
// Make newform 3525.2.a.e in Magma, downloaded from the LMFDB on 24 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_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_3525_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_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 := [[-1], [-1], [0], [3], [-2], [1], [-3], [-3], [9], [-5], [-8], [-4], [9], [-2], [-1], [-3], [-9], [1], [-8], [13], [2], [-6], [4], [10], [8], [-4], [13], [-4], [-12], [-10], [16], [8], [12], [-13], [6], [11], [-4], [-12], [-11], [3], [-24], [-24], [-16], [2], [-3], [-5], [-4], [0], [-27], [-18], [-6], [8], [-23], [11], [-20], [-12], [26], [-12], [-28], [-15], [-4], [18], [5], [0], [3], [8], [-4], [4], [28], [20], [18], [-36], [-18], [34], [22], [26], [23], [-24], [-18], [-14], [-12], [20], [21], [-9], [-30], [27], [35], [16], [-30], [-28], [-1], [-21], [-43], [-11], [-7], [-33], [1], [20], [-41], [-21], [26], [44], [28], [30], [-28], [7], [4], [-32], [-26], [29], [38], [-36], [-38], [46], [16], [18], [-37], [-24], [9], [-12], [-25], [9], [-46], [-6], [11], [11], [26], [-35], [2], [-44], [-34], [-33], [-17], [38], [18], [6], [-18], [4], [-12], [-2], [26], [13], [36], [-10], [42], [-30], [-16], [28], [-23], [-10], [17], [-51], [-23], [-24], [23], [16], [-55], [22], [-26], [24], [50], [2], [20], [8], [57], [-51], [-54], [49], [-6], [-26], [25], [-17], [-18], [1], [-5], [19], [32], [-4], [12], [-48], [64], [-10], [0], [21], [-29], [-6], [18], [-51], [-15], [-30], [58], [39], [43], [14], [-30], [42], [34], [1], [3], [-36], [-28], [2], [23], [-29], [51], [2], [-39], [24], [22], [52], [0], [-10], [40], [-5], [21], [-8], [28], [-21], [16], [-64], [62], [74], [-6], [4], [-30], [26], [62], [-34], [-8], [-56], [-9], [54], [-50], [-60], [-62], [12], [-38], [27], [-41], [-27], [5], [-22], [-42], [-52], [-2], [48], [-63], [22], [-18], [34], [47], [-12], [-16], [60], [52], [-19], [12], [22], [-38], [-8], [-1], [7], [25], [40], [50], [78], [4], [33], [64], [-18], [25], [-11], [-6], [40], [49], [16], [6], [39], [-44], [-75], [66], [63], [-44], [-50], [19], [16], [-9], [-14], [-83], [-48], [34], [-68], [60], [37], [-4], [77], [-16], [-26], [-60], [40], [1], [-12], [-50], [-60], [44], [47], [-12], [9], [-21], [88], [52], [-32], [45], [-84], [-73], [59], [-10], [-80], [21], [2], [-38], [-13], [-34], [-11], [-4], [-10], [-44], [-90], [52], [22], [43], [57], [-77], [44], [70], [-53], [-18], [21], [29], [-80], [-86], [-38], [-28], [8], [42], [-85], [4], [66], [49], [43], [16], [68], [66], [34], [-76], [-28], [80], [-57], [66], [21], [13], [-43], [78], [48], [-16], [26], [28], [89], [76], [8], [-8], [20], [14], [67], [77], [-54], [41], [-28], [30], [62], [70], [21], [-8], [-17], [76], [51], [-13], [13], [31], [80], [-40], [-57], [58], [-13], [-39], [4], [-59], [16], [-20], [55], [64], [25], [-102], [53], [-22], [-26], [26], [26], [-82], [74], [-31], [-59], [-72], [32], [-24], [-30], [-89], [-10], [54], [-78], [22], [4], [57], [32], [-71], [64], [-92], [36], [41], [36]];
    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_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_3525_2_a_e(: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_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_3525_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3525_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<7,R![-3, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;