
// Make newform 7605.2.a.n in Magma, downloaded from the LMFDB on 14 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7605_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_7605_2_a_n();".
// 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_7605_2_a_n();".
// 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_7605_a();"
function MakeCharacter_7605_a()
    N := 7605;
    order := 1;
    char_gens := [6761, 1522, 6931];
    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_7605_a_Hecke(Kf)
    return MakeCharacter_7605_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], [0], [1], [3], [3], [0], [3], [0], [-3], [6], [6], [-9], [3], [10], [12], [3], [-12], [1], [0], [-9], [6], [1], [-6], [-15], [9], [18], [4], [9], [18], [-18], [2], [-12], [-18], [-13], [-3], [6], [-4], [3], [-12], [-6], [-24], [25], [12], [3], [-6], [-16], [4], [24], [0], [6], [-15], [27], [30], [-6], [18], [-24], [24], [6], [-8], [6], [-22], [-6], [15], [-6], [-8], [-6], [-24], [14], [9], [-18], [0], [24], [26], [4], [30], [30], [30], [3], [18], [-24], [18], [18], [0], [38], [19], [21], [15], [33], [-9], [15], [-15], [-3], [-15], [12], [-24], [-36], [-15], [-12], [2], [12], [26], [-30], [9], [-30], [-31], [21], [18], [-42], [-48], [-17], [-4], [21], [18], [30], [6], [-24], [27], [39], [18], [6], [-24], [-44], [39], [30], [-30], [12], [0], [30], [26], [15], [30], [-36], [-5], [16], [18], [12], [18], [12], [-51], [-24], [-42], [-33], [-14], [-12], [2], [-9], [9], [21], [-5], [-30], [-30], [0], [-16], [-33], [8], [-18], [-11], [-27], [34], [9], [24], [27], [-48], [18], [-24], [36], [-25], [8], [42], [-33], [-12], [-12], [-18], [33], [-5], [-12], [12], [45], [16], [-26], [-33], [-6], [-14], [-48], [6], [24], [52], [-3], [-54], [-51], [-16], [-48], [-7], [-21], [0], [-9], [42], [-10], [-30], [9], [54], [-55], [-39], [-55], [-15], [-51], [54], [-12], [45], [23], [2], [-48], [-26], [42], [60], [30], [-68], [0], [42], [-42], [59], [-66], [30], [24], [33], [-71], [9], [-12], [-46], [3], [40], [11], [-18], [-30], [-46], [-66], [48], [24], [-24], [-24], [36], [-25], [-14], [-36], [6], [-72], [0], [63], [42], [60], [54], [-54], [12], [-19], [-63], [-63], [-38], [-21], [42], [33], [26], [15], [-60], [56], [66], [37], [-27], [-27], [-15], [30], [14], [-63], [-51], [40], [40], [-15], [6], [-60], [24], [6], [9], [24], [24], [0], [36], [18], [34], [48], [24], [0], [30], [-60], [-78], [60], [62], [-30], [44], [-63], [30], [-33], [-14], [12], [-47], [-33], [-76], [27], [39], [-49], [-12], [-70], [81], [-66], [6], [40], [42], [25], [-27], [87], [-30], [-6], [-44], [-30], [12], [3], [-30], [-65], [48], [39], [33], [18], [-18], [-3], [-44], [-12], [24], [-6], [-78], [-42], [0], [70], [45], [-51], [-39], [-44], [36], [-48], [-11], [-45], [48], [-87], [84], [-33], [39], [80], [-85], [54], [75], [0], [-18], [-24], [-90], [-84], [-48], [60], [34], [8], [-72], [39], [25], [-60], [-20], [-18], [-72], [-41], [38], [21], [36], [-39], [-72], [-44], [-42], [48], [0], [36], [-66], [-36], [42], [-40], [-84], [-51], [-24], [66], [87], [100], [-21], [-68], [-12], [-66], [-5], [-18], [-30], [-75], [-51], [-24], [51], [43], [-18], [-30], [81], [-51], [-62], [-27], [69], [-53], [4], [-6], [33], [28], [48], [-21], [42], [63], [42], [-42], [-6], [-12], [-15], [-57], [42], [-12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7605_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7605_2_a_n();".
// 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_7605_2_a_n(:prec:=1)
    chi := MakeCharacter_7605_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_7605_2_a_n();".
// 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_7605_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7605_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<7,R![-3, 1]>,<11,R![-3, 1]>],Snew);
    return Vf;
end function;