [10], [-50], [80], [-8], [24], [-42], [90], [16], [-90], [68], [72], [2], [18], [-46], [40], [48], [-60], [-54], [-78], [-2], [-26], [-66], [-36], [40], [4], [-58], [74], [-2], [-60], [16], [-16], [30], [-88], [38], [20], [48], [14], [-62], [-28], [78], [14], [8], [-66], [-72], [-2], [40], [-74], [-92], [22], [10], [64], [-60], [16], [60], [56], [76], [54], [4], [-10], [-82], [54], [94], [-8], [86], [-88], [-34], [-62], [26], [60], [-8], [-30], [-18], [-4], [-22], [-66], [60], [2], [106], [-38], [-58], [-56], [-8], [28], [28], [-72], [106], [78], [-100], [-56]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : 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![0, 1]>,<11,R![-4, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;