
// Make newform 7.17.b.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_7_b_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_7_17_b_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_7_17_b_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_7_b();"
function MakeCharacter_7_b()
    N := 7;
    order := 2;
    char_gens := [3];
    v := [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_7_b_Hecke();"
function MakeCharacter_7_b_Hecke(Kf)
    N := 7;
    order := 2;
    char_gens := [3];
    char_values := [[-1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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+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 := 17;
    raw_aps := [[449], [0], [0], [5764801], [-255690046], [0], [0], [0], [-156184073086], [-988786884286], [0], [-2723955766846], [0], [21863294238914], [0], [111956183305922], [0], [0], [-561251106979006], [500488282933634], [0], [1139826930254594], [0], [0], [0], [0], [0], [-18659624241902398], [-24122168572206526], [3369395461377794], [135082015853524994], [0], [176665732343629442], [0], [305537961173621954], [-511302425389177726], [0], [-257293322059872958], [0], [0], [-1555493233843615678], [0], [-50622182511836158], [1064895034175265794], [4515033229016329154], [0], [6780591652692164162], [0], [0], [0], [16549805234827377794], [-15973856160849565438], [0], [0], [0], [-39738598562698601086], [0], [0], [11711122427190125762], [-4919571391895142526], [0], [0], [0], [0], [0], [23555202056590777154], [-199417326829391546686], [-305651841020891554558], [-62978577036067802686], [0], [0], [-533001770252544993406], [0], [-722084486503103515966], [251940920804481488834], [0], [-375791849447006664766], [0], [-1163301466502923853566], [0], [0], [-33158708844228222526], [138814269653957172482], [0], [0], [-200158536290427739198], [2405159749652283251714], [-2398452643287571191166], [0], [4104095079508690103042], [0], [0], [-6167047011122541506686], [-2279664089557009689406], [-5801337283026202123198], [0], [0], [0], [0], [12530323594711931994434], [11802975303057709468994], [-16465765681288099942846], [0], [4051662701234123501954], [5153945754498189269954], [0], [0], [0], [13052484551522622675074], [0], [0], [5591269518885114052034], [-34378485919536367287166], [0], [22922567823026406229634], [46500210151777069873154], [0], [0], [-64858099690055622022078], [-51432402498300536180158], [0], [-76162431819921735267838], [0], [-4901239821188050913086], [0], [95956544725562295432002], [-29823741004764197098558], [0], [0], [0], [5033399530542623445314], [151791335850861733756802], [-41438083939933730092798], [-215453996269590368662846], [0], [0], [0], [0], [0], [238864723429866898223234], [0], [412741731332613030256322], [-420585798613187294291326], [86994549773710197861314], [0], [0], [0], [0], [0], [-354814332506173382542846], [-651791951243442543906238], [0], [-734537205318313374118846], [0], [148979947266307589841602], [-387115871411993119834366], [-469772406273163552725886], [0], [0], [0], [712280375755967189497922], [1054157955307250416785794], [277130456232367726289282], [0], [1307524164001318204568834], [0], [1859835589018145236726274], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7_17_b_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_7_17_b_a(:prec:=1)
    chi := MakeCharacter_7_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 17));
    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_7_17_b_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_7_17_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,17,sign)));
    Vf := Kernel([<2,R![-449, 1]>],Snew);
    return Vf;
end function;