
// Make newform 7650.2.a.bf in Magma, downloaded from the LMFDB on 13 January 2025.
// 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_bf();".
// 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_bf();".
// 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]];
    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], [4], [-2], [6], [1], [0], [0], [-6], [0], [8], [-6], [-6], [8], [6], [12], [10], [-10], [6], [0], [0], [12], [0], [8], [-12], [-10], [0], [-14], [2], [14], [-10], [-2], [20], [8], [12], [-10], [-16], [20], [18], [16], [-14], [-16], [-12], [-18], [8], [-4], [-2], [20], [2], [22], [4], [-14], [-24], [6], [-24], [26], [4], [28], [8], [20], [-14], [-22], [-6], [-20], [-34], [-20], [8], [-12], [18], [-6], [-12], [32], [2], [4], [-32], [-24], [32], [10], [38], [14], [26], [-18], [14], [-16], [36], [-6], [-6], [-28], [-14], [36], [6], [24], [20], [-4], [44], [-28], [10], [-38], [-22], [8], [-34], [4], [36], [-12], [34], [28], [46], [12], [30], [8], [-26], [-26], [4], [36], [-18], [16], [40], [18], [-36], [-22], [28], [38], [12], [28], [-40], [-18], [34], [2], [38], [52], [8], [-8], [-38], [-20], [-38], [14], [-48], [18], [-30], [12], [-18], [12], [32], [14], [-42], [-12], [46], [-8], [-16], [32], [18], [34], [-24], [-28], [-6], [40], [54], [18], [34], [-36], [-18], [-54], [20], [42], [48], [8], [-28], [-30], [-26], [-36], [30], [-2], [-10], [-28], [-18], [-20], [-30], [-26], [-30], [-46], [22], [-32], [22], [-40], [0], [-60], [-46], [-34], [-18], [52], [-12], [8], [36], [44], [-46], [-2], [-60], [62], [0], [26], [0], [2], [-46], [12], [-6], [-40], [4], [50], [-12], [-24], [-60], [8], [-36], [-70], [-34], [22], [-48], [4], [70], [66], [16], [24], [64], [-28], [18], [-42], [26], [42], [54], [-42], [-36], [32], [12], [-2], [8], [6], [-42], [-22], [48], [-28], [60], [14], [70], [-62], [42], [12], [14], [-28], [48], [-38], [-14], [-32], [14], [-26], [24], [10], [4], [6], [14], [4], [-52], [-58], [-12], [-18], [72], [-56], [-8], [-40], [14], [70], [-34], [-10], [-40], [-26], [26], [28], [-2], [54], [-12], [-24], [-72], [-12], [-46], [-28], [-48], [-64], [-62], [-56], [-8], [-50], [32], [34], [-30], [-40], [-66], [16], [-30], [46], [-50], [-54], [-82], [-24], [-52], [4], [-76], [-12], [-58], [12], [46], [20], [-70], [-10], [-2], [-16], [-34], [-36], [-18], [84], [48], [-20], [-8], [4], [34], [14], [-42], [4], [-4], [12], [-90], [-42], [-66], [40], [-60], [36], [-28], [74], [50], [26], [74], [-62], [-38], [-42], [-44], [-42], [-66], [-50], [6], [2], [-22], [-88], [8], [-36], [8], [-22], [42], [12], [70], [62], [0], [-76], [-38], [32], [38], [30], [-62], [-18], [30], [-26], [-72], [60], [-36], [-76], [4], [-12], [70], [86], [78], [12], [26], [-58], [-58], [40], [-26], [-76], [28], [-32], [58], [16], [48], [30], [-62], [88], [10], [40], [-4], [-8], [-8], [76], [24], [94], [-82], [38], [-22], [56], [8], [26], [-20], [-10], [6], [-16], [34], [-100], [60], [-62], [14], [-18], [-60], [82], [72], [-92], [32], [-36], [-80], [4], [-46], [-36], [66], [-68], [58], [-42], [-36], [108], [108], [-28], [76], [-6], [38], [44], [-56], [-72], [38], [38], [-88], [54], [-86], [90], [0], [-42], [-82], [42], [18], [-48], [60], [46], [-48], [-26]];
    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_bf();".
// 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_bf(: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_bf();".
// 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_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7650_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, 1]>,<11,R![2, 1]>,<13,R![-6, 1]>,<19,R![0, 1]>,<23,R![0, 1]>,<29,R![6, 1]>],Snew);
    return Vf;
end function;