LMFDB - Embedded newform 161.2.a.d.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [161,2,Mod(1,161)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("161.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(161, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 161 = 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	161.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.28559147254$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2147108.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 $ x^5 - 2x^4 - 9x^3 + 17x^2 + 16x - 27
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.11948$ of defining polynomial
Character	$\chi$	$=$	161.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.11948 q^{2} -1.84074 q^{3} +2.49221 q^{4} +2.40920 q^{5} -3.90141 q^{6} +1.00000 q^{7} +1.04322 q^{8} +0.388311 q^{9} +5.10626 q^{10} +5.87722 q^{11} -4.58750 q^{12} -6.24994 q^{13} +2.11948 q^{14} -4.43471 q^{15} -2.77332 q^{16} -5.42479 q^{17} +0.823019 q^{18} -2.23897 q^{19} +6.00423 q^{20} -1.84074 q^{21} +12.4567 q^{22} -1.00000 q^{23} -1.92030 q^{24} +0.804258 q^{25} -13.2466 q^{26} +4.80743 q^{27} +2.49221 q^{28} +0.642864 q^{29} -9.39929 q^{30} +7.84074 q^{31} -7.96445 q^{32} -10.8184 q^{33} -11.4977 q^{34} +2.40920 q^{35} +0.967751 q^{36} +0.557492 q^{37} -4.74545 q^{38} +11.5045 q^{39} +2.51334 q^{40} +2.56847 q^{41} -3.90141 q^{42} -8.81841 q^{43} +14.6472 q^{44} +0.935520 q^{45} -2.11948 q^{46} +4.26766 q^{47} +5.10495 q^{48} +1.00000 q^{49} +1.70461 q^{50} +9.98561 q^{51} -15.5761 q^{52} +3.01559 q^{53} +10.1893 q^{54} +14.1594 q^{55} +1.04322 q^{56} +4.12134 q^{57} +1.36254 q^{58} +4.17024 q^{59} -11.0522 q^{60} -0.148289 q^{61} +16.6183 q^{62} +0.388311 q^{63} -11.3339 q^{64} -15.0574 q^{65} -22.9294 q^{66} +13.3396 q^{67} -13.5197 q^{68} +1.84074 q^{69} +5.10626 q^{70} -7.93141 q^{71} +0.405095 q^{72} +4.28111 q^{73} +1.18159 q^{74} -1.48043 q^{75} -5.57996 q^{76} +5.87722 q^{77} +24.3836 q^{78} -0.861628 q^{79} -6.68149 q^{80} -10.0141 q^{81} +5.44382 q^{82} -4.81841 q^{83} -4.58750 q^{84} -13.0694 q^{85} -18.6905 q^{86} -1.18334 q^{87} +6.13125 q^{88} +6.32964 q^{89} +1.98282 q^{90} -6.24994 q^{91} -2.49221 q^{92} -14.4327 q^{93} +9.04523 q^{94} -5.39412 q^{95} +14.6605 q^{96} +10.4822 q^{97} +2.11948 q^{98} +2.28219 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 12 q^{4} - 4 q^{5} - 3 q^{6} + 5 q^{7} + 3 q^{8} + 11 q^{9} - 8 q^{10} - 4 q^{11} + 3 q^{12} - 6 q^{13} + 2 q^{14} + 10 q^{15} + 10 q^{16} - 12 q^{17} - 19 q^{18} + 6 q^{19} + 14 q^{22}+ \cdots - 8 q^{99}+O(q^{100}) $ 5 * q + 2 * q^2 + 12 * q^4 - 4 * q^5 - 3 * q^6 + 5 * q^7 + 3 * q^8 + 11 * q^9 - 8 * q^10 - 4 * q^11 + 3 * q^12 - 6 * q^13 + 2 * q^14 + 10 * q^15 + 10 * q^16 - 12 * q^17 - 19 * q^18 + 6 * q^19 + 14 * q^22 - 5 * q^23 - 36 * q^24 + 19 * q^25 + q^26 + 12 * q^28 - 4 * q^29 - 48 * q^30 + 30 * q^31 + 8 * q^32 - 22 * q^33 + 6 * q^34 - 4 * q^35 - q^36 + 4 * q^37 - 40 * q^38 + 16 * q^39 - 50 * q^40 + 6 * q^41 - 3 * q^42 - 12 * q^43 - 26 * q^44 - 12 * q^45 - 2 * q^46 + 10 * q^47 + 25 * q^48 + 5 * q^49 - 2 * q^50 - 4 * q^51 - 21 * q^52 + 16 * q^53 + 33 * q^54 + 18 * q^55 + 3 * q^56 + 6 * q^57 + 13 * q^58 + 22 * q^59 + 30 * q^60 - 18 * q^61 + 15 * q^62 + 11 * q^63 + 25 * q^64 - 26 * q^65 + 4 * q^66 - 2 * q^67 + 12 * q^68 - 8 * q^70 + 4 * q^71 - 41 * q^72 - 2 * q^73 + 38 * q^74 - 30 * q^75 + 10 * q^76 - 4 * q^77 + 41 * q^78 + 30 * q^79 - 10 * q^80 - 3 * q^81 - 7 * q^82 + 8 * q^83 + 3 * q^84 - 12 * q^85 + 8 * q^86 - 12 * q^87 + 4 * q^88 - 20 * q^89 + 34 * q^90 - 6 * q^91 - 12 * q^92 - 26 * q^93 - 25 * q^94 + 8 * q^95 - q^96 - 12 * q^97 + 2 * q^98 - 8 * q^99

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 $

$2$	2.11948	1.49870	0.749350	−	0.662174i	$-0.230366\pi$
$2$	2.11948	1.49870	0.749350	+	0.662174i	$0.230366\pi$
$3$	−1.84074	−1.06275	−0.531375	−	0.847137i	$-0.678324\pi$
$3$	−1.84074	−1.06275	−0.531375	+	0.847137i	$0.678324\pi$
$4$	2.49221	1.24610
$5$	2.40920	1.07743	0.538714	−	0.842489i	$-0.318910\pi$
$5$	2.40920	1.07743	0.538714	+	0.842489i	$0.318910\pi$
$6$	−3.90141	−1.59274
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.04322	0.368835
$9$	0.388311	0.129437
$10$	5.10626	1.61474
$11$	5.87722	1.77205	0.886024	−	0.463640i	$-0.153457\pi$
$11$	5.87722	1.77205	0.886024	+	0.463640i	$0.153457\pi$
$12$	−4.58750	−1.32430
$13$	−6.24994	−1.73342	−0.866711	−	0.498811i	$-0.833770\pi$
$13$	−6.24994	−1.73342	−0.866711	+	0.498811i	$0.833770\pi$
$14$	2.11948	0.566456
$15$	−4.43471	−1.14504
$16$	−2.77332	−0.693330
$17$	−5.42479	−1.31570	−0.657852	−	0.753147i	$-0.728535\pi$
$17$	−5.42479	−1.31570	−0.657852	+	0.753147i	$0.728535\pi$
$18$	0.823019	0.193987
$19$	−2.23897	−0.513654	−0.256827	−	0.966457i	$-0.582677\pi$
$19$	−2.23897	−0.513654	−0.256827	+	0.966457i	$0.582677\pi$
$20$	6.00423	1.34259
$21$	−1.84074	−0.401682
$22$	12.4567	2.65577
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.92030	−0.391979
$25$	0.804258	0.160852
$26$	−13.2466	−2.59788
$27$	4.80743	0.925191
$28$	2.49221	0.470983
$29$	0.642864	0.119377	0.0596884	−	0.998217i	$-0.480989\pi$
$29$	0.642864	0.119377	0.0596884	+	0.998217i	$0.480989\pi$
$30$	−9.39929	−1.71607
$31$	7.84074	1.40824	0.704119	−	0.710082i	$-0.251342\pi$
$31$	7.84074	1.40824	0.704119	+	0.710082i	$0.251342\pi$
$32$	−7.96445	−1.40793
$33$	−10.8184	−1.88324
$34$	−11.4977	−1.97185
$35$	2.40920	0.407230
$36$	0.967751	0.161292
$37$	0.557492	0.0916511	0.0458256	−	0.998949i	$-0.485408\pi$
$37$	0.557492	0.0916511	0.0458256	+	0.998949i	$0.485408\pi$
$38$	−4.74545	−0.769813
$39$	11.5045	1.84219
$40$	2.51334	0.397393
$41$	2.56847	0.401127	0.200564	−	0.979681i	$-0.435723\pi$
$41$	2.56847	0.401127	0.200564	+	0.979681i	$0.435723\pi$
$42$	−3.90141	−0.602000
$43$	−8.81841	−1.34479	−0.672397	−	0.740191i	$-0.734735\pi$
$43$	−8.81841	−1.34479	−0.672397	+	0.740191i	$0.734735\pi$
$44$	14.6472	2.20815
$45$	0.935520	0.139459
$46$	−2.11948	−0.312501
$47$	4.26766	0.622502	0.311251	−	0.950328i	$-0.399252\pi$
$47$	4.26766	0.622502	0.311251	+	0.950328i	$0.399252\pi$
$48$	5.10495	0.736836
$49$	1.00000	0.142857
$50$	1.70461	0.241068
$51$	9.98561	1.39826
$52$	−15.5761	−2.16002
$53$	3.01559	0.414223	0.207111	−	0.978317i	$-0.433594\pi$
$53$	3.01559	0.414223	0.207111	+	0.978317i	$0.433594\pi$
$54$	10.1893	1.38658
$55$	14.1594	1.90925
$56$	1.04322	0.139407
$57$	4.12134	0.545885
$58$	1.36254	0.178910
$59$	4.17024	0.542919	0.271459	−	0.962450i	$-0.412494\pi$
$59$	4.17024	0.542919	0.271459	+	0.962450i	$0.412494\pi$
$60$	−11.0522	−1.42683
$61$	−0.148289	−0.0189865	−0.00949325	−	0.999955i	$-0.503022\pi$
$61$	−0.148289	−0.0189865	−0.00949325	+	0.999955i	$0.503022\pi$
$62$	16.6183	2.11053
$63$	0.388311	0.0489226
$64$	−11.3339	−1.41673
$65$	−15.0574	−1.86764
$66$	−22.9294	−2.82242
$67$	13.3396	1.62969	0.814843	−	0.579681i	$-0.196823\pi$
$67$	13.3396	1.62969	0.814843	+	0.579681i	$0.196823\pi$
$68$	−13.5197	−1.63950
$69$	1.84074	0.221599
$70$	5.10626	0.610315
$71$	−7.93141	−0.941285	−0.470643	−	0.882324i	$-0.655978\pi$
$71$	−7.93141	−0.941285	−0.470643	+	0.882324i	$0.655978\pi$
$72$	0.405095	0.0477409
$73$	4.28111	0.501066	0.250533	−	0.968108i	$-0.419394\pi$
$73$	4.28111	0.501066	0.250533	+	0.968108i	$0.419394\pi$
$74$	1.18159	0.137358
$75$	−1.48043	−0.170945
$76$	−5.57996	−0.640066
$77$	5.87722	0.669771
$78$	24.3836	2.76090
$79$	−0.861628	−0.0969407	−0.0484704	−	0.998825i	$-0.515435\pi$
$79$	−0.861628	−0.0969407	−0.0484704	+	0.998825i	$0.515435\pi$
$80$	−6.68149	−0.747013
$81$	−10.0141	−1.11268
$82$	5.44382	0.601169
$83$	−4.81841	−0.528889	−0.264444	−	0.964401i	$-0.585189\pi$
$83$	−4.81841	−0.528889	−0.264444	+	0.964401i	$0.585189\pi$
$84$	−4.58750	−0.500537
$85$	−13.0694	−1.41758
$86$	−18.6905	−2.01544
$87$	−1.18334	−0.126868
$88$	6.13125	0.653593
$89$	6.32964	0.670941	0.335470	−	0.942051i	$-0.391105\pi$
$89$	6.32964	0.670941	0.335470	+	0.942051i	$0.391105\pi$
$90$	1.98282	0.209007
$91$	−6.24994	−0.655172
$92$	−2.49221	−0.259830
$93$	−14.4327	−1.49660
$94$	9.04523	0.932944
$95$	−5.39412	−0.553425
$96$	14.6605	1.49628
$97$	10.4822	1.06430	0.532151	−	0.846649i	$-0.321384\pi$
$97$	10.4822	1.06430	0.532151	+	0.846649i	$0.321384\pi$
$98$	2.11948	0.214100
$99$	2.28219	0.229369
$100$	2.00438	0.200438
$101$	−1.57308	−0.156527	−0.0782636	−	0.996933i	$-0.524938\pi$
$101$	−1.57308	−0.156527	−0.0782636	+	0.996933i	$0.524938\pi$
$102$	21.1643	2.09558
$103$	5.01559	0.494201	0.247100	−	0.968990i	$-0.420522\pi$
$103$	5.01559	0.494201	0.247100	+	0.968990i	$0.420522\pi$
$104$	−6.52008	−0.639346
$105$	−4.43471	−0.432783
$106$	6.39148	0.620796
$107$	−5.81204	−0.561872	−0.280936	−	0.959727i	$-0.590645\pi$
$107$	−5.81204	−0.561872	−0.280936	+	0.959727i	$0.590645\pi$
$108$	11.9811	1.15288
$109$	7.65030	0.732766	0.366383	−	0.930464i	$-0.380596\pi$
$109$	7.65030	0.732766	0.366383	+	0.930464i	$0.380596\pi$
$110$	30.0106	2.86140
$111$	−1.02620	−0.0974022
$112$	−2.77332	−0.262054
$113$	10.4999	0.987745	0.493873	−	0.869534i	$-0.335581\pi$
$113$	10.4999	0.987745	0.493873	+	0.869534i	$0.335581\pi$
$114$	8.73512	0.818119
$115$	−2.40920	−0.224659
$116$	1.60215	0.148756
$117$	−2.42692	−0.224369
$118$	8.83875	0.813673
$119$	−5.42479	−0.497290
$120$	−4.62639	−0.422329
$121$	23.5417	2.14015
$122$	−0.314297	−0.0284551
$123$	−4.72787	−0.426298
$124$	19.5407	1.75481
$125$	−10.1084	−0.904122
$126$	0.823019	0.0733203
$127$	−17.9732	−1.59486	−0.797432	−	0.603409i	$-0.793809\pi$
$127$	−17.9732	−1.59486	−0.797432	+	0.603409i	$0.793809\pi$
$128$	−8.09304	−0.715331
$129$	16.2324	1.42918
$130$	−31.9138	−2.79903
$131$	−13.0641	−1.14142	−0.570708	−	0.821153i	$-0.693331\pi$
$131$	−13.0641	−1.14142	−0.570708	+	0.821153i	$0.693331\pi$
$132$	−26.9617	−2.34671
$133$	−2.23897	−0.194143
$134$	28.2730	2.44241
$135$	11.5821	0.996826
$136$	−5.65926	−0.485278
$137$	−15.0262	−1.28377	−0.641887	−	0.766799i	$-0.721848\pi$
$137$	−15.0262	−1.28377	−0.641887	+	0.766799i	$0.721848\pi$
$138$	3.90141	0.332110
$139$	−12.7987	−1.08557	−0.542786	−	0.839871i	$-0.682631\pi$
$139$	−12.7987	−1.08557	−0.542786	+	0.839871i	$0.682631\pi$
$140$	6.00423	0.507450
$141$	−7.85563	−0.661564
$142$	−16.8105	−1.41071
$143$	−36.7322	−3.07170
$144$	−1.07691	−0.0897426
$145$	1.54879	0.128620
$146$	9.07375	0.750949
$147$	−1.84074	−0.151821
$148$	1.38939	0.114207
$149$	−21.1303	−1.73106	−0.865532	−	0.500854i	$-0.833019\pi$
$149$	−21.1303	−1.73106	−0.865532	+	0.500854i	$0.833019\pi$
$150$	−3.13774	−0.256195
$151$	19.5264	1.58904	0.794520	−	0.607239i	$-0.207723\pi$
$151$	19.5264	1.58904	0.794520	+	0.607239i	$0.207723\pi$
$152$	−2.33574	−0.189453
$153$	−2.10651	−0.170301
$154$	12.4567	1.00379
$155$	18.8899	1.51728
$156$	28.6716	2.29556
$157$	−4.60638	−0.367630	−0.183815	−	0.982961i	$-0.558845\pi$
$157$	−4.60638	−0.367630	−0.183815	+	0.982961i	$0.558845\pi$
$158$	−1.82621	−0.145285
$159$	−5.55090	−0.440215
$160$	−19.1880	−1.51694
$161$	−1.00000	−0.0788110
$162$	−21.2248	−1.66758
$163$	16.3229	1.27851	0.639254	−	0.768996i	$-0.279243\pi$
$163$	16.3229	1.27851	0.639254	+	0.768996i	$0.279243\pi$
$164$	6.40115	0.499846
$165$	−26.0637	−2.02906
$166$	−10.2125	−0.792646
$167$	16.7254	1.29425	0.647125	−	0.762384i	$-0.275971\pi$
$167$	16.7254	1.29425	0.647125	+	0.762384i	$0.275971\pi$
$168$	−1.92030	−0.148154
$169$	26.0617	2.00475
$170$	−27.7004	−2.12452
$171$	−0.869415	−0.0664858
$172$	−21.9773	−1.67575
$173$	9.27650	0.705279	0.352640	−	0.935759i	$-0.385284\pi$
$173$	9.27650	0.705279	0.352640	+	0.935759i	$0.385284\pi$
$174$	−2.50807	−0.190137
$175$	0.804258	0.0607962
$176$	−16.2994	−1.22861
$177$	−7.67631	−0.576987
$178$	13.4156	1.00554
$179$	−7.48254	−0.559272	−0.279636	−	0.960106i	$-0.590214\pi$
$179$	−7.48254	−0.559272	−0.279636	+	0.960106i	$0.590214\pi$
$180$	2.33151	0.173780
$181$	−10.6482	−0.791472	−0.395736	−	0.918364i	$-0.629510\pi$
$181$	−10.6482	−0.791472	−0.395736	+	0.918364i	$0.629510\pi$
$182$	−13.2466	−0.981906
$183$	0.272962	0.0201779
$184$	−1.04322	−0.0769074
$185$	1.34311	0.0987475
$186$	−30.5899	−2.24296
$187$	−31.8827	−2.33149
$188$	10.6359	0.775701
$189$	4.80743	0.349689
$190$	−11.4327	−0.829419
$191$	3.30294	0.238992	0.119496	−	0.992835i	$-0.461872\pi$
$191$	3.30294	0.238992	0.119496	+	0.992835i	$0.461872\pi$
$192$	20.8627	1.50563
$193$	9.38315	0.675414	0.337707	−	0.941251i	$-0.390349\pi$
$193$	9.38315	0.675414	0.337707	+	0.941251i	$0.390349\pi$
$194$	22.2168	1.59507
$195$	27.7167	1.98483
$196$	2.49221	0.178015
$197$	−23.0929	−1.64530	−0.822651	−	0.568547i	$-0.807506\pi$
$197$	−23.0929	−1.64530	−0.822651	+	0.568547i	$0.807506\pi$
$198$	4.83706	0.343755
$199$	21.6149	1.53224	0.766118	−	0.642699i	$-0.222186\pi$
$199$	21.6149	1.53224	0.766118	+	0.642699i	$0.222186\pi$
$200$	0.839020	0.0593277
$201$	−24.5546	−1.73195
$202$	−3.33411	−0.234587
$203$	0.642864	0.0451202
$204$	24.8862	1.74238
$205$	6.18796	0.432186
$206$	10.6304	0.740659
$207$	−0.388311	−0.0269895
$208$	17.3331	1.20183
$209$	−13.1589	−0.910219
$210$	−9.39929	−0.648612
$211$	15.4579	1.06416	0.532081	−	0.846693i	$-0.321410\pi$
$211$	15.4579	1.06416	0.532081	+	0.846693i	$0.321410\pi$
$212$	7.51547	0.516164
$213$	14.5996	1.00035
$214$	−12.3185	−0.842077
$215$	−21.2453	−1.44892
$216$	5.01522	0.341243
$217$	7.84074	0.532264
$218$	16.2147	1.09820
$219$	−7.88040	−0.532508
$220$	35.2882	2.37913
$221$	33.9046	2.28067
$222$	−2.17500	−0.145977
$223$	−11.0708	−0.741357	−0.370679	−	0.928761i	$-0.620875\pi$
$223$	−11.0708	−0.741357	−0.370679	+	0.928761i	$0.620875\pi$
$224$	−7.96445	−0.532147
$225$	0.312302	0.0208201
$226$	22.2543	1.48033
$227$	−2.44676	−0.162397	−0.0811984	−	0.996698i	$-0.525875\pi$
$227$	−2.44676	−0.162397	−0.0811984	+	0.996698i	$0.525875\pi$
$228$	10.2712	0.680230
$229$	−5.74097	−0.379374	−0.189687	−	0.981845i	$-0.560747\pi$
$229$	−5.74097	−0.379374	−0.189687	+	0.981845i	$0.560747\pi$
$230$	−5.10626	−0.336697
$231$	−10.8184	−0.711799
$232$	0.670650	0.0440303
$233$	8.66481	0.567651	0.283825	−	0.958876i	$-0.408396\pi$
$233$	8.66481	0.567651	0.283825	+	0.958876i	$0.408396\pi$
$234$	−5.14382	−0.336262
$235$	10.2817	0.670701
$236$	10.3931	0.676533
$237$	1.58603	0.103024
$238$	−11.4977	−0.745288
$239$	−0.995387	−0.0643862	−0.0321931	−	0.999482i	$-0.510249\pi$
$239$	−0.995387	−0.0643862	−0.0321931	+	0.999482i	$0.510249\pi$
$240$	12.2989	0.793888
$241$	−13.9247	−0.896967	−0.448483	−	0.893791i	$-0.648036\pi$
$241$	−13.9247	−0.896967	−0.448483	+	0.893791i	$0.648036\pi$
$242$	49.8961	3.20745
$243$	4.01111	0.257313
$244$	−0.369568	−0.0236591
$245$	2.40920	0.153918
$246$	−10.0206	−0.638892
$247$	13.9934	0.890378
$248$	8.17963	0.519407
$249$	8.86941	0.562076
$250$	−21.4246	−1.35501
$251$	−0.229739	−0.0145010	−0.00725049	−	0.999974i	$-0.502308\pi$
$251$	−0.229739	−0.0145010	−0.00725049	+	0.999974i	$0.502308\pi$
$252$	0.967751	0.0609626
$253$	−5.87722	−0.369497
$254$	−38.0939	−2.39022
$255$	24.0574	1.50653
$256$	5.51468	0.344667
$257$	−4.24568	−0.264838	−0.132419	−	0.991194i	$-0.542274\pi$
$257$	−4.24568	−0.264838	−0.132419	+	0.991194i	$0.542274\pi$
$258$	34.4042	2.14191
$259$	0.557492	0.0346409
$260$	−37.5261	−2.32727
$261$	0.249631	0.0154518
$262$	−27.6892	−1.71064
$263$	2.99220	0.184507	0.0922535	−	0.995736i	$-0.470593\pi$
$263$	2.99220	0.184507	0.0922535	+	0.995736i	$0.470593\pi$
$264$	−11.2860	−0.694606
$265$	7.26516	0.446295
$266$	−4.74545	−0.290962
$267$	−11.6512	−0.713042
$268$	33.2449	2.03076
$269$	−8.92878	−0.544397	−0.272199	−	0.962241i	$-0.587751\pi$
$269$	−8.92878	−0.544397	−0.272199	+	0.962241i	$0.587751\pi$
$270$	24.5480	1.49394
$271$	20.9773	1.27428	0.637140	−	0.770748i	$-0.280117\pi$
$271$	20.9773	1.27428	0.637140	+	0.770748i	$0.280117\pi$
$272$	15.0447	0.912218
$273$	11.5045	0.696284
$274$	−31.8478	−1.92399
$275$	4.72680	0.285036
$276$	4.58750	0.276135
$277$	5.82584	0.350041	0.175020	−	0.984565i	$-0.444001\pi$
$277$	5.82584	0.350041	0.175020	+	0.984565i	$0.444001\pi$
$278$	−27.1266	−1.62695
$279$	3.04464	0.182278
$280$	2.51334	0.150201
$281$	−21.1877	−1.26395	−0.631976	−	0.774988i	$-0.717756\pi$
$281$	−21.1877	−1.26395	−0.631976	+	0.774988i	$0.717756\pi$
$282$	−16.6499	−0.991486
$283$	−12.8049	−0.761173	−0.380587	−	0.924745i	$-0.624278\pi$
$283$	−12.8049	−0.761173	−0.380587	+	0.924745i	$0.624278\pi$
$284$	−19.7667	−1.17294
$285$	9.92916	0.588152
$286$	−77.8533	−4.60356
$287$	2.56847	0.151612
$288$	−3.09268	−0.182238
$289$	12.4283	0.731079
$290$	3.28263	0.192763
$291$	−19.2949	−1.13109
$292$	10.6694	0.624381
$293$	−3.11921	−0.182226	−0.0911132	−	0.995841i	$-0.529043\pi$
$293$	−3.11921	−0.182226	−0.0911132	+	0.995841i	$0.529043\pi$
$294$	−3.90141	−0.227535
$295$	10.0469	0.584956
$296$	0.581588	0.0338041
$297$	28.2543	1.63948
$298$	−44.7854	−2.59435
$299$	6.24994	0.361443
$300$	−3.68953	−0.213015
$301$	−8.81841	−0.508284
$302$	41.3859	2.38149
$303$	2.89562	0.166349
$304$	6.20937	0.356132
$305$	−0.357259	−0.0204566
$306$	−4.46470	−0.255230
$307$	−16.1152	−0.919746	−0.459873	−	0.887985i	$-0.652105\pi$
$307$	−16.1152	−0.919746	−0.459873	+	0.887985i	$0.652105\pi$
$308$	14.6472	0.834604
$309$	−9.23237	−0.525211
$310$	40.0369	2.27394
$311$	30.2428	1.71491	0.857457	−	0.514556i	$-0.172043\pi$
$311$	30.2428	1.71491	0.857457	+	0.514556i	$0.172043\pi$
$312$	12.0017	0.679465
$313$	−20.4956	−1.15848	−0.579241	−	0.815156i	$-0.696651\pi$
$313$	−20.4956	−1.15848	−0.579241	+	0.815156i	$0.696651\pi$
$314$	−9.76315	−0.550967
$315$	0.935520	0.0527106
$316$	−2.14736	−0.120798
$317$	2.95042	0.165712	0.0828559	−	0.996562i	$-0.473596\pi$
$317$	2.95042	0.165712	0.0828559	+	0.996562i	$0.473596\pi$
$318$	−11.7650	−0.659751
$319$	3.77825	0.211541
$320$	−27.3056	−1.52643
$321$	10.6984	0.597129
$322$	−2.11948	−0.118114
$323$	12.1459	0.675817
$324$	−24.9573	−1.38652
$325$	−5.02656	−0.278823
$326$	34.5961	1.91610
$327$	−14.0822	−0.778747
$328$	2.67948	0.147950
$329$	4.26766	0.235284
$330$	−55.2416	−3.04095
$331$	−21.2497	−1.16799	−0.583994	−	0.811758i	$-0.698511\pi$
$331$	−21.2497	−1.16799	−0.583994	+	0.811758i	$0.698511\pi$
$332$	−12.0085	−0.659050
$333$	0.216480	0.0118630
$334$	35.4492	1.93969
$335$	32.1377	1.75587
$336$	5.10495	0.278498
$337$	−24.8118	−1.35158	−0.675792	−	0.737092i	$-0.736198\pi$
$337$	−24.8118	−1.35158	−0.675792	+	0.737092i	$0.736198\pi$
$338$	55.2374	3.00452
$339$	−19.3275	−1.04973
$340$	−32.5717	−1.76645
$341$	46.0817	2.49546
$342$	−1.84271	−0.0996423
$343$	1.00000	0.0539949
$344$	−9.19956	−0.496007
$345$	4.43471	0.238757
$346$	19.6614	1.05700
$347$	11.8158	0.634304	0.317152	−	0.948375i	$-0.397274\pi$
$347$	11.8158	0.634304	0.317152	+	0.948375i	$0.397274\pi$
$348$	−2.94913	−0.158090
$349$	−12.1614	−0.650984	−0.325492	−	0.945545i	$-0.605530\pi$
$349$	−12.1614	−0.650984	−0.325492	+	0.945545i	$0.605530\pi$
$350$	1.70461	0.0911152
$351$	−30.0462	−1.60375
$352$	−46.8088	−2.49492
$353$	17.5264	0.932838	0.466419	−	0.884564i	$-0.345544\pi$
$353$	17.5264	0.932838	0.466419	+	0.884564i	$0.345544\pi$
$354$	−16.2698	−0.864730
$355$	−19.1084	−1.01417
$356$	15.7748	0.836061
$357$	9.98561	0.528494
$358$	−15.8591	−0.838181
$359$	7.70275	0.406535	0.203268	−	0.979123i	$-0.434844\pi$
$359$	7.70275	0.406535	0.203268	+	0.979123i	$0.434844\pi$
$360$	0.975956	0.0514374
$361$	−13.9870	−0.736160
$362$	−22.5686	−1.18618
$363$	−43.3340	−2.27445
$364$	−15.5761	−0.816411
$365$	10.3141	0.539863
$366$	0.578537	0.0302406
$367$	15.0819	0.787271	0.393635	−	0.919267i	$-0.371217\pi$
$367$	15.0819	0.787271	0.393635	+	0.919267i	$0.371217\pi$
$368$	2.77332	0.144569
$369$	0.997364	0.0519207
$370$	2.84670	0.147993
$371$	3.01559	0.156561
$372$	−35.9693	−1.86492
$373$	−21.5089	−1.11369	−0.556843	−	0.830618i	$-0.687988\pi$
$373$	−21.5089	−1.11369	−0.556843	+	0.830618i	$0.687988\pi$
$374$	−67.5747	−3.49421
$375$	18.6069	0.960856
$376$	4.45212	0.229600
$377$	−4.01786	−0.206930
$378$	10.1893	0.524079
$379$	34.7117	1.78302	0.891511	−	0.452999i	$-0.149646\pi$
$379$	34.7117	1.78302	0.891511	+	0.452999i	$0.149646\pi$
$380$	−13.4433	−0.689625
$381$	33.0839	1.69494
$382$	7.00052	0.358178
$383$	−10.2963	−0.526119	−0.263059	−	0.964780i	$-0.584732\pi$
$383$	−10.2963	−0.526119	−0.263059	+	0.964780i	$0.584732\pi$
$384$	14.8972	0.760218
$385$	14.1594	0.721630
$386$	19.8874	1.01224
$387$	−3.42428	−0.174066
$388$	26.1237	1.32623
$389$	−26.1594	−1.32633	−0.663167	−	0.748471i	$-0.730788\pi$
$389$	−26.1594	−1.32633	−0.663167	+	0.748471i	$0.730788\pi$
$390$	58.7450	2.97467
$391$	5.42479	0.274343
$392$	1.04322	0.0526907
$393$	24.0476	1.21304
$394$	−48.9450	−2.46582
$395$	−2.07584	−0.104447
$396$	5.68768	0.285817
$397$	−4.52380	−0.227043	−0.113522	−	0.993536i	$-0.536213\pi$
$397$	−4.52380	−0.227043	−0.113522	+	0.993536i	$0.536213\pi$
$398$	45.8123	2.29636
$399$	4.12134	0.206325
$400$	−2.23046	−0.111523
$401$	−18.0222	−0.899987	−0.449994	−	0.893032i	$-0.648574\pi$
$401$	−18.0222	−0.899987	−0.449994	+	0.893032i	$0.648574\pi$
$402$	−52.0431	−2.59567
$403$	−49.0041	−2.44107
$404$	−3.92044	−0.195049
$405$	−24.1261	−1.19884
$406$	1.36254	0.0676216
$407$	3.27650	0.162410
$408$	10.4172	0.515729
$409$	4.67260	0.231045	0.115523	−	0.993305i	$-0.463146\pi$
$409$	4.67260	0.231045	0.115523	+	0.993305i	$0.463146\pi$
$410$	13.1153	0.647717
$411$	27.6593	1.36433
$412$	12.4999	0.615825
$413$	4.17024	0.205204
$414$	−0.823019	−0.0404492
$415$	−11.6085	−0.569840
$416$	49.7773	2.44053
$417$	23.5591	1.15369
$418$	−27.8900	−1.36415
$419$	−12.7674	−0.623728	−0.311864	−	0.950127i	$-0.600953\pi$
$419$	−12.7674	−0.623728	−0.311864	+	0.950127i	$0.600953\pi$
$420$	−11.0522	−0.539292
$421$	28.1367	1.37130	0.685649	−	0.727932i	$-0.259518\pi$
$421$	28.1367	1.37130	0.685649	+	0.727932i	$0.259518\pi$
$422$	32.7626	1.59486
$423$	1.65718	0.0805748
$424$	3.14593	0.152780
$425$	−4.36293	−0.211633
$426$	30.9437	1.49923
$427$	−0.148289	−0.00717622
$428$	−14.4848	−0.700150
$429$	67.6144	3.26445
$430$	−45.0291	−2.17150
$431$	−5.49352	−0.264613	−0.132307	−	0.991209i	$-0.542238\pi$
$431$	−5.49352	−0.264613	−0.132307	+	0.991209i	$0.542238\pi$
$432$	−13.3325	−0.641462
$433$	−21.0177	−1.01005	−0.505023	−	0.863106i	$-0.668516\pi$
$433$	−21.0177	−1.01005	−0.505023	+	0.863106i	$0.668516\pi$
$434$	16.6183	0.797704
$435$	−2.85091	−0.136691
$436$	19.0661	0.913102
$437$	2.23897	0.107104
$438$	−16.7024	−0.798070
$439$	39.2674	1.87413	0.937066	−	0.349153i	$-0.113531\pi$
$439$	39.2674	1.87413	0.937066	+	0.349153i	$0.113531\pi$
$440$	14.7714	0.704199
$441$	0.388311	0.0184910
$442$	71.8602	3.41804
$443$	14.5489	0.691240	0.345620	−	0.938375i	$-0.387669\pi$
$443$	14.5489	0.691240	0.345620	+	0.938375i	$0.387669\pi$
$444$	−2.55749	−0.121373
$445$	15.2494	0.722890
$446$	−23.4644	−1.11107
$447$	38.8954	1.83969
$448$	−11.3339	−0.535475
$449$	11.3660	0.536395	0.268197	−	0.963364i	$-0.413572\pi$
$449$	11.3660	0.536395	0.268197	+	0.963364i	$0.413572\pi$
$450$	0.661919	0.0312032
$451$	15.0954	0.710816
$452$	26.1679	1.23083
$453$	−35.9430	−1.68875
$454$	−5.18586	−0.243384
$455$	−15.0574	−0.705900
$456$	4.29948	0.201342
$457$	−9.42744	−0.440997	−0.220499	−	0.975387i	$-0.570768\pi$
$457$	−9.42744	−0.440997	−0.220499	+	0.975387i	$0.570768\pi$
$458$	−12.1679	−0.568568
$459$	−26.0793	−1.21728
$460$	−6.00423	−0.279949
$461$	−28.7697	−1.33994	−0.669968	−	0.742390i	$-0.733692\pi$
$461$	−28.7697	−1.33994	−0.669968	+	0.742390i	$0.733692\pi$
$462$	−22.9294	−1.06677
$463$	−4.65928	−0.216535	−0.108268	−	0.994122i	$-0.534530\pi$
$463$	−4.65928	−0.216535	−0.108268	+	0.994122i	$0.534530\pi$
$464$	−1.78287	−0.0827675
$465$	−34.7714	−1.61248
$466$	18.3649	0.850738
$467$	7.82926	0.362295	0.181147	−	0.983456i	$-0.442019\pi$
$467$	7.82926	0.362295	0.181147	+	0.983456i	$0.442019\pi$
$468$	−6.04839	−0.279587
$469$	13.3396	0.615964
$470$	21.7918	1.00518
$471$	8.47914	0.390698
$472$	4.35049	0.200247
$473$	−51.8277	−2.38304
$474$	3.36156	0.154402
$475$	−1.80070	−0.0826220
$476$	−13.5197	−0.619674
$477$	1.17099	0.0536158
$478$	−2.10971	−0.0964957
$479$	−33.9427	−1.55088	−0.775440	−	0.631421i	$-0.782472\pi$
$479$	−33.9427	−1.55088	−0.775440	+	0.631421i	$0.782472\pi$
$480$	35.3200	1.61213
$481$	−3.48429	−0.158870
$482$	−29.5131	−1.34428
$483$	1.84074	0.0837564
$484$	58.6707	2.66685
$485$	25.2536	1.14671
$486$	8.50149	0.385635
$487$	−11.6880	−0.529633	−0.264817	−	0.964299i	$-0.585311\pi$
$487$	−11.6880	−0.529633	−0.264817	+	0.964299i	$0.585311\pi$
$488$	−0.154699	−0.00700289
$489$	−30.0462	−1.35873
$490$	5.10626	0.230677
$491$	−19.6742	−0.887885	−0.443943	−	0.896055i	$-0.646421\pi$
$491$	−19.6742	−0.887885	−0.443943	+	0.896055i	$0.646421\pi$
$492$	−11.7828	−0.531211
$493$	−3.48740	−0.157065
$494$	29.6588	1.33441
$495$	5.49825	0.247128
$496$	−21.7449	−0.976374
$497$	−7.93141	−0.355772
$498$	18.7986	0.842384
$499$	−30.1698	−1.35059	−0.675294	−	0.737549i	$-0.735983\pi$
$499$	−30.1698	−1.35059	−0.675294	+	0.737549i	$0.735983\pi$
$500$	−25.1922	−1.12663
$501$	−30.7870	−1.37546
$502$	−0.486927	−0.0217326
$503$	24.1110	1.07506	0.537529	−	0.843246i	$-0.319358\pi$
$503$	24.1110	1.07506	0.537529	+	0.843246i	$0.319358\pi$
$504$	0.405095	0.0180444
$505$	−3.78987	−0.168647
$506$	−12.4567	−0.553766
$507$	−47.9728	−2.13055
$508$	−44.7929	−1.98736
$509$	21.5774	0.956404	0.478202	−	0.878250i	$-0.341289\pi$
$509$	21.5774	0.956404	0.478202	+	0.878250i	$0.341289\pi$
$510$	50.9891	2.25784
$511$	4.28111	0.189385
$512$	27.8744	1.23188
$513$	−10.7637	−0.475228
$514$	−8.99864	−0.396913
$515$	12.0836	0.532466
$516$	40.4544	1.78091
$517$	25.0819	1.10310
$518$	1.18159	0.0519163
$519$	−17.0756	−0.749535
$520$	−15.7082	−0.688850
$521$	−22.0862	−0.967613	−0.483807	−	0.875175i	$-0.660746\pi$
$521$	−22.0862	−0.967613	−0.483807	+	0.875175i	$0.660746\pi$
$522$	0.529089	0.0231576
$523$	5.79856	0.253553	0.126777	−	0.991931i	$-0.459537\pi$
$523$	5.79856	0.253553	0.126777	+	0.991931i	$0.459537\pi$
$524$	−32.5585	−1.42232
$525$	−1.48043	−0.0646111
$526$	6.34191	0.276521
$527$	−42.5343	−1.85283
$528$	30.0029	1.30571
$529$	1.00000	0.0434783
$530$	15.3984	0.668863
$531$	1.61935	0.0702738
$532$	−5.57996	−0.241922
$533$	−16.0528	−0.695322
$534$	−24.6945	−1.06864
$535$	−14.0024	−0.605376
$536$	13.9161	0.601085
$537$	13.7734	0.594366
$538$	−18.9244	−0.815888
$539$	5.87722	0.253150
$540$	28.8649	1.24215
$541$	29.6582	1.27511	0.637553	−	0.770407i	$-0.279947\pi$
$541$	29.6582	1.27511	0.637553	+	0.770407i	$0.279947\pi$
$542$	44.4610	1.90976
$543$	19.6005	0.841137
$544$	43.2055	1.85242
$545$	18.4311	0.789502
$546$	24.3836	1.04352
$547$	−7.24730	−0.309872	−0.154936	−	0.987924i	$-0.549517\pi$
$547$	−7.24730	−0.309872	−0.154936	+	0.987924i	$0.549517\pi$
$548$	−37.4484	−1.59972
$549$	−0.0575824	−0.00245756
$550$	10.0184	0.427184
$551$	−1.43935	−0.0613183
$552$	1.92030	0.0817333
$553$	−0.861628	−0.0366402
$554$	12.3478	0.524606
$555$	−2.47231	−0.104944
$556$	−31.8970	−1.35274
$557$	21.7634	0.922146	0.461073	−	0.887362i	$-0.347465\pi$
$557$	21.7634	0.922146	0.461073	+	0.887362i	$0.347465\pi$
$558$	6.45307	0.273180
$559$	55.1145	2.33109
$560$	−6.68149	−0.282345
$561$	58.6876	2.47779
$562$	−44.9070	−1.89429
$563$	−26.9844	−1.13726	−0.568629	−	0.822594i	$-0.692526\pi$
$563$	−26.9844	−1.13726	−0.568629	+	0.822594i	$0.692526\pi$
$564$	−19.5779	−0.824377
$565$	25.2963	1.06422
$566$	−27.1398	−1.14077
$567$	−10.0141	−0.420555
$568$	−8.27423	−0.347179
$569$	37.4420	1.56965	0.784826	−	0.619717i	$-0.212752\pi$
$569$	37.4420	1.56965	0.784826	+	0.619717i	$0.212752\pi$
$570$	21.0447	0.881464
$571$	−3.32489	−0.139142	−0.0695711	−	0.997577i	$-0.522163\pi$
$571$	−3.32489	−0.139142	−0.0695711	+	0.997577i	$0.522163\pi$
$572$	−91.5443	−3.82766
$573$	−6.07984	−0.253989
$574$	5.44382	0.227221
$575$	−0.804258	−0.0335399
$576$	−4.40107	−0.183378
$577$	−41.0573	−1.70924	−0.854618	−	0.519257i	$-0.826209\pi$
$577$	−41.0573	−1.70924	−0.854618	+	0.519257i	$0.826209\pi$
$578$	26.3417	1.09567
$579$	−17.2719	−0.717796
$580$	3.85990	0.160274
$581$	−4.81841	−0.199901
$582$	−40.8952	−1.69516
$583$	17.7233	0.734022
$584$	4.46616	0.184811
$585$	−5.84694	−0.241741
$586$	−6.61112	−0.273103
$587$	−31.9182	−1.31741	−0.658703	−	0.752403i	$-0.728895\pi$
$587$	−31.9182	−1.31741	−0.658703	+	0.752403i	$0.728895\pi$
$588$	−4.58750	−0.189185
$589$	−17.5551	−0.723347
$590$	21.2943	0.876674
$591$	42.5080	1.74854
$592$	−1.54610	−0.0635445
$593$	36.5062	1.49913	0.749566	−	0.661930i	$-0.230262\pi$
$593$	36.5062	1.49913	0.749566	+	0.661930i	$0.230262\pi$
$594$	59.8845	2.45709
$595$	−13.0694	−0.535794
$596$	−52.6611	−2.15708
$597$	−39.7873	−1.62838
$598$	13.2466	0.541695
$599$	−7.51073	−0.306880	−0.153440	−	0.988158i	$-0.549035\pi$
$599$	−7.51073	−0.306880	−0.153440	+	0.988158i	$0.549035\pi$
$600$	−1.54441	−0.0630505
$601$	−6.05329	−0.246919	−0.123460	−	0.992350i	$-0.539399\pi$
$601$	−6.05329	−0.246919	−0.123460	+	0.992350i	$0.539399\pi$
$602$	−18.6905	−0.761766
$603$	5.17990	0.210942
$604$	48.6639	1.98011
$605$	56.7166	2.30586
$606$	6.13723	0.249308
$607$	24.4047	0.990557	0.495278	−	0.868734i	$-0.335066\pi$
$607$	24.4047	0.990557	0.495278	+	0.868734i	$0.335066\pi$
$608$	17.8321	0.723188
$609$	−1.18334	−0.0479515
$610$	−0.757204	−0.0306583
$611$	−26.6726	−1.07906
$612$	−5.24985	−0.212213
$613$	4.81180	0.194347	0.0971734	−	0.995267i	$-0.469020\pi$
$613$	4.81180	0.194347	0.0971734	+	0.995267i	$0.469020\pi$
$614$	−34.1560	−1.37842
$615$	−11.3904	−0.459305
$616$	6.13125	0.247035
$617$	−38.1414	−1.53552	−0.767758	−	0.640740i	$-0.778628\pi$
$617$	−38.1414	−1.53552	−0.767758	+	0.640740i	$0.778628\pi$
$618$	−19.5679	−0.787135
$619$	−17.8340	−0.716809	−0.358404	−	0.933566i	$-0.616679\pi$
$619$	−17.8340	−0.716809	−0.358404	+	0.933566i	$0.616679\pi$
$620$	47.0776	1.89068
$621$	−4.80743	−0.192916
$622$	64.0992	2.57014
$623$	6.32964	0.253592
$624$	−31.9056	−1.27725
$625$	−28.3745	−1.13498
$626$	−43.4402	−1.73622
$627$	24.2220	0.967335
$628$	−11.4801	−0.458104
$629$	−3.02428	−0.120586
$630$	1.98282	0.0789974
$631$	22.2906	0.887377	0.443688	−	0.896181i	$-0.353670\pi$
$631$	22.2906	0.887377	0.443688	+	0.896181i	$0.353670\pi$
$632$	−0.898870	−0.0357551
$633$	−28.4538	−1.13094
$634$	6.25336	0.248352
$635$	−43.3011	−1.71835
$636$	−13.8340	−0.548553
$637$	−6.24994	−0.247632
$638$	8.00793	0.317037
$639$	−3.07986	−0.121837
$640$	−19.4978	−0.770718
$641$	28.4927	1.12540	0.562698	−	0.826663i	$-0.309763\pi$
$641$	28.4927	1.12540	0.562698	+	0.826663i	$0.309763\pi$
$642$	22.6752	0.894917
$643$	0.975186	0.0384576	0.0192288	−	0.999815i	$-0.493879\pi$
$643$	0.975186	0.0384576	0.0192288	+	0.999815i	$0.493879\pi$
$644$	−2.49221	−0.0982067
$645$	39.1070	1.53984
$646$	25.7431	1.01285
$647$	−27.5193	−1.08190	−0.540948	−	0.841056i	$-0.681935\pi$
$647$	−27.5193	−1.08190	−0.540948	+	0.841056i	$0.681935\pi$
$648$	−10.4470	−0.410396
$649$	24.5094	0.962078
$650$	−10.6537	−0.417873
$651$	−14.4327	−0.565663
$652$	40.6800	1.59315
$653$	−26.1590	−1.02368	−0.511840	−	0.859081i	$-0.671036\pi$
$653$	−26.1590	−1.02368	−0.511840	+	0.859081i	$0.671036\pi$
$654$	−29.8469	−1.16711
$655$	−31.4741	−1.22979
$656$	−7.12318	−0.278113
$657$	1.66240	0.0648566
$658$	9.04523	0.352620
$659$	−2.67085	−0.104042	−0.0520208	−	0.998646i	$-0.516566\pi$
$659$	−2.67085	−0.104042	−0.0520208	+	0.998646i	$0.516566\pi$
$660$	−64.9562	−2.52842
$661$	−18.4928	−0.719285	−0.359643	−	0.933090i	$-0.617101\pi$
$661$	−18.4928	−0.719285	−0.359643	+	0.933090i	$0.617101\pi$
$662$	−45.0384	−1.75047
$663$	−62.4095	−2.42378
$664$	−5.02667	−0.195073
$665$	−5.39412	−0.209175
$666$	0.458826	0.0177792
$667$	−0.642864	−0.0248918
$668$	41.6831	1.61277
$669$	20.3785	0.787877
$670$	68.1153	2.63152
$671$	−0.871528	−0.0336450
$672$	14.6605	0.565539
$673$	47.5002	1.83100	0.915499	−	0.402321i	$-0.131797\pi$
$673$	47.5002	1.83100	0.915499	+	0.402321i	$0.131797\pi$
$674$	−52.5882	−2.02562
$675$	3.86641	0.148818
$676$	64.9512	2.49812
$677$	33.4288	1.28477	0.642386	−	0.766381i	$-0.277944\pi$
$677$	33.4288	1.28477	0.642386	+	0.766381i	$0.277944\pi$
$678$	−40.9643	−1.57323
$679$	10.4822	0.402268
$680$	−13.6343	−0.522852
$681$	4.50383	0.172587
$682$	97.6694	3.73995
$683$	46.7168	1.78757	0.893784	−	0.448498i	$-0.148041\pi$
$683$	46.7168	1.78757	0.893784	+	0.448498i	$0.148041\pi$
$684$	−2.16676	−0.0828482
$685$	−36.2012	−1.38317
$686$	2.11948	0.0809222
$687$	10.5676	0.403180
$688$	24.4563	0.932386
$689$	−18.8472	−0.718023
$690$	9.39929	0.357825
$691$	−22.8670	−0.869903	−0.434952	−	0.900454i	$-0.643235\pi$
$691$	−22.8670	−0.869903	−0.434952	+	0.900454i	$0.643235\pi$
$692$	23.1190	0.878851
$693$	2.28219	0.0866931
$694$	25.0433	0.950631
$695$	−30.8347	−1.16963
$696$	−1.23449	−0.0467932
$697$	−13.9334	−0.527765
$698$	−25.7758	−0.975630
$699$	−15.9496	−0.603271
$700$	2.00438	0.0757583
$701$	39.0885	1.47635	0.738177	−	0.674607i	$-0.235687\pi$
$701$	39.0885	1.47635	0.738177	+	0.674607i	$0.235687\pi$
$702$	−63.6823	−2.40353
$703$	−1.24821	−0.0470769
$704$	−66.6116	−2.51052
$705$	−18.9258	−0.712787
$706$	37.1470	1.39805
$707$	−1.57308	−0.0591617
$708$	−19.1309	−0.718985
$709$	18.3762	0.690132	0.345066	−	0.938578i	$-0.387856\pi$
$709$	18.3762	0.690132	0.345066	+	0.938578i	$0.387856\pi$
$710$	−40.4999	−1.51993
$711$	−0.334580	−0.0125477
$712$	6.60323	0.247466
$713$	−7.84074	−0.293638
$714$	21.1643	0.792055
$715$	−88.4954	−3.30954
$716$	−18.6480	−0.696910
$717$	1.83224	0.0684264
$718$	16.3258	0.609275
$719$	−11.2232	−0.418553	−0.209276	−	0.977857i	$-0.567111\pi$
$719$	−11.2232	−0.418553	−0.209276	+	0.977857i	$0.567111\pi$
$720$	−2.59450	−0.0966912
$721$	5.01559	0.186790
$722$	−29.6453	−1.10328
$723$	25.6316	0.953251
$724$	−26.5374	−0.986256
$725$	0.517028	0.0192019
$726$	−91.8457	−3.40871
$727$	7.71953	0.286302	0.143151	−	0.989701i	$-0.454277\pi$
$727$	7.71953	0.286302	0.143151	+	0.989701i	$0.454277\pi$
$728$	−6.52008	−0.241650
$729$	22.6590	0.839224
$730$	21.8605	0.809093
$731$	47.8380	1.76935
$732$	0.680277	0.0251437
$733$	−23.2083	−0.857217	−0.428609	−	0.903490i	$-0.640996\pi$
$733$	−23.2083	−0.857217	−0.428609	+	0.903490i	$0.640996\pi$
$734$	31.9659	1.17988
$735$	−4.43471	−0.163577
$736$	7.96445	0.293573
$737$	78.3995	2.88788
$738$	2.11390	0.0778136
$739$	8.36545	0.307728	0.153864	−	0.988092i	$-0.450828\pi$
$739$	8.36545	0.307728	0.153864	+	0.988092i	$0.450828\pi$
$740$	3.34731	0.123050
$741$	−25.7582	−0.946249
$742$	6.39148	0.234639
$743$	9.54735	0.350258	0.175129	−	0.984545i	$-0.443966\pi$
$743$	9.54735	0.350258	0.175129	+	0.984545i	$0.443966\pi$
$744$	−15.0566	−0.552000
$745$	−50.9072	−1.86510
$746$	−45.5877	−1.66908
$747$	−1.87104	−0.0684578
$748$	−79.4582	−2.90528
$749$	−5.81204	−0.212367
$750$	39.4370	1.44003
$751$	2.83381	0.103407	0.0517035	−	0.998662i	$-0.483535\pi$
$751$	2.83381	0.103407	0.0517035	+	0.998662i	$0.483535\pi$
$752$	−11.8356	−0.431599
$753$	0.422889	0.0154109
$754$	−8.51578	−0.310126
$755$	47.0432	1.71208
$756$	11.9811	0.435749
$757$	49.7767	1.80916	0.904582	−	0.426300i	$-0.140183\pi$
$757$	49.7767	1.80916	0.904582	+	0.426300i	$0.140183\pi$
$758$	73.5709	2.67222
$759$	10.8184	0.392683
$760$	−5.62727	−0.204123
$761$	48.7458	1.76704	0.883518	−	0.468398i	$-0.155169\pi$
$761$	48.7458	1.76704	0.883518	+	0.468398i	$0.155169\pi$
$762$	70.1208	2.54021
$763$	7.65030	0.276959
$764$	8.23161	0.297809
$765$	−5.07500	−0.183487
$766$	−21.8229	−0.788495
$767$	−26.0637	−0.941107
$768$	−10.1511	−0.366295
$769$	−13.3190	−0.480296	−0.240148	−	0.970736i	$-0.577196\pi$
$769$	−13.3190	−0.480296	−0.240148	+	0.970736i	$0.577196\pi$
$770$	30.0106	1.08151
$771$	7.81517	0.281457
$772$	23.3847	0.841635
$773$	−13.9252	−0.500855	−0.250427	−	0.968135i	$-0.580571\pi$
$773$	−13.9252	−0.500855	−0.250427	+	0.968135i	$0.580571\pi$
$774$	−7.25771	−0.260873
$775$	6.30597	0.226517
$776$	10.9352	0.392552
$777$	−1.02620	−0.0368146
$778$	−55.4444	−1.98778
$779$	−5.75071	−0.206040
$780$	69.0756	2.47330
$781$	−46.6146	−1.66800
$782$	11.4977	0.411159
$783$	3.09052	0.110446
$784$	−2.77332	−0.0990472
$785$	−11.0977	−0.396094
$786$	50.9685	1.81798
$787$	45.7185	1.62969	0.814844	−	0.579680i	$-0.196822\pi$
$787$	45.7185	1.62969	0.814844	+	0.579680i	$0.196822\pi$
$788$	−57.5523	−2.05022
$789$	−5.50785	−0.196085
$790$	−4.39970	−0.156534
$791$	10.4999	0.373333
$792$	2.38083	0.0845991
$793$	0.926799	0.0329116
$794$	−9.58812	−0.340270
$795$	−13.3732	−0.474300
$796$	53.8687	1.90933
$797$	−11.4211	−0.404555	−0.202277	−	0.979328i	$-0.564834\pi$
$797$	−11.4211	−0.404555	−0.202277	+	0.979328i	$0.564834\pi$
$798$	8.73512	0.309220
$799$	−23.1511	−0.819029
$800$	−6.40547	−0.226468
$801$	2.45787	0.0868446
$802$	−38.1978	−1.34881
$803$	25.1610	0.887913
$804$	−61.1952	−2.15819
$805$	−2.40920	−0.0849132
$806$	−103.863	−3.65843
$807$	16.4355	0.578558
$808$	−1.64107	−0.0577327
$809$	17.6307	0.619864	0.309932	−	0.950759i	$-0.399694\pi$
$809$	17.6307	0.619864	0.309932	+	0.950759i	$0.399694\pi$
$810$	−51.1349	−1.79670
$811$	30.9666	1.08738	0.543692	−	0.839285i	$-0.317026\pi$
$811$	30.9666	1.08738	0.543692	+	0.839285i	$0.317026\pi$
$812$	1.60215	0.0562244
$813$	−38.6137	−1.35424
$814$	6.94449	0.243404
$815$	39.3252	1.37750
$816$	−27.6933	−0.969459
$817$	19.7441	0.690759
$818$	9.90349	0.346267
$819$	−2.42692	−0.0848035
$820$	15.4217	0.538548
$821$	25.3060	0.883187	0.441593	−	0.897215i	$-0.354413\pi$
$821$	25.3060	0.883187	0.441593	+	0.897215i	$0.354413\pi$
$822$	58.6233	2.04472
$823$	45.6057	1.58972	0.794858	−	0.606795i	$-0.207545\pi$
$823$	45.6057	1.58972	0.794858	+	0.606795i	$0.207545\pi$
$824$	5.23237	0.182278
$825$	−8.70079	−0.302922
$826$	8.83875	0.307539
$827$	−25.4338	−0.884420	−0.442210	−	0.896912i	$-0.645805\pi$
$827$	−25.4338	−0.884420	−0.442210	+	0.896912i	$0.645805\pi$
$828$	−0.967751	−0.0336317
$829$	−5.12771	−0.178093	−0.0890463	−	0.996027i	$-0.528382\pi$
$829$	−5.12771	−0.178093	−0.0890463	+	0.996027i	$0.528382\pi$
$830$	−24.6040	−0.854019
$831$	−10.7238	−0.372006
$832$	70.8360	2.45580
$833$	−5.42479	−0.187958
$834$	49.9330	1.72904
$835$	40.2948	1.39446
$836$	−32.7946	−1.13423
$837$	37.6938	1.30289
$838$	−27.0603	−0.934782
$839$	−7.81839	−0.269921	−0.134960	−	0.990851i	$-0.543091\pi$
$839$	−7.81839	−0.269921	−0.134960	+	0.990851i	$0.543091\pi$
$840$	−4.62639	−0.159626
$841$	−28.5867	−0.985749
$842$	59.6352	2.05517
$843$	39.0010	1.34327
$844$	38.5242	1.32606
$845$	62.7880	2.15997
$846$	3.51236	0.120757
$847$	23.5417	0.808901
$848$	−8.36319	−0.287193
$849$	23.5705	0.808937
$850$	−9.24715	−0.317175
$851$	−0.557492	−0.0191106
$852$	36.3853	1.24654
$853$	−36.6540	−1.25501	−0.627505	−	0.778613i	$-0.715924\pi$
$853$	−36.6540	−1.25501	−0.627505	+	0.778613i	$0.715924\pi$
$854$	−0.314297	−0.0107550
$855$	−2.09460	−0.0716337
$856$	−6.06326	−0.207238
$857$	44.0599	1.50506	0.752529	−	0.658559i	$-0.228834\pi$
$857$	44.0599	1.50506	0.752529	+	0.658559i	$0.228834\pi$
$858$	143.307	4.89244
$859$	−6.04906	−0.206391	−0.103196	−	0.994661i	$-0.532907\pi$
$859$	−6.04906	−0.206391	−0.103196	+	0.994661i	$0.532907\pi$
$860$	−52.9477	−1.80550
$861$	−4.72787	−0.161125
$862$	−11.6434	−0.396576
$863$	−25.7890	−0.877867	−0.438933	−	0.898520i	$-0.644644\pi$
$863$	−25.7890	−0.877867	−0.438933	+	0.898520i	$0.644644\pi$
$864$	−38.2885	−1.30260
$865$	22.3490	0.759888
$866$	−44.5467	−1.51376
$867$	−22.8773	−0.776954
$868$	19.5407	0.663256
$869$	−5.06397	−0.171784
$870$	−6.04246	−0.204858
$871$	−83.3714	−2.82493
$872$	7.98097	0.270270
$873$	4.07034	0.137760
$874$	4.74545	0.160517
$875$	−10.1084	−0.341726
$876$	−19.6396	−0.663560
$877$	9.59839	0.324114	0.162057	−	0.986781i	$-0.448187\pi$
$877$	9.59839	0.324114	0.162057	+	0.986781i	$0.448187\pi$
$878$	83.2266	2.80876
$879$	5.74165	0.193661
$880$	−39.2686	−1.32374
$881$	−9.38385	−0.316150	−0.158075	−	0.987427i	$-0.550529\pi$
$881$	−9.38385	−0.316150	−0.158075	+	0.987427i	$0.550529\pi$
$882$	0.823019	0.0277125
$883$	−4.98842	−0.167874	−0.0839368	−	0.996471i	$-0.526749\pi$
$883$	−4.98842	−0.167874	−0.0839368	+	0.996471i	$0.526749\pi$
$884$	84.4973	2.84195
$885$	−18.4938	−0.621662
$886$	30.8362	1.03596
$887$	36.6169	1.22948	0.614738	−	0.788732i	$-0.289262\pi$
$887$	36.6169	1.22948	0.614738	+	0.788732i	$0.289262\pi$
$888$	−1.07055	−0.0359253
$889$	−17.9732	−0.602802
$890$	32.3208	1.08340
$891$	−58.8553	−1.97173
$892$	−27.5908	−0.923808
$893$	−9.55514	−0.319750
$894$	82.4381	2.75714
$895$	−18.0270	−0.602575
$896$	−8.09304	−0.270370
$897$	−11.5045	−0.384124
$898$	24.0900	0.803895
$899$	5.04052	0.168111
$900$	0.778321	0.0259440
$901$	−16.3589	−0.544995
$902$	31.9945	1.06530
$903$	16.2324	0.540179
$904$	10.9537	0.364315
$905$	−25.6536	−0.852754
$906$	−76.1806	−2.53093
$907$	34.2361	1.13679	0.568395	−	0.822756i	$-0.307564\pi$
$907$	34.2361	1.13679	0.568395	+	0.822756i	$0.307564\pi$
$908$	−6.09782	−0.202363
$909$	−0.610844	−0.0202604
$910$	−31.9138	−1.05793
$911$	2.10958	0.0698934	0.0349467	−	0.999389i	$-0.488874\pi$
$911$	2.10958	0.0698934	0.0349467	+	0.999389i	$0.488874\pi$
$912$	−11.4298	−0.378479
$913$	−28.3188	−0.937216
$914$	−19.9813	−0.660923
$915$	0.657620	0.0217402
$916$	−14.3077	−0.472739
$917$	−13.0641	−0.431415
$918$	−55.2746	−1.82433
$919$	49.7639	1.64156	0.820779	−	0.571245i	$-0.193539\pi$
$919$	49.7639	1.64156	0.820779	+	0.571245i	$0.193539\pi$
$920$	−2.51334	−0.0828622
$921$	29.6639	0.977460
$922$	−60.9768	−2.00816
$923$	49.5708	1.63164
$924$	−26.9617	−0.886975
$925$	0.448367	0.0147422
$926$	−9.87527	−0.324521
$927$	1.94761	0.0639678
$928$	−5.12005	−0.168074
$929$	−45.0863	−1.47923	−0.739617	−	0.673028i	$-0.764993\pi$
$929$	−45.0863	−1.47923	−0.739617	+	0.673028i	$0.764993\pi$
$930$	−73.6973	−2.41663
$931$	−2.23897	−0.0733791
$932$	21.5945	0.707351
$933$	−55.6691	−1.82252
$934$	16.5940	0.542971
$935$	−76.8118	−2.51201
$936$	−2.53182	−0.0827551
$937$	14.8779	0.486040	0.243020	−	0.970021i	$-0.421862\pi$
$937$	14.8779	0.486040	0.243020	+	0.970021i	$0.421862\pi$
$938$	28.2730	0.923145
$939$	37.7271	1.23118
$940$	25.6240	0.835763
$941$	30.3917	0.990742	0.495371	−	0.868681i	$-0.335032\pi$
$941$	30.3917	0.990742	0.495371	+	0.868681i	$0.335032\pi$
$942$	17.9714	0.585540
$943$	−2.56847	−0.0836408
$944$	−11.5654	−0.376422
$945$	11.5821	0.376765
$946$	−109.848	−3.57146
$947$	−26.5595	−0.863068	−0.431534	−	0.902097i	$-0.642028\pi$
$947$	−26.5595	−0.863068	−0.431534	+	0.902097i	$0.642028\pi$
$948$	3.95272	0.128378
$949$	−26.7567	−0.868559
$950$	−3.81656	−0.123826
$951$	−5.43094	−0.176110
$952$	−5.65926	−0.183418
$953$	8.75467	0.283592	0.141796	−	0.989896i	$-0.454712\pi$
$953$	8.75467	0.283592	0.141796	+	0.989896i	$0.454712\pi$
$954$	2.48188	0.0803540
$955$	7.95745	0.257497
$956$	−2.48071	−0.0802319
$957$	−6.95476	−0.224815
$958$	−71.9409	−2.32430
$959$	−15.0262	−0.485221
$960$	50.2624	1.62221
$961$	30.4771	0.983134
$962$	−7.38489	−0.238099
$963$	−2.25688	−0.0727270
$964$	−34.7031	−1.11771
$965$	22.6059	0.727710
$966$	3.90141	0.125526
$967$	−19.3848	−0.623372	−0.311686	−	0.950185i	$-0.600894\pi$
$967$	−19.3848	−0.623372	−0.311686	+	0.950185i	$0.600894\pi$
$968$	24.5592	0.789363
$969$	−22.3574	−0.718224
$970$	53.5247	1.71857
$971$	−17.9424	−0.575799	−0.287899	−	0.957661i	$-0.592957\pi$
$971$	−17.9424	−0.575799	−0.287899	+	0.957661i	$0.592957\pi$
$972$	9.99652	0.320639
$973$	−12.7987	−0.410308
$974$	−24.7725	−0.793762
$975$	9.25258	0.296320
$976$	0.411254	0.0131639
$977$	5.12609	0.163998	0.0819991	−	0.996632i	$-0.473870\pi$
$977$	5.12609	0.163998	0.0819991	+	0.996632i	$0.473870\pi$
$978$	−63.6823	−2.03634
$979$	37.2007	1.18894
$980$	6.00423	0.191798
$981$	2.97070	0.0948470
$982$	−41.6992	−1.33067
$983$	−48.1332	−1.53521	−0.767606	−	0.640922i	$-0.778552\pi$
$983$	−48.1332	−1.53521	−0.767606	+	0.640922i	$0.778552\pi$
$984$	−4.93222	−0.157233
$985$	−55.6355	−1.77269
$986$	−7.39148	−0.235393
$987$	−7.85563	−0.250048
$988$	34.8744	1.10950
$989$	8.81841	0.280409
$990$	11.6535	0.370371
$991$	−6.50910	−0.206769	−0.103384	−	0.994641i	$-0.532967\pi$
$991$	−6.50910	−0.206769	−0.103384	+	0.994641i	$0.532967\pi$
$992$	−62.4471	−1.98270
$993$	39.1151	1.24128
$994$	−16.8105	−0.533196
$995$	52.0746	1.65088
$996$	22.1044	0.700405
$997$	43.2321	1.36917	0.684587	−	0.728931i	$-0.259982\pi$
$997$	43.2321	1.36917	0.684587	+	0.728931i	$0.259982\pi$
$998$	−63.9445	−2.02413
$999$	2.68010	0.0847948

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.2.a.d.1.4	✓	5
3.2	odd	2		1449.2.a.r.1.2		5
4.3	odd	2		2576.2.a.bd.1.4		5
5.4	even	2		4025.2.a.p.1.2		5
7.6	odd	2		1127.2.a.h.1.4		5
23.22	odd	2		3703.2.a.j.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.d.1.4	✓	5	1.1	even	1	trivial
1127.2.a.h.1.4		5	7.6	odd	2
1449.2.a.r.1.2		5	3.2	odd	2
2576.2.a.bd.1.4		5	4.3	odd	2
3703.2.a.j.1.4		5	23.22	odd	2
4025.2.a.p.1.2		5	5.4	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 161 = 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	161.a (trivial)

\(f(q)\)	\(=\)	\(q+2.11948 q^{2} -1.84074 q^{3} +2.49221 q^{4} +2.40920 q^{5} -3.90141 q^{6} +1.00000 q^{7} +1.04322 q^{8} +0.388311 q^{9} +5.10626 q^{10} +5.87722 q^{11} -4.58750 q^{12} -6.24994 q^{13} +2.11948 q^{14} -4.43471 q^{15} -2.77332 q^{16} -5.42479 q^{17} +0.823019 q^{18} -2.23897 q^{19} +6.00423 q^{20} -1.84074 q^{21} +12.4567 q^{22} -1.00000 q^{23} -1.92030 q^{24} +0.804258 q^{25} -13.2466 q^{26} +4.80743 q^{27} +2.49221 q^{28} +0.642864 q^{29} -9.39929 q^{30} +7.84074 q^{31} -7.96445 q^{32} -10.8184 q^{33} -11.4977 q^{34} +2.40920 q^{35} +0.967751 q^{36} +0.557492 q^{37} -4.74545 q^{38} +11.5045 q^{39} +2.51334 q^{40} +2.56847 q^{41} -3.90141 q^{42} -8.81841 q^{43} +14.6472 q^{44} +0.935520 q^{45} -2.11948 q^{46} +4.26766 q^{47} +5.10495 q^{48} +1.00000 q^{49} +1.70461 q^{50} +9.98561 q^{51} -15.5761 q^{52} +3.01559 q^{53} +10.1893 q^{54} +14.1594 q^{55} +1.04322 q^{56} +4.12134 q^{57} +1.36254 q^{58} +4.17024 q^{59} -11.0522 q^{60} -0.148289 q^{61} +16.6183 q^{62} +0.388311 q^{63} -11.3339 q^{64} -15.0574 q^{65} -22.9294 q^{66} +13.3396 q^{67} -13.5197 q^{68} +1.84074 q^{69} +5.10626 q^{70} -7.93141 q^{71} +0.405095 q^{72} +4.28111 q^{73} +1.18159 q^{74} -1.48043 q^{75} -5.57996 q^{76} +5.87722 q^{77} +24.3836 q^{78} -0.861628 q^{79} -6.68149 q^{80} -10.0141 q^{81} +5.44382 q^{82} -4.81841 q^{83} -4.58750 q^{84} -13.0694 q^{85} -18.6905 q^{86} -1.18334 q^{87} +6.13125 q^{88} +6.32964 q^{89} +1.98282 q^{90} -6.24994 q^{91} -2.49221 q^{92} -14.4327 q^{93} +9.04523 q^{94} -5.39412 q^{95} +14.6605 q^{96} +10.4822 q^{97} +2.11948 q^{98} +2.28219 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 12 q^{4} - 4 q^{5} - 3 q^{6} + 5 q^{7} + 3 q^{8} + 11 q^{9} - 8 q^{10} - 4 q^{11} + 3 q^{12} - 6 q^{13} + 2 q^{14} + 10 q^{15} + 10 q^{16} - 12 q^{17} - 19 q^{18} + 6 q^{19} + 14 q^{22}+ \cdots - 8 q^{99}+O(q^{100}) \) 5 * q + 2 * q^2 + 12 * q^4 - 4 * q^5 - 3 * q^6 + 5 * q^7 + 3 * q^8 + 11 * q^9 - 8 * q^10 - 4 * q^11 + 3 * q^12 - 6 * q^13 + 2 * q^14 + 10 * q^15 + 10 * q^16 - 12 * q^17 - 19 * q^18 + 6 * q^19 + 14 * q^22 - 5 * q^23 - 36 * q^24 + 19 * q^25 + q^26 + 12 * q^28 - 4 * q^29 - 48 * q^30 + 30 * q^31 + 8 * q^32 - 22 * q^33 + 6 * q^34 - 4 * q^35 - q^36 + 4 * q^37 - 40 * q^38 + 16 * q^39 - 50 * q^40 + 6 * q^41 - 3 * q^42 - 12 * q^43 - 26 * q^44 - 12 * q^45 - 2 * q^46 + 10 * q^47 + 25 * q^48 + 5 * q^49 - 2 * q^50 - 4 * q^51 - 21 * q^52 + 16 * q^53 + 33 * q^54 + 18 * q^55 + 3 * q^56 + 6 * q^57 + 13 * q^58 + 22 * q^59 + 30 * q^60 - 18 * q^61 + 15 * q^62 + 11 * q^63 + 25 * q^64 - 26 * q^65 + 4 * q^66 - 2 * q^67 + 12 * q^68 - 8 * q^70 + 4 * q^71 - 41 * q^72 - 2 * q^73 + 38 * q^74 - 30 * q^75 + 10 * q^76 - 4 * q^77 + 41 * q^78 + 30 * q^79 - 10 * q^80 - 3 * q^81 - 7 * q^82 + 8 * q^83 + 3 * q^84 - 12 * q^85 + 8 * q^86 - 12 * q^87 + 4 * q^88 - 20 * q^89 + 34 * q^90 - 6 * q^91 - 12 * q^92 - 26 * q^93 - 25 * q^94 + 8 * q^95 - q^96 - 12 * q^97 + 2 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more