LMFDB - Embedded newform 2057.4.a.s.1.17

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			1.17
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)

$f(q)$	$=$	\(q-1.50580 q^{2} +6.30025 q^{3} -5.73256 q^{4} +3.99925 q^{5} -9.48693 q^{6} +11.1510 q^{7} +20.6785 q^{8} +12.6932 q^{9} -6.02208 q^{10} -36.1166 q^{12} -22.6601 q^{13} -16.7911 q^{14} +25.1963 q^{15} +14.7228 q^{16} -17.0000 q^{17} -19.1134 q^{18} +2.68258 q^{19} -22.9260 q^{20} +70.2539 q^{21} +79.8743 q^{23} +130.280 q^{24} -109.006 q^{25} +34.1216 q^{26} -90.1366 q^{27} -63.9236 q^{28} -170.274 q^{29} -37.9406 q^{30} +135.823 q^{31} -187.598 q^{32} +25.5986 q^{34} +44.5955 q^{35} -72.7645 q^{36} -181.035 q^{37} -4.03943 q^{38} -142.764 q^{39} +82.6985 q^{40} +172.155 q^{41} -105.788 q^{42} +200.616 q^{43} +50.7632 q^{45} -120.275 q^{46} -420.433 q^{47} +92.7574 q^{48} -218.656 q^{49} +164.141 q^{50} -107.104 q^{51} +129.900 q^{52} +756.563 q^{53} +135.728 q^{54} +230.585 q^{56} +16.9009 q^{57} +256.399 q^{58} -681.645 q^{59} -144.439 q^{60} -72.3490 q^{61} -204.522 q^{62} +141.541 q^{63} +164.702 q^{64} -90.6234 q^{65} -600.206 q^{67} +97.4536 q^{68} +503.228 q^{69} -67.1519 q^{70} -952.156 q^{71} +262.476 q^{72} -245.062 q^{73} +272.603 q^{74} -686.765 q^{75} -15.3780 q^{76} +214.975 q^{78} -292.538 q^{79} +58.8802 q^{80} -910.599 q^{81} -259.231 q^{82} +1418.56 q^{83} -402.735 q^{84} -67.9873 q^{85} -302.088 q^{86} -1072.77 q^{87} -578.399 q^{89} -76.4393 q^{90} -252.682 q^{91} -457.885 q^{92} +855.718 q^{93} +633.089 q^{94} +10.7283 q^{95} -1181.91 q^{96} +1150.30 q^{97} +329.252 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.50580	−0.532381	−0.266190	−	0.963920i	$-0.585765\pi$
$2$	−1.50580	−0.532381	−0.266190	+	0.963920i	$0.585765\pi$
$3$	6.30025	1.21248	0.606242	−	0.795280i	$-0.292676\pi$
$3$	6.30025	1.21248	0.606242	+	0.795280i	$0.292676\pi$
$4$	−5.73256	−0.716570
$5$	3.99925	0.357704	0.178852	−	0.983876i	$-0.442762\pi$
$5$	3.99925	0.357704	0.178852	+	0.983876i	$0.442762\pi$
$6$	−9.48693	−0.645503
$7$	11.1510	0.602095	0.301048	−	0.953609i	$-0.402664\pi$
$7$	11.1510	0.602095	0.301048	+	0.953609i	$0.402664\pi$
$8$	20.6785	0.913869
$9$	12.6932	0.470118
$10$	−6.02208	−0.190435
$11$	0	0
$11$	0	0
$12$	−36.1166	−0.868830
$13$	−22.6601	−0.483444	−0.241722	−	0.970346i	$-0.577712\pi$
$13$	−22.6601	−0.483444	−0.241722	+	0.970346i	$0.577712\pi$
$14$	−16.7911	−0.320544
$15$	25.1963	0.433710
$16$	14.7228	0.230044
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	−19.1134	−0.250282
$19$	2.68258	0.0323908	0.0161954	−	0.999869i	$-0.494845\pi$
$19$	2.68258	0.0323908	0.0161954	+	0.999869i	$0.494845\pi$
$20$	−22.9260	−0.256320
$21$	70.2539	0.730031
$22$	0	0
$23$	79.8743	0.724128	0.362064	−	0.932153i	$-0.382072\pi$
$23$	79.8743	0.724128	0.362064	+	0.932153i	$0.382072\pi$
$24$	130.280	1.10805
$25$	−109.006	−0.872048
$26$	34.1216	0.257377
$27$	−90.1366	−0.642474
$28$	−63.9236	−0.431444
$29$	−170.274	−1.09031	−0.545157	−	0.838334i	$-0.683530\pi$
$29$	−170.274	−1.09031	−0.545157	+	0.838334i	$0.683530\pi$
$30$	−37.9406	−0.230899
$31$	135.823	0.786919	0.393459	−	0.919342i	$-0.371278\pi$
$31$	135.823	0.786919	0.393459	+	0.919342i	$0.371278\pi$
$32$	−187.598	−1.03634
$33$	0	0
$34$	25.5986	0.129121
$35$	44.5955	0.215372
$36$	−72.7645	−0.336873
$37$	−181.035	−0.804378	−0.402189	−	0.915557i	$-0.631751\pi$
$37$	−181.035	−0.804378	−0.402189	+	0.915557i	$0.631751\pi$
$38$	−4.03943	−0.0172443
$39$	−142.764	−0.586169
$40$	82.6985	0.326895
$41$	172.155	0.655759	0.327879	−	0.944720i	$-0.393666\pi$
$41$	172.155	0.655759	0.327879	+	0.944720i	$0.393666\pi$
$42$	−105.788	−0.388655
$43$	200.616	0.711480	0.355740	−	0.934585i	$-0.384229\pi$
$43$	200.616	0.711480	0.355740	+	0.934585i	$0.384229\pi$
$44$	0	0
$45$	50.7632	0.168163
$46$	−120.275	−0.385512
$47$	−420.433	−1.30482	−0.652410	−	0.757866i	$-0.726242\pi$
$47$	−420.433	−1.30482	−0.652410	+	0.757866i	$0.726242\pi$
$48$	92.7574	0.278924
$49$	−218.656	−0.637481
$50$	164.141	0.464262
$51$	−107.104	−0.294071
$52$	129.900	0.346422
$53$	756.563	1.96079	0.980395	−	0.197043i	$-0.0631338\pi$
$53$	756.563	1.96079	0.980395	+	0.197043i	$0.0631338\pi$
$54$	135.728	0.342041
$55$	0	0
$56$	230.585	0.550237
$57$	16.9009	0.0392734
$58$	256.399	0.580462
$59$	−681.645	−1.50411	−0.752056	−	0.659099i	$-0.770938\pi$
$59$	−681.645	−1.50411	−0.752056	+	0.659099i	$0.770938\pi$
$60$	−144.439	−0.310784
$61$	−72.3490	−0.151858	−0.0759290	−	0.997113i	$-0.524192\pi$
$61$	−72.3490	−0.151858	−0.0759290	+	0.997113i	$0.524192\pi$
$62$	−204.522	−0.418941
$63$	141.541	0.283056
$64$	164.702	0.321684
$65$	−90.6234	−0.172930
$66$	0	0
$67$	−600.206	−1.09443	−0.547215	−	0.836992i	$-0.684312\pi$
$67$	−600.206	−1.09443	−0.547215	+	0.836992i	$0.684312\pi$
$68$	97.4536	0.173794
$69$	503.228	0.877994
$70$	−67.1519	−0.114660
$71$	−952.156	−1.59155	−0.795776	−	0.605592i	$-0.792936\pi$
$71$	−952.156	−1.59155	−0.795776	+	0.605592i	$0.792936\pi$
$72$	262.476	0.429626
$73$	−245.062	−0.392909	−0.196454	−	0.980513i	$-0.562943\pi$
$73$	−245.062	−0.392909	−0.196454	+	0.980513i	$0.562943\pi$
$74$	272.603	0.428236
$75$	−686.765	−1.05734
$76$	−15.3780	−0.0232103
$77$	0	0
$78$	214.975	0.312065
$79$	−292.538	−0.416621	−0.208311	−	0.978063i	$-0.566797\pi$
$79$	−292.538	−0.416621	−0.208311	+	0.978063i	$0.566797\pi$
$80$	58.8802	0.0822875
$81$	−910.599	−1.24911
$82$	−259.231	−0.349114
$83$	1418.56	1.87599	0.937993	−	0.346655i	$-0.112683\pi$
$83$	1418.56	1.87599	0.937993	+	0.346655i	$0.112683\pi$
$84$	−402.735	−0.523119
$85$	−67.9873	−0.0867559
$86$	−302.088	−0.378779
$87$	−1072.77	−1.32199
$88$	0	0
$89$	−578.399	−0.688879	−0.344439	−	0.938809i	$-0.611931\pi$
$89$	−578.399	−0.688879	−0.344439	+	0.938809i	$0.611931\pi$
$90$	−76.4393	−0.0895268
$91$	−252.682	−0.291080
$92$	−457.885	−0.518889
$93$	855.718	0.954127
$94$	633.089	0.694661
$95$	10.7283	0.0115863
$96$	−1181.91	−1.25655
$97$	1150.30	1.20408	0.602040	−	0.798466i	$-0.294355\pi$
$97$	1150.30	1.20408	0.602040	+	0.798466i	$0.294355\pi$
$98$	329.252	0.339383
$99$	0	0
$100$	624.884	0.624884
$101$	397.161	0.391277	0.195638	−	0.980676i	$-0.437322\pi$
$101$	397.161	0.391277	0.195638	+	0.980676i	$0.437322\pi$
$102$	161.278	0.156558
$103$	−626.854	−0.599668	−0.299834	−	0.953991i	$-0.596931\pi$
$103$	−626.854	−0.599668	−0.299834	+	0.953991i	$0.596931\pi$
$104$	−468.577	−0.441805
$105$	280.963	0.261135
$106$	−1139.23	−1.04389
$107$	−842.255	−0.760971	−0.380485	−	0.924787i	$-0.624243\pi$
$107$	−842.255	−0.760971	−0.380485	+	0.924787i	$0.624243\pi$
$108$	516.714	0.460378
$109$	1418.65	1.24662	0.623312	−	0.781974i	$-0.285787\pi$
$109$	1418.65	1.24662	0.623312	+	0.781974i	$0.285787\pi$
$110$	0	0
$111$	−1140.57	−0.975295
$112$	164.173	0.138508
$113$	−302.413	−0.251758	−0.125879	−	0.992046i	$-0.540175\pi$
$113$	−302.413	−0.251758	−0.125879	+	0.992046i	$0.540175\pi$
$114$	−25.4494	−0.0209084
$115$	319.437	0.259023
$116$	976.106	0.781286
$117$	−287.629	−0.227276
$118$	1026.42	0.800761
$119$	−189.566	−0.146030
$120$	521.022	0.396355
$121$	0	0
$122$	108.943	0.0808463
$123$	1084.62	0.795097
$124$	−778.613	−0.563883
$125$	−935.849	−0.669639
$126$	−213.133	−0.150693
$127$	415.092	0.290027	0.145014	−	0.989430i	$-0.453677\pi$
$127$	415.092	0.290027	0.145014	+	0.989430i	$0.453677\pi$
$128$	1252.77	0.865082
$129$	1263.93	0.862659
$130$	136.461	0.0920646
$131$	1435.89	0.957665	0.478832	−	0.877906i	$-0.341060\pi$
$131$	1435.89	0.957665	0.478832	+	0.877906i	$0.341060\pi$
$132$	0	0
$133$	29.9133	0.0195024
$134$	903.791	0.582654
$135$	−360.479	−0.229815
$136$	−351.535	−0.221646
$137$	−1024.05	−0.638615	−0.319307	−	0.947651i	$-0.603450\pi$
$137$	−1024.05	−0.638615	−0.319307	+	0.947651i	$0.603450\pi$
$138$	−757.761	−0.467427
$139$	−111.869	−0.0682632	−0.0341316	−	0.999417i	$-0.510867\pi$
$139$	−111.869	−0.0682632	−0.0341316	+	0.999417i	$0.510867\pi$
$140$	−255.647	−0.154329
$141$	−2648.84	−1.58207
$142$	1433.76	0.847312
$143$	0	0
$144$	186.879	0.108148
$145$	−680.968	−0.390009
$146$	369.014	0.209177
$147$	−1377.59	−0.772936
$148$	1037.80	0.576393
$149$	−1560.89	−0.858206	−0.429103	−	0.903256i	$-0.641170\pi$
$149$	−1560.89	−0.858206	−0.429103	+	0.903256i	$0.641170\pi$
$150$	1034.13	0.562910
$151$	905.685	0.488104	0.244052	−	0.969762i	$-0.421523\pi$
$151$	905.685	0.488104	0.244052	+	0.969762i	$0.421523\pi$
$152$	55.4717	0.0296010
$153$	−215.784	−0.114020
$154$	0	0
$155$	543.189	0.281484
$156$	818.405	0.420031
$157$	−2995.40	−1.52267	−0.761335	−	0.648359i	$-0.775456\pi$
$157$	−2995.40	−1.52267	−0.761335	+	0.648359i	$0.775456\pi$
$158$	440.504	0.221801
$159$	4766.53	2.37743
$160$	−750.250	−0.370703
$161$	890.675	0.435994
$162$	1371.18	0.665001
$163$	−1021.29	−0.490759	−0.245379	−	0.969427i	$-0.578913\pi$
$163$	−1021.29	−0.490759	−0.245379	+	0.969427i	$0.578913\pi$
$164$	−986.890	−0.469897
$165$	0	0
$166$	−2136.06	−0.998739
$167$	1442.09	0.668216	0.334108	−	0.942535i	$-0.391565\pi$
$167$	1442.09	0.668216	0.334108	+	0.942535i	$0.391565\pi$
$168$	1452.75	0.667153
$169$	−1683.52	−0.766282
$170$	102.375	0.0461872
$171$	34.0504	0.0152275
$172$	−1150.04	−0.509826
$173$	−3638.54	−1.59903	−0.799517	−	0.600644i	$-0.794911\pi$
$173$	−3638.54	−1.59903	−0.799517	+	0.600644i	$0.794911\pi$
$174$	1615.38	0.703801
$175$	−1215.52	−0.525056
$176$	0	0
$177$	−4294.54	−1.82371
$178$	870.954	0.366746
$179$	1093.04	0.456410	0.228205	−	0.973613i	$-0.426714\pi$
$179$	1093.04	0.456410	0.228205	+	0.973613i	$0.426714\pi$
$180$	−291.003	−0.120501
$181$	−813.441	−0.334047	−0.167024	−	0.985953i	$-0.553416\pi$
$181$	−813.441	−0.334047	−0.167024	+	0.985953i	$0.553416\pi$
$182$	380.488	0.154965
$183$	−455.817	−0.184125
$184$	1651.68	0.661758
$185$	−724.005	−0.287729
$186$	−1288.54	−0.507959
$187$	0	0
$188$	2410.16	0.934995
$189$	−1005.11	−0.386830
$190$	−16.1547	−0.00616834
$191$	898.735	0.340472	0.170236	−	0.985403i	$-0.445547\pi$
$191$	898.735	0.340472	0.170236	+	0.985403i	$0.445547\pi$
$192$	1037.67	0.390037
$193$	110.064	0.0410496	0.0205248	−	0.999789i	$-0.493466\pi$
$193$	110.064	0.0410496	0.0205248	+	0.999789i	$0.493466\pi$
$194$	−1732.13	−0.641029
$195$	−570.950	−0.209675
$196$	1253.46	0.456800
$197$	−4331.34	−1.56647	−0.783237	−	0.621723i	$-0.786433\pi$
$197$	−4331.34	−1.56647	−0.783237	+	0.621723i	$0.786433\pi$
$198$	0	0
$199$	1169.70	0.416674	0.208337	−	0.978057i	$-0.433195\pi$
$199$	1169.70	0.416674	0.208337	+	0.978057i	$0.433195\pi$
$200$	−2254.08	−0.796938
$201$	−3781.45	−1.32698
$202$	−598.045	−0.208308
$203$	−1898.72	−0.656472
$204$	613.982	0.210722
$205$	688.492	0.234568
$206$	943.917	0.319252
$207$	1013.86	0.340425
$208$	−333.620	−0.111213
$209$	0	0
$210$	−423.074	−0.139023
$211$	−1998.27	−0.651975	−0.325987	−	0.945374i	$-0.605697\pi$
$211$	−1998.27	−0.651975	−0.325987	+	0.945374i	$0.605697\pi$
$212$	−4337.04	−1.40504
$213$	−5998.82	−1.92973
$214$	1268.27	0.405126
$215$	802.314	0.254499
$216$	−1863.89	−0.587137
$217$	1514.55	0.473800
$218$	−2136.20	−0.663678
$219$	−1543.95	−0.476395
$220$	0	0
$221$	385.221	0.117252
$222$	1717.47	0.519229
$223$	2759.52	0.828660	0.414330	−	0.910127i	$-0.364016\pi$
$223$	2759.52	0.828660	0.414330	+	0.910127i	$0.364016\pi$
$224$	−2091.89	−0.623976
$225$	−1383.63	−0.409965
$226$	455.374	0.134031
$227$	1133.37	0.331386	0.165693	−	0.986177i	$-0.447014\pi$
$227$	1133.37	0.331386	0.165693	+	0.986177i	$0.447014\pi$
$228$	−96.8856	−0.0281421
$229$	2038.96	0.588377	0.294188	−	0.955747i	$-0.404951\pi$
$229$	2038.96	0.588377	0.294188	+	0.955747i	$0.404951\pi$
$230$	−481.009	−0.137899
$231$	0	0
$232$	−3521.01	−0.996404
$233$	−1183.12	−0.332656	−0.166328	−	0.986070i	$-0.553191\pi$
$233$	−1183.12	−0.332656	−0.166328	+	0.986070i	$0.553191\pi$
$234$	433.111	0.120997
$235$	−1681.42	−0.466739
$236$	3907.57	1.07780
$237$	−1843.06	−0.505147
$238$	285.449	0.0777434
$239$	2497.60	0.675967	0.337983	−	0.941152i	$-0.390255\pi$
$239$	2497.60	0.675967	0.337983	+	0.941152i	$0.390255\pi$
$240$	370.960	0.0997723
$241$	−2821.54	−0.754156	−0.377078	−	0.926182i	$-0.623071\pi$
$241$	−2821.54	−0.754156	−0.377078	+	0.926182i	$0.623071\pi$
$242$	0	0
$243$	−3303.32	−0.872049
$244$	414.745	0.108817
$245$	−874.460	−0.228030
$246$	−1633.22	−0.423295
$247$	−60.7874	−0.0156592
$248$	2808.61	0.719141
$249$	8937.26	2.27460
$250$	1409.20	0.356503
$251$	−2702.82	−0.679684	−0.339842	−	0.940483i	$-0.610374\pi$
$251$	−2702.82	−0.679684	−0.339842	+	0.940483i	$0.610374\pi$
$252$	−811.394	−0.202829
$253$	0	0
$254$	−625.046	−0.154405
$255$	−428.337	−0.105190
$256$	−3204.04	−0.782237
$257$	−3368.12	−0.817499	−0.408750	−	0.912647i	$-0.634035\pi$
$257$	−3368.12	−0.817499	−0.408750	+	0.912647i	$0.634035\pi$
$258$	−1903.23	−0.459263
$259$	−2018.72	−0.484312
$260$	519.504	0.123916
$261$	−2161.32	−0.512576
$262$	−2162.16	−0.509843
$263$	−7600.25	−1.78194	−0.890972	−	0.454058i	$-0.849976\pi$
$263$	−7600.25	−1.78194	−0.890972	+	0.454058i	$0.849976\pi$
$264$	0	0
$265$	3025.68	0.701382
$266$	−45.0435	−0.0103827
$267$	−3644.06	−0.835254
$268$	3440.72	0.784237
$269$	276.256	0.0626157	0.0313078	−	0.999510i	$-0.490033\pi$
$269$	276.256	0.0626157	0.0313078	+	0.999510i	$0.490033\pi$
$270$	542.809	0.122349
$271$	1881.55	0.421756	0.210878	−	0.977512i	$-0.432368\pi$
$271$	1881.55	0.421756	0.210878	+	0.977512i	$0.432368\pi$
$272$	−250.288	−0.0557938
$273$	−1591.96	−0.352929
$274$	1542.01	0.339986
$275$	0	0
$276$	−2884.79	−0.629144
$277$	6207.66	1.34651	0.673253	−	0.739412i	$-0.264896\pi$
$277$	6207.66	1.34651	0.673253	+	0.739412i	$0.264896\pi$
$278$	168.452	0.0363420
$279$	1724.02	0.369945
$280$	922.168	0.196822
$281$	6991.90	1.48435	0.742174	−	0.670207i	$-0.233795\pi$
$281$	6991.90	1.48435	0.742174	+	0.670207i	$0.233795\pi$
$282$	3988.62	0.842266
$283$	2659.40	0.558604	0.279302	−	0.960203i	$-0.409897\pi$
$283$	2659.40	0.558604	0.279302	+	0.960203i	$0.409897\pi$
$284$	5458.30	1.14046
$285$	67.5910	0.0140482
$286$	0	0
$287$	1919.70	0.394829
$288$	−2381.21	−0.487202
$289$	289.000	0.0588235
$290$	1025.40	0.207633
$291$	7247.21	1.45993
$292$	1404.83	0.281547
$293$	−6449.04	−1.28586	−0.642930	−	0.765925i	$-0.722281\pi$
$293$	−6449.04	−1.28586	−0.642930	+	0.765925i	$0.722281\pi$
$294$	2074.37	0.411496
$295$	−2726.07	−0.538027
$296$	−3743.53	−0.735096
$297$	0	0
$298$	2350.38	0.456893
$299$	−1809.96	−0.350076
$300$	3936.93	0.757662
$301$	2237.06	0.428379
$302$	−1363.78	−0.259857
$303$	2502.21	0.474417
$304$	39.4951	0.00745131
$305$	−289.342	−0.0543202
$306$	324.928	0.0607022
$307$	−5577.92	−1.03697	−0.518483	−	0.855088i	$-0.673503\pi$
$307$	−5577.92	−1.03697	−0.518483	+	0.855088i	$0.673503\pi$
$308$	0	0
$309$	−3949.34	−0.727087
$310$	−817.935	−0.149857
$311$	−4156.41	−0.757840	−0.378920	−	0.925429i	$-0.623704\pi$
$311$	−4156.41	−0.757840	−0.378920	+	0.925429i	$0.623704\pi$
$312$	−2952.15	−0.535682
$313$	−2372.20	−0.428385	−0.214193	−	0.976791i	$-0.568712\pi$
$313$	−2372.20	−0.428385	−0.214193	+	0.976791i	$0.568712\pi$
$314$	4510.48	0.810640
$315$	566.059	0.101250
$316$	1676.99	0.298539
$317$	−908.954	−0.161047	−0.0805236	−	0.996753i	$-0.525659\pi$
$317$	−908.954	−0.161047	−0.0805236	+	0.996753i	$0.525659\pi$
$318$	−7177.45	−1.26570
$319$	0	0
$320$	658.686	0.115068
$321$	−5306.42	−0.922665
$322$	−1341.18	−0.232115
$323$	−45.6038	−0.00785593
$324$	5220.07	0.895073
$325$	2470.08	0.421587
$326$	1537.86	0.261271
$327$	8937.85	1.51151
$328$	3559.91	0.599278
$329$	−4688.24	−0.785626
$330$	0	0
$331$	−7925.80	−1.31614	−0.658068	−	0.752958i	$-0.728626\pi$
$331$	−7925.80	−1.31614	−0.658068	+	0.752958i	$0.728626\pi$
$332$	−8131.96	−1.34428
$333$	−2297.91	−0.378152
$334$	−2171.50	−0.355746
$335$	−2400.38	−0.391482
$336$	1034.33	0.167939
$337$	−4354.86	−0.703930	−0.351965	−	0.936013i	$-0.614486\pi$
$337$	−4354.86	−0.703930	−0.351965	+	0.936013i	$0.614486\pi$
$338$	2535.05	0.407954
$339$	−1905.28	−0.305253
$340$	389.741	0.0621667
$341$	0	0
$342$	−51.2732	−0.00810683
$343$	−6263.00	−0.985920
$344$	4148.44	0.650200
$345$	2012.54	0.314062
$346$	5478.91	0.851295
$347$	−2614.97	−0.404550	−0.202275	−	0.979329i	$-0.564833\pi$
$347$	−2614.97	−0.404550	−0.202275	+	0.979329i	$0.564833\pi$
$348$	6149.72	0.947297
$349$	−7268.79	−1.11487	−0.557435	−	0.830221i	$-0.688214\pi$
$349$	−7268.79	−1.11487	−0.557435	+	0.830221i	$0.688214\pi$
$350$	1830.33	0.279530
$351$	2042.50	0.310600
$352$	0	0
$353$	4735.05	0.713941	0.356970	−	0.934116i	$-0.383810\pi$
$353$	4735.05	0.713941	0.356970	+	0.934116i	$0.383810\pi$
$354$	6466.72	0.970910
$355$	−3807.91	−0.569304
$356$	3315.71	0.493630
$357$	−1194.32	−0.177059
$358$	−1645.90	−0.242984
$359$	584.329	0.0859045	0.0429523	−	0.999077i	$-0.486324\pi$
$359$	584.329	0.0859045	0.0429523	+	0.999077i	$0.486324\pi$
$360$	1049.71	0.153679
$361$	−6851.80	−0.998951
$362$	1224.88	0.177840
$363$	0	0
$364$	1448.51	0.208579
$365$	−980.064	−0.140545
$366$	686.369	0.0980248
$367$	9157.19	1.30246	0.651228	−	0.758882i	$-0.274254\pi$
$367$	9157.19	1.30246	0.651228	+	0.758882i	$0.274254\pi$
$368$	1175.97	0.166581
$369$	2185.20	0.308284
$370$	1090.21	0.153182
$371$	8436.40	1.18058
$372$	−4905.46	−0.683699
$373$	4022.61	0.558399	0.279200	−	0.960233i	$-0.409931\pi$
$373$	4022.61	0.558399	0.279200	+	0.960233i	$0.409931\pi$
$374$	0	0
$375$	−5896.08	−0.811926
$376$	−8693.94	−1.19243
$377$	3858.42	0.527106
$378$	1513.49	0.205941
$379$	7595.19	1.02939	0.514695	−	0.857373i	$-0.327905\pi$
$379$	7595.19	1.02939	0.514695	+	0.857373i	$0.327905\pi$
$380$	−61.5007	−0.00830242
$381$	2615.18	0.351653
$382$	−1353.32	−0.181261
$383$	796.779	0.106302	0.0531508	−	0.998586i	$-0.483074\pi$
$383$	796.779	0.106302	0.0531508	+	0.998586i	$0.483074\pi$
$384$	7892.78	1.04890
$385$	0	0
$386$	−165.734	−0.0218540
$387$	2546.45	0.334480
$388$	−6594.19	−0.862808
$389$	−12611.6	−1.64378	−0.821892	−	0.569643i	$-0.807082\pi$
$389$	−12611.6	−1.64378	−0.821892	+	0.569643i	$0.807082\pi$
$390$	859.737	0.111627
$391$	−1357.86	−0.175627
$392$	−4521.48	−0.582575
$393$	9046.46	1.16115
$394$	6522.14	0.833961
$395$	−1169.93	−0.149027
$396$	0	0
$397$	5783.57	0.731157	0.365578	−	0.930781i	$-0.380871\pi$
$397$	5783.57	0.731157	0.365578	+	0.930781i	$0.380871\pi$
$398$	−1761.34	−0.221829
$399$	188.461	0.0236463
$400$	−1604.87	−0.200609
$401$	4826.35	0.601038	0.300519	−	0.953776i	$-0.402840\pi$
$401$	4826.35	0.601038	0.300519	+	0.953776i	$0.402840\pi$
$402$	5694.11	0.706459
$403$	−3077.75	−0.380431
$404$	−2276.75	−0.280378
$405$	−3641.71	−0.446810
$406$	2859.09	0.349493
$407$	0	0
$408$	−2214.76	−0.268742
$409$	−16074.0	−1.94329	−0.971647	−	0.236435i	$-0.924021\pi$
$409$	−16074.0	−1.94329	−0.971647	+	0.236435i	$0.924021\pi$
$410$	−1036.73	−0.124879
$411$	−6451.75	−0.774310
$412$	3593.48	0.429704
$413$	−7601.00	−0.905619
$414$	−1526.67	−0.181236
$415$	5673.16	0.671047
$416$	4250.98	0.501013
$417$	−704.802	−0.0827681
$418$	0	0
$419$	−7459.94	−0.869789	−0.434895	−	0.900481i	$-0.643214\pi$
$419$	−7459.94	−0.869789	−0.434895	+	0.900481i	$0.643214\pi$
$420$	−1610.64	−0.187122
$421$	11766.6	1.36216	0.681078	−	0.732211i	$-0.261512\pi$
$421$	11766.6	1.36216	0.681078	+	0.732211i	$0.261512\pi$
$422$	3009.00	0.347099
$423$	−5336.64	−0.613419
$424$	15644.6	1.79191
$425$	1853.10	0.211503
$426$	9033.03	1.02735
$427$	−806.761	−0.0914330
$428$	4828.28	0.545289
$429$	0	0
$430$	−1208.12	−0.135491
$431$	12685.4	1.41771	0.708856	−	0.705353i	$-0.249212\pi$
$431$	12685.4	1.41771	0.708856	+	0.705353i	$0.249212\pi$
$432$	−1327.06	−0.147797
$433$	−12744.4	−1.41445	−0.707224	−	0.706989i	$-0.750053\pi$
$433$	−12744.4	−1.41445	−0.707224	+	0.706989i	$0.750053\pi$
$434$	−2280.62	−0.252242
$435$	−4290.27	−0.472880
$436$	−8132.50	−0.893293
$437$	214.269	0.0234551
$438$	2324.88	0.253624
$439$	6537.10	0.710703	0.355351	−	0.934733i	$-0.384361\pi$
$439$	6537.10	0.710703	0.355351	+	0.934733i	$0.384361\pi$
$440$	0	0
$441$	−2775.44	−0.299691
$442$	−580.067	−0.0624230
$443$	−7378.24	−0.791311	−0.395655	−	0.918399i	$-0.629483\pi$
$443$	−7378.24	−0.791311	−0.395655	+	0.918399i	$0.629483\pi$
$444$	6538.37	0.698868
$445$	−2313.16	−0.246415
$446$	−4155.29	−0.441163
$447$	−9833.97	−1.04056
$448$	1836.59	0.193685
$449$	−5208.16	−0.547413	−0.273706	−	0.961813i	$-0.588250\pi$
$449$	−5208.16	−0.547413	−0.273706	+	0.961813i	$0.588250\pi$
$450$	2083.48	0.218258
$451$	0	0
$452$	1733.60	0.180402
$453$	5706.05	0.591818
$454$	−1706.63	−0.176424
$455$	−1010.54	−0.104120
$456$	349.486	0.0358907
$457$	−3386.48	−0.346637	−0.173318	−	0.984866i	$-0.555449\pi$
$457$	−3386.48	−0.346637	−0.173318	+	0.984866i	$0.555449\pi$
$458$	−3070.27	−0.313241
$459$	1532.32	0.155823
$460$	−1831.20	−0.185609
$461$	−16291.9	−1.64596	−0.822980	−	0.568070i	$-0.807690\pi$
$461$	−16291.9	−1.64596	−0.822980	+	0.568070i	$0.807690\pi$
$462$	0	0
$463$	9107.57	0.914179	0.457089	−	0.889421i	$-0.348892\pi$
$463$	9107.57	0.914179	0.457089	+	0.889421i	$0.348892\pi$
$464$	−2506.91	−0.250820
$465$	3422.23	0.341295
$466$	1781.55	0.177100
$467$	−11862.5	−1.17545	−0.587723	−	0.809063i	$-0.699975\pi$
$467$	−11862.5	−1.17545	−0.587723	+	0.809063i	$0.699975\pi$
$468$	1648.85	0.162859
$469$	−6692.88	−0.658952
$470$	2531.88	0.248483
$471$	−18871.8	−1.84621
$472$	−14095.4	−1.37456
$473$	0	0
$474$	2775.29	0.268931
$475$	−292.417	−0.0282463
$476$	1086.70	0.104640
$477$	9603.18	0.921802
$478$	−3760.88	−0.359872
$479$	−19704.9	−1.87962	−0.939812	−	0.341691i	$-0.889000\pi$
$479$	−19704.9	−1.87962	−0.939812	+	0.341691i	$0.889000\pi$
$480$	−4726.76	−0.449471
$481$	4102.27	0.388872
$482$	4248.68	0.401498
$483$	5611.48	0.528636
$484$	0	0
$485$	4600.36	0.430704
$486$	4974.14	0.464262
$487$	16784.5	1.56176	0.780881	−	0.624680i	$-0.214771\pi$
$487$	16784.5	1.56176	0.780881	+	0.624680i	$0.214771\pi$
$488$	−1496.07	−0.138778
$489$	−6434.39	−0.595037
$490$	1316.76	0.121399
$491$	14282.0	1.31270	0.656351	−	0.754455i	$-0.272099\pi$
$491$	14282.0	1.31270	0.656351	+	0.754455i	$0.272099\pi$
$492$	−6217.66	−0.569743
$493$	2894.66	0.264440
$494$	91.5338	0.00833664
$495$	0	0
$496$	1999.69	0.181026
$497$	−10617.5	−0.958265
$498$	−13457.7	−1.21095
$499$	−20353.7	−1.82597	−0.912983	−	0.407997i	$-0.866227\pi$
$499$	−20353.7	−1.82597	−0.912983	+	0.407997i	$0.866227\pi$
$500$	5364.81	0.479843
$501$	9085.52	0.810202
$502$	4069.91	0.361851
$503$	4074.93	0.361217	0.180609	−	0.983555i	$-0.442193\pi$
$503$	4074.93	0.361217	0.180609	+	0.983555i	$0.442193\pi$
$504$	2926.86	0.258676
$505$	1588.35	0.139961
$506$	0	0
$507$	−10606.6	−0.929104
$508$	−2379.54	−0.207825
$509$	8484.72	0.738857	0.369429	−	0.929259i	$-0.379553\pi$
$509$	8484.72	0.738857	0.369429	+	0.929259i	$0.379553\pi$
$510$	644.990	0.0560013
$511$	−2732.68	−0.236568
$512$	−5197.53	−0.448634
$513$	−241.798	−0.0208103
$514$	5071.71	0.435221
$515$	−2506.95	−0.214503
$516$	−7245.57	−0.618156
$517$	0	0
$518$	3039.78	0.257839
$519$	−22923.7	−1.93880
$520$	−1873.96	−0.158035
$521$	−8623.25	−0.725128	−0.362564	−	0.931959i	$-0.618099\pi$
$521$	−8623.25	−0.725128	−0.362564	+	0.931959i	$0.618099\pi$
$522$	3254.51	0.272885
$523$	−3442.52	−0.287822	−0.143911	−	0.989591i	$-0.545968\pi$
$523$	−3442.52	−0.287822	−0.143911	+	0.989591i	$0.545968\pi$
$524$	−8231.32	−0.686234
$525$	−7658.09	−0.636622
$526$	11444.5	0.948673
$527$	−2308.99	−0.190856
$528$	0	0
$529$	−5787.10	−0.475639
$530$	−4556.08	−0.373402
$531$	−8652.24	−0.707110
$532$	−171.480	−0.0139748
$533$	−3901.05	−0.317023
$534$	5487.23	0.444674
$535$	−3368.39	−0.272202
$536$	−12411.4	−1.00017
$537$	6886.41	0.553390
$538$	−415.986	−0.0333354
$539$	0	0
$540$	2066.47	0.164679
$541$	−16491.7	−1.31060	−0.655298	−	0.755371i	$-0.727457\pi$
$541$	−16491.7	−1.31060	−0.655298	+	0.755371i	$0.727457\pi$
$542$	−2833.23	−0.224535
$543$	−5124.88	−0.405027
$544$	3189.16	0.251349
$545$	5673.53	0.445922
$546$	2397.17	0.187893
$547$	−8902.21	−0.695852	−0.347926	−	0.937522i	$-0.613114\pi$
$547$	−8902.21	−0.695852	−0.347926	+	0.937522i	$0.613114\pi$
$548$	5870.41	0.457612
$549$	−918.338	−0.0713911
$550$	0	0
$551$	−456.773	−0.0353161
$552$	10406.0	0.802372
$553$	−3262.08	−0.250846
$554$	−9347.50	−0.716854
$555$	−4561.41	−0.348867
$556$	641.295	0.0489154
$557$	16074.9	1.22283	0.611415	−	0.791310i	$-0.290601\pi$
$557$	16074.9	1.22283	0.611415	+	0.791310i	$0.290601\pi$
$558$	−2596.03	−0.196951
$559$	−4545.98	−0.343961
$560$	656.571	0.0495449
$561$	0	0
$562$	−10528.4	−0.790239
$563$	15287.1	1.14436	0.572181	−	0.820127i	$-0.306098\pi$
$563$	15287.1	1.14436	0.572181	+	0.820127i	$0.306098\pi$
$564$	15184.6	1.13367
$565$	−1209.43	−0.0900548
$566$	−4004.53	−0.297390
$567$	−10154.1	−0.752082
$568$	−19689.2	−1.45447
$569$	3.71723	0.000273874 0	0.000136937	−	1.00000i	$-0.499956\pi$
$569$	3.71723	0.000273874 0	0.000136937		1.00000i	$0.499956\pi$
$570$	−101.779	−0.00747901
$571$	4733.15	0.346893	0.173447	−	0.984843i	$-0.444510\pi$
$571$	4733.15	0.346893	0.173447	+	0.984843i	$0.444510\pi$
$572$	0	0
$573$	5662.26	0.412817
$574$	−2890.68	−0.210200
$575$	−8706.78	−0.631474
$576$	2090.60	0.151229
$577$	−25103.3	−1.81120	−0.905602	−	0.424128i	$-0.860581\pi$
$577$	−25103.3	−1.81120	−0.905602	+	0.424128i	$0.860581\pi$
$578$	−435.176	−0.0313165
$579$	693.430	0.0497720
$580$	3903.69	0.279469
$581$	15818.3	1.12952
$582$	−10912.9	−0.777238
$583$	0	0
$584$	−5067.51	−0.359067
$585$	−1150.30	−0.0812974
$586$	9710.97	0.684567
$587$	−10840.2	−0.762220	−0.381110	−	0.924530i	$-0.624458\pi$
$587$	−10840.2	−0.762220	−0.381110	+	0.924530i	$0.624458\pi$
$588$	7897.11	0.553863
$589$	364.355	0.0254889
$590$	4104.92	0.286435
$591$	−27288.6	−1.89932
$592$	−2665.34	−0.185042
$593$	20976.0	1.45258	0.726290	−	0.687389i	$-0.241243\pi$
$593$	20976.0	1.45258	0.726290	+	0.687389i	$0.241243\pi$
$594$	0	0
$595$	−758.123	−0.0522353
$596$	8947.87	0.614965
$597$	7369.44	0.505211
$598$	2725.44	0.186374
$599$	−18749.5	−1.27894	−0.639469	−	0.768817i	$-0.720846\pi$
$599$	−18749.5	−1.27894	−0.639469	+	0.768817i	$0.720846\pi$
$600$	−14201.3	−0.966275
$601$	−20116.2	−1.36532	−0.682660	−	0.730737i	$-0.739177\pi$
$601$	−20116.2	−1.36532	−0.682660	+	0.730737i	$0.739177\pi$
$602$	−3368.57	−0.228061
$603$	−7618.52	−0.514511
$604$	−5191.90	−0.349761
$605$	0	0
$606$	−3767.83	−0.252571
$607$	26832.4	1.79422	0.897112	−	0.441803i	$-0.145661\pi$
$607$	26832.4	1.79422	0.897112	+	0.441803i	$0.145661\pi$
$608$	−503.245	−0.0335679
$609$	−11962.4	−0.795962
$610$	435.691	0.0289190
$611$	9527.06	0.630808
$612$	1237.00	0.0817036
$613$	−6053.43	−0.398851	−0.199425	−	0.979913i	$-0.563908\pi$
$613$	−6053.43	−0.398851	−0.199425	+	0.979913i	$0.563908\pi$
$614$	8399.24	0.552061
$615$	4337.67	0.284409
$616$	0	0
$617$	1053.63	0.0687483	0.0343741	−	0.999409i	$-0.489056\pi$
$617$	1053.63	0.0687483	0.0343741	+	0.999409i	$0.489056\pi$
$618$	5946.92	0.387088
$619$	−402.567	−0.0261398	−0.0130699	−	0.999915i	$-0.504160\pi$
$619$	−402.567	−0.0261398	−0.0130699	+	0.999915i	$0.504160\pi$
$620$	−3113.87	−0.201703
$621$	−7199.60	−0.465233
$622$	6258.72	0.403460
$623$	−6449.71	−0.414771
$624$	−2101.89	−0.134844
$625$	9883.06	0.632516
$626$	3572.06	0.228064
$627$	0	0
$628$	17171.3	1.09110
$629$	3077.60	0.195090
$630$	−852.372	−0.0539036
$631$	13810.4	0.871290	0.435645	−	0.900119i	$-0.356520\pi$
$631$	13810.4	0.871290	0.435645	+	0.900119i	$0.356520\pi$
$632$	−6049.25	−0.380738
$633$	−12589.6	−0.790509
$634$	1368.70	0.0857385
$635$	1660.06	0.103744
$636$	−27324.5	−1.70359
$637$	4954.76	0.308187
$638$	0	0
$639$	−12085.9	−0.748216
$640$	5010.15	0.309443
$641$	−6107.76	−0.376353	−0.188176	−	0.982135i	$-0.560258\pi$
$641$	−6107.76	−0.376353	−0.188176	+	0.982135i	$0.560258\pi$
$642$	7990.41	0.491209
$643$	18237.5	1.11853	0.559267	−	0.828988i	$-0.311083\pi$
$643$	18237.5	1.11853	0.559267	+	0.828988i	$0.311083\pi$
$644$	−5105.85	−0.312420
$645$	5054.78	0.308576
$646$	68.6703	0.00418235
$647$	−2257.88	−0.137197	−0.0685985	−	0.997644i	$-0.521853\pi$
$647$	−2257.88	−0.137197	−0.0685985	+	0.997644i	$0.521853\pi$
$648$	−18829.8	−1.14152
$649$	0	0
$650$	−3719.46	−0.224445
$651$	9542.07	0.574475
$652$	5854.62	0.351663
$653$	11855.0	0.710447	0.355224	−	0.934781i	$-0.384405\pi$
$653$	11855.0	0.710447	0.355224	+	0.934781i	$0.384405\pi$
$654$	−13458.6	−0.804700
$655$	5742.48	0.342560
$656$	2534.61	0.150853
$657$	−3110.62	−0.184713
$658$	7059.55	0.418252
$659$	23007.2	1.35999	0.679995	−	0.733217i	$-0.261982\pi$
$659$	23007.2	1.35999	0.679995	+	0.733217i	$0.261982\pi$
$660$	0	0
$661$	9957.76	0.585949	0.292974	−	0.956120i	$-0.405355\pi$
$661$	9957.76	0.585949	0.292974	+	0.956120i	$0.405355\pi$
$662$	11934.7	0.700686
$663$	2426.99	0.142167
$664$	29333.6	1.71441
$665$	119.631	0.00697607
$666$	3460.20	0.201321
$667$	−13600.5	−0.789526
$668$	−8266.86	−0.478824
$669$	17385.7	1.00474
$670$	3614.49	0.208418
$671$	0	0
$672$	−13179.5	−0.756561
$673$	16409.4	0.939875	0.469937	−	0.882700i	$-0.344277\pi$
$673$	16409.4	0.939875	0.469937	+	0.882700i	$0.344277\pi$
$674$	6557.55	0.374759
$675$	9825.43	0.560268
$676$	9650.89	0.549095
$677$	1840.32	0.104475	0.0522374	−	0.998635i	$-0.483365\pi$
$677$	1840.32	0.104475	0.0522374	+	0.998635i	$0.483365\pi$
$678$	2868.97	0.162511
$679$	12827.0	0.724971
$680$	−1405.88	−0.0792836
$681$	7140.54	0.401800
$682$	0	0
$683$	10437.7	0.584757	0.292378	−	0.956303i	$-0.405553\pi$
$683$	10437.7	0.584757	0.292378	+	0.956303i	$0.405553\pi$
$684$	−195.196	−0.0109116
$685$	−4095.42	−0.228435
$686$	9430.84	0.524885
$687$	12846.0	0.713398
$688$	2953.63	0.163672
$689$	−17143.8	−0.947933
$690$	−3030.48	−0.167200
$691$	28169.6	1.55083	0.775413	−	0.631454i	$-0.217542\pi$
$691$	28169.6	1.55083	0.775413	+	0.631454i	$0.217542\pi$
$692$	20858.1	1.14582
$693$	0	0
$694$	3937.62	0.215375
$695$	−447.391	−0.0244180
$696$	−22183.3	−1.20812
$697$	−2926.64	−0.159045
$698$	10945.4	0.593535
$699$	−7453.97	−0.403340
$700$	6968.06	0.376240
$701$	35428.2	1.90885	0.954425	−	0.298451i	$-0.0964699\pi$
$701$	35428.2	1.90885	0.954425	+	0.298451i	$0.0964699\pi$
$702$	−3075.60	−0.165358
$703$	−485.641	−0.0260545
$704$	0	0
$705$	−10593.4	−0.565914
$706$	−7130.04	−0.380088
$707$	4428.72	0.235586
$708$	24618.7	1.30682
$709$	−1867.54	−0.0989235	−0.0494618	−	0.998776i	$-0.515751\pi$
$709$	−1867.54	−0.0989235	−0.0494618	+	0.998776i	$0.515751\pi$
$710$	5733.96	0.303087
$711$	−3713.24	−0.195861
$712$	−11960.4	−0.629545
$713$	10848.7	0.569830
$714$	1798.40	0.0942626
$715$	0	0
$716$	−6265.91	−0.327050
$717$	15735.5	0.819599
$718$	−879.883	−0.0457339
$719$	4174.43	0.216523	0.108261	−	0.994122i	$-0.465472\pi$
$719$	4174.43	0.216523	0.108261	+	0.994122i	$0.465472\pi$
$720$	747.377	0.0386848
$721$	−6990.02	−0.361057
$722$	10317.5	0.531822
$723$	−17776.4	−0.914402
$724$	4663.10	0.239368
$725$	18560.9	0.950805
$726$	0	0
$727$	−13141.5	−0.670414	−0.335207	−	0.942144i	$-0.608806\pi$
$727$	−13141.5	−0.670414	−0.335207	+	0.942144i	$0.608806\pi$
$728$	−5225.08	−0.266009
$729$	3774.45	0.191762
$730$	1475.78	0.0748235
$731$	−3410.47	−0.172559
$732$	2613.00	0.131939
$733$	13987.8	0.704842	0.352421	−	0.935842i	$-0.385358\pi$
$733$	13987.8	0.704842	0.352421	+	0.935842i	$0.385358\pi$
$734$	−13788.9	−0.693403
$735$	−5509.32	−0.276482
$736$	−14984.2	−0.750443
$737$	0	0
$738$	−3290.47	−0.164124
$739$	−28875.4	−1.43735	−0.718674	−	0.695347i	$-0.755251\pi$
$739$	−28875.4	−1.43735	−0.718674	+	0.695347i	$0.755251\pi$
$740$	4150.40	0.206178
$741$	−382.976	−0.0189865
$742$	−12703.5	−0.628520
$743$	−36283.9	−1.79156	−0.895778	−	0.444501i	$-0.853381\pi$
$743$	−36283.9	−1.79156	−0.895778	+	0.444501i	$0.853381\pi$
$744$	17695.0	0.871947
$745$	−6242.37	−0.306984
$746$	−6057.25	−0.297281
$747$	18006.0	0.881934
$748$	0	0
$749$	−9391.95	−0.458177
$750$	8878.33	0.432254
$751$	−4118.63	−0.200121	−0.100060	−	0.994981i	$-0.531904\pi$
$751$	−4118.63	−0.200121	−0.100060	+	0.994981i	$0.531904\pi$
$752$	−6189.96	−0.300166
$753$	−17028.5	−0.824106
$754$	−5810.01	−0.280621
$755$	3622.06	0.174597
$756$	5761.85	0.277191
$757$	−29532.3	−1.41792	−0.708962	−	0.705246i	$-0.750836\pi$
$757$	−29532.3	−1.41792	−0.708962	+	0.705246i	$0.750836\pi$
$758$	−11436.8	−0.548028
$759$	0	0
$760$	221.845	0.0105884
$761$	−12128.1	−0.577717	−0.288858	−	0.957372i	$-0.593276\pi$
$761$	−12128.1	−0.577717	−0.288858	+	0.957372i	$0.593276\pi$
$762$	−3937.95	−0.187214
$763$	15819.3	0.750586
$764$	−5152.06	−0.243972
$765$	−862.975	−0.0407855
$766$	−1199.79	−0.0565929
$767$	15446.1	0.727155
$768$	−20186.3	−0.948450
$769$	30084.4	1.41076	0.705378	−	0.708831i	$-0.250777\pi$
$769$	30084.4	1.41076	0.705378	+	0.708831i	$0.250777\pi$
$770$	0	0
$771$	−21220.0	−0.991205
$772$	−630.948	−0.0294149
$773$	−31795.7	−1.47945	−0.739724	−	0.672911i	$-0.765044\pi$
$773$	−31795.7	−1.47945	−0.739724	+	0.672911i	$0.765044\pi$
$774$	−3834.45	−0.178071
$775$	−14805.5	−0.686231
$776$	23786.6	1.10037
$777$	−12718.4	−0.587221
$778$	18990.5	0.875120
$779$	461.820	0.0212406
$780$	3273.01	0.150247
$781$	0	0
$782$	2044.67	0.0935004
$783$	15347.9	0.700498
$784$	−3219.23	−0.146649
$785$	−11979.4	−0.544665
$786$	−13622.2	−0.618176
$787$	9408.43	0.426143	0.213071	−	0.977037i	$-0.431653\pi$
$787$	9408.43	0.426143	0.213071	+	0.977037i	$0.431653\pi$
$788$	24829.7	1.12249
$789$	−47883.5	−2.16058
$790$	1761.69	0.0793392
$791$	−3372.20	−0.151582
$792$	0	0
$793$	1639.43	0.0734149
$794$	−8708.91	−0.389254
$795$	19062.6	0.850415
$796$	−6705.41	−0.298576
$797$	14202.5	0.631213	0.315606	−	0.948890i	$-0.397792\pi$
$797$	14202.5	0.631213	0.315606	+	0.948890i	$0.397792\pi$
$798$	−283.785	−0.0125888
$799$	7147.37	0.316465
$800$	20449.3	0.903739
$801$	−7341.73	−0.323854
$802$	−7267.52	−0.319981
$803$	0	0
$804$	21677.4	0.950875
$805$	3562.03	0.155957
$806$	4634.49	0.202534
$807$	1740.48	0.0759205
$808$	8212.69	0.357576
$809$	−22964.2	−0.997994	−0.498997	−	0.866604i	$-0.666298\pi$
$809$	−22964.2	−0.997994	−0.498997	+	0.866604i	$0.666298\pi$
$810$	5483.70	0.237873
$811$	−20308.8	−0.879332	−0.439666	−	0.898161i	$-0.644903\pi$
$811$	−20308.8	−0.879332	−0.439666	+	0.898161i	$0.644903\pi$
$812$	10884.5	0.470409
$813$	11854.2	0.511372
$814$	0	0
$815$	−4084.40	−0.175546
$816$	−1576.88	−0.0676491
$817$	538.168	0.0230454
$818$	24204.2	1.03457
$819$	−3207.33	−0.136842
$820$	−3946.82	−0.168084
$821$	−21937.1	−0.932536	−0.466268	−	0.884644i	$-0.654402\pi$
$821$	−21937.1	−0.932536	−0.466268	+	0.884644i	$0.654402\pi$
$822$	9715.05	0.412228
$823$	−5914.24	−0.250495	−0.125247	−	0.992126i	$-0.539972\pi$
$823$	−5914.24	−0.250495	−0.125247	+	0.992126i	$0.539972\pi$
$824$	−12962.4	−0.548018
$825$	0	0
$826$	11445.6	0.482134
$827$	31820.1	1.33796	0.668980	−	0.743281i	$-0.266731\pi$
$827$	31820.1	1.33796	0.668980	+	0.743281i	$0.266731\pi$
$828$	−5812.01	−0.243939
$829$	11709.5	0.490577	0.245288	−	0.969450i	$-0.421117\pi$
$829$	11709.5	0.490577	0.245288	+	0.969450i	$0.421117\pi$
$830$	−8542.65	−0.357253
$831$	39109.8	1.63262
$832$	−3732.17	−0.155516
$833$	3717.15	0.154612
$834$	1061.29	0.0440641
$835$	5767.27	0.239024
$836$	0	0
$837$	−12242.6	−0.505575
$838$	11233.2	0.463059
$839$	9060.54	0.372830	0.186415	−	0.982471i	$-0.440313\pi$
$839$	9060.54	0.372830	0.186415	+	0.982471i	$0.440313\pi$
$840$	5809.89	0.238643
$841$	4604.22	0.188782
$842$	−17718.1	−0.725186
$843$	44050.8	1.79975
$844$	11455.2	0.467186
$845$	−6732.82	−0.274102
$846$	8035.91	0.326573
$847$	0	0
$848$	11138.7	0.451067
$849$	16754.9	0.677299
$850$	−2790.40	−0.112600
$851$	−14460.0	−0.582473
$852$	34388.6	1.38279
$853$	22288.1	0.894641	0.447321	−	0.894374i	$-0.352378\pi$
$853$	22288.1	0.894641	0.447321	+	0.894374i	$0.352378\pi$
$854$	1214.82	0.0486772
$855$	136.176	0.00544694
$856$	−17416.6	−0.695428
$857$	21176.9	0.844095	0.422048	−	0.906574i	$-0.361311\pi$
$857$	21176.9	0.844095	0.422048	+	0.906574i	$0.361311\pi$
$858$	0	0
$859$	46426.9	1.84408	0.922040	−	0.387094i	$-0.126521\pi$
$859$	46426.9	1.84408	0.922040	+	0.387094i	$0.126521\pi$
$860$	−4599.31	−0.182367
$861$	12094.6	0.478724
$862$	−19101.7	−0.754763
$863$	10335.0	0.407657	0.203828	−	0.979007i	$-0.434662\pi$
$863$	10335.0	0.407657	0.203828	+	0.979007i	$0.434662\pi$
$864$	16909.4	0.665822
$865$	−14551.4	−0.571980
$866$	19190.5	0.753025
$867$	1820.77	0.0713226
$868$	−8682.28	−0.339511
$869$	0	0
$870$	6460.29	0.251752
$871$	13600.7	0.529096
$872$	29335.5	1.13925
$873$	14601.0	0.566059
$874$	−322.646	−0.0124870
$875$	−10435.6	−0.403186
$876$	8850.80	0.341371
$877$	−41945.5	−1.61505	−0.807526	−	0.589832i	$-0.799194\pi$
$877$	−41945.5	−1.61505	−0.807526	+	0.589832i	$0.799194\pi$
$878$	−9843.56	−0.378365
$879$	−40630.6	−1.55908
$880$	0	0
$881$	28124.0	1.07551	0.537753	−	0.843103i	$-0.319273\pi$
$881$	28124.0	1.07551	0.537753	+	0.843103i	$0.319273\pi$
$882$	4179.26	0.159550
$883$	23637.1	0.900852	0.450426	−	0.892814i	$-0.351272\pi$
$883$	23637.1	0.900852	0.450426	+	0.892814i	$0.351272\pi$
$884$	−2208.31	−0.0840197
$885$	−17174.9	−0.652349
$886$	11110.2	0.421279
$887$	−11521.8	−0.436150	−0.218075	−	0.975932i	$-0.569978\pi$
$887$	−11521.8	−0.436150	−0.218075	+	0.975932i	$0.569978\pi$
$888$	−23585.2	−0.891293
$889$	4628.68	0.174624
$890$	3483.16	0.131186
$891$	0	0
$892$	−15819.1	−0.593793
$893$	−1127.85	−0.0422642
$894$	14808.0	0.553975
$895$	4371.33	0.163260
$896$	13969.6	0.520862
$897$	−11403.2	−0.424461
$898$	7842.45	0.291432
$899$	−23127.1	−0.857988
$900$	7931.76	0.293769
$901$	−12861.6	−0.475561
$902$	0	0
$903$	14094.0	0.519403
$904$	−6253.45	−0.230074
$905$	−3253.15	−0.119490
$906$	−8592.17	−0.315073
$907$	44016.0	1.61139	0.805693	−	0.592334i	$-0.201793\pi$
$907$	44016.0	1.61139	0.805693	+	0.592334i	$0.201793\pi$
$908$	−6497.13	−0.237461
$909$	5041.23	0.183946
$910$	1521.67	0.0554317
$911$	−23513.9	−0.855159	−0.427579	−	0.903978i	$-0.640634\pi$
$911$	−23513.9	−0.855159	−0.427579	+	0.903978i	$0.640634\pi$
$912$	248.829	0.00903459
$913$	0	0
$914$	5099.37	0.184543
$915$	−1822.93	−0.0658624
$916$	−11688.5	−0.421614
$917$	16011.5	0.576606
$918$	−2307.37	−0.0829571
$919$	37913.4	1.36088	0.680440	−	0.732804i	$-0.261789\pi$
$919$	37913.4	1.36088	0.680440	+	0.732804i	$0.261789\pi$
$920$	6605.49	0.236714
$921$	−35142.3	−1.25731
$922$	24532.3	0.876278
$923$	21575.9	0.769426
$924$	0	0
$925$	19733.9	0.701456
$926$	−13714.2	−0.486691
$927$	−7956.77	−0.281914
$928$	31943.0	1.12994
$929$	4936.27	0.174331	0.0871657	−	0.996194i	$-0.472219\pi$
$929$	4936.27	0.174331	0.0871657	+	0.996194i	$0.472219\pi$
$930$	−5153.20	−0.181699
$931$	−586.562	−0.0206485
$932$	6782.32	0.238372
$933$	−26186.4	−0.918869
$934$	17862.6	0.625785
$935$	0	0
$936$	−5947.73	−0.207700
$937$	5470.99	0.190746	0.0953732	−	0.995442i	$-0.469596\pi$
$937$	5470.99	0.190746	0.0953732	+	0.995442i	$0.469596\pi$
$938$	10078.1	0.350813
$939$	−14945.5	−0.519411
$940$	9638.84	0.334451
$941$	−6841.46	−0.237009	−0.118504	−	0.992954i	$-0.537810\pi$
$941$	−6841.46	−0.237009	−0.118504	+	0.992954i	$0.537810\pi$
$942$	28417.2	0.982889
$943$	13750.8	0.474853
$944$	−10035.7	−0.346012
$945$	−4019.69	−0.138371
$946$	0	0
$947$	41759.2	1.43294	0.716469	−	0.697619i	$-0.245757\pi$
$947$	41759.2	1.43294	0.716469	+	0.697619i	$0.245757\pi$
$948$	10565.5	0.361973
$949$	5553.12	0.189949
$950$	440.322	0.0150378
$951$	−5726.64	−0.195267
$952$	−3919.95	−0.133452
$953$	9149.56	0.311000	0.155500	−	0.987836i	$-0.450301\pi$
$953$	9149.56	0.311000	0.155500	+	0.987836i	$0.450301\pi$
$954$	−14460.5	−0.490750
$955$	3594.27	0.121788
$956$	−14317.6	−0.484378
$957$	0	0
$958$	29671.7	1.00068
$959$	−11419.1	−0.384507
$960$	4149.89	0.139518
$961$	−11343.2	−0.380759
$962$	−6177.20	−0.207028
$963$	−10690.9	−0.357746
$964$	16174.7	0.540406
$965$	440.173	0.0146836
$966$	−8449.77	−0.281436
$967$	−6328.85	−0.210467	−0.105234	−	0.994448i	$-0.533559\pi$
$967$	−6328.85	−0.210467	−0.105234	+	0.994448i	$0.533559\pi$
$968$	0	0
$969$	−287.316	−0.00952519
$970$	−6927.22	−0.229299
$971$	35693.4	1.17967	0.589834	−	0.807525i	$-0.299193\pi$
$971$	35693.4	1.17967	0.589834	+	0.807525i	$0.299193\pi$
$972$	18936.5	0.624884
$973$	−1247.44	−0.0411010
$974$	−25274.1	−0.831452
$975$	15562.2	0.511167
$976$	−1065.18	−0.0349340
$977$	48332.9	1.58271	0.791355	−	0.611357i	$-0.209376\pi$
$977$	48332.9	1.58271	0.791355	+	0.611357i	$0.209376\pi$
$978$	9688.91	0.316787
$979$	0	0
$980$	5012.90	0.163399
$981$	18007.2	0.586060
$982$	−21505.8	−0.698858
$983$	−471.913	−0.0153120	−0.00765599	−	0.999971i	$-0.502437\pi$
$983$	−471.913	−0.0153120	−0.00765599	+	0.999971i	$0.502437\pi$
$984$	22428.3	0.726615
$985$	−17322.1	−0.560334
$986$	−4358.78	−0.140783
$987$	−29537.1	−0.952559
$988$	348.468	0.0112209
$989$	16024.1	0.515203
$990$	0	0
$991$	23467.5	0.752239	0.376119	−	0.926571i	$-0.377258\pi$
$991$	23467.5	0.752239	0.376119	+	0.926571i	$0.377258\pi$
$992$	−25480.0	−0.815516
$993$	−49934.5	−1.59579
$994$	15987.8	0.510162
$995$	4677.94	0.149046
$996$	−51233.4	−1.62991
$997$	−31933.7	−1.01440	−0.507198	−	0.861830i	$-0.669319\pi$
$997$	−31933.7	−1.01440	−0.507198	+	0.861830i	$0.669319\pi$
$998$	30648.6	0.972110
$999$	16317.9	0.516792

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2057.4.a.s.1.17		44
11.2	odd	10		187.4.g.a.103.8	yes	88
11.6	odd	10		187.4.g.a.69.8	✓	88
11.10	odd	2		2057.4.a.t.1.28		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.4.g.a.69.8	✓	88	11.6	odd	10
187.4.g.a.103.8	yes	88	11.2	odd	10
2057.4.a.s.1.17		44	1.1	even	1	trivial
2057.4.a.t.1.28		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)

\(f(q)\)	\(=\)	\(q-1.50580 q^{2} +6.30025 q^{3} -5.73256 q^{4} +3.99925 q^{5} -9.48693 q^{6} +11.1510 q^{7} +20.6785 q^{8} +12.6932 q^{9} -6.02208 q^{10} -36.1166 q^{12} -22.6601 q^{13} -16.7911 q^{14} +25.1963 q^{15} +14.7228 q^{16} -17.0000 q^{17} -19.1134 q^{18} +2.68258 q^{19} -22.9260 q^{20} +70.2539 q^{21} +79.8743 q^{23} +130.280 q^{24} -109.006 q^{25} +34.1216 q^{26} -90.1366 q^{27} -63.9236 q^{28} -170.274 q^{29} -37.9406 q^{30} +135.823 q^{31} -187.598 q^{32} +25.5986 q^{34} +44.5955 q^{35} -72.7645 q^{36} -181.035 q^{37} -4.03943 q^{38} -142.764 q^{39} +82.6985 q^{40} +172.155 q^{41} -105.788 q^{42} +200.616 q^{43} +50.7632 q^{45} -120.275 q^{46} -420.433 q^{47} +92.7574 q^{48} -218.656 q^{49} +164.141 q^{50} -107.104 q^{51} +129.900 q^{52} +756.563 q^{53} +135.728 q^{54} +230.585 q^{56} +16.9009 q^{57} +256.399 q^{58} -681.645 q^{59} -144.439 q^{60} -72.3490 q^{61} -204.522 q^{62} +141.541 q^{63} +164.702 q^{64} -90.6234 q^{65} -600.206 q^{67} +97.4536 q^{68} +503.228 q^{69} -67.1519 q^{70} -952.156 q^{71} +262.476 q^{72} -245.062 q^{73} +272.603 q^{74} -686.765 q^{75} -15.3780 q^{76} +214.975 q^{78} -292.538 q^{79} +58.8802 q^{80} -910.599 q^{81} -259.231 q^{82} +1418.56 q^{83} -402.735 q^{84} -67.9873 q^{85} -302.088 q^{86} -1072.77 q^{87} -578.399 q^{89} -76.4393 q^{90} -252.682 q^{91} -457.885 q^{92} +855.718 q^{93} +633.089 q^{94} +10.7283 q^{95} -1181.91 q^{96} +1150.30 q^{97} +329.252 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - q^{2} - 28 q^{3} + 139 q^{4} - 24 q^{5} + 24 q^{6} + 2 q^{7} + 63 q^{8} + 356 q^{9} + 65 q^{10} - 337 q^{12} - 12 q^{13} - 151 q^{14} - 320 q^{15} + 311 q^{16} - 748 q^{17} + 55 q^{18} + 36 q^{19}+ \cdots - 2302 q^{98}+O(q^{100}) \) 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

Related objects

Downloads

Learn more