
// Make newform 7392.2.a.h in Magma, downloaded from the LMFDB on 15 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7392_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_7392_2_a_h();".
// 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_7392_2_a_h();".
// 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_7392_a();"
function MakeCharacter_7392_a()
    N := 7392;
    order := 1;
    char_gens := [4159, 6469, 2465, 5281, 673];
    v := [1, 1, 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_7392_a_Hecke(Kf)
    return MakeCharacter_7392_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 := [[0], [1], [-2], [-1], [1], [2], [2], [-4], [4], [-6], [-4], [-2], [-6], [12], [-4], [6], [4], [2], [-12], [-12], [2], [-8], [12], [-6], [18], [-6], [4], [-12], [-6], [2], [-16], [4], [-6], [-4], [-14], [0], [-18], [20], [16], [-6], [20], [-18], [-20], [-22], [2], [4], [-20], [-28], [4], [-18], [10], [-8], [-14], [12], [-30], [24], [-18], [-24], [-22], [10], [20], [18], [20], [20], [-22], [-10], [-28], [2], [-12], [10], [-14], [-8], [-4], [10], [-36], [12], [14], [-2], [-14], [-22], [4], [22], [-24], [-14], [-8], [4], [-30], [18], [-30], [4], [28], [24], [36], [36], [-20], [-24], [14], [10], [-4], [-22], [12], [-6], [4], [-30], [12], [-14], [-12], [26], [-12], [2], [32], [2], [26], [20], [-20], [18], [4], [28], [30], [28], [38], [-22], [18], [36], [20], [34], [-34], [-12], [28], [2], [44], [16], [4], [22], [-38], [-14], [6], [-12], [6], [18], [-36], [-30], [-4], [-4], [46], [36], [-22], [42], [36], [-20], [18], [18], [-4], [0], [-12], [-12], [-24], [-14], [34], [-22], [-36], [-46], [8], [-20], [-14], [-28], [-28], [-46], [-38], [-26], [12], [46], [16], [-38], [-4], [-54], [44], [22], [-16], [26], [-44], [44], [22], [18], [36], [-42], [2], [-4], [34], [0], [2], [12], [-20], [6], [44], [-38], [18], [54], [58], [24], [26], [-8], [46], [10], [-68], [-42], [-4], [-20], [-6], [-20], [-38], [-10], [52], [12], [-24], [-38], [24], [34], [-36], [-66], [-38], [-8], [34], [-36], [-12], [26], [-6], [-28], [-8], [36], [-2], [36], [-40], [10], [20], [72], [-14], [-46], [36], [36], [-12], [4], [36], [14], [58], [56], [-28], [36], [-52], [56], [-30], [42], [-20], [-6], [26], [-76], [-26], [-28], [-34], [-22], [24], [-60], [42], [-22], [-30], [28], [-50], [58], [20], [42], [62], [-12], [58], [8], [58], [-76], [28], [58], [26], [60], [24], [76], [-8], [34], [68], [-60], [-30], [-6], [60], [58], [-26], [-68], [-6], [-44], [50], [66], [-20], [22], [36], [-44], [-38], [-22], [0], [-4], [-12], [-78], [44], [-66], [-36], [74], [72], [78], [42], [20], [-72], [34], [-44], [-56], [2], [50], [-28], [-22], [-54], [-68], [-14], [-62], [-68], [-44], [-8], [-22], [18], [22], [-24], [76], [-28], [-12], [14], [-6], [-70], [-16], [-26], [-6], [18], [-12], [46], [60], [14], [4], [0], [-42], [-28], [-22], [62], [96], [-30], [66], [28], [-12], [58], [4], [-62], [66], [-12], [-4], [-84], [26], [42], [16], [82], [44], [68], [96], [-78], [48], [-2], [28], [-88], [-38], [50], [-54], [-42], [-6], [64], [34], [-92], [-36], [100], [86], [-12], [12], [50], [-42], [-68], [28], [-28], [-30], [24], [58], [-68], [74], [-62], [-62], [-8], [-6], [18], [24], [94], [-70], [-44], [-100], [-70], [34], [92], [-4], [90], [22], [40], [44], [66], [72], [-34], [-70], [4], [-20], [-54], [14], [-76], [-70], [20], [-96], [42], [-92], [-76], [108], [-18], [-78], [-38], [70], [28]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7392_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7392_2_a_h();".
// 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_7392_2_a_h(:prec:=1)
    chi := MakeCharacter_7392_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(3067) - 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_7392_2_a_h();".
// 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_7392_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7392_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>,<19,R![4, 1]>,<23,R![-4, 1]>],Snew);
    return Vf;
end function;