
// Make newform 3150.2.a.bn in Magma, downloaded from the LMFDB on 20 April 2024.
// 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_bn();".
// 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_bn();".
// 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)
    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+1)]];
    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], [0], [4], [-2], [-8], [8], [8], [4], [-8], [12], [8], [-4], [-6], [8], [-6], [8], [0], [-4], [8], [0], [-4], [-12], [12], [-8], [-4], [2], [2], [-8], [8], [10], [16], [0], [0], [12], [24], [-12], [6], [16], [2], [16], [-16], [22], [-12], [-12], [-8], [-24], [-2], [-26], [-16], [18], [8], [18], [0], [-28], [-12], [-32], [24], [-16], [-18], [-16], [0], [-28], [-14], [20], [-8], [36], [-6], [-18], [16], [-8], [-16], [-20], [-12], [24], [4], [-24], [22], [-40], [-26], [-16], [4], [-28], [12], [-16], [-16], [-20], [-8], [-32], [0], [40], [0], [28], [28], [-4], [12], [16], [2], [-8], [2], [-40], [24], [20], [-28], [16], [-30], [0], [-10], [8], [-24], [38], [16], [-24], [-8], [16], [-36], [18], [0], [-14], [-40], [-14], [4], [-32], [-40], [-42], [-16], [40], [20], [4], [0], [40], [24], [4], [-2], [18], [0], [-26], [-32], [24], [-48], [24], [20], [10], [32], [-28], [-58], [0], [-32], [40], [-12], [40], [12], [-40], [32], [-40], [-12], [28], [-28], [-60], [-54], [8], [24], [30], [4], [-32], [4], [-14], [46], [0], [54], [64], [40], [28], [-12], [20], [40], [24], [-10], [40], [56], [-8], [-54], [-16], [36], [48], [-32], [-26], [0], [-28], [-36], [-20], [36], [44], [10], [-34], [-24], [50], [-28], [-32], [28], [12], [-30], [-24], [-10], [12], [4], [48], [32], [-8], [-20], [-24], [72], [-32], [-58], [24], [60], [-48], [34], [22], [-4], [0], [-24], [-16], [6], [-6], [32], [-8], [48], [-40], [48], [-40], [48], [16], [-36], [-50], [6], [48], [0], [12], [16], [56], [18], [30], [48], [-8], [-72], [-12], [48], [-16], [60], [-56], [38], [-54], [-32], [26], [32], [18], [20], [40], [60], [-2], [44], [-14], [-8], [0], [-68], [24], [-6], [70], [24], [-44], [-64], [-60], [-24], [-12], [34], [-10], [72], [-60], [16], [28], [62], [32], [0], [-64], [-6], [-52], [84], [0], [-72], [6], [8], [32], [28], [52], [-50], [24], [32], [-4], [50], [-72], [36], [-44], [-56], [-44], [86], [0], [64], [4], [-72], [-24], [24], [-72], [-70], [72], [64], [-44], [-24], [-8], [-56], [20], [56], [22], [50], [-20], [-32], [-24], [-58], [-50], [-30], [52], [48], [-44], [-64], [-18], [14], [-38], [8], [64], [22], [36], [-32], [-30], [64], [-82], [-24], [-48], [62], [72], [16], [-64], [-24], [74], [-26], [16], [-24], [18], [72], [48], [-12], [16], [-16], [64], [24], [10], [-40], [54], [0], [-64], [24], [0], [28], [16], [56], [48], [-12], [-20], [-68], [12], [-38], [-72], [-66], [-40], [36], [-56], [68], [56], [36], [14], [-30], [16], [-80], [-48], [-72], [52], [-44], [-76], [-40], [-42], [-46], [-72], [-6], [-60], [92], [0], [-48], [-16], [-72], [60], [54], [-76], [60], [-88], [36], [-96], [56], [-30], [36], [-24], [68], [-64], [-88], [84], [54], [52], [24], [72], [64]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3150_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3150_2_a_bn();".
// 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_bn(: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_bn();".
// 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_bn( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3150_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![0, 1]>,<13,R![-4, 1]>,<17,R![2, 1]>,<19,R![8, 1]>,<29,R![-8, 1]>],Snew);
    return Vf;
end function;