
// Make newform 5175.2.a.e in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5175_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_5175_2_a_e();".
// 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_5175_2_a_e();".
// 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_5175_a();"
function MakeCharacter_5175_a()
    N := 5175;
    order := 1;
    char_gens := [4601, 3727, 2926];
    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_5175_a_Hecke(Kf)
    return MakeCharacter_5175_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], [1], [-1], [0], [-5], [1], [5], [-2], [4], [5], [9], [-6], [2], [-8], [-8], [-8], [10], [3], [-3], [3], [-10], [2], [-10], [17], [12], [-4], [-6], [-8], [6], [18], [20], [2], [6], [-14], [6], [18], [15], [-6], [-20], [-15], [-10], [-17], [-7], [-6], [-2], [-20], [-22], [-5], [-10], [-30], [8], [-2], [16], [-3], [-14], [33], [30], [-32], [24], [-16], [-28], [12], [-33], [14], [-22], [-24], [-25], [-11], [-33], [-17], [-20], [-12], [-19], [14], [-22], [-32], [1], [-9], [22], [8], [-32], [20], [-14], [18], [38], [-7], [28], [33], [9], [-14], [-24], [10], [-21], [-18], [0], [-5], [31], [18], [-16], [-41], [24], [20], [31], [8], [-25], [-30], [-30], [-2], [-28], [28], [36], [7], [16], [-31], [-38], [-27], [3], [22], [-19], [-12], [2], [-20], [-36], [6], [40], [32], [22], [-22], [-11], [23], [-40], [39], [-52], [20], [43], [8], [15], [12], [-1], [-20], [37], [15], [9], [-37], [-26], [-44], [6], [38], [20], [-16], [28], [-17], [55], [-16], [45], [-26], [-18], [-22], [-28], [4], [15], [-18], [39], [2], [37], [-20], [-54], [-17], [-13], [9], [14], [-54], [-44], [-6], [3], [-53], [20], [46], [47], [53], [49], [-9], [-40], [10], [-17], [25], [56], [21], [-4], [61], [-30], [39], [14], [-8], [14], [16], [-30], [2], [-26], [13], [-16], [1], [47], [0], [-44], [9], [52], [-55], [3], [29], [15], [-14], [-42], [8], [-10], [45], [62], [-11], [39], [-46], [-63], [24], [13], [-50], [72], [36], [0], [-18], [-32], [1], [-5], [12], [-45], [-16], [60], [12], [42], [67], [-52], [-14], [-14], [-26], [68], [6], [12], [5], [11], [42], [38], [33], [-12], [-59], [30], [70], [43], [-57], [-50], [-72], [-45], [-38], [6], [62], [-75], [-48], [-42], [41], [-33], [43], [76], [-36], [1], [7], [28], [-52], [29], [-76], [52], [-4], [20], [-9], [26], [44], [-30], [60], [30], [40], [39], [10], [-39], [-39], [-20], [-14], [-28], [-51], [45], [66], [18], [4], [-60], [33], [66], [36], [-71], [-6], [-6], [-40], [-5], [54], [-48], [-46], [-16], [40], [-26], [40], [-72], [66], [-54], [-8], [-58], [-53], [-14], [74], [-78], [84], [-62], [-35], [6], [-18], [-39], [-70], [-18], [52], [-6], [38], [-72], [-17], [92], [54], [72], [51], [24], [24], [8], [-66], [-86], [-43], [24], [-64], [62], [-35], [-8], [14], [21], [-57], [21], [57], [-14], [54], [-21], [-80], [-33], [22], [46], [38], [-56], [-28], [-94], [70], [34], [-30], [-59], [7], [62], [87], [-38], [-42], [-23], [54], [-84], [-98], [-82], [-95], [-92], [18], [-50], [4], [-24], [16], [67], [16], [-32], [24], [24], [-45], [66], [-37], [-14], [49], [74], [31], [34], [18], [60], [-68], [-98], [-81], [-54], [34], [-63], [103], [-81], [-2], [-39], [52], [104], [-80], [71], [96], [-58], [80], [44], [-61], [-67], [-27], [57], [-26], [16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5175_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5175_2_a_e();".
// 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_5175_2_a_e(:prec:=1)
    chi := MakeCharacter_5175_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_5175_2_a_e();".
// 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_5175_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5175_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<7,R![1, 1]>,<11,R![-1, 1]>],Snew);
    return Vf;
end function;