
// Make newform 3025.2.a.d in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3025_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_3025_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_3025_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_3025_a();"
function MakeCharacter_3025_a()
    N := 3025;
    order := 1;
    char_gens := [727, 2301];
    v := [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_3025_a_Hecke(Kf)
    return MakeCharacter_3025_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 := [[0], [1], [0], [0], [0], [0], [0], [0], [9], [0], [-5], [-7], [0], [0], [12], [-6], [-15], [0], [-13], [-3], [0], [0], [0], [-9], [-17], [0], [4], [0], [0], [-21], [0], [0], [3], [0], [0], [0], [23], [16], [0], [0], [21], [-25], [-15], [0], [0], [-20], [0], [1], [0], [-5], [0], [0], [0], [-27], [-18], [0], [30], [0], [0], [0], [0], [0], [0], [-12], [-19], [27], [35], [0], [0], [0], [9], [0], [37], [0], [-25], [-39], [-15], [-2], [30], [0], [24], [10], [0], [-29], [0], [-21], [39], [0], [0], [31], [3], [0], [-43], [0], [40], [0], [-45], [-15], [0], [0], [0], [0], [0], [0], [0], [47], [48], [0], [-36], [0], [0], [0], [42], [-1], [7], [-45], [-41], [27], [-51], [0], [13], [0], [0], [-24], [17], [0], [19], [51], [53], [0], [0], [0], [-23], [38], [0], [0], [-54], [0], [3], [0], [0], [0], [49], [0], [29], [-45], [0], [0], [-31], [36], [0], [57], [56], [0], [-8], [-60], [0], [30], [0], [0], [-57], [0], [0], [-45], [27], [-51], [-20], [0], [0], [9], [0], [50], [0], [0], [41], [-15], [0], [63], [0], [0], [-28], [0], [59], [0], [-21], [6], [0], [64], [0], [0], [-61], [0], [-47], [18], [0], [69], [0], [1], [0], [0], [0], [0], [58], [0], [-69], [-57], [65], [0], [0], [40], [0], [-45], [-19], [-72], [0], [-53], [0], [0], [12], [-39], [0], [0], [-75], [-29], [0], [0], [-54], [-60], [0], [0], [31], [0], [0], [0], [71], [0], [-59], [0], [-45], [-15], [69], [0], [-76], [61], [0], [0], [-52], [72], [0], [0], [0], [0], [-57], [10], [0], [0], [-70], [0], [27], [0], [0], [0], [0], [0], [78], [-80], [-81], [-3], [0], [0], [-67], [-83], [26], [0], [0], [-79], [-63], [0], [0], [0], [0], [85], [0], [0], [0], [-75], [34], [0], [-71], [0], [-27], [48], [0], [0], [0], [0], [73], [81], [0], [0], [0], [0], [0], [-24], [-5], [-7], [-87], [85], [84], [0], [0], [90], [0], [59], [0], [0], [39], [0], [-46], [0], [0], [23], [0], [-61], [0], [-25], [79], [89], [0], [0], [0], [-18], [0], [0], [0], [-57], [65], [0], [95], [0], [91], [93], [0], [35], [9], [0], [83], [37], [0], [-87], [0], [97], [30], [0], [0], [0], [-75], [0], [0], [-21], [0], [0], [-63], [0], [-43], [71], [0], [0], [0], [57], [-16], [0], [0], [0], [47], [96], [0], [0], [0], [0], [-78], [81], [0], [0], [0], [9], [28], [103], [0], [-87], [-91], [-39], [-15], [53], [63], [0], [0], [90], [32], [0], [0], [-21], [0], [102], [0], [0], [-98], [0], [104], [-45], [0], [0], [-96], [0], [0], [-75], [0], [-107], [-18], [0], [-69], [0], [108], [0], [-74], [93], [-51], [0], [-97], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3025_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3025_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_3025_2_a_d(:prec:=1)
    chi := MakeCharacter_3025_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_3025_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_3025_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3025_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<3,R![-1, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;