
// Make newform 400.6.a.a in Magma, downloaded from the LMFDB on 03 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_400_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_400_6_a_a();".
// 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_400_6_a_a();".
// 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_400_a();"
function MakeCharacter_400_a()
    N := 400;
    order := 1;
    char_gens := [351, 101, 177];
    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_400_a_Hecke( : Kf := Kf)
    return MakeCharacter_400_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 := 6;
    raw_aps := [[0], [-26], [0], [-22], [768], [46], [-378], [-1100], [-1986], [-5610], [3988], [142], [1542], [-5026], [24738], [14166], [-28380], [5522], [-24742], [-42372], [52126], [39640], [-59826], [57690], [144382], [-141258], [139814], [86418], [218450], [28806], [-216502], [244608], [239502], [-30860], [-100950], [-12452], [6022], [-500866], [555258], [-417354], [52380], [546662], [452028], [-485594], [-1010178], [807640], [-149552], [-443506], [420018], [1058750], [1273446], [370680], [-561298], [-577152], [651462], [917574], [-735390], [1121308], [1660342], [1452102], [309014], [1593006], [1247258], [665988], [591286], [516342], [3295768], [-1910978], [2420058], [2507270], [413166], [-1737720], [1160978], [-343754], [-573140], [-2880546], [-3085590], [-885458], [-3753438], [-1946530], [2991660], [3966602], [5171148], [4534846], [1082200], [-1080786], [2617830], [-1590458], [4254702], [3266054], [-601542], [4579320], [7052258], [2623488], [3612340], [9156294], [7261590], [5810202], [-8170666], [-817378], [-3507502], [-9614898], [2019414], [1378590], [-8542952], [2314582], [928338], [909486], [8511360], [6124982], [-4516462], [-9639794], [9926502], [-7633220], [-18031412], [9301902], [-13833226], [-14899662], [19330566], [4061100], [-13515238], [-14351954], [-1895298], [2916414], [-14427752], [-15867918], [-301810], [-21267720], [15500858], [12184966], [29028220], [16114494], [2924308], [-26032538], [16326282], [25813250], [19059246], [-17341102], [25816902], [8884890], [24639568], [11376762], [-13078306], [-35718822], [16188050], [25549680], [22295326], [-19687218], [-6775820], [-26380386], [-29516138], [-14856498], [-14534026], [-17202822], [-34443382], [983748], [-30885680], [-32087370], [-15251978], [3481662], [-25400982], [49735206], [30517298], [-31985352], [-2907858], [34962054], [-3004652], [-32078858], [14435570], [21341886], [43715820], [15677342], [4674948], [-53788514], [20155600], [-36327030], [43874968], [10153062], [-6831826], [-84190], [-11830702], [15216528], [-9914714], [-71711298], [1494894], [3684750], [72245662], [-48994186], [5969150], [61070148], [-1749674], [77006814], [-54916472], [59206002], [-64444302], [12877926], [2815982], [-42428714], [18462], [3447414], [57710550], [-63229172], [-18485018], [-47063470], [-108621900], [-18925578], [5227360], [57254214], [-48175590], [-46595672], [89063782], [-99811878], [-48793306], [-30247662], [-109157520], [-41824138], [-42589702], [120329682], [-14920902], [-14172834], [-27757558], [-61667360], [-21124590], [105733334], [-19530942], [88099610], [-97583874], [-10638720], [77851058], [-41798472], [131586526], [-60761180], [113780908], [27769782], [149988614], [-11519382], [-142206490], [136371366], [65034060], [92076948], [-14722626], [51771328], [-94613266], [157671110], [125887326], [7923720], [1548218], [-24070872], [129638260], [-49739706], [-22387778], [13238802], [66828738], [-87309490], [12248886], [34824420], [-156499018], [95922698], [105014502], [-202403738], [184751294], [-105008862], [-24924610], [-14254274], [123749142], [-107417660], [187734870], [195913482], [213373934], [61571166], [10806542], [-39202342], [-112574714], [140620240], [-57342698], [-203036866], [-65819382], [-57091450], [47660582], [79243248], [152208174], [-53071532], [-4568142], [170270582], [78629858], [-147215052], [-245601434], [269243502], [77591920], [241099830], [-106760478], [95838498], [194686926], [-48838512], [-16160234], [-291967710], [199494148], [265257246], [-24416820], [91558058], [115352686], [86104422], [240168760], [47480214], [-223835552], [-216297218], [-205354182], [274608710], [129363720], [102789646], [1164174], [112867470], [-198419358], [-205913386], [332010378], [-80479510], [-12806700], [-180022332], [-182234714], [-3752970], [-154320872], [76904542], [-88973838], [235448534], [-259822914], [6649022], [46610980], [-100465186], [109091058], [-289544634], [-318047998], [-15937338], [-32899640], [-139203426], [-16164272], [6822738], [-404834110], [461230566], [326286422], [-299125942], [79347406], [81139062], [-466732230], [490226908], [-13857954], [-26214300], [72204842], [-376935862], [-263000652], [133897662], [-272296592], [221393302], [-382025598], [384846134], [4040270], [-443370834], [-476406120], [-71790552], [-282528498], [-439534866], [-133609658], [-108601878], [417137778], [-148474860], [492332978], [354155926], [-193708338], [-366809866], [-607017418], [-250302192], [117128140], [89020614], [248126730], [-50749892], [-643150658], [218739300], [-85406652], [256531246], [-379584090], [394109662], [172168062], [92638446], [-237849262], [-122484498], [408828340], [492807294], [-363359732], [-550633058], [-156250546], [261981138], [148523990], [253350246], [519936660], [573400778], [-136399092], [132691726], [-419706920], [330492570], [-289758752], [-554314758], [-468420670], [-458011674], [-666697582], [-88638258], [-76997430], [559642588], [5960182], [-417775278], [-122013586], [-168031860], [-596754194], [439847742], [150845814], [-89614112], [-176752178], [417687822], [170399520], [617647658], [426029022], [-558811266], [146025330], [100742542], [-385405662], [-892254540], [313739926], [587619222], [-543280746], [129720510], [193284088], [-102879240]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_400_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_400_6_a_a();".
// 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_400_6_a_a(:prec:=1)
    chi := MakeCharacter_400_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_400_6_a_a();".
// 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_400_6_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![26, 1]>],Snew);
    return Vf;
end function;