
// Make newform 7650.2.a.s in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7650_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_7650_2_a_s();".
// 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_7650_2_a_s();".
// 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_7650_a();"
function MakeCharacter_7650_a()
    N := 7650;
    order := 1;
    char_gens := [5951, 6427, 5851];
    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_7650_a_Hecke(Kf)
    return MakeCharacter_7650_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 := [[-1], [0], [0], [0], [6], [-3], [1], [-7], [-8], [5], [5], [-8], [0], [4], [3], [9], [-5], [-3], [2], [15], [11], [8], [4], [1], [9], [12], [0], [-4], [9], [-13], [-1], [-6], [12], [0], [2], [-8], [-14], [4], [-6], [0], [12], [2], [0], [-26], [2], [7], [16], [15], [-9], [18], [27], [-10], [-20], [-12], [28], [27], [19], [-6], [2], [33], [-11], [-19], [24], [4], [22], [0], [7], [-13], [5], [-2], [-8], [28], [10], [14], [-20], [-27], [-30], [2], [10], [-31], [0], [20], [16], [-20], [16], [10], [6], [12], [20], [-31], [-14], [37], [2], [13], [-28], [-42], [12], [-28], [-14], [-34], [-13], [11], [32], [21], [-10], [-16], [36], [22], [32], [-2], [12], [17], [-29], [14], [-42], [-18], [44], [17], [30], [-19], [-16], [35], [-24], [45], [-16], [12], [23], [27], [29], [14], [-1], [24], [11], [45], [-14], [37], [-6], [47], [54], [-10], [-32], [27], [54], [-32], [18], [-31], [-30], [-57], [33], [-8], [10], [-38], [26], [-14], [-27], [-8], [-6], [34], [44], [5], [-9], [-44], [16], [-37], [38], [30], [-19], [-28], [-10], [-2], [-45], [2], [-57], [-24], [22], [12], [-14], [25], [29], [-14], [29], [30], [-22], [-44], [44], [-36], [12], [-44], [-12], [-8], [14], [3], [-3], [-60], [51], [-22], [-68], [-50], [-19], [17], [50], [13], [26], [-35], [-40], [-30], [30], [14], [-18], [-33], [-11], [-70], [52], [24], [51], [28], [47], [31], [12], [23], [-28], [-33], [2], [-14], [54], [-12], [-29], [15], [-15], [-26], [5], [-36], [50], [-33], [34], [4], [-4], [-24], [8], [-60], [-29], [12], [-64], [-20], [-15], [69], [-56], [-24], [8], [-7], [-79], [56], [53], [4], [-13], [-47], [-21], [52], [-46], [26], [52], [26], [38], [-68], [-21], [48], [-36], [-67], [5], [-26], [-7], [24], [-58], [-16], [-68], [65], [-68], [62], [43], [63], [36], [-63], [46], [46], [-4], [-12], [5], [38], [-70], [49], [-15], [65], [24], [-40], [6], [-33], [26], [-37], [-16], [4], [-16], [-3], [55], [-17], [74], [-18], [12], [9], [-74], [79], [-36], [-54], [-64], [-34], [45], [-85], [-35], [64], [2], [-15], [-22], [-11], [10], [68], [35], [-23], [18], [-39], [-24], [18], [90], [-4], [-4], [70], [32], [-60], [-20], [-14], [8], [-56], [-19], [-54], [33], [-22], [-66], [62], [30], [-52], [-17], [-48], [-52], [42], [54], [34], [12], [-14], [-66], [-4], [-38], [-27], [-58], [48], [16], [34], [-38], [20], [-54], [29], [52], [-63], [-50], [60], [44], [-44], [38], [44], [-27], [-24], [69], [-98], [-85], [22], [18], [53], [28], [-6], [14], [42], [-7], [-48], [-98], [-74], [31], [-92], [62], [-50], [-74], [78], [-104], [-20], [82], [54], [75], [17], [18], [28], [-67], [18], [-28], [-10], [-69], [-49], [-62], [-24], [-14], [21], [44], [38], [61], [21], [-26], [-74], [-58], [-91], [-29], [-55], [84], [90], [61], [-88], [-99], [65], [34], [30], [-12], [-39], [64], [-94], [20], [-104], [4], [-72], [-4], [-22], [16], [-52], [-40], [-6], [-38], [-64], [14], [-21], [7], [29], [-6], [46], [4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7650_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7650_2_a_s();".
// 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_7650_2_a_s(:prec:=1)
    chi := MakeCharacter_7650_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(3229) - 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_7650_2_a_s();".
// 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_7650_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7650_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![-6, 1]>,<13,R![3, 1]>,<19,R![7, 1]>,<23,R![8, 1]>,<29,R![-5, 1]>],Snew);
    return Vf;
end function;