
// Make newform 3525.2.a.o in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3525_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_3525_2_a_o();".
// 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_3525_2_a_o();".
// 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_3525_a();"
function MakeCharacter_3525_a()
    N := 3525;
    order := 1;
    char_gens := [2351, 1552, 2026];
    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_3525_a_Hecke(Kf)
    return MakeCharacter_3525_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 := [[2], [1], [0], [4], [0], [5], [6], [-2], [1], [-6], [-8], [2], [-2], [-5], [1], [12], [-3], [-1], [8], [1], [-13], [-1], [12], [-15], [-8], [13], [16], [9], [-8], [15], [-8], [15], [3], [0], [-23], [4], [-20], [-19], [19], [2], [0], [-8], [-9], [10], [-12], [-16], [-18], [7], [-20], [-10], [26], [0], [7], [-12], [-23], [-24], [-17], [16], [-6], [-6], [-8], [-17], [-26], [-6], [-26], [31], [3], [-2], [18], [8], [16], [10], [0], [33], [36], [-10], [12], [8], [-13], [0], [24], [4], [0], [-14], [35], [-20], [22], [-24], [10], [-19], [-35], [27], [42], [0], [32], [-44], [-44], [11], [22], [-25], [-20], [33], [15], [-16], [-33], [11], [27], [-18], [-10], [-35], [1], [42], [8], [-1], [10], [-36], [44], [30], [30], [-43], [-31], [41], [-3], [22], [18], [30], [-47], [12], [-5], [-52], [41], [40], [-24], [6], [-13], [31], [20], [-19], [30], [-22], [40], [-48], [-28], [-54], [2], [-6], [50], [-11], [-40], [26], [18], [-6], [26], [13], [-2], [12], [-32], [-25], [39], [10], [40], [-29], [-22], [32], [-8], [-15], [-3], [2], [54], [-3], [36], [34], [-6], [-10], [-42], [10], [3], [-38], [-49], [46], [-46], [-14], [-6], [38], [59], [50], [-6], [16], [-29], [-8], [42], [21], [-38], [15], [16], [14], [-22], [-63], [-6], [36], [-2], [33], [43], [25], [4], [6], [22], [28], [-48], [-4], [4], [26], [34], [53], [-1], [52], [-48], [4], [-18], [-3], [-10], [25], [-24], [33], [18], [-68], [21], [-30], [4], [42], [23], [19], [-37], [9], [-9], [3], [-69], [28], [45], [-35], [-21], [-44], [5], [58], [-12], [-39], [2], [-18], [-29], [34], [-46], [-5], [44], [42], [27], [-21], [-12], [33], [9], [-12], [32], [-64], [46], [-38], [18], [71], [1], [40], [21], [59], [-65], [-2], [16], [10], [-37], [73], [0], [-7], [48], [59], [30], [38], [12], [-5], [56], [-30], [-19], [-6], [-76], [23], [-30], [-60], [-61], [39], [-46], [40], [61], [11], [22], [79], [-46], [3], [56], [41], [-41], [61], [18], [54], [-51], [0], [24], [89], [-48], [-59], [9], [-3], [24], [-46], [-29], [9], [92], [87], [-40], [-34], [71], [-44], [56], [69], [87], [8], [-25], [72], [-86], [84], [71], [-3], [78], [-92], [63], [7], [-44], [18], [-26], [-88], [-47], [-1], [-2], [-37], [40], [66], [-20], [-48], [60], [52], [-58], [38], [-35], [-53], [1], [-40], [-29], [-96], [-6], [-2], [20], [-53], [-55], [-2], [-50], [32], [36], [-39], [-20], [-24], [66], [10], [-5], [70], [-51], [38], [78], [30], [23], [-80], [-43], [51], [16], [69], [-6], [26], [16], [-74], [36], [58], [-3], [16], [-60], [39], [-58], [72], [-16], [50], [67], [70], [-70], [16], [6], [-23], [100], [-72], [-6], [56], [28], [50], [-74], [58], [21], [107], [-37], [-56], [-7], [-34], [-16], [-64], [-7], [-18], [-14], [-15], [46], [-54]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3525_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3525_2_a_o();".
// 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_3525_2_a_o(:prec:=1)
    chi := MakeCharacter_3525_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_3525_2_a_o();".
// 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_3525_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3525_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 1]>,<7,R![-4, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;