LMFDB - Embedded newform 1519.4.a.h.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1519,4,Mod(1,1519)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1519, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1519.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1519 = 7^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1519.a (trivial)

Newform invariants

comment:select newform

sage:traces = [23,5,-6,91,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$89.6239012987$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	1519.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.88515 q^{2} -5.49288 q^{3} -4.44620 q^{4} +9.32401 q^{5} +10.3549 q^{6} +23.4630 q^{8} +3.17171 q^{9} -17.5772 q^{10} -66.7201 q^{11} +24.4224 q^{12} +36.7478 q^{13} -51.2156 q^{15} -8.66168 q^{16} +13.0398 q^{17} -5.97915 q^{18} +72.1659 q^{19} -41.4564 q^{20} +125.777 q^{22} +2.75035 q^{23} -128.879 q^{24} -38.0629 q^{25} -69.2752 q^{26} +130.886 q^{27} -98.2040 q^{29} +96.5492 q^{30} -31.0000 q^{31} -171.375 q^{32} +366.485 q^{33} -24.5820 q^{34} -14.1020 q^{36} -192.099 q^{37} -136.044 q^{38} -201.851 q^{39} +218.769 q^{40} -133.653 q^{41} +511.611 q^{43} +296.651 q^{44} +29.5730 q^{45} -5.18483 q^{46} -392.730 q^{47} +47.5775 q^{48} +71.7544 q^{50} -71.6260 q^{51} -163.388 q^{52} -498.188 q^{53} -246.740 q^{54} -622.098 q^{55} -396.398 q^{57} +185.130 q^{58} +567.336 q^{59} +227.715 q^{60} +814.950 q^{61} +58.4397 q^{62} +392.362 q^{64} +342.637 q^{65} -690.880 q^{66} +676.291 q^{67} -57.9775 q^{68} -15.1073 q^{69} +517.685 q^{71} +74.4177 q^{72} +492.415 q^{73} +362.136 q^{74} +209.075 q^{75} -320.864 q^{76} +380.520 q^{78} -173.504 q^{79} -80.7615 q^{80} -804.576 q^{81} +251.955 q^{82} -891.483 q^{83} +121.583 q^{85} -964.464 q^{86} +539.423 q^{87} -1565.45 q^{88} -595.383 q^{89} -55.7496 q^{90} -12.2286 q^{92} +170.279 q^{93} +740.356 q^{94} +672.875 q^{95} +941.343 q^{96} +1855.96 q^{97} -211.616 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 5 q^{2} - 6 q^{3} + 91 q^{4} - 40 q^{5} - 36 q^{6} + 39 q^{8} + 211 q^{9} - 40 q^{10} + 44 q^{11} - 414 q^{12} + 20 q^{13} + 523 q^{16} - 306 q^{17} + 51 q^{18} - 296 q^{19} - 400 q^{20} - 326 q^{22}+ \cdots - 3456 q^{99}+O(q^{100}) $ 23 * q + 5 * q^2 - 6 * q^3 + 91 * q^4 - 40 * q^5 - 36 * q^6 + 39 * q^8 + 211 * q^9 - 40 * q^10 + 44 * q^11 - 414 * q^12 + 20 * q^13 + 523 * q^16 - 306 * q^17 + 51 * q^18 - 296 * q^19 - 400 * q^20 - 326 * q^22 - 130 * q^23 - 432 * q^24 + 801 * q^25 - 474 * q^26 - 192 * q^27 - 16 * q^29 - 74 * q^30 - 713 * q^31 + 287 * q^32 - 488 * q^33 - 692 * q^34 + 623 * q^36 - 296 * q^37 + 40 * q^38 - 124 * q^39 - 48 * q^40 - 2344 * q^41 - 48 * q^43 + 1352 * q^44 - 132 * q^45 - 836 * q^46 - 436 * q^47 - 4190 * q^48 - 767 * q^50 - 872 * q^51 + 632 * q^52 - 884 * q^53 - 100 * q^54 - 3308 * q^55 + 404 * q^57 - 562 * q^58 - 400 * q^59 + 2248 * q^60 - 860 * q^61 - 155 * q^62 + 3125 * q^64 - 1676 * q^65 - 128 * q^66 - 164 * q^67 - 302 * q^68 - 2136 * q^69 + 56 * q^71 + 1613 * q^72 - 1610 * q^73 + 908 * q^74 + 910 * q^75 - 5208 * q^76 + 1424 * q^78 + 318 * q^79 - 11480 * q^80 + 231 * q^81 + 2608 * q^82 - 3166 * q^83 - 716 * q^85 - 224 * q^86 - 1248 * q^87 - 5562 * q^88 - 2854 * q^89 - 7172 * q^90 - 3558 * q^92 + 186 * q^93 - 4228 * q^94 + 984 * q^95 + 4178 * q^96 - 3332 * q^97 - 3456 * 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 $

$2$	−1.88515	−0.666502	−0.333251	−	0.942838i	$-0.608146\pi$
$2$	−1.88515	−0.666502	−0.333251	+	0.942838i	$0.608146\pi$
$3$	−5.49288	−1.05710	−0.528552	−	0.848901i	$-0.677265\pi$
$3$	−5.49288	−1.05710	−0.528552	+	0.848901i	$0.677265\pi$
$4$	−4.44620	−0.555775
$5$	9.32401	0.833964	0.416982	−	0.908915i	$-0.363088\pi$
$5$	9.32401	0.833964	0.416982	+	0.908915i	$0.363088\pi$
$6$	10.3549	0.704562
$7$	0	0
$7$	0	0
$8$	23.4630	1.03693
$9$	3.17171	0.117471
$10$	−17.5772	−0.555839
$11$	−66.7201	−1.82881	−0.914403	−	0.404806i	$-0.867339\pi$
$11$	−66.7201	−1.82881	−0.914403	+	0.404806i	$0.867339\pi$
$12$	24.4224	0.587513
$13$	36.7478	0.784001	0.392000	−	0.919965i	$-0.371783\pi$
$13$	36.7478	0.784001	0.392000	+	0.919965i	$0.371783\pi$
$14$	0	0
$15$	−51.2156	−0.881588
$16$	−8.66168	−0.135339
$17$	13.0398	0.186036	0.0930180	−	0.995664i	$-0.470349\pi$
$17$	13.0398	0.186036	0.0930180	+	0.995664i	$0.470349\pi$
$18$	−5.97915	−0.0782944
$19$	72.1659	0.871368	0.435684	−	0.900100i	$-0.356507\pi$
$19$	72.1659	0.871368	0.435684	+	0.900100i	$0.356507\pi$
$20$	−41.4564	−0.463497
$21$	0	0
$22$	125.777	1.21890
$23$	2.75035	0.0249342	0.0124671	−	0.999922i	$-0.496031\pi$
$23$	2.75035	0.0249342	0.0124671	+	0.999922i	$0.496031\pi$
$24$	−128.879	−1.09614
$25$	−38.0629	−0.304503
$26$	−69.2752	−0.522538
$27$	130.886	0.932926
$28$	0	0
$29$	−98.2040	−0.628829	−0.314414	−	0.949286i	$-0.601808\pi$
$29$	−98.2040	−0.628829	−0.314414	+	0.949286i	$0.601808\pi$
$30$	96.5492	0.587580
$31$	−31.0000	−0.179605
$31$	−31.0000	−0.179605
$32$	−171.375	−0.946724
$33$	366.485	1.93324
$34$	−24.5820	−0.123993
$35$	0	0
$36$	−14.1020	−0.0652872
$37$	−192.099	−0.853538	−0.426769	−	0.904361i	$-0.640348\pi$
$37$	−192.099	−0.853538	−0.426769	+	0.904361i	$0.640348\pi$
$38$	−136.044	−0.580768
$39$	−201.851	−0.828771
$40$	218.769	0.864760
$41$	−133.653	−0.509098	−0.254549	−	0.967060i	$-0.581927\pi$
$41$	−133.653	−0.509098	−0.254549	+	0.967060i	$0.581927\pi$
$42$	0	0
$43$	511.611	1.81442	0.907208	−	0.420681i	$-0.138209\pi$
$43$	511.611	1.81442	0.907208	+	0.420681i	$0.138209\pi$
$44$	296.651	1.01640
$45$	29.5730	0.0979663
$46$	−5.18483	−0.0166187
$47$	−392.730	−1.21884	−0.609421	−	0.792847i	$-0.708598\pi$
$47$	−392.730	−1.21884	−0.609421	+	0.792847i	$0.708598\pi$
$48$	47.5775	0.143067
$49$	0	0
$50$	71.7544	0.202952
$51$	−71.6260	−0.196660
$52$	−163.388	−0.435728
$53$	−498.188	−1.29116	−0.645579	−	0.763693i	$-0.723384\pi$
$53$	−498.188	−1.29116	−0.645579	+	0.763693i	$0.723384\pi$
$54$	−246.740	−0.621797
$55$	−622.098	−1.52516
$56$	0	0
$57$	−396.398	−0.921127
$58$	185.130	0.419116
$59$	567.336	1.25188	0.625939	−	0.779872i	$-0.284716\pi$
$59$	567.336	1.25188	0.625939	+	0.779872i	$0.284716\pi$
$60$	227.715	0.489965
$61$	814.950	1.71055	0.855276	−	0.518173i	$-0.173388\pi$
$61$	814.950	1.71055	0.855276	+	0.518173i	$0.173388\pi$
$62$	58.4397	0.119707
$63$	0	0
$64$	392.362	0.766332
$65$	342.637	0.653829
$66$	−690.880	−1.28851
$67$	676.291	1.23317	0.616583	−	0.787290i	$-0.288516\pi$
$67$	676.291	1.23317	0.616583	+	0.787290i	$0.288516\pi$
$68$	−57.9775	−0.103394
$69$	−15.1073	−0.0263581
$70$	0	0
$71$	517.685	0.865323	0.432661	−	0.901556i	$-0.357575\pi$
$71$	517.685	0.865323	0.432661	+	0.901556i	$0.357575\pi$
$72$	74.4177	0.121808
$73$	492.415	0.789491	0.394746	−	0.918790i	$-0.370833\pi$
$73$	492.415	0.789491	0.394746	+	0.918790i	$0.370833\pi$
$74$	362.136	0.568885
$75$	209.075	0.321892
$76$	−320.864	−0.484285
$77$	0	0
$78$	380.520	0.552377
$79$	−173.504	−0.247097	−0.123549	−	0.992339i	$-0.539427\pi$
$79$	−173.504	−0.247097	−0.123549	+	0.992339i	$0.539427\pi$
$80$	−80.7615	−0.112868
$81$	−804.576	−1.10367
$82$	251.955	0.339315
$83$	−891.483	−1.17895	−0.589475	−	0.807786i	$-0.700666\pi$
$83$	−891.483	−1.17895	−0.589475	+	0.807786i	$0.700666\pi$
$84$	0	0
$85$	121.583	0.155147
$86$	−964.464	−1.20931
$87$	539.423	0.664738
$88$	−1565.45	−1.89634
$89$	−595.383	−0.709107	−0.354553	−	0.935036i	$-0.615367\pi$
$89$	−595.383	−0.709107	−0.354553	+	0.935036i	$0.615367\pi$
$90$	−55.7496	−0.0652947
$91$	0	0
$92$	−12.2286	−0.0138578
$93$	170.279	0.189862
$94$	740.356	0.812361
$95$	672.875	0.726690
$96$	941.343	1.00079
$97$	1855.96	1.94273	0.971364	−	0.237596i	$-0.0763596\pi$
$97$	1855.96	1.94273	0.971364	+	0.237596i	$0.0763596\pi$
$98$	0	0
$99$	−211.616	−0.214831
$100$	169.235	0.169235
$101$	995.030	0.980289	0.490145	−	0.871641i	$-0.336944\pi$
$101$	995.030	0.980289	0.490145	+	0.871641i	$0.336944\pi$
$102$	135.026	0.131074
$103$	425.212	0.406770	0.203385	−	0.979099i	$-0.434806\pi$
$103$	425.212	0.406770	0.203385	+	0.979099i	$0.434806\pi$
$104$	862.213	0.812952
$105$	0	0
$106$	939.160	0.860559
$107$	181.535	0.164016	0.0820078	−	0.996632i	$-0.473867\pi$
$107$	181.535	0.164016	0.0820078	+	0.996632i	$0.473867\pi$
$108$	−581.945	−0.518497
$109$	460.949	0.405054	0.202527	−	0.979277i	$-0.435085\pi$
$109$	460.949	0.405054	0.202527	+	0.979277i	$0.435085\pi$
$110$	1172.75	1.01652
$111$	1055.18	0.902279
$112$	0	0
$113$	209.552	0.174451	0.0872256	−	0.996189i	$-0.472200\pi$
$113$	209.552	0.174451	0.0872256	+	0.996189i	$0.472200\pi$
$114$	747.271	0.613933
$115$	25.6443	0.0207943
$116$	436.635	0.349487
$117$	116.553	0.0920970
$118$	−1069.51	−0.834379
$119$	0	0
$120$	−1201.67	−0.914142
$121$	3120.57	2.34453
$122$	−1536.30	−1.14009
$123$	734.137	0.538170
$124$	137.832	0.0998202
$125$	−1520.40	−1.08791
$126$	0	0
$127$	722.469	0.504793	0.252397	−	0.967624i	$-0.418781\pi$
$127$	722.469	0.504793	0.252397	+	0.967624i	$0.418781\pi$
$128$	631.340	0.435962
$129$	−2810.22	−1.91803
$130$	−645.922	−0.435778
$131$	1959.74	1.30705	0.653525	−	0.756905i	$-0.273289\pi$
$131$	1959.74	1.30705	0.653525	+	0.756905i	$0.273289\pi$
$132$	−1629.47	−1.07445
$133$	0	0
$134$	−1274.91	−0.821908
$135$	1220.38	0.778027
$136$	305.952	0.192906
$137$	226.311	0.141132	0.0705660	−	0.997507i	$-0.477519\pi$
$137$	226.311	0.141132	0.0705660	+	0.997507i	$0.477519\pi$
$138$	28.4796	0.0175677
$139$	−2927.74	−1.78653	−0.893265	−	0.449531i	$-0.851591\pi$
$139$	−2927.74	−1.78653	−0.893265	+	0.449531i	$0.851591\pi$
$140$	0	0
$141$	2157.22	1.28844
$142$	−975.915	−0.576739
$143$	−2451.82	−1.43378
$144$	−27.4723	−0.0158983
$145$	−915.655	−0.524421
$146$	−928.278	−0.526197
$147$	0	0
$148$	854.112	0.474375
$149$	−1863.95	−1.02484	−0.512419	−	0.858736i	$-0.671250\pi$
$149$	−1863.95	−1.02484	−0.512419	+	0.858736i	$0.671250\pi$
$150$	−394.138	−0.214542
$151$	1135.32	0.611862	0.305931	−	0.952054i	$-0.401032\pi$
$151$	1135.32	0.611862	0.305931	+	0.952054i	$0.401032\pi$
$152$	1693.23	0.903545
$153$	41.3584	0.0218538
$154$	0	0
$155$	−289.044	−0.149784
$156$	897.471	0.460610
$157$	−1325.81	−0.673954	−0.336977	−	0.941513i	$-0.609405\pi$
$157$	−1325.81	−0.673954	−0.336977	+	0.941513i	$0.609405\pi$
$158$	327.081	0.164691
$159$	2736.49	1.36489
$160$	−1597.90	−0.789534
$161$	0	0
$162$	1516.75	0.735599
$163$	−2827.27	−1.35858	−0.679292	−	0.733868i	$-0.737713\pi$
$163$	−2827.27	−1.35858	−0.679292	+	0.733868i	$0.737713\pi$
$164$	594.246	0.282944
$165$	3417.11	1.61225
$166$	1680.58	0.785773
$167$	−858.549	−0.397824	−0.198912	−	0.980017i	$-0.563741\pi$
$167$	−858.549	−0.397824	−0.198912	+	0.980017i	$0.563741\pi$
$168$	0	0
$169$	−846.599	−0.385343
$170$	−229.203	−0.103406
$171$	228.889	0.102360
$172$	−2274.73	−1.00841
$173$	−1486.81	−0.653411	−0.326705	−	0.945126i	$-0.605939\pi$
$173$	−1486.81	−0.653411	−0.326705	+	0.945126i	$0.605939\pi$
$174$	−1016.89	−0.443049
$175$	0	0
$176$	577.908	0.247508
$177$	−3116.31	−1.32337
$178$	1122.39	0.472621
$179$	1847.83	0.771584	0.385792	−	0.922586i	$-0.373928\pi$
$179$	1847.83	0.771584	0.385792	+	0.922586i	$0.373928\pi$
$180$	−131.488	−0.0544472
$181$	−4676.40	−1.92041	−0.960205	−	0.279296i	$-0.909899\pi$
$181$	−4676.40	−1.92041	−0.960205	+	0.279296i	$0.909899\pi$
$182$	0	0
$183$	−4476.42	−1.80823
$184$	64.5314	0.0258550
$185$	−1791.13	−0.711820
$186$	−321.002	−0.126543
$187$	−870.016	−0.340224
$188$	1746.16	0.677403
$189$	0	0
$190$	−1268.47	−0.484340
$191$	2962.03	1.12212	0.561060	−	0.827775i	$-0.310394\pi$
$191$	2962.03	1.12212	0.561060	+	0.827775i	$0.310394\pi$
$192$	−2155.20	−0.810093
$193$	−543.711	−0.202783	−0.101392	−	0.994847i	$-0.532330\pi$
$193$	−543.711	−0.202783	−0.101392	+	0.994847i	$0.532330\pi$
$194$	−3498.77	−1.29483
$195$	−1882.06	−0.691165
$196$	0	0
$197$	−2923.14	−1.05718	−0.528592	−	0.848876i	$-0.677280\pi$
$197$	−2923.14	−1.05718	−0.528592	+	0.848876i	$0.677280\pi$
$198$	398.929	0.143185
$199$	2340.43	0.833712	0.416856	−	0.908973i	$-0.363132\pi$
$199$	2340.43	0.833712	0.416856	+	0.908973i	$0.363132\pi$
$200$	−893.070	−0.315748
$201$	−3714.79	−1.30359
$202$	−1875.78	−0.653365
$203$	0	0
$204$	318.464	0.109299
$205$	−1246.18	−0.424570
$206$	−801.589	−0.271113
$207$	8.72330	0.00292904
$208$	−318.298	−0.106106
$209$	−4814.91	−1.59356
$210$	0	0
$211$	−5129.22	−1.67351	−0.836754	−	0.547579i	$-0.815550\pi$
$211$	−5129.22	−1.67351	−0.836754	+	0.547579i	$0.815550\pi$
$212$	2215.04	0.717594
$213$	−2843.58	−0.914737
$214$	−342.221	−0.109317
$215$	4770.26	1.51316
$216$	3070.97	0.967376
$217$	0	0
$218$	−868.959	−0.269970
$219$	−2704.78	−0.834575
$220$	2765.97	0.847645
$221$	479.184	0.145852
$222$	−1989.17	−0.601371
$223$	−1440.31	−0.432514	−0.216257	−	0.976337i	$-0.569385\pi$
$223$	−1440.31	−0.432514	−0.216257	+	0.976337i	$0.569385\pi$
$224$	0	0
$225$	−120.724	−0.0357702
$226$	−395.037	−0.116272
$227$	−3749.47	−1.09631	−0.548153	−	0.836378i	$-0.684669\pi$
$227$	−3749.47	−1.09631	−0.548153	+	0.836378i	$0.684669\pi$
$228$	1762.47	0.511940
$229$	2418.83	0.697996	0.348998	−	0.937124i	$-0.386522\pi$
$229$	2418.83	0.697996	0.348998	+	0.937124i	$0.386522\pi$
$230$	−48.3434	−0.0138594
$231$	0	0
$232$	−2304.16	−0.652050
$233$	−4328.92	−1.21716	−0.608578	−	0.793494i	$-0.708260\pi$
$233$	−4328.92	−1.21716	−0.608578	+	0.793494i	$0.708260\pi$
$234$	−219.721	−0.0613828
$235$	−3661.82	−1.01647
$236$	−2522.49	−0.695763
$237$	953.034	0.261208
$238$	0	0
$239$	1509.57	0.408562	0.204281	−	0.978912i	$-0.434514\pi$
$239$	1509.57	0.408562	0.204281	+	0.978912i	$0.434514\pi$
$240$	443.613	0.119313
$241$	2231.17	0.596357	0.298179	−	0.954510i	$-0.403621\pi$
$241$	2231.17	0.596357	0.298179	+	0.954510i	$0.403621\pi$
$242$	−5882.74	−1.56263
$243$	885.520	0.233770
$244$	−3623.43	−0.950682
$245$	0	0
$246$	−1383.96	−0.358691
$247$	2651.94	0.683153
$248$	−727.352	−0.186238
$249$	4896.80	1.24627
$250$	2866.18	0.725094
$251$	2232.89	0.561509	0.280755	−	0.959780i	$-0.409415\pi$
$251$	2232.89	0.561509	0.280755	+	0.959780i	$0.409415\pi$
$252$	0	0
$253$	−183.504	−0.0455999
$254$	−1361.96	−0.336446
$255$	−667.841	−0.164007
$256$	−4329.07	−1.05690
$257$	−2578.99	−0.625964	−0.312982	−	0.949759i	$-0.601328\pi$
$257$	−2578.99	−0.625964	−0.312982	+	0.949759i	$0.601328\pi$
$258$	5297.68	1.27837
$259$	0	0
$260$	−1523.43	−0.363382
$261$	−311.474	−0.0738689
$262$	−3694.41	−0.871151
$263$	5906.18	1.38476	0.692378	−	0.721535i	$-0.256563\pi$
$263$	5906.18	1.38476	0.692378	+	0.721535i	$0.256563\pi$
$264$	8598.83	2.00463
$265$	−4645.11	−1.07678
$266$	0	0
$267$	3270.37	0.749600
$268$	−3006.93	−0.685363
$269$	−4279.83	−0.970059	−0.485030	−	0.874498i	$-0.661191\pi$
$269$	−4279.83	−0.970059	−0.485030	+	0.874498i	$0.661191\pi$
$270$	−2300.60	−0.518557
$271$	−594.378	−0.133232	−0.0666160	−	0.997779i	$-0.521220\pi$
$271$	−594.378	−0.133232	−0.0666160	+	0.997779i	$0.521220\pi$
$272$	−112.946	−0.0251779
$273$	0	0
$274$	−426.631	−0.0940647
$275$	2539.56	0.556877
$276$	67.1703	0.0146492
$277$	−1771.31	−0.384216	−0.192108	−	0.981374i	$-0.561532\pi$
$277$	−1771.31	−0.384216	−0.192108	+	0.981374i	$0.561532\pi$
$278$	5519.23	1.19073
$279$	−98.3229	−0.0210983
$280$	0	0
$281$	2818.49	0.598352	0.299176	−	0.954198i	$-0.403288\pi$
$281$	2818.49	0.598352	0.299176	+	0.954198i	$0.403288\pi$
$282$	−4066.69	−0.858751
$283$	−8933.66	−1.87651	−0.938253	−	0.345950i	$-0.887557\pi$
$283$	−8933.66	−1.87651	−0.938253	+	0.345950i	$0.887557\pi$
$284$	−2301.73	−0.480925
$285$	−3696.02	−0.768187
$286$	4622.05	0.955620
$287$	0	0
$288$	−543.552	−0.111212
$289$	−4742.96	−0.965391
$290$	1726.15	0.349527
$291$	−10194.6	−2.05367
$292$	−2189.38	−0.438780
$293$	148.819	0.0296727	0.0148364	−	0.999890i	$-0.495277\pi$
$293$	148.819	0.0296727	0.0148364	+	0.999890i	$0.495277\pi$
$294$	0	0
$295$	5289.84	1.04402
$296$	−4507.22	−0.885057
$297$	−8732.72	−1.70614
$298$	3513.83	0.683056
$299$	101.069	0.0195485
$300$	−929.590	−0.178900
$301$	0	0
$302$	−2140.25	−0.407807
$303$	−5465.58	−1.03627
$304$	−625.078	−0.117930
$305$	7598.60	1.42654
$306$	−77.9668	−0.0145656
$307$	5511.71	1.02466	0.512329	−	0.858789i	$-0.328783\pi$
$307$	5511.71	1.02466	0.512329	+	0.858789i	$0.328783\pi$
$308$	0	0
$309$	−2335.64	−0.429999
$310$	544.892	0.0998316
$311$	−1046.04	−0.190725	−0.0953627	−	0.995443i	$-0.530401\pi$
$311$	−1046.04	−0.190725	−0.0953627	+	0.995443i	$0.530401\pi$
$312$	−4736.03	−0.859375
$313$	−5805.52	−1.04839	−0.524197	−	0.851597i	$-0.675634\pi$
$313$	−5805.52	−1.04839	−0.524197	+	0.851597i	$0.675634\pi$
$314$	2499.35	0.449192
$315$	0	0
$316$	771.432	0.137330
$317$	−4488.12	−0.795198	−0.397599	−	0.917559i	$-0.630156\pi$
$317$	−4488.12	−0.795198	−0.397599	+	0.917559i	$0.630156\pi$
$318$	−5158.69	−0.909701
$319$	6552.18	1.15001
$320$	3658.38	0.639093
$321$	−997.150	−0.173382
$322$	0	0
$323$	941.028	0.162106
$324$	3577.31	0.613393
$325$	−1398.73	−0.238731
$326$	5329.84	0.905499
$327$	−2531.94	−0.428185
$328$	−3135.89	−0.527898
$329$	0	0
$330$	−6441.77	−1.07457
$331$	2287.24	0.379812	0.189906	−	0.981802i	$-0.439182\pi$
$331$	2287.24	0.379812	0.189906	+	0.981802i	$0.439182\pi$
$332$	3963.71	0.655232
$333$	−609.282	−0.100266
$334$	1618.50	0.265150
$335$	6305.74	1.02842
$336$	0	0
$337$	2177.51	0.351977	0.175989	−	0.984392i	$-0.443688\pi$
$337$	2177.51	0.351977	0.175989	+	0.984392i	$0.443688\pi$
$338$	1595.97	0.256832
$339$	−1151.04	−0.184413
$340$	−540.583	−0.0862271
$341$	2068.32	0.328463
$342$	−431.490	−0.0682232
$343$	0	0
$344$	12003.9	1.88142
$345$	−140.861	−0.0219817
$346$	2802.86	0.435500
$347$	160.321	0.0248025	0.0124013	−	0.999923i	$-0.496052\pi$
$347$	160.321	0.0248025	0.0124013	+	0.999923i	$0.496052\pi$
$348$	−2398.38	−0.369445
$349$	6482.45	0.994263	0.497131	−	0.867675i	$-0.334387\pi$
$349$	6482.45	0.994263	0.497131	+	0.867675i	$0.334387\pi$
$350$	0	0
$351$	4809.77	0.731415
$352$	11434.2	1.73137
$353$	−5168.68	−0.779323	−0.389662	−	0.920958i	$-0.627408\pi$
$353$	−5168.68	−0.779323	−0.389662	+	0.920958i	$0.627408\pi$
$354$	5874.71	0.882026
$355$	4826.90	0.721648
$356$	2647.19	0.394104
$357$	0	0
$358$	−3483.45	−0.514263
$359$	5121.82	0.752978	0.376489	−	0.926421i	$-0.377131\pi$
$359$	5121.82	0.752978	0.376489	+	0.926421i	$0.377131\pi$
$360$	693.871	0.101584
$361$	−1651.09	−0.240718
$362$	8815.73	1.27996
$363$	−17140.9	−2.47841
$364$	0	0
$365$	4591.28	0.658408
$366$	8438.73	1.20519
$367$	5110.55	0.726890	0.363445	−	0.931616i	$-0.381601\pi$
$367$	5110.55	0.726890	0.363445	+	0.931616i	$0.381601\pi$
$368$	−23.8226	−0.00337457
$369$	−423.907	−0.0598041
$370$	3376.56	0.474430
$371$	0	0
$372$	−757.096	−0.105520
$373$	7291.31	1.01214	0.506072	−	0.862491i	$-0.331097\pi$
$373$	7291.31	1.01214	0.506072	+	0.862491i	$0.331097\pi$
$374$	1640.11	0.226760
$375$	8351.37	1.15003
$376$	−9214.63	−1.26385
$377$	−3608.78	−0.493002
$378$	0	0
$379$	1889.91	0.256143	0.128072	−	0.991765i	$-0.459121\pi$
$379$	1889.91	0.256143	0.128072	+	0.991765i	$0.459121\pi$
$380$	−2991.74	−0.403876
$381$	−3968.43	−0.533620
$382$	−5583.88	−0.747895
$383$	−8656.98	−1.15496	−0.577482	−	0.816403i	$-0.695965\pi$
$383$	−8656.98	−1.15496	−0.577482	+	0.816403i	$0.695965\pi$
$384$	−3467.88	−0.460858
$385$	0	0
$386$	1024.98	0.135155
$387$	1622.68	0.213141
$388$	−8251.99	−1.07972
$389$	−7406.14	−0.965311	−0.482656	−	0.875810i	$-0.660328\pi$
$389$	−7406.14	−0.965311	−0.482656	+	0.875810i	$0.660328\pi$
$390$	3547.97	0.460663
$391$	35.8640	0.00463867
$392$	0	0
$393$	−10764.6	−1.38169
$394$	5510.57	0.704616
$395$	−1617.75	−0.206070
$396$	940.889	0.119398
$397$	−12868.1	−1.62678	−0.813392	−	0.581717i	$-0.802381\pi$
$397$	−12868.1	−1.62678	−0.813392	+	0.581717i	$0.802381\pi$
$398$	−4412.07	−0.555671
$399$	0	0
$400$	329.689	0.0412111
$401$	−13935.1	−1.73538	−0.867689	−	0.497108i	$-0.834395\pi$
$401$	−13935.1	−1.73538	−0.867689	+	0.497108i	$0.834395\pi$
$402$	7002.94	0.868842
$403$	−1139.18	−0.140811
$404$	−4424.11	−0.544820
$405$	−7501.87	−0.920423
$406$	0	0
$407$	12816.9	1.56096
$408$	−1680.56	−0.203922
$409$	−4223.15	−0.510566	−0.255283	−	0.966866i	$-0.582169\pi$
$409$	−4223.15	−0.510566	−0.255283	+	0.966866i	$0.582169\pi$
$410$	2349.23	0.282977
$411$	−1243.10	−0.149191
$412$	−1890.58	−0.226073
$413$	0	0
$414$	−16.4448	−0.00195221
$415$	−8312.19	−0.983203
$416$	−6297.67	−0.742232
$417$	16081.7	1.88855
$418$	9076.84	1.06211
$419$	1123.83	0.131032	0.0655162	−	0.997852i	$-0.479131\pi$
$419$	1123.83	0.131032	0.0655162	+	0.997852i	$0.479131\pi$
$420$	0	0
$421$	−5863.73	−0.678814	−0.339407	−	0.940640i	$-0.610226\pi$
$421$	−5863.73	−0.678814	−0.339407	+	0.940640i	$0.610226\pi$
$422$	9669.36	1.11540
$423$	−1245.63	−0.143178
$424$	−11689.0	−1.33884
$425$	−496.333	−0.0566486
$426$	5360.58	0.609674
$427$	0	0
$428$	−807.142	−0.0911558
$429$	13467.5	1.51566
$430$	−8992.67	−1.00852
$431$	5529.78	0.618005	0.309002	−	0.951061i	$-0.400005\pi$
$431$	5529.78	0.618005	0.309002	+	0.951061i	$0.400005\pi$
$432$	−1133.69	−0.126261
$433$	−2712.93	−0.301097	−0.150549	−	0.988603i	$-0.548104\pi$
$433$	−2712.93	−0.301097	−0.150549	+	0.988603i	$0.548104\pi$
$434$	0	0
$435$	5029.58	0.554368
$436$	−2049.47	−0.225119
$437$	198.481	0.0217269
$438$	5098.92	0.556246
$439$	−2828.22	−0.307479	−0.153740	−	0.988111i	$-0.549132\pi$
$439$	−2828.22	−0.307479	−0.153740	+	0.988111i	$0.549132\pi$
$440$	−14596.3	−1.58148
$441$	0	0
$442$	−903.334	−0.0972109
$443$	5580.20	0.598473	0.299236	−	0.954179i	$-0.403268\pi$
$443$	5580.20	0.598473	0.299236	+	0.954179i	$0.403268\pi$
$444$	−4691.53	−0.501465
$445$	−5551.36	−0.591370
$446$	2715.21	0.288271
$447$	10238.5	1.08336
$448$	0	0
$449$	14118.7	1.48397	0.741985	−	0.670417i	$-0.233885\pi$
$449$	14118.7	1.48397	0.741985	+	0.670417i	$0.233885\pi$
$450$	227.584	0.0238409
$451$	8917.31	0.931041
$452$	−931.710	−0.0969556
$453$	−6236.18	−0.646802
$454$	7068.33	0.730690
$455$	0	0
$456$	−9300.69	−0.955142
$457$	−6171.35	−0.631693	−0.315846	−	0.948810i	$-0.602288\pi$
$457$	−6171.35	−0.631693	−0.315846	+	0.948810i	$0.602288\pi$
$458$	−4559.87	−0.465215
$459$	1706.72	0.173558
$460$	−114.020	−0.0115569
$461$	3884.02	0.392401	0.196200	−	0.980564i	$-0.437140\pi$
$461$	3884.02	0.392401	0.196200	+	0.980564i	$0.437140\pi$
$462$	0	0
$463$	8129.30	0.815984	0.407992	−	0.912986i	$-0.366229\pi$
$463$	8129.30	0.815984	0.407992	+	0.912986i	$0.366229\pi$
$464$	850.612	0.0851049
$465$	1587.68	0.158338
$466$	8160.68	0.811236
$467$	2181.77	0.216189	0.108094	−	0.994141i	$-0.465525\pi$
$467$	2181.77	0.216189	0.108094	+	0.994141i	$0.465525\pi$
$468$	−518.219	−0.0511852
$469$	0	0
$470$	6903.09	0.677480
$471$	7282.49	0.712440
$472$	13311.4	1.29811
$473$	−34134.7	−3.31822
$474$	−1796.61	−0.174095
$475$	−2746.84	−0.265334
$476$	0	0
$477$	−1580.11	−0.151673
$478$	−2845.78	−0.272307
$479$	13772.5	1.31374	0.656871	−	0.754003i	$-0.271879\pi$
$479$	13772.5	1.31374	0.656871	+	0.754003i	$0.271879\pi$
$480$	8777.09	0.834620
$481$	−7059.23	−0.669175
$482$	−4206.09	−0.397473
$483$	0	0
$484$	−13874.7	−1.30303
$485$	17305.0	1.62017
$486$	−1669.34	−0.155808
$487$	−13043.6	−1.21368	−0.606838	−	0.794825i	$-0.707562\pi$
$487$	−13043.6	−1.21368	−0.606838	+	0.794825i	$0.707562\pi$
$488$	19121.2	1.77372
$489$	15529.9	1.43617
$490$	0	0
$491$	−12846.0	−1.18072	−0.590358	−	0.807141i	$-0.701014\pi$
$491$	−12846.0	−1.18072	−0.590358	+	0.807141i	$0.701014\pi$
$492$	−3264.12	−0.299102
$493$	−1280.56	−0.116985
$494$	−4999.31	−0.455323
$495$	−1973.11	−0.179161
$496$	268.512	0.0243076
$497$	0	0
$498$	−9231.22	−0.830644
$499$	−11056.3	−0.991884	−0.495942	−	0.868356i	$-0.665177\pi$
$499$	−11056.3	−0.991884	−0.495942	+	0.868356i	$0.665177\pi$
$500$	6760.00	0.604633
$501$	4715.91	0.420541
$502$	−4209.34	−0.374247
$503$	−8979.36	−0.795964	−0.397982	−	0.917393i	$-0.630289\pi$
$503$	−8979.36	−0.795964	−0.397982	+	0.917393i	$0.630289\pi$
$504$	0	0
$505$	9277.67	0.817526
$506$	345.932	0.0303924
$507$	4650.26	0.407348
$508$	−3212.24	−0.280552
$509$	−13284.8	−1.15685	−0.578426	−	0.815735i	$-0.696333\pi$
$509$	−13284.8	−1.15685	−0.578426	+	0.815735i	$0.696333\pi$
$510$	1258.98	0.109311
$511$	0	0
$512$	3110.23	0.268465
$513$	9445.50	0.812922
$514$	4861.78	0.417206
$515$	3964.68	0.339232
$516$	12494.8	1.06599
$517$	26203.0	2.22903
$518$	0	0
$519$	8166.87	0.690724
$520$	8039.28	0.677973
$521$	16425.2	1.38119	0.690596	−	0.723241i	$-0.257348\pi$
$521$	16425.2	1.38119	0.690596	+	0.723241i	$0.257348\pi$
$522$	587.176	0.0492337
$523$	1965.90	0.164365	0.0821826	−	0.996617i	$-0.473811\pi$
$523$	1965.90	0.164365	0.0821826	+	0.996617i	$0.473811\pi$
$524$	−8713.41	−0.726426
$525$	0	0
$526$	−11134.0	−0.922942
$527$	−404.233	−0.0334131
$528$	−3174.38	−0.261642
$529$	−12159.4	−0.999378
$530$	8756.73	0.717676
$531$	1799.42	0.147059
$532$	0	0
$533$	−4911.44	−0.399133
$534$	−6165.14	−0.499610
$535$	1692.63	0.136783
$536$	15867.8	1.27870
$537$	−10149.9	−0.815646
$538$	8068.13	0.646546
$539$	0	0
$540$	−5426.06	−0.432408
$541$	−10258.1	−0.815214	−0.407607	−	0.913157i	$-0.633637\pi$
$541$	−10258.1	−0.815214	−0.407607	+	0.913157i	$0.633637\pi$
$542$	1120.49	0.0887994
$543$	25686.9	2.03007
$544$	−2234.70	−0.176125
$545$	4297.89	0.337801
$546$	0	0
$547$	19483.8	1.52298	0.761488	−	0.648179i	$-0.224469\pi$
$547$	19483.8	1.52298	0.761488	+	0.648179i	$0.224469\pi$
$548$	−1006.23	−0.0784376
$549$	2584.78	0.200939
$550$	−4787.46	−0.371160
$551$	−7086.98	−0.547941
$552$	−354.463	−0.0273314
$553$	0	0
$554$	3339.19	0.256080
$555$	9838.48	0.752469
$556$	13017.3	0.992909
$557$	−22287.0	−1.69539	−0.847695	−	0.530485i	$-0.822010\pi$
$557$	−22287.0	−1.69539	−0.847695	+	0.530485i	$0.822010\pi$
$558$	185.354	0.0140621
$559$	18800.6	1.42250
$560$	0	0
$561$	4778.89	0.359652
$562$	−5313.28	−0.398802
$563$	−6643.41	−0.497312	−0.248656	−	0.968592i	$-0.579989\pi$
$563$	−6643.41	−0.497312	−0.248656	+	0.968592i	$0.579989\pi$
$564$	−9591.43	−0.716086
$565$	1953.86	0.145486
$566$	16841.3	1.25069
$567$	0	0
$568$	12146.4	0.897277
$569$	−6014.51	−0.443131	−0.221565	−	0.975146i	$-0.571117\pi$
$569$	−6014.51	−0.443131	−0.221565	+	0.975146i	$0.571117\pi$
$570$	6967.56	0.511998
$571$	−17862.5	−1.30915	−0.654574	−	0.755998i	$-0.727152\pi$
$571$	−17862.5	−1.30915	−0.654574	+	0.755998i	$0.727152\pi$
$572$	10901.3	0.796862
$573$	−16270.1	−1.18620
$574$	0	0
$575$	−104.686	−0.00759256
$576$	1244.46	0.0900214
$577$	−23123.4	−1.66835	−0.834177	−	0.551497i	$-0.814057\pi$
$577$	−23123.4	−1.66835	−0.834177	+	0.551497i	$0.814057\pi$
$578$	8941.21	0.643435
$579$	2986.54	0.214363
$580$	4071.19	0.291460
$581$	0	0
$582$	19218.3	1.36877
$583$	33239.1	2.36128
$584$	11553.5	0.818645
$585$	1086.74	0.0768056
$586$	−280.547	−0.0197769
$587$	−24310.2	−1.70935	−0.854676	−	0.519161i	$-0.826244\pi$
$587$	−24310.2	−1.70935	−0.854676	+	0.519161i	$0.826244\pi$
$588$	0	0
$589$	−2237.14	−0.156502
$590$	−9972.15	−0.695842
$591$	16056.5	1.11756
$592$	1663.90	0.115517
$593$	704.620	0.0487948	0.0243974	−	0.999702i	$-0.492233\pi$
$593$	704.620	0.0487948	0.0243974	+	0.999702i	$0.492233\pi$
$594$	16462.5	1.13715
$595$	0	0
$596$	8287.50	0.569579
$597$	−12855.7	−0.881321
$598$	−190.531	−0.0130291
$599$	−9704.05	−0.661931	−0.330966	−	0.943643i	$-0.607374\pi$
$599$	−9704.05	−0.661931	−0.330966	+	0.943643i	$0.607374\pi$
$600$	4905.52	0.333779
$601$	13971.7	0.948282	0.474141	−	0.880449i	$-0.342759\pi$
$601$	13971.7	0.948282	0.474141	+	0.880449i	$0.342759\pi$
$602$	0	0
$603$	2145.00	0.144861
$604$	−5047.86	−0.340057
$605$	29096.2	1.95525
$606$	10303.4	0.690675
$607$	−16249.2	−1.08655	−0.543273	−	0.839556i	$-0.682815\pi$
$607$	−16249.2	−1.08655	−0.543273	+	0.839556i	$0.682815\pi$
$608$	−12367.4	−0.824944
$609$	0	0
$610$	−14324.5	−0.950791
$611$	−14432.0	−0.955573
$612$	−183.888	−0.0121458
$613$	9193.37	0.605737	0.302868	−	0.953032i	$-0.402056\pi$
$613$	9193.37	0.605737	0.302868	+	0.953032i	$0.402056\pi$
$614$	−10390.4	−0.682937
$615$	6845.10	0.448815
$616$	0	0
$617$	−12376.3	−0.807537	−0.403768	−	0.914861i	$-0.632300\pi$
$617$	−12376.3	−0.807537	−0.403768	+	0.914861i	$0.632300\pi$
$618$	4403.03	0.286595
$619$	−5981.48	−0.388394	−0.194197	−	0.980963i	$-0.562210\pi$
$619$	−5981.48	−0.388394	−0.194197	+	0.980963i	$0.562210\pi$
$620$	1285.15	0.0832465
$621$	359.982	0.0232618
$622$	1971.95	0.127119
$623$	0	0
$624$	1748.37	0.112165
$625$	−9418.35	−0.602774
$626$	10944.3	0.698756
$627$	26447.7	1.68456
$628$	5894.80	0.374567
$629$	−2504.93	−0.158789
$630$	0	0
$631$	−26330.5	−1.66117	−0.830586	−	0.556891i	$-0.811994\pi$
$631$	−26330.5	−1.66117	−0.830586	+	0.556891i	$0.811994\pi$
$632$	−4070.91	−0.256222
$633$	28174.2	1.76907
$634$	8460.78	0.530001
$635$	6736.31	0.420980
$636$	−12167.0	−0.758572
$637$	0	0
$638$	−12351.9	−0.766481
$639$	1641.94	0.101650
$640$	5886.62	0.363577
$641$	−30520.0	−1.88061	−0.940303	−	0.340338i	$-0.889459\pi$
$641$	−30520.0	−1.88061	−0.940303	+	0.340338i	$0.889459\pi$
$642$	1879.78	0.115559
$643$	26976.4	1.65450	0.827250	−	0.561834i	$-0.189904\pi$
$643$	26976.4	1.65450	0.827250	+	0.561834i	$0.189904\pi$
$644$	0	0
$645$	−26202.5	−1.59957
$646$	−1773.98	−0.108044
$647$	32125.5	1.95206	0.976030	−	0.217636i	$-0.0698346\pi$
$647$	32125.5	1.95206	0.976030	+	0.217636i	$0.0698346\pi$
$648$	−18877.8	−1.14443
$649$	−37852.7	−2.28944
$650$	2636.82	0.159115
$651$	0	0
$652$	12570.6	0.755067
$653$	9507.76	0.569782	0.284891	−	0.958560i	$-0.408043\pi$
$653$	9507.76	0.569782	0.284891	+	0.958560i	$0.408043\pi$
$654$	4773.09	0.285386
$655$	18272.7	1.09003
$656$	1157.66	0.0689007
$657$	1561.80	0.0927420
$658$	0	0
$659$	357.324	0.0211220	0.0105610	−	0.999944i	$-0.496638\pi$
$659$	357.324	0.0211220	0.0105610	+	0.999944i	$0.496638\pi$
$660$	−15193.2	−0.896050
$661$	16790.4	0.988007	0.494004	−	0.869460i	$-0.335533\pi$
$661$	16790.4	0.988007	0.494004	+	0.869460i	$0.335533\pi$
$662$	−4311.79	−0.253146
$663$	−2632.10	−0.154181
$664$	−20916.8	−1.22249
$665$	0	0
$666$	1148.59	0.0668272
$667$	−270.096	−0.0156794
$668$	3817.28	0.221100
$669$	7911.47	0.457212
$670$	−11887.3	−0.685442
$671$	−54373.5	−3.12826
$672$	0	0
$673$	15476.0	0.886413	0.443207	−	0.896419i	$-0.353841\pi$
$673$	15476.0	0.886413	0.443207	+	0.896419i	$0.353841\pi$
$674$	−4104.93	−0.234593
$675$	−4981.90	−0.284079
$676$	3764.15	0.214164
$677$	6617.54	0.375676	0.187838	−	0.982200i	$-0.439852\pi$
$677$	6617.54	0.375676	0.187838	+	0.982200i	$0.439852\pi$
$678$	2169.89	0.122912
$679$	0	0
$680$	2852.70	0.160877
$681$	20595.4	1.15891
$682$	−3899.10	−0.218921
$683$	29533.9	1.65459	0.827293	−	0.561770i	$-0.189880\pi$
$683$	29533.9	1.65459	0.827293	+	0.561770i	$0.189880\pi$
$684$	−1017.69	−0.0568892
$685$	2110.13	0.117699
$686$	0	0
$687$	−13286.4	−0.737855
$688$	−4431.41	−0.245561
$689$	−18307.3	−1.01227
$690$	265.544	0.0146509
$691$	12828.8	0.706270	0.353135	−	0.935572i	$-0.385116\pi$
$691$	12828.8	0.706270	0.353135	+	0.935572i	$0.385116\pi$
$692$	6610.66	0.363150
$693$	0	0
$694$	−302.229	−0.0165309
$695$	−27298.3	−1.48990
$696$	12656.5	0.689285
$697$	−1742.80	−0.0947106
$698$	−12220.4	−0.662678
$699$	23778.2	1.28666
$700$	0	0
$701$	−5929.11	−0.319457	−0.159729	−	0.987161i	$-0.551062\pi$
$701$	−5929.11	−0.319457	−0.159729	+	0.987161i	$0.551062\pi$
$702$	−9067.15	−0.487489
$703$	−13863.0	−0.743746
$704$	−26178.4	−1.40147
$705$	20113.9	1.07452
$706$	9743.75	0.519420
$707$	0	0
$708$	13855.7	0.735494
$709$	1817.66	0.0962817	0.0481409	−	0.998841i	$-0.484670\pi$
$709$	1817.66	0.0962817	0.0481409	+	0.998841i	$0.484670\pi$
$710$	−9099.44	−0.480980
$711$	−550.302	−0.0290266
$712$	−13969.5	−0.735292
$713$	−85.2609	−0.00447832
$714$	0	0
$715$	−22860.7	−1.19573
$716$	−8215.84	−0.428828
$717$	−8291.91	−0.431892
$718$	−9655.41	−0.501862
$719$	−32965.9	−1.70990	−0.854951	−	0.518708i	$-0.826413\pi$
$719$	−32965.9	−1.70990	−0.854951	+	0.518708i	$0.826413\pi$
$720$	−256.152	−0.0132586
$721$	0	0
$722$	3112.55	0.160439
$723$	−12255.5	−0.630412
$724$	20792.2	1.06732
$725$	3737.93	0.191481
$726$	32313.2	1.65187
$727$	10984.0	0.560352	0.280176	−	0.959949i	$-0.409607\pi$
$727$	10984.0	0.560352	0.280176	+	0.959949i	$0.409607\pi$
$728$	0	0
$729$	16859.5	0.856552
$730$	−8655.27	−0.438830
$731$	6671.30	0.337547
$732$	19903.1	1.00497
$733$	36249.7	1.82662	0.913312	−	0.407261i	$-0.133516\pi$
$733$	36249.7	1.82662	0.913312	+	0.407261i	$0.133516\pi$
$734$	−9634.16	−0.484473
$735$	0	0
$736$	−471.342	−0.0236058
$737$	−45122.2	−2.25522
$738$	799.128	0.0398595
$739$	−23770.5	−1.18324	−0.591619	−	0.806218i	$-0.701511\pi$
$739$	−23770.5	−1.18324	−0.591619	+	0.806218i	$0.701511\pi$
$740$	7963.74	0.395612
$741$	−14566.8	−0.722164
$742$	0	0
$743$	−8381.03	−0.413823	−0.206911	−	0.978360i	$-0.566341\pi$
$743$	−8381.03	−0.413823	−0.206911	+	0.978360i	$0.566341\pi$
$744$	3995.26	0.196873
$745$	−17379.5	−0.854678
$746$	−13745.2	−0.674596
$747$	−2827.52	−0.138492
$748$	3868.26	0.189088
$749$	0	0
$750$	−15743.6	−0.766500
$751$	−25231.4	−1.22597	−0.612987	−	0.790093i	$-0.710032\pi$
$751$	−25231.4	−1.22597	−0.612987	+	0.790093i	$0.710032\pi$
$752$	3401.70	0.164957
$753$	−12265.0	−0.593574
$754$	6803.11	0.328587
$755$	10585.7	0.510271
$756$	0	0
$757$	407.684	0.0195740	0.00978700	−	0.999952i	$-0.496885\pi$
$757$	407.684	0.0195740	0.00978700	+	0.999952i	$0.496885\pi$
$758$	−3562.77	−0.170720
$759$	1007.96	0.0482039
$760$	15787.7	0.753524
$761$	12070.6	0.574980	0.287490	−	0.957784i	$-0.407179\pi$
$761$	12070.6	0.574980	0.287490	+	0.957784i	$0.407179\pi$
$762$	7481.10	0.355658
$763$	0	0
$764$	−13169.8	−0.623647
$765$	385.626	0.0182253
$766$	16319.7	0.769786
$767$	20848.3	0.981473
$768$	23779.0	1.11726
$769$	−12595.5	−0.590646	−0.295323	−	0.955397i	$-0.595427\pi$
$769$	−12595.5	−0.590646	−0.295323	+	0.955397i	$0.595427\pi$
$770$	0	0
$771$	14166.1	0.661709
$772$	2417.45	0.112702
$773$	8785.12	0.408770	0.204385	−	0.978891i	$-0.434481\pi$
$773$	8785.12	0.408770	0.204385	+	0.978891i	$0.434481\pi$
$774$	−3059.00	−0.142059
$775$	1179.95	0.0546904
$776$	43546.5	2.01447
$777$	0	0
$778$	13961.7	0.643382
$779$	−9645.15	−0.443612
$780$	8368.03	0.384133
$781$	−34540.0	−1.58251
$782$	−67.6091	−0.00309168
$783$	−12853.5	−0.586651
$784$	0	0
$785$	−12361.8	−0.562054
$786$	20293.0	0.920898
$787$	−38765.3	−1.75582	−0.877912	−	0.478823i	$-0.841064\pi$
$787$	−38765.3	−1.75582	−0.877912	+	0.478823i	$0.841064\pi$
$788$	12996.9	0.587557
$789$	−32441.9	−1.46383
$790$	3049.70	0.137346
$791$	0	0
$792$	−4965.15	−0.222764
$793$	29947.6	1.34107
$794$	24258.4	1.08425
$795$	25515.0	1.13827
$796$	−10406.0	−0.463357
$797$	13331.9	0.592524	0.296262	−	0.955107i	$-0.404260\pi$
$797$	13331.9	0.592524	0.296262	+	0.955107i	$0.404260\pi$
$798$	0	0
$799$	−5121.12	−0.226749
$800$	6523.04	0.288281
$801$	−1888.38	−0.0832992
$802$	26269.8	1.15663
$803$	−32854.0	−1.44383
$804$	16516.7	0.724501
$805$	0	0
$806$	2147.53	0.0938506
$807$	23508.6	1.02545
$808$	23346.4	1.01649
$809$	21033.9	0.914105	0.457052	−	0.889440i	$-0.348905\pi$
$809$	21033.9	0.914105	0.457052	+	0.889440i	$0.348905\pi$
$810$	14142.2	0.613463
$811$	37012.7	1.60258	0.801290	−	0.598276i	$-0.204147\pi$
$811$	37012.7	1.60258	0.801290	+	0.598276i	$0.204147\pi$
$812$	0	0
$813$	3264.84	0.140840
$814$	−24161.8	−1.04038
$815$	−26361.5	−1.13301
$816$	620.401	0.0266157
$817$	36920.8	1.58102
$818$	7961.29	0.340293
$819$	0	0
$820$	5540.76	0.235965
$821$	35840.7	1.52357	0.761785	−	0.647830i	$-0.224323\pi$
$821$	35840.7	1.52357	0.761785	+	0.647830i	$0.224323\pi$
$822$	2343.43	0.0994363
$823$	36223.9	1.53425	0.767123	−	0.641500i	$-0.221688\pi$
$823$	36223.9	1.53425	0.767123	+	0.641500i	$0.221688\pi$
$824$	9976.73	0.421791
$825$	−13949.5	−0.588678
$826$	0	0
$827$	3151.66	0.132520	0.0662600	−	0.997802i	$-0.478893\pi$
$827$	3151.66	0.132520	0.0662600	+	0.997802i	$0.478893\pi$
$828$	−38.7856	−0.00162789
$829$	24737.2	1.03638	0.518189	−	0.855266i	$-0.326606\pi$
$829$	24737.2	1.03638	0.518189	+	0.855266i	$0.326606\pi$
$830$	15669.7	0.655307
$831$	9729.59	0.406156
$832$	14418.4	0.600805
$833$	0	0
$834$	−30316.5	−1.25872
$835$	−8005.12	−0.331771
$836$	21408.1	0.885662
$837$	−4057.46	−0.167558
$838$	−2118.59	−0.0873333
$839$	−10665.8	−0.438883	−0.219441	−	0.975626i	$-0.570424\pi$
$839$	−10665.8	−0.438883	−0.219441	+	0.975626i	$0.570424\pi$
$840$	0	0
$841$	−14745.0	−0.604574
$842$	11054.0	0.452431
$843$	−15481.6	−0.632520
$844$	22805.6	0.930094
$845$	−7893.69	−0.321362
$846$	2348.19	0.0954285
$847$	0	0
$848$	4315.14	0.174744
$849$	49071.5	1.98366
$850$	935.662	0.0377564
$851$	−528.340	−0.0212823
$852$	12643.1	0.508388
$853$	16402.1	0.658377	0.329189	−	0.944264i	$-0.393225\pi$
$853$	16402.1	0.658377	0.329189	+	0.944264i	$0.393225\pi$
$854$	0	0
$855$	2134.16	0.0853647
$856$	4259.36	0.170072
$857$	−22835.5	−0.910203	−0.455102	−	0.890439i	$-0.650397\pi$
$857$	−22835.5	−0.910203	−0.455102	+	0.890439i	$0.650397\pi$
$858$	−25388.3	−1.01019
$859$	21492.0	0.853662	0.426831	−	0.904331i	$-0.359630\pi$
$859$	21492.0	0.853662	0.426831	+	0.904331i	$0.359630\pi$
$860$	−21209.5	−0.840976
$861$	0	0
$862$	−10424.5	−0.411901
$863$	−22420.9	−0.884376	−0.442188	−	0.896922i	$-0.645798\pi$
$863$	−22420.9	−0.884376	−0.442188	+	0.896922i	$0.645798\pi$
$864$	−22430.6	−0.883223
$865$	−13863.0	−0.544921
$866$	5114.29	0.200682
$867$	26052.5	1.02052
$868$	0	0
$869$	11576.2	0.451892
$870$	−9481.53	−0.369487
$871$	24852.2	0.966803
$872$	10815.2	0.420012
$873$	5886.57	0.228213
$874$	−374.168	−0.0144810
$875$	0	0
$876$	12026.0	0.463836
$877$	−37647.2	−1.44955	−0.724774	−	0.688986i	$-0.758056\pi$
$877$	−37647.2	−1.44955	−0.724774	+	0.688986i	$0.758056\pi$
$878$	5331.62	0.204936
$879$	−817.446	−0.0313672
$880$	5388.41	0.206413
$881$	−7657.11	−0.292820	−0.146410	−	0.989224i	$-0.546772\pi$
$881$	−7657.11	−0.292820	−0.146410	+	0.989224i	$0.546772\pi$
$882$	0	0
$883$	−14662.7	−0.558823	−0.279411	−	0.960171i	$-0.590139\pi$
$883$	−14662.7	−0.558823	−0.279411	+	0.960171i	$0.590139\pi$
$884$	−2130.55	−0.0810612
$885$	−29056.4	−1.10364
$886$	−10519.5	−0.398883
$887$	−47147.8	−1.78475	−0.892373	−	0.451299i	$-0.850961\pi$
$887$	−47147.8	−1.78475	−0.892373	+	0.451299i	$0.850961\pi$
$888$	24757.6	0.935598
$889$	0	0
$890$	10465.1	0.394149
$891$	53681.4	2.01840
$892$	6403.93	0.240380
$893$	−28341.7	−1.06206
$894$	−19301.0	−0.722062
$895$	17229.2	0.643474
$896$	0	0
$897$	−555.162	−0.0206648
$898$	−26615.9	−0.989069
$899$	3044.33	0.112941
$900$	536.765	0.0198802
$901$	−6496.27	−0.240202
$902$	−16810.5	−0.620541
$903$	0	0
$904$	4916.71	0.180893
$905$	−43602.8	−1.60155
$906$	11756.1	0.431095
$907$	6170.87	0.225910	0.112955	−	0.993600i	$-0.463968\pi$
$907$	6170.87	0.225910	0.112955	+	0.993600i	$0.463968\pi$
$908$	16670.9	0.609299
$909$	3155.94	0.115155
$910$	0	0
$911$	−50242.3	−1.82722	−0.913612	−	0.406586i	$-0.866719\pi$
$911$	−50242.3	−1.82722	−0.913612	+	0.406586i	$0.866719\pi$
$912$	3433.47	0.124664
$913$	59479.8	2.15607
$914$	11633.9	0.421024
$915$	−41738.2	−1.50800
$916$	−10754.6	−0.387929
$917$	0	0
$918$	−3217.44	−0.115677
$919$	24844.1	0.891764	0.445882	−	0.895092i	$-0.352890\pi$
$919$	24844.1	0.891764	0.445882	+	0.895092i	$0.352890\pi$
$920$	601.691	0.0215621
$921$	−30275.2	−1.08317
$922$	−7321.96	−0.261536
$923$	19023.8	0.678414
$924$	0	0
$925$	7311.86	0.259905
$926$	−15325.0	−0.543855
$927$	1348.65	0.0477835
$928$	16829.7	0.595327
$929$	24340.8	0.859630	0.429815	−	0.902917i	$-0.358579\pi$
$929$	24340.8	0.859630	0.429815	+	0.902917i	$0.358579\pi$
$930$	−2993.03	−0.105532
$931$	0	0
$932$	19247.3	0.676465
$933$	5745.78	0.201617
$934$	−4112.96	−0.144090
$935$	−8112.03	−0.283734
$936$	2734.69	0.0954979
$937$	52000.2	1.81299	0.906496	−	0.422215i	$-0.138747\pi$
$937$	52000.2	1.81299	0.906496	+	0.422215i	$0.138747\pi$
$938$	0	0
$939$	31889.0	1.10826
$940$	16281.2	0.564930
$941$	−806.468	−0.0279385	−0.0139692	−	0.999902i	$-0.504447\pi$
$941$	−806.468	−0.0279385	−0.0139692	+	0.999902i	$0.504447\pi$
$942$	−13728.6	−0.474843
$943$	−367.591	−0.0126940
$944$	−4914.08	−0.169428
$945$	0	0
$946$	64349.1	2.21160
$947$	46422.7	1.59296	0.796481	−	0.604664i	$-0.206693\pi$
$947$	46422.7	1.59296	0.796481	+	0.604664i	$0.206693\pi$
$948$	−4237.38	−0.145173
$949$	18095.2	0.618962
$950$	5178.22	0.176846
$951$	24652.7	0.840607
$952$	0	0
$953$	−21564.0	−0.732975	−0.366487	−	0.930423i	$-0.619440\pi$
$953$	−21564.0	−0.732975	−0.366487	+	0.930423i	$0.619440\pi$
$954$	2978.74	0.101090
$955$	27618.0	0.935808
$956$	−6711.87	−0.227068
$957$	−35990.3	−1.21568
$958$	−25963.3	−0.875612
$959$	0	0
$960$	−20095.1	−0.675589
$961$	961.000	0.0322581
$962$	13307.7	0.446006
$963$	575.776	0.0192670
$964$	−9920.22	−0.331441
$965$	−5069.56	−0.169114
$966$	0	0
$967$	1603.40	0.0533216	0.0266608	−	0.999645i	$-0.491513\pi$
$967$	1603.40	0.0533216	0.0266608	+	0.999645i	$0.491513\pi$
$968$	73217.8	2.43110
$969$	−5168.95	−0.171363
$970$	−32622.6	−1.07984
$971$	14645.9	0.484046	0.242023	−	0.970271i	$-0.422189\pi$
$971$	14645.9	0.484046	0.242023	+	0.970271i	$0.422189\pi$
$972$	−3937.20	−0.129924
$973$	0	0
$974$	24589.1	0.808918
$975$	7683.05	0.252364
$976$	−7058.83	−0.231504
$977$	26865.9	0.879750	0.439875	−	0.898059i	$-0.355023\pi$
$977$	26865.9	0.879750	0.439875	+	0.898059i	$0.355023\pi$
$978$	−29276.2	−0.957207
$979$	39724.0	1.29682
$980$	0	0
$981$	1462.00	0.0475820
$982$	24216.7	0.786950
$983$	−28969.3	−0.939956	−0.469978	−	0.882678i	$-0.655738\pi$
$983$	−28969.3	−0.939956	−0.469978	+	0.882678i	$0.655738\pi$
$984$	17225.0	0.558043
$985$	−27255.4	−0.881655
$986$	2414.05	0.0779706
$987$	0	0
$988$	−11791.0	−0.379679
$989$	1407.11	0.0452411
$990$	3719.62	0.119411
$991$	−50226.1	−1.60997	−0.804986	−	0.593293i	$-0.797827\pi$
$991$	−50226.1	−1.60997	−0.804986	+	0.593293i	$0.797827\pi$
$992$	5312.63	0.170037
$993$	−12563.5	−0.401501
$994$	0	0
$995$	21822.2	0.695286
$996$	−21772.2	−0.692649
$997$	24320.5	0.772555	0.386278	−	0.922383i	$-0.373761\pi$
$997$	24320.5	0.772555	0.386278	+	0.922383i	$0.373761\pi$
$998$	20842.9	0.661092
$999$	−25143.1	−0.796288

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1519.4.a.h.1.8	✓	23
7.6	odd	2		1519.4.a.i.1.8	yes	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1519.4.a.h.1.8	✓	23	1.1	even	1	trivial
1519.4.a.i.1.8	yes	23	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1519 = 7^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1519.a (trivial)

\(f(q)\)	\(=\)	\(q-1.88515 q^{2} -5.49288 q^{3} -4.44620 q^{4} +9.32401 q^{5} +10.3549 q^{6} +23.4630 q^{8} +3.17171 q^{9} -17.5772 q^{10} -66.7201 q^{11} +24.4224 q^{12} +36.7478 q^{13} -51.2156 q^{15} -8.66168 q^{16} +13.0398 q^{17} -5.97915 q^{18} +72.1659 q^{19} -41.4564 q^{20} +125.777 q^{22} +2.75035 q^{23} -128.879 q^{24} -38.0629 q^{25} -69.2752 q^{26} +130.886 q^{27} -98.2040 q^{29} +96.5492 q^{30} -31.0000 q^{31} -171.375 q^{32} +366.485 q^{33} -24.5820 q^{34} -14.1020 q^{36} -192.099 q^{37} -136.044 q^{38} -201.851 q^{39} +218.769 q^{40} -133.653 q^{41} +511.611 q^{43} +296.651 q^{44} +29.5730 q^{45} -5.18483 q^{46} -392.730 q^{47} +47.5775 q^{48} +71.7544 q^{50} -71.6260 q^{51} -163.388 q^{52} -498.188 q^{53} -246.740 q^{54} -622.098 q^{55} -396.398 q^{57} +185.130 q^{58} +567.336 q^{59} +227.715 q^{60} +814.950 q^{61} +58.4397 q^{62} +392.362 q^{64} +342.637 q^{65} -690.880 q^{66} +676.291 q^{67} -57.9775 q^{68} -15.1073 q^{69} +517.685 q^{71} +74.4177 q^{72} +492.415 q^{73} +362.136 q^{74} +209.075 q^{75} -320.864 q^{76} +380.520 q^{78} -173.504 q^{79} -80.7615 q^{80} -804.576 q^{81} +251.955 q^{82} -891.483 q^{83} +121.583 q^{85} -964.464 q^{86} +539.423 q^{87} -1565.45 q^{88} -595.383 q^{89} -55.7496 q^{90} -12.2286 q^{92} +170.279 q^{93} +740.356 q^{94} +672.875 q^{95} +941.343 q^{96} +1855.96 q^{97} -211.616 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 5 q^{2} - 6 q^{3} + 91 q^{4} - 40 q^{5} - 36 q^{6} + 39 q^{8} + 211 q^{9} - 40 q^{10} + 44 q^{11} - 414 q^{12} + 20 q^{13} + 523 q^{16} - 306 q^{17} + 51 q^{18} - 296 q^{19} - 400 q^{20} - 326 q^{22}+ \cdots - 3456 q^{99}+O(q^{100}) \) 23 * q + 5 * q^2 - 6 * q^3 + 91 * q^4 - 40 * q^5 - 36 * q^6 + 39 * q^8 + 211 * q^9 - 40 * q^10 + 44 * q^11 - 414 * q^12 + 20 * q^13 + 523 * q^16 - 306 * q^17 + 51 * q^18 - 296 * q^19 - 400 * q^20 - 326 * q^22 - 130 * q^23 - 432 * q^24 + 801 * q^25 - 474 * q^26 - 192 * q^27 - 16 * q^29 - 74 * q^30 - 713 * q^31 + 287 * q^32 - 488 * q^33 - 692 * q^34 + 623 * q^36 - 296 * q^37 + 40 * q^38 - 124 * q^39 - 48 * q^40 - 2344 * q^41 - 48 * q^43 + 1352 * q^44 - 132 * q^45 - 836 * q^46 - 436 * q^47 - 4190 * q^48 - 767 * q^50 - 872 * q^51 + 632 * q^52 - 884 * q^53 - 100 * q^54 - 3308 * q^55 + 404 * q^57 - 562 * q^58 - 400 * q^59 + 2248 * q^60 - 860 * q^61 - 155 * q^62 + 3125 * q^64 - 1676 * q^65 - 128 * q^66 - 164 * q^67 - 302 * q^68 - 2136 * q^69 + 56 * q^71 + 1613 * q^72 - 1610 * q^73 + 908 * q^74 + 910 * q^75 - 5208 * q^76 + 1424 * q^78 + 318 * q^79 - 11480 * q^80 + 231 * q^81 + 2608 * q^82 - 3166 * q^83 - 716 * q^85 - 224 * q^86 - 1248 * q^87 - 5562 * q^88 - 2854 * q^89 - 7172 * q^90 - 3558 * q^92 + 186 * q^93 - 4228 * q^94 + 984 * q^95 + 4178 * q^96 - 3332 * q^97 - 3456 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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