
// Make newform 24.11.h.a in Magma, downloaded from the LMFDB on 08 February 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_24_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_24_h_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_24_11_h_a();".
// 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_24_11_h_a();".
// 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_24_h();"
function MakeCharacter_24_h()
    N := 24;
    order := 2;
    char_gens := [7, 13, 17];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_24_h_Hecke();"
function MakeCharacter_24_h_Hecke( : Kf := false)
    N := 24;
    order := 2;
    char_gens := [7, 13, 17];
    char_values := [[1], [-1], [-1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 11;
    raw_aps := [[-32], [243], [5282], [24950], [-227050], [0], [0], [0], [0], [-36304750], [-8955802], [0], [0], [0], [0], [617985986], [-588563050], [0], [0], [0], [4081435250], [-5863410298], [-7877173786], [0], [-9031061950], [3376014050], [-5574599050], [24657472886], [0], [0], [-38407077850], [-3984276250], [0], [0], [58496336450], [-107720563498], [0], [0], [0], [-191661476686], [-124960401850], [0], [0], [-479356803550], [318952156514], [-288206813002], [0], [462290042150], [1119697159814], [0], [0], [0], [1610600868098], [-442817312650], [0], [0], [2807540044850], [-1493148594298], [0], [0], [0], [-1893610235614], [0], [0], [-5672603618350], [-2299316895214], [0], [-8605782917950], [-6980450486986], [0], [0], [0], [3790561564550], [0], [0], [0], [-16811198993950], [0], [0], [18314163740402], [21674832153350], [0], [0], [29176640679650], [32583959397398], [15941455600886], [0], [1942934157650], [3309819633650], [-41671569576250], [24680494573286], [0], [-49253614635850], [34367851503350], [0], [0], [55323954228050], [0], [0], [0], [0], [-106705095615886], [112538569458086], [0], [0], [122172580604450], [121393727484086], [0], [0], [-143946568853998], [13852934421350], [0], [0], [0], [111993443978198], [0], [0], [0], [225712449430514], [-50983652822650], [0], [-179601854933950], [243400293353186], [138953439760214], [0], [60061922575250], [0], [0], [-378536502222250], [0], [0], [0], [124677593244998], [0], [0], [-89318141150302], [-484706097662686], [0], [642983164059986], [0], [0], [-643389297716350], [749443106331350], [188456395345814], [0], [0], [0], [0], [0], [0], [0], [0], [0], [0], [0], [0], [1263000183921302], [0], [-1434725544045550], [-1348455847891150], [937214985939014], [0], [1192030799650550], [-1466189274476650], [0], [0], [-1255236354923098], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_24_h_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_24_11_h_a();".
// 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_24_11_h_a(:prec:=1)
    chi := MakeCharacter_24_h();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 11));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_24_11_h_a();".
// 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_24_11_h_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_24_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,11,sign)));
    Vf := Kernel([<5,R![-5282, 1]>],Snew);
    return Vf;
end function;