
// Make newform 3150.2.a.s in Magma, downloaded from the LMFDB on 20 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3150_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_3150_2_a_s();".
// 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_3150_2_a_s();".
// 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_3150_a();"
function MakeCharacter_3150_a()
    N := 3150;
    order := 1;
    char_gens := [2801, 127, 451];
    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_3150_a_Hecke( : Kf := Kf)
    return MakeCharacter_3150_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], [1], [4], [-6], [4], [6], [0], [6], [-4], [-8], [-10], [2], [10], [14], [4], [-8], [-6], [2], [10], [16], [-8], [-2], [-2], [6], [8], [12], [-18], [-14], [8], [-4], [6], [10], [-6], [-16], [14], [10], [6], [-2], [24], [-8], [18], [10], [22], [-16], [16], [-16], [-20], [4], [-6], [22], [10], [12], [16], [4], [-18], [-12], [12], [8], [4], [2], [-4], [12], [10], [2], [16], [26], [20], [28], [0], [-34], [-16], [-32], [0], [-30], [-22], [26], [16], [18], [36], [14], [-30], [22], [28], [-4], [12], [14], [-2], [4], [-32], [-36], [-8], [-24], [-24], [30], [-38], [38], [-16], [18], [-30], [2], [-44], [24], [-36], [2], [4], [12], [-6], [-30], [32], [-48], [10], [-10], [-40], [-4], [-16], [2], [-38], [-20], [-8], [26], [6], [52], [-26], [-2], [-14], [-36], [0], [-22], [40], [0], [-40], [-40], [-6], [-26], [-54], [-52], [-42], [24], [-22], [30], [-4], [4], [-4], [32], [-38], [-48], [30], [44], [28], [26], [-6], [-2], [10], [-30], [8], [34], [22], [-58], [4], [-30], [8], [36], [-6], [-38], [-8], [-14], [22], [14], [44], [32], [6], [-6], [-4], [-14], [4], [18], [-56], [-20], [-20], [12], [40], [-44], [-36], [-54], [-40], [-32], [-50], [60], [6], [44], [16], [-10], [12], [-36], [22], [44], [0], [-14], [58], [-64], [30], [54], [-36], [58], [-4], [-36], [36], [-34], [-6], [6], [-16], [-32], [24], [-10], [32], [30], [20], [-6], [50], [-40], [24], [-36], [52], [-70], [-40], [2], [-40], [8], [-4], [-42], [40], [-60], [-28], [6], [-38], [-74], [-16], [-8], [-28], [-10], [48], [34], [-12], [60], [-8], [60], [56], [52], [-52], [30], [12], [54], [18], [64], [-10], [-52], [42], [-18], [-12], [68], [-32], [-2], [24], [-58], [-6], [14], [14], [6], [0], [-66], [-38], [48], [-14], [-8], [28], [-46], [54], [28], [2], [80], [-42], [12], [-32], [-66], [-34], [54], [-28], [42], [42], [28], [2], [-76], [12], [-78], [28], [26], [60], [28], [54], [-38], [16], [-36], [68], [-42], [4], [0], [-6], [-32], [-6], [6], [88], [-38], [12], [38], [-84], [30], [-42], [-4], [-74], [-22], [50], [24], [54], [-70], [20], [24], [44], [6], [62], [-54], [64], [-24], [-52], [-48], [-86], [84], [-2], [-32], [32], [74], [-6], [-64], [30], [-80], [-84], [-10], [0], [-54], [-46], [54], [30], [64], [-30], [-24], [12], [-44], [18], [-36], [-32], [-22], [-68], [4], [-4], [38], [2], [44], [-62], [-12], [-26], [20], [-42], [-28], [68], [-4], [54], [46], [-18], [-42], [-14], [-46], [24], [-18], [-30], [-6], [-48], [58], [-38], [-58], [98], [86], [-92], [36], [-62], [86], [-8], [-38], [48], [-30], [12], [42], [-28], [12], [22], [40], [-92], [-36], [-84], [84], [94], [-54], [36], [52], [-62], [-62], [6], [-72], [96], [22], [-62], [-62], [48], [44], [-6], [-70], [-20], [-12], [10], [80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3150_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3150_2_a_s();".
// 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_3150_2_a_s(:prec:=1)
    chi := MakeCharacter_3150_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_3150_2_a_s();".
// 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_3150_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3150_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-4, 1]>,<13,R![6, 1]>,<17,R![-4, 1]>,<19,R![-6, 1]>,<29,R![-6, 1]>],Snew);
    return Vf;
end function;