LMFDB - Embedded newform 1859.4.a.l.1.34

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1859,4,Mod(1,1859)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1859, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1859.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1859 = 11 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1859.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$109.684550701$
Analytic rank:	$1$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 143)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.34
Character	$\chi$	$=$	1859.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.85206 q^{2} +3.92771 q^{3} +15.5425 q^{4} +7.28280 q^{5} +19.0575 q^{6} -33.6189 q^{7} +36.5968 q^{8} -11.5731 q^{9} +35.3366 q^{10} -11.0000 q^{11} +61.0464 q^{12} -163.121 q^{14} +28.6047 q^{15} +53.2297 q^{16} +19.6860 q^{17} -56.1535 q^{18} -140.469 q^{19} +113.193 q^{20} -132.045 q^{21} -53.3727 q^{22} +49.5620 q^{23} +143.741 q^{24} -71.9609 q^{25} -151.504 q^{27} -522.522 q^{28} -51.1267 q^{29} +138.792 q^{30} -28.5019 q^{31} -34.5002 q^{32} -43.2048 q^{33} +95.5175 q^{34} -244.840 q^{35} -179.875 q^{36} -157.622 q^{37} -681.564 q^{38} +266.527 q^{40} -253.996 q^{41} -640.691 q^{42} -37.2395 q^{43} -170.968 q^{44} -84.2847 q^{45} +240.478 q^{46} +579.589 q^{47} +209.071 q^{48} +787.230 q^{49} -349.159 q^{50} +77.3206 q^{51} +412.349 q^{53} -735.107 q^{54} -80.1108 q^{55} -1230.34 q^{56} -551.720 q^{57} -248.070 q^{58} -423.228 q^{59} +444.589 q^{60} +249.679 q^{61} -138.293 q^{62} +389.076 q^{63} -593.235 q^{64} -209.632 q^{66} +635.707 q^{67} +305.969 q^{68} +194.665 q^{69} -1187.98 q^{70} +718.897 q^{71} -423.539 q^{72} -1151.31 q^{73} -764.792 q^{74} -282.641 q^{75} -2183.24 q^{76} +369.808 q^{77} -388.060 q^{79} +387.661 q^{80} -282.588 q^{81} -1232.40 q^{82} -616.271 q^{83} -2052.31 q^{84} +143.369 q^{85} -180.689 q^{86} -200.811 q^{87} -402.565 q^{88} +1052.24 q^{89} -408.955 q^{90} +770.319 q^{92} -111.947 q^{93} +2812.20 q^{94} -1023.01 q^{95} -135.507 q^{96} -1116.11 q^{97} +3819.69 q^{98} +127.304 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 4 q^{2} + 12 q^{3} + 152 q^{4} - 40 q^{5} - 98 q^{6} - 56 q^{7} - 84 q^{8} + 360 q^{9} - 56 q^{10} - 396 q^{11} + 66 q^{12} + 164 q^{14} - 120 q^{15} + 644 q^{16} + 138 q^{17} + 28 q^{18} - 498 q^{19}+ \cdots - 3960 q^{99}+O(q^{100}) $ 36 * q - 4 * q^2 + 12 * q^3 + 152 * q^4 - 40 * q^5 - 98 * q^6 - 56 * q^7 - 84 * q^8 + 360 * q^9 - 56 * q^10 - 396 * q^11 + 66 * q^12 + 164 * q^14 - 120 * q^15 + 644 * q^16 + 138 * q^17 + 28 * q^18 - 498 * q^19 - 320 * q^20 - 404 * q^21 + 44 * q^22 + 46 * q^23 - 1340 * q^24 + 818 * q^25 + 24 * q^27 - 1382 * q^28 + 262 * q^29 + 104 * q^30 - 660 * q^31 - 106 * q^32 - 132 * q^33 + 640 * q^34 + 68 * q^35 + 1960 * q^36 - 1372 * q^37 + 224 * q^38 + 108 * q^40 - 1216 * q^41 + 82 * q^42 + 436 * q^43 - 1672 * q^44 - 820 * q^45 - 2512 * q^46 + 140 * q^47 - 1944 * q^48 + 1192 * q^49 - 3484 * q^50 + 712 * q^51 + 146 * q^53 - 3078 * q^54 + 440 * q^55 - 102 * q^56 - 3656 * q^57 + 324 * q^58 - 404 * q^59 + 2944 * q^60 - 590 * q^61 + 1776 * q^62 - 4364 * q^63 + 2514 * q^64 + 1078 * q^66 - 2236 * q^67 - 1530 * q^68 + 12 * q^69 - 4852 * q^70 - 292 * q^71 - 2990 * q^72 - 1936 * q^73 + 1568 * q^74 + 4688 * q^75 - 7398 * q^76 + 616 * q^77 - 1464 * q^79 + 808 * q^80 + 3780 * q^81 - 4072 * q^82 - 4422 * q^83 + 2678 * q^84 - 3700 * q^85 + 2588 * q^86 - 1832 * q^87 + 924 * q^88 - 6856 * q^89 - 3422 * q^90 - 668 * q^92 - 840 * q^93 - 4166 * q^94 - 256 * q^95 - 18290 * q^96 - 2776 * q^97 - 4608 * q^98 - 3960 * q^99

Coefficient data

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

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

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

$2$	4.85206	1.71546	0.857732	−	0.514098i	$-0.171873\pi$
$2$	4.85206	1.71546	0.857732	+	0.514098i	$0.171873\pi$
$3$	3.92771	0.755887	0.377944	−	0.925829i	$-0.376631\pi$
$3$	3.92771	0.755887	0.377944	+	0.925829i	$0.376631\pi$
$4$	15.5425	1.94281
$5$	7.28280	0.651393	0.325697	−	0.945474i	$-0.394401\pi$
$5$	7.28280	0.651393	0.325697	+	0.945474i	$0.394401\pi$
$6$	19.0575	1.29670
$7$	−33.6189	−1.81525	−0.907625	−	0.419782i	$-0.862106\pi$
$7$	−33.6189	−1.81525	−0.907625	+	0.419782i	$0.862106\pi$
$8$	36.5968	1.61736
$9$	−11.5731	−0.428634
$10$	35.3366	1.11744
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	61.0464	1.46855
$13$	0	0
$13$	0	0
$14$	−163.121	−3.11399
$15$	28.6047	0.492380
$16$	53.2297	0.831714
$17$	19.6860	0.280856	0.140428	−	0.990091i	$-0.455152\pi$
$17$	19.6860	0.280856	0.140428	+	0.990091i	$0.455152\pi$
$18$	−56.1535	−0.735306
$19$	−140.469	−1.69609	−0.848047	−	0.529922i	$-0.822221\pi$
$19$	−140.469	−1.69609	−0.848047	+	0.529922i	$0.822221\pi$
$20$	113.193	1.26554
$21$	−132.045	−1.37212
$22$	−53.3727	−0.517232
$23$	49.5620	0.449322	0.224661	−	0.974437i	$-0.427873\pi$
$23$	49.5620	0.449322	0.224661	+	0.974437i	$0.427873\pi$
$24$	143.741	1.22255
$25$	−71.9609	−0.575687
$26$	0	0
$27$	−151.504	−1.07989
$28$	−522.522	−3.52669
$29$	−51.1267	−0.327379	−0.163690	−	0.986512i	$-0.552340\pi$
$29$	−51.1267	−0.327379	−0.163690	+	0.986512i	$0.552340\pi$
$30$	138.792	0.844660
$31$	−28.5019	−0.165132	−0.0825661	−	0.996586i	$-0.526312\pi$
$31$	−28.5019	−0.165132	−0.0825661	+	0.996586i	$0.526312\pi$
$32$	−34.5002	−0.190589
$33$	−43.2048	−0.227909
$34$	95.5175	0.481797
$35$	−244.840	−1.18244
$36$	−179.875	−0.832757
$37$	−157.622	−0.700348	−0.350174	−	0.936685i	$-0.613878\pi$
$37$	−157.622	−0.700348	−0.350174	+	0.936685i	$0.613878\pi$
$38$	−681.564	−2.90959
$39$	0	0
$40$	266.527	1.05354
$41$	−253.996	−0.967499	−0.483750	−	0.875206i	$-0.660725\pi$
$41$	−253.996	−0.967499	−0.483750	+	0.875206i	$0.660725\pi$
$42$	−640.691	−2.35383
$43$	−37.2395	−0.132069	−0.0660346	−	0.997817i	$-0.521035\pi$
$43$	−37.2395	−0.132069	−0.0660346	+	0.997817i	$0.521035\pi$
$44$	−170.968	−0.585781
$45$	−84.2847	−0.279209
$46$	240.478	0.770795
$47$	579.589	1.79876	0.899380	−	0.437168i	$-0.144018\pi$
$47$	579.589	1.79876	0.899380	+	0.437168i	$0.144018\pi$
$48$	209.071	0.628683
$49$	787.230	2.29513
$50$	−349.159	−0.987570
$51$	77.3206	0.212295
$52$	0	0
$53$	412.349	1.06869	0.534344	−	0.845267i	$-0.320559\pi$
$53$	412.349	1.06869	0.534344	+	0.845267i	$0.320559\pi$
$54$	−735.107	−1.85251
$55$	−80.1108	−0.196402
$56$	−1230.34	−2.93592
$57$	−551.720	−1.28206
$58$	−248.070	−0.561607
$59$	−423.228	−0.933892	−0.466946	−	0.884286i	$-0.654646\pi$
$59$	−423.228	−0.933892	−0.466946	+	0.884286i	$0.654646\pi$
$60$	444.589	0.956603
$61$	249.679	0.524067	0.262034	−	0.965059i	$-0.415607\pi$
$61$	249.679	0.524067	0.262034	+	0.965059i	$0.415607\pi$
$62$	−138.293	−0.283278
$63$	389.076	0.778078
$64$	−593.235	−1.15866
$65$	0	0
$66$	−209.632	−0.390969
$67$	635.707	1.15916	0.579581	−	0.814914i	$-0.303216\pi$
$67$	635.707	1.15916	0.579581	+	0.814914i	$0.303216\pi$
$68$	305.969	0.545650
$69$	194.665	0.339637
$70$	−1187.98	−2.02843
$71$	718.897	1.20165	0.600826	−	0.799379i	$-0.294838\pi$
$71$	718.897	1.20165	0.600826	+	0.799379i	$0.294838\pi$
$72$	−423.539	−0.693258
$73$	−1151.31	−1.84589	−0.922946	−	0.384929i	$-0.874226\pi$
$73$	−1151.31	−1.84589	−0.922946	+	0.384929i	$0.874226\pi$
$74$	−764.792	−1.20142
$75$	−282.641	−0.435155
$76$	−2183.24	−3.29519
$77$	369.808	0.547318
$78$	0	0
$79$	−388.060	−0.552660	−0.276330	−	0.961063i	$-0.589118\pi$
$79$	−388.060	−0.552660	−0.276330	+	0.961063i	$0.589118\pi$
$80$	387.661	0.541773
$81$	−282.588	−0.387638
$82$	−1232.40	−1.65971
$83$	−616.271	−0.814994	−0.407497	−	0.913207i	$-0.633598\pi$
$83$	−616.271	−0.814994	−0.407497	+	0.913207i	$0.633598\pi$
$84$	−2052.31	−2.66578
$85$	143.369	0.182947
$86$	−180.689	−0.226560
$87$	−200.811	−0.247462
$88$	−402.565	−0.487654
$89$	1052.24	1.25323	0.626614	−	0.779330i	$-0.284440\pi$
$89$	1052.24	1.25323	0.626614	+	0.779330i	$0.284440\pi$
$90$	−408.955	−0.478974
$91$	0	0
$92$	770.319	0.872949
$93$	−111.947	−0.124821
$94$	2812.20	3.08571
$95$	−1023.01	−1.10482
$96$	−135.507	−0.144063
$97$	−1116.11	−1.16829	−0.584144	−	0.811650i	$-0.698570\pi$
$97$	−1116.11	−1.16829	−0.584144	+	0.811650i	$0.698570\pi$
$98$	3819.69	3.93721
$99$	127.304	0.129238
$100$	−1118.45	−1.11845
$101$	1743.94	1.71810	0.859051	−	0.511889i	$-0.171054\pi$
$101$	1743.94	1.71810	0.859051	+	0.511889i	$0.171054\pi$
$102$	375.165	0.364185
$103$	1242.54	1.18865	0.594326	−	0.804224i	$-0.297419\pi$
$103$	1242.54	1.18865	0.594326	+	0.804224i	$0.297419\pi$
$104$	0	0
$105$	−961.658	−0.893792
$106$	2000.74	1.83329
$107$	−1164.54	−1.05215	−0.526075	−	0.850438i	$-0.676337\pi$
$107$	−1164.54	−1.05215	−0.526075	+	0.850438i	$0.676337\pi$
$108$	−2354.75	−2.09802
$109$	−1251.73	−1.09995	−0.549973	−	0.835182i	$-0.685362\pi$
$109$	−1251.73	−1.09995	−0.549973	+	0.835182i	$0.685362\pi$
$110$	−388.703	−0.336921
$111$	−619.093	−0.529385
$112$	−1789.52	−1.50977
$113$	−1020.38	−0.849465	−0.424732	−	0.905319i	$-0.639632\pi$
$113$	−1020.38	−0.849465	−0.424732	+	0.905319i	$0.639632\pi$
$114$	−2676.98	−2.19932
$115$	360.950	0.292685
$116$	−794.638	−0.636037
$117$	0	0
$118$	−2053.53	−1.60206
$119$	−661.820	−0.509823
$120$	1046.84	0.796358
$121$	121.000	0.0909091
$122$	1211.46	0.899018
$123$	−997.621	−0.731320
$124$	−442.992	−0.320821
$125$	−1434.43	−1.02639
$126$	1887.82	1.33476
$127$	634.113	0.443059	0.221529	−	0.975154i	$-0.428895\pi$
$127$	634.113	0.443059	0.221529	+	0.975154i	$0.428895\pi$
$128$	−2602.41	−1.79705
$129$	−146.266	−0.0998295
$130$	0	0
$131$	−923.025	−0.615611	−0.307805	−	0.951449i	$-0.599595\pi$
$131$	−923.025	−0.615611	−0.307805	+	0.951449i	$0.599595\pi$
$132$	−671.511	−0.442784
$133$	4722.41	3.07883
$134$	3084.49	1.98850
$135$	−1103.37	−0.703431
$136$	720.442	0.454246
$137$	285.350	0.177950	0.0889748	−	0.996034i	$-0.471641\pi$
$137$	285.350	0.177950	0.0889748	+	0.996034i	$0.471641\pi$
$138$	944.527	0.582634
$139$	−233.049	−0.142208	−0.0711042	−	0.997469i	$-0.522652\pi$
$139$	−233.049	−0.142208	−0.0711042	+	0.997469i	$0.522652\pi$
$140$	−3805.42	−2.29726
$141$	2276.45	1.35966
$142$	3488.13	2.06139
$143$	0	0
$144$	−616.034	−0.356501
$145$	−372.346	−0.213253
$146$	−5586.21	−3.16656
$147$	3092.01	1.73486
$148$	−2449.84	−1.36065
$149$	1544.53	0.849211	0.424605	−	0.905378i	$-0.360413\pi$
$149$	1544.53	0.849211	0.424605	+	0.905378i	$0.360413\pi$
$150$	−1371.39	−0.746492
$151$	−1010.81	−0.544759	−0.272380	−	0.962190i	$-0.587811\pi$
$151$	−1010.81	−0.544759	−0.272380	+	0.962190i	$0.587811\pi$
$152$	−5140.71	−2.74320
$153$	−227.828	−0.120384
$154$	1794.33	0.938905
$155$	−207.574	−0.107566
$156$	0	0
$157$	1076.84	0.547398	0.273699	−	0.961815i	$-0.411753\pi$
$157$	1076.84	0.547398	0.273699	+	0.961815i	$0.411753\pi$
$158$	−1882.89	−0.948069
$159$	1619.58	0.807808
$160$	−251.258	−0.124148
$161$	−1666.22	−0.815631
$162$	−1371.14	−0.664980
$163$	325.227	0.156280	0.0781402	−	0.996942i	$-0.475102\pi$
$163$	325.227	0.156280	0.0781402	+	0.996942i	$0.475102\pi$
$164$	−3947.73	−1.87967
$165$	−314.652	−0.148458
$166$	−2990.18	−1.39809
$167$	129.913	0.0601975	0.0300987	−	0.999547i	$-0.490418\pi$
$167$	129.913	0.0601975	0.0300987	+	0.999547i	$0.490418\pi$
$168$	−4832.43	−2.21922
$169$	0	0
$170$	695.635	0.313840
$171$	1625.66	0.727003
$172$	−578.796	−0.256586
$173$	−4001.50	−1.75854	−0.879272	−	0.476320i	$-0.841971\pi$
$173$	−4001.50	−1.75854	−0.879272	+	0.476320i	$0.841971\pi$
$174$	−974.347	−0.424512
$175$	2419.24	1.04502
$176$	−585.527	−0.250771
$177$	−1662.32	−0.705917
$178$	5105.54	2.14987
$179$	841.267	0.351281	0.175640	−	0.984454i	$-0.443800\pi$
$179$	841.267	0.351281	0.175640	+	0.984454i	$0.443800\pi$
$180$	−1310.00	−0.542452
$181$	−3876.21	−1.59180	−0.795902	−	0.605426i	$-0.793003\pi$
$181$	−3876.21	−1.59180	−0.795902	+	0.605426i	$0.793003\pi$
$182$	0	0
$183$	980.665	0.396136
$184$	1813.81	0.726717
$185$	−1147.93	−0.456202
$186$	−543.175	−0.214127
$187$	−216.545	−0.0846811
$188$	9008.27	3.49466
$189$	5093.39	1.96026
$190$	−4963.69	−1.89528
$191$	655.840	0.248455	0.124228	−	0.992254i	$-0.460355\pi$
$191$	655.840	0.248455	0.124228	+	0.992254i	$0.460355\pi$
$192$	−2330.05	−0.875818
$193$	−562.365	−0.209741	−0.104870	−	0.994486i	$-0.533443\pi$
$193$	−562.365	−0.209741	−0.104870	+	0.994486i	$0.533443\pi$
$194$	−5415.44	−2.00415
$195$	0	0
$196$	12235.5	4.45901
$197$	756.084	0.273446	0.136723	−	0.990609i	$-0.456343\pi$
$197$	756.084	0.273446	0.136723	+	0.990609i	$0.456343\pi$
$198$	617.689	0.221703
$199$	2485.64	0.885439	0.442719	−	0.896660i	$-0.354014\pi$
$199$	2485.64	0.885439	0.442719	+	0.896660i	$0.354014\pi$
$200$	−2633.54	−0.931095
$201$	2496.87	0.876197
$202$	8461.70	2.94734
$203$	1718.82	0.594275
$204$	1201.76	0.412450
$205$	−1849.80	−0.630222
$206$	6028.88	2.03909
$207$	−573.588	−0.192595
$208$	0	0
$209$	1545.16	0.511391
$210$	−4666.03	−1.53327
$211$	−1957.58	−0.638699	−0.319350	−	0.947637i	$-0.603464\pi$
$211$	−1957.58	−0.638699	−0.319350	+	0.947637i	$0.603464\pi$
$212$	6408.94	2.07626
$213$	2823.62	0.908314
$214$	−5650.40	−1.80492
$215$	−271.208	−0.0860290
$216$	−5544.55	−1.74657
$217$	958.204	0.299756
$218$	−6073.48	−1.88692
$219$	−4521.99	−1.39529
$220$	−1245.12	−0.381574
$221$	0	0
$222$	−3003.88	−0.908140
$223$	2632.96	0.790654	0.395327	−	0.918541i	$-0.370631\pi$
$223$	2632.96	0.790654	0.395327	+	0.918541i	$0.370631\pi$
$224$	1159.86	0.345966
$225$	832.812	0.246759
$226$	−4950.96	−1.45723
$227$	−1064.02	−0.311108	−0.155554	−	0.987827i	$-0.549716\pi$
$227$	−1064.02	−0.311108	−0.155554	+	0.987827i	$0.549716\pi$
$228$	−8575.12	−2.49080
$229$	747.625	0.215740	0.107870	−	0.994165i	$-0.465597\pi$
$229$	747.625	0.215740	0.107870	+	0.994165i	$0.465597\pi$
$230$	1751.35	0.502091
$231$	1452.50	0.413711
$232$	−1871.07	−0.529491
$233$	−2448.79	−0.688522	−0.344261	−	0.938874i	$-0.611871\pi$
$233$	−2448.79	−0.688522	−0.344261	+	0.938874i	$0.611871\pi$
$234$	0	0
$235$	4221.03	1.17170
$236$	−6578.03	−1.81438
$237$	−1524.19	−0.417749
$238$	−3211.19	−0.874583
$239$	2808.34	0.760068	0.380034	−	0.924972i	$-0.375912\pi$
$239$	2808.34	0.760068	0.380034	+	0.924972i	$0.375912\pi$
$240$	1522.62	0.409519
$241$	12.2520	0.00327479	0.00163739	−	0.999999i	$-0.499479\pi$
$241$	12.2520	0.00327479	0.00163739	+	0.999999i	$0.499479\pi$
$242$	587.100	0.155951
$243$	2980.68	0.786876
$244$	3880.64	1.01817
$245$	5733.24	1.49503
$246$	−4840.52	−1.25455
$247$	0	0
$248$	−1043.08	−0.267079
$249$	−2420.53	−0.616043
$250$	−6959.93	−1.76074
$251$	−6089.82	−1.53142	−0.765709	−	0.643188i	$-0.777612\pi$
$251$	−6089.82	−1.53142	−0.765709	+	0.643188i	$0.777612\pi$
$252$	6047.21	1.51166
$253$	−545.182	−0.135476
$254$	3076.76	0.760051
$255$	563.111	0.138288
$256$	−7881.19	−1.92412
$257$	4485.77	1.08877	0.544387	−	0.838834i	$-0.316762\pi$
$257$	4485.77	1.08877	0.544387	+	0.838834i	$0.316762\pi$
$258$	−709.692	−0.171254
$259$	5299.08	1.27131
$260$	0	0
$261$	591.696	0.140326
$262$	−4478.57	−1.05606
$263$	7665.86	1.79733	0.898664	−	0.438638i	$-0.144539\pi$
$263$	7665.86	1.79733	0.898664	+	0.438638i	$0.144539\pi$
$264$	−1581.16	−0.368611
$265$	3003.05	0.696136
$266$	22913.4	5.28162
$267$	4132.89	0.947299
$268$	9880.48	2.25204
$269$	−3980.87	−0.902296	−0.451148	−	0.892449i	$-0.648985\pi$
$269$	−3980.87	−0.902296	−0.451148	+	0.892449i	$0.648985\pi$
$270$	−5353.63	−1.20671
$271$	−564.461	−0.126526	−0.0632630	−	0.997997i	$-0.520151\pi$
$271$	−564.461	−0.126526	−0.0632630	+	0.997997i	$0.520151\pi$
$272$	1047.88	0.233592
$273$	0	0
$274$	1384.54	0.305266
$275$	791.570	0.173576
$276$	3025.59	0.659851
$277$	3367.91	0.730534	0.365267	−	0.930903i	$-0.380978\pi$
$277$	3367.91	0.730534	0.365267	+	0.930903i	$0.380978\pi$
$278$	−1130.77	−0.243953
$279$	329.857	0.0707814
$280$	−8960.34	−1.91244
$281$	−2359.54	−0.500918	−0.250459	−	0.968127i	$-0.580582\pi$
$281$	−2359.54	−0.500918	−0.250459	+	0.968127i	$0.580582\pi$
$282$	11045.5	2.33245
$283$	−5644.97	−1.18572	−0.592859	−	0.805306i	$-0.702001\pi$
$283$	−5644.97	−1.18572	−0.592859	+	0.805306i	$0.702001\pi$
$284$	11173.5	2.33459
$285$	−4018.07	−0.835122
$286$	0	0
$287$	8539.05	1.75625
$288$	399.275	0.0816928
$289$	−4525.46	−0.921120
$290$	−1806.64	−0.365827
$291$	−4383.76	−0.883094
$292$	−17894.2	−3.58623
$293$	907.243	0.180893	0.0904466	−	0.995901i	$-0.471171\pi$
$293$	907.243	0.180893	0.0904466	+	0.995901i	$0.471171\pi$
$294$	15002.6	2.97609
$295$	−3082.28	−0.608331
$296$	−5768.46	−1.13272
$297$	1666.54	0.325598
$298$	7494.13	1.45679
$299$	0	0
$300$	−4392.96	−0.845425
$301$	1251.95	0.239739
$302$	−4904.52	−0.934514
$303$	6849.68	1.29869
$304$	−7477.12	−1.41067
$305$	1818.36	0.341374
$306$	−1105.44	−0.206515
$307$	4548.47	0.845585	0.422793	−	0.906227i	$-0.361050\pi$
$307$	4548.47	0.845585	0.422793	+	0.906227i	$0.361050\pi$
$308$	5747.74	1.06334
$309$	4880.33	0.898487
$310$	−1007.16	−0.184526
$311$	5264.16	0.959817	0.479908	−	0.877319i	$-0.340670\pi$
$311$	5264.16	0.959817	0.479908	+	0.877319i	$0.340670\pi$
$312$	0	0
$313$	3910.77	0.706229	0.353114	−	0.935580i	$-0.385123\pi$
$313$	3910.77	0.706229	0.353114	+	0.935580i	$0.385123\pi$
$314$	5224.91	0.939041
$315$	2833.56	0.506835
$316$	−6031.43	−1.07372
$317$	−8552.60	−1.51534	−0.757668	−	0.652640i	$-0.773662\pi$
$317$	−8552.60	−1.51534	−0.757668	+	0.652640i	$0.773662\pi$
$318$	7858.33	1.38576
$319$	562.394	0.0987085
$320$	−4320.41	−0.754745
$321$	−4573.96	−0.795306
$322$	−8084.61	−1.39919
$323$	−2765.26	−0.476357
$324$	−4392.14	−0.753110
$325$	0	0
$326$	1578.02	0.268093
$327$	−4916.43	−0.831435
$328$	−9295.42	−1.56480
$329$	−19485.1	−3.26520
$330$	−1526.71	−0.254674
$331$	−10272.2	−1.70578	−0.852888	−	0.522094i	$-0.825151\pi$
$331$	−10272.2	−1.70578	−0.852888	+	0.522094i	$0.825151\pi$
$332$	−9578.40	−1.58338
$333$	1824.18	0.300193
$334$	630.347	0.103267
$335$	4629.72	0.755071
$336$	−7028.73	−1.14122
$337$	8953.48	1.44726	0.723631	−	0.690187i	$-0.242472\pi$
$337$	8953.48	1.44726	0.723631	+	0.690187i	$0.242472\pi$
$338$	0	0
$339$	−4007.76	−0.642100
$340$	2228.31	0.355433
$341$	313.521	0.0497893
$342$	7887.82	1.24715
$343$	−14934.5	−2.35099
$344$	−1362.85	−0.213604
$345$	1417.71	0.221237
$346$	−19415.5	−3.01672
$347$	773.649	0.119688	0.0598439	−	0.998208i	$-0.480940\pi$
$347$	773.649	0.119688	0.0598439	+	0.998208i	$0.480940\pi$
$348$	−3121.11	−0.480772
$349$	−81.0078	−0.0124248	−0.00621239	−	0.999981i	$-0.501977\pi$
$349$	−81.0078	−0.0124248	−0.00621239	+	0.999981i	$0.501977\pi$
$350$	11738.3	1.79269
$351$	0	0
$352$	379.502	0.0574646
$353$	−11417.2	−1.72146	−0.860731	−	0.509061i	$-0.829993\pi$
$353$	−11417.2	−1.72146	−0.860731	+	0.509061i	$0.829993\pi$
$354$	−8065.66	−1.21097
$355$	5235.58	0.782749
$356$	16354.5	2.43479
$357$	−2599.43	−0.385369
$358$	4081.88	0.602609
$359$	1673.33	0.246003	0.123002	−	0.992406i	$-0.460748\pi$
$359$	1673.33	0.246003	0.123002	+	0.992406i	$0.460748\pi$
$360$	−3084.55	−0.451583
$361$	12872.5	1.87673
$362$	−18807.6	−2.73068
$363$	475.252	0.0687170
$364$	0	0
$365$	−8384.73	−1.20240
$366$	4758.25	0.679557
$367$	−6442.11	−0.916281	−0.458141	−	0.888880i	$-0.651484\pi$
$367$	−6442.11	−0.916281	−0.458141	+	0.888880i	$0.651484\pi$
$368$	2638.17	0.373707
$369$	2939.52	0.414703
$370$	−5569.82	−0.782598
$371$	−13862.7	−1.93994
$372$	−1739.94	−0.242505
$373$	6908.41	0.958992	0.479496	−	0.877544i	$-0.340819\pi$
$373$	6908.41	0.958992	0.479496	+	0.877544i	$0.340819\pi$
$374$	−1050.69	−0.145267
$375$	−5634.00	−0.775837
$376$	21211.1	2.90925
$377$	0	0
$378$	24713.5	3.36276
$379$	−14614.2	−1.98068	−0.990341	−	0.138653i	$-0.955723\pi$
$379$	−14614.2	−1.98068	−0.990341	+	0.138653i	$0.955723\pi$
$380$	−15900.1	−2.14647
$381$	2490.61	0.334902
$382$	3182.18	0.426216
$383$	1984.03	0.264697	0.132349	−	0.991203i	$-0.457748\pi$
$383$	1984.03	0.264697	0.132349	+	0.991203i	$0.457748\pi$
$384$	−10221.5	−1.35837
$385$	2693.24	0.356519
$386$	−2728.63	−0.359802
$387$	430.978	0.0566094
$388$	−17347.2	−2.26977
$389$	−1403.79	−0.182969	−0.0914844	−	0.995807i	$-0.529161\pi$
$389$	−1403.79	−0.182969	−0.0914844	+	0.995807i	$0.529161\pi$
$390$	0	0
$391$	975.676	0.126194
$392$	28810.1	3.71206
$393$	−3625.37	−0.465333
$394$	3668.57	0.469086
$395$	−2826.16	−0.359999
$396$	1978.63	0.251086
$397$	−15159.0	−1.91640	−0.958198	−	0.286107i	$-0.907639\pi$
$397$	−15159.0	−1.91640	−0.958198	+	0.286107i	$0.907639\pi$
$398$	12060.5	1.51894
$399$	18548.2	2.32725
$400$	−3830.46	−0.478807
$401$	−11427.3	−1.42308	−0.711538	−	0.702648i	$-0.752001\pi$
$401$	−11427.3	−1.42308	−0.711538	+	0.702648i	$0.752001\pi$
$402$	12115.0	1.50308
$403$	0	0
$404$	27105.2	3.33796
$405$	−2058.03	−0.252505
$406$	8339.84	1.01946
$407$	1733.84	0.211163
$408$	2829.69	0.343359
$409$	14770.2	1.78567	0.892836	−	0.450381i	$-0.148712\pi$
$409$	14770.2	1.78567	0.892836	+	0.450381i	$0.148712\pi$
$410$	−8975.34	−1.08112
$411$	1120.77	0.134510
$412$	19312.2	2.30933
$413$	14228.5	1.69525
$414$	−2783.08	−0.330389
$415$	−4488.17	−0.530881
$416$	0	0
$417$	−915.348	−0.107494
$418$	7497.20	0.877273
$419$	5041.53	0.587816	0.293908	−	0.955834i	$-0.405044\pi$
$419$	5041.53	0.587816	0.293908	+	0.955834i	$0.405044\pi$
$420$	−14946.6	−1.73647
$421$	−12174.1	−1.40934	−0.704669	−	0.709537i	$-0.748904\pi$
$421$	−12174.1	−1.40934	−0.704669	+	0.709537i	$0.748904\pi$
$422$	−9498.31	−1.09567
$423$	−6707.65	−0.771010
$424$	15090.6	1.72846
$425$	−1416.62	−0.161685
$426$	13700.4	1.55818
$427$	−8393.93	−0.951313
$428$	−18099.8	−2.04413
$429$	0	0
$430$	−1315.92	−0.147580
$431$	987.140	0.110322	0.0551611	−	0.998477i	$-0.482433\pi$
$431$	987.140	0.110322	0.0551611	+	0.998477i	$0.482433\pi$
$432$	−8064.51	−0.898157
$433$	16004.0	1.77622	0.888112	−	0.459627i	$-0.152017\pi$
$433$	16004.0	1.77622	0.888112	+	0.459627i	$0.152017\pi$
$434$	4649.27	0.514221
$435$	−1462.46	−0.161195
$436$	−19455.1	−2.13699
$437$	−6961.92	−0.762091
$438$	−21941.0	−2.39356
$439$	10882.5	1.18313	0.591563	−	0.806259i	$-0.298511\pi$
$439$	10882.5	1.18313	0.591563	+	0.806259i	$0.298511\pi$
$440$	−2931.80	−0.317654
$441$	−9110.71	−0.983772
$442$	0	0
$443$	−14689.4	−1.57542	−0.787712	−	0.616043i	$-0.788735\pi$
$443$	−14689.4	−1.57542	−0.787712	+	0.616043i	$0.788735\pi$
$444$	−9622.26	−1.02850
$445$	7663.25	0.816344
$446$	12775.3	1.35634
$447$	6066.44	0.641908
$448$	19943.9	2.10326
$449$	−4909.49	−0.516021	−0.258010	−	0.966142i	$-0.583067\pi$
$449$	−4909.49	−0.516021	−0.258010	+	0.966142i	$0.583067\pi$
$450$	4040.86	0.423306
$451$	2793.95	0.291712
$452$	−15859.3	−1.65035
$453$	−3970.17	−0.411777
$454$	−5162.70	−0.533695
$455$	0	0
$456$	−20191.2	−2.07355
$457$	−9752.89	−0.998295	−0.499148	−	0.866517i	$-0.666353\pi$
$457$	−9752.89	−0.998295	−0.499148	+	0.866517i	$0.666353\pi$
$458$	3627.52	0.370094
$459$	−2982.50	−0.303292
$460$	5610.08	0.568633
$461$	−5771.01	−0.583043	−0.291521	−	0.956564i	$-0.594161\pi$
$461$	−5771.01	−0.583043	−0.291521	+	0.956564i	$0.594161\pi$
$462$	7047.61	0.709706
$463$	10103.3	1.01413	0.507063	−	0.861909i	$-0.330731\pi$
$463$	10103.3	1.01413	0.507063	+	0.861909i	$0.330731\pi$
$464$	−2721.46	−0.272286
$465$	−815.289	−0.0813078
$466$	−11881.7	−1.18113
$467$	−11122.1	−1.10207	−0.551037	−	0.834481i	$-0.685768\pi$
$467$	−11122.1	−1.10207	−0.551037	+	0.834481i	$0.685768\pi$
$468$	0	0
$469$	−21371.7	−2.10417
$470$	20480.7	2.01001
$471$	4229.53	0.413771
$472$	−15488.8	−1.51044
$473$	409.635	0.0398204
$474$	−7395.45	−0.716633
$475$	10108.3	0.976419
$476$	−10286.3	−0.990492
$477$	−4772.16	−0.458076
$478$	13626.2	1.30387
$479$	12970.4	1.23723	0.618614	−	0.785695i	$-0.287694\pi$
$479$	12970.4	1.23723	0.618614	+	0.785695i	$0.287694\pi$
$480$	−986.868	−0.0938420
$481$	0	0
$482$	59.4477	0.00561778
$483$	−6544.43	−0.616525
$484$	1880.64	0.176620
$485$	−8128.41	−0.761014
$486$	14462.5	1.34986
$487$	−8900.44	−0.828167	−0.414084	−	0.910239i	$-0.635898\pi$
$487$	−8900.44	−0.828167	−0.414084	+	0.910239i	$0.635898\pi$
$488$	9137.44	0.847608
$489$	1277.39	0.118130
$490$	27818.0	2.56467
$491$	5298.03	0.486959	0.243479	−	0.969906i	$-0.421711\pi$
$491$	5298.03	0.486959	0.243479	+	0.969906i	$0.421711\pi$
$492$	−15505.5	−1.42082
$493$	−1006.48	−0.0919463
$494$	0	0
$495$	927.132	0.0841848
$496$	−1517.15	−0.137343
$497$	−24168.5	−2.18130
$498$	−11744.6	−1.05680
$499$	16775.9	1.50499	0.752496	−	0.658597i	$-0.228850\pi$
$499$	16775.9	1.50499	0.752496	+	0.658597i	$0.228850\pi$
$500$	−22294.6	−1.99409
$501$	510.261	0.0455025
$502$	−29548.2	−2.62709
$503$	637.501	0.0565105	0.0282552	−	0.999601i	$-0.491005\pi$
$503$	637.501	0.0565105	0.0282552	+	0.999601i	$0.491005\pi$
$504$	14238.9	1.25844
$505$	12700.8	1.11916
$506$	−2645.26	−0.232403
$507$	0	0
$508$	9855.72	0.860781
$509$	−9312.20	−0.810915	−0.405458	−	0.914114i	$-0.632888\pi$
$509$	−9312.20	−0.810915	−0.405458	+	0.914114i	$0.632888\pi$
$510$	2732.25	0.237227
$511$	38705.6	3.35076
$512$	−17420.7	−1.50370
$513$	21281.6	1.83159
$514$	21765.3	1.86775
$515$	9049.17	0.774280
$516$	−2273.34	−0.193950
$517$	−6375.48	−0.542346
$518$	25711.5	2.18088
$519$	−15716.7	−1.32926
$520$	0	0
$521$	2984.82	0.250993	0.125496	−	0.992094i	$-0.459948\pi$
$521$	2984.82	0.250993	0.125496	+	0.992094i	$0.459948\pi$
$522$	2870.95	0.240724
$523$	6809.82	0.569355	0.284677	−	0.958623i	$-0.408114\pi$
$523$	6809.82	0.569355	0.284677	+	0.958623i	$0.408114\pi$
$524$	−14346.1	−1.19602
$525$	9502.08	0.789914
$526$	37195.2	3.08325
$527$	−561.088	−0.0463783
$528$	−2299.78	−0.189555
$529$	−9710.61	−0.798110
$530$	14571.0	1.19420
$531$	4898.07	0.400298
$532$	73398.1	5.98160
$533$	0	0
$534$	20053.1	1.62506
$535$	−8481.08	−0.685363
$536$	23264.8	1.87479
$537$	3304.25	0.265529
$538$	−19315.4	−1.54786
$539$	−8659.53	−0.692008
$540$	−17149.2	−1.36664
$541$	−200.402	−0.0159260	−0.00796301	−	0.999968i	$-0.502535\pi$
$541$	−200.402	−0.0159260	−0.00796301	+	0.999968i	$0.502535\pi$
$542$	−2738.80	−0.217051
$543$	−15224.6	−1.20322
$544$	−679.169	−0.0535278
$545$	−9116.10	−0.716497
$546$	0	0
$547$	−9841.40	−0.769265	−0.384632	−	0.923070i	$-0.625672\pi$
$547$	−9841.40	−0.769265	−0.384632	+	0.923070i	$0.625672\pi$
$548$	4435.06	0.345723
$549$	−2889.56	−0.224633
$550$	3840.75	0.297764
$551$	7181.71	0.555266
$552$	7124.11	0.549316
$553$	13046.1	1.00322
$554$	16341.3	1.25320
$555$	−4508.73	−0.344837
$556$	−3622.17	−0.276285
$557$	−14021.7	−1.06664	−0.533320	−	0.845914i	$-0.679056\pi$
$557$	−14021.7	−1.06664	−0.533320	+	0.845914i	$0.679056\pi$
$558$	1600.49	0.121423
$559$	0	0
$560$	−13032.7	−0.983453
$561$	−850.527	−0.0640094
$562$	−11448.6	−0.859307
$563$	2138.86	0.160110	0.0800552	−	0.996790i	$-0.474490\pi$
$563$	2138.86	0.160110	0.0800552	+	0.996790i	$0.474490\pi$
$564$	35381.8	2.64157
$565$	−7431.24	−0.553335
$566$	−27389.7	−2.03406
$567$	9500.31	0.703661
$568$	26309.3	1.94351
$569$	13039.8	0.960734	0.480367	−	0.877068i	$-0.340503\pi$
$569$	13039.8	0.960734	0.480367	+	0.877068i	$0.340503\pi$
$570$	−19495.9	−1.43262
$571$	15518.7	1.13736	0.568682	−	0.822557i	$-0.307453\pi$
$571$	15518.7	1.13736	0.568682	+	0.822557i	$0.307453\pi$
$572$	0	0
$573$	2575.95	0.187804
$574$	41432.0	3.01279
$575$	−3566.53	−0.258669
$576$	6865.58	0.496642
$577$	−45.4628	−0.00328014	−0.00164007	−	0.999999i	$-0.500522\pi$
$577$	−45.4628	−0.00328014	−0.00164007	+	0.999999i	$0.500522\pi$
$578$	−21957.8	−1.58015
$579$	−2208.81	−0.158540
$580$	−5787.19	−0.414310
$581$	20718.3	1.47942
$582$	−21270.3	−1.51492
$583$	−4535.84	−0.322222
$584$	−42134.1	−2.98548
$585$	0	0
$586$	4402.00	0.310316
$587$	−27397.3	−1.92642	−0.963208	−	0.268756i	$-0.913388\pi$
$587$	−27397.3	−1.92642	−0.963208	+	0.268756i	$0.913388\pi$
$588$	48057.6	3.37051
$589$	4003.64	0.280080
$590$	−14955.4	−1.04357
$591$	2969.68	0.206694
$592$	−8390.17	−0.582490
$593$	−11744.6	−0.813310	−0.406655	−	0.913582i	$-0.633305\pi$
$593$	−11744.6	−0.813310	−0.406655	+	0.913582i	$0.633305\pi$
$594$	8086.17	0.558552
$595$	−4819.90	−0.332095
$596$	24005.8	1.64986
$597$	9762.86	0.669292
$598$	0	0
$599$	27018.7	1.84299	0.921497	−	0.388386i	$-0.126967\pi$
$599$	27018.7	1.84299	0.921497	+	0.388386i	$0.126967\pi$
$600$	−10343.8	−0.703803
$601$	−17817.6	−1.20931	−0.604656	−	0.796487i	$-0.706689\pi$
$601$	−17817.6	−1.20931	−0.604656	+	0.796487i	$0.706689\pi$
$602$	6074.55	0.411263
$603$	−7357.11	−0.496857
$604$	−15710.5	−1.05837
$605$	881.218	0.0592176
$606$	33235.1	2.22786
$607$	1888.35	0.126270	0.0631349	−	0.998005i	$-0.479890\pi$
$607$	1888.35	0.126270	0.0631349	+	0.998005i	$0.479890\pi$
$608$	4846.20	0.323256
$609$	6751.04	0.449205
$610$	8822.80	0.585614
$611$	0	0
$612$	−3541.02	−0.233884
$613$	−23090.6	−1.52140	−0.760700	−	0.649103i	$-0.775144\pi$
$613$	−23090.6	−1.52140	−0.760700	+	0.649103i	$0.775144\pi$
$614$	22069.4	1.45057
$615$	−7265.47	−0.476377
$616$	13533.8	0.885213
$617$	6821.01	0.445063	0.222531	−	0.974926i	$-0.428568\pi$
$617$	6821.01	0.445063	0.222531	+	0.974926i	$0.428568\pi$
$618$	23679.7	1.54132
$619$	−6426.72	−0.417305	−0.208652	−	0.977990i	$-0.566908\pi$
$619$	−6426.72	−0.417305	−0.208652	+	0.977990i	$0.566908\pi$
$620$	−3226.22	−0.208981
$621$	−7508.84	−0.485216
$622$	25542.0	1.64653
$623$	−35375.2	−2.27492
$624$	0	0
$625$	−1451.53	−0.0928976
$626$	18975.3	1.21151
$627$	6068.92	0.386554
$628$	16736.9	1.06349
$629$	−3102.94	−0.196697
$630$	13748.6	0.869457
$631$	24596.8	1.55180	0.775899	−	0.630858i	$-0.217297\pi$
$631$	24596.8	1.55180	0.775899	+	0.630858i	$0.217297\pi$
$632$	−14201.7	−0.893853
$633$	−7688.81	−0.482785
$634$	−41497.7	−2.59950
$635$	4618.12	0.288605
$636$	25172.4	1.56942
$637$	0	0
$638$	2728.77	0.169331
$639$	−8319.88	−0.515070
$640$	−18952.8	−1.17059
$641$	10681.7	0.658195	0.329098	−	0.944296i	$-0.393255\pi$
$641$	10681.7	0.658195	0.329098	+	0.944296i	$0.393255\pi$
$642$	−22193.1	−1.36432
$643$	16532.1	1.01394	0.506971	−	0.861963i	$-0.330765\pi$
$643$	16532.1	1.01394	0.506971	+	0.861963i	$0.330765\pi$
$644$	−25897.3	−1.58462
$645$	−1065.23	−0.0650282
$646$	−13417.2	−0.817173
$647$	16684.5	1.01381	0.506905	−	0.862002i	$-0.330789\pi$
$647$	16684.5	1.01381	0.506905	+	0.862002i	$0.330789\pi$
$648$	−10341.8	−0.626953
$649$	4655.51	0.281579
$650$	0	0
$651$	3763.54	0.226582
$652$	5054.84	0.303624
$653$	−14318.0	−0.858051	−0.429026	−	0.903292i	$-0.641143\pi$
$653$	−14318.0	−0.858051	−0.429026	+	0.903292i	$0.641143\pi$
$654$	−23854.8	−1.42630
$655$	−6722.20	−0.401005
$656$	−13520.1	−0.804683
$657$	13324.2	0.791213
$658$	−94543.1	−5.60133
$659$	2611.89	0.154392	0.0771962	−	0.997016i	$-0.475403\pi$
$659$	2611.89	0.154392	0.0771962	+	0.997016i	$0.475403\pi$
$660$	−4890.48	−0.288427
$661$	22924.1	1.34893	0.674466	−	0.738306i	$-0.264374\pi$
$661$	22924.1	1.34893	0.674466	+	0.738306i	$0.264374\pi$
$662$	−49841.4	−2.92620
$663$	0	0
$664$	−22553.5	−1.31814
$665$	34392.3	2.00553
$666$	8851.03	0.514971
$667$	−2533.94	−0.147099
$668$	2019.18	0.116953
$669$	10341.5	0.597645
$670$	22463.7	1.29530
$671$	−2746.47	−0.158012
$672$	4555.58	0.261511
$673$	13171.3	0.754408	0.377204	−	0.926130i	$-0.376886\pi$
$673$	13171.3	0.754408	0.377204	+	0.926130i	$0.376886\pi$
$674$	43442.9	2.48273
$675$	10902.4	0.621677
$676$	0	0
$677$	−14414.4	−0.818303	−0.409151	−	0.912467i	$-0.634175\pi$
$677$	−14414.4	−0.818303	−0.409151	+	0.912467i	$0.634175\pi$
$678$	−19445.9	−1.10150
$679$	37522.4	2.12073
$680$	5246.84	0.295893
$681$	−4179.16	−0.235163
$682$	1521.23	0.0854117
$683$	3456.95	0.193670	0.0968350	−	0.995300i	$-0.469128\pi$
$683$	3456.95	0.193670	0.0968350	+	0.995300i	$0.469128\pi$
$684$	25266.9	1.41243
$685$	2078.15	0.115915
$686$	−72463.2	−4.03303
$687$	2936.45	0.163075
$688$	−1982.25	−0.109844
$689$	0	0
$690$	6878.80	0.379524
$691$	16264.8	0.895430	0.447715	−	0.894176i	$-0.352238\pi$
$691$	16264.8	0.895430	0.447715	+	0.894176i	$0.352238\pi$
$692$	−62193.3	−3.41653
$693$	−4279.83	−0.234599
$694$	3753.79	0.205320
$695$	−1697.25	−0.0926336
$696$	−7349.03	−0.400236
$697$	−5000.15	−0.271728
$698$	−393.055	−0.0213142
$699$	−9618.13	−0.520445
$700$	37601.2	2.03027
$701$	−9771.57	−0.526487	−0.263243	−	0.964729i	$-0.584792\pi$
$701$	−9771.57	−0.526487	−0.263243	+	0.964729i	$0.584792\pi$
$702$	0	0
$703$	22141.0	1.18786
$704$	6525.58	0.349350
$705$	16579.0	0.885673
$706$	−55396.9	−2.95310
$707$	−58629.3	−3.11879
$708$	−25836.6	−1.37147
$709$	−5957.18	−0.315552	−0.157776	−	0.987475i	$-0.550432\pi$
$709$	−5957.18	−0.315552	−0.157776	+	0.987475i	$0.550432\pi$
$710$	25403.4	1.34278
$711$	4491.07	0.236889
$712$	38508.6	2.02693
$713$	−1412.61	−0.0741975
$714$	−12612.6	−0.661086
$715$	0	0
$716$	13075.4	0.682473
$717$	11030.3	0.574526
$718$	8119.12	0.422009
$719$	26368.5	1.36770	0.683852	−	0.729621i	$-0.260303\pi$
$719$	26368.5	1.36770	0.683852	+	0.729621i	$0.260303\pi$
$720$	−4486.45	−0.232223
$721$	−41772.8	−2.15770
$722$	62458.2	3.21946
$723$	48.1224	0.00247537
$724$	−60246.1	−3.09258
$725$	3679.12	0.188468
$726$	2305.95	0.117882
$727$	−34152.2	−1.74228	−0.871138	−	0.491038i	$-0.836618\pi$
$727$	−34152.2	−1.74228	−0.871138	+	0.491038i	$0.836618\pi$
$728$	0	0
$729$	19337.1	0.982428
$730$	−40683.2	−2.06268
$731$	−733.096	−0.0370924
$732$	15242.0	0.769619
$733$	10731.5	0.540758	0.270379	−	0.962754i	$-0.412851\pi$
$733$	10731.5	0.540758	0.270379	+	0.962754i	$0.412851\pi$
$734$	−31257.5	−1.57185
$735$	22518.5	1.13008
$736$	−1709.90	−0.0856355
$737$	−6992.77	−0.349501
$738$	14262.8	0.711408
$739$	−7371.07	−0.366914	−0.183457	−	0.983028i	$-0.558729\pi$
$739$	−7371.07	−0.366914	−0.183457	+	0.983028i	$0.558729\pi$
$740$	−17841.7	−0.886316
$741$	0	0
$742$	−67262.7	−3.32789
$743$	−27347.1	−1.35029	−0.675146	−	0.737684i	$-0.735919\pi$
$743$	−27347.1	−1.35029	−0.675146	+	0.737684i	$0.735919\pi$
$744$	−4096.91	−0.201882
$745$	11248.5	0.553170
$746$	33520.1	1.64512
$747$	7132.17	0.349334
$748$	−3365.66	−0.164520
$749$	39150.4	1.90991
$750$	−27336.5	−1.33092
$751$	−8946.91	−0.434724	−0.217362	−	0.976091i	$-0.569745\pi$
$751$	−8946.91	−0.434724	−0.217362	+	0.976091i	$0.569745\pi$
$752$	30851.4	1.49605
$753$	−23919.0	−1.15758
$754$	0	0
$755$	−7361.53	−0.354852
$756$	79164.2	3.80843
$757$	−12527.0	−0.601455	−0.300728	−	0.953710i	$-0.597230\pi$
$757$	−12527.0	−0.601455	−0.300728	+	0.953710i	$0.597230\pi$
$758$	−70908.8	−3.39779
$759$	−2141.32	−0.102404
$760$	−37438.7	−1.78690
$761$	−35342.0	−1.68351	−0.841753	−	0.539863i	$-0.818476\pi$
$761$	−35342.0	−1.68351	−0.841753	+	0.539863i	$0.818476\pi$
$762$	12084.6	0.574513
$763$	42081.8	1.99668
$764$	10193.4	0.482702
$765$	−1659.22	−0.0784175
$766$	9626.62	0.454078
$767$	0	0
$768$	−30955.0	−1.45442
$769$	10655.2	0.499657	0.249829	−	0.968290i	$-0.419626\pi$
$769$	10655.2	0.499657	0.249829	+	0.968290i	$0.419626\pi$
$770$	13067.7	0.611596
$771$	17618.8	0.822990
$772$	−8740.57	−0.407487
$773$	6327.89	0.294435	0.147217	−	0.989104i	$-0.452968\pi$
$773$	6327.89	0.294435	0.147217	+	0.989104i	$0.452968\pi$
$774$	2091.13	0.0971113
$775$	2051.02	0.0950645
$776$	−40846.1	−1.88955
$777$	20813.2	0.960965
$778$	−6811.27	−0.313876
$779$	35678.5	1.64097
$780$	0	0
$781$	−7907.87	−0.362312
$782$	4734.04	0.216482
$783$	7745.90	0.353532
$784$	41904.0	1.90889
$785$	7842.44	0.356571
$786$	−17590.5	−0.798261
$787$	17119.9	0.775423	0.387712	−	0.921781i	$-0.373266\pi$
$787$	17119.9	0.775423	0.387712	+	0.921781i	$0.373266\pi$
$788$	11751.5	0.531254
$789$	30109.2	1.35858
$790$	−13712.7	−0.617565
$791$	34304.1	1.54199
$792$	4658.93	0.209025
$793$	0	0
$794$	−73552.5	−3.28751
$795$	11795.1	0.526200
$796$	38633.1	1.72024
$797$	−24952.3	−1.10898	−0.554490	−	0.832191i	$-0.687086\pi$
$797$	−24952.3	−1.10898	−0.554490	+	0.832191i	$0.687086\pi$
$798$	89997.2	3.99231
$799$	11409.8	0.505192
$800$	2482.66	0.109719
$801$	−12177.7	−0.537176
$802$	−55446.1	−2.44123
$803$	12664.4	0.556558
$804$	38807.6	1.70229
$805$	−12134.7	−0.531296
$806$	0	0
$807$	−15635.7	−0.682035
$808$	63822.5	2.77880
$809$	24696.9	1.07329	0.536647	−	0.843807i	$-0.319691\pi$
$809$	24696.9	1.07329	0.536647	+	0.843807i	$0.319691\pi$
$810$	−9985.71	−0.433163
$811$	30317.8	1.31270	0.656352	−	0.754455i	$-0.272099\pi$
$811$	30317.8	1.31270	0.656352	+	0.754455i	$0.272099\pi$
$812$	26714.9	1.15457
$813$	−2217.04	−0.0956395
$814$	8412.71	0.362242
$815$	2368.56	0.101800
$816$	4115.76	0.176569
$817$	5231.00	0.224002
$818$	71666.0	3.06326
$819$	0	0
$820$	−28750.5	−1.22441
$821$	−21601.5	−0.918269	−0.459135	−	0.888367i	$-0.651840\pi$
$821$	−21601.5	−0.918269	−0.459135	+	0.888367i	$0.651840\pi$
$822$	5438.05	0.230747
$823$	8064.84	0.341583	0.170791	−	0.985307i	$-0.445368\pi$
$823$	8064.84	0.341583	0.170791	+	0.985307i	$0.445368\pi$
$824$	45473.0	1.92248
$825$	3109.05	0.131204
$826$	69037.4	2.90813
$827$	−28844.4	−1.21284	−0.606420	−	0.795145i	$-0.707395\pi$
$827$	−28844.4	−1.21284	−0.606420	+	0.795145i	$0.707395\pi$
$828$	−8914.99	−0.374176
$829$	−37091.0	−1.55395	−0.776974	−	0.629532i	$-0.783247\pi$
$829$	−37091.0	−1.55395	−0.776974	+	0.629532i	$0.783247\pi$
$830$	−21776.9	−0.910708
$831$	13228.2	0.552202
$832$	0	0
$833$	15497.4	0.644600
$834$	−4441.33	−0.184401
$835$	946.131	0.0392122
$836$	24015.6	0.993539
$837$	4318.16	0.178324
$838$	24461.8	1.00838
$839$	2748.30	0.113089	0.0565446	−	0.998400i	$-0.481992\pi$
$839$	2748.30	0.113089	0.0565446	+	0.998400i	$0.481992\pi$
$840$	−35193.6	−1.44559
$841$	−21775.1	−0.892823
$842$	−59069.7	−2.41767
$843$	−9267.56	−0.378638
$844$	−30425.8	−1.24087
$845$	0	0
$846$	−32546.0	−1.32264
$847$	−4067.89	−0.165023
$848$	21949.2	0.888843
$849$	−22171.8	−0.896270
$850$	−6873.52	−0.277365
$851$	−7812.06	−0.314682
$852$	43886.1	1.76469
$853$	27636.7	1.10933	0.554667	−	0.832072i	$-0.312846\pi$
$853$	27636.7	1.10933	0.554667	+	0.832072i	$0.312846\pi$
$854$	−40727.9	−1.63194
$855$	11839.4	0.473565
$856$	−42618.3	−1.70171
$857$	8533.01	0.340119	0.170060	−	0.985434i	$-0.445604\pi$
$857$	8533.01	0.340119	0.170060	+	0.985434i	$0.445604\pi$
$858$	0	0
$859$	−785.062	−0.0311828	−0.0155914	−	0.999878i	$-0.504963\pi$
$859$	−785.062	−0.0311828	−0.0155914	+	0.999878i	$0.504963\pi$
$860$	−4215.26	−0.167138
$861$	33538.9	1.32753
$862$	4789.67	0.189254
$863$	7926.87	0.312670	0.156335	−	0.987704i	$-0.450032\pi$
$863$	7926.87	0.312670	0.156335	+	0.987704i	$0.450032\pi$
$864$	5226.92	0.205814
$865$	−29142.1	−1.14550
$866$	77652.6	3.04705
$867$	−17774.7	−0.696263
$868$	14892.9	0.582371
$869$	4268.66	0.166633
$870$	−7095.97	−0.276524
$871$	0	0
$872$	−45809.3	−1.77901
$873$	12916.9	0.500768
$874$	−33779.7	−1.30734
$875$	48223.8	1.86316
$876$	−70283.1	−2.71078
$877$	34503.3	1.32850	0.664249	−	0.747512i	$-0.268752\pi$
$877$	34503.3	1.32850	0.664249	+	0.747512i	$0.268752\pi$
$878$	52802.5	2.02961
$879$	3563.38	0.136735
$880$	−4264.27	−0.163351
$881$	17746.0	0.678637	0.339319	−	0.940671i	$-0.389804\pi$
$881$	17746.0	0.678637	0.339319	+	0.940671i	$0.389804\pi$
$882$	−44205.7	−1.68762
$883$	31839.7	1.21347	0.606733	−	0.794906i	$-0.292480\pi$
$883$	31839.7	1.21347	0.606733	+	0.794906i	$0.292480\pi$
$884$	0	0
$885$	−12106.3	−0.459829
$886$	−71273.8	−2.70258
$887$	−28582.5	−1.08197	−0.540985	−	0.841032i	$-0.681949\pi$
$887$	−28582.5	−1.08197	−0.540985	+	0.841032i	$0.681949\pi$
$888$	−22656.8	−0.856208
$889$	−21318.2	−0.804262
$890$	37182.6	1.40041
$891$	3108.47	0.116877
$892$	40922.8	1.53609
$893$	−81414.2	−3.05086
$894$	29434.8	1.10117
$895$	6126.78	0.228822
$896$	87490.2	3.26210
$897$	0	0
$898$	−23821.2	−0.885215
$899$	1457.21	0.0540609
$900$	12944.0	0.479407
$901$	8117.48	0.300147
$902$	13556.4	0.500421
$903$	4917.30	0.181215
$904$	−37342.7	−1.37389
$905$	−28229.7	−1.03689
$906$	−19263.5	−0.706388
$907$	1181.49	0.0432532	0.0216266	−	0.999766i	$-0.493116\pi$
$907$	1181.49	0.0432532	0.0216266	+	0.999766i	$0.493116\pi$
$908$	−16537.6	−0.604426
$909$	−20182.8	−0.736438
$910$	0	0
$911$	5094.51	0.185279	0.0926393	−	0.995700i	$-0.470470\pi$
$911$	5094.51	0.185279	0.0926393	+	0.995700i	$0.470470\pi$
$912$	−29367.9	−1.06630
$913$	6778.98	0.245730
$914$	−47321.6	−1.71254
$915$	7141.99	0.258040
$916$	11620.0	0.419143
$917$	31031.1	1.11749
$918$	−14471.3	−0.520287
$919$	−16185.3	−0.580960	−0.290480	−	0.956881i	$-0.593815\pi$
$919$	−16185.3	−0.580960	−0.290480	+	0.956881i	$0.593815\pi$
$920$	13209.6	0.473378
$921$	17865.0	0.639167
$922$	−28001.3	−1.00019
$923$	0	0
$924$	22575.5	0.803764
$925$	11342.6	0.403181
$926$	49021.8	1.73969
$927$	−14380.1	−0.509497
$928$	1763.88	0.0623947
$929$	−25919.4	−0.915378	−0.457689	−	0.889112i	$-0.651323\pi$
$929$	−25919.4	−0.915378	−0.457689	+	0.889112i	$0.651323\pi$
$930$	−3955.84	−0.139481
$931$	−110581.	−3.89276
$932$	−38060.4	−1.33767
$933$	20676.1	0.725513
$934$	−53965.0	−1.89057
$935$	−1577.06	−0.0551607
$936$	0	0
$937$	2913.18	0.101568	0.0507842	−	0.998710i	$-0.483828\pi$
$937$	2913.18	0.101568	0.0507842	+	0.998710i	$0.483828\pi$
$938$	−103697.	−3.60963
$939$	15360.3	0.533829
$940$	65605.4	2.27640
$941$	−36762.9	−1.27358	−0.636788	−	0.771039i	$-0.719737\pi$
$941$	−36762.9	−1.27358	−0.636788	+	0.771039i	$0.719737\pi$
$942$	20521.9	0.709810
$943$	−12588.5	−0.434718
$944$	−22528.3	−0.776731
$945$	37094.2	1.27690
$946$	1987.57	0.0683104
$947$	−9413.68	−0.323024	−0.161512	−	0.986871i	$-0.551637\pi$
$947$	−9413.68	−0.323024	−0.161512	+	0.986871i	$0.551637\pi$
$948$	−23689.7	−0.811609
$949$	0	0
$950$	49045.9	1.67501
$951$	−33592.1	−1.14542
$952$	−24220.5	−0.824570
$953$	−18094.1	−0.615031	−0.307515	−	0.951543i	$-0.599498\pi$
$953$	−18094.1	−0.615031	−0.307515	+	0.951543i	$0.599498\pi$
$954$	−23154.8	−0.785813
$955$	4776.35	0.161842
$956$	43648.7	1.47667
$957$	2208.92	0.0746125
$958$	62933.2	2.12242
$959$	−9593.15	−0.323023
$960$	−16969.3	−0.570502
$961$	−28978.6	−0.972731
$962$	0	0
$963$	13477.3	0.450987
$964$	190.428	0.00636231
$965$	−4095.59	−0.136624
$966$	−31754.0	−1.05763
$967$	−28961.4	−0.963119	−0.481559	−	0.876413i	$-0.659929\pi$
$967$	−28961.4	−0.963119	−0.481559	+	0.876413i	$0.659929\pi$
$968$	4428.21	0.147033
$969$	−10861.1	−0.360072
$970$	−39439.6	−1.30549
$971$	−19721.3	−0.651790	−0.325895	−	0.945406i	$-0.605666\pi$
$971$	−19721.3	−0.651790	−0.325895	+	0.945406i	$0.605666\pi$
$972$	46327.3	1.52875
$973$	7834.85	0.258144
$974$	−43185.5	−1.42069
$975$	0	0
$976$	13290.3	0.435874
$977$	30135.2	0.986807	0.493403	−	0.869801i	$-0.335753\pi$
$977$	30135.2	0.986807	0.493403	+	0.869801i	$0.335753\pi$
$978$	6198.00	0.202648
$979$	−11574.6	−0.377862
$980$	89108.9	2.90457
$981$	14486.4	0.471474
$982$	25706.4	0.835360
$983$	25989.9	0.843285	0.421642	−	0.906762i	$-0.361454\pi$
$983$	25989.9	0.843285	0.421642	+	0.906762i	$0.361454\pi$
$984$	−36509.7	−1.18281
$985$	5506.41	0.178121
$986$	−4883.50	−0.157730
$987$	−76531.9	−2.46812
$988$	0	0
$989$	−1845.67	−0.0593416
$990$	4498.50	0.144416
$991$	−17046.1	−0.546404	−0.273202	−	0.961957i	$-0.588083\pi$
$991$	−17046.1	−0.546404	−0.273202	+	0.961957i	$0.588083\pi$
$992$	983.323	0.0314723
$993$	−40346.2	−1.28937
$994$	−117267.	−3.74194
$995$	18102.4	0.576769
$996$	−37621.1	−1.19686
$997$	−38846.7	−1.23399	−0.616995	−	0.786967i	$-0.711650\pi$
$997$	−38846.7	−1.23399	−0.616995	+	0.786967i	$0.711650\pi$
$998$	81397.5	2.58176
$999$	23880.3	0.756297

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1859.4.a.l.1.34		36
13.2	odd	12		143.4.j.a.56.34	yes	72
13.7	odd	12		143.4.j.a.23.34	✓	72
13.12	even	2		1859.4.a.m.1.3		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.4.j.a.23.34	✓	72	13.7	odd	12
143.4.j.a.56.34	yes	72	13.2	odd	12
1859.4.a.l.1.34		36	1.1	even	1	trivial
1859.4.a.m.1.3		36	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1859 = 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1859.a (trivial)

\(f(q)\)	\(=\)	\(q+4.85206 q^{2} +3.92771 q^{3} +15.5425 q^{4} +7.28280 q^{5} +19.0575 q^{6} -33.6189 q^{7} +36.5968 q^{8} -11.5731 q^{9} +35.3366 q^{10} -11.0000 q^{11} +61.0464 q^{12} -163.121 q^{14} +28.6047 q^{15} +53.2297 q^{16} +19.6860 q^{17} -56.1535 q^{18} -140.469 q^{19} +113.193 q^{20} -132.045 q^{21} -53.3727 q^{22} +49.5620 q^{23} +143.741 q^{24} -71.9609 q^{25} -151.504 q^{27} -522.522 q^{28} -51.1267 q^{29} +138.792 q^{30} -28.5019 q^{31} -34.5002 q^{32} -43.2048 q^{33} +95.5175 q^{34} -244.840 q^{35} -179.875 q^{36} -157.622 q^{37} -681.564 q^{38} +266.527 q^{40} -253.996 q^{41} -640.691 q^{42} -37.2395 q^{43} -170.968 q^{44} -84.2847 q^{45} +240.478 q^{46} +579.589 q^{47} +209.071 q^{48} +787.230 q^{49} -349.159 q^{50} +77.3206 q^{51} +412.349 q^{53} -735.107 q^{54} -80.1108 q^{55} -1230.34 q^{56} -551.720 q^{57} -248.070 q^{58} -423.228 q^{59} +444.589 q^{60} +249.679 q^{61} -138.293 q^{62} +389.076 q^{63} -593.235 q^{64} -209.632 q^{66} +635.707 q^{67} +305.969 q^{68} +194.665 q^{69} -1187.98 q^{70} +718.897 q^{71} -423.539 q^{72} -1151.31 q^{73} -764.792 q^{74} -282.641 q^{75} -2183.24 q^{76} +369.808 q^{77} -388.060 q^{79} +387.661 q^{80} -282.588 q^{81} -1232.40 q^{82} -616.271 q^{83} -2052.31 q^{84} +143.369 q^{85} -180.689 q^{86} -200.811 q^{87} -402.565 q^{88} +1052.24 q^{89} -408.955 q^{90} +770.319 q^{92} -111.947 q^{93} +2812.20 q^{94} -1023.01 q^{95} -135.507 q^{96} -1116.11 q^{97} +3819.69 q^{98} +127.304 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{2} + 12 q^{3} + 152 q^{4} - 40 q^{5} - 98 q^{6} - 56 q^{7} - 84 q^{8} + 360 q^{9} - 56 q^{10} - 396 q^{11} + 66 q^{12} + 164 q^{14} - 120 q^{15} + 644 q^{16} + 138 q^{17} + 28 q^{18} - 498 q^{19}+ \cdots - 3960 q^{99}+O(q^{100}) \) 36 * q - 4 * q^2 + 12 * q^3 + 152 * q^4 - 40 * q^5 - 98 * q^6 - 56 * q^7 - 84 * q^8 + 360 * q^9 - 56 * q^10 - 396 * q^11 + 66 * q^12 + 164 * q^14 - 120 * q^15 + 644 * q^16 + 138 * q^17 + 28 * q^18 - 498 * q^19 - 320 * q^20 - 404 * q^21 + 44 * q^22 + 46 * q^23 - 1340 * q^24 + 818 * q^25 + 24 * q^27 - 1382 * q^28 + 262 * q^29 + 104 * q^30 - 660 * q^31 - 106 * q^32 - 132 * q^33 + 640 * q^34 + 68 * q^35 + 1960 * q^36 - 1372 * q^37 + 224 * q^38 + 108 * q^40 - 1216 * q^41 + 82 * q^42 + 436 * q^43 - 1672 * q^44 - 820 * q^45 - 2512 * q^46 + 140 * q^47 - 1944 * q^48 + 1192 * q^49 - 3484 * q^50 + 712 * q^51 + 146 * q^53 - 3078 * q^54 + 440 * q^55 - 102 * q^56 - 3656 * q^57 + 324 * q^58 - 404 * q^59 + 2944 * q^60 - 590 * q^61 + 1776 * q^62 - 4364 * q^63 + 2514 * q^64 + 1078 * q^66 - 2236 * q^67 - 1530 * q^68 + 12 * q^69 - 4852 * q^70 - 292 * q^71 - 2990 * q^72 - 1936 * q^73 + 1568 * q^74 + 4688 * q^75 - 7398 * q^76 + 616 * q^77 - 1464 * q^79 + 808 * q^80 + 3780 * q^81 - 4072 * q^82 - 4422 * q^83 + 2678 * q^84 - 3700 * q^85 + 2588 * q^86 - 1832 * q^87 + 924 * q^88 - 6856 * q^89 - 3422 * q^90 - 668 * q^92 - 840 * q^93 - 4166 * q^94 - 256 * q^95 - 18290 * q^96 - 2776 * q^97 - 4608 * q^98 - 3960 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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