
// Make newform 425.4.a.d in Magma, downloaded from the LMFDB on 01 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_425_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_425_4_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_425_4_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_425_a();"
function MakeCharacter_425_a()
    N := 425;
    order := 1;
    char_gens := [52, 326];
    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_425_a_Hecke( : Kf := Kf)
    return MakeCharacter_425_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]];
    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 := 4;
    raw_aps := [[3], [8], [0], [28], [-24], [58], [-17], [116], [60], [30], [-172], [58], [-342], [148], [-288], [-318], [252], [110], [484], [-708], [-362], [-484], [-756], [-774], [382], [-210], [232], [-432], [-1186], [366], [472], [2760], [-1098], [2528], [1614], [-3328], [2458], [-272], [-3516], [1842], [-3516], [3398], [-2640], [-2882], [42], [-3220], [-2080], [-4664], [1440], [-1186], [5334], [5328], [5618], [-2028], [1902], [5472], [-3570], [272], [-3830], [8874], [2632], [6402], [8980], [-3972], [-4730], [2898], [-4564], [-722], [-5544], [11126], [-7842], [5040], [8404], [8098], [320], [10872], [1374], [7522], [2706], [266], [2688], [-13810], [3036], [11422], [-52], [-3108], [6114], [-4106], [3366], [-896], [10236], [5172], [15052], [8700], [-1168], [1740], [-12570], [11658], [-13700], [17822], [-3800], [10074], [15948], [21834], [-21208], [-12530], [-2220], [25038], [5784], [-4198], [12124], [-7454], [-28842], [-17224], [-12448], [-25182], [-17048], [-7128], [-18462], [28092], [10910], [28414], [6042], [-34752], [18320], [-22890], [22886], [-13452], [27304], [-24470], [35252], [-1548], [2948], [754], [-41574], [-15118], [-23550], [-5240], [-5526], [-438], [-30448], [-21930], [27436], [17832], [-4090], [-2508], [42442], [-32730], [-6148], [22512], [-9182], [28530], [12436], [-7404], [-15368], [27276], [-46456], [13026], [-26330], [28254], [-49272], [-32922], [1168], [-19812], [28494], [42708], [-29500], [9322], [-6190], [-25254], [9444], [27254], [-55788], [-48266], [52136], [-15174], [30992], [-66282], [5056], [-39370], [7936], [-62280], [-14726], [-70074], [5496], [56670], [34882], [-4988], [-38902], [22212], [4222], [28392], [-41068], [21222], [38088], [4662], [42002], [1618], [51774], [-35352], [-44562], [-31444], [-10574], [13106], [-4524], [1818], [52304], [31212], [-84630], [-11692], [17038], [41070], [68956], [57588], [36132], [-29878], [-28016], [-82254], [-48564], [-30006], [4670], [10172], [36546], [-12308], [55860], [20606], [1926], [44076], [58672], [87912], [-18662], [64136], [-35416], [44298], [35356], [-63864], [-89134], [91146], [-41808], [107664], [-8352], [34436], [-45992], [96662], [50958], [-5700], [17740], [-50832], [25628], [22104], [-59366], [8226], [65856], [45002], [60522], [59964], [-29050], [-96512], [-51846], [55462], [-132788], [122148], [-80890], [63634], [16062], [113276], [-127578], [-92694], [-70520], [105090], [30494], [25156], [130582], [35624], [-87122], [77632], [12372], [109526], [-63478], [-13116], [-36432], [13196], [-56004], [46478], [-32384], [101712], [-94898], [67194], [55280], [-13518], [54798], [-127464], [48294], [-42312], [-2990], [95214], [-53296], [-148062], [-84216], [-75140], [11446], [113226], [78476], [600], [40928], [130174], [-110244], [145982], [121704], [140962], [-52860], [-3690], [55746], [97900], [-63912], [30938], [10020], [-876], [22366], [39090], [-77488], [-123194], [81030], [150016], [99030], [48530], [-1096], [28024], [-9300], [-174438], [81758], [141522], [79364], [-119484], [11288], [-137112], [208694], [83454], [-129430], [8008], [127042], [-90570], [-69210], [-110248], [-63030], [149520], [-16378], [20812], [-126828], [-89094], [-162124], [-165002], [-17322], [151900], [-167506], [-30906], [140664], [155616], [124302], [-8976], [115306], [-220566], [238128], [-121800], [21724], [82150], [-71598], [-189128], [-153766], [-56988], [54248], [3348], [16350], [13160], [-41102], [-192552], [-60228], [70222], [-109422], [-1514], [-163266], [-219834], [-133244], [-24834], [28016], [229452], [-95416], [47602], [4048], [-153504], [58850], [-133398], [-92892], [222244], [-85680], [-87770], [-27904], [-50406], [47816], [-146562], [-142114], [32142], [-119648], [95142], [-149730], [54620], [-190934], [68034], [202732], [-133224], [20590], [178866], [-166644], [198632], [-66554], [-90354], [152940], [77284], [-10674], [70632], [-205674], [202186], [210492], [72492], [-128618], [35946], [171720], [-270054], [296372], [-166188]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_425_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_425_4_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_425_4_a_d(:prec:=1)
    chi := MakeCharacter_425_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_425_4_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_425_4_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_425_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-3, 1]>,<3,R![-8, 1]>],Snew);
    return Vf;
end function;