
// Make newform 7600.2.a.a 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_a();".
// 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_a();".
// 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], [-3], [0], [-5], [4], [1], [3], [-1], [7], [-3], [2], [2], [-6], [6], [0], [13], [9], [-12], [-3], [0], [-11], [2], [-10], [2], [2], [-8], [4], [-13], [19], [0], [-6], [-16], [-9], [-16], [-4], [10], [-6], [22], [-2], [14], [8], [26], [-9], [-10], [22], [15], [5], [-2], [5], [-6], [-10], [11], [-12], [12], [-22], [8], [2], [27], [8], [-18], [-2], [27], [4], [-25], [1], [-9], [7], [-6], [-6], [14], [-7], [5], [8], [23], [33], [-4], [-4], [16], [-6], [22], [-14], [1], [36], [16], [-26], [36], [-22], [29], [18], [-8], [-8], [0], [-38], [-18], [-42], [-21], [2], [24], [-9], [16], [20], [-12], [20], [24], [-6], [7], [18], [-30], [26], [42], [-26], [20], [-14], [24], [-8], [8], [26], [-21], [16], [-33], [15], [44], [39], [-44], [42], [24], [2], [27], [23], [-36], [10], [-18], [26], [6], [11], [-47], [51], [-39], [-31], [25], [-37], [-52], [-43], [-3], [35], [4], [-42], [40], [28], [56], [-33], [10], [30], [-28], [-1], [12], [5], [-3], [-47], [-51], [-24], [24], [-44], [-28], [62], [-42], [-30], [50], [-28], [22], [-33], [21], [16], [19], [6], [30], [10], [30], [26], [2], [37], [39], [-62], [28], [31], [6], [-13], [52], [4], [-33], [-20], [40], [20], [7], [-12], [-50], [33], [-4], [27], [-7], [2], [-12], [-43], [-50], [50], [52], [59], [-11], [1], [-19], [-67], [48], [-24], [23], [18], [16], [-13], [34], [38], [-22], [58], [-10], [52], [59], [-21], [6], [-22], [12], [-44], [74], [32], [5], [-44], [-14], [-30], [-63], [-14], [-22], [-42], [-74], [-4], [-40], [9], [45], [-18], [0], [-8], [64], [-40], [-57], [44], [23], [-45], [40], [18], [-40], [-2], [28], [-42], [70], [62], [20], [-50], [53], [-6], [41], [51], [43], [67], [-44], [-55], [-33], [-15], [-59], [68], [13], [52], [-51], [72], [64], [22], [15], [47], [-63], [8], [4], [-43], [10], [-83], [2], [24], [64], [24], [-4], [78], [-86], [46], [14], [-29], [-28], [-18], [-45], [-45], [29], [22], [82], [-3], [3], [72], [8], [-8], [24], [-60], [-44], [44], [38], [38], [60], [-65], [-66], [-49], [23], [33], [28], [55], [-16], [-15], [87], [74], [30], [-28], [-1], [-37], [-54], [-2], [-78], [23], [-4], [-41], [77], [-41], [-5], [69], [58], [-27], [72], [-50], [-77], [56], [0], [60], [68], [60], [8], [-58], [-56], [15], [12], [11], [58], [-2], [-14], [8], [-15], [-72], [86], [-30], [44], [50], [42], [43], [67], [65], [56], [36], [-21], [-48], [62], [17], [-22], [2], [18], [33], [30], [101], [-70], [21], [-18], [-6], [-36], [-46], [-29], [42], [100], [-14], [26], [46], [-4], [52], [30], [14], [42], [2], [-62], [-70], [-13], [12], [-72], [-85], [22], [82], [94], [-39], [40], [13], [-50], [-64], [-46], [23], [48], [62], [-45], [97], [-9], [-4], [47], [-91], [-9], [-56], [39]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<7,R![5, 1]>,<11,R![-4, 1]>,<13,R![-1, 1]>],Snew);
    return Vf;
end function;