
// Make newform 3150.2.a.v in Magma, downloaded from the LMFDB on 10 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_v();".
// 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_v();".
// 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], [-6], [2], [-2], [4], [-4], [-2], [-2], [-10], [-6], [-2], [2], [6], [4], [-12], [10], [-12], [-2], [-16], [-12], [-14], [18], [-6], [-16], [-4], [14], [14], [-4], [-20], [-2], [-8], [6], [12], [18], [6], [18], [0], [14], [8], [-16], [4], [10], [18], [-12], [8], [-24], [-8], [22], [-8], [-18], [16], [-18], [12], [-14], [-30], [2], [12], [4], [-12], [-32], [0], [34], [-26], [20], [20], [-12], [24], [-30], [-4], [8], [34], [-20], [30], [6], [14], [24], [10], [16], [-2], [4], [2], [-10], [-12], [-20], [28], [-34], [-16], [-20], [0], [-4], [6], [4], [10], [34], [42], [-4], [-22], [26], [38], [-4], [24], [-4], [-14], [28], [-18], [24], [-10], [-8], [-18], [26], [4], [-12], [-16], [-28], [-42], [-6], [6], [28], [16], [-8], [28], [-8], [22], [18], [40], [32], [-26], [28], [0], [-20], [-30], [38], [34], [-36], [-48], [40], [-52], [0], [-18], [24], [-4], [-20], [-8], [22], [-34], [-24], [-32], [-22], [-18], [-46], [-42], [-38], [16], [20], [6], [38], [-50], [12], [42], [-56], [24], [30], [-14], [4], [14], [-2], [-56], [-30], [-24], [8], [44], [10], [6], [-52], [-34], [-24], [-52], [-52], [28], [-46], [30], [20], [-54], [-18], [-28], [10], [24], [-14], [36], [-20], [-30], [44], [-2], [14], [6], [-10], [-34], [50], [-38], [-42], [-30], [-24], [-4], [18], [-68], [-48], [32], [32], [-34], [24], [12], [32], [26], [36], [-6], [-12], [34], [-38], [26], [36], [-60], [20], [18], [30], [-32], [8], [30], [70], [-36], [8], [0], [-40], [-42], [34], [-6], [-18], [-72], [12], [28], [16], [-54], [-30], [0], [-64], [-36], [36], [-72], [10], [18], [16], [38], [-36], [38], [-58], [-68], [-12], [34], [-40], [12], [28], [-2], [-38], [-56], [18], [-22], [78], [-54], [-32], [18], [-46], [-16], [-54], [-48], [60], [34], [-46], [-60], [42], [-80], [-26], [20], [4], [-12], [4], [-18], [62], [-66], [66], [24], [54], [28], [22], [-46], [34], [16], [0], [-32], [22], [-46], [24], [84], [52], [32], [52], [-28], [60], [-46], [58], [30], [-24], [-30], [72], [26], [0], [-4], [-22], [-12], [16], [-16], [-2], [-56], [-74], [-10], [-44], [-64], [0], [-34], [22], [18], [70], [24], [-92], [56], [-6], [-22], [34], [-8], [-46], [34], [54], [-32], [66], [50], [-40], [-18], [-48], [-28], [-24], [92], [22], [-48], [-94], [82], [-64], [-24], [30], [-36], [-6], [6], [-44], [-50], [56], [60], [-56], [52], [6], [-6], [8], [60], [-78], [10], [-46], [100], [84], [18], [2], [38], [78], [70], [-12], [-78], [-40], [-18], [16], [-50], [34], [-66], [14], [12], [14], [-40], [36], [-72], [-74], [38], [-28], [34], [84], [-46], [12], [42], [-90], [14], [42], [12], [-20], [60], [-18], [-18], [20], [36], [-68], [30], [24], [-8], [18], [54], [30], [-38], [8], [-104], [90], [12], [-44], [-12], [36], [-40]];
    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_v();".
// 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_v(: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_v();".
// 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_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3150_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![6, 1]>,<13,R![-2, 1]>,<17,R![2, 1]>,<19,R![-4, 1]>,<29,R![2, 1]>],Snew);
    return Vf;
end function;