
// Make newform 7406.2.a.c in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7406_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_7406_2_a_c();".
// 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_7406_2_a_c();".
// 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_7406_a();"
function MakeCharacter_7406_a()
    N := 7406;
    order := 1;
    char_gens := [2117, 3179];
    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_7406_a_Hecke(Kf)
    return MakeCharacter_7406_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], [-4], [-2], [4], [4], [0], [2], [4], [4], [-6], [-4], [4], [12], [8], [0], [4], [8], [-10], [-8], [-12], [-4], [12], [6], [-16], [-12], [-4], [16], [-8], [0], [0], [-8], [-20], [-8], [-24], [12], [-12], [-14], [-12], [-8], [8], [-2], [-6], [-24], [12], [-12], [-20], [8], [26], [0], [-28], [20], [18], [0], [18], [20], [-22], [8], [-4], [-32], [32], [28], [4], [2], [4], [8], [-36], [-30], [-2], [8], [8], [-4], [-28], [-16], [12], [-2], [0], [-26], [-12], [28], [-24], [-20], [36], [4], [-14], [-8], [2], [-8], [20], [16], [-8], [4], [20], [-24], [-30], [28], [-28], [34], [12], [20], [12], [0], [-28], [14], [-40], [2], [-24], [-26], [-4], [-20], [8], [-4], [-48], [-48], [-4], [-4], [14], [-12], [-32], [-14], [-8], [12], [40], [-4], [-28], [20], [8], [-32], [44], [-48], [16], [-36], [38], [4], [-32], [-4], [24], [-42], [24], [22], [0], [-12], [-14], [16], [22], [-6], [48], [56], [18], [-28], [44], [-28], [-20], [24], [-40], [-30], [52], [48], [-28], [56], [32], [28], [32], [-24], [-40], [26], [-40], [54], [-44], [-30], [32], [-56], [-44], [28], [4], [6], [0], [-16], [-64], [-44], [26], [-58], [24], [-40], [-34], [28], [-26], [-60], [-46], [-52], [-20], [46], [52], [28], [-56], [36], [-44], [36], [20], [-68], [42], [-44], [-36], [-14], [56], [-20], [26], [48], [34], [54], [-48], [60], [-28], [4], [0], [-2], [64], [-66], [-26], [32], [-66], [48], [-24], [-42], [-36], [40], [-16], [20], [-46], [60], [32], [38], [-4], [-8], [52], [36], [20], [-12], [-4], [64], [4], [-18], [-14], [20], [4], [-28], [-36], [16], [52], [-36], [48], [-44], [30], [-12], [-52], [52], [-22], [70], [-16], [68], [10], [16], [18], [44], [-12], [-20], [12], [-38], [-62], [12], [-4], [64], [78], [12], [44], [-34], [-64], [-4], [-76], [32], [-56], [-16], [72], [-48], [56], [-76], [76], [-18], [76], [-28], [22], [12], [50], [24], [-56], [-58], [32], [-8], [-84], [12], [72], [12], [-44], [-62], [52], [-16], [-48], [-26], [-84], [-44], [-32], [-12], [24], [-52], [8], [0], [44], [18], [-52], [-80], [14], [-32], [-24], [4], [76], [80], [48], [44], [-14], [-2], [44], [8], [44], [-24], [36], [-36], [-38], [-64], [86], [48], [54], [8], [12], [-20], [-10], [60], [-24], [64], [48], [-42], [-78], [72], [4], [22], [32], [52], [-2], [72], [-28], [-10], [48], [60], [-80], [-42], [-50], [56], [24], [44], [16], [24], [36], [-80], [-34], [-24], [24], [84], [60], [-38], [24], [-56], [72], [66], [36], [-36], [-32], [-54], [-44], [24], [-56], [26], [-60], [-16], [-24], [32], [0], [76], [-68], [-42], [-50], [78], [-40], [60], [-58], [68], [-12], [14], [-4], [8], [66], [-42], [-44], [84], [56], [-62], [24], [-28], [60], [72], [76], [-64], [96], [0], [54], [-18], [-4], [-6], [72], [-68]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7406_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7406_2_a_c();".
// 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_7406_2_a_c(:prec:=1)
    chi := MakeCharacter_7406_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_7406_2_a_c();".
// 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_7406_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7406_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![0, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;