
// Make newform 7752.2.a.d in Magma, downloaded from the LMFDB on 28 April 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_7752_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_7752_2_a_d();".
// 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_7752_2_a_d();".
// 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_7752_a();"
function MakeCharacter_7752_a()
    N := 7752;
    order := 1;
    char_gens := [5815, 3877, 2585, 2281, 6937];
    v := [1, 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_7752_a_Hecke( : Kf := Kf)
    return MakeCharacter_7752_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 := [[0], [-1], [0], [-2], [4], [2], [-1], [-1], [6], [-4], [-2], [-4], [2], [-12], [-12], [6], [12], [-8], [4], [6], [10], [6], [-4], [6], [10], [-2], [4], [8], [0], [-18], [16], [4], [-22], [-4], [2], [-16], [-2], [0], [2], [-20], [4], [-20], [12], [-14], [-12], [22], [-24], [-20], [-12], [-2], [10], [-4], [10], [20], [-22], [4], [-20], [-24], [4], [6], [4], [-14], [12], [-10], [-34], [-24], [-20], [-10], [-16], [18], [30], [-16], [-22], [2], [20], [-28], [18], [-8], [-22], [14], [0], [-14], [-30], [-26], [-14], [-12], [10], [22], [-34], [4], [36], [18], [6], [12], [28], [30], [-14], [-14], [36], [16], [-16], [-10], [-12], [-6], [-8], [2], [12], [18], [-16], [-10], [22], [-10], [-14], [0], [-20], [18], [0], [28], [-16], [-4], [34], [-6], [16], [-44], [20], [6], [-36], [-6], [0], [-22], [20], [-6], [26], [22], [-6], [42], [2], [16], [-18], [-18], [-4], [-12], [-54], [-32], [46], [-26], [4], [-22], [4], [-48], [40], [-54], [36], [-58], [24], [30], [40], [-10], [22], [16], [20], [-34], [-52], [-20], [-46], [14], [-30], [-16], [2], [4], [-36], [-30], [-46], [-26], [-32], [-30], [-24], [-44], [-36], [22], [-4], [40], [4], [-6], [4], [22], [-36], [-36], [-14], [-50], [-34], [-8], [52], [42], [-24], [-10], [50], [-64], [2], [16], [24], [62], [-26], [34], [60], [50], [-36], [60], [34], [-4], [62], [26], [38], [12], [-10], [-14], [-44], [-30], [-18], [-18], [54], [-26], [38], [10], [36], [-46], [6], [-62], [48], [48], [18], [52], [20], [38], [-52], [-56], [70], [76], [-20], [48], [36], [36], [24], [-50], [-58], [30], [-38], [20], [-4], [48], [6], [18], [-68], [62], [66], [68], [40], [-76], [-40], [26], [-26], [36], [-72], [16], [-18], [52], [70], [22], [12], [42], [64], [20], [-70], [44], [-34], [20], [-12], [2], [6], [-28], [-20], [42], [50], [-6], [-4], [24], [46], [8], [-64], [14], [0], [72], [-46], [16], [-4], [-8], [-20], [-10], [-60], [36], [38], [-30], [-18], [-44], [0], [34], [-52], [-44], [-24], [86], [-34], [-16], [-70], [-44], [48], [38], [84], [-22], [-22], [-18], [-68], [22], [66], [20], [-66], [50], [40], [16], [-18], [-76], [44], [64], [22], [36], [16], [84], [-6], [14], [26], [-68], [2], [-70], [40], [4], [-90], [-36], [72], [-4], [-30], [-76], [-12], [-70], [18], [62], [6], [-22], [-32], [32], [78], [-84], [8], [10], [12], [-68], [76], [22], [72], [-40], [98], [-20], [72], [-34], [-82], [-24], [-68], [52], [-38], [18], [30], [10], [-4], [-42], [6], [90], [-4], [-18], [-40], [-66], [-40], [88], [-26], [-16], [84], [-84], [-28], [2], [-8], [62], [0], [54], [-76], [-46], [-12], [18], [42], [-70], [22], [82], [44], [24], [-82], [-46], [4], [-20], [78], [-84], [66], [70], [14], [36], [54], [-64], [-78], [-92], [-78], [-46], [-16], [18], [92], [74]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7752_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7752_2_a_d();".
// 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_7752_2_a_d(:prec:=1)
    chi := MakeCharacter_7752_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_7752_2_a_d();".
// 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_7752_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7752_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![2, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;