LMFDB - Embedded newform 1519.4.a.h.1.15

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			1.15
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)

$f(q)$	$=$	\(q+1.28627 q^{2} +10.2631 q^{3} -6.34552 q^{4} -1.02970 q^{5} +13.2010 q^{6} -18.4521 q^{8} +78.3306 q^{9} -1.32447 q^{10} -54.1330 q^{11} -65.1246 q^{12} +43.3374 q^{13} -10.5679 q^{15} +27.0298 q^{16} -111.884 q^{17} +100.754 q^{18} +8.45276 q^{19} +6.53399 q^{20} -69.6294 q^{22} -172.189 q^{23} -189.376 q^{24} -123.940 q^{25} +55.7434 q^{26} +526.810 q^{27} -45.3527 q^{29} -13.5931 q^{30} -31.0000 q^{31} +182.385 q^{32} -555.571 q^{33} -143.913 q^{34} -497.049 q^{36} -72.9027 q^{37} +10.8725 q^{38} +444.775 q^{39} +19.0002 q^{40} -121.635 q^{41} -391.185 q^{43} +343.502 q^{44} -80.6571 q^{45} -221.480 q^{46} +329.593 q^{47} +277.409 q^{48} -159.419 q^{50} -1148.27 q^{51} -274.999 q^{52} -706.357 q^{53} +677.617 q^{54} +55.7408 q^{55} +86.7513 q^{57} -58.3357 q^{58} +629.715 q^{59} +67.0588 q^{60} +479.489 q^{61} -39.8742 q^{62} +18.3564 q^{64} -44.6246 q^{65} -714.611 q^{66} -650.441 q^{67} +709.963 q^{68} -1767.18 q^{69} -885.987 q^{71} -1445.37 q^{72} +804.668 q^{73} -93.7722 q^{74} -1272.00 q^{75} -53.6372 q^{76} +572.099 q^{78} +554.343 q^{79} -27.8326 q^{80} +3291.76 q^{81} -156.455 q^{82} +83.8433 q^{83} +115.207 q^{85} -503.168 q^{86} -465.458 q^{87} +998.870 q^{88} -204.101 q^{89} -103.746 q^{90} +1092.63 q^{92} -318.155 q^{93} +423.945 q^{94} -8.70382 q^{95} +1871.83 q^{96} +360.853 q^{97} -4240.27 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.28627	0.454763	0.227382	−	0.973806i	$-0.426984\pi$
$2$	1.28627	0.454763	0.227382	+	0.973806i	$0.426984\pi$
$3$	10.2631	1.97513	0.987565	−	0.157214i	$-0.0502513\pi$
$3$	10.2631	1.97513	0.987565	+	0.157214i	$0.0502513\pi$
$4$	−6.34552	−0.793190
$5$	−1.02970	−0.0920992	−0.0460496	−	0.998939i	$-0.514663\pi$
$5$	−1.02970	−0.0920992	−0.0460496	+	0.998939i	$0.514663\pi$
$6$	13.2010	0.898216
$7$	0	0
$7$	0	0
$8$	−18.4521	−0.815477
$9$	78.3306	2.90113
$10$	−1.32447	−0.0418834
$11$	−54.1330	−1.48379	−0.741896	−	0.670515i	$-0.766073\pi$
$11$	−54.1330	−1.48379	−0.741896	+	0.670515i	$0.766073\pi$
$12$	−65.1246	−1.56665
$13$	43.3374	0.924588	0.462294	−	0.886727i	$-0.347026\pi$
$13$	43.3374	0.924588	0.462294	+	0.886727i	$0.347026\pi$
$14$	0	0
$15$	−10.5679	−0.181908
$16$	27.0298	0.422341
$17$	−111.884	−1.59623	−0.798114	−	0.602507i	$-0.794169\pi$
$17$	−111.884	−1.59623	−0.798114	+	0.602507i	$0.794169\pi$
$18$	100.754	1.31933
$19$	8.45276	0.102063	0.0510315	−	0.998697i	$-0.483749\pi$
$19$	8.45276	0.102063	0.0510315	+	0.998697i	$0.483749\pi$
$20$	6.53399	0.0730522
$21$	0	0
$22$	−69.6294	−0.674774
$23$	−172.189	−1.56104	−0.780518	−	0.625134i	$-0.785045\pi$
$23$	−172.189	−1.56104	−0.780518	+	0.625134i	$0.785045\pi$
$24$	−189.376	−1.61067
$25$	−123.940	−0.991518
$26$	55.7434	0.420469
$27$	526.810	3.75499
$28$	0	0
$29$	−45.3527	−0.290407	−0.145203	−	0.989402i	$-0.546384\pi$
$29$	−45.3527	−0.290407	−0.145203	+	0.989402i	$0.546384\pi$
$30$	−13.5931	−0.0827250
$31$	−31.0000	−0.179605
$31$	−31.0000	−0.179605
$32$	182.385	1.00754
$33$	−555.571	−2.93068
$34$	−143.913	−0.725906
$35$	0	0
$36$	−497.049	−2.30115
$37$	−72.9027	−0.323923	−0.161961	−	0.986797i	$-0.551782\pi$
$37$	−72.9027	−0.323923	−0.161961	+	0.986797i	$0.551782\pi$
$38$	10.8725	0.0464145
$39$	444.775	1.82618
$40$	19.0002	0.0751048
$41$	−121.635	−0.463322	−0.231661	−	0.972797i	$-0.574416\pi$
$41$	−121.635	−0.463322	−0.231661	+	0.972797i	$0.574416\pi$
$42$	0	0
$43$	−391.185	−1.38733	−0.693665	−	0.720298i	$-0.744005\pi$
$43$	−391.185	−1.38733	−0.693665	+	0.720298i	$0.744005\pi$
$44$	343.502	1.17693
$45$	−80.6571	−0.267192
$46$	−221.480	−0.709902
$47$	329.593	1.02290	0.511448	−	0.859314i	$-0.329109\pi$
$47$	329.593	1.02290	0.511448	+	0.859314i	$0.329109\pi$
$48$	277.409	0.834178
$49$	0	0
$50$	−159.419	−0.450906
$51$	−1148.27	−3.15276
$52$	−274.999	−0.733374
$53$	−706.357	−1.83067	−0.915335	−	0.402693i	$-0.868074\pi$
$53$	−706.357	−1.83067	−0.915335	+	0.402693i	$0.868074\pi$
$54$	677.617	1.70763
$55$	55.7408	0.136656
$56$	0	0
$57$	86.7513	0.201588
$58$	−58.3357	−0.132066
$59$	629.715	1.38952	0.694762	−	0.719240i	$-0.255510\pi$
$59$	629.715	1.38952	0.694762	+	0.719240i	$0.255510\pi$
$60$	67.0588	0.144288
$61$	479.489	1.00643	0.503216	−	0.864161i	$-0.332150\pi$
$61$	479.489	1.00643	0.503216	+	0.864161i	$0.332150\pi$
$62$	−39.8742	−0.0816779
$63$	0	0
$64$	18.3564	0.0358523
$65$	−44.6246	−0.0851538
$66$	−714.611	−1.33277
$67$	−650.441	−1.18603	−0.593015	−	0.805191i	$-0.702063\pi$
$67$	−650.441	−1.18603	−0.593015	+	0.805191i	$0.702063\pi$
$68$	709.963	1.26611
$69$	−1767.18	−3.08325
$70$	0	0
$71$	−885.987	−1.48095	−0.740474	−	0.672085i	$-0.765399\pi$
$71$	−885.987	−1.48095	−0.740474	+	0.672085i	$0.765399\pi$
$72$	−1445.37	−2.36581
$73$	804.668	1.29013	0.645064	−	0.764129i	$-0.276831\pi$
$73$	804.668	1.29013	0.645064	+	0.764129i	$0.276831\pi$
$74$	−93.7722	−0.147308
$75$	−1272.00	−1.95838
$76$	−53.6372	−0.0809554
$77$	0	0
$78$	572.099	0.830480
$79$	554.343	0.789475	0.394737	−	0.918794i	$-0.370836\pi$
$79$	554.343	0.789475	0.394737	+	0.918794i	$0.370836\pi$
$80$	−27.8326	−0.0388973
$81$	3291.76	4.51545
$82$	−156.455	−0.210702
$83$	83.8433	0.110880	0.0554398	−	0.998462i	$-0.482344\pi$
$83$	83.8433	0.110880	0.0554398	+	0.998462i	$0.482344\pi$
$84$	0	0
$85$	115.207	0.147011
$86$	−503.168	−0.630907
$87$	−465.458	−0.573591
$88$	998.870	1.21000
$89$	−204.101	−0.243086	−0.121543	−	0.992586i	$-0.538784\pi$
$89$	−204.101	−0.243086	−0.121543	+	0.992586i	$0.538784\pi$
$90$	−103.746	−0.121509
$91$	0	0
$92$	1092.63	1.23820
$93$	−318.155	−0.354744
$94$	423.945	0.465176
$95$	−8.70382	−0.00939992
$96$	1871.83	1.99003
$97$	360.853	0.377723	0.188861	−	0.982004i	$-0.439520\pi$
$97$	360.853	0.377723	0.188861	+	0.982004i	$0.439520\pi$
$98$	0	0
$99$	−4240.27	−4.30468
$100$	786.462	0.786462
$101$	−1271.87	−1.25302	−0.626511	−	0.779412i	$-0.715518\pi$
$101$	−1271.87	−1.25302	−0.626511	+	0.779412i	$0.715518\pi$
$102$	−1476.98	−1.43376
$103$	−1340.73	−1.28258	−0.641292	−	0.767297i	$-0.721601\pi$
$103$	−1340.73	−1.28258	−0.641292	+	0.767297i	$0.721601\pi$
$104$	−799.669	−0.753980
$105$	0	0
$106$	−908.562	−0.832522
$107$	−499.473	−0.451270	−0.225635	−	0.974212i	$-0.572446\pi$
$107$	−499.473	−0.451270	−0.225635	+	0.974212i	$0.572446\pi$
$108$	−3342.88	−2.97842
$109$	833.263	0.732221	0.366111	−	0.930571i	$-0.380689\pi$
$109$	833.263	0.732221	0.366111	+	0.930571i	$0.380689\pi$
$110$	71.6974	0.0621462
$111$	−748.206	−0.639789
$112$	0	0
$113$	−244.165	−0.203267	−0.101633	−	0.994822i	$-0.532407\pi$
$113$	−244.165	−0.203267	−0.101633	+	0.994822i	$0.532407\pi$
$114$	111.585	0.0916746
$115$	177.303	0.143770
$116$	287.787	0.230348
$117$	3394.65	2.68235
$118$	809.980	0.631904
$119$	0	0
$120$	195.000	0.148342
$121$	1599.38	1.20164
$122$	616.750	0.457688
$123$	−1248.35	−0.915121
$124$	196.711	0.142461
$125$	256.333	0.183417
$126$	0	0
$127$	−21.4025	−0.0149540	−0.00747702	−	0.999972i	$-0.502380\pi$
$127$	−21.4025	−0.0149540	−0.00747702	+	0.999972i	$0.502380\pi$
$128$	−1435.47	−0.991238
$129$	−4014.76	−2.74016
$130$	−57.3991	−0.0387248
$131$	−1397.55	−0.932093	−0.466047	−	0.884760i	$-0.654322\pi$
$131$	−1397.55	−0.932093	−0.466047	+	0.884760i	$0.654322\pi$
$132$	3525.39	2.32459
$133$	0	0
$134$	−836.640	−0.539363
$135$	−542.457	−0.345831
$136$	2064.50	1.30169
$137$	−1047.90	−0.653490	−0.326745	−	0.945113i	$-0.605952\pi$
$137$	−1047.90	−0.653490	−0.326745	+	0.945113i	$0.605952\pi$
$138$	−2273.07	−1.40215
$139$	1893.75	1.15558	0.577790	−	0.816185i	$-0.303915\pi$
$139$	1893.75	1.15558	0.577790	+	0.816185i	$0.303915\pi$
$140$	0	0
$141$	3382.64	2.02035
$142$	−1139.61	−0.673481
$143$	−2345.99	−1.37190
$144$	2117.26	1.22527
$145$	46.6998	0.0267462
$146$	1035.02	0.586703
$147$	0	0
$148$	462.606	0.256932
$149$	287.025	0.157812	0.0789060	−	0.996882i	$-0.474857\pi$
$149$	287.025	0.157812	0.0789060	+	0.996882i	$0.474857\pi$
$150$	−1636.13	−0.890597
$151$	289.173	0.155845	0.0779224	−	0.996959i	$-0.475171\pi$
$151$	289.173	0.155845	0.0779224	+	0.996959i	$0.475171\pi$
$152$	−155.972	−0.0832300
$153$	−8763.95	−4.63087
$154$	0	0
$155$	31.9207	0.0165415
$156$	−2822.33	−1.44851
$157$	−418.205	−0.212589	−0.106294	−	0.994335i	$-0.533899\pi$
$157$	−418.205	−0.212589	−0.106294	+	0.994335i	$0.533899\pi$
$158$	713.033	0.359024
$159$	−7249.39	−3.61581
$160$	−187.802	−0.0927939
$161$	0	0
$162$	4234.08	2.05346
$163$	376.341	0.180842	0.0904211	−	0.995904i	$-0.471179\pi$
$163$	376.341	0.180842	0.0904211	+	0.995904i	$0.471179\pi$
$164$	771.838	0.367503
$165$	572.072	0.269914
$166$	107.845	0.0504239
$167$	−1927.01	−0.892912	−0.446456	−	0.894806i	$-0.647314\pi$
$167$	−1927.01	−0.892912	−0.446456	+	0.894806i	$0.647314\pi$
$168$	0	0
$169$	−318.866	−0.145137
$170$	148.187	0.0668554
$171$	662.110	0.296098
$172$	2482.28	1.10042
$173$	−3036.77	−1.33458	−0.667288	−	0.744800i	$-0.732545\pi$
$173$	−3036.77	−1.33458	−0.667288	+	0.744800i	$0.732545\pi$
$174$	−598.703	−0.260848
$175$	0	0
$176$	−1463.21	−0.626666
$177$	6462.81	2.74449
$178$	−262.528	−0.110547
$179$	−1448.42	−0.604804	−0.302402	−	0.953180i	$-0.597789\pi$
$179$	−1448.42	−0.604804	−0.302402	+	0.953180i	$0.597789\pi$
$180$	511.812	0.211934
$181$	1024.12	0.420564	0.210282	−	0.977641i	$-0.432562\pi$
$181$	1024.12	0.420564	0.210282	+	0.977641i	$0.432562\pi$
$182$	0	0
$183$	4921.03	1.98783
$184$	3177.25	1.27299
$185$	75.0680	0.0298330
$186$	−409.232	−0.161324
$187$	6056.62	2.36847
$188$	−2091.44	−0.811352
$189$	0	0
$190$	−11.1954	−0.00427474
$191$	−1879.59	−0.712056	−0.356028	−	0.934475i	$-0.615869\pi$
$191$	−1879.59	−0.712056	−0.356028	+	0.934475i	$0.615869\pi$
$192$	188.393	0.0708129
$193$	−3890.26	−1.45092	−0.725459	−	0.688266i	$-0.758372\pi$
$193$	−3890.26	−1.45092	−0.725459	+	0.688266i	$0.758372\pi$
$194$	464.153	0.171775
$195$	−457.985	−0.168190
$196$	0	0
$197$	−2370.53	−0.857328	−0.428664	−	0.903464i	$-0.641016\pi$
$197$	−2370.53	−0.857328	−0.428664	+	0.903464i	$0.641016\pi$
$198$	−5454.11	−1.95761
$199$	−1234.10	−0.439612	−0.219806	−	0.975544i	$-0.570542\pi$
$199$	−1234.10	−0.439612	−0.219806	+	0.975544i	$0.570542\pi$
$200$	2286.95	0.808560
$201$	−6675.52	−2.34256
$202$	−1635.96	−0.569829
$203$	0	0
$204$	7286.40	2.50073
$205$	125.248	0.0426716
$206$	−1724.53	−0.583272
$207$	−13487.6	−4.52877
$208$	1171.40	0.390491
$209$	−457.573	−0.151440
$210$	0	0
$211$	266.109	0.0868232	0.0434116	−	0.999057i	$-0.486177\pi$
$211$	266.109	0.0868232	0.0434116	+	0.999057i	$0.486177\pi$
$212$	4482.20	1.45207
$213$	−9092.95	−2.92506
$214$	−642.455	−0.205221
$215$	402.804	0.127772
$216$	−9720.77	−3.06211
$217$	0	0
$218$	1071.80	0.332987
$219$	8258.37	2.54817
$220$	−353.704	−0.108394
$221$	−4848.77	−1.47585
$222$	−962.391	−0.290953
$223$	−3204.05	−0.962148	−0.481074	−	0.876680i	$-0.659753\pi$
$223$	−3204.05	−0.962148	−0.481074	+	0.876680i	$0.659753\pi$
$224$	0	0
$225$	−9708.28	−2.87653
$226$	−314.061	−0.0924382
$227$	5854.81	1.71188	0.855942	−	0.517072i	$-0.172978\pi$
$227$	5854.81	1.71188	0.855942	+	0.517072i	$0.172978\pi$
$228$	−550.482	−0.159897
$229$	6063.97	1.74986	0.874931	−	0.484247i	$-0.160906\pi$
$229$	6063.97	1.74986	0.874931	+	0.484247i	$0.160906\pi$
$230$	228.058	0.0653814
$231$	0	0
$232$	836.855	0.236820
$233$	3647.51	1.02556	0.512782	−	0.858519i	$-0.328615\pi$
$233$	3647.51	1.02556	0.512782	+	0.858519i	$0.328615\pi$
$234$	4366.42	1.21984
$235$	−339.383	−0.0942080
$236$	−3995.87	−1.10216
$237$	5689.27	1.55931
$238$	0	0
$239$	2873.95	0.777827	0.388913	−	0.921274i	$-0.372851\pi$
$239$	2873.95	0.777827	0.388913	+	0.921274i	$0.372851\pi$
$240$	−285.648	−0.0768272
$241$	569.470	0.152211	0.0761054	−	0.997100i	$-0.475751\pi$
$241$	569.470	0.152211	0.0761054	+	0.997100i	$0.475751\pi$
$242$	2057.23	0.546462
$243$	19559.7	5.16361
$244$	−3042.61	−0.798291
$245$	0	0
$246$	−1605.71	−0.416164
$247$	366.321	0.0943662
$248$	572.016	0.146464
$249$	860.490	0.219001
$250$	329.713	0.0834115
$251$	4763.52	1.19789	0.598946	−	0.800789i	$-0.295586\pi$
$251$	4763.52	1.19789	0.598946	+	0.800789i	$0.295586\pi$
$252$	0	0
$253$	9321.09	2.31625
$254$	−27.5293	−0.00680055
$255$	1182.38	0.290366
$256$	−1993.24	−0.486631
$257$	5739.65	1.39311	0.696555	−	0.717503i	$-0.254715\pi$
$257$	5739.65	1.39311	0.696555	+	0.717503i	$0.254715\pi$
$258$	−5164.05	−1.24612
$259$	0	0
$260$	283.166	0.0675432
$261$	−3552.51	−0.842509
$262$	−1797.62	−0.423882
$263$	6408.29	1.50248	0.751239	−	0.660030i	$-0.229456\pi$
$263$	6408.29	1.50248	0.751239	+	0.660030i	$0.229456\pi$
$264$	10251.5	2.38990
$265$	727.336	0.168603
$266$	0	0
$267$	−2094.70	−0.480126
$268$	4127.39	0.940748
$269$	−5582.27	−1.26527	−0.632634	−	0.774451i	$-0.718026\pi$
$269$	−5582.27	−1.26527	−0.632634	+	0.774451i	$0.718026\pi$
$270$	−697.743	−0.157271
$271$	6099.57	1.36724	0.683621	−	0.729837i	$-0.260404\pi$
$271$	6099.57	1.36724	0.683621	+	0.729837i	$0.260404\pi$
$272$	−3024.21	−0.674152
$273$	0	0
$274$	−1347.88	−0.297183
$275$	6709.23	1.47121
$276$	11213.7	2.44560
$277$	5005.22	1.08568	0.542842	−	0.839835i	$-0.317348\pi$
$277$	5005.22	1.08568	0.542842	+	0.839835i	$0.317348\pi$
$278$	2435.86	0.525516
$279$	−2428.25	−0.521059
$280$	0	0
$281$	−2764.32	−0.586852	−0.293426	−	0.955982i	$-0.594795\pi$
$281$	−2764.32	−0.586852	−0.293426	+	0.955982i	$0.594795\pi$
$282$	4350.97	0.918783
$283$	−1342.81	−0.282057	−0.141028	−	0.990006i	$-0.545041\pi$
$283$	−1342.81	−0.282057	−0.141028	+	0.990006i	$0.545041\pi$
$284$	5622.05	1.17467
$285$	−89.3279	−0.0185661
$286$	−3017.56	−0.623888
$287$	0	0
$288$	14286.3	2.92302
$289$	7605.04	1.54794
$290$	60.0683	0.0121632
$291$	3703.47	0.746051
$292$	−5106.04	−1.02332
$293$	345.135	0.0688158	0.0344079	−	0.999408i	$-0.489045\pi$
$293$	345.135	0.0688158	0.0344079	+	0.999408i	$0.489045\pi$
$294$	0	0
$295$	−648.418	−0.127974
$296$	1345.21	0.264151
$297$	−28517.8	−5.57162
$298$	369.190	0.0717671
$299$	−7462.22	−1.44331
$300$	8071.52	1.55336
$301$	0	0
$302$	371.953	0.0708725
$303$	−13053.2	−2.47488
$304$	228.477	0.0431054
$305$	−493.730	−0.0926915
$306$	−11272.8	−2.10595
$307$	−1474.88	−0.274189	−0.137095	−	0.990558i	$-0.543776\pi$
$307$	−1474.88	−0.274189	−0.137095	+	0.990558i	$0.543776\pi$
$308$	0	0
$309$	−13760.0	−2.53327
$310$	41.0585	0.00752247
$311$	7374.52	1.34460	0.672300	−	0.740279i	$-0.265307\pi$
$311$	7374.52	1.34460	0.672300	+	0.740279i	$0.265307\pi$
$312$	−8207.06	−1.48921
$313$	177.313	0.0320201	0.0160101	−	0.999872i	$-0.494904\pi$
$313$	177.313	0.0320201	0.0160101	+	0.999872i	$0.494904\pi$
$314$	−537.923	−0.0966775
$315$	0	0
$316$	−3517.60	−0.626204
$317$	4631.31	0.820569	0.410284	−	0.911958i	$-0.365429\pi$
$317$	4631.31	0.820569	0.410284	+	0.911958i	$0.365429\pi$
$318$	−9324.63	−1.64434
$319$	2455.08	0.430903
$320$	−18.9016	−0.00330197
$321$	−5126.13	−0.891316
$322$	0	0
$323$	−945.729	−0.162916
$324$	−20887.9	−3.58161
$325$	−5371.23	−0.916745
$326$	484.074	0.0822404
$327$	8551.84	1.44623
$328$	2244.43	0.377829
$329$	0	0
$330$	735.836	0.122747
$331$	3936.31	0.653653	0.326827	−	0.945084i	$-0.394021\pi$
$331$	3936.31	0.653653	0.326827	+	0.945084i	$0.394021\pi$
$332$	−532.030	−0.0879486
$333$	−5710.52	−0.939743
$334$	−2478.64	−0.406064
$335$	669.760	0.109232
$336$	0	0
$337$	5.47071	0.000884299 0	0.000442150	−	1.00000i	$-0.499859\pi$
$337$	5.47071	0.000884299 0	0.000442150		1.00000i	$0.499859\pi$
$338$	−410.147	−0.0660031
$339$	−2505.89	−0.401478
$340$	−731.049	−0.116608
$341$	1678.12	0.266497
$342$	851.649	0.134655
$343$	0	0
$344$	7218.21	1.13134
$345$	1819.67	0.283965
$346$	−3906.10	−0.606916
$347$	−6665.73	−1.03123	−0.515613	−	0.856822i	$-0.672436\pi$
$347$	−6665.73	−1.03123	−0.515613	+	0.856822i	$0.672436\pi$
$348$	2953.58	0.454967
$349$	−1931.57	−0.296259	−0.148129	−	0.988968i	$-0.547325\pi$
$349$	−1931.57	−0.296259	−0.148129	+	0.988968i	$0.547325\pi$
$350$	0	0
$351$	22830.6	3.47182
$352$	−9873.03	−1.49498
$353$	7767.89	1.17123	0.585614	−	0.810590i	$-0.300854\pi$
$353$	7767.89	1.17123	0.585614	+	0.810590i	$0.300854\pi$
$354$	8312.88	1.24809
$355$	912.302	0.136394
$356$	1295.13	0.192814
$357$	0	0
$358$	−1863.05	−0.275043
$359$	5564.98	0.818129	0.409065	−	0.912505i	$-0.365855\pi$
$359$	5564.98	0.818129	0.409065	+	0.912505i	$0.365855\pi$
$360$	1488.30	0.217889
$361$	−6787.55	−0.989583
$362$	1317.29	0.191257
$363$	16414.6	2.37339
$364$	0	0
$365$	−828.567	−0.118820
$366$	6329.75	0.903993
$367$	−3611.04	−0.513609	−0.256805	−	0.966463i	$-0.582670\pi$
$367$	−3611.04	−0.513609	−0.256805	+	0.966463i	$0.582670\pi$
$368$	−4654.23	−0.659290
$369$	−9527.76	−1.34416
$370$	96.5574	0.0135670
$371$	0	0
$372$	2018.86	0.281379
$373$	514.912	0.0714775	0.0357388	−	0.999361i	$-0.488622\pi$
$373$	514.912	0.0714775	0.0357388	+	0.999361i	$0.488622\pi$
$374$	7790.42	1.07709
$375$	2630.77	0.362273
$376$	−6081.71	−0.834149
$377$	−1965.47	−0.268506
$378$	0	0
$379$	−6352.13	−0.860916	−0.430458	−	0.902611i	$-0.641648\pi$
$379$	−6352.13	−0.860916	−0.430458	+	0.902611i	$0.641648\pi$
$380$	55.2303	0.00745593
$381$	−219.655	−0.0295362
$382$	−2417.66	−0.323817
$383$	−7547.21	−1.00690	−0.503452	−	0.864023i	$-0.667937\pi$
$383$	−7547.21	−1.00690	−0.503452	+	0.864023i	$0.667937\pi$
$384$	−14732.3	−1.95782
$385$	0	0
$386$	−5003.90	−0.659824
$387$	−30641.8	−4.02483
$388$	−2289.80	−0.299606
$389$	−2193.93	−0.285956	−0.142978	−	0.989726i	$-0.545668\pi$
$389$	−2193.93	−0.285956	−0.142978	+	0.989726i	$0.545668\pi$
$390$	−589.091	−0.0764866
$391$	19265.2	2.49177
$392$	0	0
$393$	−14343.1	−1.84100
$394$	−3049.13	−0.389881
$395$	−570.808	−0.0727100
$396$	26906.7	3.41443
$397$	3816.01	0.482418	0.241209	−	0.970473i	$-0.422456\pi$
$397$	3816.01	0.482418	0.241209	+	0.970473i	$0.422456\pi$
$398$	−1587.37	−0.199919
$399$	0	0
$400$	−3350.07	−0.418759
$401$	−4494.18	−0.559672	−0.279836	−	0.960048i	$-0.590280\pi$
$401$	−4494.18	−0.559672	−0.279836	+	0.960048i	$0.590280\pi$
$402$	−8586.49	−1.06531
$403$	−1343.46	−0.166061
$404$	8070.65	0.993886
$405$	−3389.53	−0.415869
$406$	0	0
$407$	3946.44	0.480634
$408$	21188.1	2.57100
$409$	−7270.60	−0.878993	−0.439496	−	0.898244i	$-0.644843\pi$
$409$	−7270.60	−0.878993	−0.439496	+	0.898244i	$0.644843\pi$
$410$	161.102	0.0194055
$411$	−10754.7	−1.29073
$412$	8507.63	1.01733
$413$	0	0
$414$	−17348.7	−2.05952
$415$	−86.3335	−0.0102119
$416$	7904.08	0.931562
$417$	19435.7	2.28242
$418$	−588.561	−0.0688695
$419$	−7712.89	−0.899283	−0.449641	−	0.893209i	$-0.648448\pi$
$419$	−7712.89	−0.899283	−0.449641	+	0.893209i	$0.648448\pi$
$420$	0	0
$421$	7128.52	0.825232	0.412616	−	0.910905i	$-0.364615\pi$
$421$	7128.52	0.825232	0.412616	+	0.910905i	$0.364615\pi$
$422$	342.286	0.0394840
$423$	25817.3	2.96756
$424$	13033.8	1.49287
$425$	13866.9	1.58269
$426$	−11695.9	−1.33021
$427$	0	0
$428$	3169.42	0.357943
$429$	−24077.0	−2.70967
$430$	518.113	0.0581061
$431$	3661.65	0.409224	0.204612	−	0.978843i	$-0.434407\pi$
$431$	3661.65	0.409224	0.204612	+	0.978843i	$0.434407\pi$
$432$	14239.6	1.58589
$433$	8571.29	0.951293	0.475646	−	0.879637i	$-0.342214\pi$
$433$	8571.29	0.951293	0.475646	+	0.879637i	$0.342214\pi$
$434$	0	0
$435$	479.283	0.0528273
$436$	−5287.49	−0.580791
$437$	−1455.47	−0.159324
$438$	10622.4	1.15881
$439$	−16167.4	−1.75770	−0.878849	−	0.477100i	$-0.841688\pi$
$439$	−16167.4	−1.75770	−0.878849	+	0.477100i	$0.841688\pi$
$440$	−1028.54	−0.111440
$441$	0	0
$442$	−6236.80	−0.671164
$443$	9825.17	1.05374	0.526871	−	0.849945i	$-0.323365\pi$
$443$	9825.17	1.05374	0.526871	+	0.849945i	$0.323365\pi$
$444$	4747.76	0.507474
$445$	210.163	0.0223880
$446$	−4121.26	−0.437550
$447$	2945.76	0.311699
$448$	0	0
$449$	−18256.8	−1.91892	−0.959458	−	0.281850i	$-0.909052\pi$
$449$	−18256.8	−1.91892	−0.959458	+	0.281850i	$0.909052\pi$
$450$	−12487.4	−1.30814
$451$	6584.47	0.687474
$452$	1549.36	0.161229
$453$	2967.80	0.307814
$454$	7530.84	0.778502
$455$	0	0
$456$	−1600.75	−0.164390
$457$	−7388.48	−0.756277	−0.378139	−	0.925749i	$-0.623436\pi$
$457$	−7388.48	−0.756277	−0.378139	+	0.925749i	$0.623436\pi$
$458$	7799.87	0.795773
$459$	−58941.6	−5.99381
$460$	−1125.08	−0.114037
$461$	−17402.8	−1.75820	−0.879100	−	0.476637i	$-0.841856\pi$
$461$	−17402.8	−1.75820	−0.879100	+	0.476637i	$0.841856\pi$
$462$	0	0
$463$	4741.12	0.475893	0.237946	−	0.971278i	$-0.423526\pi$
$463$	4741.12	0.475893	0.237946	+	0.971278i	$0.423526\pi$
$464$	−1225.88	−0.122651
$465$	327.605	0.0326716
$466$	4691.66	0.466389
$467$	14000.0	1.38725	0.693624	−	0.720338i	$-0.256013\pi$
$467$	14000.0	1.38725	0.693624	+	0.720338i	$0.256013\pi$
$468$	−21540.8	−2.12762
$469$	0	0
$470$	−436.536	−0.0428424
$471$	−4292.07	−0.419890
$472$	−11619.6	−1.13312
$473$	21176.0	2.05851
$474$	7317.90	0.709119
$475$	−1047.63	−0.101197
$476$	0	0
$477$	−55329.4	−5.31102
$478$	3696.67	0.353727
$479$	9622.70	0.917896	0.458948	−	0.888463i	$-0.348226\pi$
$479$	9622.70	0.917896	0.458948	+	0.888463i	$0.348226\pi$
$480$	−1927.42	−0.183280
$481$	−3159.42	−0.299495
$482$	732.490	0.0692199
$483$	0	0
$484$	−10148.9	−0.953129
$485$	−371.571	−0.0347880
$486$	25159.0	2.34822
$487$	−9512.27	−0.885097	−0.442548	−	0.896745i	$-0.645926\pi$
$487$	−9512.27	−0.885097	−0.442548	+	0.896745i	$0.645926\pi$
$488$	−8847.60	−0.820722
$489$	3862.41	0.357187
$490$	0	0
$491$	5570.18	0.511973	0.255986	−	0.966680i	$-0.417600\pi$
$491$	5570.18	0.511973	0.255986	+	0.966680i	$0.417600\pi$
$492$	7921.43	0.725865
$493$	5074.25	0.463555
$494$	471.186	0.0429143
$495$	4366.21	0.396458
$496$	−837.925	−0.0758547
$497$	0	0
$498$	1106.82	0.0995938
$499$	−740.318	−0.0664152	−0.0332076	−	0.999448i	$-0.510572\pi$
$499$	−740.318	−0.0664152	−0.0332076	+	0.999448i	$0.510572\pi$
$500$	−1626.57	−0.145485
$501$	−19777.0	−1.76362
$502$	6127.15	0.544757
$503$	−4786.98	−0.424335	−0.212168	−	0.977233i	$-0.568052\pi$
$503$	−4786.98	−0.424335	−0.212168	+	0.977233i	$0.568052\pi$
$504$	0	0
$505$	1309.64	0.115402
$506$	11989.4	1.05335
$507$	−3272.55	−0.286665
$508$	135.810	0.0118614
$509$	−891.162	−0.0776032	−0.0388016	−	0.999247i	$-0.512354\pi$
$509$	−891.162	−0.0776032	−0.0388016	+	0.999247i	$0.512354\pi$
$510$	1520.85	0.132048
$511$	0	0
$512$	8919.89	0.769936
$513$	4453.00	0.383245
$514$	7382.71	0.633536
$515$	1380.55	0.118125
$516$	25475.8	2.17347
$517$	−17841.9	−1.51777
$518$	0	0
$519$	−31166.6	−2.63596
$520$	823.419	0.0694410
$521$	−17140.9	−1.44138	−0.720688	−	0.693259i	$-0.756174\pi$
$521$	−17140.9	−1.44138	−0.720688	+	0.693259i	$0.756174\pi$
$522$	−4569.47	−0.383142
$523$	−73.8293	−0.00617271	−0.00308636	−	0.999995i	$-0.500982\pi$
$523$	−73.8293	−0.00617271	−0.00308636	+	0.999995i	$0.500982\pi$
$524$	8868.16	0.739327
$525$	0	0
$526$	8242.75	0.683272
$527$	3468.41	0.286691
$528$	−15017.0	−1.23775
$529$	17481.9	1.43683
$530$	935.547	0.0766746
$531$	49326.0	4.03119
$532$	0	0
$533$	−5271.35	−0.428382
$534$	−2694.34	−0.218344
$535$	514.308	0.0415616
$536$	12002.0	0.967181
$537$	−14865.2	−1.19457
$538$	−7180.28	−0.575398
$539$	0	0
$540$	3442.17	0.274310
$541$	−1609.86	−0.127936	−0.0639680	−	0.997952i	$-0.520376\pi$
$541$	−1609.86	−0.127936	−0.0639680	+	0.997952i	$0.520376\pi$
$542$	7845.66	0.621771
$543$	10510.6	0.830669
$544$	−20405.9	−1.60827
$545$	−858.012	−0.0674370
$546$	0	0
$547$	1373.01	0.107323	0.0536616	−	0.998559i	$-0.482911\pi$
$547$	1373.01	0.107323	0.0536616	+	0.998559i	$0.482911\pi$
$548$	6649.47	0.518342
$549$	37558.7	2.91979
$550$	8629.85	0.669051
$551$	−383.356	−0.0296398
$552$	32608.3	2.51432
$553$	0	0
$554$	6438.04	0.493730
$555$	770.428	0.0589241
$556$	−12016.8	−0.916595
$557$	−7155.45	−0.544320	−0.272160	−	0.962252i	$-0.587738\pi$
$557$	−7155.45	−0.544320	−0.272160	+	0.962252i	$0.587738\pi$
$558$	−3123.37	−0.236959
$559$	−16953.0	−1.28271
$560$	0	0
$561$	62159.5	4.67803
$562$	−3555.64	−0.266879
$563$	6327.94	0.473696	0.236848	−	0.971547i	$-0.423886\pi$
$563$	6327.94	0.473696	0.236848	+	0.971547i	$0.423886\pi$
$564$	−21464.6	−1.60252
$565$	251.417	0.0187207
$566$	−1727.22	−0.128269
$567$	0	0
$568$	16348.4	1.20768
$569$	−10823.8	−0.797466	−0.398733	−	0.917067i	$-0.630550\pi$
$569$	−10823.8	−0.797466	−0.398733	+	0.917067i	$0.630550\pi$
$570$	−114.899	−0.00844316
$571$	4257.42	0.312027	0.156014	−	0.987755i	$-0.450136\pi$
$571$	4257.42	0.312027	0.156014	+	0.987755i	$0.450136\pi$
$572$	14886.5	1.08817
$573$	−19290.4	−1.40640
$574$	0	0
$575$	21341.0	1.54779
$576$	1437.87	0.104012
$577$	−9652.95	−0.696460	−0.348230	−	0.937409i	$-0.613217\pi$
$577$	−9652.95	−0.696460	−0.348230	+	0.937409i	$0.613217\pi$
$578$	9782.09	0.703947
$579$	−39926.0	−2.86575
$580$	−296.334	−0.0212149
$581$	0	0
$582$	4763.64	0.339277
$583$	38237.2	2.71633
$584$	−14847.9	−1.05207
$585$	−3495.47	−0.247043
$586$	443.936	0.0312949
$587$	1159.02	0.0814954	0.0407477	−	0.999169i	$-0.487026\pi$
$587$	1159.02	0.0814954	0.0407477	+	0.999169i	$0.487026\pi$
$588$	0	0
$589$	−262.036	−0.0183310
$590$	−834.037	−0.0581979
$591$	−24329.0	−1.69333
$592$	−1970.55	−0.136806
$593$	12071.9	0.835973	0.417986	−	0.908453i	$-0.362736\pi$
$593$	12071.9	0.835973	0.417986	+	0.908453i	$0.362736\pi$
$594$	−36681.5	−2.53377
$595$	0	0
$596$	−1821.32	−0.125175
$597$	−12665.6	−0.868290
$598$	−9598.39	−0.656367
$599$	10108.0	0.689486	0.344743	−	0.938697i	$-0.387966\pi$
$599$	10108.0	0.689486	0.344743	+	0.938697i	$0.387966\pi$
$600$	23471.2	1.59701
$601$	25132.8	1.70581	0.852903	−	0.522069i	$-0.174840\pi$
$601$	25132.8	1.70581	0.852903	+	0.522069i	$0.174840\pi$
$602$	0	0
$603$	−50949.5	−3.44083
$604$	−1834.95	−0.123615
$605$	−1646.89	−0.110670
$606$	−16789.9	−1.12549
$607$	17536.9	1.17265	0.586326	−	0.810075i	$-0.300574\pi$
$607$	17536.9	1.17265	0.586326	+	0.810075i	$0.300574\pi$
$608$	1541.65	0.102833
$609$	0	0
$610$	−635.068	−0.0421527
$611$	14283.7	0.945758
$612$	55611.8	3.67316
$613$	24094.7	1.58756	0.793780	−	0.608205i	$-0.208110\pi$
$613$	24094.7	1.58756	0.793780	+	0.608205i	$0.208110\pi$
$614$	−1897.09	−0.124691
$615$	1285.43	0.0842820
$616$	0	0
$617$	14811.1	0.966408	0.483204	−	0.875508i	$-0.339473\pi$
$617$	14811.1	0.966408	0.483204	+	0.875508i	$0.339473\pi$
$618$	−17699.0	−1.15204
$619$	5388.94	0.349919	0.174959	−	0.984576i	$-0.444021\pi$
$619$	5388.94	0.349919	0.174959	+	0.984576i	$0.444021\pi$
$620$	−202.554	−0.0131206
$621$	−90710.7	−5.86167
$622$	9485.59	0.611475
$623$	0	0
$624$	12022.2	0.771271
$625$	15228.5	0.974625
$626$	228.071	0.0145616
$627$	−4696.11	−0.299114
$628$	2653.73	0.168623
$629$	8156.65	0.517054
$630$	0	0
$631$	−14417.6	−0.909599	−0.454799	−	0.890594i	$-0.650289\pi$
$631$	−14417.6	−0.909599	−0.454799	+	0.890594i	$0.650289\pi$
$632$	−10228.8	−0.643799
$633$	2731.09	0.171487
$634$	5957.09	0.373165
$635$	22.0382	0.00137726
$636$	46001.2	2.86803
$637$	0	0
$638$	3157.88	0.195959
$639$	−69399.9	−4.29643
$640$	1478.10	0.0912923
$641$	−14713.8	−0.906646	−0.453323	−	0.891346i	$-0.649762\pi$
$641$	−14713.8	−0.906646	−0.453323	+	0.891346i	$0.649762\pi$
$642$	−6593.56	−0.405338
$643$	−18089.3	−1.10944	−0.554721	−	0.832037i	$-0.687175\pi$
$643$	−18089.3	−1.10944	−0.554721	+	0.832037i	$0.687175\pi$
$644$	0	0
$645$	4134.01	0.252366
$646$	−1216.46	−0.0740881
$647$	13410.8	0.814891	0.407446	−	0.913229i	$-0.366420\pi$
$647$	13410.8	0.814891	0.407446	+	0.913229i	$0.366420\pi$
$648$	−60740.1	−3.68225
$649$	−34088.4	−2.06176
$650$	−6908.82	−0.416902
$651$	0	0
$652$	−2388.08	−0.143442
$653$	−644.781	−0.0386405	−0.0193203	−	0.999813i	$-0.506150\pi$
$653$	−644.781	−0.0386405	−0.0193203	+	0.999813i	$0.506150\pi$
$654$	10999.9	0.657693
$655$	1439.06	0.0858451
$656$	−3287.78	−0.195680
$657$	63030.2	3.74283
$658$	0	0
$659$	−15096.9	−0.892401	−0.446200	−	0.894933i	$-0.647223\pi$
$659$	−15096.9	−0.892401	−0.446200	+	0.894933i	$0.647223\pi$
$660$	−3630.09	−0.214093
$661$	−5811.60	−0.341974	−0.170987	−	0.985273i	$-0.554696\pi$
$661$	−5811.60	−0.341974	−0.170987	+	0.985273i	$0.554696\pi$
$662$	5063.14	0.297258
$663$	−49763.2	−2.91500
$664$	−1547.09	−0.0904197
$665$	0	0
$666$	−7345.24	−0.427361
$667$	7809.23	0.453335
$668$	12227.9	0.708249
$669$	−32883.4	−1.90037
$670$	861.489	0.0496749
$671$	−25956.2	−1.49333
$672$	0	0
$673$	21598.4	1.23708	0.618542	−	0.785751i	$-0.287723\pi$
$673$	21598.4	1.23708	0.618542	+	0.785751i	$0.287723\pi$
$674$	7.03679	0.000402147 0
$675$	−65292.7	−3.72314
$676$	2023.37	0.115121
$677$	−7224.50	−0.410133	−0.205067	−	0.978748i	$-0.565741\pi$
$677$	−7224.50	−0.410133	−0.205067	+	0.978748i	$0.565741\pi$
$678$	−3223.23	−0.182577
$679$	0	0
$680$	−2125.82	−0.119884
$681$	60088.4	3.38119
$682$	2158.51	0.121193
$683$	−7695.88	−0.431149	−0.215574	−	0.976487i	$-0.569162\pi$
$683$	−7695.88	−0.431149	−0.215574	+	0.976487i	$0.569162\pi$
$684$	−4201.43	−0.234862
$685$	1079.02	0.0601859
$686$	0	0
$687$	62235.0	3.45620
$688$	−10573.7	−0.585927
$689$	−30611.7	−1.69262
$690$	2340.58	0.129137
$691$	−20573.2	−1.13262	−0.566309	−	0.824193i	$-0.691629\pi$
$691$	−20573.2	−1.13262	−0.566309	+	0.824193i	$0.691629\pi$
$692$	19269.9	1.05857
$693$	0	0
$694$	−8573.90	−0.468964
$695$	−1950.00	−0.106428
$696$	8588.71	0.467750
$697$	13609.0	0.739568
$698$	−2484.51	−0.134728
$699$	37434.7	2.02562
$700$	0	0
$701$	−15490.7	−0.834631	−0.417315	−	0.908762i	$-0.637029\pi$
$701$	−15490.7	−0.834631	−0.417315	+	0.908762i	$0.637029\pi$
$702$	29366.2	1.57885
$703$	−616.229	−0.0330605
$704$	−993.685	−0.0531973
$705$	−3483.11	−0.186073
$706$	9991.57	0.532631
$707$	0	0
$708$	−41009.9	−2.17690
$709$	33182.7	1.75769	0.878846	−	0.477105i	$-0.158314\pi$
$709$	33182.7	1.75769	0.878846	+	0.477105i	$0.158314\pi$
$710$	1173.46	0.0620271
$711$	43422.1	2.29037
$712$	3766.10	0.198231
$713$	5337.85	0.280370
$714$	0	0
$715$	2415.66	0.126351
$716$	9190.97	0.479725
$717$	29495.6	1.53631
$718$	7158.04	0.372055
$719$	13090.2	0.678972	0.339486	−	0.940611i	$-0.389747\pi$
$719$	13090.2	0.678972	0.339486	+	0.940611i	$0.389747\pi$
$720$	−2180.15	−0.112846
$721$	0	0
$722$	−8730.59	−0.450026
$723$	5844.52	0.300636
$724$	−6498.57	−0.333588
$725$	5621.01	0.287943
$726$	21113.5	1.07933
$727$	19101.2	0.974449	0.487225	−	0.873277i	$-0.338009\pi$
$727$	19101.2	0.974449	0.487225	+	0.873277i	$0.338009\pi$
$728$	0	0
$729$	111865.	5.68334
$730$	−1065.76	−0.0540349
$731$	43767.4	2.21449
$732$	−31226.5	−1.57673
$733$	21913.8	1.10423	0.552117	−	0.833767i	$-0.313820\pi$
$733$	21913.8	1.10423	0.552117	+	0.833767i	$0.313820\pi$
$734$	−4644.75	−0.233571
$735$	0	0
$736$	−31404.6	−1.57281
$737$	35210.3	1.75982
$738$	−12255.2	−0.611275
$739$	−20105.9	−1.00082	−0.500412	−	0.865788i	$-0.666818\pi$
$739$	−20105.9	−1.00082	−0.500412	+	0.865788i	$0.666818\pi$
$740$	−476.346	−0.0236633
$741$	3759.58	0.186385
$742$	0	0
$743$	3302.81	0.163080	0.0815400	−	0.996670i	$-0.474016\pi$
$743$	3302.81	0.163080	0.0815400	+	0.996670i	$0.474016\pi$
$744$	5870.65	0.289285
$745$	−295.550	−0.0145344
$746$	662.313	0.0325054
$747$	6567.50	0.321676
$748$	−38432.4	−1.87865
$749$	0	0
$750$	3383.87	0.164748
$751$	14554.4	0.707188	0.353594	−	0.935399i	$-0.384959\pi$
$751$	14554.4	0.707188	0.353594	+	0.935399i	$0.384959\pi$
$752$	8908.86	0.432011
$753$	48888.4	2.36599
$754$	−2528.12	−0.122107
$755$	−297.762	−0.0143532
$756$	0	0
$757$	−2978.50	−0.143006	−0.0715030	−	0.997440i	$-0.522780\pi$
$757$	−2978.50	−0.143006	−0.0715030	+	0.997440i	$0.522780\pi$
$758$	−8170.52	−0.391513
$759$	95663.0	4.57490
$760$	160.604	0.00766542
$761$	−9887.45	−0.470985	−0.235493	−	0.971876i	$-0.575670\pi$
$761$	−9887.45	−0.470985	−0.235493	+	0.971876i	$0.575670\pi$
$762$	−282.535	−0.0134320
$763$	0	0
$764$	11927.0	0.564796
$765$	9024.25	0.426500
$766$	−9707.71	−0.457903
$767$	27290.2	1.28474
$768$	−20456.8	−0.961159
$769$	−37520.2	−1.75944	−0.879722	−	0.475489i	$-0.842271\pi$
$769$	−37520.2	−1.75944	−0.879722	+	0.475489i	$0.842271\pi$
$770$	0	0
$771$	58906.4	2.75157
$772$	24685.7	1.15085
$773$	28965.3	1.34775	0.673874	−	0.738847i	$-0.264629\pi$
$773$	28965.3	1.34775	0.673874	+	0.738847i	$0.264629\pi$
$774$	−39413.5	−1.83035
$775$	3842.13	0.178082
$776$	−6658.52	−0.308024
$777$	0	0
$778$	−2821.98	−0.130042
$779$	−1028.15	−0.0472880
$780$	2906.16	0.133407
$781$	47961.2	2.19742
$782$	24780.1	1.13316
$783$	−23892.3	−1.09047
$784$	0	0
$785$	430.626	0.0195793
$786$	−18449.1	−0.837222
$787$	−41230.6	−1.86749	−0.933744	−	0.357943i	$-0.883478\pi$
$787$	−41230.6	−1.86749	−0.933744	+	0.357943i	$0.883478\pi$
$788$	15042.3	0.680024
$789$	65768.7	2.96759
$790$	−734.210	−0.0330659
$791$	0	0
$792$	78242.1	3.51037
$793$	20779.8	0.930534
$794$	4908.40	0.219386
$795$	7464.70	0.333013
$796$	7830.98	0.348696
$797$	6124.65	0.272203	0.136102	−	0.990695i	$-0.456543\pi$
$797$	6124.65	0.272203	0.136102	+	0.990695i	$0.456543\pi$
$798$	0	0
$799$	−36876.3	−1.63278
$800$	−22604.7	−0.998996
$801$	−15987.4	−0.705226
$802$	−5780.70	−0.254518
$803$	−43559.1	−1.91428
$804$	42359.7	1.85810
$805$	0	0
$806$	−1728.05	−0.0755184
$807$	−57291.3	−2.49907
$808$	23468.6	1.02181
$809$	−28936.0	−1.25752	−0.628760	−	0.777599i	$-0.716437\pi$
$809$	−28936.0	−1.25752	−0.628760	+	0.777599i	$0.716437\pi$
$810$	−4359.83	−0.189122
$811$	24351.4	1.05437	0.527184	−	0.849751i	$-0.323248\pi$
$811$	24351.4	1.05437	0.527184	+	0.849751i	$0.323248\pi$
$812$	0	0
$813$	62600.3	2.70048
$814$	5076.17	0.218575
$815$	−387.518	−0.0166554
$816$	−31037.7	−1.33154
$817$	−3306.60	−0.141595
$818$	−9351.92	−0.399734
$819$	0	0
$820$	−794.763	−0.0338467
$821$	−29891.6	−1.27068	−0.635338	−	0.772234i	$-0.719139\pi$
$821$	−29891.6	−1.27068	−0.635338	+	0.772234i	$0.719139\pi$
$822$	−13833.4	−0.586975
$823$	−2102.48	−0.0890495	−0.0445248	−	0.999008i	$-0.514177\pi$
$823$	−2102.48	−0.0890495	−0.0445248	+	0.999008i	$0.514177\pi$
$824$	24739.3	1.04592
$825$	68857.3	2.90582
$826$	0	0
$827$	−38717.1	−1.62796	−0.813981	−	0.580892i	$-0.802704\pi$
$827$	−38717.1	−1.62796	−0.813981	+	0.580892i	$0.802704\pi$
$828$	85586.2	3.59218
$829$	−7832.41	−0.328143	−0.164072	−	0.986448i	$-0.552463\pi$
$829$	−7832.41	−0.328143	−0.164072	+	0.986448i	$0.552463\pi$
$830$	−111.048	−0.00464401
$831$	51369.0	2.14437
$832$	795.518	0.0331486
$833$	0	0
$834$	24999.4	1.03796
$835$	1984.24	0.0822365
$836$	2903.54	0.120121
$837$	−16331.1	−0.674416
$838$	−9920.82	−0.408961
$839$	106.194	0.00436977	0.00218488	−	0.999998i	$-0.499305\pi$
$839$	106.194	0.00436977	0.00218488	+	0.999998i	$0.499305\pi$
$840$	0	0
$841$	−22332.1	−0.915664
$842$	9169.16	0.375285
$843$	−28370.4	−1.15911
$844$	−1688.60	−0.0688673
$845$	328.337	0.0133670
$846$	33207.8	1.34954
$847$	0	0
$848$	−19092.7	−0.773167
$849$	−13781.4	−0.557098
$850$	17836.5	0.719748
$851$	12553.0	0.505655
$852$	57699.5	2.32013
$853$	20215.0	0.811430	0.405715	−	0.914000i	$-0.367022\pi$
$853$	20215.0	0.811430	0.405715	+	0.914000i	$0.367022\pi$
$854$	0	0
$855$	−681.775	−0.0272704
$856$	9216.35	0.368000
$857$	39365.0	1.56906	0.784529	−	0.620093i	$-0.212905\pi$
$857$	39365.0	1.56906	0.784529	+	0.620093i	$0.212905\pi$
$858$	−30969.4	−1.23226
$859$	−7930.06	−0.314983	−0.157491	−	0.987520i	$-0.550341\pi$
$859$	−7930.06	−0.314983	−0.157491	+	0.987520i	$0.550341\pi$
$860$	−2556.00	−0.101348
$861$	0	0
$862$	4709.86	0.186100
$863$	30214.9	1.19180	0.595902	−	0.803057i	$-0.296795\pi$
$863$	30214.9	1.19180	0.595902	+	0.803057i	$0.296795\pi$
$864$	96082.1	3.78331
$865$	3126.97	0.122913
$866$	11024.9	0.432613
$867$	78051.1	3.05738
$868$	0	0
$869$	−30008.3	−1.17142
$870$	616.485	0.0240239
$871$	−28188.4	−1.09659
$872$	−15375.5	−0.597110
$873$	28265.9	1.09583
$874$	−1872.12	−0.0724547
$875$	0	0
$876$	−52403.6	−2.02118
$877$	−5708.88	−0.219812	−0.109906	−	0.993942i	$-0.535055\pi$
$877$	−5708.88	−0.219812	−0.109906	+	0.993942i	$0.535055\pi$
$878$	−20795.6	−0.799337
$879$	3542.15	0.135920
$880$	1506.66	0.0577155
$881$	−13656.9	−0.522260	−0.261130	−	0.965304i	$-0.584095\pi$
$881$	−13656.9	−0.522260	−0.261130	+	0.965304i	$0.584095\pi$
$882$	0	0
$883$	31457.6	1.19891	0.599453	−	0.800410i	$-0.295385\pi$
$883$	31457.6	1.19891	0.599453	+	0.800410i	$0.295385\pi$
$884$	30768.0	1.17063
$885$	−6654.76	−0.252765
$886$	12637.8	0.479203
$887$	−21708.7	−0.821765	−0.410882	−	0.911688i	$-0.634779\pi$
$887$	−21708.7	−0.821765	−0.410882	+	0.911688i	$0.634779\pi$
$888$	13806.0	0.521733
$889$	0	0
$890$	270.325	0.0101813
$891$	−178193.	−6.69999
$892$	20331.4	0.763166
$893$	2785.97	0.104400
$894$	3789.02	0.141749
$895$	1491.44	0.0557020
$896$	0	0
$897$	−76585.3	−2.85073
$898$	−23483.1	−0.872653
$899$	1405.94	0.0521586
$900$	61604.1	2.28163
$901$	79030.0	2.92217
$902$	8469.38	0.312638
$903$	0	0
$904$	4505.37	0.165759
$905$	−1054.54	−0.0387337
$906$	3817.38	0.139982
$907$	−16914.6	−0.619227	−0.309614	−	0.950862i	$-0.600200\pi$
$907$	−16914.6	−0.619227	−0.309614	+	0.950862i	$0.600200\pi$
$908$	−37151.8	−1.35785
$909$	−99626.0	−3.63519
$910$	0	0
$911$	27177.4	0.988395	0.494198	−	0.869350i	$-0.335462\pi$
$911$	27177.4	0.988395	0.494198	+	0.869350i	$0.335462\pi$
$912$	2344.87	0.0851387
$913$	−4538.69	−0.164522
$914$	−9503.55	−0.343927
$915$	−5067.19	−0.183078
$916$	−38479.1	−1.38797
$917$	0	0
$918$	−75814.6	−2.72577
$919$	−6795.11	−0.243906	−0.121953	−	0.992536i	$-0.538916\pi$
$919$	−6795.11	−0.243906	−0.121953	+	0.992536i	$0.538916\pi$
$920$	−3271.62	−0.117241
$921$	−15136.8	−0.541559
$922$	−22384.6	−0.799565
$923$	−38396.4	−1.36927
$924$	0	0
$925$	9035.54	0.321175
$926$	6098.33	0.216419
$927$	−105020.	−3.72095
$928$	−8271.65	−0.292597
$929$	−38256.1	−1.35107	−0.675535	−	0.737328i	$-0.736087\pi$
$929$	−38256.1	−1.35107	−0.675535	+	0.737328i	$0.736087\pi$
$930$	421.386	0.0148579
$931$	0	0
$932$	−23145.4	−0.813467
$933$	75685.2	2.65576
$934$	18007.8	0.630869
$935$	−6236.51	−0.218134
$936$	−62638.6	−2.18740
$937$	−38604.1	−1.34593	−0.672967	−	0.739672i	$-0.734981\pi$
$937$	−38604.1	−1.34593	−0.672967	+	0.739672i	$0.734981\pi$
$938$	0	0
$939$	1819.77	0.0632439
$940$	2153.56	0.0747249
$941$	−45329.0	−1.57033	−0.785166	−	0.619285i	$-0.787423\pi$
$941$	−45329.0	−1.57033	−0.785166	+	0.619285i	$0.787423\pi$
$942$	−5520.74	−0.190951
$943$	20944.2	0.723263
$944$	17021.1	0.586853
$945$	0	0
$946$	27238.0	0.936135
$947$	34584.3	1.18673	0.593367	−	0.804932i	$-0.297798\pi$
$947$	34584.3	1.18673	0.593367	+	0.804932i	$0.297798\pi$
$948$	−36101.4	−1.23683
$949$	34872.3	1.19284
$950$	−1347.53	−0.0460208
$951$	47531.5	1.62073
$952$	0	0
$953$	−32454.6	−1.10316	−0.551578	−	0.834123i	$-0.685974\pi$
$953$	−32454.6	−1.10316	−0.551578	+	0.834123i	$0.685974\pi$
$954$	−71168.2	−2.41526
$955$	1935.42	0.0655798
$956$	−18236.7	−0.616965
$957$	25196.7	0.851089
$958$	12377.3	0.417426
$959$	0	0
$960$	−193.988	−0.00652181
$961$	961.000	0.0322581
$962$	−4063.85	−0.136199
$963$	−39124.0	−1.30920
$964$	−3613.59	−0.120732
$965$	4005.80	0.133628
$966$	0	0
$967$	−7167.37	−0.238353	−0.119176	−	0.992873i	$-0.538025\pi$
$967$	−7167.37	−0.238353	−0.119176	+	0.992873i	$0.538025\pi$
$968$	−29512.0	−0.979910
$969$	−9706.08	−0.321780
$970$	−477.939	−0.0158203
$971$	−42836.2	−1.41574	−0.707868	−	0.706345i	$-0.750343\pi$
$971$	−42836.2	−1.41574	−0.707868	+	0.706345i	$0.750343\pi$
$972$	−124117.	−4.09572
$973$	0	0
$974$	−12235.3	−0.402510
$975$	−55125.3	−1.81069
$976$	12960.5	0.425057
$977$	31612.8	1.03519	0.517596	−	0.855625i	$-0.326827\pi$
$977$	31612.8	1.03519	0.517596	+	0.855625i	$0.326827\pi$
$978$	4968.09	0.162435
$979$	11048.6	0.360689
$980$	0	0
$981$	65270.0	2.12427
$982$	7164.72	0.232826
$983$	5057.04	0.164084	0.0820421	−	0.996629i	$-0.473856\pi$
$983$	5057.04	0.164084	0.0820421	+	0.996629i	$0.473856\pi$
$984$	23034.7	0.746261
$985$	2440.94	0.0789592
$986$	6526.83	0.210808
$987$	0	0
$988$	−2324.50	−0.0748503
$989$	67357.7	2.16567
$990$	5616.11	0.180295
$991$	−33211.6	−1.06458	−0.532292	−	0.846561i	$-0.678669\pi$
$991$	−33211.6	−1.06458	−0.532292	+	0.846561i	$0.678669\pi$
$992$	−5653.92	−0.180960
$993$	40398.7	1.29105
$994$	0	0
$995$	1270.75	0.0404879
$996$	−5460.26	−0.173710
$997$	−10915.8	−0.346746	−0.173373	−	0.984856i	$-0.555467\pi$
$997$	−10915.8	−0.346746	−0.173373	+	0.984856i	$0.555467\pi$
$998$	−952.245	−0.0302032
$999$	−38405.9	−1.21633

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1519.4.a.h.1.15	✓	23
7.6	odd	2		1519.4.a.i.1.15	yes	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1519.4.a.h.1.15	✓	23	1.1	even	1	trivial
1519.4.a.i.1.15	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)

\(f(q)\)	\(=\)	\(q+1.28627 q^{2} +10.2631 q^{3} -6.34552 q^{4} -1.02970 q^{5} +13.2010 q^{6} -18.4521 q^{8} +78.3306 q^{9} -1.32447 q^{10} -54.1330 q^{11} -65.1246 q^{12} +43.3374 q^{13} -10.5679 q^{15} +27.0298 q^{16} -111.884 q^{17} +100.754 q^{18} +8.45276 q^{19} +6.53399 q^{20} -69.6294 q^{22} -172.189 q^{23} -189.376 q^{24} -123.940 q^{25} +55.7434 q^{26} +526.810 q^{27} -45.3527 q^{29} -13.5931 q^{30} -31.0000 q^{31} +182.385 q^{32} -555.571 q^{33} -143.913 q^{34} -497.049 q^{36} -72.9027 q^{37} +10.8725 q^{38} +444.775 q^{39} +19.0002 q^{40} -121.635 q^{41} -391.185 q^{43} +343.502 q^{44} -80.6571 q^{45} -221.480 q^{46} +329.593 q^{47} +277.409 q^{48} -159.419 q^{50} -1148.27 q^{51} -274.999 q^{52} -706.357 q^{53} +677.617 q^{54} +55.7408 q^{55} +86.7513 q^{57} -58.3357 q^{58} +629.715 q^{59} +67.0588 q^{60} +479.489 q^{61} -39.8742 q^{62} +18.3564 q^{64} -44.6246 q^{65} -714.611 q^{66} -650.441 q^{67} +709.963 q^{68} -1767.18 q^{69} -885.987 q^{71} -1445.37 q^{72} +804.668 q^{73} -93.7722 q^{74} -1272.00 q^{75} -53.6372 q^{76} +572.099 q^{78} +554.343 q^{79} -27.8326 q^{80} +3291.76 q^{81} -156.455 q^{82} +83.8433 q^{83} +115.207 q^{85} -503.168 q^{86} -465.458 q^{87} +998.870 q^{88} -204.101 q^{89} -103.746 q^{90} +1092.63 q^{92} -318.155 q^{93} +423.945 q^{94} -8.70382 q^{95} +1871.83 q^{96} +360.853 q^{97} -4240.27 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 5 q^{2} - 6 q^{3} + 91 q^{4} - 40 q^{5} - 36 q^{6} + 39 q^{8} + 211 q^{9} - 40 q^{10} + 44 q^{11} - 414 q^{12} + 20 q^{13} + 523 q^{16} - 306 q^{17} + 51 q^{18} - 296 q^{19} - 400 q^{20} - 326 q^{22}+ \cdots - 3456 q^{99}+O(q^{100}) \) 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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