
// Make newform 2601.2.a.h in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2601_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_2601_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_2601_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_2601_a();"
function MakeCharacter_2601_a()
    N := 2601;
    order := 1;
    char_gens := [290, 2026];
    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_2601_a_Hecke(Kf)
    return MakeCharacter_2601_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], [-2], [-1], [6], [1], [0], [5], [-2], [-6], [-7], [-7], [-6], [7], [12], [-12], [6], [-11], [-11], [2], [6], [0], [-2], [-16], [-17], [4], [-7], [-16], [-6], [4], [5], [10], [6], [-13], [-14], [5], [-17], [-12], [8], [16], [18], [-13], [10], [1], [12], [-16], [1], [0], [18], [6], [-12], [-20], [18], [6], [24], [16], [-14], [-13], [-5], [-18], [-11], [0], [12], [4], [1], [2], [19], [14], [26], [-30], [-24], [12], [-7], [-6], [-25], [-18], [-30], [-30], [-2], [5], [2], [13], [12], [-1], [-1], [26], [-20], [18], [-40], [41], [12], [-6], [-5], [4], [13], [6], [-28], [14], [-36], [30], [41], [12], [-32], [20], [23], [-5], [-34], [6], [-6], [-19], [31], [19], [-24], [-24], [-17], [30], [-17], [4], [24], [20], [-25], [1], [-42], [-28], [35], [-6], [-41], [-32], [13], [37], [19], [-30], [-17], [13], [34], [22], [-4], [35], [-16], [-10], [23], [18], [-11], [-18], [17], [20], [13], [-32], [-11], [-4], [54], [32], [-36], [48], [-11], [50], [0], [-12], [-17], [36], [-28], [34], [48], [0], [18], [-36], [0], [29], [35], [18], [0], [-37], [-18], [61], [43], [44], [-47], [0], [24], [-29], [-36], [-36], [43], [6], [-56], [24], [-13], [12], [-25], [66], [19], [50], [25], [-14], [12], [-64], [-67], [46], [54], [-4], [-58], [-24], [29], [-23], [14], [22], [-17], [42], [6], [23], [37], [-24], [48], [-48], [52], [-55], [-59], [-42], [-54], [26], [-31], [-24], [30], [-11], [-2], [-5], [12], [0], [-43], [-12], [-55], [-56], [35], [-12], [-17], [28], [22], [-16], [18], [-10], [-8], [35], [-36], [18], [42], [-8], [-77], [-32], [-71], [36], [-47], [-56], [20], [-53], [60], [56], [11], [77], [16], [-61], [24], [18], [-13], [13], [-56], [12], [-76], [-58], [59], [-6], [54], [25], [-23], [-23], [-14], [41], [0], [19], [31], [68], [-30], [12], [-74], [-1], [-36], [-42], [6], [6], [-49], [-46], [66], [48], [-56], [-76], [6], [-54], [-24], [78], [52], [65], [30], [0], [-23], [30], [13], [34], [-40], [11], [24], [-13], [-38], [-16], [-48], [-76], [46], [-59], [56], [88], [35], [0], [55], [66], [-20], [-83], [-90], [-54], [36], [8], [18], [6], [43], [42], [0], [6], [19], [24], [-41], [-36], [-13], [56], [-6], [24], [54], [44], [-66], [-24], [2], [-7], [-48], [60], [0], [-37], [30], [31], [35], [-30], [70], [-86], [18], [24], [-6], [20], [-78], [90], [-84], [6], [-50], [60], [-23], [-8], [41], [60], [96], [-12], [19], [-12], [62], [-65], [52], [53], [-66], [-66], [89], [-44], [-13], [-6], [-31], [-59], [12], [28], [55], [14], [-32], [12], [42], [-1], [49], [58], [67], [8], [-7], [-10], [-72], [70], [-58], [-85], [-31], [36], [0], [-60], [42], [102], [-50], [31], [-19], [84], [46], [-25], [90], [-18], [50], [29], [-78], [-16], [-91], [-32], [-36], [-20], [37], [-60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2601_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2601_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_2601_2_a_h(:prec:=1)
    chi := MakeCharacter_2601_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_2601_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_2601_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2601_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<5,R![2, 1]>,<7,R![1, 1]>],Snew);
    return Vf;
end function;