LMFDB - Embedded newform 2057.4.a.s.1.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2057,4,Mod(1,2057)] 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("2057.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2057, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2057 = 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2057.a (trivial)

Newform invariants

comment:select newform

sage:traces = [44,-1,-28,139,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$121.366928882$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 187)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	2057.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.49542 q^{2} -4.41660 q^{3} -5.76373 q^{4} -15.6043 q^{5} +6.60466 q^{6} +31.8747 q^{7} +20.5825 q^{8} -7.49362 q^{9} +23.3349 q^{10} +25.4561 q^{12} +21.7566 q^{13} -47.6660 q^{14} +68.9180 q^{15} +15.3304 q^{16} -17.0000 q^{17} +11.2061 q^{18} -106.640 q^{19} +89.9389 q^{20} -140.778 q^{21} -202.129 q^{23} -90.9048 q^{24} +118.494 q^{25} -32.5353 q^{26} +152.345 q^{27} -183.717 q^{28} +133.356 q^{29} -103.061 q^{30} +96.7146 q^{31} -187.585 q^{32} +25.4221 q^{34} -497.382 q^{35} +43.1912 q^{36} -223.110 q^{37} +159.472 q^{38} -96.0905 q^{39} -321.176 q^{40} +230.504 q^{41} +210.522 q^{42} -419.986 q^{43} +116.933 q^{45} +302.267 q^{46} -136.012 q^{47} -67.7082 q^{48} +672.997 q^{49} -177.198 q^{50} +75.0822 q^{51} -125.399 q^{52} +685.851 q^{53} -227.819 q^{54} +656.062 q^{56} +470.987 q^{57} -199.423 q^{58} -155.901 q^{59} -397.224 q^{60} +377.256 q^{61} -144.629 q^{62} -238.857 q^{63} +157.875 q^{64} -339.497 q^{65} +5.55804 q^{67} +97.9834 q^{68} +892.722 q^{69} +743.794 q^{70} +394.434 q^{71} -154.238 q^{72} +333.509 q^{73} +333.642 q^{74} -523.341 q^{75} +614.645 q^{76} +143.695 q^{78} -449.839 q^{79} -239.220 q^{80} -470.518 q^{81} -344.700 q^{82} -200.319 q^{83} +811.406 q^{84} +265.273 q^{85} +628.054 q^{86} -588.981 q^{87} +1554.53 q^{89} -174.863 q^{90} +693.487 q^{91} +1165.01 q^{92} -427.150 q^{93} +203.395 q^{94} +1664.04 q^{95} +828.490 q^{96} -1688.97 q^{97} -1006.41 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - q^{2} - 28 q^{3} + 139 q^{4} - 24 q^{5} + 24 q^{6} + 2 q^{7} + 63 q^{8} + 356 q^{9} + 65 q^{10} - 337 q^{12} - 12 q^{13} - 151 q^{14} - 320 q^{15} + 311 q^{16} - 748 q^{17} + 55 q^{18} + 36 q^{19}+ \cdots - 2302 q^{98}+O(q^{100}) $ 44 * q - q^2 - 28 * q^3 + 139 * q^4 - 24 * q^5 + 24 * q^6 + 2 * q^7 + 63 * q^8 + 356 * q^9 + 65 * q^10 - 337 * q^12 - 12 * q^13 - 151 * q^14 - 320 * q^15 + 311 * q^16 - 748 * q^17 + 55 * q^18 + 36 * q^19 - 209 * q^20 + 240 * q^21 - 656 * q^23 + 104 * q^24 + 458 * q^25 - 455 * q^26 - 538 * q^27 - 63 * q^28 + 698 * q^29 - 304 * q^30 - 1418 * q^31 + 83 * q^32 + 17 * q^34 - 102 * q^35 - 85 * q^36 - 1358 * q^37 - 568 * q^38 - 114 * q^39 + 281 * q^40 - 54 * q^41 - 1943 * q^42 - 382 * q^43 - 1226 * q^45 - 232 * q^46 - 544 * q^47 - 2743 * q^48 + 864 * q^49 - 1307 * q^50 + 476 * q^51 - 3112 * q^52 - 2492 * q^53 - 2400 * q^54 + 27 * q^56 - 56 * q^57 - 3746 * q^58 - 662 * q^59 - 1392 * q^60 + 964 * q^61 + 3256 * q^62 - 2322 * q^63 - 3653 * q^64 + 152 * q^65 - 3442 * q^67 - 2363 * q^68 - 1664 * q^69 - 3130 * q^70 - 2936 * q^71 + 1281 * q^72 + 2240 * q^73 - 3494 * q^74 - 1892 * q^75 - 1261 * q^76 + 1120 * q^78 + 1016 * q^79 - 3280 * q^80 + 2140 * q^81 - 3985 * q^82 + 1656 * q^83 + 5401 * q^84 + 408 * q^85 - 1373 * q^86 - 1968 * q^87 - 2992 * q^89 - 1287 * q^90 - 8310 * q^91 - 2442 * q^92 - 4058 * q^93 + 694 * q^94 + 8280 * q^95 - 2203 * q^96 - 4746 * q^97 - 2302 * q^98

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 $

$2$	−1.49542	−0.528710	−0.264355	−	0.964425i	$-0.585159\pi$
$2$	−1.49542	−0.528710	−0.264355	+	0.964425i	$0.585159\pi$
$3$	−4.41660	−0.849976	−0.424988	−	0.905199i	$-0.639722\pi$
$3$	−4.41660	−0.849976	−0.424988	+	0.905199i	$0.639722\pi$
$4$	−5.76373	−0.720466
$5$	−15.6043	−1.39569	−0.697845	−	0.716249i	$-0.745858\pi$
$5$	−15.6043	−1.39569	−0.697845	+	0.716249i	$0.745858\pi$
$6$	6.60466	0.449390
$7$	31.8747	1.72107	0.860536	−	0.509389i	$-0.170129\pi$
$7$	31.8747	1.72107	0.860536	+	0.509389i	$0.170129\pi$
$8$	20.5825	0.909627
$9$	−7.49362	−0.277542
$10$	23.3349	0.737915
$11$	0	0
$11$	0	0
$12$	25.4561	0.612378
$13$	21.7566	0.464170	0.232085	−	0.972696i	$-0.425445\pi$
$13$	21.7566	0.464170	0.232085	+	0.972696i	$0.425445\pi$
$14$	−47.6660	−0.909948
$15$	68.9180	1.18630
$16$	15.3304	0.239537
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	11.2061	0.146739
$19$	−106.640	−1.28763	−0.643814	−	0.765182i	$-0.722649\pi$
$19$	−106.640	−1.28763	−0.643814	+	0.765182i	$0.722649\pi$
$20$	89.9389	1.00555
$21$	−140.778	−1.46287
$22$	0	0
$23$	−202.129	−1.83247	−0.916233	−	0.400645i	$-0.868786\pi$
$23$	−202.129	−1.83247	−0.916233	+	0.400645i	$0.868786\pi$
$24$	−90.9048	−0.773161
$25$	118.494	0.947951
$26$	−32.5353	−0.245411
$27$	152.345	1.08588
$28$	−183.717	−1.23997
$29$	133.356	0.853917	0.426959	−	0.904271i	$-0.359585\pi$
$29$	133.356	0.853917	0.426959	+	0.904271i	$0.359585\pi$
$30$	−103.061	−0.627210
$31$	96.7146	0.560337	0.280169	−	0.959951i	$-0.409610\pi$
$31$	96.7146	0.560337	0.280169	+	0.959951i	$0.409610\pi$
$32$	−187.585	−1.03627
$33$	0	0
$34$	25.4221	0.128231
$35$	−497.382	−2.40208
$36$	43.1912	0.199959
$37$	−223.110	−0.991325	−0.495662	−	0.868515i	$-0.665075\pi$
$37$	−223.110	−0.991325	−0.495662	+	0.868515i	$0.665075\pi$
$38$	159.472	0.680782
$39$	−96.0905	−0.394533
$40$	−321.176	−1.26956
$41$	230.504	0.878018	0.439009	−	0.898483i	$-0.355330\pi$
$41$	230.504	0.878018	0.439009	+	0.898483i	$0.355330\pi$
$42$	210.522	0.773434
$43$	−419.986	−1.48947	−0.744736	−	0.667360i	$-0.767424\pi$
$43$	−419.986	−1.48947	−0.744736	+	0.667360i	$0.767424\pi$
$44$	0	0
$45$	116.933	0.387362
$46$	302.267	0.968843
$47$	−136.012	−0.422116	−0.211058	−	0.977474i	$-0.567691\pi$
$47$	−136.012	−0.422116	−0.211058	+	0.977474i	$0.567691\pi$
$48$	−67.7082	−0.203601
$49$	672.997	1.96209
$50$	−177.198	−0.501191
$51$	75.0822	0.206149
$52$	−125.399	−0.334419
$53$	685.851	1.77753	0.888763	−	0.458367i	$-0.151565\pi$
$53$	685.851	1.77753	0.888763	+	0.458367i	$0.151565\pi$
$54$	−227.819	−0.574115
$55$	0	0
$56$	656.062	1.56553
$57$	470.987	1.09445
$58$	−199.423	−0.451474
$59$	−155.901	−0.344009	−0.172005	−	0.985096i	$-0.555024\pi$
$59$	−155.901	−0.344009	−0.172005	+	0.985096i	$0.555024\pi$
$60$	−397.224	−0.854691
$61$	377.256	0.791847	0.395924	−	0.918283i	$-0.370424\pi$
$61$	377.256	0.791847	0.395924	+	0.918283i	$0.370424\pi$
$62$	−144.629	−0.296256
$63$	−238.857	−0.477669
$64$	157.875	0.308350
$65$	−339.497	−0.647837
$66$	0	0
$67$	5.55804	0.0101347	0.00506733	−	0.999987i	$-0.498387\pi$
$67$	5.55804	0.0101347	0.00506733	+	0.999987i	$0.498387\pi$
$68$	97.9834	0.174739
$69$	892.722	1.55755
$70$	743.794	1.27001
$71$	394.434	0.659305	0.329653	−	0.944102i	$-0.393068\pi$
$71$	394.434	0.659305	0.329653	+	0.944102i	$0.393068\pi$
$72$	−154.238	−0.252459
$73$	333.509	0.534716	0.267358	−	0.963597i	$-0.413849\pi$
$73$	333.509	0.534716	0.267358	+	0.963597i	$0.413849\pi$
$74$	333.642	0.524123
$75$	−523.341	−0.805735
$76$	614.645	0.927692
$77$	0	0
$78$	143.695	0.208593
$79$	−449.839	−0.640644	−0.320322	−	0.947309i	$-0.603791\pi$
$79$	−449.839	−0.640644	−0.320322	+	0.947309i	$0.603791\pi$
$80$	−239.220	−0.334320
$81$	−470.518	−0.645429
$82$	−344.700	−0.464217
$83$	−200.319	−0.264915	−0.132457	−	0.991189i	$-0.542287\pi$
$83$	−200.319	−0.264915	−0.132457	+	0.991189i	$0.542287\pi$
$84$	811.406	1.05395
$85$	265.273	0.338505
$86$	628.054	0.787498
$87$	−588.981	−0.725809
$88$	0	0
$89$	1554.53	1.85146	0.925731	−	0.378183i	$-0.123451\pi$
$89$	1554.53	1.85146	0.925731	+	0.378183i	$0.123451\pi$
$90$	−174.863	−0.204802
$91$	693.487	0.798870
$92$	1165.01	1.32023
$93$	−427.150	−0.476273
$94$	203.395	0.223177
$95$	1664.04	1.79713
$96$	828.490	0.880807
$97$	−1688.97	−1.76793	−0.883963	−	0.467556i	$-0.845135\pi$
$97$	−1688.97	−1.76793	−0.883963	+	0.467556i	$0.845135\pi$
$98$	−1006.41	−1.03738
$99$	0	0
$100$	−682.967	−0.682967
$101$	1452.13	1.43062	0.715308	−	0.698810i	$-0.246287\pi$
$101$	1452.13	1.43062	0.715308	+	0.698810i	$0.246287\pi$
$102$	−112.279	−0.108993
$103$	−1068.18	−1.02185	−0.510925	−	0.859625i	$-0.670697\pi$
$103$	−1068.18	−1.02185	−0.510925	+	0.859625i	$0.670697\pi$
$104$	447.806	0.422222
$105$	2196.74	2.04171
$106$	−1025.63	−0.939795
$107$	473.070	0.427415	0.213707	−	0.976898i	$-0.431446\pi$
$107$	473.070	0.427415	0.213707	+	0.976898i	$0.431446\pi$
$108$	−878.073	−0.782339
$109$	1151.12	1.01153	0.505767	−	0.862670i	$-0.331209\pi$
$109$	1151.12	1.01153	0.505767	+	0.862670i	$0.331209\pi$
$110$	0	0
$111$	985.387	0.842602
$112$	488.652	0.412261
$113$	871.025	0.725125	0.362563	−	0.931959i	$-0.381902\pi$
$113$	871.025	0.725125	0.362563	+	0.931959i	$0.381902\pi$
$114$	−704.322	−0.578648
$115$	3154.07	2.55756
$116$	−768.628	−0.615218
$117$	−163.036	−0.128826
$118$	233.137	0.181881
$119$	−541.870	−0.417421
$120$	1418.50	1.07909
$121$	0	0
$122$	−564.155	−0.418657
$123$	−1018.05	−0.746294
$124$	−557.437	−0.403704
$125$	101.523	0.0726438
$126$	357.191	0.252548
$127$	1229.37	0.858967	0.429483	−	0.903075i	$-0.358696\pi$
$127$	1229.37	0.858967	0.429483	+	0.903075i	$0.358696\pi$
$128$	1264.59	0.873245
$129$	1854.91	1.26601
$130$	507.690	0.342518
$131$	−2071.99	−1.38191	−0.690956	−	0.722897i	$-0.742810\pi$
$131$	−2071.99	−1.38191	−0.690956	+	0.722897i	$0.742810\pi$
$132$	0	0
$133$	−3399.13	−2.21610
$134$	−8.31159	−0.00535830
$135$	−2377.23	−1.51555
$136$	−349.903	−0.220617
$137$	2397.90	1.49538	0.747688	−	0.664050i	$-0.231164\pi$
$137$	2397.90	1.49538	0.747688	+	0.664050i	$0.231164\pi$
$138$	−1334.99	−0.823493
$139$	1097.10	0.669461	0.334730	−	0.942314i	$-0.391355\pi$
$139$	1097.10	0.669461	0.334730	+	0.942314i	$0.391355\pi$
$140$	2866.78	1.73062
$141$	600.712	0.358788
$142$	−589.843	−0.348581
$143$	0	0
$144$	−114.880	−0.0664816
$145$	−2080.93	−1.19180
$146$	−498.735	−0.282709
$147$	−2972.36	−1.66773
$148$	1285.94	0.714216
$149$	2879.54	1.58323	0.791614	−	0.611021i	$-0.209241\pi$
$149$	2879.54	1.58323	0.791614	+	0.611021i	$0.209241\pi$
$150$	782.612	0.426000
$151$	−1623.91	−0.875178	−0.437589	−	0.899175i	$-0.644167\pi$
$151$	−1623.91	−0.875178	−0.437589	+	0.899175i	$0.644167\pi$
$152$	−2194.92	−1.17126
$153$	127.392	0.0673137
$154$	0	0
$155$	−1509.16	−0.782057
$156$	553.839	0.284248
$157$	1411.85	0.717696	0.358848	−	0.933396i	$-0.383170\pi$
$157$	1411.85	0.717696	0.358848	+	0.933396i	$0.383170\pi$
$158$	672.697	0.338715
$159$	−3029.13	−1.51085
$160$	2927.14	1.44632
$161$	−6442.79	−3.15381
$162$	703.620	0.341245
$163$	49.2889	0.0236847	0.0118423	−	0.999930i	$-0.496230\pi$
$163$	49.2889	0.0236847	0.0118423	+	0.999930i	$0.496230\pi$
$164$	−1328.57	−0.632582
$165$	0	0
$166$	299.561	0.140063
$167$	−2190.40	−1.01496	−0.507479	−	0.861664i	$-0.669423\pi$
$167$	−2190.40	−1.01496	−0.507479	+	0.861664i	$0.669423\pi$
$168$	−2897.56	−1.33067
$169$	−1723.65	−0.784546
$170$	−396.694	−0.178971
$171$	799.121	0.357370
$172$	2420.68	1.07311
$173$	−283.327	−0.124514	−0.0622571	−	0.998060i	$-0.519830\pi$
$173$	−283.327	−0.124514	−0.0622571	+	0.998060i	$0.519830\pi$
$174$	880.772	0.383742
$175$	3776.96	1.63149
$176$	0	0
$177$	688.552	0.292400
$178$	−2324.67	−0.978886
$179$	1177.37	0.491625	0.245813	−	0.969317i	$-0.420945\pi$
$179$	1177.37	0.491625	0.245813	+	0.969317i	$0.420945\pi$
$180$	−673.968	−0.279081
$181$	−1795.12	−0.737183	−0.368592	−	0.929591i	$-0.620160\pi$
$181$	−1795.12	−0.737183	−0.368592	+	0.929591i	$0.620160\pi$
$182$	−1037.05	−0.422370
$183$	−1666.19	−0.673051
$184$	−4160.31	−1.66686
$185$	3481.47	1.38358
$186$	638.767	0.251810
$187$	0	0
$188$	783.938	0.304120
$189$	4855.94	1.86888
$190$	−2488.44	−0.950160
$191$	−1754.23	−0.664565	−0.332282	−	0.943180i	$-0.607819\pi$
$191$	−1754.23	−0.664565	−0.332282	+	0.943180i	$0.607819\pi$
$192$	−697.273	−0.262090
$193$	−2883.02	−1.07526	−0.537628	−	0.843182i	$-0.680680\pi$
$193$	−2883.02	−1.07526	−0.537628	+	0.843182i	$0.680680\pi$
$194$	2525.71	0.934720
$195$	1499.42	0.550646
$196$	−3878.97	−1.41362
$197$	−554.915	−0.200691	−0.100345	−	0.994953i	$-0.531995\pi$
$197$	−554.915	−0.200691	−0.100345	+	0.994953i	$0.531995\pi$
$198$	0	0
$199$	2232.48	0.795259	0.397629	−	0.917546i	$-0.369833\pi$
$199$	2232.48	0.795259	0.397629	+	0.917546i	$0.369833\pi$
$200$	2438.90	0.862282
$201$	−24.5477	−0.00861422
$202$	−2171.54	−0.756380
$203$	4250.69	1.46965
$204$	−432.754	−0.148524
$205$	−3596.86	−1.22544
$206$	1597.37	0.540262
$207$	1514.68	0.508586
$208$	333.538	0.111186
$209$	0	0
$210$	−3285.04	−1.07947
$211$	−1394.14	−0.454866	−0.227433	−	0.973794i	$-0.573033\pi$
$211$	−1394.14	−0.454866	−0.227433	+	0.973794i	$0.573033\pi$
$212$	−3953.06	−1.28065
$213$	−1742.06	−0.560393
$214$	−707.436	−0.225978
$215$	6553.58	2.07884
$216$	3135.64	0.987745
$217$	3082.75	0.964381
$218$	−1721.40	−0.534808
$219$	−1472.98	−0.454495
$220$	0	0
$221$	−369.863	−0.112578
$222$	−1473.56	−0.445492
$223$	1233.42	0.370386	0.185193	−	0.982702i	$-0.440709\pi$
$223$	1233.42	0.370386	0.185193	+	0.982702i	$0.440709\pi$
$224$	−5979.23	−1.78350
$225$	−887.949	−0.263096
$226$	−1302.55	−0.383381
$227$	−261.485	−0.0764555	−0.0382277	−	0.999269i	$-0.512171\pi$
$227$	−261.485	−0.0764555	−0.0382277	+	0.999269i	$0.512171\pi$
$228$	−2714.64	−0.788516
$229$	−713.661	−0.205939	−0.102970	−	0.994685i	$-0.532834\pi$
$229$	−713.661	−0.205939	−0.102970	+	0.994685i	$0.532834\pi$
$230$	−4716.66	−1.35220
$231$	0	0
$232$	2744.80	0.776746
$233$	1065.51	0.299587	0.149794	−	0.988717i	$-0.452139\pi$
$233$	1065.51	0.299587	0.149794	+	0.988717i	$0.452139\pi$
$234$	243.807	0.0681118
$235$	2122.38	0.589143
$236$	898.570	0.247847
$237$	1986.76	0.544532
$238$	810.322	0.220695
$239$	−2975.70	−0.805364	−0.402682	−	0.915340i	$-0.631922\pi$
$239$	−2975.70	−0.805364	−0.402682	+	0.915340i	$0.631922\pi$
$240$	1056.54	0.284164
$241$	−4302.38	−1.14996	−0.574980	−	0.818167i	$-0.694990\pi$
$241$	−4302.38	−1.14996	−0.574980	+	0.818167i	$0.694990\pi$
$242$	0	0
$243$	−2035.21	−0.537280
$244$	−2174.40	−0.570499
$245$	−10501.6	−2.73847
$246$	1522.40	0.394573
$247$	−2320.13	−0.597678
$248$	1990.63	0.509698
$249$	884.731	0.225171
$250$	−151.819	−0.0384075
$251$	−1156.91	−0.290930	−0.145465	−	0.989363i	$-0.546468\pi$
$251$	−1156.91	−0.290930	−0.145465	+	0.989363i	$0.546468\pi$
$252$	1376.71	0.344144
$253$	0	0
$254$	−1838.42	−0.454144
$255$	−1171.61	−0.287721
$256$	−3154.10	−0.770043
$257$	3387.17	0.822125	0.411062	−	0.911607i	$-0.365158\pi$
$257$	3387.17	0.822125	0.411062	+	0.911607i	$0.365158\pi$
$258$	−2773.87	−0.669354
$259$	−7111.56	−1.70614
$260$	1956.77	0.466745
$261$	−999.320	−0.236998
$262$	3098.48	0.730630
$263$	−2296.19	−0.538361	−0.269180	−	0.963090i	$-0.586753\pi$
$263$	−2296.19	−0.538361	−0.269180	+	0.963090i	$0.586753\pi$
$264$	0	0
$265$	−10702.2	−2.48088
$266$	5083.11	1.17167
$267$	−6865.75	−1.57370
$268$	−32.0350	−0.00730169
$269$	−718.997	−0.162967	−0.0814833	−	0.996675i	$-0.525966\pi$
$269$	−718.997	−0.162967	−0.0814833	+	0.996675i	$0.525966\pi$
$270$	3554.95	0.801287
$271$	−6404.99	−1.43570	−0.717852	−	0.696196i	$-0.754875\pi$
$271$	−6404.99	−1.43570	−0.717852	+	0.696196i	$0.754875\pi$
$272$	−260.617	−0.0580963
$273$	−3062.86	−0.679020
$274$	−3585.86	−0.790620
$275$	0	0
$276$	−5145.41	−1.12216
$277$	−2852.91	−0.618825	−0.309413	−	0.950928i	$-0.600132\pi$
$277$	−2852.91	−0.618825	−0.309413	+	0.950928i	$0.600132\pi$
$278$	−1640.63	−0.353950
$279$	−724.743	−0.155517
$280$	−10237.4	−2.18500
$281$	4307.78	0.914522	0.457261	−	0.889332i	$-0.348830\pi$
$281$	4307.78	0.914522	0.457261	+	0.889332i	$0.348830\pi$
$282$	−898.315	−0.189695
$283$	1991.75	0.418365	0.209183	−	0.977877i	$-0.432920\pi$
$283$	1991.75	0.418365	0.209183	+	0.977877i	$0.432920\pi$
$284$	−2273.41	−0.475007
$285$	−7349.42	−1.52752
$286$	0	0
$287$	7347.26	1.51113
$288$	1405.69	0.287609
$289$	289.000	0.0588235
$290$	3111.85	0.630118
$291$	7459.51	1.50269
$292$	−1922.25	−0.385245
$293$	597.874	0.119209	0.0596044	−	0.998222i	$-0.481016\pi$
$293$	597.874	0.119209	0.0596044	+	0.998222i	$0.481016\pi$
$294$	4444.92	0.881745
$295$	2432.72	0.480131
$296$	−4592.16	−0.901736
$297$	0	0
$298$	−4306.11	−0.837068
$299$	−4397.64	−0.850576
$300$	3016.39	0.580505
$301$	−13386.9	−2.56349
$302$	2428.42	0.462715
$303$	−6413.47	−1.21599
$304$	−1634.84	−0.308435
$305$	−5886.81	−1.10517
$306$	−190.504	−0.0355894
$307$	3912.03	0.727268	0.363634	−	0.931542i	$-0.381536\pi$
$307$	3912.03	0.727268	0.363634	+	0.931542i	$0.381536\pi$
$308$	0	0
$309$	4717.71	0.868548
$310$	2256.83	0.413481
$311$	5524.46	1.00728	0.503639	−	0.863914i	$-0.331994\pi$
$311$	5524.46	1.00728	0.503639	+	0.863914i	$0.331994\pi$
$312$	−1977.78	−0.358878
$313$	−846.446	−0.152856	−0.0764280	−	0.997075i	$-0.524352\pi$
$313$	−846.446	−0.152856	−0.0764280	+	0.997075i	$0.524352\pi$
$314$	−2111.31	−0.379453
$315$	3727.20	0.666678
$316$	2592.75	0.461562
$317$	6260.91	1.10930	0.554649	−	0.832084i	$-0.312853\pi$
$317$	6260.91	1.10930	0.554649	+	0.832084i	$0.312853\pi$
$318$	4529.81	0.798803
$319$	0	0
$320$	−2463.53	−0.430361
$321$	−2089.36	−0.363292
$322$	9634.66	1.66745
$323$	1812.88	0.312296
$324$	2711.94	0.465010
$325$	2578.03	0.440010
$326$	−73.7075	−0.0125223
$327$	−5084.04	−0.859780
$328$	4744.36	0.798669
$329$	−4335.35	−0.726492
$330$	0	0
$331$	104.588	0.0173676	0.00868380	−	0.999962i	$-0.497236\pi$
$331$	104.588	0.0173676	0.00868380	+	0.999962i	$0.497236\pi$
$332$	1154.59	0.190862
$333$	1671.90	0.275134
$334$	3275.56	0.536618
$335$	−86.7293	−0.0141449
$336$	−2158.18	−0.350412
$337$	−10467.5	−1.69199	−0.845997	−	0.533188i	$-0.820994\pi$
$337$	−10467.5	−1.69199	−0.845997	+	0.533188i	$0.820994\pi$
$338$	2577.57	0.414797
$339$	−3846.97	−0.616339
$340$	−1528.96	−0.243881
$341$	0	0
$342$	−1195.02	−0.188945
$343$	10518.6	1.65583
$344$	−8644.37	−1.35486
$345$	−13930.3	−2.17386
$346$	423.692	0.0658319
$347$	8711.96	1.34779	0.673894	−	0.738828i	$-0.264620\pi$
$347$	8711.96	1.34779	0.673894	+	0.738828i	$0.264620\pi$
$348$	3394.73	0.522921
$349$	−4643.42	−0.712196	−0.356098	−	0.934449i	$-0.615893\pi$
$349$	−4643.42	−0.712196	−0.356098	+	0.934449i	$0.615893\pi$
$350$	−5648.13	−0.862586
$351$	3314.51	0.504032
$352$	0	0
$353$	−7914.58	−1.19334	−0.596672	−	0.802485i	$-0.703511\pi$
$353$	−7914.58	−1.19334	−0.596672	+	0.802485i	$0.703511\pi$
$354$	−1029.67	−0.154595
$355$	−6154.86	−0.920186
$356$	−8959.90	−1.33392
$357$	2393.23	0.354798
$358$	−1760.66	−0.259927
$359$	11442.2	1.68217	0.841083	−	0.540906i	$-0.181918\pi$
$359$	11442.2	1.68217	0.841083	+	0.540906i	$0.181918\pi$
$360$	2406.77	0.352355
$361$	4513.13	0.657987
$362$	2684.45	0.389756
$363$	0	0
$364$	−3997.07	−0.575559
$365$	−5204.17	−0.746298
$366$	2491.65	0.355849
$367$	8895.20	1.26519	0.632596	−	0.774482i	$-0.281989\pi$
$367$	8895.20	1.26519	0.632596	+	0.774482i	$0.281989\pi$
$368$	−3098.71	−0.438944
$369$	−1727.31	−0.243687
$370$	−5206.25	−0.731513
$371$	21861.3	3.05925
$372$	2461.98	0.343138
$373$	11644.0	1.61636	0.808179	−	0.588937i	$-0.200453\pi$
$373$	11644.0	1.61636	0.808179	+	0.588937i	$0.200453\pi$
$374$	0	0
$375$	−448.386	−0.0617455
$376$	−2799.47	−0.383968
$377$	2901.38	0.396363
$378$	−7261.66	−0.988093
$379$	7181.59	0.973334	0.486667	−	0.873588i	$-0.338213\pi$
$379$	7181.59	0.973334	0.486667	+	0.873588i	$0.338213\pi$
$380$	−9591.10	−1.29477
$381$	−5429.63	−0.730101
$382$	2623.31	0.351362
$383$	−12489.8	−1.66631	−0.833157	−	0.553037i	$-0.813469\pi$
$383$	−12489.8	−1.66631	−0.833157	+	0.553037i	$0.813469\pi$
$384$	−5585.21	−0.742237
$385$	0	0
$386$	4311.32	0.568498
$387$	3147.22	0.413390
$388$	9734.76	1.27373
$389$	9213.37	1.20087	0.600433	−	0.799675i	$-0.294995\pi$
$389$	9213.37	1.20087	0.600433	+	0.799675i	$0.294995\pi$
$390$	−2242.26	−0.291132
$391$	3436.19	0.444438
$392$	13852.0	1.78477
$393$	9151.14	1.17459
$394$	829.830	0.106107
$395$	7019.42	0.894141
$396$	0	0
$397$	640.374	0.0809558	0.0404779	−	0.999180i	$-0.487112\pi$
$397$	640.374	0.0809558	0.0404779	+	0.999180i	$0.487112\pi$
$398$	−3338.49	−0.420461
$399$	15012.6	1.88363
$400$	1816.56	0.227070
$401$	4465.93	0.556155	0.278077	−	0.960559i	$-0.410303\pi$
$401$	4465.93	0.556155	0.278077	+	0.960559i	$0.410303\pi$
$402$	36.7090	0.00455442
$403$	2104.19	0.260092
$404$	−8369.67	−1.03071
$405$	7342.10	0.900819
$406$	−6356.55	−0.777020
$407$	0	0
$408$	1545.38	0.187519
$409$	−7112.64	−0.859896	−0.429948	−	0.902854i	$-0.641468\pi$
$409$	−7112.64	−0.859896	−0.429948	+	0.902854i	$0.641468\pi$
$410$	5378.80	0.647903
$411$	−10590.6	−1.27103
$412$	6156.68	0.736208
$413$	−4969.29	−0.592065
$414$	−2265.07	−0.268894
$415$	3125.84	0.369739
$416$	−4081.23	−0.481007
$417$	−4845.47	−0.569025
$418$	0	0
$419$	−8658.88	−1.00958	−0.504790	−	0.863242i	$-0.668430\pi$
$419$	−8658.88	−1.00958	−0.504790	+	0.863242i	$0.668430\pi$
$420$	−12661.4	−1.47098
$421$	9273.85	1.07359	0.536793	−	0.843714i	$-0.319636\pi$
$421$	9273.85	1.07359	0.536793	+	0.843714i	$0.319636\pi$
$422$	2084.83	0.240492
$423$	1019.22	0.117155
$424$	14116.5	1.61689
$425$	−2014.40	−0.229912
$426$	2605.10	0.296285
$427$	12024.9	1.36283
$428$	−2726.65	−0.307938
$429$	0	0
$430$	−9800.34	−1.09910
$431$	−3728.56	−0.416701	−0.208351	−	0.978054i	$-0.566810\pi$
$431$	−3728.56	−0.416701	−0.208351	+	0.978054i	$0.566810\pi$
$432$	2335.50	0.260109
$433$	−4907.52	−0.544666	−0.272333	−	0.962203i	$-0.587795\pi$
$433$	−4907.52	−0.544666	−0.272333	+	0.962203i	$0.587795\pi$
$434$	−4610.00	−0.509878
$435$	9190.63	1.01300
$436$	−6634.74	−0.728776
$437$	21555.0	2.35954
$438$	2202.71	0.240296
$439$	84.3252	0.00916770	0.00458385	−	0.999989i	$-0.498541\pi$
$439$	84.3252	0.00916770	0.00458385	+	0.999989i	$0.498541\pi$
$440$	0	0
$441$	−5043.19	−0.544562
$442$	553.099	0.0595209
$443$	−12903.7	−1.38391	−0.691957	−	0.721939i	$-0.743251\pi$
$443$	−12903.7	−1.38391	−0.691957	+	0.721939i	$0.743251\pi$
$444$	−5679.50	−0.607066
$445$	−24257.4	−2.58407
$446$	−1844.48	−0.195826
$447$	−12717.8	−1.34571
$448$	5032.23	0.530693
$449$	8605.38	0.904483	0.452241	−	0.891896i	$-0.350625\pi$
$449$	8605.38	0.904483	0.452241	+	0.891896i	$0.350625\pi$
$450$	1327.85	0.139101
$451$	0	0
$452$	−5020.35	−0.522428
$453$	7172.16	0.743880
$454$	391.030	0.0404228
$455$	−10821.4	−1.11498
$456$	9694.10	0.995544
$457$	−6565.81	−0.672069	−0.336034	−	0.941850i	$-0.609086\pi$
$457$	−6565.81	−0.672069	−0.336034	+	0.941850i	$0.609086\pi$
$458$	1067.22	0.108882
$459$	−2589.86	−0.263364
$460$	−18179.2	−1.84263
$461$	−2492.28	−0.251794	−0.125897	−	0.992043i	$-0.540181\pi$
$461$	−2492.28	−0.251794	−0.125897	+	0.992043i	$0.540181\pi$
$462$	0	0
$463$	−17573.5	−1.76396	−0.881978	−	0.471290i	$-0.843788\pi$
$463$	−17573.5	−1.76396	−0.881978	+	0.471290i	$0.843788\pi$
$464$	2044.40	0.204545
$465$	6665.37	0.664729
$466$	−1593.38	−0.158395
$467$	3282.78	0.325287	0.162644	−	0.986685i	$-0.447998\pi$
$467$	3282.78	0.325287	0.162644	+	0.986685i	$0.447998\pi$
$468$	939.696	0.0928151
$469$	177.161	0.0174425
$470$	−3173.84	−0.311485
$471$	−6235.60	−0.610024
$472$	−3208.83	−0.312920
$473$	0	0
$474$	−2971.04	−0.287899
$475$	−12636.2	−1.22061
$476$	3123.19	0.300738
$477$	−5139.51	−0.493337
$478$	4449.91	0.425804
$479$	−2403.66	−0.229282	−0.114641	−	0.993407i	$-0.536572\pi$
$479$	−2403.66	−0.229282	−0.114641	+	0.993407i	$0.536572\pi$
$480$	−12928.0	−1.22933
$481$	−4854.12	−0.460143
$482$	6433.85	0.607995
$483$	28455.3	2.68066
$484$	0	0
$485$	26355.2	2.46748
$486$	3043.49	0.284065
$487$	−8497.33	−0.790659	−0.395329	−	0.918539i	$-0.629370\pi$
$487$	−8497.33	−0.790659	−0.395329	+	0.918539i	$0.629370\pi$
$488$	7764.88	0.720286
$489$	−217.689	−0.0201314
$490$	15704.3	1.44786
$491$	−13855.9	−1.27354	−0.636771	−	0.771053i	$-0.719730\pi$
$491$	−13855.9	−1.27354	−0.636771	+	0.771053i	$0.719730\pi$
$492$	5867.74	0.537680
$493$	−2267.05	−0.207105
$494$	3469.57	0.315998
$495$	0	0
$496$	1482.67	0.134222
$497$	12572.5	1.13471
$498$	−1323.04	−0.119050
$499$	3506.86	0.314607	0.157303	−	0.987550i	$-0.449720\pi$
$499$	3506.86	0.314607	0.157303	+	0.987550i	$0.449720\pi$
$500$	−585.150	−0.0523374
$501$	9674.11	0.862690
$502$	1730.06	0.153818
$503$	−2426.32	−0.215078	−0.107539	−	0.994201i	$-0.534297\pi$
$503$	−2426.32	−0.215078	−0.107539	+	0.994201i	$0.534297\pi$
$504$	−4916.28	−0.434501
$505$	−22659.4	−1.99670
$506$	0	0
$507$	7612.67	0.666845
$508$	−7085.74	−0.618856
$509$	−10845.0	−0.944397	−0.472198	−	0.881492i	$-0.656539\pi$
$509$	−10845.0	−0.944397	−0.472198	+	0.881492i	$0.656539\pi$
$510$	1752.04	0.152121
$511$	10630.5	0.920285
$512$	−5400.06	−0.466116
$513$	−16246.1	−1.39821
$514$	−5065.23	−0.434665
$515$	16668.1	1.42619
$516$	−10691.2	−0.912120
$517$	0	0
$518$	10634.7	0.902054
$519$	1251.34	0.105834
$520$	−6987.70	−0.589290
$521$	7344.97	0.617637	0.308819	−	0.951121i	$-0.400066\pi$
$521$	7344.97	0.617637	0.308819	+	0.951121i	$0.400066\pi$
$522$	1494.40	0.125303
$523$	1641.60	0.137251	0.0686255	−	0.997642i	$-0.478139\pi$
$523$	1641.60	0.137251	0.0686255	+	0.997642i	$0.478139\pi$
$524$	11942.4	0.995620
$525$	−16681.3	−1.38673
$526$	3433.76	0.284637
$527$	−1644.15	−0.135902
$528$	0	0
$529$	28689.0	2.35793
$530$	16004.3	1.31166
$531$	1168.26	0.0954769
$532$	19591.6	1.59663
$533$	5015.00	0.407550
$534$	10267.2	0.832029
$535$	−7381.92	−0.596539
$536$	114.398	0.00921877
$537$	−5199.98	−0.417869
$538$	1075.20	0.0861620
$539$	0	0
$540$	13701.7	1.09190
$541$	−10867.7	−0.863661	−0.431831	−	0.901955i	$-0.642132\pi$
$541$	−10867.7	−0.863661	−0.431831	+	0.901955i	$0.642132\pi$
$542$	9578.14	0.759071
$543$	7928.33	0.626588
$544$	3188.95	0.251333
$545$	−17962.4	−1.41179
$546$	4580.25	0.359005
$547$	−1011.33	−0.0790516	−0.0395258	−	0.999219i	$-0.512585\pi$
$547$	−1011.33	−0.0790516	−0.0395258	+	0.999219i	$0.512585\pi$
$548$	−13820.9	−1.07737
$549$	−2827.01	−0.219771
$550$	0	0
$551$	−14221.1	−1.09953
$552$	18374.5	1.41679
$553$	−14338.5	−1.10260
$554$	4266.29	0.327179
$555$	−15376.3	−1.17601
$556$	−6323.40	−0.482324
$557$	13427.8	1.02146	0.510732	−	0.859740i	$-0.329374\pi$
$557$	13427.8	1.02146	0.510732	+	0.859740i	$0.329374\pi$
$558$	1083.79	0.0822233
$559$	−9137.49	−0.691368
$560$	−7625.06	−0.575389
$561$	0	0
$562$	−6441.93	−0.483517
$563$	−16146.6	−1.20870	−0.604349	−	0.796720i	$-0.706567\pi$
$563$	−16146.6	−1.20870	−0.604349	+	0.796720i	$0.706567\pi$
$564$	−3462.34	−0.258494
$565$	−13591.7	−1.01205
$566$	−2978.50	−0.221194
$567$	−14997.6	−1.11083
$568$	8118.44	0.599722
$569$	−14175.9	−1.04444	−0.522219	−	0.852812i	$-0.674895\pi$
$569$	−14175.9	−1.04444	−0.522219	+	0.852812i	$0.674895\pi$
$570$	10990.5	0.807613
$571$	−7696.76	−0.564097	−0.282049	−	0.959400i	$-0.591014\pi$
$571$	−7696.76	−0.564097	−0.282049	+	0.959400i	$0.591014\pi$
$572$	0	0
$573$	7747.75	0.564864
$574$	−10987.2	−0.798951
$575$	−23951.0	−1.73709
$576$	−1183.06	−0.0855800
$577$	−21496.6	−1.55098	−0.775490	−	0.631360i	$-0.782497\pi$
$577$	−21496.6	−1.55098	−0.775490	+	0.631360i	$0.782497\pi$
$578$	−432.175	−0.0311006
$579$	12733.2	0.913942
$580$	11993.9	0.858654
$581$	−6385.12	−0.455937
$582$	−11155.1	−0.794489
$583$	0	0
$584$	6864.45	0.486392
$585$	2544.06	0.179802
$586$	−894.072	−0.0630269
$587$	24560.0	1.72692	0.863459	−	0.504419i	$-0.168293\pi$
$587$	24560.0	1.72692	0.863459	+	0.504419i	$0.168293\pi$
$588$	17131.9	1.20154
$589$	−10313.7	−0.721506
$590$	−3637.93	−0.253850
$591$	2450.84	0.170582
$592$	−3420.36	−0.237459
$593$	−22430.8	−1.55333	−0.776663	−	0.629917i	$-0.783089\pi$
$593$	−22430.8	−1.55333	−0.776663	+	0.629917i	$0.783089\pi$
$594$	0	0
$595$	8455.50	0.582591
$596$	−16596.9	−1.14066
$597$	−9859.99	−0.675951
$598$	6576.31	0.449708
$599$	−12888.5	−0.879145	−0.439573	−	0.898207i	$-0.644870\pi$
$599$	−12888.5	−0.879145	−0.439573	+	0.898207i	$0.644870\pi$
$600$	−10771.7	−0.732919
$601$	20398.3	1.38447	0.692234	−	0.721673i	$-0.256626\pi$
$601$	20398.3	1.38447	0.692234	+	0.721673i	$0.256626\pi$
$602$	20019.0	1.35534
$603$	−41.6499	−0.00281279
$604$	9359.77	0.630536
$605$	0	0
$606$	9590.82	0.642905
$607$	−20652.1	−1.38096	−0.690481	−	0.723351i	$-0.742601\pi$
$607$	−20652.1	−1.38096	−0.690481	+	0.723351i	$0.742601\pi$
$608$	20004.1	1.33433
$609$	−18773.6	−1.24917
$610$	8803.24	0.584316
$611$	−2959.17	−0.195933
$612$	−734.250	−0.0484973
$613$	21523.8	1.41817	0.709085	−	0.705123i	$-0.249108\pi$
$613$	21523.8	1.41817	0.709085	+	0.705123i	$0.249108\pi$
$614$	−5850.11	−0.384513
$615$	15885.9	1.04160
$616$	0	0
$617$	−7872.87	−0.513695	−0.256847	−	0.966452i	$-0.582684\pi$
$617$	−7872.87	−0.513695	−0.256847	+	0.966452i	$0.582684\pi$
$618$	−7054.95	−0.459210
$619$	−5264.88	−0.341864	−0.170932	−	0.985283i	$-0.554678\pi$
$619$	−5264.88	−0.341864	−0.170932	+	0.985283i	$0.554678\pi$
$620$	8698.40	0.563446
$621$	−30793.2	−1.98984
$622$	−8261.37	−0.532558
$623$	49550.3	3.18650
$624$	−1473.10	−0.0945054
$625$	−16395.9	−1.04934
$626$	1265.79	0.0808165
$627$	0	0
$628$	−8137.54	−0.517075
$629$	3792.86	0.240432
$630$	−5573.71	−0.352479
$631$	26907.2	1.69756	0.848779	−	0.528748i	$-0.177338\pi$
$631$	26907.2	1.69756	0.848779	+	0.528748i	$0.177338\pi$
$632$	−9258.82	−0.582747
$633$	6157.38	0.386625
$634$	−9362.66	−0.586497
$635$	−19183.4	−1.19885
$636$	17459.1	1.08852
$637$	14642.2	0.910744
$638$	0	0
$639$	−2955.74	−0.182985
$640$	−19733.1	−1.21878
$641$	−10649.6	−0.656215	−0.328108	−	0.944640i	$-0.606411\pi$
$641$	−10649.6	−0.656215	−0.328108	+	0.944640i	$0.606411\pi$
$642$	3124.47	0.192076
$643$	−26955.1	−1.65320	−0.826598	−	0.562792i	$-0.809727\pi$
$643$	−26955.1	−1.65320	−0.826598	+	0.562792i	$0.809727\pi$
$644$	37134.5	2.27221
$645$	−28944.6	−1.76696
$646$	−2711.02	−0.165114
$647$	7272.21	0.441886	0.220943	−	0.975287i	$-0.429087\pi$
$647$	7272.21	0.441886	0.220943	+	0.975287i	$0.429087\pi$
$648$	−9684.44	−0.587100
$649$	0	0
$650$	−3855.23	−0.232638
$651$	−13615.3	−0.819700
$652$	−284.088	−0.0170640
$653$	−10555.2	−0.632555	−0.316277	−	0.948667i	$-0.602433\pi$
$653$	−10555.2	−0.632555	−0.316277	+	0.948667i	$0.602433\pi$
$654$	7602.76	0.454574
$655$	32331.9	1.92872
$656$	3533.72	0.210318
$657$	−2499.19	−0.148406
$658$	6483.16	0.384103
$659$	−27154.5	−1.60514	−0.802572	−	0.596555i	$-0.796536\pi$
$659$	−27154.5	−1.60514	−0.802572	+	0.596555i	$0.796536\pi$
$660$	0	0
$661$	−15024.1	−0.884069	−0.442035	−	0.896998i	$-0.645743\pi$
$661$	−15024.1	−0.884069	−0.442035	+	0.896998i	$0.645743\pi$
$662$	−156.403	−0.00918242
$663$	1633.54	0.0956883
$664$	−4123.08	−0.240973
$665$	53040.9	3.09299
$666$	−2500.19	−0.145466
$667$	−26955.1	−1.56477
$668$	12624.9	0.731243
$669$	−5447.53	−0.314819
$670$	129.696	0.00747853
$671$	0	0
$672$	26407.9	1.51593
$673$	−2397.60	−0.137327	−0.0686633	−	0.997640i	$-0.521873\pi$
$673$	−2397.60	−0.137327	−0.0686633	+	0.997640i	$0.521873\pi$
$674$	15653.3	0.894573
$675$	18051.9	1.02936
$676$	9934.64	0.565239
$677$	11268.4	0.639707	0.319853	−	0.947467i	$-0.396366\pi$
$677$	11268.4	0.639707	0.319853	+	0.947467i	$0.396366\pi$
$678$	5752.83	0.325864
$679$	−53835.4	−3.04273
$680$	5459.98	0.307913
$681$	1154.88	0.0649853
$682$	0	0
$683$	−10787.5	−0.604350	−0.302175	−	0.953252i	$-0.597713\pi$
$683$	−10787.5	−0.604350	−0.302175	+	0.953252i	$0.597713\pi$
$684$	−4605.92	−0.257473
$685$	−37417.6	−2.08708
$686$	−15729.7	−0.875453
$687$	3151.96	0.175043
$688$	−6438.55	−0.356784
$689$	14921.8	0.825074
$690$	20831.6	1.14934
$691$	−9034.83	−0.497397	−0.248698	−	0.968581i	$-0.580003\pi$
$691$	−9034.83	−0.497397	−0.248698	+	0.968581i	$0.580003\pi$
$692$	1633.02	0.0897083
$693$	0	0
$694$	−13028.0	−0.712589
$695$	−17119.5	−0.934360
$696$	−12122.7	−0.660215
$697$	−3918.58	−0.212951
$698$	6943.85	0.376545
$699$	−4705.93	−0.254642
$700$	−21769.4	−1.17544
$701$	−34213.5	−1.84340	−0.921701	−	0.387901i	$-0.873200\pi$
$701$	−34213.5	−1.84340	−0.921701	+	0.387901i	$0.873200\pi$
$702$	−4956.57	−0.266487
$703$	23792.5	1.27646
$704$	0	0
$705$	−9373.69	−0.500757
$706$	11835.6	0.630933
$707$	46286.2	2.46219
$708$	−3968.63	−0.210664
$709$	12764.6	0.676142	0.338071	−	0.941121i	$-0.390226\pi$
$709$	12764.6	0.676142	0.338071	+	0.941121i	$0.390226\pi$
$710$	9204.08	0.486511
$711$	3370.93	0.177805
$712$	31996.2	1.68414
$713$	−19548.8	−1.02680
$714$	−3578.87	−0.187585
$715$	0	0
$716$	−6786.05	−0.354199
$717$	13142.5	0.684540
$718$	−17110.9	−0.889378
$719$	−17617.5	−0.913800	−0.456900	−	0.889518i	$-0.651040\pi$
$719$	−17617.5	−0.913800	−0.456900	+	0.889518i	$0.651040\pi$
$720$	1792.62	0.0927877
$721$	−34047.8	−1.75868
$722$	−6749.01	−0.347884
$723$	19001.9	0.977438
$724$	10346.6	0.531116
$725$	15801.9	0.809472
$726$	0	0
$727$	−33812.5	−1.72494	−0.862472	−	0.506105i	$-0.831085\pi$
$727$	−33812.5	−1.72494	−0.862472	+	0.506105i	$0.831085\pi$
$728$	14273.7	0.726674
$729$	21692.7	1.10210
$730$	7782.40	0.394575
$731$	7139.76	0.361250
$732$	9603.47	0.484910
$733$	13072.0	0.658700	0.329350	−	0.944208i	$-0.393170\pi$
$733$	13072.0	0.658700	0.329350	+	0.944208i	$0.393170\pi$
$734$	−13302.0	−0.668920
$735$	46381.6	2.32763
$736$	37916.4	1.89894
$737$	0	0
$738$	2583.05	0.128839
$739$	26183.9	1.30337	0.651685	−	0.758489i	$-0.274062\pi$
$739$	26183.9	1.30337	0.651685	+	0.758489i	$0.274062\pi$
$740$	−20066.2	−0.996824
$741$	10247.1	0.508012
$742$	−32691.8	−1.61746
$743$	−17676.4	−0.872793	−0.436396	−	0.899755i	$-0.643745\pi$
$743$	−17676.4	−0.872793	−0.436396	+	0.899755i	$0.643745\pi$
$744$	−8791.82	−0.433231
$745$	−44933.2	−2.20970
$746$	−17412.6	−0.854584
$747$	1501.12	0.0735248
$748$	0	0
$749$	15079.0	0.735612
$750$	670.524	0.0326454
$751$	12798.7	0.621879	0.310939	−	0.950430i	$-0.399356\pi$
$751$	12798.7	0.621879	0.310939	+	0.950430i	$0.399356\pi$
$752$	−2085.12	−0.101112
$753$	5109.61	0.247284
$754$	−4338.77	−0.209561
$755$	25339.9	1.22148
$756$	−27988.3	−1.34646
$757$	19686.2	0.945190	0.472595	−	0.881280i	$-0.343317\pi$
$757$	19686.2	0.945190	0.472595	+	0.881280i	$0.343317\pi$
$758$	−10739.5	−0.514611
$759$	0	0
$760$	34250.2	1.63472
$761$	−6660.94	−0.317291	−0.158646	−	0.987336i	$-0.550713\pi$
$761$	−6660.94	−0.317291	−0.158646	+	0.987336i	$0.550713\pi$
$762$	8119.56	0.386011
$763$	36691.6	1.74093
$764$	10110.9	0.478796
$765$	−1987.86	−0.0939491
$766$	18677.4	0.880996
$767$	−3391.88	−0.159679
$768$	13930.4	0.654518
$769$	22328.3	1.04705	0.523523	−	0.852011i	$-0.324617\pi$
$769$	22328.3	1.04705	0.523523	+	0.852011i	$0.324617\pi$
$770$	0	0
$771$	−14959.8	−0.698786
$772$	16617.0	0.774686
$773$	−3418.53	−0.159063	−0.0795317	−	0.996832i	$-0.525342\pi$
$773$	−3418.53	−0.159063	−0.0795317	+	0.996832i	$0.525342\pi$
$774$	−4706.40	−0.218563
$775$	11460.1	0.531172
$776$	−34763.2	−1.60815
$777$	31408.9	1.45018
$778$	−13777.8	−0.634909
$779$	−24581.0	−1.13056
$780$	−8642.27	−0.396722
$781$	0	0
$782$	−5138.53	−0.234979
$783$	20316.1	0.927251
$784$	10317.3	0.469994
$785$	−22031.0	−1.00168
$786$	−13684.8	−0.621017
$787$	7877.99	0.356824	0.178412	−	0.983956i	$-0.442904\pi$
$787$	7877.99	0.356824	0.178412	+	0.983956i	$0.442904\pi$
$788$	3198.38	0.144591
$789$	10141.3	0.457594
$790$	−10497.0	−0.472741
$791$	27763.7	1.24799
$792$	0	0
$793$	8207.83	0.367552
$794$	−957.627	−0.0428021
$795$	47267.5	2.10868
$796$	−12867.4	−0.572957
$797$	13082.6	0.581444	0.290722	−	0.956808i	$-0.406104\pi$
$797$	13082.6	0.581444	0.290722	+	0.956808i	$0.406104\pi$
$798$	−22450.1	−0.995895
$799$	2312.21	0.102378
$800$	−22227.7	−0.982336
$801$	−11649.1	−0.513858
$802$	−6678.43	−0.294044
$803$	0	0
$804$	141.486	0.00620625
$805$	100535.	4.40174
$806$	−3146.63	−0.137513
$807$	3175.52	0.138518
$808$	29888.4	1.30133
$809$	−15905.3	−0.691223	−0.345611	−	0.938378i	$-0.612328\pi$
$809$	−15905.3	−0.691223	−0.345611	+	0.938378i	$0.612328\pi$
$810$	−10979.5	−0.476272
$811$	−3286.15	−0.142284	−0.0711421	−	0.997466i	$-0.522664\pi$
$811$	−3286.15	−0.142284	−0.0711421	+	0.997466i	$0.522664\pi$
$812$	−24499.8	−1.05884
$813$	28288.3	1.22031
$814$	0	0
$815$	−769.118	−0.0330565
$816$	1151.04	0.0493805
$817$	44787.4	1.91788
$818$	10636.4	0.454635
$819$	−5196.73	−0.221720
$820$	20731.3	0.882889
$821$	−31145.2	−1.32396	−0.661982	−	0.749520i	$-0.730284\pi$
$821$	−31145.2	−1.32396	−0.661982	+	0.749520i	$0.730284\pi$
$822$	15837.3	0.672008
$823$	−47102.2	−1.99499	−0.997497	−	0.0707041i	$-0.977475\pi$
$823$	−47102.2	−1.99499	−0.997497	+	0.0707041i	$0.977475\pi$
$824$	−21985.8	−0.929503
$825$	0	0
$826$	7431.17	0.313031
$827$	8462.69	0.355836	0.177918	−	0.984045i	$-0.443064\pi$
$827$	8462.69	0.355836	0.177918	+	0.984045i	$0.443064\pi$
$828$	−8730.18	−0.366419
$829$	16497.7	0.691181	0.345590	−	0.938385i	$-0.387679\pi$
$829$	16497.7	0.691181	0.345590	+	0.938385i	$0.387679\pi$
$830$	−4674.44	−0.195484
$831$	12600.2	0.525986
$832$	3434.84	0.143127
$833$	−11441.0	−0.475877
$834$	7245.99	0.300849
$835$	34179.6	1.41657
$836$	0	0
$837$	14733.9	0.608458
$838$	12948.6	0.533775
$839$	8646.29	0.355784	0.177892	−	0.984050i	$-0.443072\pi$
$839$	8646.29	0.355784	0.177892	+	0.984050i	$0.443072\pi$
$840$	45214.4	1.85720
$841$	−6605.16	−0.270825
$842$	−13868.3	−0.567616
$843$	−19025.8	−0.777322
$844$	8035.46	0.327716
$845$	26896.3	1.09498
$846$	−1524.17	−0.0619408
$847$	0	0
$848$	10514.4	0.425784
$849$	−8796.78	−0.355600
$850$	3012.36	0.121557
$851$	45096.9	1.81657
$852$	10040.7	0.403744
$853$	5835.31	0.234229	0.117114	−	0.993118i	$-0.462636\pi$
$853$	5835.31	0.234229	0.117114	+	0.993118i	$0.462636\pi$
$854$	−17982.3	−0.720540
$855$	−12469.7	−0.498778
$856$	9736.96	0.388788
$857$	−6819.10	−0.271804	−0.135902	−	0.990722i	$-0.543393\pi$
$857$	−6819.10	−0.271804	−0.135902	+	0.990722i	$0.543393\pi$
$858$	0	0
$859$	28500.5	1.13204	0.566020	−	0.824391i	$-0.308482\pi$
$859$	28500.5	1.13204	0.566020	+	0.824391i	$0.308482\pi$
$860$	−37773.1	−1.49773
$861$	−32449.9	−1.28443
$862$	5575.75	0.220314
$863$	−10383.3	−0.409562	−0.204781	−	0.978808i	$-0.565648\pi$
$863$	−10383.3	−0.409562	−0.204781	+	0.978808i	$0.565648\pi$
$864$	−28577.6	−1.12527
$865$	4421.12	0.173783
$866$	7338.80	0.287970
$867$	−1276.40	−0.0499986
$868$	−17768.1	−0.694804
$869$	0	0
$870$	−13743.8	−0.535585
$871$	120.924	0.00470421
$872$	23692.9	0.920119
$873$	12656.5	0.490673
$874$	−32233.8	−1.24751
$875$	3236.01	0.125025
$876$	8489.83	0.327448
$877$	−9730.29	−0.374651	−0.187325	−	0.982298i	$-0.559982\pi$
$877$	−9730.29	−0.374651	−0.187325	+	0.982298i	$0.559982\pi$
$878$	−126.101	−0.00484705
$879$	−2640.57	−0.101325
$880$	0	0
$881$	−37327.4	−1.42746	−0.713730	−	0.700421i	$-0.752995\pi$
$881$	−37327.4	−1.42746	−0.713730	+	0.700421i	$0.752995\pi$
$882$	7541.67	0.287915
$883$	−14389.4	−0.548407	−0.274203	−	0.961672i	$-0.588414\pi$
$883$	−14389.4	−0.548407	−0.274203	+	0.961672i	$0.588414\pi$
$884$	2131.79	0.0811084
$885$	−10744.4	−0.408099
$886$	19296.4	0.731689
$887$	38356.7	1.45196	0.725982	−	0.687714i	$-0.241386\pi$
$887$	38356.7	1.45196	0.725982	+	0.687714i	$0.241386\pi$
$888$	20281.7	0.766453
$889$	39185.8	1.47834
$890$	36274.9	1.36622
$891$	0	0
$892$	−7109.10	−0.266850
$893$	14504.4	0.543528
$894$	19018.4	0.711488
$895$	−18372.1	−0.686156
$896$	40308.6	1.50292
$897$	19422.6	0.722969
$898$	−12868.6	−0.478209
$899$	12897.5	0.478482
$900$	5117.90	0.189552
$901$	−11659.5	−0.431113
$902$	0	0
$903$	59124.8	2.17890
$904$	17927.9	0.659593
$905$	28011.6	1.02888
$906$	−10725.4	−0.393296
$907$	−19562.8	−0.716177	−0.358089	−	0.933688i	$-0.616571\pi$
$907$	−19562.8	−0.716177	−0.358089	+	0.933688i	$0.616571\pi$
$908$	1507.13	0.0550836
$909$	−10881.7	−0.397055
$910$	16182.5	0.589498
$911$	−1194.41	−0.0434387	−0.0217194	−	0.999764i	$-0.506914\pi$
$911$	−1194.41	−0.0434387	−0.0217194	+	0.999764i	$0.506914\pi$
$912$	7220.42	0.262162
$913$	0	0
$914$	9818.62	0.355329
$915$	25999.7	0.939370
$916$	4113.35	0.148372
$917$	−66044.0	−2.37837
$918$	3872.92	0.139243
$919$	−3059.04	−0.109802	−0.0549012	−	0.998492i	$-0.517484\pi$
$919$	−3059.04	−0.109802	−0.0549012	+	0.998492i	$0.517484\pi$
$920$	64918.8	2.32642
$921$	−17277.9	−0.618160
$922$	3727.00	0.133126
$923$	8581.56	0.306030
$924$	0	0
$925$	−26437.1	−0.939727
$926$	26279.8	0.932621
$927$	8004.51	0.283606
$928$	−25015.7	−0.884891
$929$	16418.2	0.579832	0.289916	−	0.957052i	$-0.406373\pi$
$929$	16418.2	0.579832	0.289916	+	0.957052i	$0.406373\pi$
$930$	−9967.51	−0.351449
$931$	−71768.6	−2.52644
$932$	−6141.30	−0.215842
$933$	−24399.3	−0.856161
$934$	−4909.13	−0.171983
$935$	0	0
$936$	−3355.69	−0.117184
$937$	21079.0	0.734922	0.367461	−	0.930039i	$-0.380227\pi$
$937$	21079.0	0.734922	0.367461	+	0.930039i	$0.380227\pi$
$938$	−264.930	−0.00922202
$939$	3738.42	0.129924
$940$	−12232.8	−0.424457
$941$	−2769.28	−0.0959362	−0.0479681	−	0.998849i	$-0.515275\pi$
$941$	−2769.28	−0.0959362	−0.0479681	+	0.998849i	$0.515275\pi$
$942$	9324.82	0.322525
$943$	−46591.6	−1.60894
$944$	−2390.02	−0.0824031
$945$	−75773.5	−2.60837
$946$	0	0
$947$	2386.23	0.0818818	0.0409409	−	0.999162i	$-0.486964\pi$
$947$	2386.23	0.0818818	0.0409409	+	0.999162i	$0.486964\pi$
$948$	−11451.2	−0.392317
$949$	7256.03	0.248199
$950$	18896.4	0.645348
$951$	−27651.9	−0.942876
$952$	−11153.0	−0.379698
$953$	13711.6	0.466066	0.233033	−	0.972469i	$-0.425135\pi$
$953$	13711.6	0.466066	0.233033	+	0.972469i	$0.425135\pi$
$954$	7685.71	0.260832
$955$	27373.6	0.927527
$956$	17151.1	0.580237
$957$	0	0
$958$	3594.47	0.121223
$959$	76432.4	2.57365
$960$	10880.4	0.365797
$961$	−20437.3	−0.686022
$962$	7258.93	0.243282
$963$	−3545.01	−0.118625
$964$	24797.7	0.828508
$965$	44987.5	1.50072
$966$	−42552.5	−1.41729
$967$	35220.4	1.17126	0.585632	−	0.810577i	$-0.300847\pi$
$967$	35220.4	1.17126	0.585632	+	0.810577i	$0.300847\pi$
$968$	0	0
$969$	−8006.78	−0.265444
$970$	−39412.0	−1.30458
$971$	−15030.9	−0.496771	−0.248386	−	0.968661i	$-0.579900\pi$
$971$	−15030.9	−0.496771	−0.248386	+	0.968661i	$0.579900\pi$
$972$	11730.4	0.387092
$973$	34969.8	1.15219
$974$	12707.1	0.418029
$975$	−11386.1	−0.373998
$976$	5783.48	0.189677
$977$	28323.6	0.927486	0.463743	−	0.885970i	$-0.346506\pi$
$977$	28323.6	0.927486	0.463743	+	0.885970i	$0.346506\pi$
$978$	325.537	0.0106437
$979$	0	0
$980$	60528.6	1.97298
$981$	−8626.06	−0.280743
$982$	20720.4	0.673334
$983$	32925.5	1.06832	0.534161	−	0.845383i	$-0.320628\pi$
$983$	32925.5	1.06832	0.534161	+	0.845383i	$0.320628\pi$
$984$	−20954.0	−0.678849
$985$	8659.06	0.280102
$986$	3390.19	0.109499
$987$	19147.5	0.617500
$988$	13372.6	0.430607
$989$	84891.2	2.72941
$990$	0	0
$991$	43658.0	1.39944	0.699719	−	0.714419i	$-0.253309\pi$
$991$	43658.0	1.39944	0.699719	+	0.714419i	$0.253309\pi$
$992$	−18142.2	−0.580662
$993$	−461.924	−0.0147620
$994$	−18801.1	−0.599934
$995$	−34836.3	−1.10994
$996$	−5099.35	−0.162228
$997$	−23471.4	−0.745585	−0.372792	−	0.927915i	$-0.621600\pi$
$997$	−23471.4	−0.745585	−0.372792	+	0.927915i	$0.621600\pi$
$998$	−5244.22	−0.166336
$999$	−33989.6	−1.07646

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2057.4.a.s.1.18		44
11.2	odd	10		187.4.g.a.103.9	yes	88
11.6	odd	10		187.4.g.a.69.9	✓	88
11.10	odd	2		2057.4.a.t.1.27		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.4.g.a.69.9	✓	88	11.6	odd	10
187.4.g.a.103.9	yes	88	11.2	odd	10
2057.4.a.s.1.18		44	1.1	even	1	trivial
2057.4.a.t.1.27		44	11.10	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2057 = 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2057.a (trivial)

\(f(q)\)	\(=\)	\(q-1.49542 q^{2} -4.41660 q^{3} -5.76373 q^{4} -15.6043 q^{5} +6.60466 q^{6} +31.8747 q^{7} +20.5825 q^{8} -7.49362 q^{9} +23.3349 q^{10} +25.4561 q^{12} +21.7566 q^{13} -47.6660 q^{14} +68.9180 q^{15} +15.3304 q^{16} -17.0000 q^{17} +11.2061 q^{18} -106.640 q^{19} +89.9389 q^{20} -140.778 q^{21} -202.129 q^{23} -90.9048 q^{24} +118.494 q^{25} -32.5353 q^{26} +152.345 q^{27} -183.717 q^{28} +133.356 q^{29} -103.061 q^{30} +96.7146 q^{31} -187.585 q^{32} +25.4221 q^{34} -497.382 q^{35} +43.1912 q^{36} -223.110 q^{37} +159.472 q^{38} -96.0905 q^{39} -321.176 q^{40} +230.504 q^{41} +210.522 q^{42} -419.986 q^{43} +116.933 q^{45} +302.267 q^{46} -136.012 q^{47} -67.7082 q^{48} +672.997 q^{49} -177.198 q^{50} +75.0822 q^{51} -125.399 q^{52} +685.851 q^{53} -227.819 q^{54} +656.062 q^{56} +470.987 q^{57} -199.423 q^{58} -155.901 q^{59} -397.224 q^{60} +377.256 q^{61} -144.629 q^{62} -238.857 q^{63} +157.875 q^{64} -339.497 q^{65} +5.55804 q^{67} +97.9834 q^{68} +892.722 q^{69} +743.794 q^{70} +394.434 q^{71} -154.238 q^{72} +333.509 q^{73} +333.642 q^{74} -523.341 q^{75} +614.645 q^{76} +143.695 q^{78} -449.839 q^{79} -239.220 q^{80} -470.518 q^{81} -344.700 q^{82} -200.319 q^{83} +811.406 q^{84} +265.273 q^{85} +628.054 q^{86} -588.981 q^{87} +1554.53 q^{89} -174.863 q^{90} +693.487 q^{91} +1165.01 q^{92} -427.150 q^{93} +203.395 q^{94} +1664.04 q^{95} +828.490 q^{96} -1688.97 q^{97} -1006.41 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - q^{2} - 28 q^{3} + 139 q^{4} - 24 q^{5} + 24 q^{6} + 2 q^{7} + 63 q^{8} + 356 q^{9} + 65 q^{10} - 337 q^{12} - 12 q^{13} - 151 q^{14} - 320 q^{15} + 311 q^{16} - 748 q^{17} + 55 q^{18} + 36 q^{19}+ \cdots - 2302 q^{98}+O(q^{100}) \) 44 * q - q^2 - 28 * q^3 + 139 * q^4 - 24 * q^5 + 24 * q^6 + 2 * q^7 + 63 * q^8 + 356 * q^9 + 65 * q^10 - 337 * q^12 - 12 * q^13 - 151 * q^14 - 320 * q^15 + 311 * q^16 - 748 * q^17 + 55 * q^18 + 36 * q^19 - 209 * q^20 + 240 * q^21 - 656 * q^23 + 104 * q^24 + 458 * q^25 - 455 * q^26 - 538 * q^27 - 63 * q^28 + 698 * q^29 - 304 * q^30 - 1418 * q^31 + 83 * q^32 + 17 * q^34 - 102 * q^35 - 85 * q^36 - 1358 * q^37 - 568 * q^38 - 114 * q^39 + 281 * q^40 - 54 * q^41 - 1943 * q^42 - 382 * q^43 - 1226 * q^45 - 232 * q^46 - 544 * q^47 - 2743 * q^48 + 864 * q^49 - 1307 * q^50 + 476 * q^51 - 3112 * q^52 - 2492 * q^53 - 2400 * q^54 + 27 * q^56 - 56 * q^57 - 3746 * q^58 - 662 * q^59 - 1392 * q^60 + 964 * q^61 + 3256 * q^62 - 2322 * q^63 - 3653 * q^64 + 152 * q^65 - 3442 * q^67 - 2363 * q^68 - 1664 * q^69 - 3130 * q^70 - 2936 * q^71 + 1281 * q^72 + 2240 * q^73 - 3494 * q^74 - 1892 * q^75 - 1261 * q^76 + 1120 * q^78 + 1016 * q^79 - 3280 * q^80 + 2140 * q^81 - 3985 * q^82 + 1656 * q^83 + 5401 * q^84 + 408 * q^85 - 1373 * q^86 - 1968 * q^87 - 2992 * q^89 - 1287 * q^90 - 8310 * q^91 - 2442 * q^92 - 4058 * q^93 + 694 * q^94 + 8280 * q^95 - 2203 * q^96 - 4746 * q^97 - 2302 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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