
// Make newform 7600.2.a.e in Magma, downloaded from the LMFDB on 23 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_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_7600_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_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], [-4], [0], [-8], [1], [6], [2], [8], [0], [-6], [10], [-6], [0], [4], [6], [2], [-16], [16], [-8], [10], [-10], [-4], [6], [-14], [2], [2], [12], [6], [-4], [-16], [4], [-14], [8], [-4], [18], [6], [-24], [12], [-26], [-16], [-4], [12], [-8], [-20], [26], [10], [26], [-8], [0], [2], [-28], [20], [6], [18], [-8], [-12], [-6], [-14], [16], [-6], [0], [8], [-8], [-12], [-20], [-2], [-30], [-8], [0], [-6], [-16], [-4], [-14], [-30], [-20], [-30], [22], [-12], [22], [0], [-20], [-32], [-30], [-18], [-8], [14], [-18], [-26], [8], [-10], [36], [-4], [-26], [2], [42], [-2], [-34], [-6], [28], [6], [-42], [-4], [32], [-18], [0], [16], [-38], [-2], [-4], [-16], [36], [0], [34], [26], [-38], [-28], [-36], [38], [-28], [-24], [30], [-12], [22], [-38], [-40], [2], [-4], [4], [-6], [48], [-36], [-22], [-18], [40], [-22], [16], [-42], [12], [30], [-2], [-6], [2], [0], [-44], [12], [12], [50], [-32], [-6], [34], [30], [2], [48], [-8], [6], [48], [-50], [-26], [12], [-22], [-4], [52], [-6], [0], [-28], [14], [12], [-4], [14], [0], [-24], [-32], [54], [-28], [6], [42], [-22], [2], [-12], [8], [12], [-18], [10], [-64], [54], [-10], [32], [-60], [-46], [52], [-18], [62], [52], [26], [-52], [0], [-18], [34], [-32], [48], [-18], [-60], [60], [-56], [46], [54], [60], [-16], [62], [22], [-54], [24], [-22], [-54], [-62], [-2], [-20], [-10], [56], [-50], [46], [74], [-14], [12], [-48], [62], [68], [28], [52], [-48], [-38], [-62], [-54], [-26], [-12], [-28], [-72], [54], [20], [30], [2], [-36], [24], [-38], [-36], [44], [-58], [-44], [-30], [42], [70], [-4], [-44], [22], [-70], [72], [-16], [26], [-62], [-70], [-8], [-32], [28], [-62], [-22], [-58], [20], [-2], [-46], [0], [-56], [28], [14], [38], [2], [-22], [52], [2], [48], [42], [22], [-50], [-24], [-80], [48], [72], [-66], [54], [30], [84], [-60], [8], [-38], [8], [4], [60], [46], [-64], [-48], [8], [-18], [20], [-4], [-38], [-30], [24], [52], [6], [10], [66], [-10], [74], [38], [12], [48], [-16], [-26], [84], [-32], [-18], [-14], [16], [-62], [-4], [62], [6], [-36], [-10], [0], [56], [-46], [-20], [-42], [82], [36], [2], [82], [-32], [-48], [58], [-16], [-24], [-36], [30], [10], [-72], [-28], [68], [28], [-10], [-46], [74], [-84], [48], [-28], [8], [-54], [4], [-70], [-34], [12], [-26], [4], [44], [10], [42], [36], [-44], [22], [-22], [40], [-52], [84], [-64], [64], [54], [-44], [14], [8], [50], [-32], [-20], [-78], [40], [-92], [10], [62], [-50], [32], [-36], [-42], [-72], [28], [-72], [-42], [-4], [-18], [-46], [48], [-18], [76], [-6], [40], [60], [-62], [86], [-12], [20], [-68], [-10], [36], [80], [-82], [72], [46], [72], [-86], [82], [104], [58], [-20], [4], [32], [-42], [-18], [-28], [-48]];
    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_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_7600_2_a_e(: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_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_7600_2_a_e( : 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![4, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;