LMFDB - Embedded newform 8993.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8993,2,Mod(1,8993)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8993, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8993.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8993 = 17 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8993.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:	$71.8094665377$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8993.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.42689 q^{2} +0.360604 q^{3} +3.88978 q^{4} -1.70693 q^{5} -0.875146 q^{6} -4.05794 q^{7} -4.58629 q^{8} -2.86996 q^{9} +O(q^{10})$	\(q-2.42689 q^{2} +0.360604 q^{3} +3.88978 q^{4} -1.70693 q^{5} -0.875146 q^{6} -4.05794 q^{7} -4.58629 q^{8} -2.86996 q^{9} +4.14252 q^{10} -5.15668 q^{11} +1.40267 q^{12} -1.58498 q^{13} +9.84816 q^{14} -0.615525 q^{15} +3.35085 q^{16} -1.00000 q^{17} +6.96508 q^{18} -2.03460 q^{19} -6.63957 q^{20} -1.46331 q^{21} +12.5147 q^{22} -1.65384 q^{24} -2.08641 q^{25} +3.84656 q^{26} -2.11674 q^{27} -15.7845 q^{28} +7.82832 q^{29} +1.49381 q^{30} +6.63057 q^{31} +1.04045 q^{32} -1.85952 q^{33} +2.42689 q^{34} +6.92660 q^{35} -11.1635 q^{36} +1.52074 q^{37} +4.93774 q^{38} -0.571550 q^{39} +7.82846 q^{40} +1.29576 q^{41} +3.55129 q^{42} -9.98186 q^{43} -20.0584 q^{44} +4.89882 q^{45} +2.10644 q^{47} +1.20833 q^{48} +9.46685 q^{49} +5.06347 q^{50} -0.360604 q^{51} -6.16522 q^{52} +1.99816 q^{53} +5.13708 q^{54} +8.80207 q^{55} +18.6109 q^{56} -0.733685 q^{57} -18.9984 q^{58} -6.72985 q^{59} -2.39426 q^{60} +10.3703 q^{61} -16.0917 q^{62} +11.6461 q^{63} -9.22676 q^{64} +2.70544 q^{65} +4.51285 q^{66} +13.4813 q^{67} -3.88978 q^{68} -16.8101 q^{70} -3.26310 q^{71} +13.1625 q^{72} +8.64554 q^{73} -3.69066 q^{74} -0.752367 q^{75} -7.91414 q^{76} +20.9255 q^{77} +1.38709 q^{78} -10.8691 q^{79} -5.71965 q^{80} +7.84659 q^{81} -3.14466 q^{82} -2.39925 q^{83} -5.69196 q^{84} +1.70693 q^{85} +24.2249 q^{86} +2.82293 q^{87} +23.6500 q^{88} +10.9691 q^{89} -11.8889 q^{90} +6.43174 q^{91} +2.39101 q^{93} -5.11210 q^{94} +3.47290 q^{95} +0.375192 q^{96} -12.5988 q^{97} -22.9750 q^{98} +14.7995 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 4 q^{3} + 24 q^{4} - 4 q^{5} - 2 q^{6} - 20 q^{7} + 6 q^{8} + 24 q^{9}+O(q^{10}) $ 28 * q + 4 * q^3 + 24 * q^4 - 4 * q^5 - 2 * q^6 - 20 * q^7 + 6 * q^8 + 24 * q^9	\( 28 q + 4 q^{3} + 24 q^{4} - 4 q^{5} - 2 q^{6} - 20 q^{7} + 6 q^{8} + 24 q^{9} + 2 q^{10} - 8 q^{11} + 6 q^{12} - 8 q^{14} - 24 q^{15} + 32 q^{16} - 28 q^{17} - 6 q^{18} - 16 q^{19} - 12 q^{20} - 40 q^{21} - 20 q^{22} - 32 q^{24} + 20 q^{25} + 26 q^{26} + 4 q^{27} - 40 q^{28} + 12 q^{29} - 16 q^{30} + 4 q^{31} - 8 q^{32} - 4 q^{33} + 2 q^{36} - 36 q^{37} - 32 q^{38} + 24 q^{39} + 48 q^{40} + 16 q^{41} - 24 q^{42} - 48 q^{43} - 62 q^{44} + 60 q^{45} - 6 q^{48} + 24 q^{49} + 12 q^{50} - 4 q^{51} - 18 q^{52} - 24 q^{53} + 30 q^{54} + 12 q^{55} + 46 q^{56} - 24 q^{57} - 14 q^{58} - 12 q^{59} - 48 q^{60} - 36 q^{61} - 6 q^{62} + 20 q^{63} + 2 q^{64} - 16 q^{65} - 16 q^{66} - 68 q^{67} - 24 q^{68} - 4 q^{70} + 24 q^{71} + 30 q^{72} + 44 q^{73} - 68 q^{74} - 20 q^{75} - 40 q^{76} - 28 q^{77} - 38 q^{78} - 60 q^{79} - 28 q^{80} - 4 q^{81} - 34 q^{82} - 64 q^{83} - 48 q^{84} + 4 q^{85} + 96 q^{86} - 52 q^{87} - 60 q^{88} - 16 q^{89} - 126 q^{90} + 4 q^{91} - 24 q^{93} - 50 q^{94} + 16 q^{95} - 26 q^{96} - 104 q^{97} + 14 q^{98} - 40 q^{99}+O(q^{100}) \) 28 * q + 4 * q^3 + 24 * q^4 - 4 * q^5 - 2 * q^6 - 20 * q^7 + 6 * q^8 + 24 * q^9 + 2 * q^10 - 8 * q^11 + 6 * q^12 - 8 * q^14 - 24 * q^15 + 32 * q^16 - 28 * q^17 - 6 * q^18 - 16 * q^19 - 12 * q^20 - 40 * q^21 - 20 * q^22 - 32 * q^24 + 20 * q^25 + 26 * q^26 + 4 * q^27 - 40 * q^28 + 12 * q^29 - 16 * q^30 + 4 * q^31 - 8 * q^32 - 4 * q^33 + 2 * q^36 - 36 * q^37 - 32 * q^38 + 24 * q^39 + 48 * q^40 + 16 * q^41 - 24 * q^42 - 48 * q^43 - 62 * q^44 + 60 * q^45 - 6 * q^48 + 24 * q^49 + 12 * q^50 - 4 * q^51 - 18 * q^52 - 24 * q^53 + 30 * q^54 + 12 * q^55 + 46 * q^56 - 24 * q^57 - 14 * q^58 - 12 * q^59 - 48 * q^60 - 36 * q^61 - 6 * q^62 + 20 * q^63 + 2 * q^64 - 16 * q^65 - 16 * q^66 - 68 * q^67 - 24 * q^68 - 4 * q^70 + 24 * q^71 + 30 * q^72 + 44 * q^73 - 68 * q^74 - 20 * q^75 - 40 * q^76 - 28 * q^77 - 38 * q^78 - 60 * q^79 - 28 * q^80 - 4 * q^81 - 34 * q^82 - 64 * q^83 - 48 * q^84 + 4 * q^85 + 96 * q^86 - 52 * q^87 - 60 * q^88 - 16 * q^89 - 126 * q^90 + 4 * q^91 - 24 * q^93 - 50 * q^94 + 16 * q^95 - 26 * q^96 - 104 * q^97 + 14 * q^98 - 40 * 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 $

$2$	−2.42689	−1.71607	−0.858034	−	0.513592i	$-0.828314\pi$
$2$	−2.42689	−1.71607	−0.858034	+	0.513592i	$0.828314\pi$
$3$	0.360604	0.208195	0.104098	−	0.994567i	$-0.466805\pi$
$3$	0.360604	0.208195	0.104098	+	0.994567i	$0.466805\pi$
$4$	3.88978	1.94489
$5$	−1.70693	−0.763360	−0.381680	−	0.924294i	$-0.624654\pi$
$5$	−1.70693	−0.763360	−0.381680	+	0.924294i	$0.624654\pi$
$6$	−0.875146	−0.357277
$7$	−4.05794	−1.53376	−0.766878	−	0.641793i	$-0.778191\pi$
$7$	−4.05794	−1.53376	−0.766878	+	0.641793i	$0.778191\pi$
$8$	−4.58629	−1.62150
$9$	−2.86996	−0.956655
$10$	4.14252	1.30998
$11$	−5.15668	−1.55480	−0.777399	−	0.629008i	$-0.783461\pi$
$11$	−5.15668	−1.55480	−0.777399	+	0.629008i	$0.783461\pi$
$12$	1.40267	0.404917
$13$	−1.58498	−0.439594	−0.219797	−	0.975546i	$-0.570539\pi$
$13$	−1.58498	−0.439594	−0.219797	+	0.975546i	$0.570539\pi$
$14$	9.84816	2.63203
$15$	−0.615525	−0.158928
$16$	3.35085	0.837712
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	6.96508	1.64169
$19$	−2.03460	−0.466768	−0.233384	−	0.972385i	$-0.574980\pi$
$19$	−2.03460	−0.466768	−0.233384	+	0.972385i	$0.574980\pi$
$20$	−6.63957	−1.48465
$21$	−1.46331	−0.319320
$22$	12.5147	2.66814
$23$	0	0
$23$	0	0
$24$	−1.65384	−0.337588
$25$	−2.08641	−0.417281
$26$	3.84656	0.754373
$27$	−2.11674	−0.407366
$28$	−15.7845	−2.98299
$29$	7.82832	1.45368	0.726841	−	0.686806i	$-0.240988\pi$
$29$	7.82832	1.45368	0.726841	+	0.686806i	$0.240988\pi$
$30$	1.49381	0.272731
$31$	6.63057	1.19089	0.595443	−	0.803398i	$-0.296977\pi$
$31$	6.63057	1.19089	0.595443	+	0.803398i	$0.296977\pi$
$32$	1.04045	0.183928
$33$	−1.85952	−0.323701
$34$	2.42689	0.416208
$35$	6.92660	1.17081
$36$	−11.1635	−1.86059
$37$	1.52074	0.250008	0.125004	−	0.992156i	$-0.460106\pi$
$37$	1.52074	0.250008	0.125004	+	0.992156i	$0.460106\pi$
$38$	4.93774	0.801007
$39$	−0.571550	−0.0915212
$40$	7.82846	1.23779
$41$	1.29576	0.202363	0.101182	−	0.994868i	$-0.467738\pi$
$41$	1.29576	0.202363	0.101182	+	0.994868i	$0.467738\pi$
$42$	3.55129	0.547976
$43$	−9.98186	−1.52222	−0.761110	−	0.648623i	$-0.775345\pi$
$43$	−9.98186	−1.52222	−0.761110	+	0.648623i	$0.775345\pi$
$44$	−20.0584	−3.02391
$45$	4.89882	0.730272
$46$	0	0
$47$	2.10644	0.307256	0.153628	−	0.988129i	$-0.450904\pi$
$47$	2.10644	0.307256	0.153628	+	0.988129i	$0.450904\pi$
$48$	1.20833	0.174407
$49$	9.46685	1.35241
$50$	5.06347	0.716083
$51$	−0.360604	−0.0504947
$52$	−6.16522	−0.854962
$53$	1.99816	0.274468	0.137234	−	0.990539i	$-0.456179\pi$
$53$	1.99816	0.274468	0.137234	+	0.990539i	$0.456179\pi$
$54$	5.13708	0.699068
$55$	8.80207	1.18687
$56$	18.6109	2.48698
$57$	−0.733685	−0.0971789
$58$	−18.9984	−2.49462
$59$	−6.72985	−0.876152	−0.438076	−	0.898938i	$-0.644340\pi$
$59$	−6.72985	−0.876152	−0.438076	+	0.898938i	$0.644340\pi$
$60$	−2.39426	−0.309097
$61$	10.3703	1.32777	0.663887	−	0.747832i	$-0.268905\pi$
$61$	10.3703	1.32777	0.663887	+	0.747832i	$0.268905\pi$
$62$	−16.0917	−2.04364
$63$	11.6461	1.46728
$64$	−9.22676	−1.15334
$65$	2.70544	0.335568
$66$	4.51285	0.555494
$67$	13.4813	1.64700	0.823499	−	0.567317i	$-0.192019\pi$
$67$	13.4813	1.64700	0.823499	+	0.567317i	$0.192019\pi$
$68$	−3.88978	−0.471706
$69$	0	0
$70$	−16.8101	−2.00919
$71$	−3.26310	−0.387258	−0.193629	−	0.981075i	$-0.562026\pi$
$71$	−3.26310	−0.387258	−0.193629	+	0.981075i	$0.562026\pi$
$72$	13.1625	1.55121
$73$	8.64554	1.01188	0.505942	−	0.862567i	$-0.331145\pi$
$73$	8.64554	1.01188	0.505942	+	0.862567i	$0.331145\pi$
$74$	−3.69066	−0.429030
$75$	−0.752367	−0.0868759
$76$	−7.91414	−0.907814
$77$	20.9255	2.38468
$78$	1.38709	0.157057
$79$	−10.8691	−1.22287	−0.611437	−	0.791293i	$-0.709408\pi$
$79$	−10.8691	−1.22287	−0.611437	+	0.791293i	$0.709408\pi$
$80$	−5.71965	−0.639476
$81$	7.84659	0.871843
$82$	−3.14466	−0.347270
$83$	−2.39925	−0.263352	−0.131676	−	0.991293i	$-0.542036\pi$
$83$	−2.39925	−0.263352	−0.131676	+	0.991293i	$0.542036\pi$
$84$	−5.69196	−0.621044
$85$	1.70693	0.185142
$86$	24.2249	2.61223
$87$	2.82293	0.302649
$88$	23.6500	2.52110
$89$	10.9691	1.16272	0.581360	−	0.813647i	$-0.302521\pi$
$89$	10.9691	1.16272	0.581360	+	0.813647i	$0.302521\pi$
$90$	−11.8889	−1.25320
$91$	6.43174	0.674230
$92$	0	0
$93$	2.39101	0.247937
$94$	−5.11210	−0.527273
$95$	3.47290	0.356312
$96$	0.375192	0.0382929
$97$	−12.5988	−1.27921	−0.639607	−	0.768702i	$-0.720903\pi$
$97$	−12.5988	−1.27921	−0.639607	+	0.768702i	$0.720903\pi$
$98$	−22.9750	−2.32082
$99$	14.7995	1.48741
$100$	−8.11567	−0.811567
$101$	15.9950	1.59156	0.795779	−	0.605588i	$-0.207062\pi$
$101$	15.9950	1.59156	0.795779	+	0.605588i	$0.207062\pi$
$102$	0.875146	0.0866524
$103$	−5.53038	−0.544925	−0.272462	−	0.962166i	$-0.587838\pi$
$103$	−5.53038	−0.544925	−0.272462	+	0.962166i	$0.587838\pi$
$104$	7.26917	0.712801
$105$	2.49776	0.243757
$106$	−4.84931	−0.471006
$107$	−13.9111	−1.34484	−0.672421	−	0.740169i	$-0.734745\pi$
$107$	−13.9111	−1.34484	−0.672421	+	0.740169i	$0.734745\pi$
$108$	−8.23364	−0.792283
$109$	8.85360	0.848021	0.424010	−	0.905657i	$-0.360622\pi$
$109$	8.85360	0.848021	0.424010	+	0.905657i	$0.360622\pi$
$110$	−21.3616	−2.03675
$111$	0.548385	0.0520504
$112$	−13.5975	−1.28485
$113$	1.33534	0.125618	0.0628089	−	0.998026i	$-0.479994\pi$
$113$	1.33534	0.125618	0.0628089	+	0.998026i	$0.479994\pi$
$114$	1.78057	0.166766
$115$	0	0
$116$	30.4505	2.82725
$117$	4.54883	0.420539
$118$	16.3326	1.50354
$119$	4.05794	0.371990
$120$	2.82298	0.257701
$121$	15.5914	1.41740
$122$	−25.1674	−2.27855
$123$	0.467256	0.0421311
$124$	25.7915	2.31614
$125$	12.0960	1.08190
$126$	−28.2639	−2.51794
$127$	−10.5108	−0.932678	−0.466339	−	0.884606i	$-0.654427\pi$
$127$	−10.5108	−0.932678	−0.466339	+	0.884606i	$0.654427\pi$
$128$	20.3114	1.79529
$129$	−3.59950	−0.316919
$130$	−6.56580	−0.575858
$131$	−19.1648	−1.67444	−0.837218	−	0.546869i	$-0.815820\pi$
$131$	−19.1648	−1.67444	−0.837218	+	0.546869i	$0.815820\pi$
$132$	−7.23314	−0.629564
$133$	8.25626	0.715909
$134$	−32.7175	−2.82636
$135$	3.61311	0.310967
$136$	4.58629	0.393271
$137$	10.6171	0.907078	0.453539	−	0.891236i	$-0.350161\pi$
$137$	10.6171	0.907078	0.453539	+	0.891236i	$0.350161\pi$
$138$	0	0
$139$	13.4623	1.14185	0.570927	−	0.821001i	$-0.306584\pi$
$139$	13.4623	1.14185	0.570927	+	0.821001i	$0.306584\pi$
$140$	26.9430	2.27710
$141$	0.759592	0.0639692
$142$	7.91917	0.664562
$143$	8.17323	0.683480
$144$	−9.61681	−0.801401
$145$	−13.3624	−1.10968
$146$	−20.9818	−1.73646
$147$	3.41379	0.281565
$148$	5.91534	0.486238
$149$	17.3271	1.41949	0.709746	−	0.704458i	$-0.248810\pi$
$149$	17.3271	1.41949	0.709746	+	0.704458i	$0.248810\pi$
$150$	1.82591	0.149085
$151$	5.36141	0.436305	0.218153	−	0.975915i	$-0.429997\pi$
$151$	5.36141	0.436305	0.218153	+	0.975915i	$0.429997\pi$
$152$	9.33125	0.756864
$153$	2.86996	0.232023
$154$	−50.7838	−4.09228
$155$	−11.3179	−0.909075
$156$	−2.22321	−0.177999
$157$	3.23004	0.257785	0.128893	−	0.991659i	$-0.458858\pi$
$157$	3.23004	0.257785	0.128893	+	0.991659i	$0.458858\pi$
$158$	26.3782	2.09853
$159$	0.720545	0.0571429
$160$	−1.77598	−0.140403
$161$	0	0
$162$	−19.0428	−1.49614
$163$	7.59562	0.594935	0.297468	−	0.954732i	$-0.403858\pi$
$163$	7.59562	0.594935	0.297468	+	0.954732i	$0.403858\pi$
$164$	5.04022	0.393575
$165$	3.17407	0.247101
$166$	5.82272	0.451931
$167$	20.0865	1.55434	0.777170	−	0.629290i	$-0.216654\pi$
$167$	20.0865	1.55434	0.777170	+	0.629290i	$0.216654\pi$
$168$	6.71117	0.517778
$169$	−10.4878	−0.806757
$170$	−4.14252	−0.317716
$171$	5.83922	0.446536
$172$	−38.8273	−2.96055
$173$	13.7680	1.04676	0.523380	−	0.852100i	$-0.324671\pi$
$173$	13.7680	1.04676	0.523380	+	0.852100i	$0.324671\pi$
$174$	−6.85092	−0.519367
$175$	8.46650	0.640008
$176$	−17.2793	−1.30247
$177$	−2.42682	−0.182411
$178$	−26.6207	−1.99531
$179$	−21.6271	−1.61648	−0.808241	−	0.588851i	$-0.799580\pi$
$179$	−21.6271	−1.61648	−0.808241	+	0.588851i	$0.799580\pi$
$180$	19.0553	1.42030
$181$	−16.1795	−1.20261	−0.601305	−	0.799020i	$-0.705352\pi$
$181$	−16.1795	−1.20261	−0.601305	+	0.799020i	$0.705352\pi$
$182$	−15.6091	−1.15702
$183$	3.73956	0.276436
$184$	0	0
$185$	−2.59578	−0.190846
$186$	−5.80272	−0.425476
$187$	5.15668	0.377094
$188$	8.19360	0.597580
$189$	8.58958	0.624800
$190$	−8.42835	−0.611457
$191$	17.2210	1.24607	0.623035	−	0.782194i	$-0.285899\pi$
$191$	17.2210	1.24607	0.623035	+	0.782194i	$0.285899\pi$
$192$	−3.32721	−0.240121
$193$	1.44485	0.104003	0.0520014	−	0.998647i	$-0.483440\pi$
$193$	1.44485	0.104003	0.0520014	+	0.998647i	$0.483440\pi$
$194$	30.5759	2.19522
$195$	0.975593	0.0698637
$196$	36.8240	2.63029
$197$	−21.7467	−1.54939	−0.774693	−	0.632338i	$-0.782095\pi$
$197$	−21.7467	−1.54939	−0.774693	+	0.632338i	$0.782095\pi$
$198$	−35.9167	−2.55249
$199$	−24.7748	−1.75624	−0.878121	−	0.478439i	$-0.841203\pi$
$199$	−24.7748	−1.75624	−0.878121	+	0.478439i	$0.841203\pi$
$200$	9.56887	0.676621
$201$	4.86140	0.342897
$202$	−38.8180	−2.73122
$203$	−31.7668	−2.22959
$204$	−1.40267	−0.0982068
$205$	−2.21176	−0.154476
$206$	13.4216	0.935129
$207$	0	0
$208$	−5.31102	−0.368253
$209$	10.4918	0.725731
$210$	−6.06179	−0.418303
$211$	9.92000	0.682921	0.341461	−	0.939896i	$-0.389078\pi$
$211$	9.92000	0.682921	0.341461	+	0.939896i	$0.389078\pi$
$212$	7.77240	0.533811
$213$	−1.17669	−0.0806253
$214$	33.7608	2.30784
$215$	17.0383	1.16200
$216$	9.70796	0.660543
$217$	−26.9064	−1.82653
$218$	−21.4867	−1.45526
$219$	3.11762	0.210669
$220$	34.2382	2.30834
$221$	1.58498	0.106617
$222$	−1.33087	−0.0893220
$223$	7.47155	0.500332	0.250166	−	0.968203i	$-0.419515\pi$
$223$	7.47155	0.500332	0.250166	+	0.968203i	$0.419515\pi$
$224$	−4.22210	−0.282101
$225$	5.98791	0.399194
$226$	−3.24071	−0.215569
$227$	−16.8976	−1.12153	−0.560767	−	0.827974i	$-0.689494\pi$
$227$	−16.8976	−1.12153	−0.560767	+	0.827974i	$0.689494\pi$
$228$	−2.85387	−0.189002
$229$	−16.7812	−1.10893	−0.554466	−	0.832206i	$-0.687078\pi$
$229$	−16.7812	−1.10893	−0.554466	+	0.832206i	$0.687078\pi$
$230$	0	0
$231$	7.54583	0.496479
$232$	−35.9029	−2.35714
$233$	18.1117	1.18654	0.593270	−	0.805004i	$-0.297837\pi$
$233$	18.1117	1.18654	0.593270	+	0.805004i	$0.297837\pi$
$234$	−11.0395	−0.721675
$235$	−3.59554	−0.234547
$236$	−26.1777	−1.70402
$237$	−3.91946	−0.254596
$238$	−9.84816	−0.638361
$239$	8.59588	0.556021	0.278011	−	0.960578i	$-0.410325\pi$
$239$	8.59588	0.556021	0.278011	+	0.960578i	$0.410325\pi$
$240$	−2.06253	−0.133136
$241$	−20.1867	−1.30034	−0.650168	−	0.759790i	$-0.725302\pi$
$241$	−20.1867	−1.30034	−0.650168	+	0.759790i	$0.725302\pi$
$242$	−37.8385	−2.43235
$243$	9.17972	0.588879
$244$	40.3380	2.58238
$245$	−16.1592	−1.03237
$246$	−1.13398	−0.0722998
$247$	3.22479	0.205188
$248$	−30.4097	−1.93102
$249$	−0.865182	−0.0548287
$250$	−29.3555	−1.85661
$251$	18.1661	1.14663	0.573317	−	0.819334i	$-0.305656\pi$
$251$	18.1661	1.14663	0.573317	+	0.819334i	$0.305656\pi$
$252$	45.3009	2.85369
$253$	0	0
$254$	25.5084	1.60054
$255$	0.615525	0.0385457
$256$	−30.8400	−1.92750
$257$	−1.78680	−0.111458	−0.0557288	−	0.998446i	$-0.517748\pi$
$257$	−1.78680	−0.111458	−0.0557288	+	0.998446i	$0.517748\pi$
$258$	8.73559	0.543854
$259$	−6.17105	−0.383451
$260$	10.5236	0.652644
$261$	−22.4670	−1.39067
$262$	46.5108	2.87345
$263$	18.5896	1.14629	0.573143	−	0.819456i	$-0.305724\pi$
$263$	18.5896	1.14629	0.573143	+	0.819456i	$0.305724\pi$
$264$	8.52831	0.524881
$265$	−3.41071	−0.209518
$266$	−20.0370	−1.22855
$267$	3.95550	0.242072
$268$	52.4392	3.20323
$269$	6.97889	0.425511	0.212755	−	0.977106i	$-0.431756\pi$
$269$	6.97889	0.425511	0.212755	+	0.977106i	$0.431756\pi$
$270$	−8.76861	−0.533641
$271$	−11.7711	−0.715044	−0.357522	−	0.933905i	$-0.616378\pi$
$271$	−11.7711	−0.715044	−0.357522	+	0.933905i	$0.616378\pi$
$272$	−3.35085	−0.203175
$273$	2.31931	0.140371
$274$	−25.7665	−1.55661
$275$	10.7589	0.648788
$276$	0	0
$277$	−22.6181	−1.35899	−0.679494	−	0.733681i	$-0.737800\pi$
$277$	−22.6181	−1.35899	−0.679494	+	0.733681i	$0.737800\pi$
$278$	−32.6714	−1.95950
$279$	−19.0295	−1.13927
$280$	−31.7674	−1.89846
$281$	−22.2086	−1.32486	−0.662428	−	0.749126i	$-0.730474\pi$
$281$	−22.2086	−1.32486	−0.662428	+	0.749126i	$0.730474\pi$
$282$	−1.84345	−0.109776
$283$	−16.5173	−0.981852	−0.490926	−	0.871201i	$-0.663342\pi$
$283$	−16.5173	−0.981852	−0.490926	+	0.871201i	$0.663342\pi$
$284$	−12.6927	−0.753176
$285$	1.25234	0.0741825
$286$	−19.8355	−1.17290
$287$	−5.25811	−0.310376
$288$	−2.98606	−0.175956
$289$	1.00000	0.0588235
$290$	32.4289	1.90429
$291$	−4.54319	−0.266326
$292$	33.6293	1.96801
$293$	−4.82228	−0.281721	−0.140860	−	0.990029i	$-0.544987\pi$
$293$	−4.82228	−0.281721	−0.140860	+	0.990029i	$0.544987\pi$
$294$	−8.28488	−0.483184
$295$	11.4874	0.668820
$296$	−6.97454	−0.405387
$297$	10.9153	0.633372
$298$	−42.0510	−2.43595
$299$	0	0
$300$	−2.92655	−0.168964
$301$	40.5058	2.33471
$302$	−13.0115	−0.748730
$303$	5.76785	0.331354
$304$	−6.81762	−0.391017
$305$	−17.7012	−1.01357
$306$	−6.96508	−0.398167
$307$	−15.6995	−0.896017	−0.448008	−	0.894029i	$-0.647867\pi$
$307$	−15.6995	−0.896017	−0.448008	+	0.894029i	$0.647867\pi$
$308$	81.3956	4.63795
$309$	−1.99428	−0.113451
$310$	27.4673	1.56004
$311$	−2.76784	−0.156950	−0.0784748	−	0.996916i	$-0.525005\pi$
$311$	−2.76784	−0.156950	−0.0784748	+	0.996916i	$0.525005\pi$
$312$	2.62129	0.148402
$313$	−18.5547	−1.04877	−0.524387	−	0.851480i	$-0.675705\pi$
$313$	−18.5547	−1.04877	−0.524387	+	0.851480i	$0.675705\pi$
$314$	−7.83894	−0.442377
$315$	−19.8791	−1.12006
$316$	−42.2786	−2.37836
$317$	23.3959	1.31404	0.657022	−	0.753872i	$-0.271816\pi$
$317$	23.3959	1.31404	0.657022	+	0.753872i	$0.271816\pi$
$318$	−1.74868	−0.0980612
$319$	−40.3681	−2.26018
$320$	15.7494	0.880418
$321$	−5.01642	−0.279989
$322$	0	0
$323$	2.03460	0.113208
$324$	30.5215	1.69564
$325$	3.30691	0.183434
$326$	−18.4337	−1.02095
$327$	3.19265	0.176554
$328$	−5.94272	−0.328132
$329$	−8.54781	−0.471256
$330$	−7.70310	−0.424042
$331$	−28.3543	−1.55849	−0.779245	−	0.626719i	$-0.784397\pi$
$331$	−28.3543	−1.55849	−0.779245	+	0.626719i	$0.784397\pi$
$332$	−9.33258	−0.512192
$333$	−4.36446	−0.239171
$334$	−48.7477	−2.66736
$335$	−23.0115	−1.25725
$336$	−4.90333	−0.267499
$337$	−5.81441	−0.316731	−0.158366	−	0.987381i	$-0.550622\pi$
$337$	−5.81441	−0.316731	−0.158366	+	0.987381i	$0.550622\pi$
$338$	25.4528	1.38445
$339$	0.481528	0.0261530
$340$	6.63957	0.360081
$341$	−34.1918	−1.85159
$342$	−14.1711	−0.766287
$343$	−10.0103	−0.540507
$344$	45.7797	2.46828
$345$	0	0
$346$	−33.4133	−1.79631
$347$	−9.45648	−0.507651	−0.253825	−	0.967250i	$-0.581689\pi$
$347$	−9.45648	−0.507651	−0.253825	+	0.967250i	$0.581689\pi$
$348$	10.9806	0.588620
$349$	24.8712	1.33133	0.665663	−	0.746253i	$-0.268149\pi$
$349$	24.8712	1.33133	0.665663	+	0.746253i	$0.268149\pi$
$350$	−20.5473	−1.09830
$351$	3.35498	0.179075
$352$	−5.36529	−0.285971
$353$	−25.5257	−1.35859	−0.679297	−	0.733864i	$-0.737715\pi$
$353$	−25.5257	−1.35859	−0.679297	+	0.733864i	$0.737715\pi$
$354$	5.88961	0.313029
$355$	5.56986	0.295618
$356$	42.6673	2.26136
$357$	1.46331	0.0774466
$358$	52.4865	2.77400
$359$	21.6024	1.14013	0.570066	−	0.821599i	$-0.306918\pi$
$359$	21.6024	1.14013	0.570066	+	0.821599i	$0.306918\pi$
$360$	−22.4674	−1.18414
$361$	−14.8604	−0.782127
$362$	39.2657	2.06376
$363$	5.62232	0.295095
$364$	25.0181	1.31130
$365$	−14.7573	−0.772432
$366$	−9.07549	−0.474383
$367$	16.0989	0.840357	0.420178	−	0.907442i	$-0.361967\pi$
$367$	16.0989	0.840357	0.420178	+	0.907442i	$0.361967\pi$
$368$	0	0
$369$	−3.71878	−0.193592
$370$	6.29968	0.327505
$371$	−8.10840	−0.420967
$372$	9.30053	0.482210
$373$	−4.21024	−0.217998	−0.108999	−	0.994042i	$-0.534765\pi$
$373$	−4.21024	−0.217998	−0.108999	+	0.994042i	$0.534765\pi$
$374$	−12.5147	−0.647119
$375$	4.36186	0.225245
$376$	−9.66076	−0.498216
$377$	−12.4077	−0.639030
$378$	−20.8459	−1.07220
$379$	30.1891	1.55071	0.775356	−	0.631525i	$-0.217571\pi$
$379$	30.1891	1.55071	0.775356	+	0.631525i	$0.217571\pi$
$380$	13.5088	0.692989
$381$	−3.79022	−0.194179
$382$	−41.7935	−2.13834
$383$	6.12931	0.313193	0.156597	−	0.987663i	$-0.449948\pi$
$383$	6.12931	0.313193	0.156597	+	0.987663i	$0.449948\pi$
$384$	7.32438	0.373771
$385$	−35.7183	−1.82037
$386$	−3.50649	−0.178476
$387$	28.6476	1.45624
$388$	−49.0066	−2.48793
$389$	12.9170	0.654917	0.327459	−	0.944866i	$-0.393808\pi$
$389$	12.9170	0.654917	0.327459	+	0.944866i	$0.393808\pi$
$390$	−2.36765	−0.119891
$391$	0	0
$392$	−43.4177	−2.19293
$393$	−6.91091	−0.348609
$394$	52.7767	2.65885
$395$	18.5528	0.933493
$396$	57.5668	2.89284
$397$	−19.8694	−0.997215	−0.498608	−	0.866828i	$-0.666155\pi$
$397$	−19.8694	−0.997215	−0.498608	+	0.866828i	$0.666155\pi$
$398$	60.1257	3.01383
$399$	2.97725	0.149049
$400$	−6.99123	−0.349561
$401$	−15.2269	−0.760395	−0.380197	−	0.924905i	$-0.624144\pi$
$401$	−15.2269	−0.760395	−0.380197	+	0.924905i	$0.624144\pi$
$402$	−11.7981	−0.588435
$403$	−10.5093	−0.523506
$404$	62.2169	3.09541
$405$	−13.3935	−0.665530
$406$	77.0945	3.82614
$407$	−7.84196	−0.388711
$408$	1.65384	0.0818771
$409$	31.8273	1.57376	0.786879	−	0.617107i	$-0.211695\pi$
$409$	31.8273	1.57376	0.786879	+	0.617107i	$0.211695\pi$
$410$	5.36770	0.265092
$411$	3.82857	0.188849
$412$	−21.5120	−1.05982
$413$	27.3093	1.34380
$414$	0	0
$415$	4.09535	0.201033
$416$	−1.64910	−0.0808536
$417$	4.85455	0.237728
$418$	−25.4623	−1.24540
$419$	−10.6826	−0.521877	−0.260939	−	0.965355i	$-0.584032\pi$
$419$	−10.6826	−0.521877	−0.260939	+	0.965355i	$0.584032\pi$
$420$	9.71575	0.474080
$421$	−0.807315	−0.0393461	−0.0196731	−	0.999806i	$-0.506263\pi$
$421$	−0.807315	−0.0393461	−0.0196731	+	0.999806i	$0.506263\pi$
$422$	−24.0747	−1.17194
$423$	−6.04541	−0.293938
$424$	−9.16414	−0.445050
$425$	2.08641	0.101206
$426$	2.85569	0.138359
$427$	−42.0818	−2.03648
$428$	−54.1114	−2.61557
$429$	2.94730	0.142297
$430$	−41.3500	−1.99407
$431$	16.3580	0.787936	0.393968	−	0.919124i	$-0.371102\pi$
$431$	16.3580	0.787936	0.393968	+	0.919124i	$0.371102\pi$
$432$	−7.09286	−0.341255
$433$	1.54291	0.0741474	0.0370737	−	0.999313i	$-0.488196\pi$
$433$	1.54291	0.0741474	0.0370737	+	0.999313i	$0.488196\pi$
$434$	65.2989	3.13445
$435$	−4.81852	−0.231031
$436$	34.4386	1.64931
$437$	0	0
$438$	−7.56612	−0.361523
$439$	26.4540	1.26258	0.631291	−	0.775546i	$-0.282525\pi$
$439$	26.4540	1.26258	0.631291	+	0.775546i	$0.282525\pi$
$440$	−40.3689	−1.92451
$441$	−27.1695	−1.29379
$442$	−3.84656	−0.182962
$443$	−33.1607	−1.57551	−0.787756	−	0.615988i	$-0.788757\pi$
$443$	−33.1607	−1.57551	−0.787756	+	0.615988i	$0.788757\pi$
$444$	2.13310	0.101232
$445$	−18.7234	−0.887574
$446$	−18.1326	−0.858604
$447$	6.24823	0.295531
$448$	37.4416	1.76895
$449$	−11.7119	−0.552718	−0.276359	−	0.961054i	$-0.589128\pi$
$449$	−11.7119	−0.552718	−0.276359	+	0.961054i	$0.589128\pi$
$450$	−14.5320	−0.685044
$451$	−6.68181	−0.314634
$452$	5.19417	0.244313
$453$	1.93335	0.0908366
$454$	41.0086	1.92463
$455$	−10.9785	−0.514680
$456$	3.36489	0.157575
$457$	24.6137	1.15138	0.575690	−	0.817668i	$-0.304733\pi$
$457$	24.6137	1.15138	0.575690	+	0.817668i	$0.304733\pi$
$458$	40.7261	1.90300
$459$	2.11674	0.0988007
$460$	0	0
$461$	25.2427	1.17567	0.587836	−	0.808980i	$-0.299980\pi$
$461$	25.2427	1.17567	0.587836	+	0.808980i	$0.299980\pi$
$462$	−18.3129	−0.851992
$463$	−4.71145	−0.218960	−0.109480	−	0.993989i	$-0.534918\pi$
$463$	−4.71145	−0.218960	−0.109480	+	0.993989i	$0.534918\pi$
$464$	26.2315	1.21777
$465$	−4.08128	−0.189265
$466$	−43.9552	−2.03618
$467$	34.7974	1.61023	0.805116	−	0.593117i	$-0.202103\pi$
$467$	34.7974	1.61023	0.805116	+	0.593117i	$0.202103\pi$
$468$	17.6940	0.817904
$469$	−54.7061	−2.52609
$470$	8.72597	0.402499
$471$	1.16477	0.0536696
$472$	30.8651	1.42068
$473$	51.4733	2.36674
$474$	9.51208	0.436904
$475$	4.24499	0.194774
$476$	15.7845	0.723481
$477$	−5.73464	−0.262571
$478$	−20.8612	−0.954170
$479$	5.52837	0.252598	0.126299	−	0.991992i	$-0.459690\pi$
$479$	5.52837	0.252598	0.126299	+	0.991992i	$0.459690\pi$
$480$	−0.640425	−0.0292313
$481$	−2.41033	−0.109902
$482$	48.9907	2.23147
$483$	0	0
$484$	60.6471	2.75668
$485$	21.5052	0.976502
$486$	−22.2782	−1.01056
$487$	−18.4533	−0.836201	−0.418100	−	0.908401i	$-0.637304\pi$
$487$	−18.4533	−0.836201	−0.418100	+	0.908401i	$0.637304\pi$
$488$	−47.5610	−2.15299
$489$	2.73902	0.123863
$490$	39.2166	1.77163
$491$	−10.8182	−0.488221	−0.244110	−	0.969747i	$-0.578496\pi$
$491$	−10.8182	−0.488221	−0.244110	+	0.969747i	$0.578496\pi$
$492$	1.81753	0.0819404
$493$	−7.82832	−0.352570
$494$	−7.82620	−0.352117
$495$	−25.2616	−1.13543
$496$	22.2180	0.997619
$497$	13.2414	0.593960
$498$	2.09970	0.0940898
$499$	27.8639	1.24736	0.623679	−	0.781681i	$-0.285637\pi$
$499$	27.8639	1.24736	0.623679	+	0.781681i	$0.285637\pi$
$500$	47.0507	2.10417
$501$	7.24328	0.323606
$502$	−44.0871	−1.96770
$503$	12.2148	0.544633	0.272316	−	0.962208i	$-0.412210\pi$
$503$	12.2148	0.544633	0.272316	+	0.962208i	$0.412210\pi$
$504$	−53.4126	−2.37918
$505$	−27.3022	−1.21493
$506$	0	0
$507$	−3.78196	−0.167963
$508$	−40.8845	−1.81396
$509$	15.6090	0.691858	0.345929	−	0.938261i	$-0.387564\pi$
$509$	15.6090	0.691858	0.345929	+	0.938261i	$0.387564\pi$
$510$	−1.49381	−0.0661470
$511$	−35.0831	−1.55198
$512$	34.2223	1.51243
$513$	4.30670	0.190146
$514$	4.33636	0.191269
$515$	9.43995	0.415974
$516$	−14.0013	−0.616372
$517$	−10.8623	−0.477721
$518$	14.9765	0.658028
$519$	4.96479	0.217930
$520$	−12.4079	−0.544124
$521$	40.1313	1.75819	0.879093	−	0.476651i	$-0.158149\pi$
$521$	40.1313	1.75819	0.879093	+	0.476651i	$0.158149\pi$
$522$	54.5249	2.38649
$523$	−28.8505	−1.26154	−0.630771	−	0.775969i	$-0.717261\pi$
$523$	−28.8505	−1.26154	−0.630771	+	0.775969i	$0.717261\pi$
$524$	−74.5469	−3.25660
$525$	3.05306	0.133246
$526$	−45.1149	−1.96710
$527$	−6.63057	−0.288832
$528$	−6.23098	−0.271168
$529$	0	0
$530$	8.27740	0.359547
$531$	19.3144	0.838175
$532$	32.1151	1.39237
$533$	−2.05375	−0.0889577
$534$	−9.59954	−0.415413
$535$	23.7453	1.02660
$536$	−61.8290	−2.67061
$537$	−7.79882	−0.336544
$538$	−16.9370	−0.730205
$539$	−48.8176	−2.10272
$540$	14.0542	0.604797
$541$	−17.5502	−0.754542	−0.377271	−	0.926103i	$-0.623137\pi$
$541$	−17.5502	−0.754542	−0.377271	+	0.926103i	$0.623137\pi$
$542$	28.5672	1.22706
$543$	−5.83438	−0.250377
$544$	−1.04045	−0.0446091
$545$	−15.1124	−0.647345
$546$	−5.62871	−0.240887
$547$	9.31045	0.398086	0.199043	−	0.979991i	$-0.436217\pi$
$547$	9.31045	0.398086	0.199043	+	0.979991i	$0.436217\pi$
$548$	41.2981	1.76417
$549$	−29.7623	−1.27022
$550$	−26.1107	−1.11336
$551$	−15.9275	−0.678533
$552$	0	0
$553$	44.1063	1.87559
$554$	54.8915	2.33212
$555$	−0.936051	−0.0397332
$556$	52.3653	2.22078
$557$	13.0830	0.554343	0.277171	−	0.960820i	$-0.410603\pi$
$557$	13.0830	0.554343	0.277171	+	0.960820i	$0.410603\pi$
$558$	46.1825	1.95506
$559$	15.8210	0.669158
$560$	23.2100	0.980800
$561$	1.85952	0.0785091
$562$	53.8978	2.27354
$563$	23.1322	0.974908	0.487454	−	0.873149i	$-0.337926\pi$
$563$	23.1322	0.974908	0.487454	+	0.873149i	$0.337926\pi$
$564$	2.95465	0.124413
$565$	−2.27932	−0.0958916
$566$	40.0857	1.68493
$567$	−31.8410	−1.33719
$568$	14.9655	0.627939
$569$	−10.4990	−0.440139	−0.220069	−	0.975484i	$-0.570628\pi$
$569$	−10.4990	−0.440139	−0.220069	+	0.975484i	$0.570628\pi$
$570$	−3.03930	−0.127302
$571$	38.4701	1.60992	0.804962	−	0.593327i	$-0.202186\pi$
$571$	38.4701	1.60992	0.804962	+	0.593327i	$0.202186\pi$
$572$	31.7921	1.32929
$573$	6.20998	0.259426
$574$	12.7608	0.532627
$575$	0	0
$576$	26.4805	1.10335
$577$	14.3871	0.598945	0.299472	−	0.954105i	$-0.403189\pi$
$577$	14.3871	0.598945	0.299472	+	0.954105i	$0.403189\pi$
$578$	−2.42689	−0.100945
$579$	0.521020	0.0216529
$580$	−51.9767	−2.15821
$581$	9.73602	0.403918
$582$	11.0258	0.457034
$583$	−10.3039	−0.426743
$584$	−39.6510	−1.64077
$585$	−7.76451	−0.321023
$586$	11.7031	0.483452
$587$	13.0100	0.536982	0.268491	−	0.963282i	$-0.413475\pi$
$587$	13.0100	0.536982	0.268491	+	0.963282i	$0.413475\pi$
$588$	13.2789	0.547613
$589$	−13.4905	−0.555868
$590$	−27.8785	−1.14774
$591$	−7.84194	−0.322574
$592$	5.09576	0.209434
$593$	35.7377	1.46757	0.733785	−	0.679382i	$-0.237752\pi$
$593$	35.7377	1.46757	0.733785	+	0.679382i	$0.237752\pi$
$594$	−26.4903	−1.08691
$595$	−6.92660	−0.283963
$596$	67.3987	2.76076
$597$	−8.93392	−0.365641
$598$	0	0
$599$	33.0762	1.35146	0.675728	−	0.737151i	$-0.263829\pi$
$599$	33.0762	1.35146	0.675728	+	0.737151i	$0.263829\pi$
$600$	3.45058	0.140869
$601$	22.4734	0.916708	0.458354	−	0.888770i	$-0.348439\pi$
$601$	22.4734	0.916708	0.458354	+	0.888770i	$0.348439\pi$
$602$	−98.3029	−4.00653
$603$	−38.6907	−1.57561
$604$	20.8547	0.848566
$605$	−26.6133	−1.08198
$606$	−13.9979	−0.568627
$607$	−14.0141	−0.568816	−0.284408	−	0.958703i	$-0.591797\pi$
$607$	−14.0141	−0.568816	−0.284408	+	0.958703i	$0.591797\pi$
$608$	−2.11690	−0.0858518
$609$	−11.4553	−0.464190
$610$	42.9589	1.73936
$611$	−3.33866	−0.135068
$612$	11.1635	0.451259
$613$	−23.3178	−0.941796	−0.470898	−	0.882188i	$-0.656070\pi$
$613$	−23.3178	−0.941796	−0.470898	+	0.882188i	$0.656070\pi$
$614$	38.1009	1.53763
$615$	−0.797571	−0.0321612
$616$	−95.9704	−3.86676
$617$	−11.8159	−0.475690	−0.237845	−	0.971303i	$-0.576441\pi$
$617$	−11.8159	−0.475690	−0.237845	+	0.971303i	$0.576441\pi$
$618$	4.83990	0.194689
$619$	28.5796	1.14871	0.574356	−	0.818605i	$-0.305252\pi$
$619$	28.5796	1.14871	0.574356	+	0.818605i	$0.305252\pi$
$620$	−44.0242	−1.76805
$621$	0	0
$622$	6.71723	0.269336
$623$	−44.5118	−1.78333
$624$	−1.91518	−0.0766684
$625$	−10.2149	−0.408595
$626$	45.0302	1.79977
$627$	3.78338	0.151094
$628$	12.5642	0.501364
$629$	−1.52074	−0.0606358
$630$	48.2443	1.92210
$631$	−32.2666	−1.28451	−0.642257	−	0.766490i	$-0.722002\pi$
$631$	−32.2666	−1.28451	−0.642257	+	0.766490i	$0.722002\pi$
$632$	49.8490	1.98289
$633$	3.57720	0.142181
$634$	−56.7792	−2.25499
$635$	17.9411	0.711970
$636$	2.80276	0.111137
$637$	−15.0048	−0.594510
$638$	97.9690	3.87863
$639$	9.36498	0.370473
$640$	−34.6700	−1.37045
$641$	29.8241	1.17798	0.588990	−	0.808141i	$-0.299526\pi$
$641$	29.8241	1.17798	0.588990	+	0.808141i	$0.299526\pi$
$642$	12.1743	0.480481
$643$	−19.0006	−0.749311	−0.374656	−	0.927164i	$-0.622239\pi$
$643$	−19.0006	−0.749311	−0.374656	+	0.927164i	$0.622239\pi$
$644$	0	0
$645$	6.14408	0.241923
$646$	−4.93774	−0.194273
$647$	−10.2754	−0.403968	−0.201984	−	0.979389i	$-0.564739\pi$
$647$	−10.2754	−0.403968	−0.201984	+	0.979389i	$0.564739\pi$
$648$	−35.9867	−1.41369
$649$	34.7037	1.36224
$650$	−8.02549	−0.314786
$651$	−9.70259	−0.380274
$652$	29.5453	1.15708
$653$	10.7833	0.421982	0.210991	−	0.977488i	$-0.432331\pi$
$653$	10.7833	0.421982	0.210991	+	0.977488i	$0.432331\pi$
$654$	−7.74820	−0.302978
$655$	32.7129	1.27820
$656$	4.34189	0.169522
$657$	−24.8124	−0.968024
$658$	20.7446	0.808708
$659$	−36.2715	−1.41294	−0.706469	−	0.707744i	$-0.749713\pi$
$659$	−36.2715	−1.41294	−0.706469	+	0.707744i	$0.749713\pi$
$660$	12.3464	0.480584
$661$	37.5971	1.46236	0.731178	−	0.682186i	$-0.238971\pi$
$661$	37.5971	1.46236	0.731178	+	0.682186i	$0.238971\pi$
$662$	68.8126	2.67448
$663$	0.571550	0.0221972
$664$	11.0037	0.427026
$665$	−14.0928	−0.546496
$666$	10.5921	0.410434
$667$	0	0
$668$	78.1322	3.02302
$669$	2.69427	0.104167
$670$	55.8463	2.15753
$671$	−53.4761	−2.06442
$672$	−1.52251	−0.0587320
$673$	18.8678	0.727302	0.363651	−	0.931535i	$-0.381530\pi$
$673$	18.8678	0.727302	0.363651	+	0.931535i	$0.381530\pi$
$674$	14.1109	0.543533
$675$	4.41637	0.169986
$676$	−40.7954	−1.56906
$677$	10.1835	0.391385	0.195693	−	0.980665i	$-0.437304\pi$
$677$	10.1835	0.391385	0.195693	+	0.980665i	$0.437304\pi$
$678$	−1.16861	−0.0448804
$679$	51.1252	1.96200
$680$	−7.82846	−0.300208
$681$	−6.09336	−0.233498
$682$	82.9796	3.17745
$683$	−1.41600	−0.0541816	−0.0270908	−	0.999633i	$-0.508624\pi$
$683$	−1.41600	−0.0541816	−0.0270908	+	0.999633i	$0.508624\pi$
$684$	22.7133	0.868465
$685$	−18.1226	−0.692427
$686$	24.2940	0.927548
$687$	−6.05137	−0.230874
$688$	−33.4477	−1.27518
$689$	−3.16704	−0.120654
$690$	0	0
$691$	−25.4333	−0.967529	−0.483764	−	0.875198i	$-0.660731\pi$
$691$	−25.4333	−0.967529	−0.483764	+	0.875198i	$0.660731\pi$
$692$	53.5544	2.03583
$693$	−60.0554	−2.28132
$694$	22.9498	0.871163
$695$	−22.9791	−0.871646
$696$	−12.9468	−0.490746
$697$	−1.29576	−0.0490803
$698$	−60.3596	−2.28465
$699$	6.53118	0.247032
$700$	32.9329	1.24475
$701$	29.0795	1.09832	0.549159	−	0.835718i	$-0.314948\pi$
$701$	29.0795	1.09832	0.549159	+	0.835718i	$0.314948\pi$
$702$	−8.14215	−0.307306
$703$	−3.09409	−0.116696
$704$	47.5795	1.79322
$705$	−1.29657	−0.0488316
$706$	61.9479	2.33144
$707$	−64.9065	−2.44106
$708$	−9.43979	−0.354769
$709$	27.6246	1.03746	0.518732	−	0.854937i	$-0.326404\pi$
$709$	27.6246	1.03746	0.518732	+	0.854937i	$0.326404\pi$
$710$	−13.5174	−0.507300
$711$	31.1940	1.16987
$712$	−50.3074	−1.88535
$713$	0	0
$714$	−3.55129	−0.132904
$715$	−13.9511	−0.521741
$716$	−84.1246	−3.14388
$717$	3.09971	0.115761
$718$	−52.4266	−1.95654
$719$	−21.2521	−0.792570	−0.396285	−	0.918128i	$-0.629701\pi$
$719$	−21.2521	−0.792570	−0.396285	+	0.918128i	$0.629701\pi$
$720$	16.4152	0.611758
$721$	22.4419	0.835782
$722$	36.0646	1.34218
$723$	−7.27940	−0.270724
$724$	−62.9346	−2.33894
$725$	−16.3330	−0.606594
$726$	−13.6447	−0.506404
$727$	−22.5534	−0.836460	−0.418230	−	0.908341i	$-0.637349\pi$
$727$	−22.5534	−0.836460	−0.418230	+	0.908341i	$0.637349\pi$
$728$	−29.4978	−1.09326
$729$	−20.2295	−0.749241
$730$	35.8143	1.32555
$731$	9.98186	0.369192
$732$	14.5461	0.537638
$733$	−9.21438	−0.340341	−0.170170	−	0.985415i	$-0.554432\pi$
$733$	−9.21438	−0.340341	−0.170170	+	0.985415i	$0.554432\pi$
$734$	−39.0703	−1.44211
$735$	−5.82708	−0.214935
$736$	0	0
$737$	−69.5186	−2.56075
$738$	9.02506	0.332217
$739$	25.7313	0.946542	0.473271	−	0.880917i	$-0.343073\pi$
$739$	25.7313	0.946542	0.473271	+	0.880917i	$0.343073\pi$
$740$	−10.0970	−0.371175
$741$	1.16287	0.0427192
$742$	19.6782	0.722409
$743$	24.2677	0.890297	0.445149	−	0.895457i	$-0.353151\pi$
$743$	24.2677	0.890297	0.445149	+	0.895457i	$0.353151\pi$
$744$	−10.9659	−0.402029
$745$	−29.5761	−1.08358
$746$	10.2178	0.374100
$747$	6.88578	0.251937
$748$	20.0584	0.733407
$749$	56.4506	2.06266
$750$	−10.5857	−0.386537
$751$	−13.0112	−0.474787	−0.237394	−	0.971414i	$-0.576293\pi$
$751$	−13.0112	−0.474787	−0.237394	+	0.971414i	$0.576293\pi$
$752$	7.05837	0.257392
$753$	6.55077	0.238723
$754$	30.1121	1.09662
$755$	−9.15153	−0.333058
$756$	33.4116	1.21517
$757$	−41.6052	−1.51217	−0.756083	−	0.654476i	$-0.772889\pi$
$757$	−41.6052	−1.51217	−0.756083	+	0.654476i	$0.772889\pi$
$758$	−73.2656	−2.66113
$759$	0	0
$760$	−15.9278	−0.577760
$761$	−31.0921	−1.12709	−0.563544	−	0.826086i	$-0.690562\pi$
$761$	−31.0921	−1.12709	−0.563544	+	0.826086i	$0.690562\pi$
$762$	9.19845	0.333225
$763$	−35.9273	−1.30066
$764$	66.9861	2.42347
$765$	−4.89882	−0.177117
$766$	−14.8752	−0.537461
$767$	10.6667	0.385151
$768$	−11.1210	−0.401296
$769$	−35.2823	−1.27231	−0.636155	−	0.771561i	$-0.719476\pi$
$769$	−35.2823	−1.27231	−0.636155	+	0.771561i	$0.719476\pi$
$770$	86.6842	3.12388
$771$	−0.644328	−0.0232049
$772$	5.62016	0.202274
$773$	11.3462	0.408093	0.204046	−	0.978961i	$-0.434591\pi$
$773$	11.3462	0.408093	0.204046	+	0.978961i	$0.434591\pi$
$774$	−69.5245	−2.49901
$775$	−13.8341	−0.496934
$776$	57.7818	2.07425
$777$	−2.22531	−0.0798325
$778$	−31.3481	−1.12388
$779$	−2.63634	−0.0944569
$780$	3.79485	0.135877
$781$	16.8268	0.602109
$782$	0	0
$783$	−16.5705	−0.592181
$784$	31.7220	1.13293
$785$	−5.51344	−0.196783
$786$	16.7720	0.598238
$787$	−19.2655	−0.686739	−0.343370	−	0.939200i	$-0.611568\pi$
$787$	−19.2655	−0.686739	−0.343370	+	0.939200i	$0.611568\pi$
$788$	−84.5898	−3.01339
$789$	6.70350	0.238651
$790$	−45.0256	−1.60194
$791$	−5.41871	−0.192667
$792$	−67.8748	−2.41183
$793$	−16.4366	−0.583681
$794$	48.2207	1.71129
$795$	−1.22992	−0.0436206
$796$	−96.3687	−3.41570
$797$	6.11077	0.216454	0.108227	−	0.994126i	$-0.465483\pi$
$797$	6.11077	0.216454	0.108227	+	0.994126i	$0.465483\pi$
$798$	−7.22544	−0.255778
$799$	−2.10644	−0.0745206
$800$	−2.17081	−0.0767497
$801$	−31.4808	−1.11232
$802$	36.9540	1.30489
$803$	−44.5823	−1.57328
$804$	18.9098	0.666897
$805$	0	0
$806$	25.5049	0.898372
$807$	2.51662	0.0885892
$808$	−73.3575	−2.58071
$809$	14.2319	0.500368	0.250184	−	0.968198i	$-0.419509\pi$
$809$	14.2319	0.500368	0.250184	+	0.968198i	$0.419509\pi$
$810$	32.5046	1.14210
$811$	35.4733	1.24564	0.622818	−	0.782367i	$-0.285988\pi$
$811$	35.4733	1.24564	0.622818	+	0.782367i	$0.285988\pi$
$812$	−123.566	−4.33632
$813$	−4.24471	−0.148869
$814$	19.0315	0.667055
$815$	−12.9652	−0.454150
$816$	−1.20833	−0.0423000
$817$	20.3091	0.710524
$818$	−77.2413	−2.70068
$819$	−18.4589	−0.645005
$820$	−8.60328	−0.300439
$821$	12.1221	0.423065	0.211533	−	0.977371i	$-0.432155\pi$
$821$	12.1221	0.423065	0.211533	+	0.977371i	$0.432155\pi$
$822$	−9.29150	−0.324078
$823$	3.46044	0.120623	0.0603117	−	0.998180i	$-0.480791\pi$
$823$	3.46044	0.120623	0.0603117	+	0.998180i	$0.480791\pi$
$824$	25.3640	0.883595
$825$	3.87972	0.135074
$826$	−66.2767	−2.30606
$827$	11.6763	0.406024	0.203012	−	0.979176i	$-0.434927\pi$
$827$	11.6763	0.406024	0.203012	+	0.979176i	$0.434927\pi$
$828$	0	0
$829$	−36.2430	−1.25877	−0.629386	−	0.777093i	$-0.716693\pi$
$829$	−36.2430	−1.25877	−0.629386	+	0.777093i	$0.716693\pi$
$830$	−9.93895	−0.344986
$831$	−8.15617	−0.282934
$832$	14.6242	0.507003
$833$	−9.46685	−0.328007
$834$	−11.7815	−0.407958
$835$	−34.2862	−1.18652
$836$	40.8107	1.41147
$837$	−14.0352	−0.485126
$838$	25.9254	0.895577
$839$	−4.66067	−0.160904	−0.0804521	−	0.996758i	$-0.525636\pi$
$839$	−4.66067	−0.160904	−0.0804521	+	0.996758i	$0.525636\pi$
$840$	−11.4555	−0.395251
$841$	32.2826	1.11319
$842$	1.95926	0.0675207
$843$	−8.00853	−0.275828
$844$	38.5867	1.32821
$845$	17.9020	0.615846
$846$	14.6715	0.504418
$847$	−63.2688	−2.17394
$848$	6.69552	0.229925
$849$	−5.95622	−0.204417
$850$	−5.06347	−0.173676
$851$	0	0
$852$	−4.57706	−0.156807
$853$	−7.47924	−0.256084	−0.128042	−	0.991769i	$-0.540869\pi$
$853$	−7.47924	−0.256084	−0.128042	+	0.991769i	$0.540869\pi$
$854$	102.128	3.49474
$855$	−9.96711	−0.340868
$856$	63.8006	2.18066
$857$	−30.4703	−1.04084	−0.520422	−	0.853909i	$-0.674226\pi$
$857$	−30.4703	−1.04084	−0.520422	+	0.853909i	$0.674226\pi$
$858$	−7.15277	−0.244192
$859$	−36.9564	−1.26094	−0.630468	−	0.776215i	$-0.717137\pi$
$859$	−36.9564	−1.26094	−0.630468	+	0.776215i	$0.717137\pi$
$860$	66.2753	2.25997
$861$	−1.89610	−0.0646188
$862$	−39.6989	−1.35215
$863$	−27.6149	−0.940024	−0.470012	−	0.882660i	$-0.655750\pi$
$863$	−27.6149	−0.940024	−0.470012	+	0.882660i	$0.655750\pi$
$864$	−2.20236	−0.0749260
$865$	−23.5009	−0.799054
$866$	−3.74446	−0.127242
$867$	0.360604	0.0122468
$868$	−104.660	−3.55240
$869$	56.0487	1.90132
$870$	11.6940	0.396464
$871$	−21.3675	−0.724010
$872$	−40.6052	−1.37507
$873$	36.1581	1.22377
$874$	0	0
$875$	−49.0847	−1.65936
$876$	12.1269	0.409729
$877$	48.6840	1.64394	0.821971	−	0.569529i	$-0.192874\pi$
$877$	48.6840	1.64394	0.821971	+	0.569529i	$0.192874\pi$
$878$	−64.2009	−2.16668
$879$	−1.73894	−0.0586529
$880$	29.4944	0.994256
$881$	−49.9816	−1.68392	−0.841961	−	0.539538i	$-0.818599\pi$
$881$	−49.9816	−1.68392	−0.841961	+	0.539538i	$0.818599\pi$
$882$	65.9374	2.22023
$883$	−18.1140	−0.609586	−0.304793	−	0.952419i	$-0.598587\pi$
$883$	−18.1140	−0.609586	−0.304793	+	0.952419i	$0.598587\pi$
$884$	6.16522	0.207359
$885$	4.14239	0.139245
$886$	80.4772	2.70369
$887$	−14.7548	−0.495416	−0.247708	−	0.968835i	$-0.579677\pi$
$887$	−14.7548	−0.495416	−0.247708	+	0.968835i	$0.579677\pi$
$888$	−2.51505	−0.0843996
$889$	42.6520	1.43050
$890$	45.4395	1.52314
$891$	−40.4624	−1.35554
$892$	29.0627	0.973091
$893$	−4.28576	−0.143417
$894$	−15.1638	−0.507152
$895$	36.9158	1.23396
$896$	−82.4224	−2.75354
$897$	0	0
$898$	28.4234	0.948502
$899$	51.9062	1.73117
$900$	23.2917	0.776389
$901$	−1.99816	−0.0665683
$902$	16.2160	0.539934
$903$	14.6066	0.486076
$904$	−6.12424	−0.203689
$905$	27.6171	0.918024
$906$	−4.69202	−0.155882
$907$	9.43285	0.313213	0.156606	−	0.987661i	$-0.449945\pi$
$907$	9.43285	0.313213	0.156606	+	0.987661i	$0.449945\pi$
$908$	−65.7281	−2.18126
$909$	−45.9049	−1.52257
$910$	26.6436	0.883226
$911$	16.4074	0.543603	0.271801	−	0.962353i	$-0.412381\pi$
$911$	16.4074	0.543603	0.271801	+	0.962353i	$0.412381\pi$
$912$	−2.45846	−0.0814079
$913$	12.3722	0.409460
$914$	−59.7347	−1.97585
$915$	−6.38315	−0.211020
$916$	−65.2752	−2.15675
$917$	77.7695	2.56818
$918$	−5.13708	−0.169549
$919$	24.8213	0.818778	0.409389	−	0.912360i	$-0.365742\pi$
$919$	24.8213	0.818778	0.409389	+	0.912360i	$0.365742\pi$
$920$	0	0
$921$	−5.66130	−0.186546
$922$	−61.2613	−2.01753
$923$	5.17194	0.170236
$924$	29.3516	0.965598
$925$	−3.17287	−0.104323
$926$	11.4342	0.375750
$927$	15.8720	0.521305
$928$	8.14500	0.267373
$929$	38.7314	1.27074	0.635369	−	0.772209i	$-0.280848\pi$
$929$	38.7314	1.27074	0.635369	+	0.772209i	$0.280848\pi$
$930$	9.90481	0.324792
$931$	−19.2612	−0.631261
$932$	70.4508	2.30769
$933$	−0.998094	−0.0326761
$934$	−84.4494	−2.76327
$935$	−8.80207	−0.287859
$936$	−20.8623	−0.681904
$937$	−52.8759	−1.72738	−0.863691	−	0.504022i	$-0.831853\pi$
$937$	−52.8759	−1.72738	−0.863691	+	0.504022i	$0.831853\pi$
$938$	132.766	4.33495
$939$	−6.69091	−0.218350
$940$	−13.9859	−0.456169
$941$	−45.4963	−1.48314	−0.741569	−	0.670877i	$-0.765918\pi$
$941$	−45.4963	−1.48314	−0.741569	+	0.670877i	$0.765918\pi$
$942$	−2.82676	−0.0921007
$943$	0	0
$944$	−22.5507	−0.733963
$945$	−14.6618	−0.476947
$946$	−124.920	−4.06150
$947$	54.9952	1.78710	0.893551	−	0.448961i	$-0.148206\pi$
$947$	54.9952	1.78710	0.893551	+	0.448961i	$0.148206\pi$
$948$	−15.2458	−0.495162
$949$	−13.7030	−0.444818
$950$	−10.3021	−0.334245
$951$	8.43666	0.273577
$952$	−18.6109	−0.603182
$953$	−42.6574	−1.38181	−0.690904	−	0.722947i	$-0.742787\pi$
$953$	−42.6574	−1.38181	−0.690904	+	0.722947i	$0.742787\pi$
$954$	13.9173	0.450590
$955$	−29.3950	−0.951201
$956$	33.4361	1.08140
$957$	−14.5569	−0.470559
$958$	−13.4167	−0.433475
$959$	−43.0834	−1.39124
$960$	5.67930	0.183299
$961$	12.9645	0.418210
$962$	5.84961	0.188599
$963$	39.9245	1.28655
$964$	−78.5217	−2.52901
$965$	−2.46625	−0.0793915
$966$	0	0
$967$	30.9418	0.995020	0.497510	−	0.867458i	$-0.334248\pi$
$967$	30.9418	0.995020	0.497510	+	0.867458i	$0.334248\pi$
$968$	−71.5066	−2.29831
$969$	0.733685	0.0235693
$970$	−52.1908	−1.67574
$971$	31.8284	1.02142	0.510711	−	0.859752i	$-0.329382\pi$
$971$	31.8284	1.02142	0.510711	+	0.859752i	$0.329382\pi$
$972$	35.7071	1.14531
$973$	−54.6290	−1.75133
$974$	44.7842	1.43498
$975$	1.19249	0.0381901
$976$	34.7491	1.11229
$977$	−5.11490	−0.163640	−0.0818201	−	0.996647i	$-0.526073\pi$
$977$	−5.11490	−0.163640	−0.0818201	+	0.996647i	$0.526073\pi$
$978$	−6.64728	−0.212557
$979$	−56.5640	−1.80779
$980$	−62.8558	−2.00786
$981$	−25.4095	−0.811263
$982$	26.2547	0.837820
$983$	−39.1149	−1.24757	−0.623786	−	0.781595i	$-0.714406\pi$
$983$	−39.1149	−1.24757	−0.623786	+	0.781595i	$0.714406\pi$
$984$	−2.14297	−0.0683155
$985$	37.1199	1.18274
$986$	18.9984	0.605034
$987$	−3.08238	−0.0981132
$988$	12.5437	0.399069
$989$	0	0
$990$	61.3071	1.94847
$991$	−39.2405	−1.24652	−0.623258	−	0.782017i	$-0.714191\pi$
$991$	−39.2405	−1.24652	−0.623258	+	0.782017i	$0.714191\pi$
$992$	6.89880	0.219037
$993$	−10.2247	−0.324470
$994$	−32.1355	−1.01928
$995$	42.2888	1.34064
$996$	−3.36537	−0.106636
$997$	0.0477605	0.00151259	0.000756294	−	1.00000i	$-0.499759\pi$
$997$	0.0477605	0.00151259	0.000756294		1.00000i	$0.499759\pi$
$998$	−67.6225	−2.14055
$999$	−3.21900	−0.101845

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8993.2.a.o.1.3	✓	28
23.22	odd	2		8993.2.a.p.1.3	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8993.2.a.o.1.3	✓	28	1.1	even	1	trivial
8993.2.a.p.1.3	yes	28	23.22	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8993 = 17 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8993.a (trivial)

\(f(q)\)	\(=\)	\(q-2.42689 q^{2} +0.360604 q^{3} +3.88978 q^{4} -1.70693 q^{5} -0.875146 q^{6} -4.05794 q^{7} -4.58629 q^{8} -2.86996 q^{9} +O(q^{10})\)	\(q-2.42689 q^{2} +0.360604 q^{3} +3.88978 q^{4} -1.70693 q^{5} -0.875146 q^{6} -4.05794 q^{7} -4.58629 q^{8} -2.86996 q^{9} +4.14252 q^{10} -5.15668 q^{11} +1.40267 q^{12} -1.58498 q^{13} +9.84816 q^{14} -0.615525 q^{15} +3.35085 q^{16} -1.00000 q^{17} +6.96508 q^{18} -2.03460 q^{19} -6.63957 q^{20} -1.46331 q^{21} +12.5147 q^{22} -1.65384 q^{24} -2.08641 q^{25} +3.84656 q^{26} -2.11674 q^{27} -15.7845 q^{28} +7.82832 q^{29} +1.49381 q^{30} +6.63057 q^{31} +1.04045 q^{32} -1.85952 q^{33} +2.42689 q^{34} +6.92660 q^{35} -11.1635 q^{36} +1.52074 q^{37} +4.93774 q^{38} -0.571550 q^{39} +7.82846 q^{40} +1.29576 q^{41} +3.55129 q^{42} -9.98186 q^{43} -20.0584 q^{44} +4.89882 q^{45} +2.10644 q^{47} +1.20833 q^{48} +9.46685 q^{49} +5.06347 q^{50} -0.360604 q^{51} -6.16522 q^{52} +1.99816 q^{53} +5.13708 q^{54} +8.80207 q^{55} +18.6109 q^{56} -0.733685 q^{57} -18.9984 q^{58} -6.72985 q^{59} -2.39426 q^{60} +10.3703 q^{61} -16.0917 q^{62} +11.6461 q^{63} -9.22676 q^{64} +2.70544 q^{65} +4.51285 q^{66} +13.4813 q^{67} -3.88978 q^{68} -16.8101 q^{70} -3.26310 q^{71} +13.1625 q^{72} +8.64554 q^{73} -3.69066 q^{74} -0.752367 q^{75} -7.91414 q^{76} +20.9255 q^{77} +1.38709 q^{78} -10.8691 q^{79} -5.71965 q^{80} +7.84659 q^{81} -3.14466 q^{82} -2.39925 q^{83} -5.69196 q^{84} +1.70693 q^{85} +24.2249 q^{86} +2.82293 q^{87} +23.6500 q^{88} +10.9691 q^{89} -11.8889 q^{90} +6.43174 q^{91} +2.39101 q^{93} -5.11210 q^{94} +3.47290 q^{95} +0.375192 q^{96} -12.5988 q^{97} -22.9750 q^{98} +14.7995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 4 q^{3} + 24 q^{4} - 4 q^{5} - 2 q^{6} - 20 q^{7} + 6 q^{8} + 24 q^{9}+O(q^{10}) \) 28 * q + 4 * q^3 + 24 * q^4 - 4 * q^5 - 2 * q^6 - 20 * q^7 + 6 * q^8 + 24 * q^9	\( 28 q + 4 q^{3} + 24 q^{4} - 4 q^{5} - 2 q^{6} - 20 q^{7} + 6 q^{8} + 24 q^{9} + 2 q^{10} - 8 q^{11} + 6 q^{12} - 8 q^{14} - 24 q^{15} + 32 q^{16} - 28 q^{17} - 6 q^{18} - 16 q^{19} - 12 q^{20} - 40 q^{21} - 20 q^{22} - 32 q^{24} + 20 q^{25} + 26 q^{26} + 4 q^{27} - 40 q^{28} + 12 q^{29} - 16 q^{30} + 4 q^{31} - 8 q^{32} - 4 q^{33} + 2 q^{36} - 36 q^{37} - 32 q^{38} + 24 q^{39} + 48 q^{40} + 16 q^{41} - 24 q^{42} - 48 q^{43} - 62 q^{44} + 60 q^{45} - 6 q^{48} + 24 q^{49} + 12 q^{50} - 4 q^{51} - 18 q^{52} - 24 q^{53} + 30 q^{54} + 12 q^{55} + 46 q^{56} - 24 q^{57} - 14 q^{58} - 12 q^{59} - 48 q^{60} - 36 q^{61} - 6 q^{62} + 20 q^{63} + 2 q^{64} - 16 q^{65} - 16 q^{66} - 68 q^{67} - 24 q^{68} - 4 q^{70} + 24 q^{71} + 30 q^{72} + 44 q^{73} - 68 q^{74} - 20 q^{75} - 40 q^{76} - 28 q^{77} - 38 q^{78} - 60 q^{79} - 28 q^{80} - 4 q^{81} - 34 q^{82} - 64 q^{83} - 48 q^{84} + 4 q^{85} + 96 q^{86} - 52 q^{87} - 60 q^{88} - 16 q^{89} - 126 q^{90} + 4 q^{91} - 24 q^{93} - 50 q^{94} + 16 q^{95} - 26 q^{96} - 104 q^{97} + 14 q^{98} - 40 q^{99}+O(q^{100}) \) 28 * q + 4 * q^3 + 24 * q^4 - 4 * q^5 - 2 * q^6 - 20 * q^7 + 6 * q^8 + 24 * q^9 + 2 * q^10 - 8 * q^11 + 6 * q^12 - 8 * q^14 - 24 * q^15 + 32 * q^16 - 28 * q^17 - 6 * q^18 - 16 * q^19 - 12 * q^20 - 40 * q^21 - 20 * q^22 - 32 * q^24 + 20 * q^25 + 26 * q^26 + 4 * q^27 - 40 * q^28 + 12 * q^29 - 16 * q^30 + 4 * q^31 - 8 * q^32 - 4 * q^33 + 2 * q^36 - 36 * q^37 - 32 * q^38 + 24 * q^39 + 48 * q^40 + 16 * q^41 - 24 * q^42 - 48 * q^43 - 62 * q^44 + 60 * q^45 - 6 * q^48 + 24 * q^49 + 12 * q^50 - 4 * q^51 - 18 * q^52 - 24 * q^53 + 30 * q^54 + 12 * q^55 + 46 * q^56 - 24 * q^57 - 14 * q^58 - 12 * q^59 - 48 * q^60 - 36 * q^61 - 6 * q^62 + 20 * q^63 + 2 * q^64 - 16 * q^65 - 16 * q^66 - 68 * q^67 - 24 * q^68 - 4 * q^70 + 24 * q^71 + 30 * q^72 + 44 * q^73 - 68 * q^74 - 20 * q^75 - 40 * q^76 - 28 * q^77 - 38 * q^78 - 60 * q^79 - 28 * q^80 - 4 * q^81 - 34 * q^82 - 64 * q^83 - 48 * q^84 + 4 * q^85 + 96 * q^86 - 52 * q^87 - 60 * q^88 - 16 * q^89 - 126 * q^90 + 4 * q^91 - 24 * q^93 - 50 * q^94 + 16 * q^95 - 26 * q^96 - 104 * q^97 + 14 * q^98 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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