
// Make newform 1150.2.a.h in Magma, downloaded from the LMFDB on 05 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1150_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_1150_2_a_h();".
// 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_1150_2_a_h();".
// 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_1150_a();"
function MakeCharacter_1150_a()
    N := 1150;
    order := 1;
    char_gens := [277, 51];
    v := [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_1150_a_Hecke( : Kf := Kf)
    return MakeCharacter_1150_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], [2], [2], [-2], [-1], [2], [0], [4], [6], [-10], [0], [4], [12], [-8], [10], [0], [-6], [-12], [-14], [-6], [-6], [-10], [8], [10], [0], [14], [-16], [12], [-6], [-4], [-4], [-8], [-12], [8], [-16], [6], [-16], [-24], [20], [-26], [-18], [-4], [-20], [16], [6], [8], [6], [-24], [22], [14], [-22], [16], [-14], [8], [14], [-6], [14], [-4], [16], [32], [-22], [-2], [-4], [-10], [-8], [26], [-2], [-12], [-12], [4], [22], [24], [0], [-14], [-18], [34], [-18], [28], [-12], [26], [8], [-8], [14], [10], [18], [-40], [2], [-8], [8], [24], [16], [-12], [18], [38], [34], [-6], [36], [-16], [14], [14], [14], [-30], [28], [-6], [16], [-26], [-32], [16], [-18], [-14], [-40], [14], [-10], [40], [42], [30], [-24], [2], [12], [-12], [-4], [0], [-28], [48], [36], [44], [-32], [24], [48], [48], [-30], [22], [40], [-18], [-8], [-30], [20], [30], [-24], [-6], [-30], [24], [-46], [-42], [40], [8], [-18], [18], [36], [-24], [-10], [20], [-4], [-18], [-58], [36], [-48], [-46], [16], [-38], [-6], [36], [-56], [-10], [-50], [10], [-50], [-14], [8], [-10], [56], [-6], [-16], [-42], [-32], [-28], [16], [14], [14], [6], [-44], [24], [-38], [30], [-58], [40], [10], [28], [50], [46], [26], [-18], [-22], [-20], [22], [-16], [24], [-24], [46], [2], [42], [38], [-28], [-4], [-18], [8], [18], [-50], [8], [-26], [-8], [30], [8], [-6], [-24], [-30], [30], [16], [-54], [-24], [-24], [54], [70], [24], [40], [56], [-2], [26], [40], [-58], [-10], [12], [-2], [8], [76], [-16], [-34], [-40], [32], [-2], [-18], [24], [-32], [-50], [-62], [-24], [48], [58], [72], [10], [66], [-56], [-16], [-22], [22], [46], [16], [42], [-22], [-32], [-66], [18], [56], [50], [-58], [-50], [34], [22], [30], [72], [30], [24], [-12], [14], [18], [-78], [0], [8], [-12], [-8], [72], [-48], [34], [64], [56], [30], [-28], [66], [-6], [-78], [22], [12], [-20], [18], [24], [-12], [-54], [-48], [52], [76], [14], [-82], [24], [-16], [48], [26], [56], [52], [78], [4], [4], [-70], [40], [-32], [62], [-26], [-14], [2], [-50], [-88], [42], [30], [10], [44], [-16], [88], [26], [18], [8], [12], [50], [-8], [-28], [-50], [-70], [16], [-46], [26], [-82], [-68], [8], [32], [86], [72], [-84], [-20], [48], [26], [-78], [68], [32], [54], [-32], [-18], [-58], [-64], [-16], [22], [-80], [-54], [-52], [-54], [10], [36], [14], [-80], [48], [-32], [-24], [88], [66], [4], [-36], [22], [-42], [-42], [-48], [-78], [-40], [90], [62], [16], [-8], [22], [34], [-68], [-22], [46], [-36], [-28], [84], [-22], [32], [30], [74], [22], [46], [-94], [12], [58], [22], [0], [-8], [-2], [22], [-24], [-22], [66], [-102], [-10], [78], [18], [-24], [40], [18], [-44], [88], [-60], [48], [-60], [62], [-38], [-66], [26], [84], [-64]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1150_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1150_2_a_h();".
// 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_1150_2_a_h(:prec:=1)
    chi := MakeCharacter_1150_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_1150_2_a_h();".
// 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_1150_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1150_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<7,R![-4, 1]>,<11,R![-2, 1]>],Snew);
    return Vf;
end function;