LMFDB - Embedded newform 539.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,4,Mod(1,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, 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("539.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	539.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:	$31.8020294931$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 42x^{3} + 18x^{2} + 368x + 352 $ x^5 - x^4 - 42x^3 + 18x^2 + 368*x + 352
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.18888$ of defining polynomial
Character	$\chi$	$=$	539.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.18888 q^{2} -6.48496 q^{3} -3.20880 q^{4} -7.60736 q^{5} +14.1948 q^{6} +24.5347 q^{8} +15.0547 q^{9} +16.6516 q^{10} +11.0000 q^{11} +20.8090 q^{12} -0.174238 q^{13} +49.3335 q^{15} -28.0332 q^{16} -128.863 q^{17} -32.9530 q^{18} -141.685 q^{19} +24.4105 q^{20} -24.0777 q^{22} -133.369 q^{23} -159.107 q^{24} -67.1280 q^{25} +0.381386 q^{26} +77.4647 q^{27} -177.002 q^{29} -107.985 q^{30} -48.2757 q^{31} -134.917 q^{32} -71.3346 q^{33} +282.065 q^{34} -48.3076 q^{36} +161.625 q^{37} +310.131 q^{38} +1.12993 q^{39} -186.645 q^{40} +195.689 q^{41} -488.447 q^{43} -35.2968 q^{44} -114.527 q^{45} +291.928 q^{46} +171.705 q^{47} +181.794 q^{48} +146.935 q^{50} +835.671 q^{51} +0.559095 q^{52} -431.477 q^{53} -169.561 q^{54} -83.6810 q^{55} +918.821 q^{57} +387.437 q^{58} -194.176 q^{59} -158.301 q^{60} -585.008 q^{61} +105.670 q^{62} +519.582 q^{64} +1.32549 q^{65} +156.143 q^{66} +155.905 q^{67} +413.496 q^{68} +864.891 q^{69} -374.994 q^{71} +369.364 q^{72} -210.419 q^{73} -353.778 q^{74} +435.323 q^{75} +454.639 q^{76} -2.47327 q^{78} -7.00618 q^{79} +213.258 q^{80} -908.833 q^{81} -428.339 q^{82} +93.6417 q^{83} +980.307 q^{85} +1069.15 q^{86} +1147.85 q^{87} +269.882 q^{88} +307.119 q^{89} +250.685 q^{90} +427.954 q^{92} +313.066 q^{93} -375.842 q^{94} +1077.85 q^{95} +874.929 q^{96} -965.991 q^{97} +165.602 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} - 2 q^{3} + 45 q^{4} + 24 q^{5} - 4 q^{6} + 57 q^{8} + 63 q^{9} + 10 q^{10} + 55 q^{11} - 24 q^{12} + 50 q^{13} - 146 q^{15} + 433 q^{16} - 222 q^{17} + 245 q^{18} - 160 q^{19} + 430 q^{20}+ \cdots + 693 q^{99}+O(q^{100}) $ 5 * q + q^2 - 2 * q^3 + 45 * q^4 + 24 * q^5 - 4 * q^6 + 57 * q^8 + 63 * q^9 + 10 * q^10 + 55 * q^11 - 24 * q^12 + 50 * q^13 - 146 * q^15 + 433 * q^16 - 222 * q^17 + 245 * q^18 - 160 * q^19 + 430 * q^20 + 11 * q^22 + 54 * q^23 + 220 * q^24 + 125 * q^25 + 1026 * q^26 - 110 * q^27 + 14 * q^29 - 1080 * q^30 + 34 * q^31 - 583 * q^32 - 22 * q^33 + 750 * q^34 - 411 * q^36 + 1044 * q^37 + 156 * q^38 - 124 * q^39 - 1158 * q^40 + 114 * q^41 + 672 * q^43 + 495 * q^44 + 530 * q^45 - 1224 * q^46 + 292 * q^47 + 1652 * q^48 + 1143 * q^50 + 768 * q^51 + 914 * q^52 - 710 * q^53 - 1608 * q^54 + 264 * q^55 + 2012 * q^57 - 810 * q^58 - 270 * q^59 - 3068 * q^60 - 138 * q^61 + 480 * q^62 + 3001 * q^64 - 196 * q^65 - 44 * q^66 + 1942 * q^67 - 3130 * q^68 + 1306 * q^69 - 278 * q^71 + 3565 * q^72 + 338 * q^73 + 462 * q^74 - 3960 * q^75 + 1332 * q^76 + 280 * q^78 + 576 * q^79 + 1602 * q^80 - 1439 * q^81 - 1386 * q^82 - 1644 * q^83 + 360 * q^85 + 2020 * q^86 + 1956 * q^87 + 627 * q^88 + 3656 * q^89 + 6162 * q^90 - 748 * q^92 - 1442 * q^93 - 976 * q^94 - 1276 * q^95 + 4508 * q^96 - 692 * q^97 + 693 * q^99

Coefficient data

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

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

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

$2$	−2.18888	−0.773886	−0.386943	−	0.922104i	$-0.626469\pi$
$2$	−2.18888	−0.773886	−0.386943	+	0.922104i	$0.626469\pi$
$3$	−6.48496	−1.24803	−0.624016	−	0.781412i	$-0.714500\pi$
$3$	−6.48496	−1.24803	−0.624016	+	0.781412i	$0.714500\pi$
$4$	−3.20880	−0.401100
$5$	−7.60736	−0.680423	−0.340212	−	0.940349i	$-0.610499\pi$
$5$	−7.60736	−0.680423	−0.340212	+	0.940349i	$0.610499\pi$
$6$	14.1948	0.965834
$7$	0	0
$7$	0	0
$8$	24.5347	1.08429
$9$	15.0547	0.557582
$10$	16.6516	0.526570
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	20.8090	0.500586
$13$	−0.174238	−0.00371730	−0.00185865	−	0.999998i	$-0.500592\pi$
$13$	−0.174238	−0.00371730	−0.00185865	+	0.999998i	$0.500592\pi$
$14$	0	0
$15$	49.3335	0.849190
$16$	−28.0332	−0.438018
$17$	−128.863	−1.83846	−0.919231	−	0.393719i	$-0.871188\pi$
$17$	−128.863	−1.83846	−0.919231	+	0.393719i	$0.871188\pi$
$18$	−32.9530	−0.431505
$19$	−141.685	−1.71078	−0.855388	−	0.517988i	$-0.826681\pi$
$19$	−141.685	−1.71078	−0.855388	+	0.517988i	$0.826681\pi$
$20$	24.4105	0.272918
$21$	0	0
$22$	−24.0777	−0.233335
$23$	−133.369	−1.20910	−0.604550	−	0.796567i	$-0.706647\pi$
$23$	−133.369	−1.20910	−0.604550	+	0.796567i	$0.706647\pi$
$24$	−159.107	−1.35323
$25$	−67.1280	−0.537024
$26$	0.381386	0.00287677
$27$	77.4647	0.552151
$28$	0	0
$29$	−177.002	−1.13340	−0.566698	−	0.823925i	$-0.691779\pi$
$29$	−177.002	−1.13340	−0.566698	+	0.823925i	$0.691779\pi$
$30$	−107.985	−0.657176
$31$	−48.2757	−0.279696	−0.139848	−	0.990173i	$-0.544661\pi$
$31$	−48.2757	−0.279696	−0.139848	+	0.990173i	$0.544661\pi$
$32$	−134.917	−0.745316
$33$	−71.3346	−0.376296
$34$	282.065	1.42276
$35$	0	0
$36$	−48.3076	−0.223647
$37$	161.625	0.718136	0.359068	−	0.933311i	$-0.383095\pi$
$37$	161.625	0.718136	0.359068	+	0.933311i	$0.383095\pi$
$38$	310.131	1.32394
$39$	1.12993	0.00463931
$40$	−186.645	−0.737778
$41$	195.689	0.745402	0.372701	−	0.927952i	$-0.378432\pi$
$41$	195.689	0.745402	0.372701	+	0.927952i	$0.378432\pi$
$42$	0	0
$43$	−488.447	−1.73227	−0.866133	−	0.499814i	$-0.833402\pi$
$43$	−488.447	−1.73227	−0.866133	+	0.499814i	$0.833402\pi$
$44$	−35.2968	−0.120936
$45$	−114.527	−0.379392
$46$	291.928	0.935705
$47$	171.705	0.532889	0.266444	−	0.963850i	$-0.414151\pi$
$47$	171.705	0.532889	0.266444	+	0.963850i	$0.414151\pi$
$48$	181.794	0.546660
$49$	0	0
$50$	146.935	0.415596
$51$	835.671	2.29446
$52$	0.559095	0.00149101
$53$	−431.477	−1.11826	−0.559131	−	0.829079i	$-0.688865\pi$
$53$	−431.477	−1.11826	−0.559131	+	0.829079i	$0.688865\pi$
$54$	−169.561	−0.427302
$55$	−83.6810	−0.205155
$56$	0	0
$57$	918.821	2.13510
$58$	387.437	0.877120
$59$	−194.176	−0.428468	−0.214234	−	0.976782i	$-0.568725\pi$
$59$	−194.176	−0.428468	−0.214234	+	0.976782i	$0.568725\pi$
$60$	−158.301	−0.340610
$61$	−585.008	−1.22791	−0.613955	−	0.789341i	$-0.710423\pi$
$61$	−585.008	−1.22791	−0.613955	+	0.789341i	$0.710423\pi$
$62$	105.670	0.216453
$63$	0	0
$64$	519.582	1.01481
$65$	1.32549	0.00252934
$66$	156.143	0.291210
$67$	155.905	0.284281	0.142140	−	0.989847i	$-0.454602\pi$
$67$	155.905	0.284281	0.142140	+	0.989847i	$0.454602\pi$
$68$	413.496	0.737408
$69$	864.891	1.50899
$70$	0	0
$71$	−374.994	−0.626811	−0.313405	−	0.949619i	$-0.601470\pi$
$71$	−374.994	−0.626811	−0.313405	+	0.949619i	$0.601470\pi$
$72$	369.364	0.604582
$73$	−210.419	−0.337365	−0.168683	−	0.985670i	$-0.553951\pi$
$73$	−210.419	−0.337365	−0.168683	+	0.985670i	$0.553951\pi$
$74$	−353.778	−0.555755
$75$	435.323	0.670223
$76$	454.639	0.686193
$77$	0	0
$78$	−2.47327	−0.00359029
$79$	−7.00618	−0.00997793	−0.00498897	−	0.999988i	$-0.501588\pi$
$79$	−7.00618	−0.00997793	−0.00498897	+	0.999988i	$0.501588\pi$
$80$	213.258	0.298038
$81$	−908.833	−1.24668
$82$	−428.339	−0.576856
$83$	93.6417	0.123837	0.0619187	−	0.998081i	$-0.480278\pi$
$83$	93.6417	0.123837	0.0619187	+	0.998081i	$0.480278\pi$
$84$	0	0
$85$	980.307	1.25093
$86$	1069.15	1.34058
$87$	1147.85	1.41451
$88$	269.882	0.326926
$89$	307.119	0.365781	0.182891	−	0.983133i	$-0.441455\pi$
$89$	307.119	0.365781	0.182891	+	0.983133i	$0.441455\pi$
$90$	250.685	0.293606
$91$	0	0
$92$	427.954	0.484970
$93$	313.066	0.349069
$94$	−375.842	−0.412395
$95$	1077.85	1.16405
$96$	874.929	0.930178
$97$	−965.991	−1.01115	−0.505575	−	0.862783i	$-0.668720\pi$
$97$	−965.991	−1.01115	−0.505575	+	0.862783i	$0.668720\pi$
$98$	0	0
$99$	165.602	0.168117
$100$	215.401	0.215401
$101$	577.986	0.569424	0.284712	−	0.958613i	$-0.408102\pi$
$101$	577.986	0.569424	0.284712	+	0.958613i	$0.408102\pi$
$102$	−1829.18	−1.77565
$103$	−133.232	−0.127453	−0.0637267	−	0.997967i	$-0.520299\pi$
$103$	−133.232	−0.127453	−0.0637267	+	0.997967i	$0.520299\pi$
$104$	−4.27488	−0.00403064
$105$	0	0
$106$	944.451	0.865408
$107$	8.33627	0.00753175	0.00376587	−	0.999993i	$-0.498801\pi$
$107$	8.33627	0.00753175	0.00376587	+	0.999993i	$0.498801\pi$
$108$	−248.569	−0.221468
$109$	−317.052	−0.278606	−0.139303	−	0.990250i	$-0.544486\pi$
$109$	−317.052	−0.278606	−0.139303	+	0.990250i	$0.544486\pi$
$110$	183.168	0.158767
$111$	−1048.13	−0.896256
$112$	0	0
$113$	−2170.00	−1.80652	−0.903260	−	0.429093i	$-0.858833\pi$
$113$	−2170.00	−1.80652	−0.903260	+	0.429093i	$0.858833\pi$
$114$	−2011.19	−1.65232
$115$	1014.58	0.822700
$116$	567.966	0.454606
$117$	−2.62310	−0.00207270
$118$	425.028	0.331585
$119$	0	0
$120$	1210.38	0.920769
$121$	121.000	0.0909091
$122$	1280.51	0.950263
$123$	−1269.03	−0.930285
$124$	154.907	0.112186
$125$	1461.59	1.04583
$126$	0	0
$127$	1301.24	0.909184	0.454592	−	0.890700i	$-0.349785\pi$
$127$	1301.24	0.909184	0.454592	+	0.890700i	$0.349785\pi$
$128$	−57.9688	−0.0400294
$129$	3167.56	2.16192
$130$	−2.90134	−0.00195742
$131$	−2522.22	−1.68220	−0.841098	−	0.540883i	$-0.818090\pi$
$131$	−2522.22	−1.68220	−0.841098	+	0.540883i	$0.818090\pi$
$132$	228.899	0.150932
$133$	0	0
$134$	−341.257	−0.220001
$135$	−589.302	−0.375696
$136$	−3161.62	−1.99343
$137$	2574.70	1.60563	0.802815	−	0.596228i	$-0.203334\pi$
$137$	2574.70	1.60563	0.802815	+	0.596228i	$0.203334\pi$
$138$	−1893.14	−1.16779
$139$	2741.49	1.67288	0.836438	−	0.548061i	$-0.184634\pi$
$139$	2741.49	1.67288	0.836438	+	0.548061i	$0.184634\pi$
$140$	0	0
$141$	−1113.50	−0.665062
$142$	820.816	0.485080
$143$	−1.91662	−0.00112081
$144$	−422.031	−0.244231
$145$	1346.52	0.771189
$146$	460.582	0.261082
$147$	0	0
$148$	−518.624	−0.288045
$149$	−1021.98	−0.561907	−0.280954	−	0.959721i	$-0.590651\pi$
$149$	−1021.98	−0.561907	−0.280954	+	0.959721i	$0.590651\pi$
$150$	−952.869	−0.518676
$151$	1016.40	0.547769	0.273885	−	0.961763i	$-0.411691\pi$
$151$	1016.40	0.547769	0.273885	+	0.961763i	$0.411691\pi$
$152$	−3476.20	−1.85498
$153$	−1940.00	−1.02509
$154$	0	0
$155$	367.251	0.190312
$156$	−3.62571	−0.00186083
$157$	1580.49	0.803422	0.401711	−	0.915767i	$-0.368416\pi$
$157$	1580.49	0.803422	0.401711	+	0.915767i	$0.368416\pi$
$158$	15.3357	0.00772178
$159$	2798.11	1.39563
$160$	1026.36	0.507130
$161$	0	0
$162$	1989.33	0.964792
$163$	768.358	0.369218	0.184609	−	0.982812i	$-0.440898\pi$
$163$	768.358	0.369218	0.184609	+	0.982812i	$0.440898\pi$
$164$	−627.927	−0.298981
$165$	542.668	0.256040
$166$	−204.970	−0.0958361
$167$	−3092.73	−1.43307	−0.716534	−	0.697552i	$-0.754273\pi$
$167$	−3092.73	−1.43307	−0.716534	+	0.697552i	$0.754273\pi$
$168$	0	0
$169$	−2196.97	−0.999986
$170$	−2145.77	−0.968079
$171$	−2133.03	−0.953898
$172$	1567.33	0.694812
$173$	1327.46	0.583380	0.291690	−	0.956513i	$-0.405782\pi$
$173$	1327.46	0.583380	0.291690	+	0.956513i	$0.405782\pi$
$174$	−2512.51	−1.09467
$175$	0	0
$176$	−308.365	−0.132067
$177$	1259.23	0.534741
$178$	−672.247	−0.283073
$179$	3141.37	1.31171	0.655857	−	0.754885i	$-0.272307\pi$
$179$	3141.37	1.31171	0.655857	+	0.754885i	$0.272307\pi$
$180$	367.494	0.152174
$181$	3683.36	1.51261	0.756303	−	0.654221i	$-0.227003\pi$
$181$	3683.36	1.51261	0.756303	+	0.654221i	$0.227003\pi$
$182$	0	0
$183$	3793.75	1.53247
$184$	−3272.16	−1.31102
$185$	−1229.54	−0.488636
$186$	−685.264	−0.270140
$187$	−1417.49	−0.554317
$188$	−550.968	−0.213742
$189$	0	0
$190$	−2359.28	−0.900843
$191$	−2862.38	−1.08437	−0.542185	−	0.840259i	$-0.682403\pi$
$191$	−2862.38	−1.08437	−0.542185	+	0.840259i	$0.682403\pi$
$192$	−3369.47	−1.26651
$193$	2023.91	0.754839	0.377420	−	0.926042i	$-0.376811\pi$
$193$	2023.91	0.754839	0.377420	+	0.926042i	$0.376811\pi$
$194$	2114.44	0.782515
$195$	−8.59576	−0.00315669
$196$	0	0
$197$	−4767.82	−1.72433	−0.862165	−	0.506628i	$-0.830892\pi$
$197$	−4767.82	−1.72433	−0.862165	+	0.506628i	$0.830892\pi$
$198$	−362.483	−0.130104
$199$	−3546.72	−1.26342	−0.631709	−	0.775205i	$-0.717646\pi$
$199$	−3546.72	−1.26342	−0.631709	+	0.775205i	$0.717646\pi$
$200$	−1646.97	−0.582291
$201$	−1011.04	−0.354791
$202$	−1265.14	−0.440669
$203$	0	0
$204$	−2681.50	−0.920308
$205$	−1488.68	−0.507189
$206$	291.628	0.0986343
$207$	−2007.83	−0.674173
$208$	4.88444	0.00162824
$209$	−1558.53	−0.515818
$210$	0	0
$211$	1453.49	0.474230	0.237115	−	0.971482i	$-0.423798\pi$
$211$	1453.49	0.474230	0.237115	+	0.971482i	$0.423798\pi$
$212$	1384.52	0.448536
$213$	2431.82	0.782279
$214$	−18.2471	−0.00582872
$215$	3715.79	1.17867
$216$	1900.57	0.598693
$217$	0	0
$218$	693.989	0.215610
$219$	1364.56	0.421042
$220$	268.516	0.0822879
$221$	22.4528	0.00683411
$222$	2294.24	0.693600
$223$	−6440.10	−1.93391	−0.966953	−	0.254955i	$-0.917939\pi$
$223$	−6440.10	−1.93391	−0.966953	+	0.254955i	$0.917939\pi$
$224$	0	0
$225$	−1010.59	−0.299435
$226$	4749.88	1.39804
$227$	1313.21	0.383970	0.191985	−	0.981398i	$-0.438508\pi$
$227$	1313.21	0.383970	0.191985	+	0.981398i	$0.438508\pi$
$228$	−2948.31	−0.856390
$229$	3780.00	1.09078	0.545392	−	0.838181i	$-0.316381\pi$
$229$	3780.00	1.09078	0.545392	+	0.838181i	$0.316381\pi$
$230$	−2220.80	−0.636676
$231$	0	0
$232$	−4342.70	−1.22893
$233$	2322.65	0.653055	0.326527	−	0.945188i	$-0.394121\pi$
$233$	2322.65	0.653055	0.326527	+	0.945188i	$0.394121\pi$
$234$	5.74166	0.00160403
$235$	−1306.22	−0.362590
$236$	623.073	0.171859
$237$	45.4348	0.0124528
$238$	0	0
$239$	2204.79	0.596718	0.298359	−	0.954454i	$-0.403561\pi$
$239$	2204.79	0.596718	0.298359	+	0.954454i	$0.403561\pi$
$240$	−1382.97	−0.371960
$241$	4610.39	1.23229	0.616144	−	0.787634i	$-0.288694\pi$
$241$	4610.39	1.23229	0.616144	+	0.787634i	$0.288694\pi$
$242$	−264.855	−0.0703533
$243$	3802.20	1.00375
$244$	1877.17	0.492516
$245$	0	0
$246$	2777.76	0.719934
$247$	24.6869	0.00635946
$248$	−1184.43	−0.303272
$249$	−607.263	−0.154553
$250$	−3199.24	−0.809351
$251$	−4981.12	−1.25261	−0.626306	−	0.779577i	$-0.715434\pi$
$251$	−4981.12	−1.25261	−0.626306	+	0.779577i	$0.715434\pi$
$252$	0	0
$253$	−1467.06	−0.364557
$254$	−2848.26	−0.703605
$255$	−6357.25	−1.56120
$256$	−4029.77	−0.983829
$257$	−2726.08	−0.661665	−0.330833	−	0.943689i	$-0.607330\pi$
$257$	−2726.08	−0.661665	−0.330833	+	0.943689i	$0.607330\pi$
$258$	−6933.40	−1.67308
$259$	0	0
$260$	−4.25324	−0.00101452
$261$	−2664.72	−0.631962
$262$	5520.84	1.30183
$263$	−4228.84	−0.991489	−0.495744	−	0.868468i	$-0.665105\pi$
$263$	−4228.84	−0.991489	−0.495744	+	0.868468i	$0.665105\pi$
$264$	−1750.17	−0.408014
$265$	3282.40	0.760892
$266$	0	0
$267$	−1991.65	−0.456507
$268$	−500.268	−0.114025
$269$	2396.04	0.543081	0.271541	−	0.962427i	$-0.412467\pi$
$269$	2396.04	0.543081	0.271541	+	0.962427i	$0.412467\pi$
$270$	1289.91	0.290746
$271$	2461.68	0.551795	0.275897	−	0.961187i	$-0.411025\pi$
$271$	2461.68	0.551795	0.275897	+	0.961187i	$0.411025\pi$
$272$	3612.43	0.805279
$273$	0	0
$274$	−5635.71	−1.24258
$275$	−738.408	−0.161919
$276$	−2775.26	−0.605258
$277$	5890.54	1.27772	0.638860	−	0.769323i	$-0.279406\pi$
$277$	5890.54	1.27772	0.638860	+	0.769323i	$0.279406\pi$
$278$	−6000.79	−1.29462
$279$	−726.777	−0.155953
$280$	0	0
$281$	−1964.26	−0.417004	−0.208502	−	0.978022i	$-0.566859\pi$
$281$	−1964.26	−0.417004	−0.208502	+	0.978022i	$0.566859\pi$
$282$	2437.32	0.514682
$283$	−8513.60	−1.78827	−0.894136	−	0.447796i	$-0.852209\pi$
$283$	−8513.60	−1.78827	−0.894136	+	0.447796i	$0.852209\pi$
$284$	1203.28	0.251414
$285$	−6989.80	−1.45277
$286$	4.19524	0.000867378 0
$287$	0	0
$288$	−2031.13	−0.415575
$289$	11692.6	2.37994
$290$	−2947.37	−0.596813
$291$	6264.42	1.26195
$292$	675.193	0.135317
$293$	−5219.58	−1.04072	−0.520360	−	0.853947i	$-0.674202\pi$
$293$	−5219.58	−1.04072	−0.520360	+	0.853947i	$0.674202\pi$
$294$	0	0
$295$	1477.17	0.291539
$296$	3965.43	0.778669
$297$	852.111	0.166480
$298$	2237.00	0.434852
$299$	23.2379	0.00449459
$300$	−1396.86	−0.268827
$301$	0	0
$302$	−2224.77	−0.423911
$303$	−3748.22	−0.710659
$304$	3971.87	0.749350
$305$	4450.37	0.835499
$306$	4246.42	0.793306
$307$	−2887.29	−0.536764	−0.268382	−	0.963313i	$-0.586489\pi$
$307$	−2887.29	−0.536764	−0.268382	+	0.963313i	$0.586489\pi$
$308$	0	0
$309$	864.001	0.159066
$310$	−803.868	−0.147279
$311$	5876.90	1.07154	0.535769	−	0.844364i	$-0.320022\pi$
$311$	5876.90	1.07154	0.535769	+	0.844364i	$0.320022\pi$
$312$	27.7224	0.00503036
$313$	7683.14	1.38747	0.693733	−	0.720233i	$-0.255965\pi$
$313$	7683.14	1.38747	0.693733	+	0.720233i	$0.255965\pi$
$314$	−3459.51	−0.621757
$315$	0	0
$316$	22.4814	0.00400215
$317$	−9642.33	−1.70841	−0.854207	−	0.519933i	$-0.825957\pi$
$317$	−9642.33	−1.70841	−0.854207	+	0.519933i	$0.825957\pi$
$318$	−6124.73	−1.08006
$319$	−1947.03	−0.341732
$320$	−3952.65	−0.690499
$321$	−54.0604	−0.00939986
$322$	0	0
$323$	18257.9	3.14519
$324$	2916.27	0.500046
$325$	11.6962	0.00199628
$326$	−1681.84	−0.285732
$327$	2056.07	0.347710
$328$	4801.17	0.808233
$329$	0	0
$330$	−1187.84	−0.198146
$331$	−2069.87	−0.343717	−0.171859	−	0.985122i	$-0.554977\pi$
$331$	−2069.87	−0.343717	−0.171859	+	0.985122i	$0.554977\pi$
$332$	−300.478	−0.0496713
$333$	2433.22	0.400420
$334$	6769.61	1.10903
$335$	−1186.03	−0.193431
$336$	0	0
$337$	7547.59	1.22001	0.610005	−	0.792397i	$-0.291167\pi$
$337$	7547.59	1.22001	0.610005	+	0.792397i	$0.291167\pi$
$338$	4808.90	0.773875
$339$	14072.4	2.25459
$340$	−3145.61	−0.501749
$341$	−531.033	−0.0843315
$342$	4668.94	0.738208
$343$	0	0
$344$	−11983.9	−1.87828
$345$	−6579.54	−1.02675
$346$	−2905.65	−0.451469
$347$	−6353.22	−0.982877	−0.491439	−	0.870912i	$-0.663529\pi$
$347$	−6353.22	−0.982877	−0.491439	+	0.870912i	$0.663529\pi$
$348$	−3683.23	−0.567362
$349$	1352.47	0.207439	0.103719	−	0.994607i	$-0.466926\pi$
$349$	1352.47	0.207439	0.103719	+	0.994607i	$0.466926\pi$
$350$	0	0
$351$	−13.4973	−0.00205251
$352$	−1484.08	−0.224721
$353$	−4203.80	−0.633840	−0.316920	−	0.948452i	$-0.602649\pi$
$353$	−4203.80	−0.633840	−0.316920	+	0.948452i	$0.602649\pi$
$354$	−2756.29	−0.413829
$355$	2852.71	0.426497
$356$	−985.484	−0.146715
$357$	0	0
$358$	−6876.08	−1.01512
$359$	−11100.7	−1.63195	−0.815976	−	0.578086i	$-0.803800\pi$
$359$	−11100.7	−1.63195	−0.815976	+	0.578086i	$0.803800\pi$
$360$	−2809.88	−0.411372
$361$	13215.6	1.92675
$362$	−8062.43	−1.17059
$363$	−784.680	−0.113457
$364$	0	0
$365$	1600.73	0.229551
$366$	−8304.07	−1.18596
$367$	4963.35	0.705953	0.352977	−	0.935632i	$-0.385170\pi$
$367$	4963.35	0.705953	0.352977	+	0.935632i	$0.385170\pi$
$368$	3738.74	0.529607
$369$	2946.04	0.415623
$370$	2691.32	0.378149
$371$	0	0
$372$	−1004.57	−0.140012
$373$	−8882.54	−1.23303	−0.616515	−	0.787343i	$-0.711456\pi$
$373$	−8882.54	−1.23303	−0.616515	+	0.787343i	$0.711456\pi$
$374$	3102.72	0.428978
$375$	−9478.34	−1.30522
$376$	4212.74	0.577807
$377$	30.8405	0.00421317
$378$	0	0
$379$	−1218.83	−0.165190	−0.0825952	−	0.996583i	$-0.526321\pi$
$379$	−1218.83	−0.165190	−0.0825952	+	0.996583i	$0.526321\pi$
$380$	−3458.60	−0.466901
$381$	−8438.49	−1.13469
$382$	6265.41	0.839179
$383$	−112.614	−0.0150243	−0.00751214	−	0.999972i	$-0.502391\pi$
$383$	−112.614	−0.0150243	−0.00751214	+	0.999972i	$0.502391\pi$
$384$	375.926	0.0499580
$385$	0	0
$386$	−4430.09	−0.584159
$387$	−7353.43	−0.965880
$388$	3099.68	0.405573
$389$	12656.2	1.64961	0.824804	−	0.565419i	$-0.191286\pi$
$389$	12656.2	1.64961	0.824804	+	0.565419i	$0.191286\pi$
$390$	18.8151	0.00244292
$391$	17186.3	2.22288
$392$	0	0
$393$	16356.5	2.09943
$394$	10436.2	1.33443
$395$	53.2985	0.00678922
$396$	−531.384	−0.0674320
$397$	−3707.14	−0.468655	−0.234327	−	0.972158i	$-0.575289\pi$
$397$	−3707.14	−0.468655	−0.234327	+	0.972158i	$0.575289\pi$
$398$	7763.35	0.977742
$399$	0	0
$400$	1881.81	0.235226
$401$	5671.06	0.706232	0.353116	−	0.935579i	$-0.385122\pi$
$401$	5671.06	0.706232	0.353116	+	0.935579i	$0.385122\pi$
$402$	2213.04	0.274568
$403$	8.41146	0.00103971
$404$	−1854.64	−0.228396
$405$	6913.82	0.848273
$406$	0	0
$407$	1777.88	0.216526
$408$	20503.0	2.48786
$409$	−260.465	−0.0314894	−0.0157447	−	0.999876i	$-0.505012\pi$
$409$	−260.465	−0.0314894	−0.0157447	+	0.999876i	$0.505012\pi$
$410$	3258.53	0.392506
$411$	−16696.8	−2.00388
$412$	427.514	0.0511216
$413$	0	0
$414$	4394.90	0.521733
$415$	−712.366	−0.0842619
$416$	23.5076	0.00277056
$417$	−17778.4	−2.08780
$418$	3411.44	0.399184
$419$	−4153.96	−0.484330	−0.242165	−	0.970235i	$-0.577858\pi$
$419$	−4153.96	−0.484330	−0.242165	+	0.970235i	$0.577858\pi$
$420$	0	0
$421$	4240.11	0.490856	0.245428	−	0.969415i	$-0.421072\pi$
$421$	4240.11	0.490856	0.245428	+	0.969415i	$0.421072\pi$
$422$	−3181.52	−0.367000
$423$	2584.97	0.297129
$424$	−10586.2	−1.21252
$425$	8650.31	0.987298
$426$	−5322.96	−0.605395
$427$	0	0
$428$	−26.7494	−0.00302099
$429$	12.4292	0.00139880
$430$	−8133.42	−0.912159
$431$	1012.17	0.113119	0.0565595	−	0.998399i	$-0.481987\pi$
$431$	1012.17	0.113119	0.0565595	+	0.998399i	$0.481987\pi$
$432$	−2171.58	−0.241852
$433$	−1604.50	−0.178077	−0.0890386	−	0.996028i	$-0.528379\pi$
$433$	−1604.50	−0.178077	−0.0890386	+	0.996028i	$0.528379\pi$
$434$	0	0
$435$	−8732.13	−0.962469
$436$	1017.36	0.111749
$437$	18896.3	2.06850
$438$	−2986.85	−0.325839
$439$	−10671.6	−1.16020	−0.580099	−	0.814546i	$-0.696986\pi$
$439$	−10671.6	−1.16020	−0.580099	+	0.814546i	$0.696986\pi$
$440$	−2053.09	−0.222448
$441$	0	0
$442$	−49.1465	−0.00528882
$443$	2193.74	0.235277	0.117638	−	0.993056i	$-0.462468\pi$
$443$	2193.74	0.235277	0.117638	+	0.993056i	$0.462468\pi$
$444$	3363.25	0.359489
$445$	−2336.37	−0.248886
$446$	14096.6	1.49662
$447$	6627.52	0.701278
$448$	0	0
$449$	−7070.36	−0.743143	−0.371571	−	0.928404i	$-0.621181\pi$
$449$	−7070.36	−0.743143	−0.371571	+	0.928404i	$0.621181\pi$
$450$	2212.07	0.231729
$451$	2152.58	0.224747
$452$	6963.12	0.724596
$453$	−6591.29	−0.683633
$454$	−2874.47	−0.297149
$455$	0	0
$456$	22543.0	2.31507
$457$	16245.8	1.66290	0.831451	−	0.555599i	$-0.187511\pi$
$457$	16245.8	1.66290	0.831451	+	0.555599i	$0.187511\pi$
$458$	−8273.98	−0.844143
$459$	−9982.32	−1.01511
$460$	−3255.60	−0.329985
$461$	5235.18	0.528908	0.264454	−	0.964398i	$-0.414808\pi$
$461$	5235.18	0.528908	0.264454	+	0.964398i	$0.414808\pi$
$462$	0	0
$463$	−11093.9	−1.11355	−0.556777	−	0.830662i	$-0.687962\pi$
$463$	−11093.9	−1.11355	−0.556777	+	0.830662i	$0.687962\pi$
$464$	4961.93	0.496448
$465$	−2381.61	−0.237515
$466$	−5084.00	−0.505390
$467$	7813.14	0.774195	0.387097	−	0.922039i	$-0.373478\pi$
$467$	7813.14	0.774195	0.387097	+	0.922039i	$0.373478\pi$
$468$	8.41702	0.000831361 0
$469$	0	0
$470$	2859.17	0.280603
$471$	−10249.4	−1.00270
$472$	−4764.06	−0.464584
$473$	−5372.91	−0.522298
$474$	−99.4513	−0.00963703
$475$	9511.02	0.918728
$476$	0	0
$477$	−6495.76	−0.623523
$478$	−4826.01	−0.461792
$479$	−5298.54	−0.505420	−0.252710	−	0.967542i	$-0.581322\pi$
$479$	−5298.54	−0.505420	−0.252710	+	0.967542i	$0.581322\pi$
$480$	−6655.90	−0.632915
$481$	−28.1612	−0.00266953
$482$	−10091.6	−0.953650
$483$	0	0
$484$	−388.265	−0.0364637
$485$	7348.65	0.688010
$486$	−8322.56	−0.776788
$487$	5580.89	0.519290	0.259645	−	0.965704i	$-0.416394\pi$
$487$	5580.89	0.519290	0.259645	+	0.965704i	$0.416394\pi$
$488$	−14353.0	−1.33141
$489$	−4982.77	−0.460795
$490$	0	0
$491$	16718.8	1.53668	0.768341	−	0.640041i	$-0.221083\pi$
$491$	16718.8	1.53668	0.768341	+	0.640041i	$0.221083\pi$
$492$	4072.08	0.373138
$493$	22809.0	2.08371
$494$	−54.0366	−0.00492150
$495$	−1259.79	−0.114391
$496$	1353.32	0.122512
$497$	0	0
$498$	1329.23	0.119606
$499$	19594.4	1.75784	0.878922	−	0.476965i	$-0.158263\pi$
$499$	19594.4	1.75784	0.878922	+	0.476965i	$0.158263\pi$
$500$	−4689.95	−0.419482
$501$	20056.2	1.78851
$502$	10903.1	0.969379
$503$	−9008.04	−0.798506	−0.399253	−	0.916841i	$-0.630731\pi$
$503$	−9008.04	−0.798506	−0.399253	+	0.916841i	$0.630731\pi$
$504$	0	0
$505$	−4396.95	−0.387449
$506$	3211.21	0.282126
$507$	14247.3	1.24801
$508$	−4175.42	−0.364674
$509$	3379.88	0.294324	0.147162	−	0.989112i	$-0.452986\pi$
$509$	3379.88	0.294324	0.147162	+	0.989112i	$0.452986\pi$
$510$	13915.3	1.20819
$511$	0	0
$512$	9284.42	0.801401
$513$	−10975.6	−0.944606
$514$	5967.05	0.512053
$515$	1013.54	0.0867222
$516$	−10164.1	−0.867148
$517$	1888.76	0.160672
$518$	0	0
$519$	−8608.51	−0.728076
$520$	32.5206	0.00274254
$521$	736.886	0.0619646	0.0309823	−	0.999520i	$-0.490136\pi$
$521$	736.886	0.0619646	0.0309823	+	0.999520i	$0.490136\pi$
$522$	5832.75	0.489066
$523$	−9541.00	−0.797703	−0.398852	−	0.917015i	$-0.630591\pi$
$523$	−9541.00	−0.797703	−0.398852	+	0.917015i	$0.630591\pi$
$524$	8093.32	0.674729
$525$	0	0
$526$	9256.43	0.767299
$527$	6220.95	0.514210
$528$	1999.73	0.164824
$529$	5620.20	0.461922
$530$	−7184.78	−0.588843
$531$	−2923.27	−0.238906
$532$	0	0
$533$	−34.0964	−0.00277088
$534$	4359.49	0.353284
$535$	−63.4170	−0.00512478
$536$	3825.08	0.308243
$537$	−20371.7	−1.63706
$538$	−5244.64	−0.420283
$539$	0	0
$540$	1890.95	0.150692
$541$	16721.0	1.32882	0.664411	−	0.747368i	$-0.268683\pi$
$541$	16721.0	1.32882	0.664411	+	0.747368i	$0.268683\pi$
$542$	−5388.32	−0.427026
$543$	−23886.4	−1.88778
$544$	17385.7	1.37023
$545$	2411.93	0.189570
$546$	0	0
$547$	8958.59	0.700259	0.350129	−	0.936701i	$-0.386138\pi$
$547$	8958.59	0.700259	0.350129	+	0.936701i	$0.386138\pi$
$548$	−8261.70	−0.644019
$549$	−8807.13	−0.684661
$550$	1616.29	0.125307
$551$	25078.5	1.93899
$552$	21219.9	1.63619
$553$	0	0
$554$	−12893.7	−0.988809
$555$	7973.53	0.609834
$556$	−8796.89	−0.670992
$557$	−14329.6	−1.09006	−0.545031	−	0.838416i	$-0.683482\pi$
$557$	−14329.6	−1.09006	−0.545031	+	0.838416i	$0.683482\pi$
$558$	1590.83	0.120690
$559$	85.1059	0.00643935
$560$	0	0
$561$	9192.38	0.691805
$562$	4299.54	0.322713
$563$	−15791.2	−1.18209	−0.591047	−	0.806637i	$-0.701285\pi$
$563$	−15791.2	−1.18209	−0.591047	+	0.806637i	$0.701285\pi$
$564$	3573.01	0.266757
$565$	16508.0	1.22920
$566$	18635.2	1.38392
$567$	0	0
$568$	−9200.37	−0.679646
$569$	−13996.4	−1.03121	−0.515605	−	0.856826i	$-0.672433\pi$
$569$	−13996.4	−1.03121	−0.515605	+	0.856826i	$0.672433\pi$
$570$	15299.8	1.12428
$571$	−6642.54	−0.486833	−0.243417	−	0.969922i	$-0.578268\pi$
$571$	−6642.54	−0.486833	−0.243417	+	0.969922i	$0.578268\pi$
$572$	6.15005	0.000449557 0
$573$	18562.4	1.35333
$574$	0	0
$575$	8952.78	0.649316
$576$	7822.16	0.565839
$577$	85.5448	0.00617206	0.00308603	−	0.999995i	$-0.499018\pi$
$577$	85.5448	0.00617206	0.00308603	+	0.999995i	$0.499018\pi$
$578$	−25593.8	−1.84180
$579$	−13125.0	−0.942063
$580$	−4320.72	−0.309324
$581$	0	0
$582$	−13712.1	−0.976603
$583$	−4746.25	−0.337169
$584$	−5162.57	−0.365802
$585$	19.9549	0.00141031
$586$	11425.0	0.805399
$587$	−11205.0	−0.787869	−0.393934	−	0.919139i	$-0.628886\pi$
$587$	−11205.0	−0.787869	−0.393934	+	0.919139i	$0.628886\pi$
$588$	0	0
$589$	6839.93	0.478497
$590$	−3233.35	−0.225618
$591$	30919.1	2.15202
$592$	−4530.87	−0.314556
$593$	−11664.6	−0.807773	−0.403886	−	0.914809i	$-0.632341\pi$
$593$	−11664.6	−0.807773	−0.403886	+	0.914809i	$0.632341\pi$
$594$	−1865.17	−0.128836
$595$	0	0
$596$	3279.34	0.225381
$597$	23000.3	1.57679
$598$	−50.8649	−0.00347830
$599$	−20545.1	−1.40142	−0.700711	−	0.713445i	$-0.747134\pi$
$599$	−20545.1	−1.40142	−0.700711	+	0.713445i	$0.747134\pi$
$600$	10680.5	0.726718
$601$	−3885.01	−0.263682	−0.131841	−	0.991271i	$-0.542089\pi$
$601$	−3885.01	−0.263682	−0.131841	+	0.991271i	$0.542089\pi$
$602$	0	0
$603$	2347.11	0.158510
$604$	−3261.42	−0.219710
$605$	−920.491	−0.0618567
$606$	8204.40	0.549969
$607$	−8439.80	−0.564351	−0.282175	−	0.959363i	$-0.591056\pi$
$607$	−8439.80	−0.564351	−0.282175	+	0.959363i	$0.591056\pi$
$608$	19115.6	1.27507
$609$	0	0
$610$	−9741.32	−0.646581
$611$	−29.9175	−0.00198091
$612$	6225.06	0.411165
$613$	17997.8	1.18584	0.592922	−	0.805260i	$-0.297974\pi$
$613$	17997.8	1.18584	0.592922	+	0.805260i	$0.297974\pi$
$614$	6319.94	0.415394
$615$	9654.01	0.632987
$616$	0	0
$617$	−26336.6	−1.71843	−0.859214	−	0.511616i	$-0.829047\pi$
$617$	−26336.6	−1.71843	−0.859214	+	0.511616i	$0.829047\pi$
$618$	−1891.20	−0.123099
$619$	−27875.5	−1.81003	−0.905016	−	0.425376i	$-0.860142\pi$
$619$	−27875.5	−1.81003	−0.905016	+	0.425376i	$0.860142\pi$
$620$	−1178.44	−0.0763341
$621$	−10331.4	−0.667606
$622$	−12863.8	−0.829249
$623$	0	0
$624$	−31.6754	−0.00203210
$625$	−2727.83	−0.174581
$626$	−16817.5	−1.07374
$627$	10107.0	0.643757
$628$	−5071.50	−0.322253
$629$	−20827.5	−1.32027
$630$	0	0
$631$	8242.15	0.519992	0.259996	−	0.965610i	$-0.416279\pi$
$631$	8242.15	0.519992	0.259996	+	0.965610i	$0.416279\pi$
$632$	−171.895	−0.0108190
$633$	−9425.84	−0.591854
$634$	21105.9	1.32212
$635$	−9899.00	−0.618630
$636$	−8978.59	−0.559786
$637$	0	0
$638$	4261.80	0.264462
$639$	−5645.43	−0.349499
$640$	440.990	0.0272370
$641$	−7001.97	−0.431453	−0.215726	−	0.976454i	$-0.569212\pi$
$641$	−7001.97	−0.431453	−0.215726	+	0.976454i	$0.569212\pi$
$642$	118.332	0.00727442
$643$	−2473.25	−0.151688	−0.0758442	−	0.997120i	$-0.524165\pi$
$643$	−2473.25	−0.151688	−0.0758442	+	0.997120i	$0.524165\pi$
$644$	0	0
$645$	−24096.8	−1.47102
$646$	−39964.4	−2.43402
$647$	−9153.21	−0.556182	−0.278091	−	0.960555i	$-0.589702\pi$
$647$	−9153.21	−0.556182	−0.278091	+	0.960555i	$0.589702\pi$
$648$	−22298.0	−1.35177
$649$	−2135.94	−0.129188
$650$	−25.6017	−0.00154489
$651$	0	0
$652$	−2465.51	−0.148093
$653$	725.254	0.0434630	0.0217315	−	0.999764i	$-0.493082\pi$
$653$	725.254	0.0434630	0.0217315	+	0.999764i	$0.493082\pi$
$654$	−4500.49	−0.269088
$655$	19187.5	1.14461
$656$	−5485.78	−0.326499
$657$	−3167.80	−0.188109
$658$	0	0
$659$	14332.9	0.847242	0.423621	−	0.905840i	$-0.360759\pi$
$659$	14332.9	0.847242	0.423621	+	0.905840i	$0.360759\pi$
$660$	−1741.32	−0.102698
$661$	−26773.3	−1.57543	−0.787715	−	0.616040i	$-0.788736\pi$
$661$	−26773.3	−1.57543	−0.787715	+	0.616040i	$0.788736\pi$
$662$	4530.70	0.265998
$663$	−145.606	−0.00852919
$664$	2297.47	0.134276
$665$	0	0
$666$	−5326.04	−0.309879
$667$	23606.6	1.37039
$668$	9923.95	0.574804
$669$	41763.8	2.41358
$670$	2596.07	0.149694
$671$	−6435.08	−0.370229
$672$	0	0
$673$	−25423.9	−1.45619	−0.728096	−	0.685475i	$-0.759595\pi$
$673$	−25423.9	−1.45619	−0.728096	+	0.685475i	$0.759595\pi$
$674$	−16520.8	−0.944149
$675$	−5200.05	−0.296519
$676$	7049.64	0.401095
$677$	−9287.48	−0.527248	−0.263624	−	0.964625i	$-0.584918\pi$
$677$	−9287.48	−0.527248	−0.263624	+	0.964625i	$0.584918\pi$
$678$	−30802.8	−1.74480
$679$	0	0
$680$	24051.6	1.35638
$681$	−8516.15	−0.479206
$682$	1162.37	0.0652629
$683$	−31888.8	−1.78652	−0.893258	−	0.449544i	$-0.851586\pi$
$683$	−31888.8	−1.78652	−0.893258	+	0.449544i	$0.851586\pi$
$684$	6844.46	0.382609
$685$	−19586.7	−1.09251
$686$	0	0
$687$	−24513.2	−1.36133
$688$	13692.7	0.758763
$689$	75.1796	0.00415692
$690$	14401.8	0.794591
$691$	18650.3	1.02676	0.513379	−	0.858162i	$-0.328394\pi$
$691$	18650.3	1.02676	0.513379	+	0.858162i	$0.328394\pi$
$692$	−4259.55	−0.233994
$693$	0	0
$694$	13906.4	0.760635
$695$	−20855.5	−1.13826
$696$	28162.3	1.53375
$697$	−25217.0	−1.37039
$698$	−2960.40	−0.160534
$699$	−15062.3	−0.815033
$700$	0	0
$701$	8566.01	0.461532	0.230766	−	0.973009i	$-0.425877\pi$
$701$	8566.01	0.461532	0.230766	+	0.973009i	$0.425877\pi$
$702$	29.5439	0.00158841
$703$	−22899.8	−1.22857
$704$	5715.40	0.305976
$705$	8470.81	0.452524
$706$	9201.61	0.490520
$707$	0	0
$708$	−4040.61	−0.214485
$709$	680.116	0.0360258	0.0180129	−	0.999838i	$-0.494266\pi$
$709$	680.116	0.0360258	0.0180129	+	0.999838i	$0.494266\pi$
$710$	−6244.25	−0.330060
$711$	−105.476	−0.00556352
$712$	7535.08	0.396614
$713$	6438.47	0.338180
$714$	0	0
$715$	14.5804	0.000762624 0
$716$	−10080.0	−0.526129
$717$	−14297.9	−0.744723
$718$	24298.0	1.26294
$719$	−12557.6	−0.651346	−0.325673	−	0.945482i	$-0.605591\pi$
$719$	−12557.6	−0.651346	−0.325673	+	0.945482i	$0.605591\pi$
$720$	3210.55	0.166181
$721$	0	0
$722$	−28927.3	−1.49109
$723$	−29898.2	−1.53793
$724$	−11819.2	−0.606707
$725$	11881.8	0.608661
$726$	1717.57	0.0878031
$727$	−22253.6	−1.13527	−0.567635	−	0.823281i	$-0.692141\pi$
$727$	−22253.6	−1.13527	−0.567635	+	0.823281i	$0.692141\pi$
$728$	0	0
$729$	−118.633	−0.00602716
$730$	−3503.81	−0.177646
$731$	62942.6	3.18470
$732$	−12173.4	−0.614675
$733$	−6852.72	−0.345308	−0.172654	−	0.984982i	$-0.555234\pi$
$733$	−6852.72	−0.345308	−0.172654	+	0.984982i	$0.555234\pi$
$734$	−10864.2	−0.546327
$735$	0	0
$736$	17993.6	0.901161
$737$	1714.95	0.0857139
$738$	−6448.53	−0.321645
$739$	−9133.20	−0.454628	−0.227314	−	0.973821i	$-0.572994\pi$
$739$	−9133.20	−0.454628	−0.227314	+	0.973821i	$0.572994\pi$
$740$	3945.36	0.195992
$741$	−160.093	−0.00793681
$742$	0	0
$743$	−33458.7	−1.65206	−0.826031	−	0.563624i	$-0.809407\pi$
$743$	−33458.7	−1.65206	−0.826031	+	0.563624i	$0.809407\pi$
$744$	7680.99	0.378493
$745$	7774.60	0.382335
$746$	19442.8	0.954225
$747$	1409.75	0.0690496
$748$	4548.45	0.222337
$749$	0	0
$750$	20746.9	1.01010
$751$	−37530.0	−1.82355	−0.911776	−	0.410688i	$-0.865289\pi$
$751$	−37530.0	−1.82355	−0.911776	+	0.410688i	$0.865289\pi$
$752$	−4813.44	−0.233415
$753$	32302.4	1.56330
$754$	−67.5062	−0.00326052
$755$	−7732.10	−0.372715
$756$	0	0
$757$	32174.8	1.54480	0.772399	−	0.635138i	$-0.219057\pi$
$757$	32174.8	1.54480	0.772399	+	0.635138i	$0.219057\pi$
$758$	2667.88	0.127839
$759$	9513.80	0.454979
$760$	26444.7	1.26217
$761$	−4457.43	−0.212328	−0.106164	−	0.994349i	$-0.533857\pi$
$761$	−4457.43	−0.212328	−0.106164	+	0.994349i	$0.533857\pi$
$762$	18470.8	0.878121
$763$	0	0
$764$	9184.82	0.434941
$765$	14758.2	0.697497
$766$	246.498	0.0116271
$767$	33.8329	0.00159274
$768$	26132.9	1.22785
$769$	38329.0	1.79737	0.898687	−	0.438591i	$-0.144522\pi$
$769$	38329.0	1.79737	0.898687	+	0.438591i	$0.144522\pi$
$770$	0	0
$771$	17678.5	0.825779
$772$	−6494.32	−0.302766
$773$	9529.23	0.443393	0.221696	−	0.975116i	$-0.428841\pi$
$773$	9529.23	0.443393	0.221696	+	0.975116i	$0.428841\pi$
$774$	16095.8	0.747481
$775$	3240.65	0.150203
$776$	−23700.3	−1.09638
$777$	0	0
$778$	−27703.0	−1.27661
$779$	−27726.1	−1.27521
$780$	27.5821	0.00126615
$781$	−4124.93	−0.188991
$782$	−37618.7	−1.72026
$783$	−13711.4	−0.625806
$784$	0	0
$785$	−12023.4	−0.546667
$786$	−35802.5	−1.62472
$787$	−29705.1	−1.34545	−0.672726	−	0.739892i	$-0.734877\pi$
$787$	−29705.1	−1.34545	−0.672726	+	0.739892i	$0.734877\pi$
$788$	15299.0	0.691629
$789$	27423.9	1.23741
$790$	−116.664	−0.00525408
$791$	0	0
$792$	4063.00	0.182288
$793$	101.931	0.00456451
$794$	8114.48	0.362685
$795$	−21286.2	−0.949617
$796$	11380.7	0.506758
$797$	−8596.99	−0.382084	−0.191042	−	0.981582i	$-0.561187\pi$
$797$	−8596.99	−0.382084	−0.191042	+	0.981582i	$0.561187\pi$
$798$	0	0
$799$	−22126.4	−0.979695
$800$	9056.69	0.400253
$801$	4623.59	0.203953
$802$	−12413.3	−0.546543
$803$	−2314.61	−0.101719
$804$	3244.22	0.142307
$805$	0	0
$806$	−18.4117	−0.000804620 0
$807$	−15538.2	−0.677783
$808$	14180.7	0.617422
$809$	16630.2	0.722727	0.361364	−	0.932425i	$-0.382311\pi$
$809$	16630.2	0.722727	0.361364	+	0.932425i	$0.382311\pi$
$810$	−15133.5	−0.656467
$811$	−18584.2	−0.804660	−0.402330	−	0.915495i	$-0.631799\pi$
$811$	−18584.2	−0.804660	−0.402330	+	0.915495i	$0.631799\pi$
$812$	0	0
$813$	−15963.9	−0.688657
$814$	−3891.56	−0.167567
$815$	−5845.18	−0.251224
$816$	−23426.5	−1.00501
$817$	69205.5	2.96352
$818$	570.127	0.0243692
$819$	0	0
$820$	4776.87	0.203434
$821$	−4088.88	−0.173816	−0.0869079	−	0.996216i	$-0.527699\pi$
$821$	−4088.88	−0.173816	−0.0869079	+	0.996216i	$0.527699\pi$
$822$	36547.3	1.55077
$823$	−12830.0	−0.543410	−0.271705	−	0.962381i	$-0.587587\pi$
$823$	−12830.0	−0.543410	−0.271705	+	0.962381i	$0.587587\pi$
$824$	−3268.80	−0.138197
$825$	4788.55	0.202080
$826$	0	0
$827$	−13768.7	−0.578942	−0.289471	−	0.957187i	$-0.593479\pi$
$827$	−13768.7	−0.578942	−0.289471	+	0.957187i	$0.593479\pi$
$828$	6442.73	0.270411
$829$	10907.8	0.456987	0.228494	−	0.973545i	$-0.426620\pi$
$829$	10907.8	0.456987	0.228494	+	0.973545i	$0.426620\pi$
$830$	1559.28	0.0652091
$831$	−38199.9	−1.59463
$832$	−90.5308	−0.00377234
$833$	0	0
$834$	38914.9	1.61572
$835$	23527.5	0.975093
$836$	5001.03	0.206895
$837$	−3739.66	−0.154434
$838$	9092.53	0.374817
$839$	27174.8	1.11821	0.559105	−	0.829097i	$-0.311145\pi$
$839$	27174.8	1.11821	0.559105	+	0.829097i	$0.311145\pi$
$840$	0	0
$841$	6940.81	0.284588
$842$	−9281.09	−0.379866
$843$	12738.2	0.520434
$844$	−4663.97	−0.190214
$845$	16713.1	0.680414
$846$	−5658.20	−0.229944
$847$	0	0
$848$	12095.7	0.489819
$849$	55210.3	2.23182
$850$	−18934.5	−0.764056
$851$	−21555.7	−0.868298
$852$	−7803.23	−0.313773
$853$	33975.1	1.36376	0.681878	−	0.731466i	$-0.261163\pi$
$853$	33975.1	1.36376	0.681878	+	0.731466i	$0.261163\pi$
$854$	0	0
$855$	16226.7	0.649054
$856$	204.528	0.00816662
$857$	−20423.7	−0.814073	−0.407037	−	0.913412i	$-0.633438\pi$
$857$	−20423.7	−0.814073	−0.407037	+	0.913412i	$0.633438\pi$
$858$	−27.2060	−0.00108251
$859$	36704.2	1.45789	0.728947	−	0.684570i	$-0.240010\pi$
$859$	36704.2	1.45789	0.728947	+	0.684570i	$0.240010\pi$
$860$	−11923.2	−0.472766
$861$	0	0
$862$	−2215.51	−0.0875413
$863$	26222.8	1.03434	0.517170	−	0.855883i	$-0.326985\pi$
$863$	26222.8	1.03434	0.517170	+	0.855883i	$0.326985\pi$
$864$	−10451.3	−0.411527
$865$	−10098.5	−0.396945
$866$	3512.06	0.137811
$867$	−75826.4	−2.97024
$868$	0	0
$869$	−77.0680	−0.00300846
$870$	19113.6	0.744841
$871$	−27.1645	−0.00105676
$872$	−7778.79	−0.302091
$873$	−14542.7	−0.563799
$874$	−41361.8	−1.60078
$875$	0	0
$876$	−4378.60	−0.168880
$877$	44950.8	1.73076	0.865382	−	0.501113i	$-0.167076\pi$
$877$	44950.8	1.73076	0.865382	+	0.501113i	$0.167076\pi$
$878$	23358.8	0.897861
$879$	33848.8	1.29885
$880$	2345.84	0.0898617
$881$	−3896.87	−0.149023	−0.0745113	−	0.997220i	$-0.523740\pi$
$881$	−3896.87	−0.149023	−0.0745113	+	0.997220i	$0.523740\pi$
$882$	0	0
$883$	6257.41	0.238481	0.119240	−	0.992865i	$-0.461954\pi$
$883$	6257.41	0.238481	0.119240	+	0.992865i	$0.461954\pi$
$884$	−72.0466	−0.00274117
$885$	−9579.38	−0.363850
$886$	−4801.83	−0.182077
$887$	−40626.2	−1.53787	−0.768937	−	0.639324i	$-0.779214\pi$
$887$	−40626.2	−1.53787	−0.768937	+	0.639324i	$0.779214\pi$
$888$	−25715.7	−0.971803
$889$	0	0
$890$	5114.02	0.192610
$891$	−9997.16	−0.375889
$892$	20665.0	0.775691
$893$	−24328.0	−0.911653
$894$	−14506.9	−0.542709
$895$	−23897.5	−0.892521
$896$	0	0
$897$	−150.697	−0.00560938
$898$	15476.2	0.575108
$899$	8544.91	0.317006
$900$	3242.80	0.120104
$901$	55601.4	2.05588
$902$	−4711.73	−0.173929
$903$	0	0
$904$	−53240.5	−1.95880
$905$	−28020.6	−1.02921
$906$	14427.5	0.529054
$907$	42366.9	1.55101	0.775506	−	0.631340i	$-0.217495\pi$
$907$	42366.9	1.55101	0.775506	+	0.631340i	$0.217495\pi$
$908$	−4213.85	−0.154010
$909$	8701.42	0.317501
$910$	0	0
$911$	26231.4	0.953992	0.476996	−	0.878906i	$-0.341726\pi$
$911$	26231.4	0.953992	0.476996	+	0.878906i	$0.341726\pi$
$912$	−25757.4	−0.935213
$913$	1030.06	0.0373384
$914$	−35560.1	−1.28690
$915$	−28860.5	−1.04273
$916$	−12129.3	−0.437514
$917$	0	0
$918$	21850.1	0.785578
$919$	2542.24	0.0912522	0.0456261	−	0.998959i	$-0.485472\pi$
$919$	2542.24	0.0912522	0.0456261	+	0.998959i	$0.485472\pi$
$920$	24892.5	0.892047
$921$	18724.0	0.669898
$922$	−11459.2	−0.409314
$923$	65.3381	0.00233004
$924$	0	0
$925$	−10849.6	−0.385656
$926$	24283.2	0.861764
$927$	−2005.76	−0.0710657
$928$	23880.5	0.844739
$929$	20097.8	0.709782	0.354891	−	0.934908i	$-0.384518\pi$
$929$	20097.8	0.709782	0.354891	+	0.934908i	$0.384518\pi$
$930$	5213.05	0.183809
$931$	0	0
$932$	−7452.92	−0.261941
$933$	−38111.5	−1.33731
$934$	−17102.0	−0.599138
$935$	10783.4	0.377170
$936$	−64.3571	−0.00224741
$937$	202.788	0.00707023	0.00353511	−	0.999994i	$-0.498875\pi$
$937$	202.788	0.00707023	0.00353511	+	0.999994i	$0.498875\pi$
$938$	0	0
$939$	−49824.9	−1.73160
$940$	4191.41	0.145435
$941$	−12419.1	−0.430235	−0.215118	−	0.976588i	$-0.569013\pi$
$941$	−12419.1	−0.430235	−0.215118	+	0.976588i	$0.569013\pi$
$942$	22434.8	0.775972
$943$	−26098.8	−0.901265
$944$	5443.37	0.187677
$945$	0	0
$946$	11760.7	0.404199
$947$	369.282	0.0126717	0.00633583	−	0.999980i	$-0.497983\pi$
$947$	369.282	0.0126717	0.00633583	+	0.999980i	$0.497983\pi$
$948$	−145.791	−0.00499481
$949$	36.6629	0.00125409
$950$	−20818.5	−0.710990
$951$	62530.1	2.13215
$952$	0	0
$953$	25994.1	0.883558	0.441779	−	0.897124i	$-0.354348\pi$
$953$	25994.1	0.883558	0.441779	+	0.897124i	$0.354348\pi$
$954$	14218.5	0.482536
$955$	21775.2	0.737830
$956$	−7074.72	−0.239344
$957$	12626.4	0.426492
$958$	11597.9	0.391138
$959$	0	0
$960$	25632.8	0.861764
$961$	−27460.5	−0.921770
$962$	61.6416	0.00206591
$963$	125.500	0.00419957
$964$	−14793.8	−0.494271
$965$	−15396.6	−0.513610
$966$	0	0
$967$	−53830.0	−1.79013	−0.895065	−	0.445937i	$-0.852871\pi$
$967$	−53830.0	−1.79013	−0.895065	+	0.445937i	$0.852871\pi$
$968$	2968.70	0.0985720
$969$	−118402.	−3.92530
$970$	−16085.3	−0.532441
$971$	1349.54	0.0446022	0.0223011	−	0.999751i	$-0.492901\pi$
$971$	1349.54	0.0446022	0.0223011	+	0.999751i	$0.492901\pi$
$972$	−12200.5	−0.402605
$973$	0	0
$974$	−12215.9	−0.401872
$975$	−75.8497	−0.00249142
$976$	16399.6	0.537847
$977$	6611.71	0.216507	0.108253	−	0.994123i	$-0.465474\pi$
$977$	6611.71	0.216507	0.108253	+	0.994123i	$0.465474\pi$
$978$	10906.7	0.356603
$979$	3378.31	0.110287
$980$	0	0
$981$	−4773.13	−0.155346
$982$	−36595.5	−1.18922
$983$	−40595.9	−1.31720	−0.658601	−	0.752493i	$-0.728851\pi$
$983$	−40595.9	−1.31720	−0.658601	+	0.752493i	$0.728851\pi$
$984$	−31135.4	−1.00870
$985$	36270.5	1.17327
$986$	−49926.2	−1.61255
$987$	0	0
$988$	−79.2153	−0.00255078
$989$	65143.5	2.09448
$990$	2757.54	0.0885256
$991$	−48877.7	−1.56675	−0.783376	−	0.621548i	$-0.786504\pi$
$991$	−48877.7	−1.56675	−0.783376	+	0.621548i	$0.786504\pi$
$992$	6513.19	0.208462
$993$	13423.0	0.428970
$994$	0	0
$995$	26981.2	0.859660
$996$	1948.59	0.0619913
$997$	−13127.4	−0.417000	−0.208500	−	0.978022i	$-0.566858\pi$
$997$	−13127.4	−0.417000	−0.208500	+	0.978022i	$0.566858\pi$
$998$	−42889.7	−1.36037
$999$	12520.2	0.396520

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.4.a.h.1.2		5
7.6	odd	2		77.4.a.e.1.2	✓	5
21.20	even	2		693.4.a.o.1.4		5
28.27	even	2		1232.4.a.y.1.2		5
35.34	odd	2		1925.4.a.r.1.4		5
77.76	even	2		847.4.a.f.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.e.1.2	✓	5	7.6	odd	2
539.4.a.h.1.2		5	1.1	even	1	trivial
693.4.a.o.1.4		5	21.20	even	2
847.4.a.f.1.4		5	77.76	even	2
1232.4.a.y.1.2		5	28.27	even	2
1925.4.a.r.1.4		5	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q-2.18888 q^{2} -6.48496 q^{3} -3.20880 q^{4} -7.60736 q^{5} +14.1948 q^{6} +24.5347 q^{8} +15.0547 q^{9} +16.6516 q^{10} +11.0000 q^{11} +20.8090 q^{12} -0.174238 q^{13} +49.3335 q^{15} -28.0332 q^{16} -128.863 q^{17} -32.9530 q^{18} -141.685 q^{19} +24.4105 q^{20} -24.0777 q^{22} -133.369 q^{23} -159.107 q^{24} -67.1280 q^{25} +0.381386 q^{26} +77.4647 q^{27} -177.002 q^{29} -107.985 q^{30} -48.2757 q^{31} -134.917 q^{32} -71.3346 q^{33} +282.065 q^{34} -48.3076 q^{36} +161.625 q^{37} +310.131 q^{38} +1.12993 q^{39} -186.645 q^{40} +195.689 q^{41} -488.447 q^{43} -35.2968 q^{44} -114.527 q^{45} +291.928 q^{46} +171.705 q^{47} +181.794 q^{48} +146.935 q^{50} +835.671 q^{51} +0.559095 q^{52} -431.477 q^{53} -169.561 q^{54} -83.6810 q^{55} +918.821 q^{57} +387.437 q^{58} -194.176 q^{59} -158.301 q^{60} -585.008 q^{61} +105.670 q^{62} +519.582 q^{64} +1.32549 q^{65} +156.143 q^{66} +155.905 q^{67} +413.496 q^{68} +864.891 q^{69} -374.994 q^{71} +369.364 q^{72} -210.419 q^{73} -353.778 q^{74} +435.323 q^{75} +454.639 q^{76} -2.47327 q^{78} -7.00618 q^{79} +213.258 q^{80} -908.833 q^{81} -428.339 q^{82} +93.6417 q^{83} +980.307 q^{85} +1069.15 q^{86} +1147.85 q^{87} +269.882 q^{88} +307.119 q^{89} +250.685 q^{90} +427.954 q^{92} +313.066 q^{93} -375.842 q^{94} +1077.85 q^{95} +874.929 q^{96} -965.991 q^{97} +165.602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} - 2 q^{3} + 45 q^{4} + 24 q^{5} - 4 q^{6} + 57 q^{8} + 63 q^{9} + 10 q^{10} + 55 q^{11} - 24 q^{12} + 50 q^{13} - 146 q^{15} + 433 q^{16} - 222 q^{17} + 245 q^{18} - 160 q^{19} + 430 q^{20}+ \cdots + 693 q^{99}+O(q^{100}) \) 5 * q + q^2 - 2 * q^3 + 45 * q^4 + 24 * q^5 - 4 * q^6 + 57 * q^8 + 63 * q^9 + 10 * q^10 + 55 * q^11 - 24 * q^12 + 50 * q^13 - 146 * q^15 + 433 * q^16 - 222 * q^17 + 245 * q^18 - 160 * q^19 + 430 * q^20 + 11 * q^22 + 54 * q^23 + 220 * q^24 + 125 * q^25 + 1026 * q^26 - 110 * q^27 + 14 * q^29 - 1080 * q^30 + 34 * q^31 - 583 * q^32 - 22 * q^33 + 750 * q^34 - 411 * q^36 + 1044 * q^37 + 156 * q^38 - 124 * q^39 - 1158 * q^40 + 114 * q^41 + 672 * q^43 + 495 * q^44 + 530 * q^45 - 1224 * q^46 + 292 * q^47 + 1652 * q^48 + 1143 * q^50 + 768 * q^51 + 914 * q^52 - 710 * q^53 - 1608 * q^54 + 264 * q^55 + 2012 * q^57 - 810 * q^58 - 270 * q^59 - 3068 * q^60 - 138 * q^61 + 480 * q^62 + 3001 * q^64 - 196 * q^65 - 44 * q^66 + 1942 * q^67 - 3130 * q^68 + 1306 * q^69 - 278 * q^71 + 3565 * q^72 + 338 * q^73 + 462 * q^74 - 3960 * q^75 + 1332 * q^76 + 280 * q^78 + 576 * q^79 + 1602 * q^80 - 1439 * q^81 - 1386 * q^82 - 1644 * q^83 + 360 * q^85 + 2020 * q^86 + 1956 * q^87 + 627 * q^88 + 3656 * q^89 + 6162 * q^90 - 748 * q^92 - 1442 * q^93 - 976 * q^94 - 1276 * q^95 + 4508 * q^96 - 692 * q^97 + 693 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more