LMFDB - Embedded newform 1127.2.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1127,2,Mod(1,1127)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1127, 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("1127.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1127 = 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1127.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:	$8.99914030780$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 10x^{5} - x^{4} + 29x^{3} + 9x^{2} - 24x - 11 $ x^7 - 10x^5 - x^4 + 29x^3 + 9x^2 - 24x - 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.17057$ of defining polynomial
Character	$\chi$	$=$	1127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.17057 q^{2} -3.34828 q^{3} -0.629767 q^{4} +0.135011 q^{5} -3.91940 q^{6} -3.07833 q^{8} +8.21099 q^{9} +0.158040 q^{10} +4.30224 q^{11} +2.10864 q^{12} +1.09269 q^{13} -0.452054 q^{15} -2.34386 q^{16} -4.41415 q^{17} +9.61154 q^{18} +2.11300 q^{19} -0.0850253 q^{20} +5.03607 q^{22} +1.00000 q^{23} +10.3071 q^{24} -4.98177 q^{25} +1.27907 q^{26} -17.4479 q^{27} -2.57118 q^{29} -0.529161 q^{30} +2.12825 q^{31} +3.41300 q^{32} -14.4051 q^{33} -5.16707 q^{34} -5.17101 q^{36} -10.7042 q^{37} +2.47341 q^{38} -3.65863 q^{39} -0.415607 q^{40} -4.16343 q^{41} +0.313612 q^{43} -2.70941 q^{44} +1.10857 q^{45} +1.17057 q^{46} +0.456666 q^{47} +7.84790 q^{48} -5.83151 q^{50} +14.7798 q^{51} -0.688138 q^{52} -1.47165 q^{53} -20.4240 q^{54} +0.580848 q^{55} -7.07490 q^{57} -3.00974 q^{58} -12.3961 q^{59} +0.284689 q^{60} +4.69307 q^{61} +2.49126 q^{62} +8.68287 q^{64} +0.147525 q^{65} -16.8622 q^{66} -8.88599 q^{67} +2.77989 q^{68} -3.34828 q^{69} -5.37228 q^{71} -25.2761 q^{72} -12.6459 q^{73} -12.5300 q^{74} +16.6804 q^{75} -1.33069 q^{76} -4.28268 q^{78} -3.64944 q^{79} -0.316446 q^{80} +33.7874 q^{81} -4.87358 q^{82} +9.83902 q^{83} -0.595958 q^{85} +0.367104 q^{86} +8.60903 q^{87} -13.2437 q^{88} -0.699096 q^{89} +1.29766 q^{90} -0.629767 q^{92} -7.12597 q^{93} +0.534559 q^{94} +0.285277 q^{95} -11.4277 q^{96} +8.12610 q^{97} +35.3256 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 5 q^{3} + 6 q^{4} - 4 q^{5} - 6 q^{6} - 3 q^{8} + 4 q^{9} + 2 q^{10} + 4 q^{11} - 9 q^{12} - 14 q^{13} - 3 q^{15} - 8 q^{16} - 4 q^{17} + 19 q^{18} - 9 q^{19} - 12 q^{20} - 10 q^{22} + 7 q^{23} + 4 q^{24}+ \cdots + 18 q^{99}+O(q^{100}) $ 7 * q - 5 * q^3 + 6 * q^4 - 4 * q^5 - 6 * q^6 - 3 * q^8 + 4 * q^9 + 2 * q^10 + 4 * q^11 - 9 * q^12 - 14 * q^13 - 3 * q^15 - 8 * q^16 - 4 * q^17 + 19 * q^18 - 9 * q^19 - 12 * q^20 - 10 * q^22 + 7 * q^23 + 4 * q^24 + 3 * q^25 - 14 * q^26 - 20 * q^27 - 5 * q^29 - 17 * q^30 - 10 * q^31 + 19 * q^32 - 3 * q^33 - 20 * q^34 + 4 * q^36 - 5 * q^37 + 17 * q^38 + 18 * q^39 - 4 * q^40 - 23 * q^41 - 11 * q^43 - 34 * q^44 - 6 * q^45 - 4 * q^47 - 11 * q^48 - 9 * q^50 + 7 * q^51 - 12 * q^52 + 2 * q^53 - 15 * q^54 - 17 * q^55 - 24 * q^57 - 7 * q^58 - 7 * q^59 + 32 * q^60 - 12 * q^61 + 4 * q^62 - 9 * q^64 + 9 * q^65 + 27 * q^66 + 2 * q^67 + 12 * q^68 - 5 * q^69 - 28 * q^71 - 20 * q^72 - 59 * q^73 + 2 * q^74 + 22 * q^75 - q^76 - 17 * q^78 - 4 * q^79 + 22 * q^80 + 11 * q^81 - 54 * q^82 + 9 * q^83 + 9 * q^85 - 15 * q^86 - 11 * q^87 - 33 * q^88 - q^89 - 2 * q^90 + 6 * q^92 + 15 * q^93 + 14 * q^94 - 16 * q^95 - 20 * q^96 - 35 * q^97 + 18 * 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 $

$2$	1.17057	0.827718	0.413859	−	0.910341i	$-0.364181\pi$
$2$	1.17057	0.827718	0.413859	+	0.910341i	$0.364181\pi$
$3$	−3.34828	−1.93313	−0.966566	−	0.256419i	$-0.917457\pi$
$3$	−3.34828	−1.93313	−0.966566	+	0.256419i	$0.917457\pi$
$4$	−0.629767	−0.314883
$5$	0.135011	0.0603787	0.0301893	−	0.999544i	$-0.490389\pi$
$5$	0.135011	0.0603787	0.0301893	+	0.999544i	$0.490389\pi$
$6$	−3.91940	−1.60009
$7$	0	0
$7$	0	0
$8$	−3.07833	−1.08835
$9$	8.21099	2.73700
$10$	0.158040	0.0499765
$11$	4.30224	1.29717	0.648586	−	0.761141i	$-0.275360\pi$
$11$	4.30224	1.29717	0.648586	+	0.761141i	$0.275360\pi$
$12$	2.10864	0.608711
$13$	1.09269	0.303057	0.151528	−	0.988453i	$-0.451580\pi$
$13$	1.09269	0.303057	0.151528	+	0.988453i	$0.451580\pi$
$14$	0	0
$15$	−0.452054	−0.116720
$16$	−2.34386	−0.585965
$17$	−4.41415	−1.07059	−0.535295	−	0.844665i	$-0.679799\pi$
$17$	−4.41415	−1.07059	−0.535295	+	0.844665i	$0.679799\pi$
$18$	9.61154	2.26546
$19$	2.11300	0.484754	0.242377	−	0.970182i	$-0.422073\pi$
$19$	2.11300	0.484754	0.242377	+	0.970182i	$0.422073\pi$
$20$	−0.0850253	−0.0190122
$21$	0	0
$22$	5.03607	1.07369
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	10.3071	2.10393
$25$	−4.98177	−0.996354
$26$	1.27907	0.250846
$27$	−17.4479	−3.35785
$28$	0	0
$29$	−2.57118	−0.477456	−0.238728	−	0.971086i	$-0.576730\pi$
$29$	−2.57118	−0.477456	−0.238728	+	0.971086i	$0.576730\pi$
$30$	−0.529161	−0.0966111
$31$	2.12825	0.382244	0.191122	−	0.981566i	$-0.438787\pi$
$31$	2.12825	0.382244	0.191122	+	0.981566i	$0.438787\pi$
$32$	3.41300	0.603339
$33$	−14.4051	−2.50761
$34$	−5.16707	−0.886146
$35$	0	0
$36$	−5.17101	−0.861835
$37$	−10.7042	−1.75976	−0.879882	−	0.475192i	$-0.842379\pi$
$37$	−10.7042	−1.75976	−0.879882	+	0.475192i	$0.842379\pi$
$38$	2.47341	0.401240
$39$	−3.65863	−0.585849
$40$	−0.415607	−0.0657133
$41$	−4.16343	−0.650218	−0.325109	−	0.945677i	$-0.605401\pi$
$41$	−4.16343	−0.650218	−0.325109	+	0.945677i	$0.605401\pi$
$42$	0	0
$43$	0.313612	0.0478254	0.0239127	−	0.999714i	$-0.492388\pi$
$43$	0.313612	0.0478254	0.0239127	+	0.999714i	$0.492388\pi$
$44$	−2.70941	−0.408458
$45$	1.10857	0.165256
$46$	1.17057	0.172591
$47$	0.456666	0.0666115	0.0333058	−	0.999445i	$-0.489396\pi$
$47$	0.456666	0.0666115	0.0333058	+	0.999445i	$0.489396\pi$
$48$	7.84790	1.13275
$49$	0	0
$50$	−5.83151	−0.824700
$51$	14.7798	2.06959
$52$	−0.688138	−0.0954276
$53$	−1.47165	−0.202147	−0.101073	−	0.994879i	$-0.532228\pi$
$53$	−1.47165	−0.202147	−0.101073	+	0.994879i	$0.532228\pi$
$54$	−20.4240	−2.77935
$55$	0.580848	0.0783216
$56$	0	0
$57$	−7.07490	−0.937094
$58$	−3.00974	−0.395199
$59$	−12.3961	−1.61384	−0.806920	−	0.590661i	$-0.798867\pi$
$59$	−12.3961	−1.61384	−0.806920	+	0.590661i	$0.798867\pi$
$60$	0.284689	0.0367532
$61$	4.69307	0.600886	0.300443	−	0.953800i	$-0.402866\pi$
$61$	4.69307	0.600886	0.300443	+	0.953800i	$0.402866\pi$
$62$	2.49126	0.316390
$63$	0	0
$64$	8.68287	1.08536
$65$	0.147525	0.0182982
$66$	−16.8622	−2.07559
$67$	−8.88599	−1.08560	−0.542798	−	0.839863i	$-0.682635\pi$
$67$	−8.88599	−1.08560	−0.542798	+	0.839863i	$0.682635\pi$
$68$	2.77989	0.337111
$69$	−3.34828	−0.403086
$70$	0	0
$71$	−5.37228	−0.637572	−0.318786	−	0.947827i	$-0.603275\pi$
$71$	−5.37228	−0.637572	−0.318786	+	0.947827i	$0.603275\pi$
$72$	−25.2761	−2.97882
$73$	−12.6459	−1.48010	−0.740048	−	0.672554i	$-0.765197\pi$
$73$	−12.6459	−1.48010	−0.740048	+	0.672554i	$0.765197\pi$
$74$	−12.5300	−1.45659
$75$	16.6804	1.92608
$76$	−1.33069	−0.152641
$77$	0	0
$78$	−4.28268	−0.484918
$79$	−3.64944	−0.410594	−0.205297	−	0.978700i	$-0.565816\pi$
$79$	−3.64944	−0.410594	−0.205297	+	0.978700i	$0.565816\pi$
$80$	−0.316446	−0.0353798
$81$	33.7874	3.75416
$82$	−4.87358	−0.538197
$83$	9.83902	1.07997	0.539986	−	0.841674i	$-0.318429\pi$
$83$	9.83902	1.07997	0.539986	+	0.841674i	$0.318429\pi$
$84$	0	0
$85$	−0.595958	−0.0646408
$86$	0.367104	0.0395859
$87$	8.60903	0.922985
$88$	−13.2437	−1.41178
$89$	−0.699096	−0.0741040	−0.0370520	−	0.999313i	$-0.511797\pi$
$89$	−0.699096	−0.0741040	−0.0370520	+	0.999313i	$0.511797\pi$
$90$	1.29766	0.136786
$91$	0	0
$92$	−0.629767	−0.0656577
$93$	−7.12597	−0.738928
$94$	0.534559	0.0551355
$95$	0.285277	0.0292688
$96$	−11.4277	−1.16633
$97$	8.12610	0.825081	0.412540	−	0.910939i	$-0.364642\pi$
$97$	8.12610	0.825081	0.412540	+	0.910939i	$0.364642\pi$
$98$	0	0
$99$	35.3256	3.55036
$100$	3.13736	0.313736
$101$	5.56455	0.553693	0.276847	−	0.960914i	$-0.410711\pi$
$101$	5.56455	0.553693	0.276847	+	0.960914i	$0.410711\pi$
$102$	17.3008	1.71304
$103$	0.608326	0.0599402	0.0299701	−	0.999551i	$-0.490459\pi$
$103$	0.608326	0.0599402	0.0299701	+	0.999551i	$0.490459\pi$
$104$	−3.36365	−0.329833
$105$	0	0
$106$	−1.72267	−0.167321
$107$	1.00450	0.0971091	0.0485546	−	0.998821i	$-0.484539\pi$
$107$	1.00450	0.0971091	0.0485546	+	0.998821i	$0.484539\pi$
$108$	10.9881	1.05733
$109$	−12.2530	−1.17363	−0.586813	−	0.809723i	$-0.699618\pi$
$109$	−12.2530	−1.17363	−0.586813	+	0.809723i	$0.699618\pi$
$110$	0.679923	0.0648282
$111$	35.8408	3.40186
$112$	0	0
$113$	−10.4710	−0.985029	−0.492514	−	0.870304i	$-0.663922\pi$
$113$	−10.4710	−0.985029	−0.492514	+	0.870304i	$0.663922\pi$
$114$	−8.28167	−0.775649
$115$	0.135011	0.0125898
$116$	1.61924	0.150343
$117$	8.97205	0.829466
$118$	−14.5105	−1.33580
$119$	0	0
$120$	1.39157	0.127032
$121$	7.50923	0.682657
$122$	5.49356	0.497364
$123$	13.9403	1.25696
$124$	−1.34030	−0.120362
$125$	−1.34765	−0.120537
$126$	0	0
$127$	−7.02640	−0.623492	−0.311746	−	0.950165i	$-0.600914\pi$
$127$	−7.02640	−0.623492	−0.311746	+	0.950165i	$0.600914\pi$
$128$	3.33791	0.295032
$129$	−1.05006	−0.0924527
$130$	0.172688	0.0151457
$131$	11.5317	1.00753	0.503763	−	0.863842i	$-0.331948\pi$
$131$	11.5317	1.00753	0.503763	+	0.863842i	$0.331948\pi$
$132$	9.07186	0.789604
$133$	0	0
$134$	−10.4017	−0.898568
$135$	−2.35565	−0.202742
$136$	13.5882	1.16518
$137$	−10.6639	−0.911080	−0.455540	−	0.890215i	$-0.650554\pi$
$137$	−10.6639	−0.911080	−0.455540	+	0.890215i	$0.650554\pi$
$138$	−3.91940	−0.333641
$139$	17.4719	1.48195	0.740975	−	0.671532i	$-0.234364\pi$
$139$	17.4719	1.48195	0.740975	+	0.671532i	$0.234364\pi$
$140$	0	0
$141$	−1.52905	−0.128769
$142$	−6.28862	−0.527729
$143$	4.70100	0.393117
$144$	−19.2454	−1.60378
$145$	−0.347137	−0.0288282
$146$	−14.8030	−1.22510
$147$	0	0
$148$	6.74117	0.554121
$149$	−18.0975	−1.48261	−0.741303	−	0.671171i	$-0.765792\pi$
$149$	−18.0975	−1.48261	−0.741303	+	0.671171i	$0.765792\pi$
$150$	19.5255	1.59425
$151$	13.9238	1.13310	0.566550	−	0.824027i	$-0.308278\pi$
$151$	13.9238	1.13310	0.566550	+	0.824027i	$0.308278\pi$
$152$	−6.50449	−0.527584
$153$	−36.2446	−2.93020
$154$	0	0
$155$	0.287336	0.0230794
$156$	2.30408	0.184474
$157$	−10.6893	−0.853102	−0.426551	−	0.904464i	$-0.640272\pi$
$157$	−10.6893	−0.853102	−0.426551	+	0.904464i	$0.640272\pi$
$158$	−4.27192	−0.339856
$159$	4.92750	0.390777
$160$	0.460792	0.0364288
$161$	0	0
$162$	39.5505	3.10738
$163$	2.69426	0.211030	0.105515	−	0.994418i	$-0.466351\pi$
$163$	2.69426	0.211030	0.105515	+	0.994418i	$0.466351\pi$
$164$	2.62199	0.204743
$165$	−1.94484	−0.151406
$166$	11.5173	0.893913
$167$	−23.2708	−1.80075	−0.900373	−	0.435118i	$-0.856707\pi$
$167$	−23.2708	−1.80075	−0.900373	+	0.435118i	$0.856707\pi$
$168$	0	0
$169$	−11.8060	−0.908156
$170$	−0.697611	−0.0535043
$171$	17.3498	1.32677
$172$	−0.197502	−0.0150594
$173$	−11.5097	−0.875063	−0.437532	−	0.899203i	$-0.644147\pi$
$173$	−11.5097	−0.875063	−0.437532	+	0.899203i	$0.644147\pi$
$174$	10.0775	0.763971
$175$	0	0
$176$	−10.0838	−0.760098
$177$	41.5058	3.11977
$178$	−0.818340	−0.0613372
$179$	−6.86421	−0.513055	−0.256528	−	0.966537i	$-0.582578\pi$
$179$	−6.86421	−0.513055	−0.256528	+	0.966537i	$0.582578\pi$
$180$	−0.698143	−0.0520365
$181$	−15.6248	−1.16138	−0.580690	−	0.814125i	$-0.697217\pi$
$181$	−15.6248	−1.16138	−0.580690	+	0.814125i	$0.697217\pi$
$182$	0	0
$183$	−15.7137	−1.16159
$184$	−3.07833	−0.226937
$185$	−1.44519	−0.106252
$186$	−8.34144	−0.611624
$187$	−18.9907	−1.38874
$188$	−0.287593	−0.0209749
$189$	0	0
$190$	0.333937	0.0242263
$191$	2.20523	0.159565	0.0797826	−	0.996812i	$-0.474577\pi$
$191$	2.20523	0.159565	0.0797826	+	0.996812i	$0.474577\pi$
$192$	−29.0727	−2.09814
$193$	−4.51678	−0.325125	−0.162562	−	0.986698i	$-0.551976\pi$
$193$	−4.51678	−0.325125	−0.162562	+	0.986698i	$0.551976\pi$
$194$	9.51217	0.682934
$195$	−0.493954	−0.0353728
$196$	0	0
$197$	21.2845	1.51646	0.758230	−	0.651987i	$-0.226064\pi$
$197$	21.2845	1.51646	0.758230	+	0.651987i	$0.226064\pi$
$198$	41.3511	2.93870
$199$	−12.4057	−0.879414	−0.439707	−	0.898141i	$-0.644918\pi$
$199$	−12.4057	−0.879414	−0.439707	+	0.898141i	$0.644918\pi$
$200$	15.3355	1.08438
$201$	29.7528	2.09860
$202$	6.51369	0.458302
$203$	0	0
$204$	−9.30785	−0.651680
$205$	−0.562108	−0.0392593
$206$	0.712088	0.0496135
$207$	8.21099	0.570704
$208$	−2.56111	−0.177581
$209$	9.09060	0.628810
$210$	0	0
$211$	17.5494	1.20815	0.604076	−	0.796927i	$-0.293542\pi$
$211$	17.5494	1.20815	0.604076	+	0.796927i	$0.293542\pi$
$212$	0.926797	0.0636527
$213$	17.9879	1.23251
$214$	1.17584	0.0803789
$215$	0.0423410	0.00288763
$216$	53.7102	3.65452
$217$	0	0
$218$	−14.3430	−0.971431
$219$	42.3422	2.86122
$220$	−0.365799	−0.0246622
$221$	−4.82329	−0.324450
$222$	41.9541	2.81578
$223$	−5.67894	−0.380290	−0.190145	−	0.981756i	$-0.560896\pi$
$223$	−5.67894	−0.380290	−0.190145	+	0.981756i	$0.560896\pi$
$224$	0	0
$225$	−40.9053	−2.72702
$226$	−12.2570	−0.815326
$227$	19.6293	1.30284	0.651422	−	0.758716i	$-0.274173\pi$
$227$	19.6293	1.30284	0.651422	+	0.758716i	$0.274173\pi$
$228$	4.45554	0.295075
$229$	16.1934	1.07009	0.535045	−	0.844824i	$-0.320295\pi$
$229$	16.1934	1.07009	0.535045	+	0.844824i	$0.320295\pi$
$230$	0.158040	0.0104208
$231$	0	0
$232$	7.91493	0.519640
$233$	−6.95138	−0.455400	−0.227700	−	0.973731i	$-0.573121\pi$
$233$	−6.95138	−0.455400	−0.227700	+	0.973731i	$0.573121\pi$
$234$	10.5024	0.686564
$235$	0.0616548	0.00402192
$236$	7.80668	0.508172
$237$	12.2194	0.793732
$238$	0	0
$239$	6.20904	0.401629	0.200815	−	0.979629i	$-0.435641\pi$
$239$	6.20904	0.401629	0.200815	+	0.979629i	$0.435641\pi$
$240$	1.05955	0.0683938
$241$	−19.2279	−1.23858	−0.619290	−	0.785162i	$-0.712580\pi$
$241$	−19.2279	−1.23858	−0.619290	+	0.785162i	$0.712580\pi$
$242$	8.79008	0.565048
$243$	−60.7862	−3.89944
$244$	−2.95554	−0.189209
$245$	0	0
$246$	16.3181	1.04041
$247$	2.30884	0.146908
$248$	−6.55143	−0.416016
$249$	−32.9438	−2.08773
$250$	−1.57751	−0.0997708
$251$	−18.9705	−1.19740	−0.598702	−	0.800972i	$-0.704317\pi$
$251$	−18.9705	−1.19740	−0.598702	+	0.800972i	$0.704317\pi$
$252$	0	0
$253$	4.30224	0.270479
$254$	−8.22489	−0.516076
$255$	1.99544	0.124959
$256$	−13.4585	−0.841156
$257$	−7.49033	−0.467234	−0.233617	−	0.972329i	$-0.575056\pi$
$257$	−7.49033	−0.467234	−0.233617	+	0.972329i	$0.575056\pi$
$258$	−1.22917	−0.0765247
$259$	0	0
$260$	−0.0929061	−0.00576179
$261$	−21.1119	−1.30680
$262$	13.4986	0.833948
$263$	28.5655	1.76142	0.880712	−	0.473651i	$-0.157064\pi$
$263$	28.5655	1.76142	0.880712	+	0.473651i	$0.157064\pi$
$264$	44.3436	2.72916
$265$	−0.198689	−0.0122054
$266$	0	0
$267$	2.34077	0.143253
$268$	5.59611	0.341837
$269$	−11.3097	−0.689562	−0.344781	−	0.938683i	$-0.612047\pi$
$269$	−11.3097	−0.689562	−0.344781	+	0.938683i	$0.612047\pi$
$270$	−2.75745	−0.167813
$271$	−10.7327	−0.651967	−0.325984	−	0.945375i	$-0.605695\pi$
$271$	−10.7327	−0.651967	−0.325984	+	0.945375i	$0.605695\pi$
$272$	10.3462	0.627328
$273$	0	0
$274$	−12.4829	−0.754117
$275$	−21.4328	−1.29244
$276$	2.10864	0.126925
$277$	14.3115	0.859896	0.429948	−	0.902854i	$-0.358532\pi$
$277$	14.3115	0.859896	0.429948	+	0.902854i	$0.358532\pi$
$278$	20.4521	1.22664
$279$	17.4750	1.04620
$280$	0	0
$281$	24.7009	1.47353	0.736766	−	0.676148i	$-0.236352\pi$
$281$	24.7009	1.47353	0.736766	+	0.676148i	$0.236352\pi$
$282$	−1.78985	−0.106584
$283$	2.68343	0.159513	0.0797567	−	0.996814i	$-0.474586\pi$
$283$	2.68343	0.159513	0.0797567	+	0.996814i	$0.474586\pi$
$284$	3.38328	0.200761
$285$	−0.955189	−0.0565805
$286$	5.50285	0.325390
$287$	0	0
$288$	28.0241	1.65134
$289$	2.48475	0.146162
$290$	−0.406348	−0.0238616
$291$	−27.2085	−1.59499
$292$	7.96400	0.466058
$293$	5.42744	0.317074	0.158537	−	0.987353i	$-0.449322\pi$
$293$	5.42744	0.317074	0.158537	+	0.987353i	$0.449322\pi$
$294$	0	0
$295$	−1.67361	−0.0974415
$296$	32.9511	1.91524
$297$	−75.0649	−4.35571
$298$	−21.1844	−1.22718
$299$	1.09269	0.0631917
$300$	−10.5048	−0.606492
$301$	0	0
$302$	16.2987	0.937887
$303$	−18.6317	−1.07036
$304$	−4.95256	−0.284049
$305$	0.633615	0.0362807
$306$	−42.4268	−2.42538
$307$	4.16822	0.237893	0.118947	−	0.992901i	$-0.462048\pi$
$307$	4.16822	0.237893	0.118947	+	0.992901i	$0.462048\pi$
$308$	0	0
$309$	−2.03685	−0.115872
$310$	0.336347	0.0191032
$311$	−3.34567	−0.189716	−0.0948579	−	0.995491i	$-0.530240\pi$
$311$	−3.34567	−0.189716	−0.0948579	+	0.995491i	$0.530240\pi$
$312$	11.2624	0.637610
$313$	−20.4088	−1.15358	−0.576788	−	0.816894i	$-0.695694\pi$
$313$	−20.4088	−1.15358	−0.576788	+	0.816894i	$0.695694\pi$
$314$	−12.5126	−0.706127
$315$	0	0
$316$	2.29830	0.129289
$317$	14.1307	0.793660	0.396830	−	0.917892i	$-0.370110\pi$
$317$	14.1307	0.793660	0.396830	+	0.917892i	$0.370110\pi$
$318$	5.76799	0.323453
$319$	−11.0618	−0.619343
$320$	1.17228	0.0655325
$321$	−3.36336	−0.187725
$322$	0	0
$323$	−9.32709	−0.518973
$324$	−21.2782	−1.18212
$325$	−5.44352	−0.301952
$326$	3.15381	0.174674
$327$	41.0265	2.26877
$328$	12.8164	0.707666
$329$	0	0
$330$	−2.27658	−0.125321
$331$	15.8790	0.872790	0.436395	−	0.899755i	$-0.356255\pi$
$331$	15.8790	0.872790	0.436395	+	0.899755i	$0.356255\pi$
$332$	−6.19629	−0.340066
$333$	−87.8923	−4.81647
$334$	−27.2401	−1.49051
$335$	−1.19971	−0.0655469
$336$	0	0
$337$	0.919792	0.0501043	0.0250521	−	0.999686i	$-0.492025\pi$
$337$	0.919792	0.0501043	0.0250521	+	0.999686i	$0.492025\pi$
$338$	−13.8198	−0.751697
$339$	35.0599	1.90419
$340$	0.375315	0.0203543
$341$	9.15622	0.495837
$342$	20.3091	1.09819
$343$	0	0
$344$	−0.965399	−0.0520508
$345$	−0.452054	−0.0243378
$346$	−13.4729	−0.724305
$347$	19.1019	1.02544	0.512721	−	0.858555i	$-0.328637\pi$
$347$	19.1019	1.02544	0.512721	+	0.858555i	$0.328637\pi$
$348$	−5.42169	−0.290633
$349$	22.5135	1.20512	0.602559	−	0.798074i	$-0.294148\pi$
$349$	22.5135	1.20512	0.602559	+	0.798074i	$0.294148\pi$
$350$	0	0
$351$	−19.0651	−1.01762
$352$	14.6835	0.782635
$353$	9.49999	0.505634	0.252817	−	0.967514i	$-0.418643\pi$
$353$	9.49999	0.505634	0.252817	+	0.967514i	$0.418643\pi$
$354$	48.5854	2.58228
$355$	−0.725315	−0.0384957
$356$	0.440267	0.0233341
$357$	0	0
$358$	−8.03504	−0.424665
$359$	−20.9436	−1.10536	−0.552681	−	0.833393i	$-0.686395\pi$
$359$	−20.9436	−1.10536	−0.552681	+	0.833393i	$0.686395\pi$
$360$	−3.41255	−0.179857
$361$	−14.5353	−0.765013
$362$	−18.2899	−0.961295
$363$	−25.1430	−1.31967
$364$	0	0
$365$	−1.70734	−0.0893662
$366$	−18.3940	−0.961470
$367$	−16.5334	−0.863039	−0.431519	−	0.902104i	$-0.642022\pi$
$367$	−16.5334	−0.863039	−0.431519	+	0.902104i	$0.642022\pi$
$368$	−2.34386	−0.122182
$369$	−34.1859	−1.77965
$370$	−1.69169	−0.0879468
$371$	0	0
$372$	4.48770	0.232676
$373$	0.112469	0.00582340	0.00291170	−	0.999996i	$-0.499073\pi$
$373$	0.112469	0.00582340	0.00291170	+	0.999996i	$0.499073\pi$
$374$	−22.2300	−1.14948
$375$	4.51230	0.233014
$376$	−1.40577	−0.0724968
$377$	−2.80950	−0.144696
$378$	0	0
$379$	−26.8045	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$379$	−26.8045	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$380$	−0.179658	−0.00921627
$381$	23.5264	1.20529
$382$	2.58138	0.132075
$383$	−9.99417	−0.510678	−0.255339	−	0.966852i	$-0.582187\pi$
$383$	−9.99417	−0.510678	−0.255339	+	0.966852i	$0.582187\pi$
$384$	−11.1763	−0.570336
$385$	0	0
$386$	−5.28720	−0.269111
$387$	2.57506	0.130898
$388$	−5.11755	−0.259804
$389$	−25.0768	−1.27145	−0.635723	−	0.771918i	$-0.719298\pi$
$389$	−25.0768	−1.27145	−0.635723	+	0.771918i	$0.719298\pi$
$390$	−0.578208	−0.0292787
$391$	−4.41415	−0.223233
$392$	0	0
$393$	−38.6113	−1.94768
$394$	24.9150	1.25520
$395$	−0.492714	−0.0247911
$396$	−22.2469	−1.11795
$397$	22.0042	1.10436	0.552178	−	0.833726i	$-0.313797\pi$
$397$	22.0042	1.10436	0.552178	+	0.833726i	$0.313797\pi$
$398$	−14.5217	−0.727906
$399$	0	0
$400$	11.6766	0.583829
$401$	12.0137	0.599934	0.299967	−	0.953950i	$-0.403024\pi$
$401$	12.0137	0.599934	0.299967	+	0.953950i	$0.403024\pi$
$402$	34.8277	1.73705
$403$	2.32551	0.115842
$404$	−3.50437	−0.174349
$405$	4.56167	0.226671
$406$	0	0
$407$	−46.0521	−2.28272
$408$	−45.4971	−2.25244
$409$	24.3764	1.20534	0.602669	−	0.797991i	$-0.294104\pi$
$409$	24.3764	1.20534	0.602669	+	0.797991i	$0.294104\pi$
$410$	−0.657986	−0.0324956
$411$	35.7058	1.76124
$412$	−0.383104	−0.0188742
$413$	0	0
$414$	9.61154	0.472381
$415$	1.32837	0.0652073
$416$	3.72934	0.182846
$417$	−58.5010	−2.86480
$418$	10.6412	0.520477
$419$	−4.41927	−0.215896	−0.107948	−	0.994157i	$-0.534428\pi$
$419$	−4.41927	−0.215896	−0.107948	+	0.994157i	$0.534428\pi$
$420$	0	0
$421$	32.4943	1.58368	0.791838	−	0.610731i	$-0.209124\pi$
$421$	32.4943	1.58368	0.791838	+	0.610731i	$0.209124\pi$
$422$	20.5428	1.00001
$423$	3.74968	0.182316
$424$	4.53022	0.220007
$425$	21.9903	1.06669
$426$	21.0561	1.02017
$427$	0	0
$428$	−0.632604	−0.0305781
$429$	−15.7403	−0.759947
$430$	0.0495631	0.00239014
$431$	2.70686	0.130385	0.0651925	−	0.997873i	$-0.479234\pi$
$431$	2.70686	0.130385	0.0651925	+	0.997873i	$0.479234\pi$
$432$	40.8954	1.96758
$433$	−29.1434	−1.40054	−0.700271	−	0.713877i	$-0.746937\pi$
$433$	−29.1434	−1.40054	−0.700271	+	0.713877i	$0.746937\pi$
$434$	0	0
$435$	1.16231	0.0557286
$436$	7.71654	0.369555
$437$	2.11300	0.101078
$438$	49.5645	2.36828
$439$	−32.9158	−1.57099	−0.785494	−	0.618869i	$-0.787591\pi$
$439$	−32.9158	−1.57099	−0.785494	+	0.618869i	$0.787591\pi$
$440$	−1.78804	−0.0852415
$441$	0	0
$442$	−5.64600	−0.268553
$443$	4.48315	0.213001	0.106500	−	0.994313i	$-0.466035\pi$
$443$	4.48315	0.213001	0.106500	+	0.994313i	$0.466035\pi$
$444$	−22.5713	−1.07119
$445$	−0.0943855	−0.00447430
$446$	−6.64759	−0.314773
$447$	60.5956	2.86607
$448$	0	0
$449$	6.66482	0.314532	0.157266	−	0.987556i	$-0.449732\pi$
$449$	6.66482	0.314532	0.157266	+	0.987556i	$0.449732\pi$
$450$	−47.8825	−2.25720
$451$	−17.9120	−0.843445
$452$	6.59429	0.310169
$453$	−46.6207	−2.19043
$454$	22.9775	1.07839
$455$	0	0
$456$	21.7789	1.01989
$457$	−25.0493	−1.17176	−0.585878	−	0.810400i	$-0.699250\pi$
$457$	−25.0493	−1.17176	−0.585878	+	0.810400i	$0.699250\pi$
$458$	18.9555	0.885732
$459$	77.0176	3.59487
$460$	−0.0850253	−0.00396433
$461$	4.90547	0.228471	0.114235	−	0.993454i	$-0.463558\pi$
$461$	4.90547	0.228471	0.114235	+	0.993454i	$0.463558\pi$
$462$	0	0
$463$	7.60392	0.353384	0.176692	−	0.984266i	$-0.443460\pi$
$463$	7.60392	0.353384	0.176692	+	0.984266i	$0.443460\pi$
$464$	6.02648	0.279772
$465$	−0.962083	−0.0446155
$466$	−8.13708	−0.376943
$467$	−9.87500	−0.456961	−0.228480	−	0.973549i	$-0.573376\pi$
$467$	−9.87500	−0.456961	−0.228480	+	0.973549i	$0.573376\pi$
$468$	−5.65030	−0.261185
$469$	0	0
$470$	0.0721712	0.00332901
$471$	35.7909	1.64916
$472$	38.1594	1.75643
$473$	1.34923	0.0620378
$474$	14.3036	0.656986
$475$	−10.5265	−0.482987
$476$	0	0
$477$	−12.0837	−0.553276
$478$	7.26811	0.332436
$479$	−10.8277	−0.494731	−0.247365	−	0.968922i	$-0.579565\pi$
$479$	−10.8277	−0.494731	−0.247365	+	0.968922i	$0.579565\pi$
$480$	−1.54286	−0.0704217
$481$	−11.6964	−0.533309
$482$	−22.5076	−1.02520
$483$	0	0
$484$	−4.72907	−0.214958
$485$	1.09711	0.0498173
$486$	−71.1545	−3.22763
$487$	5.33842	0.241907	0.120953	−	0.992658i	$-0.461405\pi$
$487$	5.33842	0.241907	0.120953	+	0.992658i	$0.461405\pi$
$488$	−14.4468	−0.653975
$489$	−9.02113	−0.407950
$490$	0	0
$491$	12.2646	0.553492	0.276746	−	0.960943i	$-0.410744\pi$
$491$	12.2646	0.553492	0.276746	+	0.960943i	$0.410744\pi$
$492$	−8.77916	−0.395795
$493$	11.3496	0.511159
$494$	2.70266	0.121598
$495$	4.76934	0.214366
$496$	−4.98831	−0.223982
$497$	0	0
$498$	−38.5630	−1.72805
$499$	−42.4267	−1.89928	−0.949641	−	0.313341i	$-0.898552\pi$
$499$	−42.4267	−1.89928	−0.949641	+	0.313341i	$0.898552\pi$
$500$	0.848704	0.0379552
$501$	77.9171	3.48108
$502$	−22.2062	−0.991113
$503$	24.5776	1.09586	0.547931	−	0.836523i	$-0.315416\pi$
$503$	24.5776	1.09586	0.547931	+	0.836523i	$0.315416\pi$
$504$	0	0
$505$	0.751274	0.0334313
$506$	5.03607	0.223880
$507$	39.5299	1.75559
$508$	4.42500	0.196327
$509$	31.1040	1.37866	0.689330	−	0.724447i	$-0.257905\pi$
$509$	31.1040	1.37866	0.689330	+	0.724447i	$0.257905\pi$
$510$	2.33580	0.103431
$511$	0	0
$512$	−22.4299	−0.991272
$513$	−36.8673	−1.62773
$514$	−8.76796	−0.386738
$515$	0.0821306	0.00361911
$516$	0.661294	0.0291118
$517$	1.96468	0.0864067
$518$	0	0
$519$	38.5376	1.69161
$520$	−0.454129	−0.0199149
$521$	31.6355	1.38598	0.692989	−	0.720949i	$-0.256294\pi$
$521$	31.6355	1.38598	0.692989	+	0.720949i	$0.256294\pi$
$522$	−24.7130	−1.08166
$523$	6.34825	0.277590	0.138795	−	0.990321i	$-0.455677\pi$
$523$	6.34825	0.277590	0.138795	+	0.990321i	$0.455677\pi$
$524$	−7.26227	−0.317254
$525$	0	0
$526$	33.4379	1.45796
$527$	−9.39440	−0.409227
$528$	33.7635	1.46937
$529$	1.00000	0.0434783
$530$	−0.232579	−0.0101026
$531$	−101.785	−4.41708
$532$	0	0
$533$	−4.54932	−0.197053
$534$	2.74003	0.118573
$535$	0.135619	0.00586332
$536$	27.3540	1.18151
$537$	22.9833	0.991803
$538$	−13.2387	−0.570762
$539$	0	0
$540$	1.48351	0.0638402
$541$	−8.87187	−0.381431	−0.190716	−	0.981645i	$-0.561081\pi$
$541$	−8.87187	−0.381431	−0.190716	+	0.981645i	$0.561081\pi$
$542$	−12.5634	−0.539645
$543$	52.3162	2.24510
$544$	−15.0655	−0.645928
$545$	−1.65429	−0.0708620
$546$	0	0
$547$	15.0130	0.641909	0.320955	−	0.947095i	$-0.395996\pi$
$547$	15.0130	0.641909	0.320955	+	0.947095i	$0.395996\pi$
$548$	6.71578	0.286884
$549$	38.5348	1.64462
$550$	−25.0885	−1.06978
$551$	−5.43289	−0.231449
$552$	10.3071	0.438699
$553$	0	0
$554$	16.7526	0.711751
$555$	4.83889	0.205400
$556$	−11.0032	−0.466642
$557$	5.28320	0.223856	0.111928	−	0.993716i	$-0.464297\pi$
$557$	5.28320	0.223856	0.111928	+	0.993716i	$0.464297\pi$
$558$	20.4557	0.865960
$559$	0.342680	0.0144938
$560$	0	0
$561$	63.5863	2.68462
$562$	28.9141	1.21967
$563$	−17.2543	−0.727183	−0.363592	−	0.931558i	$-0.618450\pi$
$563$	−17.2543	−0.727183	−0.363592	+	0.931558i	$0.618450\pi$
$564$	0.962942	0.0405472
$565$	−1.41370	−0.0594747
$566$	3.14114	0.132032
$567$	0	0
$568$	16.5376	0.693903
$569$	39.9893	1.67644	0.838220	−	0.545332i	$-0.183597\pi$
$569$	39.9893	1.67644	0.838220	+	0.545332i	$0.183597\pi$
$570$	−1.11811	−0.0468327
$571$	25.3546	1.06106	0.530528	−	0.847667i	$-0.321994\pi$
$571$	25.3546	1.06106	0.530528	+	0.847667i	$0.321994\pi$
$572$	−2.96053	−0.123786
$573$	−7.38375	−0.308461
$574$	0	0
$575$	−4.98177	−0.207754
$576$	71.2950	2.97063
$577$	−22.4174	−0.933250	−0.466625	−	0.884455i	$-0.654530\pi$
$577$	−22.4174	−0.933250	−0.466625	+	0.884455i	$0.654530\pi$
$578$	2.90857	0.120981
$579$	15.1234	0.628509
$580$	0.218615	0.00907751
$581$	0	0
$582$	−31.8494	−1.32020
$583$	−6.33139	−0.262219
$584$	38.9283	1.61087
$585$	1.21132	0.0500821
$586$	6.35319	0.262448
$587$	0.831262	0.0343098	0.0171549	−	0.999853i	$-0.494539\pi$
$587$	0.831262	0.0343098	0.0171549	+	0.999853i	$0.494539\pi$
$588$	0	0
$589$	4.49697	0.185295
$590$	−1.95908	−0.0806541
$591$	−71.2666	−2.93152
$592$	25.0892	1.03116
$593$	11.7503	0.482525	0.241263	−	0.970460i	$-0.422438\pi$
$593$	11.7503	0.482525	0.241263	+	0.970460i	$0.422438\pi$
$594$	−87.8687	−3.60529
$595$	0	0
$596$	11.3972	0.466848
$597$	41.5376	1.70002
$598$	1.27907	0.0523049
$599$	6.03510	0.246588	0.123294	−	0.992370i	$-0.460654\pi$
$599$	6.03510	0.246588	0.123294	+	0.992370i	$0.460654\pi$
$600$	−51.3476	−2.09626
$601$	−9.15850	−0.373583	−0.186792	−	0.982400i	$-0.559809\pi$
$601$	−9.15850	−0.373583	−0.186792	+	0.982400i	$0.559809\pi$
$602$	0	0
$603$	−72.9628	−2.97128
$604$	−8.76873	−0.356794
$605$	1.01383	0.0412180
$606$	−21.8097	−0.885958
$607$	−14.2443	−0.578157	−0.289079	−	0.957305i	$-0.593349\pi$
$607$	−14.2443	−0.578157	−0.289079	+	0.957305i	$0.593349\pi$
$608$	7.21165	0.292471
$609$	0	0
$610$	0.741690	0.0300302
$611$	0.498993	0.0201871
$612$	22.8256	0.922672
$613$	27.0567	1.09281	0.546406	−	0.837520i	$-0.315996\pi$
$613$	27.0567	1.09281	0.546406	+	0.837520i	$0.315996\pi$
$614$	4.87919	0.196908
$615$	1.88209	0.0758934
$616$	0	0
$617$	−1.73182	−0.0697206	−0.0348603	−	0.999392i	$-0.511099\pi$
$617$	−1.73182	−0.0697206	−0.0348603	+	0.999392i	$0.511099\pi$
$618$	−2.38427	−0.0959095
$619$	8.78839	0.353235	0.176618	−	0.984280i	$-0.443484\pi$
$619$	8.78839	0.353235	0.176618	+	0.984280i	$0.443484\pi$
$620$	−0.180955	−0.00726732
$621$	−17.4479	−0.700159
$622$	−3.91635	−0.157031
$623$	0	0
$624$	8.57530	0.343287
$625$	24.7269	0.989077
$626$	−23.8900	−0.954835
$627$	−30.4379	−1.21557
$628$	6.73179	0.268628
$629$	47.2501	1.88398
$630$	0	0
$631$	41.9660	1.67064	0.835320	−	0.549763i	$-0.185282\pi$
$631$	41.9660	1.67064	0.835320	+	0.549763i	$0.185282\pi$
$632$	11.2342	0.446871
$633$	−58.7604	−2.33552
$634$	16.5410	0.656926
$635$	−0.948640	−0.0376456
$636$	−3.10318	−0.123049
$637$	0	0
$638$	−12.9486	−0.512641
$639$	−44.1117	−1.74503
$640$	0.450654	0.0178136
$641$	−32.3613	−1.27819	−0.639097	−	0.769126i	$-0.720692\pi$
$641$	−32.3613	−1.27819	−0.639097	+	0.769126i	$0.720692\pi$
$642$	−3.93705	−0.155383
$643$	5.60532	0.221052	0.110526	−	0.993873i	$-0.464746\pi$
$643$	5.60532	0.221052	0.110526	+	0.993873i	$0.464746\pi$
$644$	0	0
$645$	−0.141770	−0.00558217
$646$	−10.9180	−0.429563
$647$	34.9767	1.37507	0.687537	−	0.726149i	$-0.258692\pi$
$647$	34.9767	1.37507	0.687537	+	0.726149i	$0.258692\pi$
$648$	−104.009	−4.08585
$649$	−53.3311	−2.09343
$650$	−6.37202	−0.249931
$651$	0	0
$652$	−1.69675	−0.0664500
$653$	20.9953	0.821608	0.410804	−	0.911724i	$-0.365248\pi$
$653$	20.9953	0.821608	0.410804	+	0.911724i	$0.365248\pi$
$654$	48.0244	1.87790
$655$	1.55690	0.0608331
$656$	9.75849	0.381005
$657$	−103.836	−4.05102
$658$	0	0
$659$	−12.8839	−0.501884	−0.250942	−	0.968002i	$-0.580740\pi$
$659$	−12.8839	−0.501884	−0.250942	+	0.968002i	$0.580740\pi$
$660$	1.22480	0.0476752
$661$	−49.1551	−1.91191	−0.955957	−	0.293507i	$-0.905177\pi$
$661$	−49.1551	−1.91191	−0.955957	+	0.293507i	$0.905177\pi$
$662$	18.5875	0.722423
$663$	16.1497	0.627204
$664$	−30.2877	−1.17539
$665$	0	0
$666$	−102.884	−3.98668
$667$	−2.57118	−0.0995565
$668$	14.6552	0.567025
$669$	19.0147	0.735151
$670$	−1.40434	−0.0542543
$671$	20.1907	0.779453
$672$	0	0
$673$	22.4105	0.863862	0.431931	−	0.901907i	$-0.357832\pi$
$673$	22.4105	0.863862	0.431931	+	0.901907i	$0.357832\pi$
$674$	1.07668	0.0414722
$675$	86.9213	3.34560
$676$	7.43505	0.285963
$677$	−5.02556	−0.193148	−0.0965740	−	0.995326i	$-0.530788\pi$
$677$	−5.02556	−0.193148	−0.0965740	+	0.995326i	$0.530788\pi$
$678$	41.0400	1.57613
$679$	0	0
$680$	1.83455	0.0703519
$681$	−65.7245	−2.51857
$682$	10.7180	0.410413
$683$	−12.6505	−0.484059	−0.242030	−	0.970269i	$-0.577813\pi$
$683$	−12.6505	−0.484059	−0.242030	+	0.970269i	$0.577813\pi$
$684$	−10.9263	−0.417778
$685$	−1.43974	−0.0550098
$686$	0	0
$687$	−54.2201	−2.06862
$688$	−0.735062	−0.0280240
$689$	−1.60805	−0.0612620
$690$	−0.529161	−0.0201448
$691$	−38.4578	−1.46300	−0.731502	−	0.681839i	$-0.761180\pi$
$691$	−38.4578	−1.46300	−0.731502	+	0.681839i	$0.761180\pi$
$692$	7.24840	0.275543
$693$	0	0
$694$	22.3601	0.848777
$695$	2.35890	0.0894782
$696$	−26.5014	−1.00453
$697$	18.3780	0.696116
$698$	26.3536	0.997497
$699$	23.2752	0.880349
$700$	0	0
$701$	38.2430	1.44442	0.722210	−	0.691674i	$-0.243127\pi$
$701$	38.2430	1.44442	0.722210	+	0.691674i	$0.243127\pi$
$702$	−22.3170	−0.842301
$703$	−22.6180	−0.853053
$704$	37.3558	1.40790
$705$	−0.206438	−0.00777489
$706$	11.1204	0.418522
$707$	0	0
$708$	−26.1390	−0.982363
$709$	−38.7483	−1.45522	−0.727612	−	0.685989i	$-0.759370\pi$
$709$	−38.7483	−1.45522	−0.727612	+	0.685989i	$0.759370\pi$
$710$	−0.849032	−0.0318636
$711$	−29.9655	−1.12380
$712$	2.15204	0.0806512
$713$	2.12825	0.0797034
$714$	0	0
$715$	0.634686	0.0237359
$716$	4.32285	0.161553
$717$	−20.7896	−0.776402
$718$	−24.5160	−0.914928
$719$	24.1328	0.900001	0.450001	−	0.893028i	$-0.351424\pi$
$719$	24.1328	0.900001	0.450001	+	0.893028i	$0.351424\pi$
$720$	−2.59834	−0.0968344
$721$	0	0
$722$	−17.0145	−0.633215
$723$	64.3806	2.39434
$724$	9.83997	0.365700
$725$	12.8090	0.475715
$726$	−29.4317	−1.09231
$727$	6.95322	0.257881	0.128940	−	0.991652i	$-0.458842\pi$
$727$	6.95322	0.257881	0.128940	+	0.991652i	$0.458842\pi$
$728$	0	0
$729$	102.167	3.78397
$730$	−1.99856	−0.0739700
$731$	−1.38433	−0.0512013
$732$	9.89598	0.365766
$733$	16.5243	0.610340	0.305170	−	0.952298i	$-0.401287\pi$
$733$	16.5243	0.610340	0.305170	+	0.952298i	$0.401287\pi$
$734$	−19.3536	−0.714353
$735$	0	0
$736$	3.41300	0.125805
$737$	−38.2296	−1.40821
$738$	−40.0169	−1.47304
$739$	−8.51415	−0.313198	−0.156599	−	0.987662i	$-0.550053\pi$
$739$	−8.51415	−0.313198	−0.156599	+	0.987662i	$0.550053\pi$
$740$	0.910131	0.0334571
$741$	−7.73066	−0.283993
$742$	0	0
$743$	−2.89579	−0.106236	−0.0531181	−	0.998588i	$-0.516916\pi$
$743$	−2.89579	−0.106236	−0.0531181	+	0.998588i	$0.516916\pi$
$744$	21.9360	0.804214
$745$	−2.44336	−0.0895178
$746$	0.131652	0.00482013
$747$	80.7881	2.95588
$748$	11.9597	0.437291
$749$	0	0
$750$	5.28196	0.192870
$751$	−46.1759	−1.68498	−0.842490	−	0.538711i	$-0.818911\pi$
$751$	−46.1759	−1.68498	−0.842490	+	0.538711i	$0.818911\pi$
$752$	−1.07036	−0.0390320
$753$	63.5185	2.31474
$754$	−3.28871	−0.119768
$755$	1.87986	0.0684151
$756$	0	0
$757$	−0.314284	−0.0114228	−0.00571142	−	0.999984i	$-0.501818\pi$
$757$	−0.314284	−0.0114228	−0.00571142	+	0.999984i	$0.501818\pi$
$758$	−31.3766	−1.13965
$759$	−14.4051	−0.522872
$760$	−0.878176	−0.0318548
$761$	12.4125	0.449954	0.224977	−	0.974364i	$-0.427769\pi$
$761$	12.4125	0.449954	0.224977	+	0.974364i	$0.427769\pi$
$762$	27.5393	0.997642
$763$	0	0
$764$	−1.38878	−0.0502444
$765$	−4.89341	−0.176922
$766$	−11.6989	−0.422698
$767$	−13.5451	−0.489085
$768$	45.0628	1.62606
$769$	−17.8921	−0.645206	−0.322603	−	0.946534i	$-0.604558\pi$
$769$	−17.8921	−0.645206	−0.322603	+	0.946534i	$0.604558\pi$
$770$	0	0
$771$	25.0798	0.903225
$772$	2.84452	0.102376
$773$	43.4194	1.56169	0.780844	−	0.624726i	$-0.214789\pi$
$773$	43.4194	1.56169	0.780844	+	0.624726i	$0.214789\pi$
$774$	3.01429	0.108347
$775$	−10.6024	−0.380851
$776$	−25.0148	−0.897978
$777$	0	0
$778$	−29.3542	−1.05240
$779$	−8.79730	−0.315196
$780$	0.311076	0.0111383
$781$	−23.1128	−0.827041
$782$	−5.16707	−0.184774
$783$	44.8616	1.60322
$784$	0	0
$785$	−1.44318	−0.0515092
$786$	−45.1972	−1.61213
$787$	32.1022	1.14432	0.572160	−	0.820142i	$-0.306105\pi$
$787$	32.1022	1.14432	0.572160	+	0.820142i	$0.306105\pi$
$788$	−13.4043	−0.477508
$789$	−95.6454	−3.40507
$790$	−0.576756	−0.0205201
$791$	0	0
$792$	−108.744	−3.86404
$793$	5.12806	0.182103
$794$	25.7574	0.914096
$795$	0.665266	0.0235946
$796$	7.81267	0.276913
$797$	49.8013	1.76405	0.882027	−	0.471199i	$-0.156179\pi$
$797$	49.8013	1.76405	0.882027	+	0.471199i	$0.156179\pi$
$798$	0	0
$799$	−2.01579	−0.0713136
$800$	−17.0028	−0.601139
$801$	−5.74027	−0.202822
$802$	14.0628	0.496576
$803$	−54.4058	−1.91994
$804$	−18.7373	−0.660815
$805$	0	0
$806$	2.72217	0.0958843
$807$	37.8679	1.33301
$808$	−17.1295	−0.602613
$809$	18.6808	0.656781	0.328391	−	0.944542i	$-0.393494\pi$
$809$	18.6808	0.656781	0.328391	+	0.944542i	$0.393494\pi$
$810$	5.33975	0.187620
$811$	30.9877	1.08813	0.544063	−	0.839045i	$-0.316885\pi$
$811$	30.9877	1.08813	0.544063	+	0.839045i	$0.316885\pi$
$812$	0	0
$813$	35.9362	1.26034
$814$	−53.9072	−1.88945
$815$	0.363754	0.0127417
$816$	−34.6418	−1.21271
$817$	0.662660	0.0231836
$818$	28.5343	0.997679
$819$	0	0
$820$	0.353997	0.0123621
$821$	−15.4959	−0.540810	−0.270405	−	0.962747i	$-0.587158\pi$
$821$	−15.4959	−0.540810	−0.270405	+	0.962747i	$0.587158\pi$
$822$	41.7961	1.45781
$823$	−43.2149	−1.50638	−0.753189	−	0.657805i	$-0.771485\pi$
$823$	−43.2149	−1.50638	−0.753189	+	0.657805i	$0.771485\pi$
$824$	−1.87263	−0.0652360
$825$	71.7629	2.49846
$826$	0	0
$827$	9.36433	0.325630	0.162815	−	0.986657i	$-0.447943\pi$
$827$	9.36433	0.325630	0.162815	+	0.986657i	$0.447943\pi$
$828$	−5.17101	−0.179705
$829$	−39.7104	−1.37920	−0.689600	−	0.724190i	$-0.742214\pi$
$829$	−39.7104	−1.37920	−0.689600	+	0.724190i	$0.742214\pi$
$830$	1.55495	0.0539733
$831$	−47.9190	−1.66229
$832$	9.48766	0.328926
$833$	0	0
$834$	−68.4795	−2.37125
$835$	−3.14181	−0.108727
$836$	−5.72496	−0.198002
$837$	−37.1334	−1.28352
$838$	−5.17306	−0.178701
$839$	34.8783	1.20413	0.602066	−	0.798446i	$-0.294344\pi$
$839$	34.8783	1.20413	0.602066	+	0.798446i	$0.294344\pi$
$840$	0	0
$841$	−22.3890	−0.772036
$842$	38.0369	1.31084
$843$	−82.7055	−2.84853
$844$	−11.0520	−0.380427
$845$	−1.59394	−0.0548333
$846$	4.38926	0.150906
$847$	0	0
$848$	3.44934	0.118451
$849$	−8.98488	−0.308360
$850$	25.7412	0.882915
$851$	−10.7042	−0.366936
$852$	−11.3282	−0.388097
$853$	−54.4858	−1.86556	−0.932779	−	0.360449i	$-0.882624\pi$
$853$	−54.4858	−1.86556	−0.932779	+	0.360449i	$0.882624\pi$
$854$	0	0
$855$	2.34241	0.0801087
$856$	−3.09219	−0.105689
$857$	−38.2357	−1.30611	−0.653053	−	0.757313i	$-0.726512\pi$
$857$	−38.2357	−1.30611	−0.653053	+	0.757313i	$0.726512\pi$
$858$	−18.4251	−0.629022
$859$	11.5889	0.395407	0.197704	−	0.980262i	$-0.436652\pi$
$859$	11.5889	0.395407	0.197704	+	0.980262i	$0.436652\pi$
$860$	−0.0266650	−0.000909268 0
$861$	0	0
$862$	3.16857	0.107922
$863$	−39.5284	−1.34556	−0.672781	−	0.739841i	$-0.734901\pi$
$863$	−39.5284	−1.34556	−0.672781	+	0.739841i	$0.734901\pi$
$864$	−59.5496	−2.02592
$865$	−1.55393	−0.0528351
$866$	−34.1144	−1.15925
$867$	−8.31964	−0.282550
$868$	0	0
$869$	−15.7008	−0.532612
$870$	1.36057	0.0461276
$871$	−9.70961	−0.328998
$872$	37.7188	1.27732
$873$	66.7234	2.25824
$874$	2.47341	0.0836643
$875$	0	0
$876$	−26.6657	−0.900951
$877$	18.1114	0.611578	0.305789	−	0.952099i	$-0.401080\pi$
$877$	18.1114	0.611578	0.305789	+	0.952099i	$0.401080\pi$
$878$	−38.5303	−1.30033
$879$	−18.1726	−0.612946
$880$	−1.36143	−0.0458937
$881$	49.0582	1.65281	0.826407	−	0.563073i	$-0.190381\pi$
$881$	49.0582	1.65281	0.826407	+	0.563073i	$0.190381\pi$
$882$	0	0
$883$	−26.9501	−0.906945	−0.453472	−	0.891270i	$-0.649815\pi$
$883$	−26.9501	−0.906945	−0.453472	+	0.891270i	$0.649815\pi$
$884$	3.03755	0.102164
$885$	5.60373	0.188367
$886$	5.24783	0.176304
$887$	0.766912	0.0257504	0.0128752	−	0.999917i	$-0.495902\pi$
$887$	0.766912	0.0257504	0.0128752	+	0.999917i	$0.495902\pi$
$888$	−110.330	−3.70242
$889$	0	0
$890$	−0.110485	−0.00370346
$891$	145.362	4.86979
$892$	3.57641	0.119747
$893$	0.964932	0.0322902
$894$	70.9313	2.37230
$895$	−0.926743	−0.0309776
$896$	0	0
$897$	−3.65863	−0.122158
$898$	7.80164	0.260344
$899$	−5.47210	−0.182505
$900$	25.7608	0.858694
$901$	6.49609	0.216416
$902$	−20.9673	−0.698134
$903$	0	0
$904$	32.2331	1.07206
$905$	−2.10951	−0.0701226
$906$	−54.5728	−1.81306
$907$	17.1978	0.571042	0.285521	−	0.958372i	$-0.407833\pi$
$907$	17.1978	0.571042	0.285521	+	0.958372i	$0.407833\pi$
$908$	−12.3619	−0.410244
$909$	45.6905	1.51546
$910$	0	0
$911$	−9.40906	−0.311736	−0.155868	−	0.987778i	$-0.549817\pi$
$911$	−9.40906	−0.311736	−0.155868	+	0.987778i	$0.549817\pi$
$912$	16.5826	0.549104
$913$	42.3298	1.40091
$914$	−29.3219	−0.969883
$915$	−2.12152	−0.0701353
$916$	−10.1981	−0.336954
$917$	0	0
$918$	90.1545	2.97554
$919$	41.0878	1.35536	0.677681	−	0.735356i	$-0.262985\pi$
$919$	41.0878	1.35536	0.677681	+	0.735356i	$0.262985\pi$
$920$	−0.415607	−0.0137022
$921$	−13.9564	−0.459878
$922$	5.74220	0.189109
$923$	−5.87022	−0.193221
$924$	0	0
$925$	53.3260	1.75335
$926$	8.90091	0.292502
$927$	4.99496	0.164056
$928$	−8.77543	−0.288068
$929$	−46.9779	−1.54129	−0.770647	−	0.637262i	$-0.780067\pi$
$929$	−46.9779	−1.54129	−0.770647	+	0.637262i	$0.780067\pi$
$930$	−1.12618	−0.0369291
$931$	0	0
$932$	4.37775	0.143398
$933$	11.2023	0.366746
$934$	−11.5594	−0.378234
$935$	−2.56395	−0.0838502
$936$	−27.6189	−0.902751
$937$	−40.9884	−1.33903	−0.669517	−	0.742797i	$-0.733499\pi$
$937$	−40.9884	−1.33903	−0.669517	+	0.742797i	$0.733499\pi$
$938$	0	0
$939$	68.3346	2.23001
$940$	−0.0388282	−0.00126643
$941$	26.7987	0.873612	0.436806	−	0.899556i	$-0.356110\pi$
$941$	26.7987	0.873612	0.436806	+	0.899556i	$0.356110\pi$
$942$	41.8958	1.36504
$943$	−4.16343	−0.135580
$944$	29.0548	0.945654
$945$	0	0
$946$	1.57937	0.0513497
$947$	44.9744	1.46147	0.730735	−	0.682661i	$-0.239177\pi$
$947$	44.9744	1.46147	0.730735	+	0.682661i	$0.239177\pi$
$948$	−7.69535	−0.249933
$949$	−13.8181	−0.448553
$950$	−12.3220	−0.399777
$951$	−47.3136	−1.53425
$952$	0	0
$953$	32.3616	1.04829	0.524147	−	0.851628i	$-0.324384\pi$
$953$	32.3616	1.04829	0.524147	+	0.851628i	$0.324384\pi$
$954$	−14.1448	−0.457956
$955$	0.297731	0.00963433
$956$	−3.91025	−0.126466
$957$	37.0381	1.19727
$958$	−12.6746	−0.409497
$959$	0	0
$960$	−3.92513	−0.126683
$961$	−26.4706	−0.853889
$962$	−13.6914	−0.441429
$963$	8.24798	0.265787
$964$	12.1091	0.390009
$965$	−0.609814	−0.0196306
$966$	0	0
$967$	−33.1219	−1.06513	−0.532564	−	0.846390i	$-0.678771\pi$
$967$	−33.1219	−1.06513	−0.532564	+	0.846390i	$0.678771\pi$
$968$	−23.1159	−0.742972
$969$	31.2297	1.00324
$970$	1.28425	0.0412346
$971$	47.7581	1.53263	0.766316	−	0.642464i	$-0.222088\pi$
$971$	47.7581	1.53263	0.766316	+	0.642464i	$0.222088\pi$
$972$	38.2812	1.22787
$973$	0	0
$974$	6.24899	0.200230
$975$	18.2264	0.583713
$976$	−10.9999	−0.352098
$977$	3.51776	0.112543	0.0562716	−	0.998416i	$-0.482079\pi$
$977$	3.51776	0.112543	0.0562716	+	0.998416i	$0.482079\pi$
$978$	−10.5599	−0.337667
$979$	−3.00767	−0.0961257
$980$	0	0
$981$	−100.609	−3.21221
$982$	14.3565	0.458135
$983$	44.3828	1.41559	0.707796	−	0.706417i	$-0.249690\pi$
$983$	44.3828	1.41559	0.707796	+	0.706417i	$0.249690\pi$
$984$	−42.9129	−1.36801
$985$	2.87364	0.0915618
$986$	13.2855	0.423096
$987$	0	0
$988$	−1.45403	−0.0462590
$989$	0.313612	0.00997228
$990$	5.58285	0.177435
$991$	−2.95536	−0.0938800	−0.0469400	−	0.998898i	$-0.514947\pi$
$991$	−2.95536	−0.0938800	−0.0469400	+	0.998898i	$0.514947\pi$
$992$	7.26370	0.230623
$993$	−53.1674	−1.68722
$994$	0	0
$995$	−1.67490	−0.0530978
$996$	20.7469	0.657392
$997$	−48.4248	−1.53363	−0.766815	−	0.641869i	$-0.778159\pi$
$997$	−48.4248	−1.53363	−0.766815	+	0.641869i	$0.778159\pi$
$998$	−49.6634	−1.57207
$999$	186.766	5.90902

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1127.2.a.k.1.5		7
7.3	odd	6		161.2.e.a.93.3	✓	14
7.5	odd	6		161.2.e.a.116.3	yes	14
7.6	odd	2		1127.2.a.n.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.e.a.93.3	✓	14	7.3	odd	6
161.2.e.a.116.3	yes	14	7.5	odd	6
1127.2.a.k.1.5		7	1.1	even	1	trivial
1127.2.a.n.1.5		7	7.6	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:	\( N \)	\(=\)	\( 1127 = 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1127.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17057 q^{2} -3.34828 q^{3} -0.629767 q^{4} +0.135011 q^{5} -3.91940 q^{6} -3.07833 q^{8} +8.21099 q^{9} +0.158040 q^{10} +4.30224 q^{11} +2.10864 q^{12} +1.09269 q^{13} -0.452054 q^{15} -2.34386 q^{16} -4.41415 q^{17} +9.61154 q^{18} +2.11300 q^{19} -0.0850253 q^{20} +5.03607 q^{22} +1.00000 q^{23} +10.3071 q^{24} -4.98177 q^{25} +1.27907 q^{26} -17.4479 q^{27} -2.57118 q^{29} -0.529161 q^{30} +2.12825 q^{31} +3.41300 q^{32} -14.4051 q^{33} -5.16707 q^{34} -5.17101 q^{36} -10.7042 q^{37} +2.47341 q^{38} -3.65863 q^{39} -0.415607 q^{40} -4.16343 q^{41} +0.313612 q^{43} -2.70941 q^{44} +1.10857 q^{45} +1.17057 q^{46} +0.456666 q^{47} +7.84790 q^{48} -5.83151 q^{50} +14.7798 q^{51} -0.688138 q^{52} -1.47165 q^{53} -20.4240 q^{54} +0.580848 q^{55} -7.07490 q^{57} -3.00974 q^{58} -12.3961 q^{59} +0.284689 q^{60} +4.69307 q^{61} +2.49126 q^{62} +8.68287 q^{64} +0.147525 q^{65} -16.8622 q^{66} -8.88599 q^{67} +2.77989 q^{68} -3.34828 q^{69} -5.37228 q^{71} -25.2761 q^{72} -12.6459 q^{73} -12.5300 q^{74} +16.6804 q^{75} -1.33069 q^{76} -4.28268 q^{78} -3.64944 q^{79} -0.316446 q^{80} +33.7874 q^{81} -4.87358 q^{82} +9.83902 q^{83} -0.595958 q^{85} +0.367104 q^{86} +8.60903 q^{87} -13.2437 q^{88} -0.699096 q^{89} +1.29766 q^{90} -0.629767 q^{92} -7.12597 q^{93} +0.534559 q^{94} +0.285277 q^{95} -11.4277 q^{96} +8.12610 q^{97} +35.3256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 5 q^{3} + 6 q^{4} - 4 q^{5} - 6 q^{6} - 3 q^{8} + 4 q^{9} + 2 q^{10} + 4 q^{11} - 9 q^{12} - 14 q^{13} - 3 q^{15} - 8 q^{16} - 4 q^{17} + 19 q^{18} - 9 q^{19} - 12 q^{20} - 10 q^{22} + 7 q^{23} + 4 q^{24}+ \cdots + 18 q^{99}+O(q^{100}) \) 7 * q - 5 * q^3 + 6 * q^4 - 4 * q^5 - 6 * q^6 - 3 * q^8 + 4 * q^9 + 2 * q^10 + 4 * q^11 - 9 * q^12 - 14 * q^13 - 3 * q^15 - 8 * q^16 - 4 * q^17 + 19 * q^18 - 9 * q^19 - 12 * q^20 - 10 * q^22 + 7 * q^23 + 4 * q^24 + 3 * q^25 - 14 * q^26 - 20 * q^27 - 5 * q^29 - 17 * q^30 - 10 * q^31 + 19 * q^32 - 3 * q^33 - 20 * q^34 + 4 * q^36 - 5 * q^37 + 17 * q^38 + 18 * q^39 - 4 * q^40 - 23 * q^41 - 11 * q^43 - 34 * q^44 - 6 * q^45 - 4 * q^47 - 11 * q^48 - 9 * q^50 + 7 * q^51 - 12 * q^52 + 2 * q^53 - 15 * q^54 - 17 * q^55 - 24 * q^57 - 7 * q^58 - 7 * q^59 + 32 * q^60 - 12 * q^61 + 4 * q^62 - 9 * q^64 + 9 * q^65 + 27 * q^66 + 2 * q^67 + 12 * q^68 - 5 * q^69 - 28 * q^71 - 20 * q^72 - 59 * q^73 + 2 * q^74 + 22 * q^75 - q^76 - 17 * q^78 - 4 * q^79 + 22 * q^80 + 11 * q^81 - 54 * q^82 + 9 * q^83 + 9 * q^85 - 15 * q^86 - 11 * q^87 - 33 * q^88 - q^89 - 2 * q^90 + 6 * q^92 + 15 * q^93 + 14 * q^94 - 16 * q^95 - 20 * q^96 - 35 * q^97 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more