LMFDB - Embedded newform 1127.4.a.e.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1127,4,Mod(1,1127)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1127.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1127, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1127 = 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1127.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.4951525765$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 44x^{6} - 23x^{5} + 587x^{4} + 594x^{3} - 2430x^{2} - 3403x + 110 $ x^8 - 44x^6 - 23x^5 + 587x^4 + 594x^3 - 2430x^2 - 3403x + 110
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.07589$ 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)

$f(q)$	$=$	\(q+2.07589 q^{2} -3.18430 q^{3} -3.69068 q^{4} +8.32072 q^{5} -6.61027 q^{6} -24.2686 q^{8} -16.8602 q^{9} +17.2729 q^{10} -61.6700 q^{11} +11.7522 q^{12} -62.4385 q^{13} -26.4957 q^{15} -20.8535 q^{16} -82.6471 q^{17} -34.9999 q^{18} +129.351 q^{19} -30.7091 q^{20} -128.020 q^{22} +23.0000 q^{23} +77.2785 q^{24} -55.7657 q^{25} -129.615 q^{26} +139.664 q^{27} +219.702 q^{29} -55.0022 q^{30} -4.07623 q^{31} +150.859 q^{32} +196.376 q^{33} -171.566 q^{34} +62.2256 q^{36} +68.0277 q^{37} +268.518 q^{38} +198.823 q^{39} -201.932 q^{40} +110.320 q^{41} -461.171 q^{43} +227.604 q^{44} -140.289 q^{45} +47.7455 q^{46} +288.004 q^{47} +66.4038 q^{48} -115.763 q^{50} +263.173 q^{51} +230.440 q^{52} -351.353 q^{53} +289.928 q^{54} -513.139 q^{55} -411.893 q^{57} +456.078 q^{58} +169.163 q^{59} +97.7871 q^{60} -342.218 q^{61} -8.46182 q^{62} +479.995 q^{64} -519.533 q^{65} +407.655 q^{66} +709.425 q^{67} +305.024 q^{68} -73.2390 q^{69} +372.456 q^{71} +409.173 q^{72} +361.462 q^{73} +141.218 q^{74} +177.575 q^{75} -477.392 q^{76} +412.735 q^{78} -574.150 q^{79} -173.516 q^{80} +10.4919 q^{81} +229.013 q^{82} -1129.75 q^{83} -687.683 q^{85} -957.341 q^{86} -699.600 q^{87} +1496.64 q^{88} +16.9865 q^{89} -291.225 q^{90} -84.8856 q^{92} +12.9800 q^{93} +597.865 q^{94} +1076.29 q^{95} -480.381 q^{96} +1743.62 q^{97} +1039.77 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} + 24 q^{4} + 24 q^{5} + 41 q^{6} - 69 q^{8} + 95 q^{9} + 30 q^{10} - 98 q^{11} + 131 q^{12} + 145 q^{13} - 232 q^{15} - 76 q^{16} + 96 q^{17} - 69 q^{18} + 226 q^{19} + 22 q^{20} - 98 q^{22}+ \cdots - 1676 q^{99}+O(q^{100}) $ 8 * q + 3 * q^3 + 24 * q^4 + 24 * q^5 + 41 * q^6 - 69 * q^8 + 95 * q^9 + 30 * q^10 - 98 * q^11 + 131 * q^12 + 145 * q^13 - 232 * q^15 - 76 * q^16 + 96 * q^17 - 69 * q^18 + 226 * q^19 + 22 * q^20 - 98 * q^22 + 184 * q^23 + 216 * q^24 + 60 * q^25 + 185 * q^26 + 531 * q^27 + 73 * q^29 - 416 * q^30 + 987 * q^31 + 118 * q^32 - 16 * q^33 + 846 * q^34 - 69 * q^36 - 2 * q^37 + 454 * q^38 - 675 * q^39 + 640 * q^40 + 907 * q^41 + 636 * q^43 + 126 * q^44 + 126 * q^45 - 421 * q^47 - 2165 * q^48 + 2308 * q^50 - 408 * q^51 + 931 * q^52 + 138 * q^53 - 1717 * q^54 + 90 * q^55 + 984 * q^57 + 1329 * q^58 + 924 * q^59 + 2642 * q^60 + 890 * q^61 - 2175 * q^62 + 303 * q^64 - 392 * q^65 - 408 * q^66 - 1392 * q^67 - 2592 * q^68 + 69 * q^69 + 259 * q^71 + 141 * q^72 + 297 * q^73 + 2314 * q^74 + 851 * q^75 + 428 * q^76 + 6789 * q^78 - 1788 * q^79 + 2104 * q^80 + 2300 * q^81 + 219 * q^82 - 2114 * q^83 + 528 * q^85 - 2360 * q^86 + 2441 * q^87 + 2144 * q^88 + 1040 * q^89 - 4872 * q^90 + 552 * q^92 - 621 * q^93 + 1009 * q^94 + 1268 * q^95 + 1163 * q^96 + 1452 * q^97 - 1676 * q^99

Coefficient data

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

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

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

$2$	2.07589	0.733938	0.366969	−	0.930233i	$-0.380395\pi$
$2$	2.07589	0.733938	0.366969	+	0.930233i	$0.380395\pi$
$3$	−3.18430	−0.612820	−0.306410	−	0.951900i	$-0.599128\pi$
$3$	−3.18430	−0.612820	−0.306410	+	0.951900i	$0.599128\pi$
$4$	−3.69068	−0.461335
$5$	8.32072	0.744227	0.372114	−	0.928187i	$-0.378633\pi$
$5$	8.32072	0.744227	0.372114	+	0.928187i	$0.378633\pi$
$6$	−6.61027	−0.449772
$7$	0	0
$7$	0	0
$8$	−24.2686	−1.07253
$9$	−16.8602	−0.624452
$10$	17.2729	0.546217
$11$	−61.6700	−1.69038	−0.845191	−	0.534464i	$-0.820513\pi$
$11$	−61.6700	−1.69038	−0.845191	+	0.534464i	$0.820513\pi$
$12$	11.7522	0.282715
$13$	−62.4385	−1.33210	−0.666051	−	0.745907i	$-0.732017\pi$
$13$	−62.4385	−1.33210	−0.666051	+	0.745907i	$0.732017\pi$
$14$	0	0
$15$	−26.4957	−0.456077
$16$	−20.8535	−0.325836
$17$	−82.6471	−1.17911	−0.589555	−	0.807728i	$-0.700697\pi$
$17$	−82.6471	−1.17911	−0.589555	+	0.807728i	$0.700697\pi$
$18$	−34.9999	−0.458309
$19$	129.351	1.56185	0.780924	−	0.624626i	$-0.214748\pi$
$19$	129.351	1.56185	0.780924	+	0.624626i	$0.214748\pi$
$20$	−30.7091	−0.343338
$21$	0	0
$22$	−128.020	−1.24064
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	77.2785	0.657267
$25$	−55.7657	−0.446125
$26$	−129.615	−0.977680
$27$	139.664	0.995496
$28$	0	0
$29$	219.702	1.40682	0.703409	−	0.710785i	$-0.251660\pi$
$29$	219.702	1.40682	0.703409	+	0.710785i	$0.251660\pi$
$30$	−55.0022	−0.334733
$31$	−4.07623	−0.0236166	−0.0118083	−	0.999930i	$-0.503759\pi$
$31$	−4.07623	−0.0236166	−0.0118083	+	0.999930i	$0.503759\pi$
$32$	150.859	0.833386
$33$	196.376	1.03590
$34$	−171.566	−0.865394
$35$	0	0
$36$	62.2256	0.288081
$37$	68.0277	0.302262	0.151131	−	0.988514i	$-0.451709\pi$
$37$	68.0277	0.302262	0.151131	+	0.988514i	$0.451709\pi$
$38$	268.518	1.14630
$39$	198.823	0.816338
$40$	−201.932	−0.798206
$41$	110.320	0.420222	0.210111	−	0.977678i	$-0.432617\pi$
$41$	110.320	0.420222	0.210111	+	0.977678i	$0.432617\pi$
$42$	0	0
$43$	−461.171	−1.63553	−0.817767	−	0.575550i	$-0.804788\pi$
$43$	−461.171	−1.63553	−0.817767	+	0.575550i	$0.804788\pi$
$44$	227.604	0.779832
$45$	−140.289	−0.464734
$46$	47.7455	0.153037
$47$	288.004	0.893824	0.446912	−	0.894578i	$-0.352524\pi$
$47$	288.004	0.893824	0.446912	+	0.894578i	$0.352524\pi$
$48$	66.4038	0.199679
$49$	0	0
$50$	−115.763	−0.327429
$51$	263.173	0.722582
$52$	230.440	0.614545
$53$	−351.353	−0.910606	−0.455303	−	0.890337i	$-0.650469\pi$
$53$	−351.353	−0.910606	−0.455303	+	0.890337i	$0.650469\pi$
$54$	289.928	0.730633
$55$	−513.139	−1.25803
$56$	0	0
$57$	−411.893	−0.957132
$58$	456.078	1.03252
$59$	169.163	0.373274	0.186637	−	0.982429i	$-0.440241\pi$
$59$	169.163	0.373274	0.186637	+	0.982429i	$0.440241\pi$
$60$	97.7871	0.210404
$61$	−342.218	−0.718304	−0.359152	−	0.933279i	$-0.616934\pi$
$61$	−342.218	−0.718304	−0.359152	+	0.933279i	$0.616934\pi$
$62$	−8.46182	−0.0173331
$63$	0	0
$64$	479.995	0.937490
$65$	−519.533	−0.991386
$66$	407.655	0.760286
$67$	709.425	1.29358	0.646792	−	0.762667i	$-0.276110\pi$
$67$	709.425	1.29358	0.646792	+	0.762667i	$0.276110\pi$
$68$	305.024	0.543964
$69$	−73.2390	−0.127782
$70$	0	0
$71$	372.456	0.622569	0.311285	−	0.950317i	$-0.399241\pi$
$71$	372.456	0.622569	0.311285	+	0.950317i	$0.399241\pi$
$72$	409.173	0.669743
$73$	361.462	0.579533	0.289766	−	0.957097i	$-0.406422\pi$
$73$	361.462	0.579533	0.289766	+	0.957097i	$0.406422\pi$
$74$	141.218	0.221841
$75$	177.575	0.273394
$76$	−477.392	−0.720535
$77$	0	0
$78$	412.735	0.599142
$79$	−574.150	−0.817683	−0.408841	−	0.912605i	$-0.634067\pi$
$79$	−574.150	−0.817683	−0.408841	+	0.912605i	$0.634067\pi$
$80$	−173.516	−0.242496
$81$	10.4919	0.0143922
$82$	229.013	0.308417
$83$	−1129.75	−1.49405	−0.747026	−	0.664795i	$-0.768519\pi$
$83$	−1129.75	−1.49405	−0.747026	+	0.664795i	$0.768519\pi$
$84$	0	0
$85$	−687.683	−0.877526
$86$	−957.341	−1.20038
$87$	−699.600	−0.862126
$88$	1496.64	1.81298
$89$	16.9865	0.0202310	0.0101155	−	0.999949i	$-0.496780\pi$
$89$	16.9865	0.0202310	0.0101155	+	0.999949i	$0.496780\pi$
$90$	−291.225	−0.341086
$91$	0	0
$92$	−84.8856	−0.0961949
$93$	12.9800	0.0144727
$94$	597.865	0.656012
$95$	1076.29	1.16237
$96$	−480.381	−0.510715
$97$	1743.62	1.82514	0.912568	−	0.408925i	$-0.134096\pi$
$97$	1743.62	1.82514	0.912568	+	0.408925i	$0.134096\pi$
$98$	0	0
$99$	1039.77	1.05556
$100$	205.813	0.205813
$101$	−175.263	−0.172667	−0.0863333	−	0.996266i	$-0.527515\pi$
$101$	−175.263	−0.172667	−0.0863333	+	0.996266i	$0.527515\pi$
$102$	546.319	0.530330
$103$	1409.19	1.34807	0.674035	−	0.738699i	$-0.264560\pi$
$103$	1409.19	1.34807	0.674035	+	0.738699i	$0.264560\pi$
$104$	1515.29	1.42872
$105$	0	0
$106$	−729.371	−0.668328
$107$	−1233.58	−1.11453	−0.557267	−	0.830333i	$-0.688150\pi$
$107$	−1233.58	−1.11453	−0.557267	+	0.830333i	$0.688150\pi$
$108$	−515.456	−0.459257
$109$	−470.461	−0.413413	−0.206706	−	0.978403i	$-0.566274\pi$
$109$	−470.461	−0.413413	−0.206706	+	0.978403i	$0.566274\pi$
$110$	−1065.22	−0.923315
$111$	−216.621	−0.185232
$112$	0	0
$113$	−441.191	−0.367290	−0.183645	−	0.982993i	$-0.558790\pi$
$113$	−441.191	−0.367290	−0.183645	+	0.982993i	$0.558790\pi$
$114$	−855.044	−0.702475
$115$	191.376	0.155182
$116$	−810.851	−0.649014
$117$	1052.73	0.831833
$118$	351.165	0.273960
$119$	0	0
$120$	643.013	0.489156
$121$	2472.19	1.85739
$122$	−710.407	−0.527190
$123$	−351.293	−0.257520
$124$	15.0441	0.0108951
$125$	−1504.10	−1.07625
$126$	0	0
$127$	1195.34	0.835191	0.417596	−	0.908633i	$-0.362873\pi$
$127$	1195.34	0.835191	0.417596	+	0.908633i	$0.362873\pi$
$128$	−210.456	−0.145327
$129$	1468.51	1.00229
$130$	−1078.49	−0.727616
$131$	218.106	0.145466	0.0727330	−	0.997351i	$-0.476828\pi$
$131$	218.106	0.145466	0.0727330	+	0.997351i	$0.476828\pi$
$132$	−724.761	−0.477896
$133$	0	0
$134$	1472.69	0.949410
$135$	1162.11	0.740876
$136$	2005.73	1.26463
$137$	2259.86	1.40929	0.704645	−	0.709560i	$-0.251106\pi$
$137$	2259.86	1.40929	0.704645	+	0.709560i	$0.251106\pi$
$138$	−152.036	−0.0937839
$139$	1332.93	0.813365	0.406682	−	0.913570i	$-0.366686\pi$
$139$	1332.93	0.813365	0.406682	+	0.913570i	$0.366686\pi$
$140$	0	0
$141$	−917.093	−0.547753
$142$	773.178	0.456927
$143$	3850.58	2.25176
$144$	351.594	0.203469
$145$	1828.08	1.04699
$146$	750.355	0.425341
$147$	0	0
$148$	−251.068	−0.139444
$149$	−2302.18	−1.26578	−0.632892	−	0.774240i	$-0.718132\pi$
$149$	−2302.18	−1.26578	−0.632892	+	0.774240i	$0.718132\pi$
$150$	368.626	0.200655
$151$	−2259.46	−1.21769	−0.608847	−	0.793287i	$-0.708368\pi$
$151$	−2259.46	−1.21769	−0.608847	+	0.793287i	$0.708368\pi$
$152$	−3139.16	−1.67513
$153$	1393.45	0.736297
$154$	0	0
$155$	−33.9172	−0.0175761
$156$	−733.792	−0.376605
$157$	698.918	0.355285	0.177642	−	0.984095i	$-0.443153\pi$
$157$	698.918	0.355285	0.177642	+	0.984095i	$0.443153\pi$
$158$	−1191.87	−0.600129
$159$	1118.82	0.558037
$160$	1255.26	0.620229
$161$	0	0
$162$	21.7801	0.0105630
$163$	−1586.37	−0.762293	−0.381147	−	0.924515i	$-0.624471\pi$
$163$	−1586.37	−0.762293	−0.381147	+	0.924515i	$0.624471\pi$
$164$	−407.156	−0.193863
$165$	1633.99	0.770945
$166$	−2345.24	−1.09654
$167$	634.994	0.294235	0.147118	−	0.989119i	$-0.453000\pi$
$167$	634.994	0.294235	0.147118	+	0.989119i	$0.453000\pi$
$168$	0	0
$169$	1701.56	0.774494
$170$	−1427.55	−0.644050
$171$	−2180.88	−0.975299
$172$	1702.03	0.754528
$173$	3551.23	1.56066	0.780331	−	0.625366i	$-0.215050\pi$
$173$	3551.23	1.56066	0.780331	+	0.625366i	$0.215050\pi$
$174$	−1452.29	−0.632747
$175$	0	0
$176$	1286.03	0.550787
$177$	−538.668	−0.228750
$178$	35.2621	0.0148483
$179$	2087.76	0.871769	0.435885	−	0.900003i	$-0.356436\pi$
$179$	2087.76	0.871769	0.435885	+	0.900003i	$0.356436\pi$
$180$	517.761	0.214398
$181$	−2020.11	−0.829580	−0.414790	−	0.909917i	$-0.636145\pi$
$181$	−2020.11	−0.829580	−0.414790	+	0.909917i	$0.636145\pi$
$182$	0	0
$183$	1089.73	0.440191
$184$	−558.177	−0.223638
$185$	566.039	0.224952
$186$	26.9450	0.0106221
$187$	5096.85	1.99315
$188$	−1062.93	−0.412352
$189$	0	0
$190$	2234.26	0.853108
$191$	−2291.62	−0.868145	−0.434072	−	0.900878i	$-0.642924\pi$
$191$	−2291.62	−0.868145	−0.434072	+	0.900878i	$0.642924\pi$
$192$	−1528.45	−0.574512
$193$	−4627.40	−1.72584	−0.862921	−	0.505339i	$-0.831367\pi$
$193$	−4627.40	−1.72584	−0.862921	+	0.505339i	$0.831367\pi$
$194$	3619.57	1.33954
$195$	1654.35	0.607541
$196$	0	0
$197$	3333.27	1.20551	0.602755	−	0.797926i	$-0.294070\pi$
$197$	3333.27	1.20551	0.602755	+	0.797926i	$0.294070\pi$
$198$	2158.45	0.774718
$199$	4201.92	1.49681	0.748407	−	0.663240i	$-0.230819\pi$
$199$	4201.92	1.49681	0.748407	+	0.663240i	$0.230819\pi$
$200$	1353.35	0.478483
$201$	−2259.03	−0.792734
$202$	−363.827	−0.126727
$203$	0	0
$204$	−971.288	−0.333352
$205$	917.942	0.312741
$206$	2925.32	0.989401
$207$	−387.785	−0.130207
$208$	1302.06	0.434046
$209$	−7977.07	−2.64012
$210$	0	0
$211$	5157.06	1.68259	0.841296	−	0.540575i	$-0.181793\pi$
$211$	5157.06	1.68259	0.841296	+	0.540575i	$0.181793\pi$
$212$	1296.73	0.420094
$213$	−1186.01	−0.381523
$214$	−2560.79	−0.817999
$215$	−3837.28	−1.21721
$216$	−3389.45	−1.06770
$217$	0	0
$218$	−976.625	−0.303419
$219$	−1151.00	−0.355149
$220$	1893.83	0.580372
$221$	5160.36	1.57069
$222$	−449.681	−0.135949
$223$	−4084.70	−1.22660	−0.613301	−	0.789850i	$-0.710159\pi$
$223$	−4084.70	−1.22660	−0.613301	+	0.789850i	$0.710159\pi$
$224$	0	0
$225$	940.221	0.278584
$226$	−915.864	−0.269568
$227$	4912.15	1.43626	0.718130	−	0.695909i	$-0.244998\pi$
$227$	4912.15	1.43626	0.718130	+	0.695909i	$0.244998\pi$
$228$	1520.16	0.441558
$229$	2228.04	0.642938	0.321469	−	0.946920i	$-0.395823\pi$
$229$	2228.04	0.642938	0.321469	+	0.946920i	$0.395823\pi$
$230$	397.277	0.113894
$231$	0	0
$232$	−5331.87	−1.50885
$233$	2691.78	0.756842	0.378421	−	0.925634i	$-0.376467\pi$
$233$	2691.78	0.756842	0.378421	+	0.925634i	$0.376467\pi$
$234$	2185.34	0.610514
$235$	2396.40	0.665208
$236$	−624.327	−0.172204
$237$	1828.27	0.501092
$238$	0	0
$239$	6025.81	1.63087	0.815433	−	0.578851i	$-0.196499\pi$
$239$	6025.81	1.63087	0.815433	+	0.578851i	$0.196499\pi$
$240$	552.528	0.148606
$241$	−2852.32	−0.762381	−0.381191	−	0.924496i	$-0.624486\pi$
$241$	−2852.32	−0.762381	−0.381191	+	0.924496i	$0.624486\pi$
$242$	5131.99	1.36321
$243$	−3804.34	−1.00432
$244$	1263.02	0.331378
$245$	0	0
$246$	−729.246	−0.189004
$247$	−8076.47	−2.08054
$248$	98.9244	0.0253295
$249$	3597.47	0.915585
$250$	−3122.35	−0.789898
$251$	−4810.04	−1.20959	−0.604794	−	0.796382i	$-0.706745\pi$
$251$	−4810.04	−1.20959	−0.604794	+	0.796382i	$0.706745\pi$
$252$	0	0
$253$	−1418.41	−0.352469
$254$	2481.40	0.612979
$255$	2189.79	0.537765
$256$	−4276.84	−1.04415
$257$	−3084.01	−0.748541	−0.374271	−	0.927320i	$-0.622107\pi$
$257$	−3084.01	−0.748541	−0.374271	+	0.927320i	$0.622107\pi$
$258$	3048.47	0.735617
$259$	0	0
$260$	1917.43	0.457361
$261$	−3704.23	−0.878490
$262$	452.765	0.106763
$263$	1049.01	0.245951	0.122975	−	0.992410i	$-0.460756\pi$
$263$	1049.01	0.245951	0.122975	+	0.992410i	$0.460756\pi$
$264$	−4765.77	−1.11103
$265$	−2923.51	−0.677698
$266$	0	0
$267$	−54.0901	−0.0123980
$268$	−2618.26	−0.596775
$269$	−1379.71	−0.312722	−0.156361	−	0.987700i	$-0.549976\pi$
$269$	−1379.71	−0.312722	−0.156361	+	0.987700i	$0.549976\pi$
$270$	2412.41	0.543757
$271$	−4912.15	−1.10108	−0.550538	−	0.834810i	$-0.685577\pi$
$271$	−4912.15	−1.10108	−0.550538	+	0.834810i	$0.685577\pi$
$272$	1723.48	0.384196
$273$	0	0
$274$	4691.22	1.03433
$275$	3439.07	0.754122
$276$	270.302	0.0589502
$277$	1755.35	0.380754	0.190377	−	0.981711i	$-0.439029\pi$
$277$	1755.35	0.380754	0.190377	+	0.981711i	$0.439029\pi$
$278$	2767.02	0.596960
$279$	68.7261	0.0147474
$280$	0	0
$281$	−5879.83	−1.24826	−0.624130	−	0.781321i	$-0.714546\pi$
$281$	−5879.83	−1.24826	−0.624130	+	0.781321i	$0.714546\pi$
$282$	−1903.79	−0.402017
$283$	270.228	0.0567611	0.0283805	−	0.999597i	$-0.490965\pi$
$283$	270.228	0.0567611	0.0283805	+	0.999597i	$0.490965\pi$
$284$	−1374.62	−0.287213
$285$	−3427.24	−0.712324
$286$	7993.38	1.65265
$287$	0	0
$288$	−2543.51	−0.520410
$289$	1917.54	0.390299
$290$	3794.90	0.768428
$291$	−5552.23	−1.11848
$292$	−1334.04	−0.267359
$293$	1311.50	0.261498	0.130749	−	0.991416i	$-0.458262\pi$
$293$	1311.50	0.261498	0.130749	+	0.991416i	$0.458262\pi$
$294$	0	0
$295$	1407.56	0.277801
$296$	−1650.94	−0.324185
$297$	−8613.09	−1.68277
$298$	−4779.07	−0.929007
$299$	−1436.08	−0.277762
$300$	−655.372	−0.126126
$301$	0	0
$302$	−4690.38	−0.893713
$303$	558.091	0.105814
$304$	−2697.42	−0.508906
$305$	−2847.50	−0.534581
$306$	2892.64	0.540397
$307$	4267.18	0.793292	0.396646	−	0.917972i	$-0.370174\pi$
$307$	4267.18	0.793292	0.396646	+	0.917972i	$0.370174\pi$
$308$	0	0
$309$	−4487.28	−0.826124
$310$	−70.4084	−0.0128998
$311$	−6964.77	−1.26989	−0.634945	−	0.772557i	$-0.718977\pi$
$311$	−6964.77	−1.26989	−0.634945	+	0.772557i	$0.718977\pi$
$312$	−4825.15	−0.875546
$313$	−3102.54	−0.560274	−0.280137	−	0.959960i	$-0.590380\pi$
$313$	−3102.54	−0.560274	−0.280137	+	0.959960i	$0.590380\pi$
$314$	1450.88	0.260757
$315$	0	0
$316$	2119.00	0.377225
$317$	4078.25	0.722579	0.361289	−	0.932454i	$-0.382337\pi$
$317$	4078.25	0.722579	0.361289	+	0.932454i	$0.382337\pi$
$318$	2322.54	0.409565
$319$	−13549.1	−2.37806
$320$	3993.90	0.697706
$321$	3928.11	0.683008
$322$	0	0
$323$	−10690.5	−1.84159
$324$	−38.7223	−0.00663963
$325$	3481.92	0.594284
$326$	−3293.12	−0.559476
$327$	1498.09	0.253347
$328$	−2677.31	−0.450701
$329$	0	0
$330$	3391.98	0.565826
$331$	−3138.25	−0.521129	−0.260565	−	0.965456i	$-0.583909\pi$
$331$	−3138.25	−0.521129	−0.260565	+	0.965456i	$0.583909\pi$
$332$	4169.55	0.689258
$333$	−1146.96	−0.188748
$334$	1318.18	0.215951
$335$	5902.93	0.962721
$336$	0	0
$337$	−2545.47	−0.411455	−0.205728	−	0.978609i	$-0.565956\pi$
$337$	−2545.47	−0.411455	−0.205728	+	0.978609i	$0.565956\pi$
$338$	3532.26	0.568431
$339$	1404.89	0.225082
$340$	2538.02	0.404833
$341$	251.381	0.0399210
$342$	−4527.27	−0.715809
$343$	0	0
$344$	11192.0	1.75416
$345$	−609.401	−0.0950987
$346$	7371.96	1.14543
$347$	6093.52	0.942702	0.471351	−	0.881946i	$-0.343767\pi$
$347$	6093.52	0.942702	0.471351	+	0.881946i	$0.343767\pi$
$348$	2582.00	0.397729
$349$	1519.06	0.232990	0.116495	−	0.993191i	$-0.462834\pi$
$349$	1519.06	0.232990	0.116495	+	0.993191i	$0.462834\pi$
$350$	0	0
$351$	−8720.42	−1.32610
$352$	−9303.47	−1.40874
$353$	−7226.42	−1.08959	−0.544793	−	0.838571i	$-0.683392\pi$
$353$	−7226.42	−1.08959	−0.544793	+	0.838571i	$0.683392\pi$
$354$	−1118.22	−0.167888
$355$	3099.10	0.463333
$356$	−62.6916	−0.00933328
$357$	0	0
$358$	4333.97	0.639825
$359$	−8031.14	−1.18069	−0.590344	−	0.807151i	$-0.701008\pi$
$359$	−8031.14	−1.18069	−0.590344	+	0.807151i	$0.701008\pi$
$360$	3404.61	0.498441
$361$	9872.64	1.43937
$362$	−4193.54	−0.608860
$363$	−7872.20	−1.13825
$364$	0	0
$365$	3007.62	0.431304
$366$	2262.15	0.323073
$367$	−5319.27	−0.756576	−0.378288	−	0.925688i	$-0.623487\pi$
$367$	−5319.27	−0.756576	−0.378288	+	0.925688i	$0.623487\pi$
$368$	−479.630	−0.0679414
$369$	−1860.02	−0.262409
$370$	1175.04	0.165101
$371$	0	0
$372$	−47.9049	−0.00667675
$373$	9009.21	1.25061	0.625307	−	0.780379i	$-0.284974\pi$
$373$	9009.21	1.25061	0.625307	+	0.780379i	$0.284974\pi$
$374$	10580.5	1.46285
$375$	4789.51	0.659545
$376$	−6989.45	−0.958653
$377$	−13717.9	−1.87402
$378$	0	0
$379$	−8651.75	−1.17259	−0.586293	−	0.810099i	$-0.699413\pi$
$379$	−8651.75	−1.17259	−0.586293	+	0.810099i	$0.699413\pi$
$380$	−3972.25	−0.536242
$381$	−3806.33	−0.511822
$382$	−4757.15	−0.637165
$383$	12220.3	1.63036	0.815182	−	0.579205i	$-0.196637\pi$
$383$	12220.3	1.63036	0.815182	+	0.579205i	$0.196637\pi$
$384$	670.155	0.0890591
$385$	0	0
$386$	−9605.97	−1.26666
$387$	7775.44	1.02131
$388$	−6435.15	−0.841999
$389$	4707.72	0.613601	0.306801	−	0.951774i	$-0.400742\pi$
$389$	4707.72	0.613601	0.306801	+	0.951774i	$0.400742\pi$
$390$	3434.25	0.445898
$391$	−1900.88	−0.245861
$392$	0	0
$393$	−694.517	−0.0891445
$394$	6919.50	0.884770
$395$	−4777.34	−0.608542
$396$	−3837.45	−0.486968
$397$	7998.65	1.01119	0.505593	−	0.862772i	$-0.331274\pi$
$397$	7998.65	1.01119	0.505593	+	0.862772i	$0.331274\pi$
$398$	8722.72	1.09857
$399$	0	0
$400$	1162.91	0.145364
$401$	14770.0	1.83935	0.919673	−	0.392685i	$-0.128454\pi$
$401$	14770.0	1.83935	0.919673	+	0.392685i	$0.128454\pi$
$402$	−4689.49	−0.581817
$403$	254.514	0.0314596
$404$	646.840	0.0796571
$405$	87.3004	0.0107111
$406$	0	0
$407$	−4195.27	−0.510938
$408$	−6386.84	−0.774990
$409$	7284.57	0.880682	0.440341	−	0.897831i	$-0.354858\pi$
$409$	7284.57	0.880682	0.440341	+	0.897831i	$0.354858\pi$
$410$	1905.55	0.229533
$411$	−7196.08	−0.863641
$412$	−5200.85	−0.621912
$413$	0	0
$414$	−804.999	−0.0955641
$415$	−9400.34	−1.11191
$416$	−9419.41	−1.11015
$417$	−4244.46	−0.498446
$418$	−16559.5	−1.93769
$419$	−14446.4	−1.68438	−0.842189	−	0.539183i	$-0.818733\pi$
$419$	−14446.4	−1.68438	−0.842189	+	0.539183i	$0.818733\pi$
$420$	0	0
$421$	3858.77	0.446710	0.223355	−	0.974737i	$-0.428299\pi$
$421$	3858.77	0.446710	0.223355	+	0.974737i	$0.428299\pi$
$422$	10705.5	1.23492
$423$	−4855.81	−0.558150
$424$	8526.84	0.976651
$425$	4608.87	0.526031
$426$	−2462.03	−0.280014
$427$	0	0
$428$	4552.76	0.514173
$429$	−12261.4	−1.37992
$430$	−7965.76	−0.893356
$431$	−5718.65	−0.639113	−0.319556	−	0.947567i	$-0.603534\pi$
$431$	−5718.65	−0.639113	−0.319556	+	0.947567i	$0.603534\pi$
$432$	−2912.49	−0.324368
$433$	16875.8	1.87297	0.936486	−	0.350704i	$-0.114058\pi$
$433$	16875.8	1.87297	0.936486	+	0.350704i	$0.114058\pi$
$434$	0	0
$435$	−5821.17	−0.641618
$436$	1736.32	0.190722
$437$	2975.07	0.325668
$438$	−2389.36	−0.260658
$439$	11618.0	1.26310	0.631548	−	0.775337i	$-0.282420\pi$
$439$	11618.0	1.26310	0.631548	+	0.775337i	$0.282420\pi$
$440$	12453.1	1.34927
$441$	0	0
$442$	10712.3	1.15279
$443$	792.562	0.0850017	0.0425008	−	0.999096i	$-0.486467\pi$
$443$	792.562	0.0850017	0.0425008	+	0.999096i	$0.486467\pi$
$444$	799.478	0.0854539
$445$	141.340	0.0150565
$446$	−8479.40	−0.900250
$447$	7330.83	0.775697
$448$	0	0
$449$	−12469.3	−1.31061	−0.655303	−	0.755366i	$-0.727459\pi$
$449$	−12469.3	−1.31061	−0.655303	+	0.755366i	$0.727459\pi$
$450$	1951.80	0.204463
$451$	−6803.44	−0.710336
$452$	1628.29	0.169444
$453$	7194.80	0.746227
$454$	10197.1	1.05413
$455$	0	0
$456$	9996.04	1.02655
$457$	15603.8	1.59719	0.798596	−	0.601867i	$-0.205576\pi$
$457$	15603.8	1.59719	0.798596	+	0.601867i	$0.205576\pi$
$458$	4625.16	0.471877
$459$	−11542.8	−1.17380
$460$	−706.309	−0.0715909
$461$	13552.3	1.36919	0.684594	−	0.728925i	$-0.259980\pi$
$461$	13552.3	1.36919	0.684594	+	0.728925i	$0.259980\pi$
$462$	0	0
$463$	−3954.05	−0.396890	−0.198445	−	0.980112i	$-0.563589\pi$
$463$	−3954.05	−0.396890	−0.198445	+	0.980112i	$0.563589\pi$
$464$	−4581.56	−0.458391
$465$	108.003	0.0107710
$466$	5587.84	0.555475
$467$	−7116.40	−0.705155	−0.352578	−	0.935783i	$-0.614695\pi$
$467$	−7116.40	−0.705155	−0.352578	+	0.935783i	$0.614695\pi$
$468$	−3885.27	−0.383754
$469$	0	0
$470$	4974.67	0.488222
$471$	−2225.57	−0.217725
$472$	−4105.35	−0.400348
$473$	28440.4	2.76468
$474$	3795.29	0.367771
$475$	−7213.34	−0.696780
$476$	0	0
$477$	5923.89	0.568629
$478$	12508.9	1.19695
$479$	−2061.91	−0.196683	−0.0983416	−	0.995153i	$-0.531354\pi$
$479$	−2061.91	−0.196683	−0.0983416	+	0.995153i	$0.531354\pi$
$480$	−3997.11	−0.380089
$481$	−4247.55	−0.402643
$482$	−5921.10	−0.559541
$483$	0	0
$484$	−9124.05	−0.856879
$485$	14508.2	1.35832
$486$	−7897.40	−0.737106
$487$	−20011.5	−1.86203	−0.931014	−	0.364984i	$-0.881075\pi$
$487$	−20011.5	−1.86203	−0.931014	+	0.364984i	$0.881075\pi$
$488$	8305.14	0.770402
$489$	5051.47	0.467148
$490$	0	0
$491$	10813.2	0.993877	0.496939	−	0.867786i	$-0.334457\pi$
$491$	10813.2	0.993877	0.496939	+	0.867786i	$0.334457\pi$
$492$	1296.51	0.118803
$493$	−18157.8	−1.65879
$494$	−16765.9	−1.52699
$495$	8651.62	0.785579
$496$	85.0037	0.00769512
$497$	0	0
$498$	7467.96	0.671983
$499$	−6222.30	−0.558213	−0.279107	−	0.960260i	$-0.590038\pi$
$499$	−6222.30	−0.558213	−0.279107	+	0.960260i	$0.590038\pi$
$500$	5551.15	0.496510
$501$	−2022.01	−0.180313
$502$	−9985.11	−0.887764
$503$	14068.1	1.24705	0.623524	−	0.781805i	$-0.285701\pi$
$503$	14068.1	1.24705	0.623524	+	0.781805i	$0.285701\pi$
$504$	0	0
$505$	−1458.31	−0.128503
$506$	−2944.46	−0.258691
$507$	−5418.30	−0.474625
$508$	−4411.62	−0.385303
$509$	11273.1	0.981671	0.490835	−	0.871252i	$-0.336692\pi$
$509$	11273.1	0.981671	0.490835	+	0.871252i	$0.336692\pi$
$510$	4545.77	0.394686
$511$	0	0
$512$	−7194.61	−0.621015
$513$	18065.7	1.55481
$514$	−6402.06	−0.549383
$515$	11725.4	1.00327
$516$	−5419.80	−0.462390
$517$	−17761.2	−1.51090
$518$	0	0
$519$	−11308.2	−0.956405
$520$	12608.3	1.06329
$521$	8687.87	0.730561	0.365281	−	0.930897i	$-0.380973\pi$
$521$	8687.87	0.730561	0.365281	+	0.930897i	$0.380973\pi$
$522$	−7689.57	−0.644758
$523$	19685.6	1.64587	0.822934	−	0.568137i	$-0.192336\pi$
$523$	19685.6	1.64587	0.822934	+	0.568137i	$0.192336\pi$
$524$	−804.961	−0.0671085
$525$	0	0
$526$	2177.64	0.180513
$527$	336.889	0.0278465
$528$	−4095.12	−0.337533
$529$	529.000	0.0434783
$530$	−6068.89	−0.497388
$531$	−2852.13	−0.233092
$532$	0	0
$533$	−6888.22	−0.559779
$534$	−112.285	−0.00909935
$535$	−10264.3	−0.829467
$536$	−17216.7	−1.38741
$537$	−6648.07	−0.534237
$538$	−2864.12	−0.229519
$539$	0	0
$540$	−4288.96	−0.341792
$541$	10418.4	0.827951	0.413975	−	0.910288i	$-0.364140\pi$
$541$	10418.4	0.827951	0.413975	+	0.910288i	$0.364140\pi$
$542$	−10197.1	−0.808122
$543$	6432.66	0.508383
$544$	−12468.1	−0.982654
$545$	−3914.57	−0.307673
$546$	0	0
$547$	3640.95	0.284600	0.142300	−	0.989824i	$-0.454550\pi$
$547$	3640.95	0.284600	0.142300	+	0.989824i	$0.454550\pi$
$548$	−8340.41	−0.650154
$549$	5769.86	0.448546
$550$	7139.13	0.553479
$551$	28418.7	2.19724
$552$	1777.41	0.137050
$553$	0	0
$554$	3643.92	0.279450
$555$	−1802.44	−0.137855
$556$	−4919.42	−0.375233
$557$	−5035.32	−0.383040	−0.191520	−	0.981489i	$-0.561342\pi$
$557$	−5035.32	−0.383040	−0.191520	+	0.981489i	$0.561342\pi$
$558$	142.668	0.0108237
$559$	28794.8	2.17870
$560$	0	0
$561$	−16229.9	−1.22144
$562$	−12205.9	−0.916145
$563$	−10851.4	−0.812309	−0.406155	−	0.913804i	$-0.633131\pi$
$563$	−10851.4	−0.812309	−0.406155	+	0.913804i	$0.633131\pi$
$564$	3384.70	0.252697
$565$	−3671.02	−0.273347
$566$	560.964	0.0416591
$567$	0	0
$568$	−9038.98	−0.667724
$569$	1829.57	0.134797	0.0673987	−	0.997726i	$-0.478530\pi$
$569$	1829.57	0.134797	0.0673987	+	0.997726i	$0.478530\pi$
$570$	−7114.58	−0.522802
$571$	1621.95	0.118873	0.0594365	−	0.998232i	$-0.481070\pi$
$571$	1621.95	0.118873	0.0594365	+	0.998232i	$0.481070\pi$
$572$	−14211.3	−1.03882
$573$	7297.21	0.532016
$574$	0	0
$575$	−1282.61	−0.0930236
$576$	−8092.81	−0.585417
$577$	7394.69	0.533527	0.266763	−	0.963762i	$-0.414046\pi$
$577$	7394.69	0.533527	0.266763	+	0.963762i	$0.414046\pi$
$578$	3980.60	0.286455
$579$	14735.0	1.05763
$580$	−6746.86	−0.483014
$581$	0	0
$582$	−11525.8	−0.820895
$583$	21668.0	1.53927
$584$	−8772.16	−0.621566
$585$	8759.43	0.619073
$586$	2722.54	0.191923
$587$	−9876.78	−0.694478	−0.347239	−	0.937777i	$-0.612881\pi$
$587$	−9876.78	−0.694478	−0.347239	+	0.937777i	$0.612881\pi$
$588$	0	0
$589$	−527.264	−0.0368855
$590$	2921.94	0.203889
$591$	−10614.1	−0.738760
$592$	−1418.61	−0.0984877
$593$	10088.3	0.698612	0.349306	−	0.937009i	$-0.386417\pi$
$593$	10088.3	0.698612	0.349306	+	0.937009i	$0.386417\pi$
$594$	−17879.8	−1.23505
$595$	0	0
$596$	8496.59	0.583950
$597$	−13380.2	−0.917277
$598$	−2981.16	−0.203860
$599$	7199.57	0.491096	0.245548	−	0.969384i	$-0.421032\pi$
$599$	7199.57	0.491096	0.245548	+	0.969384i	$0.421032\pi$
$600$	−4309.49	−0.293224
$601$	11500.4	0.780551	0.390275	−	0.920698i	$-0.372380\pi$
$601$	11500.4	0.780551	0.390275	+	0.920698i	$0.372380\pi$
$602$	0	0
$603$	−11961.1	−0.807781
$604$	8338.92	0.561765
$605$	20570.4	1.38232
$606$	1158.54	0.0776606
$607$	−28576.9	−1.91088	−0.955439	−	0.295190i	$-0.904617\pi$
$607$	−28576.9	−1.91088	−0.955439	+	0.295190i	$0.904617\pi$
$608$	19513.7	1.30162
$609$	0	0
$610$	−5911.10	−0.392350
$611$	−17982.5	−1.19066
$612$	−5142.76	−0.339679
$613$	−19708.3	−1.29855	−0.649276	−	0.760553i	$-0.724928\pi$
$613$	−19708.3	−1.29855	−0.649276	+	0.760553i	$0.724928\pi$
$614$	8858.20	0.582228
$615$	−2923.01	−0.191654
$616$	0	0
$617$	10092.5	0.658520	0.329260	−	0.944239i	$-0.393201\pi$
$617$	10092.5	0.658520	0.329260	+	0.944239i	$0.393201\pi$
$618$	−9315.10	−0.606324
$619$	10262.2	0.666353	0.333176	−	0.942864i	$-0.391880\pi$
$619$	10262.2	0.666353	0.333176	+	0.942864i	$0.391880\pi$
$620$	125.177	0.00810846
$621$	3212.28	0.207575
$622$	−14458.1	−0.932021
$623$	0	0
$624$	−4146.15	−0.265992
$625$	−5544.48	−0.354847
$626$	−6440.53	−0.411206
$627$	25401.4	1.61792
$628$	−2579.48	−0.163905
$629$	−5622.29	−0.356400
$630$	0	0
$631$	−6672.46	−0.420961	−0.210480	−	0.977598i	$-0.567503\pi$
$631$	−6672.46	−0.420961	−0.210480	+	0.977598i	$0.567503\pi$
$632$	13933.8	0.876989
$633$	−16421.7	−1.03113
$634$	8466.01	0.530328
$635$	9946.09	0.621572
$636$	−4129.19	−0.257442
$637$	0	0
$638$	−28126.4	−1.74535
$639$	−6279.68	−0.388764
$640$	−1751.14	−0.108156
$641$	1384.37	0.0853032	0.0426516	−	0.999090i	$-0.486419\pi$
$641$	1384.37	0.0853032	0.0426516	+	0.999090i	$0.486419\pi$
$642$	8154.32	0.501286
$643$	−8819.65	−0.540922	−0.270461	−	0.962731i	$-0.587176\pi$
$643$	−8819.65	−0.540922	−0.270461	+	0.962731i	$0.587176\pi$
$644$	0	0
$645$	12219.1	0.745930
$646$	−22192.2	−1.35161
$647$	−9653.15	−0.586560	−0.293280	−	0.956027i	$-0.594747\pi$
$647$	−9653.15	−0.586560	−0.293280	+	0.956027i	$0.594747\pi$
$648$	−254.624	−0.0154361
$649$	−10432.3	−0.630977
$650$	7228.09	0.436168
$651$	0	0
$652$	5854.77	0.351672
$653$	−15167.6	−0.908963	−0.454481	−	0.890756i	$-0.650175\pi$
$653$	−15167.6	−0.908963	−0.454481	+	0.890756i	$0.650175\pi$
$654$	3109.87	0.185941
$655$	1814.80	0.108260
$656$	−2300.56	−0.136923
$657$	−6094.32	−0.361890
$658$	0	0
$659$	−25102.5	−1.48385	−0.741924	−	0.670483i	$-0.766087\pi$
$659$	−25102.5	−1.48385	−0.741924	+	0.670483i	$0.766087\pi$
$660$	−6030.53	−0.355664
$661$	−773.543	−0.0455179	−0.0227590	−	0.999741i	$-0.507245\pi$
$661$	−773.543	−0.0455179	−0.0227590	+	0.999741i	$0.507245\pi$
$662$	−6514.66	−0.382477
$663$	−16432.2	−0.962552
$664$	27417.5	1.60241
$665$	0	0
$666$	−2380.97	−0.138529
$667$	5053.16	0.293342
$668$	−2343.56	−0.135741
$669$	13006.9	0.751685
$670$	12253.8	0.706577
$671$	21104.6	1.21421
$672$	0	0
$673$	−15576.5	−0.892169	−0.446084	−	0.894991i	$-0.647182\pi$
$673$	−15576.5	−0.892169	−0.446084	+	0.894991i	$0.647182\pi$
$674$	−5284.11	−0.301983
$675$	−7788.47	−0.444116
$676$	−6279.92	−0.357301
$677$	2931.31	0.166409	0.0832047	−	0.996532i	$-0.473484\pi$
$677$	2931.31	0.166409	0.0832047	+	0.996532i	$0.473484\pi$
$678$	2916.39	0.165197
$679$	0	0
$680$	16689.1	0.941172
$681$	−15641.8	−0.880169
$682$	521.840	0.0292996
$683$	−927.185	−0.0519440	−0.0259720	−	0.999663i	$-0.508268\pi$
$683$	−927.185	−0.0519440	−0.0259720	+	0.999663i	$0.508268\pi$
$684$	8048.93	0.449939
$685$	18803.6	1.04883
$686$	0	0
$687$	−7094.75	−0.394005
$688$	9617.03	0.532915
$689$	21938.0	1.21302
$690$	−1265.05	−0.0697966
$691$	9100.13	0.500991	0.250496	−	0.968118i	$-0.419406\pi$
$691$	9100.13	0.500991	0.250496	+	0.968118i	$0.419406\pi$
$692$	−13106.4	−0.719988
$693$	0	0
$694$	12649.5	0.691885
$695$	11090.9	0.605328
$696$	16978.3	0.924655
$697$	−9117.64	−0.495488
$698$	3153.40	0.171000
$699$	−8571.44	−0.463808
$700$	0	0
$701$	35113.0	1.89187	0.945935	−	0.324356i	$-0.105148\pi$
$701$	35113.0	1.89187	0.945935	+	0.324356i	$0.105148\pi$
$702$	−18102.6	−0.973277
$703$	8799.44	0.472087
$704$	−29601.3	−1.58472
$705$	−7630.87	−0.407653
$706$	−15001.3	−0.799688
$707$	0	0
$708$	1988.05	0.105530
$709$	33963.4	1.79905	0.899523	−	0.436874i	$-0.143914\pi$
$709$	33963.4	1.79905	0.899523	+	0.436874i	$0.143914\pi$
$710$	6433.40	0.340058
$711$	9680.29	0.510604
$712$	−412.238	−0.0216984
$713$	−93.7534	−0.00492439
$714$	0	0
$715$	32039.6	1.67582
$716$	−7705.26	−0.402177
$717$	−19188.0	−0.999427
$718$	−16671.8	−0.866553
$719$	−31871.1	−1.65312	−0.826558	−	0.562852i	$-0.809704\pi$
$719$	−31871.1	−1.65312	−0.826558	+	0.562852i	$0.809704\pi$
$720$	2925.51	0.151427
$721$	0	0
$722$	20494.5	1.05641
$723$	9082.64	0.467202
$724$	7455.59	0.382714
$725$	−12251.9	−0.627617
$726$	−16341.8	−0.835403
$727$	−4704.97	−0.240024	−0.120012	−	0.992772i	$-0.538293\pi$
$727$	−4704.97	−0.240024	−0.120012	+	0.992772i	$0.538293\pi$
$728$	0	0
$729$	11830.9	0.601072
$730$	6243.49	0.316551
$731$	38114.5	1.92847
$732$	−4021.83	−0.203075
$733$	13887.8	0.699808	0.349904	−	0.936786i	$-0.386214\pi$
$733$	13887.8	0.699808	0.349904	+	0.936786i	$0.386214\pi$
$734$	−11042.2	−0.555280
$735$	0	0
$736$	3469.76	0.173773
$737$	−43750.3	−2.18665
$738$	−3861.20	−0.192592
$739$	−3740.36	−0.186186	−0.0930929	−	0.995657i	$-0.529675\pi$
$739$	−3740.36	−0.186186	−0.0930929	+	0.995657i	$0.529675\pi$
$740$	−2089.07	−0.103778
$741$	25717.9	1.27500
$742$	0	0
$743$	−9215.34	−0.455018	−0.227509	−	0.973776i	$-0.573058\pi$
$743$	−9215.34	−0.455018	−0.227509	+	0.973776i	$0.573058\pi$
$744$	−315.005	−0.0155224
$745$	−19155.8	−0.942031
$746$	18702.1	0.917874
$747$	19047.8	0.932964
$748$	−18810.8	−0.919507
$749$	0	0
$750$	9942.51	0.484065
$751$	12001.6	0.583150	0.291575	−	0.956548i	$-0.405821\pi$
$751$	12001.6	0.583150	0.291575	+	0.956548i	$0.405821\pi$
$752$	−6005.89	−0.291240
$753$	15316.6	0.741260
$754$	−28476.8	−1.37542
$755$	−18800.3	−0.906242
$756$	0	0
$757$	−18753.4	−0.900400	−0.450200	−	0.892928i	$-0.648647\pi$
$757$	−18753.4	−0.900400	−0.450200	+	0.892928i	$0.648647\pi$
$758$	−17960.1	−0.860606
$759$	4516.65	0.216000
$760$	−26120.1	−1.24668
$761$	−2081.91	−0.0991709	−0.0495855	−	0.998770i	$-0.515790\pi$
$761$	−2081.91	−0.0991709	−0.0495855	+	0.998770i	$0.515790\pi$
$762$	−7901.52	−0.375646
$763$	0	0
$764$	8457.62	0.400505
$765$	11594.5	0.547973
$766$	25368.1	1.19659
$767$	−10562.3	−0.497239
$768$	13618.8	0.639876
$769$	−10435.2	−0.489339	−0.244669	−	0.969607i	$-0.578679\pi$
$769$	−10435.2	−0.489339	−0.244669	+	0.969607i	$0.578679\pi$
$770$	0	0
$771$	9820.42	0.458721
$772$	17078.2	0.796191
$773$	−2424.71	−0.112821	−0.0564107	−	0.998408i	$-0.517966\pi$
$773$	−2424.71	−0.112821	−0.0564107	+	0.998408i	$0.517966\pi$
$774$	16141.0	0.749580
$775$	227.314	0.0105359
$776$	−42315.3	−1.95751
$777$	0	0
$778$	9772.71	0.450345
$779$	14270.0	0.656323
$780$	−6105.68	−0.280280
$781$	−22969.4	−1.05238
$782$	−3946.03	−0.180447
$783$	30684.6	1.40048
$784$	0	0
$785$	5815.49	0.264413
$786$	−1441.74	−0.0654265
$787$	463.211	0.0209806	0.0104903	−	0.999945i	$-0.496661\pi$
$787$	463.211	0.0209806	0.0104903	+	0.999945i	$0.496661\pi$
$788$	−12302.0	−0.556143
$789$	−3340.38	−0.150723
$790$	−9917.24	−0.446632
$791$	0	0
$792$	−25233.7	−1.13212
$793$	21367.6	0.956853
$794$	16604.3	0.742148
$795$	9309.35	0.415307
$796$	−15507.9	−0.690532
$797$	−8642.97	−0.384128	−0.192064	−	0.981382i	$-0.561518\pi$
$797$	−8642.97	−0.384128	−0.192064	+	0.981382i	$0.561518\pi$
$798$	0	0
$799$	−23802.7	−1.05392
$800$	−8412.76	−0.371795
$801$	−286.395	−0.0126333
$802$	30660.9	1.34997
$803$	−22291.3	−0.979632
$804$	8337.34	0.365715
$805$	0	0
$806$	528.343	0.0230894
$807$	4393.41	0.191642
$808$	4253.38	0.185190
$809$	−34851.3	−1.51459	−0.757296	−	0.653071i	$-0.773480\pi$
$809$	−34851.3	−1.51459	−0.757296	+	0.653071i	$0.773480\pi$
$810$	181.226	0.00786128
$811$	21683.0	0.938831	0.469416	−	0.882977i	$-0.344465\pi$
$811$	21683.0	0.938831	0.469416	+	0.882977i	$0.344465\pi$
$812$	0	0
$813$	15641.8	0.674761
$814$	−8708.92	−0.374997
$815$	−13199.7	−0.567320
$816$	−5488.08	−0.235443
$817$	−59652.9	−2.55446
$818$	15122.0	0.646366
$819$	0	0
$820$	−3387.83	−0.144278
$821$	24106.9	1.02477	0.512385	−	0.858756i	$-0.328762\pi$
$821$	24106.9	1.02477	0.512385	+	0.858756i	$0.328762\pi$
$822$	−14938.3	−0.633859
$823$	40470.8	1.71412	0.857062	−	0.515213i	$-0.172287\pi$
$823$	40470.8	1.71412	0.857062	+	0.515213i	$0.172287\pi$
$824$	−34198.9	−1.44585
$825$	−10951.0	−0.462141
$826$	0	0
$827$	−7861.00	−0.330537	−0.165268	−	0.986249i	$-0.552849\pi$
$827$	−7861.00	−0.330537	−0.165268	+	0.986249i	$0.552849\pi$
$828$	1431.19	0.0600691
$829$	−21327.3	−0.893522	−0.446761	−	0.894653i	$-0.647423\pi$
$829$	−21327.3	−0.893522	−0.446761	+	0.894653i	$0.647423\pi$
$830$	−19514.1	−0.816077
$831$	−5589.58	−0.233334
$832$	−29970.1	−1.24883
$833$	0	0
$834$	−8811.03	−0.365829
$835$	5283.61	0.218978
$836$	29440.8	1.21798
$837$	−569.304	−0.0235102
$838$	−29989.2	−1.23623
$839$	−5562.15	−0.228876	−0.114438	−	0.993430i	$-0.536507\pi$
$839$	−5562.15	−0.228876	−0.114438	+	0.993430i	$0.536507\pi$
$840$	0	0
$841$	23880.2	0.979137
$842$	8010.39	0.327858
$843$	18723.2	0.764958
$844$	−19033.1	−0.776238
$845$	14158.2	0.576400
$846$	−10080.1	−0.409648
$847$	0	0
$848$	7326.94	0.296708
$849$	−860.488	−0.0347843
$850$	9567.51	0.386074
$851$	1564.64	0.0630259
$852$	4377.19	0.176010
$853$	2170.43	0.0871207	0.0435603	−	0.999051i	$-0.486130\pi$
$853$	2170.43	0.0871207	0.0435603	+	0.999051i	$0.486130\pi$
$854$	0	0
$855$	−18146.5	−0.725845
$856$	29937.3	1.19537
$857$	26643.0	1.06197	0.530985	−	0.847381i	$-0.321822\pi$
$857$	26643.0	1.06197	0.530985	+	0.847381i	$0.321822\pi$
$858$	−25453.4	−1.01278
$859$	25875.8	1.02779	0.513895	−	0.857853i	$-0.328202\pi$
$859$	25875.8	1.02779	0.513895	+	0.857853i	$0.328202\pi$
$860$	14162.1	0.561541
$861$	0	0
$862$	−11871.3	−0.469069
$863$	−23408.2	−0.923320	−0.461660	−	0.887057i	$-0.652746\pi$
$863$	−23408.2	−0.923320	−0.461660	+	0.887057i	$0.652746\pi$
$864$	21069.6	0.829633
$865$	29548.7	1.16149
$866$	35032.2	1.37465
$867$	−6106.03	−0.239183
$868$	0	0
$869$	35407.8	1.38220
$870$	−12084.1	−0.470908
$871$	−44295.4	−1.72318
$872$	11417.4	0.443397
$873$	−29397.9	−1.13971
$874$	6175.92	0.239020
$875$	0	0
$876$	4247.99	0.163843
$877$	767.049	0.0295341	0.0147671	−	0.999891i	$-0.495299\pi$
$877$	767.049	0.0295341	0.0147671	+	0.999891i	$0.495299\pi$
$878$	24117.8	0.927034
$879$	−4176.22	−0.160251
$880$	10700.7	0.409911
$881$	−10365.6	−0.396396	−0.198198	−	0.980162i	$-0.563509\pi$
$881$	−10365.6	−0.396396	−0.198198	+	0.980162i	$0.563509\pi$
$882$	0	0
$883$	−31628.8	−1.20543	−0.602714	−	0.797957i	$-0.705914\pi$
$883$	−31628.8	−1.20543	−0.602714	+	0.797957i	$0.705914\pi$
$884$	−19045.2	−0.724615
$885$	−4482.10	−0.170242
$886$	1645.27	0.0623860
$887$	4816.21	0.182314	0.0911571	−	0.995837i	$-0.470943\pi$
$887$	4816.21	0.182314	0.0911571	+	0.995837i	$0.470943\pi$
$888$	5257.08	0.198667
$889$	0	0
$890$	293.406	0.0110505
$891$	−647.037	−0.0243284
$892$	15075.3	0.565874
$893$	37253.6	1.39602
$894$	15218.0	0.569314
$895$	17371.7	0.648795
$896$	0	0
$897$	4572.93	0.170218
$898$	−25884.9	−0.961904
$899$	−895.559	−0.0332242
$900$	−3470.05	−0.128520
$901$	29038.3	1.07370
$902$	−14123.2	−0.521343
$903$	0	0
$904$	10707.1	0.393929
$905$	−16808.8	−0.617396
$906$	14935.6	0.547685
$907$	−14685.0	−0.537605	−0.268803	−	0.963195i	$-0.586628\pi$
$907$	−14685.0	−0.537605	−0.268803	+	0.963195i	$0.586628\pi$
$908$	−18129.2	−0.662597
$909$	2954.97	0.107822
$910$	0	0
$911$	9884.25	0.359473	0.179737	−	0.983715i	$-0.442475\pi$
$911$	9884.25	0.359473	0.179737	+	0.983715i	$0.442475\pi$
$912$	8589.39	0.311868
$913$	69671.8	2.52552
$914$	32391.9	1.17224
$915$	9067.30	0.327602
$916$	−8222.96	−0.296610
$917$	0	0
$918$	−23961.7	−0.861496
$919$	−2164.32	−0.0776869	−0.0388434	−	0.999245i	$-0.512367\pi$
$919$	−2164.32	−0.0776869	−0.0388434	+	0.999245i	$0.512367\pi$
$920$	−4644.43	−0.166437
$921$	−13588.0	−0.486145
$922$	28133.2	1.00490
$923$	−23255.6	−0.829325
$924$	0	0
$925$	−3793.61	−0.134847
$926$	−8208.17	−0.291293
$927$	−23759.2	−0.841805
$928$	33144.1	1.17242
$929$	52478.9	1.85336	0.926682	−	0.375845i	$-0.122648\pi$
$929$	52478.9	1.85336	0.926682	+	0.375845i	$0.122648\pi$
$930$	224.202	0.00790523
$931$	0	0
$932$	−9934.48	−0.349157
$933$	22178.0	0.778214
$934$	−14772.9	−0.517540
$935$	42409.4	1.48335
$936$	−25548.1	−0.892166
$937$	53501.9	1.86535	0.932674	−	0.360720i	$-0.117469\pi$
$937$	53501.9	1.86535	0.932674	+	0.360720i	$0.117469\pi$
$938$	0	0
$939$	9879.42	0.343347
$940$	−8844.34	−0.306884
$941$	32401.0	1.12247	0.561235	−	0.827657i	$-0.310327\pi$
$941$	32401.0	1.12247	0.561235	+	0.827657i	$0.310327\pi$
$942$	−4620.03	−0.159797
$943$	2537.36	0.0876224
$944$	−3527.65	−0.121626
$945$	0	0
$946$	59039.2	2.02910
$947$	26682.1	0.915577	0.457789	−	0.889061i	$-0.348642\pi$
$947$	26682.1	0.915577	0.457789	+	0.889061i	$0.348642\pi$
$948$	−6747.55	−0.231171
$949$	−22569.1	−0.771997
$950$	−14974.1	−0.511394
$951$	−12986.4	−0.442811
$952$	0	0
$953$	−10364.6	−0.352300	−0.176150	−	0.984363i	$-0.556364\pi$
$953$	−10364.6	−0.352300	−0.176150	+	0.984363i	$0.556364\pi$
$954$	12297.3	0.417339
$955$	−19067.9	−0.646097
$956$	−22239.3	−0.752375
$957$	43144.3	1.45732
$958$	−4280.31	−0.144353
$959$	0	0
$960$	−12717.8	−0.427568
$961$	−29774.4	−0.999442
$962$	−8817.44	−0.295515
$963$	20798.5	0.695973
$964$	10527.0	0.351713
$965$	−38503.3	−1.28442
$966$	0	0
$967$	41802.6	1.39016	0.695079	−	0.718933i	$-0.255369\pi$
$967$	41802.6	1.39016	0.695079	+	0.718933i	$0.255369\pi$
$968$	−59996.5	−1.99211
$969$	34041.7	1.12856
$970$	30117.4	0.996920
$971$	−13823.4	−0.456865	−0.228432	−	0.973560i	$-0.573360\pi$
$971$	−13823.4	−0.456865	−0.228432	+	0.973560i	$0.573360\pi$
$972$	14040.6	0.463326
$973$	0	0
$974$	−41541.7	−1.36661
$975$	−11087.5	−0.364189
$976$	7136.44	0.234049
$977$	−851.189	−0.0278730	−0.0139365	−	0.999903i	$-0.504436\pi$
$977$	−851.189	−0.0278730	−0.0139365	+	0.999903i	$0.504436\pi$
$978$	10486.3	0.342858
$979$	−1047.56	−0.0341982
$980$	0	0
$981$	7932.07	0.258156
$982$	22447.1	0.729444
$983$	38567.2	1.25138	0.625689	−	0.780073i	$-0.284818\pi$
$983$	38567.2	1.25138	0.625689	+	0.780073i	$0.284818\pi$
$984$	8525.38	0.276198
$985$	27735.2	0.897174
$986$	−37693.5	−1.21745
$987$	0	0
$988$	29807.6	0.959825
$989$	−10606.9	−0.341032
$990$	17959.8	0.576566
$991$	−28327.4	−0.908023	−0.454012	−	0.890996i	$-0.650008\pi$
$991$	−28327.4	−0.908023	−0.454012	+	0.890996i	$0.650008\pi$
$992$	−614.937	−0.0196817
$993$	9993.14	0.319358
$994$	0	0
$995$	34962.9	1.11397
$996$	−13277.1	−0.422391
$997$	−796.270	−0.0252940	−0.0126470	−	0.999920i	$-0.504026\pi$
$997$	−796.270	−0.0252940	−0.0126470	+	0.999920i	$0.504026\pi$
$998$	−12916.8	−0.409694
$999$	9501.04	0.300900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1127.4.a.e.1.5		8
7.6	odd	2		161.4.a.b.1.5	✓	8
21.20	even	2		1449.4.a.i.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.4.a.b.1.5	✓	8	7.6	odd	2
1127.4.a.e.1.5		8	1.1	even	1	trivial
1449.4.a.i.1.4		8	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1127 = 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1127.a (trivial)

\(f(q)\)	\(=\)	\(q+2.07589 q^{2} -3.18430 q^{3} -3.69068 q^{4} +8.32072 q^{5} -6.61027 q^{6} -24.2686 q^{8} -16.8602 q^{9} +17.2729 q^{10} -61.6700 q^{11} +11.7522 q^{12} -62.4385 q^{13} -26.4957 q^{15} -20.8535 q^{16} -82.6471 q^{17} -34.9999 q^{18} +129.351 q^{19} -30.7091 q^{20} -128.020 q^{22} +23.0000 q^{23} +77.2785 q^{24} -55.7657 q^{25} -129.615 q^{26} +139.664 q^{27} +219.702 q^{29} -55.0022 q^{30} -4.07623 q^{31} +150.859 q^{32} +196.376 q^{33} -171.566 q^{34} +62.2256 q^{36} +68.0277 q^{37} +268.518 q^{38} +198.823 q^{39} -201.932 q^{40} +110.320 q^{41} -461.171 q^{43} +227.604 q^{44} -140.289 q^{45} +47.7455 q^{46} +288.004 q^{47} +66.4038 q^{48} -115.763 q^{50} +263.173 q^{51} +230.440 q^{52} -351.353 q^{53} +289.928 q^{54} -513.139 q^{55} -411.893 q^{57} +456.078 q^{58} +169.163 q^{59} +97.7871 q^{60} -342.218 q^{61} -8.46182 q^{62} +479.995 q^{64} -519.533 q^{65} +407.655 q^{66} +709.425 q^{67} +305.024 q^{68} -73.2390 q^{69} +372.456 q^{71} +409.173 q^{72} +361.462 q^{73} +141.218 q^{74} +177.575 q^{75} -477.392 q^{76} +412.735 q^{78} -574.150 q^{79} -173.516 q^{80} +10.4919 q^{81} +229.013 q^{82} -1129.75 q^{83} -687.683 q^{85} -957.341 q^{86} -699.600 q^{87} +1496.64 q^{88} +16.9865 q^{89} -291.225 q^{90} -84.8856 q^{92} +12.9800 q^{93} +597.865 q^{94} +1076.29 q^{95} -480.381 q^{96} +1743.62 q^{97} +1039.77 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} + 24 q^{4} + 24 q^{5} + 41 q^{6} - 69 q^{8} + 95 q^{9} + 30 q^{10} - 98 q^{11} + 131 q^{12} + 145 q^{13} - 232 q^{15} - 76 q^{16} + 96 q^{17} - 69 q^{18} + 226 q^{19} + 22 q^{20} - 98 q^{22}+ \cdots - 1676 q^{99}+O(q^{100}) \) 8 * q + 3 * q^3 + 24 * q^4 + 24 * q^5 + 41 * q^6 - 69 * q^8 + 95 * q^9 + 30 * q^10 - 98 * q^11 + 131 * q^12 + 145 * q^13 - 232 * q^15 - 76 * q^16 + 96 * q^17 - 69 * q^18 + 226 * q^19 + 22 * q^20 - 98 * q^22 + 184 * q^23 + 216 * q^24 + 60 * q^25 + 185 * q^26 + 531 * q^27 + 73 * q^29 - 416 * q^30 + 987 * q^31 + 118 * q^32 - 16 * q^33 + 846 * q^34 - 69 * q^36 - 2 * q^37 + 454 * q^38 - 675 * q^39 + 640 * q^40 + 907 * q^41 + 636 * q^43 + 126 * q^44 + 126 * q^45 - 421 * q^47 - 2165 * q^48 + 2308 * q^50 - 408 * q^51 + 931 * q^52 + 138 * q^53 - 1717 * q^54 + 90 * q^55 + 984 * q^57 + 1329 * q^58 + 924 * q^59 + 2642 * q^60 + 890 * q^61 - 2175 * q^62 + 303 * q^64 - 392 * q^65 - 408 * q^66 - 1392 * q^67 - 2592 * q^68 + 69 * q^69 + 259 * q^71 + 141 * q^72 + 297 * q^73 + 2314 * q^74 + 851 * q^75 + 428 * q^76 + 6789 * q^78 - 1788 * q^79 + 2104 * q^80 + 2300 * q^81 + 219 * q^82 - 2114 * q^83 + 528 * q^85 - 2360 * q^86 + 2441 * q^87 + 2144 * q^88 + 1040 * q^89 - 4872 * q^90 + 552 * q^92 - 621 * q^93 + 1009 * q^94 + 1268 * q^95 + 1163 * q^96 + 1452 * q^97 - 1676 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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