
// Make newform 3525.2.a.n in Magma, downloaded from the LMFDB on 19 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_n();".
// 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_n();".
// 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], [1], [4], [-1], [4], [5], [2], [-2], [-5], [6], [-2], [-1], [1], [-4], [6], [3], [7], [-2], [14], [8], [-12], [6], [9], [0], [-8], [4], [-1], [0], [16], [-10], [-12], [-4], [22], [-15], [-17], [-7], [-10], [-4], [2], [-13], [-12], [-27], [0], [-27], [-15], [15], [-6], [-5], [8], [0], [-7], [-6], [24], [0], [18], [8], [-11], [-22], [-21], [-24], [7], [-16], [25], [-14], [-4], [29], [16], [-30], [26], [-30], [9], [-7], [-15], [0], [-12], [3], [4], [-5], [-6], [6], [-4], [17], [-19], [24], [-2], [18], [0], [-20], [22], [-30], [11], [14], [-7], [42], [8], [24], [-5], [33], [-20], [-8], [14], [-24], [35], [-23], [8], [2], [30], [23], [0], [-22], [42], [-29], [19], [2], [-4], [12], [-6], [-40], [34], [-6], [-40], [4], [-12], [42], [-39], [42], [3], [-10], [44], [-36], [-32], [-29], [12], [31], [-18], [25], [38], [-30], [-41], [24], [-31], [56], [-18], [12], [11], [-2], [4], [18], [-31], [18], [21], [36], [16], [10], [25], [50], [41], [-24], [-38], [-18], [4], [6], [-26], [24], [-55], [-2], [-29], [56], [40], [15], [22], [-57], [9], [-36], [19], [0], [19], [-19], [25], [-26], [13], [-8], [-4], [8], [57], [35], [-43], [-10], [49], [54], [55], [20], [36], [-66], [2], [-34], [-40], [66], [-32], [-44], [-54], [-18], [-4], [-18], [-19], [20], [-16], [-1], [11], [48], [-23], [-62], [-24], [18], [-37], [50], [10], [24], [45], [63], [6], [0], [44], [-31], [26], [-16], [-29], [22], [-57], [-20], [-17], [42], [16], [-38], [34], [54], [24], [30], [-54], [-43], [-59], [-47], [44], [-54], [43], [-48], [-75], [-24], [-46], [10], [-6], [-69], [66], [66], [-13], [-24], [38], [-34], [-67], [60], [-2], [11], [48], [52], [20], [-24], [12], [72], [7], [32], [74], [-5], [71], [31], [-18], [-73], [41], [-12], [14], [8], [22], [-41], [55], [-54], [-37], [-24], [0], [-84], [-48], [26], [18], [24], [11], [6], [69], [-74], [-30], [68], [-74], [-48], [37], [-56], [76], [-1], [-48], [73], [48], [26], [64], [72], [78], [-5], [-36], [85], [6], [-14], [-17], [74], [20], [45], [-34], [65], [-66], [6], [87], [-23], [2], [-38], [-35], [-66], [11], [70], [47], [26], [53], [28], [-19], [56], [-41], [54], [28], [71], [-62], [16], [-1], [-23], [-54], [28], [35], [-58], [54], [17], [41], [-12], [84], [-70], [48], [30], [-51], [-78], [10], [26], [-20], [-43], [52], [64], [-91], [60], [-95], [-36], [-36], [56], [7], [84], [-78], [-57], [90], [17], [72], [-50], [52], [36], [36], [74], [-21], [38], [-4], [0], [35], [-36], [-98], [85], [48], [-95], [47], [50], [-9], [-92], [50], [-42], [33], [-92], [30], [64], [37], [62], [61], [36], [-103], [-102], [-84], [-11], [-95], [-84], [12], [-63], [-50], [-52], [30], [-13], [14], [-86], [-6], [-24], [-30], [10], [-77], [106]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : 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![-1, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;