LMFDB - Embedded newform 16.48.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,48,Mod(1,16)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 48, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 48);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 48 $
Character orbit:	$[\chi]$	$=$	16.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$223.852260248$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5897345978580 $ x^2 - x - 5897345978580
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.42845e6$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70963e11 q^{3} -1.02521e16 q^{5} -1.08843e20 q^{7} +4.68319e22 q^{9} +O(q^{10})$	$q-2.70963e11 q^{3} -1.02521e16 q^{5} -1.08843e20 q^{7} +4.68319e22 q^{9} -3.92772e24 q^{11} -2.33551e26 q^{13} +2.77793e27 q^{15} +4.05682e28 q^{17} +1.56322e29 q^{19} +2.94925e31 q^{21} -3.32892e31 q^{23} -6.05438e32 q^{25} -5.48512e33 q^{27} +9.92617e32 q^{29} -1.71111e35 q^{31} +1.06427e36 q^{33} +1.11587e36 q^{35} +1.08691e37 q^{37} +6.32835e37 q^{39} -7.95054e37 q^{41} -3.70345e38 q^{43} -4.80124e38 q^{45} +6.30322e38 q^{47} +6.60357e39 q^{49} -1.09925e40 q^{51} +2.32864e40 q^{53} +4.02673e40 q^{55} -4.23574e40 q^{57} +3.80182e41 q^{59} +1.07627e42 q^{61} -5.09735e42 q^{63} +2.39438e42 q^{65} +1.46013e42 q^{67} +9.02012e42 q^{69} +1.66419e43 q^{71} -5.99210e42 q^{73} +1.64051e44 q^{75} +4.27507e44 q^{77} +1.80688e44 q^{79} +2.41058e44 q^{81} +2.37314e45 q^{83} -4.15908e44 q^{85} -2.68962e44 q^{87} +1.07890e45 q^{89} +2.54205e46 q^{91} +4.63646e46 q^{93} -1.60263e45 q^{95} -5.98242e45 q^{97} -1.83943e47 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 122289844824 q^{3} + 18\!\cdots\!40 q^{5}+ \cdots + 42\!\cdots\!14 q^{9}+O(q^{10}) $ 2 * q - 122289844824 * q^3 + 18178497839194140 * q^5 - 161018545456026827632 * q^7 + 42346678028148839299914 * q^9	$ 2 q - 122289844824 q^{3} + 18\!\cdots\!40 q^{5}+ \cdots - 19\!\cdots\!88 q^{99}+O(q^{100}) $ 2 * q - 122289844824 * q^3 + 18178497839194140 * q^5 - 161018545456026827632 * q^7 + 42346678028148839299914 * q^9 - 1148414200601353937498184 * q^11 - 91147460388086216079419636 * q^13 + 7004781615634981008503062320 * q^15 - 102841736272610788276309928988 * q^17 - 1021269902513170084335814340920 * q^19 + 21735508585019662606113256728384 * q^21 - 8761047668524985915668413593424 * q^23 - 507682648872986273192302737941650 * q^25 - 10104984547772788958914840573828080 * q^27 + 44866738270461738647546902365332460 * q^29 - 207627610718284877414441666427987904 * q^31 + 1477472372445089531927832872318579808 * q^33 - 367494533980940143565872352084574240 * q^35 + 15454503825836181610788445034143257052 * q^37 + 84454902625144773203023026649833922032 * q^39 - 183644075219801436287443355635335693516 * q^41 - 251792238934935641547265550793932228104 * q^43 - 607642114901245348425108558238563820020 * q^45 - 2877259596300283378159171629753736631712 * q^47 + 4082466065158778742105339967177123972626 * q^49 - 32313595089705813278113192123424261680944 * q^51 + 36237003736012302609225684974144068811964 * q^53 + 119284580676988289643910520626568801967120 * q^55 - 217433238554190330563790441928328986980960 * q^57 - 216631413518842627379871131067900548684520 * q^59 + 2343345794410501150323838534047225181209004 * q^61 - 4863331655836708395434595438453038464171824 * q^63 + 6442979197855565506819083799260229348557480 * q^65 - 9679359518522646693994124509775830904939992 * q^67 + 12666786020488197907659082189490927173898688 * q^69 + 34699689890171309465034074226599979171319056 * q^71 - 44919947676390510667705279890416107043412236 * q^73 + 178584439183245940201476654984555123803179800 * q^75 + 282496402053252599536812437150318196880350144 * q^77 - 459051391676893300046865817924586171479631200 * q^79 - 326533112556292583027849224141868597638781358 * q^81 + 1357110492338922730441331533770737425416063816 * q^83 - 4493133982253416774682211563698820690839841160 * q^85 + 6253923718333989640702287995671643099683866480 * q^87 + 4559737044037194815580171398227946101567633620 * q^89 + 17990568925125012147062510675834829593464410976 * q^91 + 40935592303229143353065306088384219268477064448 * q^93 - 35082243421784097273165847234698748298121392400 * q^95 + 53827580940974934190192198959367945947750881092 * q^97 - 196408474482694469089639477350277927611611498088 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.70963e11	−1.66173	−0.830864	−	0.556476i	$-0.812153\pi$
$3$	−2.70963e11	−1.66173	−0.830864	+	0.556476i	$0.812153\pi$
$4$	0	0
$5$	−1.02521e16	−0.384607	−0.192303	−	0.981336i	$-0.561596\pi$
$5$	−1.02521e16	−0.384607	−0.192303	+	0.981336i	$0.561596\pi$
$6$	0	0
$7$	−1.08843e20	−1.50314	−0.751568	−	0.659655i	$-0.770702\pi$
$7$	−1.08843e20	−1.50314	−0.751568	+	0.659655i	$0.770702\pi$
$8$	0	0
$9$	4.68319e22	1.76134
$10$	0	0
$11$	−3.92772e24	−1.32255	−0.661276	−	0.750143i	$-0.729985\pi$
$11$	−3.92772e24	−1.32255	−0.661276	+	0.750143i	$0.729985\pi$
$12$	0	0
$13$	−2.33551e26	−1.55136	−0.775679	−	0.631128i	$-0.782592\pi$
$13$	−2.33551e26	−1.55136	−0.775679	+	0.631128i	$0.782592\pi$
$14$	0	0
$15$	2.77793e27	0.639112
$16$	0	0
$17$	4.05682e28	0.492760	0.246380	−	0.969173i	$-0.420759\pi$
$17$	4.05682e28	0.492760	0.246380	+	0.969173i	$0.420759\pi$
$18$	0	0
$19$	1.56322e29	0.139095	0.0695473	−	0.997579i	$-0.477845\pi$
$19$	1.56322e29	0.139095	0.0695473	+	0.997579i	$0.477845\pi$
$20$	0	0
$21$	2.94925e31	2.49780
$22$	0	0
$23$	−3.32892e31	−0.332429	−0.166215	−	0.986090i	$-0.553154\pi$
$23$	−3.32892e31	−0.332429	−0.166215	+	0.986090i	$0.553154\pi$
$24$	0	0
$25$	−6.05438e32	−0.852078
$26$	0	0
$27$	−5.48512e33	−1.26514
$28$	0	0
$29$	9.92617e32	0.0427001	0.0213500	−	0.999772i	$-0.493204\pi$
$29$	9.92617e32	0.0427001	0.0213500	+	0.999772i	$0.493204\pi$
$30$	0	0
$31$	−1.71111e35	−1.53560	−0.767799	−	0.640690i	$-0.778648\pi$
$31$	−1.71111e35	−1.53560	−0.767799	+	0.640690i	$0.778648\pi$
$32$	0	0
$33$	1.06427e36	2.19772
$34$	0	0
$35$	1.11587e36	0.578117
$36$	0	0
$37$	1.08691e37	1.52564	0.762819	−	0.646612i	$-0.223815\pi$
$37$	1.08691e37	1.52564	0.762819	+	0.646612i	$0.223815\pi$
$38$	0	0
$39$	6.32835e37	2.57793
$40$	0	0
$41$	−7.95054e37	−0.999945	−0.499972	−	0.866041i	$-0.666656\pi$
$41$	−7.95054e37	−0.999945	−0.499972	+	0.866041i	$0.666656\pi$
$42$	0	0
$43$	−3.70345e38	−1.52089	−0.760446	−	0.649402i	$-0.775019\pi$
$43$	−3.70345e38	−1.52089	−0.760446	+	0.649402i	$0.775019\pi$
$44$	0	0
$45$	−4.80124e38	−0.677423
$46$	0	0
$47$	6.30322e38	0.320083	0.160041	−	0.987110i	$-0.448837\pi$
$47$	6.30322e38	0.320083	0.160041	+	0.987110i	$0.448837\pi$
$48$	0	0
$49$	6.60357e39	1.25942
$50$	0	0
$51$	−1.09925e40	−0.818833
$52$	0	0
$53$	2.32864e40	0.702456	0.351228	−	0.936290i	$-0.385764\pi$
$53$	2.32864e40	0.702456	0.351228	+	0.936290i	$0.385764\pi$
$54$	0	0
$55$	4.02673e40	0.508662
$56$	0	0
$57$	−4.23574e40	−0.231137
$58$	0	0
$59$	3.80182e41	0.922507	0.461253	−	0.887268i	$-0.347400\pi$
$59$	3.80182e41	0.922507	0.461253	+	0.887268i	$0.347400\pi$
$60$	0	0
$61$	1.07627e42	1.19308	0.596541	−	0.802583i	$-0.296541\pi$
$61$	1.07627e42	1.19308	0.596541	+	0.802583i	$0.296541\pi$
$62$	0	0
$63$	−5.09735e42	−2.64753
$64$	0	0
$65$	2.39438e42	0.596663
$66$	0	0
$67$	1.46013e42	0.178498	0.0892490	−	0.996009i	$-0.471553\pi$
$67$	1.46013e42	0.178498	0.0892490	+	0.996009i	$0.471553\pi$
$68$	0	0
$69$	9.02012e42	0.552406
$70$	0	0
$71$	1.66419e43	0.520752	0.260376	−	0.965507i	$-0.416153\pi$
$71$	1.66419e43	0.520752	0.260376	+	0.965507i	$0.416153\pi$
$72$	0	0
$73$	−5.99210e42	−0.0976095	−0.0488047	−	0.998808i	$-0.515541\pi$
$73$	−5.99210e42	−0.0976095	−0.0488047	+	0.998808i	$0.515541\pi$
$74$	0	0
$75$	1.64051e44	1.41592
$76$	0	0
$77$	4.27507e44	1.98798
$78$	0	0
$79$	1.80688e44	0.459931	0.229966	−	0.973199i	$-0.426139\pi$
$79$	1.80688e44	0.459931	0.229966	+	0.973199i	$0.426139\pi$
$80$	0	0
$81$	2.41058e44	0.340975
$82$	0	0
$83$	2.37314e45	1.89229	0.946144	−	0.323745i	$-0.104942\pi$
$83$	2.37314e45	1.89229	0.946144	+	0.323745i	$0.104942\pi$
$84$	0	0
$85$	−4.15908e44	−0.189519
$86$	0	0
$87$	−2.68962e44	−0.0709559
$88$	0	0
$89$	1.07890e45	0.166846	0.0834232	−	0.996514i	$-0.473415\pi$
$89$	1.07890e45	0.166846	0.0834232	+	0.996514i	$0.473415\pi$
$90$	0	0
$91$	2.54205e46	2.33190
$92$	0	0
$93$	4.63646e46	2.55175
$94$	0	0
$95$	−1.60263e45	−0.0534967
$96$	0	0
$97$	−5.98242e45	−0.122389	−0.0611943	−	0.998126i	$-0.519491\pi$
$97$	−5.98242e45	−0.122389	−0.0611943	+	0.998126i	$0.519491\pi$
$98$	0	0
$99$	−1.83943e47	−2.32946
$100$	0	0
$101$	−6.21755e46	−0.492115	−0.246058	−	0.969255i	$-0.579135\pi$
$101$	−6.21755e46	−0.492115	−0.246058	+	0.969255i	$0.579135\pi$
$102$	0	0
$103$	−1.44306e45	−0.00720458	−0.00360229	−	0.999994i	$-0.501147\pi$
$103$	−1.44306e45	−0.00720458	−0.00360229	+	0.999994i	$0.501147\pi$
$104$	0	0
$105$	−3.02360e47	−0.960672
$106$	0	0
$107$	8.78783e47	1.79210	0.896051	−	0.443951i	$-0.146423\pi$
$107$	8.78783e47	1.79210	0.896051	+	0.443951i	$0.146423\pi$
$108$	0	0
$109$	−1.77433e45	−0.00234159	−0.00117080	−	0.999999i	$-0.500373\pi$
$109$	−1.77433e45	−0.00234159	−0.00117080	+	0.999999i	$0.500373\pi$
$110$	0	0
$111$	−2.94511e48	−2.53520
$112$	0	0
$113$	−6.05109e46	−0.0342365	−0.0171183	−	0.999853i	$-0.505449\pi$
$113$	−6.05109e46	−0.0342365	−0.0171183	+	0.999853i	$0.505449\pi$
$114$	0	0
$115$	3.41283e47	0.127854
$116$	0	0
$117$	−1.09376e49	−2.73247
$118$	0	0
$119$	−4.41558e48	−0.740686
$120$	0	0
$121$	6.60724e48	0.749141
$122$	0	0
$123$	2.15430e49	1.66164
$124$	0	0
$125$	1.34915e49	0.712321
$126$	0	0
$127$	−8.82915e48	−0.321020	−0.160510	−	0.987034i	$-0.551314\pi$
$127$	−8.82915e48	−0.321020	−0.160510	+	0.987034i	$0.551314\pi$
$128$	0	0
$129$	1.00350e50	2.52731
$130$	0	0
$131$	1.59705e49	0.280183	0.140092	−	0.990139i	$-0.455260\pi$
$131$	1.59705e49	0.280183	0.140092	+	0.990139i	$0.455260\pi$
$132$	0	0
$133$	−1.70146e49	−0.209078
$134$	0	0
$135$	5.62339e49	0.486580
$136$	0	0
$137$	−6.03702e49	−0.369734	−0.184867	−	0.982764i	$-0.559185\pi$
$137$	−6.03702e49	−0.369734	−0.184867	+	0.982764i	$0.559185\pi$
$138$	0	0
$139$	−1.21646e50	−0.529969	−0.264985	−	0.964253i	$-0.585367\pi$
$139$	−1.21646e50	−0.529969	−0.264985	+	0.964253i	$0.585367\pi$
$140$	0	0
$141$	−1.70794e50	−0.531891
$142$	0	0
$143$	9.17321e50	2.05175
$144$	0	0
$145$	−1.01764e49	−0.0164227
$146$	0	0
$147$	−1.78932e51	−2.09281
$148$	0	0
$149$	−1.57670e51	−1.34237	−0.671185	−	0.741290i	$-0.734214\pi$
$149$	−1.57670e51	−1.34237	−0.671185	+	0.741290i	$0.734214\pi$
$150$	0	0
$151$	−1.05031e51	−0.653674	−0.326837	−	0.945081i	$-0.605983\pi$
$151$	−1.05031e51	−0.653674	−0.326837	+	0.945081i	$0.605983\pi$
$152$	0	0
$153$	1.89988e51	0.867918
$154$	0	0
$155$	1.75424e51	0.590602
$156$	0	0
$157$	−3.24813e51	−0.809081	−0.404541	−	0.914520i	$-0.632569\pi$
$157$	−3.24813e51	−0.809081	−0.404541	+	0.914520i	$0.632569\pi$
$158$	0	0
$159$	−6.30974e51	−1.16729
$160$	0	0
$161$	3.62331e51	0.499686
$162$	0	0
$163$	4.29859e51	0.443524	0.221762	−	0.975101i	$-0.428819\pi$
$163$	4.29859e51	0.443524	0.221762	+	0.975101i	$0.428819\pi$
$164$	0	0
$165$	−1.09109e52	−0.845258
$166$	0	0
$167$	6.41790e50	0.0374590	0.0187295	−	0.999825i	$-0.494038\pi$
$167$	6.41790e50	0.0374590	0.0187295	+	0.999825i	$0.494038\pi$
$168$	0	0
$169$	3.18818e52	1.40671
$170$	0	0
$171$	7.32086e51	0.244993
$172$	0	0
$173$	6.69399e52	1.70452	0.852259	−	0.523120i	$-0.175232\pi$
$173$	6.69399e52	1.70452	0.852259	+	0.523120i	$0.175232\pi$
$174$	0	0
$175$	6.58979e52	1.28079
$176$	0	0
$177$	−1.03015e53	−1.53296
$178$	0	0
$179$	1.38049e53	1.57757	0.788784	−	0.614671i	$-0.210711\pi$
$179$	1.38049e53	1.57757	0.788784	+	0.614671i	$0.210711\pi$
$180$	0	0
$181$	−9.26435e52	−0.815396	−0.407698	−	0.913117i	$-0.633668\pi$
$181$	−9.26435e52	−0.815396	−0.407698	+	0.913117i	$0.633668\pi$
$182$	0	0
$183$	−2.91629e53	−1.98258
$184$	0	0
$185$	−1.11431e53	−0.586771
$186$	0	0
$187$	−1.59340e53	−0.651701
$188$	0	0
$189$	5.97020e53	1.90168
$190$	0	0
$191$	−3.78107e53	−0.940439	−0.470220	−	0.882549i	$-0.655825\pi$
$191$	−3.78107e53	−0.940439	−0.470220	+	0.882549i	$0.655825\pi$
$192$	0	0
$193$	−6.95903e53	−1.35504	−0.677520	−	0.735504i	$-0.736945\pi$
$193$	−6.95903e53	−1.35504	−0.677520	+	0.735504i	$0.736945\pi$
$194$	0	0
$195$	−6.48787e53	−0.991491
$196$	0	0
$197$	−7.80244e53	−0.938155	−0.469077	−	0.883157i	$-0.655413\pi$
$197$	−7.80244e53	−0.938155	−0.469077	+	0.883157i	$0.655413\pi$
$198$	0	0
$199$	1.09549e54	1.03887	0.519434	−	0.854510i	$-0.326143\pi$
$199$	1.09549e54	1.03887	0.519434	+	0.854510i	$0.326143\pi$
$200$	0	0
$201$	−3.95640e53	−0.296615
$202$	0	0
$203$	−1.08040e53	−0.0641841
$204$	0	0
$205$	8.15095e53	0.384585
$206$	0	0
$207$	−1.55900e54	−0.585520
$208$	0	0
$209$	−6.13989e53	−0.183960
$210$	0	0
$211$	1.93427e54	0.463317	0.231659	−	0.972797i	$-0.425585\pi$
$211$	1.93427e54	0.463317	0.231659	+	0.972797i	$0.425585\pi$
$212$	0	0
$213$	−4.50932e54	−0.865348
$214$	0	0
$215$	3.79681e54	0.584945
$216$	0	0
$217$	1.86243e55	2.30822
$218$	0	0
$219$	1.62364e54	0.162200
$220$	0	0
$221$	−9.47471e54	−0.764448
$222$	0	0
$223$	−7.23600e54	−0.472426	−0.236213	−	0.971701i	$-0.575906\pi$
$223$	−7.23600e54	−0.472426	−0.236213	+	0.971701i	$0.575906\pi$
$224$	0	0
$225$	−2.83538e55	−1.50080
$226$	0	0
$227$	−2.17948e55	−0.937013	−0.468507	−	0.883460i	$-0.655208\pi$
$227$	−2.17948e55	−0.937013	−0.468507	+	0.883460i	$0.655208\pi$
$228$	0	0
$229$	−1.86932e55	−0.653957	−0.326979	−	0.945032i	$-0.606031\pi$
$229$	−1.86932e55	−0.653957	−0.326979	+	0.945032i	$0.606031\pi$
$230$	0	0
$231$	−1.15838e56	−3.30347
$232$	0	0
$233$	−5.14721e55	−1.19869	−0.599346	−	0.800490i	$-0.704573\pi$
$233$	−5.14721e55	−1.19869	−0.599346	+	0.800490i	$0.704573\pi$
$234$	0	0
$235$	−6.46211e54	−0.123106
$236$	0	0
$237$	−4.89597e55	−0.764281
$238$	0	0
$239$	2.55198e55	0.326984	0.163492	−	0.986545i	$-0.447724\pi$
$239$	2.55198e55	0.326984	0.163492	+	0.986545i	$0.447724\pi$
$240$	0	0
$241$	6.85848e55	0.722483	0.361241	−	0.932472i	$-0.382353\pi$
$241$	6.85848e55	0.722483	0.361241	+	0.932472i	$0.382353\pi$
$242$	0	0
$243$	8.05253e55	0.698529
$244$	0	0
$245$	−6.77003e55	−0.484382
$246$	0	0
$247$	−3.65091e55	−0.215786
$248$	0	0
$249$	−6.43031e56	−3.14447
$250$	0	0
$251$	−4.26400e56	−1.72776	−0.863882	−	0.503694i	$-0.831974\pi$
$251$	−4.26400e56	−1.72776	−0.863882	+	0.503694i	$0.831974\pi$
$252$	0	0
$253$	1.30751e56	0.439654
$254$	0	0
$255$	1.12695e56	0.314929
$256$	0	0
$257$	5.48104e56	1.27478	0.637389	−	0.770543i	$-0.280015\pi$
$257$	5.48104e56	1.27478	0.637389	+	0.770543i	$0.280015\pi$
$258$	0	0
$259$	−1.18303e57	−2.29324
$260$	0	0
$261$	4.64862e55	0.0752093
$262$	0	0
$263$	1.18349e57	1.60032	0.800158	−	0.599789i	$-0.204749\pi$
$263$	1.18349e57	1.60032	0.800158	+	0.599789i	$0.204749\pi$
$264$	0	0
$265$	−2.38734e56	−0.270169
$266$	0	0
$267$	−2.92342e56	−0.277253
$268$	0	0
$269$	−2.06922e55	−0.0164675	−0.00823375	−	0.999966i	$-0.502621\pi$
$269$	−2.06922e55	−0.0164675	−0.00823375	+	0.999966i	$0.502621\pi$
$270$	0	0
$271$	−2.88469e57	−1.92895	−0.964473	−	0.264181i	$-0.914898\pi$
$271$	−2.88469e57	−1.92895	−0.964473	+	0.264181i	$0.914898\pi$
$272$	0	0
$273$	−6.88799e57	−3.87499
$274$	0	0
$275$	2.37799e57	1.12692
$276$	0	0
$277$	−4.45744e55	−0.0178160	−0.00890801	−	0.999960i	$-0.502836\pi$
$277$	−4.45744e55	−0.0178160	−0.00890801	+	0.999960i	$0.502836\pi$
$278$	0	0
$279$	−8.01345e57	−2.70471
$280$	0	0
$281$	6.47805e56	0.184861	0.0924306	−	0.995719i	$-0.470536\pi$
$281$	6.47805e56	0.184861	0.0924306	+	0.995719i	$0.470536\pi$
$282$	0	0
$283$	−4.33569e57	−1.04731	−0.523657	−	0.851929i	$-0.675433\pi$
$283$	−4.33569e57	−1.04731	−0.523657	+	0.851929i	$0.675433\pi$
$284$	0	0
$285$	4.34252e56	0.0888970
$286$	0	0
$287$	8.65364e57	1.50305
$288$	0	0
$289$	−5.13219e57	−0.757187
$290$	0	0
$291$	1.62101e57	0.203377
$292$	0	0
$293$	−1.05782e58	−1.12986	−0.564929	−	0.825140i	$-0.691096\pi$
$293$	−1.05782e58	−1.12986	−0.564929	+	0.825140i	$0.691096\pi$
$294$	0	0
$295$	−3.89766e57	−0.354802
$296$	0	0
$297$	2.15440e58	1.67321
$298$	0	0
$299$	7.77470e57	0.515716
$300$	0	0
$301$	4.03097e58	2.28611
$302$	0	0
$303$	1.68472e58	0.817762
$304$	0	0
$305$	−1.10340e58	−0.458867
$306$	0	0
$307$	−2.06877e58	−0.737837	−0.368919	−	0.929462i	$-0.620272\pi$
$307$	−2.06877e58	−0.737837	−0.368919	+	0.929462i	$0.620272\pi$
$308$	0	0
$309$	3.91015e56	0.0119721
$310$	0	0
$311$	2.39439e58	0.629978	0.314989	−	0.949095i	$-0.397999\pi$
$311$	2.39439e58	0.629978	0.314989	+	0.949095i	$0.397999\pi$
$312$	0	0
$313$	−9.48940e57	−0.214757	−0.107378	−	0.994218i	$-0.534246\pi$
$313$	−9.48940e57	−0.214757	−0.107378	+	0.994218i	$0.534246\pi$
$314$	0	0
$315$	5.22584e58	1.01826
$316$	0	0
$317$	9.13985e58	1.53478	0.767391	−	0.641180i	$-0.221555\pi$
$317$	9.13985e58	1.53478	0.767391	+	0.641180i	$0.221555\pi$
$318$	0	0
$319$	−3.89872e57	−0.0564731
$320$	0	0
$321$	−2.38117e59	−2.97799
$322$	0	0
$323$	6.34170e57	0.0685403
$324$	0	0
$325$	1.41400e59	1.32188
$326$	0	0
$327$	4.80776e56	0.00389108
$328$	0	0
$329$	−6.86064e58	−0.481129
$330$	0	0
$331$	1.08997e59	0.662911	0.331456	−	0.943471i	$-0.392460\pi$
$331$	1.08997e59	0.662911	0.331456	+	0.943471i	$0.392460\pi$
$332$	0	0
$333$	5.09019e59	2.68717
$334$	0	0
$335$	−1.49694e58	−0.0686516
$336$	0	0
$337$	−2.00408e59	−0.799120	−0.399560	−	0.916707i	$-0.630837\pi$
$337$	−2.00408e59	−0.799120	−0.399560	+	0.916707i	$0.630837\pi$
$338$	0	0
$339$	1.63962e58	0.0568918
$340$	0	0
$341$	6.72076e59	2.03091
$342$	0	0
$343$	−1.48052e59	−0.389945
$344$	0	0
$345$	−9.24750e58	−0.212459
$346$	0	0
$347$	8.91525e59	1.78810	0.894048	−	0.447971i	$-0.147853\pi$
$347$	8.91525e59	1.78810	0.894048	+	0.447971i	$0.147853\pi$
$348$	0	0
$349$	−2.64940e59	−0.464248	−0.232124	−	0.972686i	$-0.574567\pi$
$349$	−2.64940e59	−0.464248	−0.232124	+	0.972686i	$0.574567\pi$
$350$	0	0
$351$	1.28105e60	1.96268
$352$	0	0
$353$	6.11122e59	0.819261	0.409630	−	0.912252i	$-0.365658\pi$
$353$	6.11122e59	0.819261	0.409630	+	0.912252i	$0.365658\pi$
$354$	0	0
$355$	−1.70614e59	−0.200285
$356$	0	0
$357$	1.19646e60	1.23082
$358$	0	0
$359$	2.33703e59	0.210836	0.105418	−	0.994428i	$-0.466382\pi$
$359$	2.33703e59	0.210836	0.105418	+	0.994428i	$0.466382\pi$
$360$	0	0
$361$	−1.23861e60	−0.980653
$362$	0	0
$363$	−1.79031e60	−1.24487
$364$	0	0
$365$	6.14315e58	0.0375413
$366$	0	0
$367$	−7.93168e59	−0.426298	−0.213149	−	0.977020i	$-0.568372\pi$
$367$	−7.93168e59	−0.426298	−0.213149	+	0.977020i	$0.568372\pi$
$368$	0	0
$369$	−3.72339e60	−1.76124
$370$	0	0
$371$	−2.53457e60	−1.05589
$372$	0	0
$373$	−2.14688e60	−0.788224	−0.394112	−	0.919062i	$-0.628948\pi$
$373$	−2.14688e60	−0.788224	−0.394112	+	0.919062i	$0.628948\pi$
$374$	0	0
$375$	−3.65570e60	−1.18368
$376$	0	0
$377$	−2.31826e59	−0.0662431
$378$	0	0
$379$	5.45651e60	1.37687	0.688436	−	0.725297i	$-0.258298\pi$
$379$	5.45651e60	1.37687	0.688436	+	0.725297i	$0.258298\pi$
$380$	0	0
$381$	2.39237e60	0.533447
$382$	0	0
$383$	2.64466e60	0.521434	0.260717	−	0.965415i	$-0.416041\pi$
$383$	2.64466e60	0.521434	0.260717	+	0.965415i	$0.416041\pi$
$384$	0	0
$385$	−4.38283e60	−0.764589
$386$	0	0
$387$	−1.73440e61	−2.67880
$388$	0	0
$389$	8.47705e60	1.15992	0.579962	−	0.814644i	$-0.303068\pi$
$389$	8.47705e60	1.15992	0.579962	+	0.814644i	$0.303068\pi$
$390$	0	0
$391$	−1.35048e60	−0.163808
$392$	0	0
$393$	−4.32740e60	−0.465588
$394$	0	0
$395$	−1.85243e60	−0.176893
$396$	0	0
$397$	−1.01947e61	−0.864568	−0.432284	−	0.901738i	$-0.642292\pi$
$397$	−1.01947e61	−0.864568	−0.432284	+	0.901738i	$0.642292\pi$
$398$	0	0
$399$	4.61033e60	0.347431
$400$	0	0
$401$	−2.08710e61	−1.39846	−0.699230	−	0.714897i	$-0.746474\pi$
$401$	−2.08710e61	−1.39846	−0.699230	+	0.714897i	$0.746474\pi$
$402$	0	0
$403$	3.99630e61	2.38226
$404$	0	0
$405$	−2.47134e60	−0.131141
$406$	0	0
$407$	−4.26907e61	−2.01774
$408$	0	0
$409$	2.90356e61	1.22302	0.611509	−	0.791238i	$-0.290563\pi$
$409$	2.90356e61	1.22302	0.611509	+	0.791238i	$0.290563\pi$
$410$	0	0
$411$	1.63581e61	0.614397
$412$	0	0
$413$	−4.13804e61	−1.38665
$414$	0	0
$415$	−2.43296e61	−0.727787
$416$	0	0
$417$	3.29616e61	0.880664
$418$	0	0
$419$	−4.51827e61	−1.07880	−0.539402	−	0.842048i	$-0.681350\pi$
$419$	−4.51827e61	−1.07880	−0.539402	+	0.842048i	$0.681350\pi$
$420$	0	0
$421$	6.03083e61	1.28750	0.643751	−	0.765235i	$-0.277377\pi$
$421$	6.03083e61	1.28750	0.643751	+	0.765235i	$0.277377\pi$
$422$	0	0
$423$	2.95192e61	0.563775
$424$	0	0
$425$	−2.45615e61	−0.419870
$426$	0	0
$427$	−1.17145e62	−1.79336
$428$	0	0
$429$	−2.48560e62	−3.40945
$430$	0	0
$431$	2.84185e61	0.349450	0.174725	−	0.984617i	$-0.444096\pi$
$431$	2.84185e61	0.349450	0.174725	+	0.984617i	$0.444096\pi$
$432$	0	0
$433$	9.49921e61	1.04766	0.523832	−	0.851822i	$-0.324502\pi$
$433$	9.49921e61	1.04766	0.523832	+	0.851822i	$0.324502\pi$
$434$	0	0
$435$	2.75742e60	0.0272901
$436$	0	0
$437$	−5.20383e60	−0.0462391
$438$	0	0
$439$	8.51566e61	0.679676	0.339838	−	0.940484i	$-0.389628\pi$
$439$	8.51566e61	0.679676	0.339838	+	0.940484i	$0.389628\pi$
$440$	0	0
$441$	3.09258e62	2.21827
$442$	0	0
$443$	2.00324e62	1.29195	0.645974	−	0.763360i	$-0.276452\pi$
$443$	2.00324e62	1.29195	0.645974	+	0.763360i	$0.276452\pi$
$444$	0	0
$445$	−1.10610e61	−0.0641702
$446$	0	0
$447$	4.27227e62	2.23065
$448$	0	0
$449$	1.70791e62	0.802927	0.401463	−	0.915875i	$-0.368502\pi$
$449$	1.70791e62	0.802927	0.401463	+	0.915875i	$0.368502\pi$
$450$	0	0
$451$	3.12275e62	1.32248
$452$	0	0
$453$	2.84596e62	1.08623
$454$	0	0
$455$	−2.60613e62	−0.896866
$456$	0	0
$457$	6.09529e62	1.89219	0.946093	−	0.323895i	$-0.104993\pi$
$457$	6.09529e62	1.89219	0.946093	+	0.323895i	$0.104993\pi$
$458$	0	0
$459$	−2.22521e62	−0.623409
$460$	0	0
$461$	−6.69058e62	−1.69236	−0.846179	−	0.532898i	$-0.821103\pi$
$461$	−6.69058e62	−1.69236	−0.846179	+	0.532898i	$0.821103\pi$
$462$	0	0
$463$	−4.30318e62	−0.983187	−0.491593	−	0.870825i	$-0.663585\pi$
$463$	−4.30318e62	−0.983187	−0.491593	+	0.870825i	$0.663585\pi$
$464$	0	0
$465$	−4.75334e62	−0.981419
$466$	0	0
$467$	5.60104e62	1.04550	0.522748	−	0.852487i	$-0.324907\pi$
$467$	5.60104e62	1.04550	0.522748	+	0.852487i	$0.324907\pi$
$468$	0	0
$469$	−1.58926e62	−0.268307
$470$	0	0
$471$	8.80123e62	1.34447
$472$	0	0
$473$	1.45461e63	2.01146
$474$	0	0
$475$	−9.46432e61	−0.118519
$476$	0	0
$477$	1.09055e63	1.23726
$478$	0	0
$479$	−1.45521e63	−1.49638	−0.748190	−	0.663484i	$-0.769077\pi$
$479$	−1.45521e63	−1.49638	−0.748190	+	0.663484i	$0.769077\pi$
$480$	0	0
$481$	−2.53848e63	−2.36681
$482$	0	0
$483$	−9.81782e62	−0.830343
$484$	0	0
$485$	6.13322e61	0.0470715
$486$	0	0
$487$	1.24716e62	0.0868944	0.0434472	−	0.999056i	$-0.486166\pi$
$487$	1.24716e62	0.0868944	0.0434472	+	0.999056i	$0.486166\pi$
$488$	0	0
$489$	−1.16476e63	−0.737017
$490$	0	0
$491$	−9.69339e62	−0.557263	−0.278632	−	0.960398i	$-0.589881\pi$
$491$	−9.69339e62	−0.557263	−0.278632	+	0.960398i	$0.589881\pi$
$492$	0	0
$493$	4.02687e61	0.0210409
$494$	0	0
$495$	1.88579e63	0.895926
$496$	0	0
$497$	−1.81136e63	−0.782762
$498$	0	0
$499$	3.24859e63	1.27742	0.638708	−	0.769449i	$-0.279469\pi$
$499$	3.24859e63	1.27742	0.638708	+	0.769449i	$0.279469\pi$
$500$	0	0
$501$	−1.73901e62	−0.0622466
$502$	0	0
$503$	2.69774e63	0.879330	0.439665	−	0.898162i	$-0.355097\pi$
$503$	2.69774e63	0.879330	0.439665	+	0.898162i	$0.355097\pi$
$504$	0	0
$505$	6.37428e62	0.189271
$506$	0	0
$507$	−8.63877e63	−2.33757
$508$	0	0
$509$	6.02700e63	1.48673	0.743366	−	0.668884i	$-0.233228\pi$
$509$	6.02700e63	1.48673	0.743366	+	0.668884i	$0.233228\pi$
$510$	0	0
$511$	6.52201e62	0.146720
$512$	0	0
$513$	−8.57445e62	−0.175974
$514$	0	0
$515$	1.47943e61	0.00277093
$516$	0	0
$517$	−2.47573e63	−0.423326
$518$	0	0
$519$	−1.81382e64	−2.83245
$520$	0	0
$521$	1.33678e63	0.190710	0.0953548	−	0.995443i	$-0.469601\pi$
$521$	1.33678e63	0.190710	0.0953548	+	0.995443i	$0.469601\pi$
$522$	0	0
$523$	8.20599e63	1.06989	0.534947	−	0.844885i	$-0.320331\pi$
$523$	8.20599e63	1.06989	0.534947	+	0.844885i	$0.320331\pi$
$524$	0	0
$525$	−1.78559e64	−2.12832
$526$	0	0
$527$	−6.94165e63	−0.756682
$528$	0	0
$529$	−8.91969e63	−0.889491
$530$	0	0
$531$	1.78047e64	1.62485
$532$	0	0
$533$	1.85685e64	1.55127
$534$	0	0
$535$	−9.00935e63	−0.689255
$536$	0	0
$537$	−3.74062e64	−2.62149
$538$	0	0
$539$	−2.59370e64	−1.66565
$540$	0	0
$541$	−6.23992e62	−0.0367319	−0.0183660	−	0.999831i	$-0.505846\pi$
$541$	−6.23992e62	−0.0367319	−0.0183660	+	0.999831i	$0.505846\pi$
$542$	0	0
$543$	2.51029e64	1.35497
$544$	0	0
$545$	1.81905e61	0.000900591 0
$546$	0	0
$547$	−3.05397e64	−1.38727	−0.693636	−	0.720325i	$-0.743993\pi$
$547$	−3.05397e64	−1.38727	−0.693636	+	0.720325i	$0.743993\pi$
$548$	0	0
$549$	5.04038e64	2.10142
$550$	0	0
$551$	1.55168e62	0.00593936
$552$	0	0
$553$	−1.96667e64	−0.691340
$554$	0	0
$555$	3.01935e64	0.975053
$556$	0	0
$557$	3.06228e64	0.908758	0.454379	−	0.890809i	$-0.349861\pi$
$557$	3.06228e64	0.908758	0.454379	+	0.890809i	$0.349861\pi$
$558$	0	0
$559$	8.64943e64	2.35945
$560$	0	0
$561$	4.31753e64	1.08295
$562$	0	0
$563$	1.53750e64	0.354705	0.177353	−	0.984147i	$-0.443247\pi$
$563$	1.53750e64	0.354705	0.177353	+	0.984147i	$0.443247\pi$
$564$	0	0
$565$	6.20362e62	0.0131676
$566$	0	0
$567$	−2.62376e64	−0.512533
$568$	0	0
$569$	−9.13382e63	−0.164253	−0.0821266	−	0.996622i	$-0.526171\pi$
$569$	−9.13382e63	−0.164253	−0.0821266	+	0.996622i	$0.526171\pi$
$570$	0	0
$571$	−4.20849e64	−0.696911	−0.348456	−	0.937325i	$-0.613294\pi$
$571$	−4.20849e64	−0.696911	−0.348456	+	0.937325i	$0.613294\pi$
$572$	0	0
$573$	1.02453e65	1.56275
$574$	0	0
$575$	2.01545e64	0.283255
$576$	0	0
$577$	1.16840e65	1.51342	0.756709	−	0.653752i	$-0.226806\pi$
$577$	1.16840e65	1.51342	0.756709	+	0.653752i	$0.226806\pi$
$578$	0	0
$579$	1.88564e65	2.25171
$580$	0	0
$581$	−2.58300e65	−2.84437
$582$	0	0
$583$	−9.14625e64	−0.929035
$584$	0	0
$585$	1.12133e65	1.05093
$586$	0	0
$587$	−1.22823e65	−1.06239	−0.531197	−	0.847248i	$-0.678258\pi$
$587$	−1.22823e65	−1.06239	−0.531197	+	0.847248i	$0.678258\pi$
$588$	0	0
$589$	−2.67484e64	−0.213594
$590$	0	0
$591$	2.11417e65	1.55896
$592$	0	0
$593$	2.18247e65	1.48649	0.743245	−	0.669019i	$-0.233286\pi$
$593$	2.18247e65	1.48649	0.743245	+	0.669019i	$0.233286\pi$
$594$	0	0
$595$	4.52689e64	0.284873
$596$	0	0
$597$	−2.96836e65	−1.72632
$598$	0	0
$599$	−1.76834e65	−0.950684	−0.475342	−	0.879801i	$-0.657676\pi$
$599$	−1.76834e65	−0.950684	−0.475342	+	0.879801i	$0.657676\pi$
$600$	0	0
$601$	−1.41803e65	−0.704912	−0.352456	−	0.935828i	$-0.614653\pi$
$601$	−1.41803e65	−0.704912	−0.352456	+	0.935828i	$0.614653\pi$
$602$	0	0
$603$	6.83807e64	0.314396
$604$	0	0
$605$	−6.77379e64	−0.288125
$606$	0	0
$607$	−1.14990e65	−0.452610	−0.226305	−	0.974056i	$-0.572665\pi$
$607$	−1.14990e65	−0.452610	−0.226305	+	0.974056i	$0.572665\pi$
$608$	0	0
$609$	2.92748e64	0.106656
$610$	0	0
$611$	−1.47212e65	−0.496563
$612$	0	0
$613$	−9.28200e64	−0.289948	−0.144974	−	0.989435i	$-0.546310\pi$
$613$	−9.28200e64	−0.289948	−0.144974	+	0.989435i	$0.546310\pi$
$614$	0	0
$615$	−2.20860e65	−0.639076
$616$	0	0
$617$	5.27493e65	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$617$	5.27493e65	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$618$	0	0
$619$	1.15047e65	0.285854	0.142927	−	0.989733i	$-0.454349\pi$
$619$	1.15047e65	0.285854	0.142927	+	0.989733i	$0.454349\pi$
$620$	0	0
$621$	1.82595e65	0.420568
$622$	0	0
$623$	−1.17431e65	−0.250793
$624$	0	0
$625$	2.91873e65	0.578114
$626$	0	0
$627$	1.66368e65	0.305691
$628$	0	0
$629$	4.40938e65	0.751774
$630$	0	0
$631$	4.57574e65	0.724054	0.362027	−	0.932168i	$-0.382085\pi$
$631$	4.57574e65	0.724054	0.362027	+	0.932168i	$0.382085\pi$
$632$	0	0
$633$	−5.24114e65	−0.769907
$634$	0	0
$635$	9.05172e64	0.123466
$636$	0	0
$637$	−1.54227e66	−1.95381
$638$	0	0
$639$	7.79371e65	0.917221
$640$	0	0
$641$	−9.71867e65	−1.06278	−0.531390	−	0.847128i	$-0.678330\pi$
$641$	−9.71867e65	−1.06278	−0.531390	+	0.847128i	$0.678330\pi$
$642$	0	0
$643$	−2.57162e65	−0.261366	−0.130683	−	0.991424i	$-0.541717\pi$
$643$	−2.57162e65	−0.261366	−0.130683	+	0.991424i	$0.541717\pi$
$644$	0	0
$645$	−1.02879e66	−0.972019
$646$	0	0
$647$	−2.26973e66	−1.99400	−0.996998	−	0.0774314i	$-0.975328\pi$
$647$	−2.26973e66	−1.99400	−0.996998	+	0.0774314i	$0.975328\pi$
$648$	0	0
$649$	−1.49325e66	−1.22006
$650$	0	0
$651$	−5.04649e66	−3.83563
$652$	0	0
$653$	1.60777e66	1.13701	0.568504	−	0.822681i	$-0.307522\pi$
$653$	1.60777e66	1.13701	0.568504	+	0.822681i	$0.307522\pi$
$654$	0	0
$655$	−1.63730e65	−0.107760
$656$	0	0
$657$	−2.80622e65	−0.171923
$658$	0	0
$659$	3.20448e66	1.82789	0.913946	−	0.405835i	$-0.133019\pi$
$659$	3.20448e66	1.82789	0.913946	+	0.405835i	$0.133019\pi$
$660$	0	0
$661$	6.38586e65	0.339223	0.169611	−	0.985511i	$-0.445749\pi$
$661$	6.38586e65	0.339223	0.169611	+	0.985511i	$0.445749\pi$
$662$	0	0
$663$	2.56729e66	1.27030
$664$	0	0
$665$	1.74435e65	0.0804129
$666$	0	0
$667$	−3.30434e64	−0.0141948
$668$	0	0
$669$	1.96068e66	0.785044
$670$	0	0
$671$	−4.22729e66	−1.57791
$672$	0	0
$673$	−2.24978e66	−0.783043	−0.391521	−	0.920169i	$-0.628051\pi$
$673$	−2.24978e66	−0.783043	−0.391521	+	0.920169i	$0.628051\pi$
$674$	0	0
$675$	3.32090e66	1.07800
$676$	0	0
$677$	−4.79933e65	−0.145327	−0.0726636	−	0.997357i	$-0.523150\pi$
$677$	−4.79933e65	−0.145327	−0.0726636	+	0.997357i	$0.523150\pi$
$678$	0	0
$679$	6.51147e65	0.183967
$680$	0	0
$681$	5.90558e66	1.55706
$682$	0	0
$683$	4.30311e66	1.05900	0.529500	−	0.848310i	$-0.322379\pi$
$683$	4.30311e66	1.05900	0.529500	+	0.848310i	$0.322379\pi$
$684$	0	0
$685$	6.18920e65	0.142202
$686$	0	0
$687$	5.06515e66	1.08670
$688$	0	0
$689$	−5.43855e66	−1.08976
$690$	0	0
$691$	−8.37771e66	−1.56816	−0.784081	−	0.620658i	$-0.786865\pi$
$691$	−8.37771e66	−1.56816	−0.784081	+	0.620658i	$0.786865\pi$
$692$	0	0
$693$	2.00210e67	3.50150
$694$	0	0
$695$	1.24713e66	0.203830
$696$	0	0
$697$	−3.22539e66	−0.492733
$698$	0	0
$699$	1.39470e67	1.99190
$700$	0	0
$701$	−1.18236e67	−1.57898	−0.789489	−	0.613765i	$-0.789654\pi$
$701$	−1.18236e67	−1.57898	−0.789489	+	0.613765i	$0.789654\pi$
$702$	0	0
$703$	1.69907e66	0.212208
$704$	0	0
$705$	1.75099e66	0.204569
$706$	0	0
$707$	6.76740e66	0.739717
$708$	0	0
$709$	8.35736e66	0.854836	0.427418	−	0.904054i	$-0.359423\pi$
$709$	8.35736e66	0.854836	0.427418	+	0.904054i	$0.359423\pi$
$710$	0	0
$711$	8.46197e66	0.810095
$712$	0	0
$713$	5.69614e66	0.510478
$714$	0	0
$715$	−9.40445e66	−0.789117
$716$	0	0
$717$	−6.91492e66	−0.543359
$718$	0	0
$719$	−1.29585e67	−0.953731	−0.476865	−	0.878976i	$-0.658227\pi$
$719$	−1.29585e67	−0.953731	−0.476865	+	0.878976i	$0.658227\pi$
$720$	0	0
$721$	1.57067e65	0.0108295
$722$	0	0
$723$	−1.85839e67	−1.20057
$724$	0	0
$725$	−6.00968e65	−0.0363838
$726$	0	0
$727$	−6.75407e66	−0.383272	−0.191636	−	0.981466i	$-0.561379\pi$
$727$	−6.75407e66	−0.383272	−0.191636	+	0.981466i	$0.561379\pi$
$728$	0	0
$729$	−2.82288e67	−1.50174
$730$	0	0
$731$	−1.50242e67	−0.749435
$732$	0	0
$733$	3.49174e67	1.63342	0.816711	−	0.577047i	$-0.195795\pi$
$733$	3.49174e67	1.63342	0.816711	+	0.577047i	$0.195795\pi$
$734$	0	0
$735$	1.83443e67	0.804910
$736$	0	0
$737$	−5.73498e66	−0.236073
$738$	0	0
$739$	−2.87820e66	−0.111167	−0.0555835	−	0.998454i	$-0.517702\pi$
$739$	−2.87820e66	−0.111167	−0.0555835	+	0.998454i	$0.517702\pi$
$740$	0	0
$741$	9.89260e66	0.358577
$742$	0	0
$743$	1.55726e66	0.0529814	0.0264907	−	0.999649i	$-0.491567\pi$
$743$	1.55726e66	0.0529814	0.0264907	+	0.999649i	$0.491567\pi$
$744$	0	0
$745$	1.61645e67	0.516284
$746$	0	0
$747$	1.11138e68	3.33296
$748$	0	0
$749$	−9.56498e67	−2.69378
$750$	0	0
$751$	−4.80070e67	−1.26989	−0.634946	−	0.772557i	$-0.718977\pi$
$751$	−4.80070e67	−1.26989	−0.634946	+	0.772557i	$0.718977\pi$
$752$	0	0
$753$	1.15538e68	2.87107
$754$	0	0
$755$	1.07679e67	0.251407
$756$	0	0
$757$	−3.04494e67	−0.668076	−0.334038	−	0.942560i	$-0.608411\pi$
$757$	−3.04494e67	−0.668076	−0.334038	+	0.942560i	$0.608411\pi$
$758$	0	0
$759$	−3.54285e67	−0.730586
$760$	0	0
$761$	−8.19495e67	−1.58857	−0.794286	−	0.607544i	$-0.792155\pi$
$761$	−8.19495e67	−1.58857	−0.794286	+	0.607544i	$0.792155\pi$
$762$	0	0
$763$	1.93124e65	0.00351973
$764$	0	0
$765$	−1.94778e67	−0.333807
$766$	0	0
$767$	−8.87918e67	−1.43114
$768$	0	0
$769$	−8.58726e67	−1.30192	−0.650962	−	0.759110i	$-0.725634\pi$
$769$	−8.58726e67	−1.30192	−0.650962	+	0.759110i	$0.725634\pi$
$770$	0	0
$771$	−1.48516e68	−2.11833
$772$	0	0
$773$	−5.74418e67	−0.770921	−0.385460	−	0.922724i	$-0.625957\pi$
$773$	−5.74418e67	−0.770921	−0.385460	+	0.922724i	$0.625957\pi$
$774$	0	0
$775$	1.03597e68	1.30845
$776$	0	0
$777$	3.20556e68	3.81075
$778$	0	0
$779$	−1.24284e67	−0.139087
$780$	0	0
$781$	−6.53646e67	−0.688721
$782$	0	0
$783$	−5.44463e66	−0.0540215
$784$	0	0
$785$	3.33001e67	0.311178
$786$	0	0
$787$	−1.78281e68	−1.56928	−0.784638	−	0.619954i	$-0.787151\pi$
$787$	−1.78281e68	−1.56928	−0.784638	+	0.619954i	$0.787151\pi$
$788$	0	0
$789$	−3.20683e68	−2.65929
$790$	0	0
$791$	6.58621e66	0.0514622
$792$	0	0
$793$	−2.51363e68	−1.85090
$794$	0	0
$795$	6.46880e67	0.448948
$796$	0	0
$797$	7.67558e67	0.502158	0.251079	−	0.967967i	$-0.419215\pi$
$797$	7.67558e67	0.502158	0.251079	+	0.967967i	$0.419215\pi$
$798$	0	0
$799$	2.55710e67	0.157724
$800$	0	0
$801$	5.05270e67	0.293873
$802$	0	0
$803$	2.35353e67	0.129094
$804$	0	0
$805$	−3.71465e67	−0.192183
$806$	0	0
$807$	5.60681e66	0.0273645
$808$	0	0
$809$	1.39305e68	0.641472	0.320736	−	0.947169i	$-0.396070\pi$
$809$	1.39305e68	0.641472	0.320736	+	0.947169i	$0.396070\pi$
$810$	0	0
$811$	−6.57381e67	−0.285645	−0.142823	−	0.989748i	$-0.545618\pi$
$811$	−6.57381e67	−0.285645	−0.142823	+	0.989748i	$0.545618\pi$
$812$	0	0
$813$	7.81642e68	3.20538
$814$	0	0
$815$	−4.40695e67	−0.170582
$816$	0	0
$817$	−5.78931e67	−0.211548
$818$	0	0
$819$	1.19049e69	4.10727
$820$	0	0
$821$	2.60915e68	0.850030	0.425015	−	0.905186i	$-0.360269\pi$
$821$	2.60915e68	0.850030	0.425015	+	0.905186i	$0.360269\pi$
$822$	0	0
$823$	−2.86763e68	−0.882322	−0.441161	−	0.897428i	$-0.645433\pi$
$823$	−2.86763e68	−0.882322	−0.441161	+	0.897428i	$0.645433\pi$
$824$	0	0
$825$	−6.44346e68	−1.87263
$826$	0	0
$827$	2.35931e68	0.647747	0.323873	−	0.946100i	$-0.395015\pi$
$827$	2.35931e68	0.647747	0.323873	+	0.946100i	$0.395015\pi$
$828$	0	0
$829$	8.80819e67	0.228484	0.114242	−	0.993453i	$-0.463556\pi$
$829$	8.80819e67	0.228484	0.114242	+	0.993453i	$0.463556\pi$
$830$	0	0
$831$	1.20780e67	0.0296054
$832$	0	0
$833$	2.67895e68	0.620592
$834$	0	0
$835$	−6.57968e66	−0.0144070
$836$	0	0
$837$	9.38564e68	1.94274
$838$	0	0
$839$	2.99608e68	0.586337	0.293168	−	0.956061i	$-0.405290\pi$
$839$	2.99608e68	0.586337	0.293168	+	0.956061i	$0.405290\pi$
$840$	0	0
$841$	−5.39403e68	−0.998177
$842$	0	0
$843$	−1.75531e68	−0.307189
$844$	0	0
$845$	−3.26855e68	−0.541031
$846$	0	0
$847$	−7.19155e68	−1.12606
$848$	0	0
$849$	1.17481e69	1.74035
$850$	0	0
$851$	−3.61822e68	−0.507167
$852$	0	0
$853$	9.47738e68	1.25715	0.628573	−	0.777750i	$-0.283639\pi$
$853$	9.47738e68	1.25715	0.628573	+	0.777750i	$0.283639\pi$
$854$	0	0
$855$	−7.50540e67	−0.0942259
$856$	0	0
$857$	−9.55995e68	−1.13607	−0.568037	−	0.823003i	$-0.692297\pi$
$857$	−9.55995e68	−1.13607	−0.568037	+	0.823003i	$0.692297\pi$
$858$	0	0
$859$	4.21759e68	0.474488	0.237244	−	0.971450i	$-0.423756\pi$
$859$	4.21759e68	0.474488	0.237244	+	0.971450i	$0.423756\pi$
$860$	0	0
$861$	−2.34481e69	−2.49767
$862$	0	0
$863$	−9.19093e68	−0.927055	−0.463528	−	0.886082i	$-0.653417\pi$
$863$	−9.19093e68	−0.927055	−0.463528	+	0.886082i	$0.653417\pi$
$864$	0	0
$865$	−6.86273e68	−0.655569
$866$	0	0
$867$	1.39063e69	1.25824
$868$	0	0
$869$	−7.09693e68	−0.608283
$870$	0	0
$871$	−3.41014e68	−0.276914
$872$	0	0
$873$	−2.80168e68	−0.215568
$874$	0	0
$875$	−1.46847e69	−1.07072
$876$	0	0
$877$	1.86631e69	1.28971	0.644855	−	0.764305i	$-0.276918\pi$
$877$	1.86631e69	1.28971	0.644855	+	0.764305i	$0.276918\pi$
$878$	0	0
$879$	2.86630e69	1.87752
$880$	0	0
$881$	6.60423e68	0.410098	0.205049	−	0.978752i	$-0.434265\pi$
$881$	6.60423e68	0.410098	0.205049	+	0.978752i	$0.434265\pi$
$882$	0	0
$883$	8.14396e68	0.479467	0.239734	−	0.970839i	$-0.422940\pi$
$883$	8.14396e68	0.479467	0.239734	+	0.970839i	$0.422940\pi$
$884$	0	0
$885$	1.05612e69	0.589585
$886$	0	0
$887$	1.87041e68	0.0990221	0.0495110	−	0.998774i	$-0.484234\pi$
$887$	1.87041e68	0.0990221	0.0495110	+	0.998774i	$0.484234\pi$
$888$	0	0
$889$	9.60996e68	0.482537
$890$	0	0
$891$	−9.46807e68	−0.450957
$892$	0	0
$893$	9.85332e67	0.0445218
$894$	0	0
$895$	−1.41529e69	−0.606743
$896$	0	0
$897$	−2.10665e69	−0.856980
$898$	0	0
$899$	−1.69848e68	−0.0655702
$900$	0	0
$901$	9.44686e68	0.346143
$902$	0	0
$903$	−1.09224e70	−3.79889
$904$	0	0
$905$	9.49788e68	0.313607
$906$	0	0
$907$	6.31631e69	1.98012	0.990062	−	0.140631i	$-0.0449131\pi$
$907$	6.31631e69	1.98012	0.990062	+	0.140631i	$0.0449131\pi$
$908$	0	0
$909$	−2.91180e69	−0.866782
$910$	0	0
$911$	−2.58489e69	−0.730736	−0.365368	−	0.930863i	$-0.619057\pi$
$911$	−2.58489e69	−0.730736	−0.365368	+	0.930863i	$0.619057\pi$
$912$	0	0
$913$	−9.32101e69	−2.50265
$914$	0	0
$915$	2.98980e69	0.762512
$916$	0	0
$917$	−1.73828e69	−0.421154
$918$	0	0
$919$	−8.65693e68	−0.199273	−0.0996367	−	0.995024i	$-0.531768\pi$
$919$	−8.65693e68	−0.199273	−0.0996367	+	0.995024i	$0.531768\pi$
$920$	0	0
$921$	5.60559e69	1.22608
$922$	0	0
$923$	−3.88672e69	−0.807873
$924$	0	0
$925$	−6.58054e69	−1.29996
$926$	0	0
$927$	−6.75811e67	−0.0126897
$928$	0	0
$929$	7.30808e69	1.30447	0.652235	−	0.758017i	$-0.273831\pi$
$929$	7.30808e69	1.30447	0.652235	+	0.758017i	$0.273831\pi$
$930$	0	0
$931$	1.03228e69	0.175179
$932$	0	0
$933$	−6.48789e69	−1.04685
$934$	0	0
$935$	1.63357e69	0.250648
$936$	0	0
$937$	8.40248e69	1.22610	0.613052	−	0.790043i	$-0.289942\pi$
$937$	8.40248e69	1.22610	0.613052	+	0.790043i	$0.289942\pi$
$938$	0	0
$939$	2.57127e69	0.356867
$940$	0	0
$941$	−4.67075e69	−0.616638	−0.308319	−	0.951283i	$-0.599766\pi$
$941$	−4.67075e69	−0.616638	−0.308319	+	0.951283i	$0.599766\pi$
$942$	0	0
$943$	2.64667e69	0.332411
$944$	0	0
$945$	−6.12069e69	−0.731397
$946$	0	0
$947$	1.91649e69	0.217912	0.108956	−	0.994047i	$-0.465249\pi$
$947$	1.91649e69	0.217912	0.108956	+	0.994047i	$0.465249\pi$
$948$	0	0
$949$	1.39946e69	0.151427
$950$	0	0
$951$	−2.47656e70	−2.55039
$952$	0	0
$953$	8.83831e69	0.866336	0.433168	−	0.901313i	$-0.357396\pi$
$953$	8.83831e69	0.866336	0.433168	+	0.901313i	$0.357396\pi$
$954$	0	0
$955$	3.87638e69	0.361699
$956$	0	0
$957$	1.05641e69	0.0938428
$958$	0	0
$959$	6.57090e69	0.555761
$960$	0	0
$961$	1.68624e70	1.35806
$962$	0	0
$963$	4.11551e70	3.15650
$964$	0	0
$965$	7.13445e69	0.521157
$966$	0	0
$967$	1.45957e70	1.01556	0.507778	−	0.861488i	$-0.330467\pi$
$967$	1.45957e70	1.01556	0.507778	+	0.861488i	$0.330467\pi$
$968$	0	0
$969$	−1.71836e69	−0.113895
$970$	0	0
$971$	1.63771e70	1.03415	0.517077	−	0.855939i	$-0.327020\pi$
$971$	1.63771e70	1.03415	0.517077	+	0.855939i	$0.327020\pi$
$972$	0	0
$973$	1.32404e70	0.796616
$974$	0	0
$975$	−3.83142e70	−2.19660
$976$	0	0
$977$	2.81412e70	1.53752	0.768758	−	0.639540i	$-0.220875\pi$
$977$	2.81412e70	1.53752	0.768758	+	0.639540i	$0.220875\pi$
$978$	0	0
$979$	−4.23762e69	−0.220663
$980$	0	0
$981$	−8.30951e67	−0.00412433
$982$	0	0
$983$	2.50839e70	1.18683	0.593414	−	0.804898i	$-0.297780\pi$
$983$	2.50839e70	1.18683	0.593414	+	0.804898i	$0.297780\pi$
$984$	0	0
$985$	7.99913e69	0.360821
$986$	0	0
$987$	1.85898e70	0.799505
$988$	0	0
$989$	1.23285e70	0.505588
$990$	0	0
$991$	−1.15349e70	−0.451111	−0.225556	−	0.974230i	$-0.572420\pi$
$991$	−1.15349e70	−0.451111	−0.225556	+	0.974230i	$0.572420\pi$
$992$	0	0
$993$	−2.95340e70	−1.10158
$994$	0	0
$995$	−1.12310e70	−0.399556
$996$	0	0
$997$	−1.95619e70	−0.663858	−0.331929	−	0.943304i	$-0.607699\pi$
$997$	−1.95619e70	−0.663858	−0.331929	+	0.943304i	$0.607699\pi$
$998$	0	0
$999$	−5.96181e70	−1.93014

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.48.a.b.1.1		2
4.3	odd	2		2.48.a.b.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.48.a.b.1.2	✓	2	4.3	odd	2
16.48.a.b.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 48 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70963e11 q^{3} -1.02521e16 q^{5} -1.08843e20 q^{7} +4.68319e22 q^{9} +O(q^{10})\)	\(q-2.70963e11 q^{3} -1.02521e16 q^{5} -1.08843e20 q^{7} +4.68319e22 q^{9} -3.92772e24 q^{11} -2.33551e26 q^{13} +2.77793e27 q^{15} +4.05682e28 q^{17} +1.56322e29 q^{19} +2.94925e31 q^{21} -3.32892e31 q^{23} -6.05438e32 q^{25} -5.48512e33 q^{27} +9.92617e32 q^{29} -1.71111e35 q^{31} +1.06427e36 q^{33} +1.11587e36 q^{35} +1.08691e37 q^{37} +6.32835e37 q^{39} -7.95054e37 q^{41} -3.70345e38 q^{43} -4.80124e38 q^{45} +6.30322e38 q^{47} +6.60357e39 q^{49} -1.09925e40 q^{51} +2.32864e40 q^{53} +4.02673e40 q^{55} -4.23574e40 q^{57} +3.80182e41 q^{59} +1.07627e42 q^{61} -5.09735e42 q^{63} +2.39438e42 q^{65} +1.46013e42 q^{67} +9.02012e42 q^{69} +1.66419e43 q^{71} -5.99210e42 q^{73} +1.64051e44 q^{75} +4.27507e44 q^{77} +1.80688e44 q^{79} +2.41058e44 q^{81} +2.37314e45 q^{83} -4.15908e44 q^{85} -2.68962e44 q^{87} +1.07890e45 q^{89} +2.54205e46 q^{91} +4.63646e46 q^{93} -1.60263e45 q^{95} -5.98242e45 q^{97} -1.83943e47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 122289844824 q^{3} + 18\!\cdots\!40 q^{5}+ \cdots + 42\!\cdots\!14 q^{9}+O(q^{10}) \) 2 * q - 122289844824 * q^3 + 18178497839194140 * q^5 - 161018545456026827632 * q^7 + 42346678028148839299914 * q^9	\( 2 q - 122289844824 q^{3} + 18\!\cdots\!40 q^{5}+ \cdots - 19\!\cdots\!88 q^{99}+O(q^{100}) \) 2 * q - 122289844824 * q^3 + 18178497839194140 * q^5 - 161018545456026827632 * q^7 + 42346678028148839299914 * q^9 - 1148414200601353937498184 * q^11 - 91147460388086216079419636 * q^13 + 7004781615634981008503062320 * q^15 - 102841736272610788276309928988 * q^17 - 1021269902513170084335814340920 * q^19 + 21735508585019662606113256728384 * q^21 - 8761047668524985915668413593424 * q^23 - 507682648872986273192302737941650 * q^25 - 10104984547772788958914840573828080 * q^27 + 44866738270461738647546902365332460 * q^29 - 207627610718284877414441666427987904 * q^31 + 1477472372445089531927832872318579808 * q^33 - 367494533980940143565872352084574240 * q^35 + 15454503825836181610788445034143257052 * q^37 + 84454902625144773203023026649833922032 * q^39 - 183644075219801436287443355635335693516 * q^41 - 251792238934935641547265550793932228104 * q^43 - 607642114901245348425108558238563820020 * q^45 - 2877259596300283378159171629753736631712 * q^47 + 4082466065158778742105339967177123972626 * q^49 - 32313595089705813278113192123424261680944 * q^51 + 36237003736012302609225684974144068811964 * q^53 + 119284580676988289643910520626568801967120 * q^55 - 217433238554190330563790441928328986980960 * q^57 - 216631413518842627379871131067900548684520 * q^59 + 2343345794410501150323838534047225181209004 * q^61 - 4863331655836708395434595438453038464171824 * q^63 + 6442979197855565506819083799260229348557480 * q^65 - 9679359518522646693994124509775830904939992 * q^67 + 12666786020488197907659082189490927173898688 * q^69 + 34699689890171309465034074226599979171319056 * q^71 - 44919947676390510667705279890416107043412236 * q^73 + 178584439183245940201476654984555123803179800 * q^75 + 282496402053252599536812437150318196880350144 * q^77 - 459051391676893300046865817924586171479631200 * q^79 - 326533112556292583027849224141868597638781358 * q^81 + 1357110492338922730441331533770737425416063816 * q^83 - 4493133982253416774682211563698820690839841160 * q^85 + 6253923718333989640702287995671643099683866480 * q^87 + 4559737044037194815580171398227946101567633620 * q^89 + 17990568925125012147062510675834829593464410976 * q^91 + 40935592303229143353065306088384219268477064448 * q^93 - 35082243421784097273165847234698748298121392400 * q^95 + 53827580940974934190192198959367945947750881092 * q^97 - 196408474482694469089639477350277927611611498088 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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