
// Make newform 7650.2.a.bx in Magma, downloaded from the LMFDB on 24 August 2026.
// 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_bx();".
// 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_bx();".
// 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 := 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], [0], [-4], [2], [1], [4], [0], [2], [8], [-6], [6], [4], [0], [-10], [4], [-2], [-4], [0], [6], [8], [-12], [6], [14], [10], [8], [4], [14], [-14], [0], [20], [10], [20], [-6], [8], [18], [-12], [0], [14], [12], [22], [0], [-18], [6], [-16], [12], [0], [12], [6], [-22], [-16], [18], [-12], [2], [-24], [18], [-16], [10], [6], [-4], [6], [-20], [-16], [22], [30], [-4], [30], [4], [-2], [34], [-24], [-8], [-6], [4], [-32], [-22], [18], [30], [-38], [36], [6], [-24], [-18], [-32], [12], [-18], [22], [-30], [0], [-12], [40], [-32], [20], [-20], [0], [18], [6], [-44], [-2], [-28], [14], [-44], [6], [-44], [30], [12], [18], [40], [26], [-8], [-22], [-22], [4], [40], [-18], [-28], [24], [14], [-4], [-10], [-50], [6], [4], [12], [18], [-26], [24], [-8], [-14], [20], [0], [24], [-38], [6], [2], [38], [-12], [-2], [6], [20], [-54], [-16], [4], [30], [-16], [10], [-6], [-20], [48], [-14], [30], [-52], [-48], [12], [40], [-40], [-18], [-10], [-46], [-20], [58], [24], [20], [-46], [48], [-56], [-38], [18], [-10], [-44], [30], [-48], [-42], [0], [-10], [4], [-6], [-8], [14], [-48], [52], [26], [10], [-16], [-38], [34], [60], [10], [-24], [-50], [-44], [-60], [50], [-4], [-54], [-46], [34], [-62], [24], [-46], [40], [58], [34], [-28], [-66], [-64], [4], [38], [44], [-2], [-6], [16], [60], [-16], [10], [32], [-50], [32], [62], [38], [64], [-66], [-24], [4], [-10], [-6], [56], [-8], [-4], [2], [28], [-48], [54], [-12], [-48], [-46], [-10], [-36], [8], [-36], [-36], [8], [-50], [-14], [-16], [-40], [-60], [28], [-16], [50], [46], [72], [74], [-18], [60], [-10], [12], [6], [38], [24], [-12], [-58], [-62], [2], [-28], [-30], [70], [-36], [-58], [46], [-52], [70], [-16], [-18], [-56], [-4], [62], [10], [-20], [48], [-16], [32], [70], [-68], [-80], [-34], [-42], [40], [30], [50], [44], [-6], [-4], [50], [34], [-64], [-10], [44], [-52], [-74], [-18], [-56], [76], [36], [46], [-36], [-82], [-56], [42], [-8], [42], [-18], [76], [24], [-22], [-20], [-56], [14], [-18], [44], [22], [50], [16], [-54], [50], [-20], [-68], [24], [-58], [78], [30], [72], [20], [36], [-28], [30], [66], [-54], [-48], [26], [-6], [-6], [8], [-66], [36], [38], [-76], [-40], [54], [-76], [70], [2], [-24], [-42], [90], [-80], [-84], [18], [24], [-6], [-58], [0], [-20], [44], [-10], [78], [-88], [-6], [60], [4], [24], [-6], [88], [34], [-28], [-72], [-34], [14], [38], [34], [74], [-32], [2], [92], [-48], [-32], [-38], [-20], [-80], [34], [-58], [4], [12], [-8], [-42], [-64], [-42], [-60], [-6], [30], [2], [96], [-38], [10], [16], [-94], [78], [28], [52], [62], [22], [44], [12], [22], [-78], [-72], [80], [-14], [8], [82], [90], [-24], [68], [70], [-82], [12], [54], [-68], [0], [-6], [-36], [36], [-24], [66], [-2], [10], [-74], [28], [-40], [20], [94], [-58], [-64], [-46], [66], [68], [-40], [-94], [14], [60], [-48], [-84], [-10], [14], [58], [-34]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7650_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7650_2_a_bx();".
// 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_bx(: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_bx();".
// 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_bx( : 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![4, 1]>,<13,R![-2, 1]>,<19,R![-4, 1]>,<23,R![0, 1]>,<29,R![-2, 1]>],Snew);
    return Vf;
end function;