
// Make newform 6762.2.a.bh in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6762_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_6762_2_a_bh();".
// 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_6762_2_a_bh();".
// 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_6762_a();"
function MakeCharacter_6762_a()
    N := 6762;
    order := 1;
    char_gens := [2255, 3727, 3823];
    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_6762_a_Hecke(Kf)
    return MakeCharacter_6762_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], [1], [-1], [0], [-3], [-2], [2], [2], [1], [-7], [-7], [-2], [2], [-6], [-6], [-9], [9], [-6], [10], [-8], [10], [-15], [-1], [0], [-5], [10], [8], [3], [10], [10], [-7], [-7], [-6], [-14], [-22], [-5], [-6], [2], [-6], [6], [-12], [-16], [10], [-23], [14], [4], [6], [-13], [9], [-22], [-6], [-12], [27], [3], [4], [-2], [27], [9], [-8], [-4], [-6], [-19], [2], [2], [25], [-31], [10], [-9], [36], [-16], [26], [30], [-7], [32], [-4], [-8], [38], [38], [4], [29], [36], [-18], [0], [-38], [-35], [-29], [-30], [1], [-2], [-8], [-4], [-28], [-15], [19], [6], [28], [-17], [18], [12], [-8], [-38], [33], [3], [-36], [20], [33], [3], [12], [24], [-23], [-21], [36], [-6], [16], [31], [-12], [-4], [-38], [11], [28], [-38], [25], [35], [-23], [22], [-1], [-28], [30], [-5], [28], [-14], [48], [11], [8], [-18], [-15], [14], [14], [15], [-34], [-46], [17], [16], [5], [20], [26], [-54], [-16], [-40], [0], [50], [18], [2], [36], [46], [22], [32], [-32], [-29], [15], [-24], [-26], [-13], [17], [30], [50], [3], [-36], [17], [-35], [-25], [26], [-42], [-5], [-29], [-10], [-6], [11], [32], [16], [19], [9], [-28], [-24], [-60], [-1], [20], [-50], [-5], [-26], [-21], [53], [-42], [34], [25], [2], [26], [56], [-56], [-48], [53], [-60], [-38], [-6], [-40], [-53], [-40], [-51], [22], [-64], [-6], [57], [13], [17], [-4], [-42], [1], [42], [22], [-66], [-40], [32], [46], [-7], [11], [-46], [56], [22], [-17], [44], [-58], [-4], [13], [-62], [60], [-60], [-41], [-39], [44], [-18], [-4], [14], [23], [58], [6], [26], [5], [-12], [34], [-62], [16], [-32], [-10], [38], [-21], [27], [-4], [-10], [11], [47], [48], [-27], [-68], [-46], [72], [62], [29], [22], [74], [51], [-16], [-32], [-79], [-21], [-21], [-8], [-35], [22], [-22], [60], [-28], [33], [-68], [-70], [50], [20], [-59], [13], [55], [-28], [-18], [53], [-6], [52], [-60], [35], [28], [74], [-12], [-62], [-62], [-39], [-41], [3], [18], [-18], [67], [-18], [-24], [-64], [82], [-30], [-50], [66], [-10], [-86], [77], [-50], [-67], [-54], [-34], [-42], [-5], [80], [-24], [47], [24], [20], [-54], [33], [-28], [-9], [48], [12], [40], [76], [-80], [42], [-5], [29], [-68], [-60], [51], [-57], [-49], [-31], [-38], [-8], [-54], [21], [-74], [-41], [-14], [61], [-40], [86], [0], [-77], [-58], [-48], [62], [24], [-84], [44], [68], [-53], [42], [-72], [55], [-63], [-24], [50], [-57], [44], [-48], [0], [32], [15], [-58], [98], [-62], [6], [-52], [72], [-12], [-94], [40], [22], [-66], [-42], [-9], [-26], [23], [-98], [-80], [-79], [-83], [42], [-90], [67], [-28], [84], [72], [-36], [-29], [35], [62], [20], [38], [-75], [-35], [87], [-57], [-78], [-42], [-45], [-26], [64], [-8], [54], [69], [-80], [-12], [33], [-50], [90], [27], [6], [36], [36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6762_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6762_2_a_bh();".
// 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_6762_2_a_bh(:prec:=1)
    chi := MakeCharacter_6762_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_6762_2_a_bh();".
// 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_6762_2_a_bh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 1]>,<11,R![3, 1]>,<13,R![2, 1]>,<17,R![-2, 1]>],Snew);
    return Vf;
end function;