
// Make newform 3525.2.a.d in Magma, downloaded from the LMFDB on 16 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_d();".
// 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_d();".
// 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], [0], [4], [-2], [-6], [0], [0], [-2], [4], [2], [-6], [4], [-1], [6], [0], [-2], [4], [4], [2], [0], [4], [-14], [-10], [14], [-8], [20], [6], [14], [-8], [-16], [6], [8], [-18], [-4], [-22], [-12], [-8], [6], [-12], [-18], [20], [-22], [6], [-20], [8], [0], [-12], [6], [6], [-28], [-14], [8], [22], [-24], [-10], [0], [2], [-6], [-4], [-6], [-4], [0], [-6], [-22], [-4], [-2], [-20], [14], [18], [24], [-24], [-2], [4], [-16], [-10], [-6], [18], [10], [36], [-10], [-36], [-30], [0], [36], [2], [22], [-18], [-40], [-28], [12], [8], [-8], [8], [0], [-2], [-22], [-20], [-18], [20], [2], [-44], [42], [-4], [34], [4], [-2], [16], [-22], [32], [-6], [34], [-44], [-20], [18], [20], [24], [-18], [-48], [-10], [-6], [26], [-36], [8], [-34], [38], [-20], [-16], [-38], [20], [-24], [52], [38], [-6], [-30], [54], [-20], [42], [10], [-52], [22], [-48], [44], [-42], [24], [2], [22], [40], [32], [-34], [18], [-20], [-48], [-28], [28], [-28], [-14], [-38], [-42], [-28], [-58], [8], [-28], [42], [24], [48], [22], [-6], [34], [40], [46], [24], [-62], [28], [10], [44], [14], [-24], [54], [-32], [-28], [42], [18], [-32], [6], [-54], [36], [42], [48], [-26], [-36], [-32], [-34], [-36], [-6], [58], [-2], [18], [0], [62], [-16], [14], [34], [0], [62], [-60], [-36], [10], [-32], [-42], [-10], [-8], [4], [36], [-14], [16], [-30], [-32], [-46], [38], [-64], [18], [40], [-12], [62], [14], [-40], [40], [-20], [6], [-36], [40], [-30], [-20], [-24], [34], [30], [-8], [12], [44], [20], [24], [14], [-14], [60], [-48], [28], [-60], [16], [-10], [18], [-64], [-6], [10], [8], [30], [28], [-38], [-50], [8], [28], [70], [-38], [-58], [12], [-50], [-6], [4], [18], [-50], [52], [-42], [4], [58], [16], [-60], [30], [82], [-48], [24], [12], [-32], [-26], [4], [64], [-30], [-62], [28], [18], [-50], [-20], [-66], [64], [-46], [-34], [8], [58], [12], [52], [82], [-6], [40], [84], [20], [2], [-12], [78], [36], [-62], [64], [-74], [66], [12], [56], [-70], [-76], [-8], [-54], [-46], [-20], [74], [-34], [-8], [-82], [74], [-68], [36], [-32], [22], [46], [-18], [-44], [44], [-44], [52], [6], [90], [-22], [-32], [10], [-74], [14], [-16], [66], [-88], [22], [36], [28], [46], [28], [38], [78], [-32], [-58], [-34], [68], [-24], [6], [48], [-26], [-22], [-72], [36], [-52], [-70], [-14], [80], [-62], [44], [28], [32], [86], [100], [14], [-84], [-4], [38], [66], [-82], [70], [66], [-8], [82], [-68], [96], [44], [22], [4], [56], [-46], [30], [-20], [-76], [-36], [-2], [28], [82], [-8], [-74], [70], [-86], [96], [50], [-82], [-32], [50], [50], [-52], [20], [-22], [22], [84], [32], [30], [30], [-20], [80], [30], [0], [70], [82], [-24], [-16], [-62], [-14], [76], [-54], [80], [0]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : 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![0, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;