LMFDB - Embedded newform 1519.4.a.h.1.11

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			1.11
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)

$f(q)$	$=$	\(q+0.189863 q^{2} -5.02070 q^{3} -7.96395 q^{4} -18.3852 q^{5} -0.953245 q^{6} -3.03096 q^{8} -1.79257 q^{9} -3.49067 q^{10} -26.1525 q^{11} +39.9846 q^{12} -59.6686 q^{13} +92.3065 q^{15} +63.1361 q^{16} -108.280 q^{17} -0.340344 q^{18} +66.4797 q^{19} +146.419 q^{20} -4.96539 q^{22} -18.4646 q^{23} +15.2176 q^{24} +213.015 q^{25} -11.3289 q^{26} +144.559 q^{27} +34.6654 q^{29} +17.5256 q^{30} -31.0000 q^{31} +36.2349 q^{32} +131.304 q^{33} -20.5583 q^{34} +14.2760 q^{36} +302.708 q^{37} +12.6220 q^{38} +299.578 q^{39} +55.7248 q^{40} +97.3925 q^{41} -237.966 q^{43} +208.277 q^{44} +32.9568 q^{45} -3.50575 q^{46} -53.6367 q^{47} -316.988 q^{48} +40.4436 q^{50} +543.640 q^{51} +475.198 q^{52} -199.618 q^{53} +27.4464 q^{54} +480.818 q^{55} -333.775 q^{57} +6.58168 q^{58} -475.892 q^{59} -735.124 q^{60} +171.908 q^{61} -5.88575 q^{62} -498.210 q^{64} +1097.02 q^{65} +24.9297 q^{66} +883.588 q^{67} +862.335 q^{68} +92.7054 q^{69} +934.857 q^{71} +5.43323 q^{72} -627.052 q^{73} +57.4731 q^{74} -1069.48 q^{75} -529.441 q^{76} +56.8788 q^{78} +146.960 q^{79} -1160.77 q^{80} -677.387 q^{81} +18.4912 q^{82} -382.128 q^{83} +1990.74 q^{85} -45.1809 q^{86} -174.045 q^{87} +79.2672 q^{88} +398.297 q^{89} +6.25728 q^{90} +147.051 q^{92} +155.642 q^{93} -10.1836 q^{94} -1222.24 q^{95} -181.925 q^{96} +882.691 q^{97} +46.8803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.189863	0.0671267	0.0335634	−	0.999437i	$-0.489314\pi$
$2$	0.189863	0.0671267	0.0335634	+	0.999437i	$0.489314\pi$
$3$	−5.02070	−0.966234	−0.483117	−	0.875556i	$-0.660495\pi$
$3$	−5.02070	−0.966234	−0.483117	+	0.875556i	$0.660495\pi$
$4$	−7.96395	−0.995494
$5$	−18.3852	−1.64442	−0.822210	−	0.569184i	$-0.807259\pi$
$5$	−18.3852	−1.64442	−0.822210	+	0.569184i	$0.807259\pi$
$6$	−0.953245	−0.0648601
$7$	0	0
$7$	0	0
$8$	−3.03096	−0.133951
$9$	−1.79257	−0.0663917
$10$	−3.49067	−0.110385
$11$	−26.1525	−0.716843	−0.358421	−	0.933560i	$-0.616685\pi$
$11$	−26.1525	−0.716843	−0.358421	+	0.933560i	$0.616685\pi$
$12$	39.9846	0.961880
$13$	−59.6686	−1.27301	−0.636503	−	0.771274i	$-0.719620\pi$
$13$	−59.6686	−1.27301	−0.636503	+	0.771274i	$0.719620\pi$
$14$	0	0
$15$	92.3065	1.58890
$16$	63.1361	0.986502
$17$	−108.280	−1.54481	−0.772403	−	0.635132i	$-0.780946\pi$
$17$	−108.280	−1.54481	−0.772403	+	0.635132i	$0.780946\pi$
$18$	−0.340344	−0.00445665
$19$	66.4797	0.802710	0.401355	−	0.915923i	$-0.368539\pi$
$19$	66.4797	0.802710	0.401355	+	0.915923i	$0.368539\pi$
$20$	146.419	1.63701
$21$	0	0
$22$	−4.96539	−0.0481193
$23$	−18.4646	−0.167398	−0.0836988	−	0.996491i	$-0.526673\pi$
$23$	−18.4646	−0.167398	−0.0836988	+	0.996491i	$0.526673\pi$
$24$	15.2176	0.129428
$25$	213.015	1.70412
$26$	−11.3289	−0.0854528
$27$	144.559	1.03038
$28$	0	0
$29$	34.6654	0.221973	0.110986	−	0.993822i	$-0.464599\pi$
$29$	34.6654	0.221973	0.110986	+	0.993822i	$0.464599\pi$
$30$	17.5256	0.106657
$31$	−31.0000	−0.179605
$31$	−31.0000	−0.179605
$32$	36.2349	0.200172
$33$	131.304	0.692638
$34$	−20.5583	−0.103698
$35$	0	0
$36$	14.2760	0.0660925
$37$	302.708	1.34500	0.672499	−	0.740098i	$-0.265221\pi$
$37$	302.708	1.34500	0.672499	+	0.740098i	$0.265221\pi$
$38$	12.6220	0.0538833
$39$	299.578	1.23002
$40$	55.7248	0.220272
$41$	97.3925	0.370979	0.185490	−	0.982646i	$-0.440613\pi$
$41$	97.3925	0.370979	0.185490	+	0.982646i	$0.440613\pi$
$42$	0	0
$43$	−237.966	−0.843940	−0.421970	−	0.906610i	$-0.638661\pi$
$43$	−237.966	−0.843940	−0.421970	+	0.906610i	$0.638661\pi$
$44$	208.277	0.713613
$45$	32.9568	0.109176
$46$	−3.50575	−0.0112368
$47$	−53.6367	−0.166462	−0.0832310	−	0.996530i	$-0.526524\pi$
$47$	−53.6367	−0.166462	−0.0832310	+	0.996530i	$0.526524\pi$
$48$	−316.988	−0.953192
$49$	0	0
$50$	40.4436	0.114392
$51$	543.640	1.49264
$52$	475.198	1.26727
$53$	−199.618	−0.517353	−0.258676	−	0.965964i	$-0.583286\pi$
$53$	−199.618	−0.517353	−0.258676	+	0.965964i	$0.583286\pi$
$54$	27.4464	0.0691663
$55$	480.818	1.17879
$56$	0	0
$57$	−333.775	−0.775606
$58$	6.58168	0.0149003
$59$	−475.892	−1.05010	−0.525049	−	0.851072i	$-0.675953\pi$
$59$	−475.892	−1.05010	−0.525049	+	0.851072i	$0.675953\pi$
$60$	−735.124	−1.58174
$61$	171.908	0.360828	0.180414	−	0.983591i	$-0.442256\pi$
$61$	171.908	0.360828	0.180414	+	0.983591i	$0.442256\pi$
$62$	−5.88575	−0.0120563
$63$	0	0
$64$	−498.210	−0.973065
$65$	1097.02	2.09336
$66$	24.9297	0.0464945
$67$	883.588	1.61116	0.805578	−	0.592490i	$-0.201855\pi$
$67$	883.588	1.61116	0.805578	+	0.592490i	$0.201855\pi$
$68$	862.335	1.53785
$69$	92.7054	0.161745
$70$	0	0
$71$	934.857	1.56264	0.781318	−	0.624133i	$-0.214548\pi$
$71$	934.857	1.56264	0.781318	+	0.624133i	$0.214548\pi$
$72$	5.43323	0.00889323
$73$	−627.052	−1.00535	−0.502677	−	0.864474i	$-0.667651\pi$
$73$	−627.052	−1.00535	−0.502677	+	0.864474i	$0.667651\pi$
$74$	57.4731	0.0902853
$75$	−1069.48	−1.64658
$76$	−529.441	−0.799093
$77$	0	0
$78$	56.8788	0.0825674
$79$	146.960	0.209294	0.104647	−	0.994509i	$-0.466629\pi$
$79$	146.960	0.209294	0.104647	+	0.994509i	$0.466629\pi$
$80$	−1160.77	−1.62222
$81$	−677.387	−0.929200
$82$	18.4912	0.0249026
$83$	−382.128	−0.505349	−0.252675	−	0.967551i	$-0.581310\pi$
$83$	−382.128	−0.505349	−0.252675	+	0.967551i	$0.581310\pi$
$84$	0	0
$85$	1990.74	2.54031
$86$	−45.1809	−0.0566509
$87$	−174.045	−0.214477
$88$	79.2672	0.0960218
$89$	398.297	0.474376	0.237188	−	0.971464i	$-0.423774\pi$
$89$	398.297	0.474376	0.237188	+	0.971464i	$0.423774\pi$
$90$	6.25728	0.00732861
$91$	0	0
$92$	147.051	0.166643
$93$	155.642	0.173541
$94$	−10.1836	−0.0111741
$95$	−1222.24	−1.31999
$96$	−181.925	−0.193413
$97$	882.691	0.923956	0.461978	−	0.886891i	$-0.347140\pi$
$97$	882.691	0.923956	0.461978	+	0.886891i	$0.347140\pi$
$98$	0	0
$99$	46.8803	0.0475924
$100$	−1696.44	−1.69644
$101$	−1365.23	−1.34501	−0.672503	−	0.740094i	$-0.734781\pi$
$101$	−1365.23	−1.34501	−0.672503	+	0.740094i	$0.734781\pi$
$102$	103.217	0.100196
$103$	1838.30	1.75857	0.879287	−	0.476293i	$-0.158020\pi$
$103$	1838.30	1.75857	0.879287	+	0.476293i	$0.158020\pi$
$104$	180.853	0.170520
$105$	0	0
$106$	−37.9002	−0.0347282
$107$	−99.6301	−0.0900150	−0.0450075	−	0.998987i	$-0.514331\pi$
$107$	−99.6301	−0.0900150	−0.0450075	+	0.998987i	$0.514331\pi$
$108$	−1151.26	−1.02574
$109$	1801.25	1.58283	0.791417	−	0.611277i	$-0.209344\pi$
$109$	1801.25	1.58283	0.791417	+	0.611277i	$0.209344\pi$
$110$	91.2896	0.0791284
$111$	−1519.81	−1.29958
$112$	0	0
$113$	1755.44	1.46140	0.730699	−	0.682700i	$-0.239194\pi$
$113$	1755.44	1.46140	0.730699	+	0.682700i	$0.239194\pi$
$114$	−63.3715	−0.0520639
$115$	339.476	0.275272
$116$	−276.074	−0.220972
$117$	106.960	0.0845170
$118$	−90.3542	−0.0704897
$119$	0	0
$120$	−279.778	−0.212834
$121$	−647.048	−0.486136
$122$	32.6389	0.0242212
$123$	−488.979	−0.358453
$124$	246.883	0.178796
$125$	−1618.17	−1.15787
$126$	0	0
$127$	−1013.76	−0.708317	−0.354159	−	0.935185i	$-0.615233\pi$
$127$	−1013.76	−0.708317	−0.354159	+	0.935185i	$0.615233\pi$
$128$	−384.471	−0.265490
$129$	1194.75	0.815444
$130$	208.283	0.140520
$131$	−1425.98	−0.951058	−0.475529	−	0.879700i	$-0.657743\pi$
$131$	−1425.98	−0.951058	−0.475529	+	0.879700i	$0.657743\pi$
$132$	−1045.70	−0.689517
$133$	0	0
$134$	167.761	0.108152
$135$	−2657.74	−1.69438
$136$	328.192	0.206928
$137$	−2405.43	−1.50007	−0.750035	−	0.661399i	$-0.769963\pi$
$137$	−2405.43	−1.50007	−0.750035	+	0.661399i	$0.769963\pi$
$138$	17.6013	0.0108574
$139$	75.0616	0.0458032	0.0229016	−	0.999738i	$-0.492710\pi$
$139$	75.0616	0.0458032	0.0229016	+	0.999738i	$0.492710\pi$
$140$	0	0
$141$	269.294	0.160841
$142$	177.495	0.104895
$143$	1560.48	0.912546
$144$	−113.176	−0.0654955
$145$	−637.330	−0.365016
$146$	−119.054	−0.0674861
$147$	0	0
$148$	−2410.75	−1.33894
$149$	1313.77	0.722335	0.361168	−	0.932501i	$-0.382378\pi$
$149$	1313.77	0.722335	0.361168	+	0.932501i	$0.382378\pi$
$150$	−203.055	−0.110529
$151$	−2478.46	−1.33572	−0.667862	−	0.744285i	$-0.732791\pi$
$151$	−2478.46	−1.33572	−0.667862	+	0.744285i	$0.732791\pi$
$152$	−201.498	−0.107524
$153$	194.100	0.102562
$154$	0	0
$155$	569.941	0.295347
$156$	−2385.83	−1.22448
$157$	−2223.43	−1.13025	−0.565125	−	0.825006i	$-0.691172\pi$
$157$	−2223.43	−1.13025	−0.565125	+	0.825006i	$0.691172\pi$
$158$	27.9022	0.0140492
$159$	1002.22	0.499884
$160$	−666.186	−0.329166
$161$	0	0
$162$	−128.611	−0.0623742
$163$	371.074	0.178311	0.0891556	−	0.996018i	$-0.471583\pi$
$163$	371.074	0.178311	0.0891556	+	0.996018i	$0.471583\pi$
$164$	−775.629	−0.369308
$165$	−2414.04	−1.13899
$166$	−72.5519	−0.0339224
$167$	−1843.30	−0.854123	−0.427062	−	0.904222i	$-0.640451\pi$
$167$	−1843.30	−0.854123	−0.427062	+	0.904222i	$0.640451\pi$
$168$	0	0
$169$	1363.34	0.620546
$170$	377.969	0.170523
$171$	−119.170	−0.0532933
$172$	1895.15	0.840137
$173$	−304.551	−0.133841	−0.0669207	−	0.997758i	$-0.521317\pi$
$173$	−304.551	−0.133841	−0.0669207	+	0.997758i	$0.521317\pi$
$174$	−33.0446	−0.0143972
$175$	0	0
$176$	−1651.17	−0.707167
$177$	2389.31	1.01464
$178$	75.6219	0.0318433
$179$	−1881.45	−0.785622	−0.392811	−	0.919619i	$-0.628497\pi$
$179$	−1881.45	−0.785622	−0.392811	+	0.919619i	$0.628497\pi$
$180$	−262.466	−0.108684
$181$	−1931.68	−0.793264	−0.396632	−	0.917978i	$-0.629821\pi$
$181$	−1931.68	−0.793264	−0.396632	+	0.917978i	$0.629821\pi$
$182$	0	0
$183$	−863.096	−0.348644
$184$	55.9656	0.0224231
$185$	−5565.34	−2.21174
$186$	29.5506	0.0116492
$187$	2831.79	1.10738
$188$	427.160	0.165712
$189$	0	0
$190$	−232.058	−0.0886068
$191$	−2416.72	−0.915538	−0.457769	−	0.889071i	$-0.651351\pi$
$191$	−2416.72	−0.915538	−0.457769	+	0.889071i	$0.651351\pi$
$192$	2501.36	0.940209
$193$	4419.01	1.64812	0.824061	−	0.566501i	$-0.191703\pi$
$193$	4419.01	1.64812	0.824061	+	0.566501i	$0.191703\pi$
$194$	167.590	0.0620221
$195$	−5507.80	−2.02267
$196$	0	0
$197$	2230.28	0.806602	0.403301	−	0.915067i	$-0.367863\pi$
$197$	2230.28	0.806602	0.403301	+	0.915067i	$0.367863\pi$
$198$	8.90083	0.00319472
$199$	−3692.67	−1.31541	−0.657705	−	0.753275i	$-0.728473\pi$
$199$	−3692.67	−1.31541	−0.657705	+	0.753275i	$0.728473\pi$
$200$	−645.640	−0.228268
$201$	−4436.23	−1.55675
$202$	−259.207	−0.0902859
$203$	0	0
$204$	−4329.53	−1.48592
$205$	−1790.58	−0.610046
$206$	349.025	0.118047
$207$	33.0992	0.0111138
$208$	−3767.24	−1.25582
$209$	−1738.61	−0.575417
$210$	0	0
$211$	1668.46	0.544369	0.272184	−	0.962245i	$-0.412254\pi$
$211$	1668.46	0.544369	0.272184	+	0.962245i	$0.412254\pi$
$212$	1589.75	0.515022
$213$	−4693.64	−1.50987
$214$	−18.9161	−0.00604241
$215$	4375.04	1.38779
$216$	−438.153	−0.138021
$217$	0	0
$218$	341.992	0.106250
$219$	3148.24	0.971407
$220$	−3829.21	−1.17348
$221$	6460.90	1.96655
$222$	−288.555	−0.0872367
$223$	−5800.22	−1.74176	−0.870878	−	0.491499i	$-0.836449\pi$
$223$	−5800.22	−1.74176	−0.870878	+	0.491499i	$0.836449\pi$
$224$	0	0
$225$	−381.845	−0.113139
$226$	333.293	0.0980988
$227$	4935.64	1.44313	0.721563	−	0.692349i	$-0.243424\pi$
$227$	4935.64	1.44313	0.721563	+	0.692349i	$0.243424\pi$
$228$	2658.17	0.772111
$229$	586.318	0.169192	0.0845961	−	0.996415i	$-0.473040\pi$
$229$	586.318	0.169192	0.0845961	+	0.996415i	$0.473040\pi$
$230$	64.4539	0.0184781
$231$	0	0
$232$	−105.070	−0.0297334
$233$	−3351.45	−0.942320	−0.471160	−	0.882048i	$-0.656165\pi$
$233$	−3351.45	−0.942320	−0.471160	+	0.882048i	$0.656165\pi$
$234$	20.3078	0.00567335
$235$	986.120	0.273734
$236$	3789.98	1.04537
$237$	−737.840	−0.202227
$238$	0	0
$239$	−5140.39	−1.39123	−0.695615	−	0.718414i	$-0.744868\pi$
$239$	−5140.39	−1.39123	−0.695615	+	0.718414i	$0.744868\pi$
$240$	5827.87	1.56745
$241$	1681.41	0.449416	0.224708	−	0.974426i	$-0.427857\pi$
$241$	1681.41	0.449416	0.224708	+	0.974426i	$0.427857\pi$
$242$	−122.850	−0.0326327
$243$	−502.132	−0.132559
$244$	−1369.06	−0.359202
$245$	0	0
$246$	−92.8389	−0.0240618
$247$	−3966.75	−1.02186
$248$	93.9599	0.0240583
$249$	1918.55	0.488286
$250$	−307.230	−0.0777238
$251$	−698.576	−0.175672	−0.0878362	−	0.996135i	$-0.527995\pi$
$251$	−698.576	−0.175672	−0.0878362	+	0.996135i	$0.527995\pi$
$252$	0	0
$253$	482.896	0.119998
$254$	−192.475	−0.0475470
$255$	−9994.93	−2.45454
$256$	3912.68	0.955244
$257$	747.627	0.181462	0.0907309	−	0.995875i	$-0.471080\pi$
$257$	747.627	0.181462	0.0907309	+	0.995875i	$0.471080\pi$
$258$	226.840	0.0547380
$259$	0	0
$260$	−8736.60	−2.08393
$261$	−62.1403	−0.0147371
$262$	−270.741	−0.0638414
$263$	3081.65	0.722519	0.361260	−	0.932465i	$-0.382347\pi$
$263$	3081.65	0.722519	0.361260	+	0.932465i	$0.382347\pi$
$264$	−397.977	−0.0927795
$265$	3670.02	0.850746
$266$	0	0
$267$	−1999.73	−0.458358
$268$	−7036.85	−1.60390
$269$	−1793.21	−0.406445	−0.203223	−	0.979133i	$-0.565142\pi$
$269$	−1793.21	−0.406445	−0.203223	+	0.979133i	$0.565142\pi$
$270$	−504.607	−0.113738
$271$	1801.22	0.403750	0.201875	−	0.979411i	$-0.435297\pi$
$271$	1801.22	0.403750	0.201875	+	0.979411i	$0.435297\pi$
$272$	−6836.37	−1.52396
$273$	0	0
$274$	−456.702	−0.100695
$275$	−5570.87	−1.22158
$276$	−738.301	−0.161016
$277$	8815.52	1.91218	0.956089	−	0.293078i	$-0.0946795\pi$
$277$	8815.52	1.91218	0.956089	+	0.293078i	$0.0946795\pi$
$278$	14.2514	0.00307462
$279$	55.5698	0.0119243
$280$	0	0
$281$	4440.87	0.942777	0.471389	−	0.881926i	$-0.343753\pi$
$281$	4440.87	0.942777	0.471389	+	0.881926i	$0.343753\pi$
$282$	51.1289	0.0107967
$283$	1478.74	0.310607	0.155303	−	0.987867i	$-0.450365\pi$
$283$	1478.74	0.310607	0.155303	+	0.987867i	$0.450365\pi$
$284$	−7445.16	−1.55559
$285$	6136.51	1.27542
$286$	296.278	0.0612562
$287$	0	0
$288$	−64.9538	−0.0132897
$289$	6811.52	1.38643
$290$	−121.005	−0.0245023
$291$	−4431.73	−0.892757
$292$	4993.81	1.00082
$293$	7529.68	1.50133	0.750663	−	0.660685i	$-0.229734\pi$
$293$	7529.68	1.50133	0.750663	+	0.660685i	$0.229734\pi$
$294$	0	0
$295$	8749.36	1.72680
$296$	−917.498	−0.180164
$297$	−3780.57	−0.738623
$298$	249.436	0.0484880
$299$	1101.76	0.213098
$300$	8517.31	1.63916
$301$	0	0
$302$	−470.568	−0.0896628
$303$	6854.42	1.29959
$304$	4197.27	0.791875
$305$	−3160.55	−0.593353
$306$	36.8523	0.00688467
$307$	9346.97	1.73765	0.868827	−	0.495116i	$-0.164875\pi$
$307$	9346.97	1.73765	0.868827	+	0.495116i	$0.164875\pi$
$308$	0	0
$309$	−9229.55	−1.69919
$310$	108.211	0.0198256
$311$	7064.98	1.28816	0.644081	−	0.764957i	$-0.277240\pi$
$311$	7064.98	1.28816	0.644081	+	0.764957i	$0.277240\pi$
$312$	−908.010	−0.164763
$313$	8784.58	1.58637	0.793185	−	0.608981i	$-0.208421\pi$
$313$	8784.58	1.58637	0.793185	+	0.608981i	$0.208421\pi$
$314$	−422.147	−0.0758699
$315$	0	0
$316$	−1170.38	−0.208351
$317$	9121.55	1.61614	0.808071	−	0.589085i	$-0.200512\pi$
$317$	9121.55	1.61614	0.808071	+	0.589085i	$0.200512\pi$
$318$	190.285	0.0335556
$319$	−906.586	−0.159119
$320$	9159.67	1.60013
$321$	500.213	0.0869755
$322$	0	0
$323$	−7198.41	−1.24003
$324$	5394.68	0.925014
$325$	−12710.3	−2.16935
$326$	70.4532	0.0119694
$327$	−9043.56	−1.52939
$328$	−295.193	−0.0496930
$329$	0	0
$330$	−458.337	−0.0764565
$331$	−1438.77	−0.238918	−0.119459	−	0.992839i	$-0.538116\pi$
$331$	−1438.77	−0.238918	−0.119459	+	0.992839i	$0.538116\pi$
$332$	3043.25	0.503072
$333$	−542.627	−0.0892966
$334$	−349.974	−0.0573345
$335$	−16244.9	−2.64942
$336$	0	0
$337$	11050.8	1.78628	0.893140	−	0.449779i	$-0.148497\pi$
$337$	11050.8	1.78628	0.893140	+	0.449779i	$0.148497\pi$
$338$	258.848	0.0416552
$339$	−8813.54	−1.41205
$340$	−15854.2	−2.52886
$341$	810.727	0.128749
$342$	−22.6260	−0.00357740
$343$	0	0
$344$	721.265	0.113047
$345$	−1704.41	−0.265977
$346$	−57.8229	−0.00898434
$347$	9118.03	1.41061	0.705305	−	0.708904i	$-0.250810\pi$
$347$	9118.03	1.41061	0.705305	+	0.708904i	$0.250810\pi$
$348$	1386.08	0.213511
$349$	−2409.57	−0.369574	−0.184787	−	0.982779i	$-0.559160\pi$
$349$	−2409.57	−0.369574	−0.184787	+	0.982779i	$0.559160\pi$
$350$	0	0
$351$	−8625.62	−1.31169
$352$	−947.633	−0.143492
$353$	−1260.02	−0.189983	−0.0949915	−	0.995478i	$-0.530282\pi$
$353$	−1260.02	−0.189983	−0.0949915	+	0.995478i	$0.530282\pi$
$354$	453.642	0.0681095
$355$	−17187.5	−2.56963
$356$	−3172.02	−0.472238
$357$	0	0
$358$	−357.218	−0.0527362
$359$	3985.73	0.585958	0.292979	−	0.956119i	$-0.405353\pi$
$359$	3985.73	0.585958	0.292979	+	0.956119i	$0.405353\pi$
$360$	−99.8909	−0.0146242
$361$	−2439.45	−0.355656
$362$	−366.755	−0.0532492
$363$	3248.63	0.469722
$364$	0	0
$365$	11528.5	1.65322
$366$	−163.870	−0.0234033
$367$	−10185.2	−1.44868	−0.724338	−	0.689445i	$-0.757855\pi$
$367$	−10185.2	−1.44868	−0.724338	+	0.689445i	$0.757855\pi$
$368$	−1165.79	−0.165138
$369$	−174.583	−0.0246299
$370$	−1056.65	−0.148467
$371$	0	0
$372$	−1239.52	−0.172759
$373$	−11667.5	−1.61963	−0.809814	−	0.586687i	$-0.800432\pi$
$373$	−11667.5	−1.61963	−0.809814	+	0.586687i	$0.800432\pi$
$374$	537.651	0.0743350
$375$	8124.34	1.11877
$376$	162.571	0.0222978
$377$	−2068.44	−0.282573
$378$	0	0
$379$	5670.62	0.768549	0.384274	−	0.923219i	$-0.374452\pi$
$379$	5670.62	0.768549	0.384274	+	0.923219i	$0.374452\pi$
$380$	9733.87	1.31405
$381$	5089.76	0.684400
$382$	−458.846	−0.0614570
$383$	2928.46	0.390698	0.195349	−	0.980734i	$-0.437416\pi$
$383$	2928.46	0.390698	0.195349	+	0.980734i	$0.437416\pi$
$384$	1930.31	0.256526
$385$	0	0
$386$	839.007	0.110633
$387$	426.571	0.0560306
$388$	−7029.71	−0.919792
$389$	4274.20	0.557097	0.278549	−	0.960422i	$-0.410147\pi$
$389$	4274.20	0.557097	0.278549	+	0.960422i	$0.410147\pi$
$390$	−1045.73	−0.135775
$391$	1999.35	0.258597
$392$	0	0
$393$	7159.43	0.918945
$394$	423.447	0.0541445
$395$	−2701.88	−0.344168
$396$	−373.352	−0.0473779
$397$	5235.93	0.661924	0.330962	−	0.943644i	$-0.392627\pi$
$397$	5235.93	0.661924	0.330962	+	0.943644i	$0.392627\pi$
$398$	−701.102	−0.0882992
$399$	0	0
$400$	13448.9	1.68112
$401$	−2588.32	−0.322330	−0.161165	−	0.986927i	$-0.551525\pi$
$401$	−2588.32	−0.322330	−0.161165	+	0.986927i	$0.551525\pi$
$402$	−842.276	−0.104500
$403$	1849.73	0.228639
$404$	10872.6	1.33895
$405$	12453.9	1.52800
$406$	0	0
$407$	−7916.57	−0.964152
$408$	−1647.75	−0.199941
$409$	16041.5	1.93937	0.969686	−	0.244353i	$-0.0785756\pi$
$409$	16041.5	1.93937	0.969686	+	0.244353i	$0.0785756\pi$
$410$	−339.965	−0.0409504
$411$	12076.9	1.44942
$412$	−14640.1	−1.75065
$413$	0	0
$414$	6.28432	0.000746033 0
$415$	7025.49	0.831006
$416$	−2162.09	−0.254820
$417$	−376.862	−0.0442566
$418$	−330.098	−0.0386258
$419$	4649.11	0.542062	0.271031	−	0.962571i	$-0.412635\pi$
$419$	4649.11	0.542062	0.271031	+	0.962571i	$0.412635\pi$
$420$	0	0
$421$	2949.68	0.341469	0.170734	−	0.985317i	$-0.445386\pi$
$421$	2949.68	0.341469	0.170734	+	0.985317i	$0.445386\pi$
$422$	316.780	0.0365417
$423$	96.1478	0.0110517
$424$	605.036	0.0692999
$425$	−23065.2	−2.63253
$426$	−891.148	−0.101353
$427$	0	0
$428$	793.449	0.0896094
$429$	−7834.71	−0.881733
$430$	830.658	0.0931579
$431$	−13667.1	−1.52743	−0.763713	−	0.645556i	$-0.776626\pi$
$431$	−13667.1	−1.52743	−0.763713	+	0.645556i	$0.776626\pi$
$432$	9126.89	1.01648
$433$	−9160.63	−1.01670	−0.508351	−	0.861150i	$-0.669745\pi$
$433$	−9160.63	−1.01670	−0.508351	+	0.861150i	$0.669745\pi$
$434$	0	0
$435$	3199.84	0.352691
$436$	−14345.1	−1.57570
$437$	−1227.52	−0.134372
$438$	597.734	0.0652074
$439$	−3674.59	−0.399496	−0.199748	−	0.979847i	$-0.564012\pi$
$439$	−3674.59	−0.399496	−0.199748	+	0.979847i	$0.564012\pi$
$440$	−1457.34	−0.157900
$441$	0	0
$442$	1226.69	0.132008
$443$	−18179.6	−1.94975	−0.974875	−	0.222751i	$-0.928496\pi$
$443$	−18179.6	−1.94975	−0.974875	+	0.222751i	$0.928496\pi$
$444$	12103.7	1.29373
$445$	−7322.77	−0.780073
$446$	−1101.25	−0.116918
$447$	−6596.03	−0.697945
$448$	0	0
$449$	−4008.20	−0.421289	−0.210644	−	0.977563i	$-0.567556\pi$
$449$	−4008.20	−0.421289	−0.210644	+	0.977563i	$0.567556\pi$
$450$	−72.4982	−0.00759467
$451$	−2547.06	−0.265934
$452$	−13980.2	−1.45481
$453$	12443.6	1.29062
$454$	937.095	0.0968723
$455$	0	0
$456$	1011.66	0.103893
$457$	−1793.39	−0.183570	−0.0917850	−	0.995779i	$-0.529257\pi$
$457$	−1793.39	−0.183570	−0.0917850	+	0.995779i	$0.529257\pi$
$458$	111.320	0.0113573
$459$	−15652.8	−1.59174
$460$	−2703.57	−0.274032
$461$	17027.5	1.72028	0.860140	−	0.510057i	$-0.170376\pi$
$461$	17027.5	1.72028	0.860140	+	0.510057i	$0.170376\pi$
$462$	0	0
$463$	833.738	0.0836870	0.0418435	−	0.999124i	$-0.486677\pi$
$463$	833.738	0.0836870	0.0418435	+	0.999124i	$0.486677\pi$
$464$	2188.64	0.218976
$465$	−2861.50	−0.285374
$466$	−636.316	−0.0632549
$467$	−6510.98	−0.645165	−0.322583	−	0.946541i	$-0.604551\pi$
$467$	−6510.98	−0.645165	−0.322583	+	0.946541i	$0.604551\pi$
$468$	−851.828	−0.0841362
$469$	0	0
$470$	187.228	0.0183748
$471$	11163.2	1.09209
$472$	1442.41	0.140662
$473$	6223.39	0.604972
$474$	−140.088	−0.0135748
$475$	14161.2	1.36791
$476$	0	0
$477$	357.831	0.0343479
$478$	−975.970	−0.0933887
$479$	16296.9	1.55454	0.777270	−	0.629167i	$-0.216604\pi$
$479$	16296.9	1.55454	0.777270	+	0.629167i	$0.216604\pi$
$480$	3344.72	0.318052
$481$	−18062.2	−1.71219
$482$	319.238	0.0301678
$483$	0	0
$484$	5153.06	0.483946
$485$	−16228.4	−1.51937
$486$	−95.3363	−0.00889824
$487$	6650.41	0.618807	0.309403	−	0.950931i	$-0.399871\pi$
$487$	6650.41	0.618807	0.309403	+	0.950931i	$0.399871\pi$
$488$	−521.046	−0.0483332
$489$	−1863.05	−0.172290
$490$	0	0
$491$	−2309.74	−0.212295	−0.106148	−	0.994350i	$-0.533852\pi$
$491$	−2309.74	−0.212295	−0.106148	+	0.994350i	$0.533852\pi$
$492$	3894.20	0.356838
$493$	−3753.56	−0.342905
$494$	−753.139	−0.0685938
$495$	−861.902	−0.0782619
$496$	−1957.22	−0.177181
$497$	0	0
$498$	364.261	0.0327770
$499$	8593.83	0.770967	0.385484	−	0.922715i	$-0.374035\pi$
$499$	8593.83	0.770967	0.385484	+	0.922715i	$0.374035\pi$
$500$	12887.0	1.15265
$501$	9254.64	0.825283
$502$	−132.634	−0.0117923
$503$	−1288.24	−0.114194	−0.0570970	−	0.998369i	$-0.518184\pi$
$503$	−1288.24	−0.114194	−0.0570970	+	0.998369i	$0.518184\pi$
$504$	0	0
$505$	25100.0	2.21176
$506$	91.6841	0.00805505
$507$	−6844.92	−0.599593
$508$	8073.50	0.705126
$509$	−12839.1	−1.11804	−0.559020	−	0.829154i	$-0.688822\pi$
$509$	−12839.1	−1.11804	−0.559020	+	0.829154i	$0.688822\pi$
$510$	−1897.67	−0.164765
$511$	0	0
$512$	3818.64	0.329613
$513$	9610.23	0.827100
$514$	141.947	0.0121809
$515$	−33797.5	−2.89183
$516$	−9514.96	−0.811769
$517$	1402.73	0.119327
$518$	0	0
$519$	1529.06	0.129322
$520$	−3325.02	−0.280407
$521$	17105.0	1.43835	0.719177	−	0.694827i	$-0.244519\pi$
$521$	17105.0	1.43835	0.719177	+	0.694827i	$0.244519\pi$
$522$	−11.7981	−0.000989255 0
$523$	17844.2	1.49191	0.745957	−	0.665994i	$-0.231992\pi$
$523$	17844.2	1.49191	0.745957	+	0.665994i	$0.231992\pi$
$524$	11356.5	0.946773
$525$	0	0
$526$	585.091	0.0485003
$527$	3356.67	0.277455
$528$	8290.01	0.683289
$529$	−11826.1	−0.971978
$530$	696.801	0.0571078
$531$	853.072	0.0697178
$532$	0	0
$533$	−5811.27	−0.472259
$534$	−379.675	−0.0307681
$535$	1831.72	0.148022
$536$	−2678.12	−0.215816
$537$	9446.20	0.759095
$538$	−340.464	−0.0272833
$539$	0	0
$540$	21166.1	1.68675
$541$	−15463.3	−1.22887	−0.614437	−	0.788966i	$-0.710617\pi$
$541$	−15463.3	−1.22887	−0.614437	+	0.788966i	$0.710617\pi$
$542$	341.985	0.0271024
$543$	9698.40	0.766479
$544$	−3923.51	−0.309226
$545$	−33116.4	−2.60284
$546$	0	0
$547$	−11655.3	−0.911054	−0.455527	−	0.890222i	$-0.650549\pi$
$547$	−11655.3	−0.911054	−0.455527	+	0.890222i	$0.650549\pi$
$548$	19156.7	1.49331
$549$	−308.157	−0.0239560
$550$	−1057.70	−0.0820010
$551$	2304.55	0.178180
$552$	−280.987	−0.0216659
$553$	0	0
$554$	1673.74	0.128358
$555$	27941.9	2.13706
$556$	−597.787	−0.0455968
$557$	6751.54	0.513594	0.256797	−	0.966465i	$-0.417333\pi$
$557$	6751.54	0.513594	0.256797	+	0.966465i	$0.417333\pi$
$558$	10.5507	0.000800439 0
$559$	14199.1	1.07434
$560$	0	0
$561$	−14217.5	−1.06999
$562$	843.158	0.0632855
$563$	5964.59	0.446496	0.223248	−	0.974762i	$-0.428334\pi$
$563$	5964.59	0.446496	0.223248	+	0.974762i	$0.428334\pi$
$564$	−2144.64	−0.160117
$565$	−32274.1	−2.40315
$566$	280.757	0.0208500
$567$	0	0
$568$	−2833.52	−0.209317
$569$	−24472.5	−1.80306	−0.901529	−	0.432719i	$-0.857554\pi$
$569$	−24472.5	−1.80306	−0.901529	+	0.432719i	$0.857554\pi$
$570$	1165.10	0.0856149
$571$	14479.7	1.06122	0.530611	−	0.847615i	$-0.321962\pi$
$571$	14479.7	1.06122	0.530611	+	0.847615i	$0.321962\pi$
$572$	−12427.6	−0.908434
$573$	12133.6	0.884624
$574$	0	0
$575$	−3933.24	−0.285265
$576$	893.078	0.0646034
$577$	8850.73	0.638580	0.319290	−	0.947657i	$-0.396556\pi$
$577$	8850.73	0.638580	0.319290	+	0.947657i	$0.396556\pi$
$578$	1293.26	0.0930663
$579$	−22186.5	−1.59247
$580$	5075.66	0.363371
$581$	0	0
$582$	−841.421	−0.0599279
$583$	5220.52	0.370861
$584$	1900.57	0.134668
$585$	−1966.49	−0.138982
$586$	1429.61	0.100779
$587$	8602.94	0.604908	0.302454	−	0.953164i	$-0.402194\pi$
$587$	8602.94	0.604908	0.302454	+	0.953164i	$0.402194\pi$
$588$	0	0
$589$	−2060.87	−0.144171
$590$	1661.18	0.115915
$591$	−11197.5	−0.779366
$592$	19111.8	1.32684
$593$	6629.29	0.459077	0.229538	−	0.973300i	$-0.426278\pi$
$593$	6629.29	0.459077	0.229538	+	0.973300i	$0.426278\pi$
$594$	−717.791	−0.0495814
$595$	0	0
$596$	−10462.8	−0.719081
$597$	18539.8	1.27099
$598$	209.183	0.0143046
$599$	18626.8	1.27057	0.635284	−	0.772279i	$-0.280883\pi$
$599$	18626.8	1.27057	0.635284	+	0.772279i	$0.280883\pi$
$600$	3241.57	0.220561
$601$	−8726.15	−0.592258	−0.296129	−	0.955148i	$-0.595696\pi$
$601$	−8726.15	−0.592258	−0.296129	+	0.955148i	$0.595696\pi$
$602$	0	0
$603$	−1583.90	−0.106967
$604$	19738.4	1.32971
$605$	11896.1	0.799413
$606$	1301.40	0.0872373
$607$	−3181.48	−0.212739	−0.106369	−	0.994327i	$-0.533923\pi$
$607$	−3181.48	−0.212739	−0.106369	+	0.994327i	$0.533923\pi$
$608$	2408.89	0.160680
$609$	0	0
$610$	−600.072	−0.0398298
$611$	3200.43	0.211907
$612$	−1545.80	−0.102100
$613$	19500.2	1.28484	0.642419	−	0.766353i	$-0.277931\pi$
$613$	19500.2	1.28484	0.642419	+	0.766353i	$0.277931\pi$
$614$	1774.64	0.116643
$615$	8989.96	0.589447
$616$	0	0
$617$	−18196.3	−1.18729	−0.593643	−	0.804728i	$-0.702311\pi$
$617$	−18196.3	−1.18729	−0.593643	+	0.804728i	$0.702311\pi$
$618$	−1752.35	−0.114061
$619$	−11420.7	−0.741578	−0.370789	−	0.928717i	$-0.620913\pi$
$619$	−11420.7	−0.741578	−0.370789	+	0.928717i	$0.620913\pi$
$620$	−4538.98	−0.294016
$621$	−2669.23	−0.172484
$622$	1341.38	0.0864701
$623$	0	0
$624$	18914.2	1.21342
$625$	3123.46	0.199901
$626$	1667.87	0.106488
$627$	8729.04	0.555988
$628$	17707.3	1.12516
$629$	−32777.2	−2.07776
$630$	0	0
$631$	−21082.7	−1.33009	−0.665047	−	0.746802i	$-0.731589\pi$
$631$	−21082.7	−1.33009	−0.665047	+	0.746802i	$0.731589\pi$
$632$	−445.429	−0.0280352
$633$	−8376.86	−0.525988
$634$	1731.84	0.108486
$635$	18638.1	1.16477
$636$	−7981.67	−0.497632
$637$	0	0
$638$	−172.127	−0.0106812
$639$	−1675.80	−0.103746
$640$	7068.57	0.436578
$641$	−26067.7	−1.60626	−0.803128	−	0.595806i	$-0.796833\pi$
$641$	−26067.7	−1.60626	−0.803128	+	0.595806i	$0.796833\pi$
$642$	94.9719	0.00583838
$643$	30550.6	1.87371	0.936857	−	0.349712i	$-0.113721\pi$
$643$	30550.6	1.87371	0.936857	+	0.349712i	$0.113721\pi$
$644$	0	0
$645$	−21965.8	−1.34093
$646$	−1366.71	−0.0832393
$647$	−7326.36	−0.445176	−0.222588	−	0.974913i	$-0.571451\pi$
$647$	−7326.36	−0.445176	−0.222588	+	0.974913i	$0.571451\pi$
$648$	2053.14	0.124467
$649$	12445.8	0.752756
$650$	−2413.21	−0.145622
$651$	0	0
$652$	−2955.21	−0.177508
$653$	6410.45	0.384166	0.192083	−	0.981379i	$-0.438476\pi$
$653$	6410.45	0.384166	0.192083	+	0.981379i	$0.438476\pi$
$654$	−1717.04	−0.102663
$655$	26216.9	1.56394
$656$	6148.99	0.365972
$657$	1124.04	0.0667471
$658$	0	0
$659$	−8128.08	−0.480463	−0.240231	−	0.970716i	$-0.577223\pi$
$659$	−8128.08	−0.480463	−0.240231	+	0.970716i	$0.577223\pi$
$660$	19225.3	1.13386
$661$	4380.63	0.257771	0.128886	−	0.991659i	$-0.458860\pi$
$661$	4380.63	0.257771	0.128886	+	0.991659i	$0.458860\pi$
$662$	−273.169	−0.0160378
$663$	−32438.3	−1.90015
$664$	1158.22	0.0676920
$665$	0	0
$666$	−103.025	−0.00599419
$667$	−640.084	−0.0371576
$668$	14679.9	0.850275
$669$	29121.2	1.68294
$670$	−3084.31	−0.177847
$671$	−4495.81	−0.258657
$672$	0	0
$673$	−26482.7	−1.51684	−0.758419	−	0.651767i	$-0.774028\pi$
$673$	−26482.7	−1.51684	−0.758419	+	0.651767i	$0.774028\pi$
$674$	2098.14	0.119907
$675$	30793.2	1.75590
$676$	−10857.6	−0.617750
$677$	−12797.0	−0.726483	−0.363241	−	0.931695i	$-0.618330\pi$
$677$	−12797.0	−0.726483	−0.363241	+	0.931695i	$0.618330\pi$
$678$	−1673.37	−0.0947864
$679$	0	0
$680$	−6033.87	−0.340277
$681$	−24780.3	−1.39440
$682$	153.927	0.00864248
$683$	−23096.6	−1.29395	−0.646975	−	0.762511i	$-0.723966\pi$
$683$	−23096.6	−1.29395	−0.646975	+	0.762511i	$0.723966\pi$
$684$	949.063	0.0530531
$685$	44224.2	2.46674
$686$	0	0
$687$	−2943.73	−0.163479
$688$	−15024.2	−0.832549
$689$	11911.0	0.658594
$690$	−323.604	−0.0178542
$691$	−27750.3	−1.52774	−0.763871	−	0.645369i	$-0.776704\pi$
$691$	−27750.3	−1.52774	−0.763871	+	0.645369i	$0.776704\pi$
$692$	2425.43	0.133238
$693$	0	0
$694$	1731.18	0.0946896
$695$	−1380.02	−0.0753197
$696$	527.523	0.0287295
$697$	−10545.6	−0.573091
$698$	−457.488	−0.0248083
$699$	16826.6	0.910502
$700$	0	0
$701$	14392.6	0.775462	0.387731	−	0.921772i	$-0.373259\pi$
$701$	14392.6	0.775462	0.387731	+	0.921772i	$0.373259\pi$
$702$	−1637.69	−0.0880491
$703$	20124.0	1.07964
$704$	13029.4	0.697535
$705$	−4951.01	−0.264491
$706$	−239.231	−0.0127529
$707$	0	0
$708$	−19028.3	−1.01007
$709$	−24582.0	−1.30211	−0.651055	−	0.759030i	$-0.725673\pi$
$709$	−24582.0	−1.30211	−0.651055	+	0.759030i	$0.725673\pi$
$710$	−3263.27	−0.172491
$711$	−263.436	−0.0138954
$712$	−1207.22	−0.0635431
$713$	572.404	0.0300655
$714$	0	0
$715$	−28689.7	−1.50061
$716$	14983.8	0.782082
$717$	25808.4	1.34425
$718$	756.743	0.0393334
$719$	10805.2	0.560454	0.280227	−	0.959934i	$-0.409590\pi$
$719$	10805.2	0.560454	0.280227	+	0.959934i	$0.409590\pi$
$720$	2080.77	0.107702
$721$	0	0
$722$	−463.161	−0.0238740
$723$	−8441.87	−0.434241
$724$	15383.8	0.789690
$725$	7384.24	0.378267
$726$	616.795	0.0315309
$727$	−3603.08	−0.183811	−0.0919056	−	0.995768i	$-0.529296\pi$
$727$	−3603.08	−0.183811	−0.0919056	+	0.995768i	$0.529296\pi$
$728$	0	0
$729$	20810.5	1.05728
$730$	2188.83	0.110976
$731$	25766.9	1.30372
$732$	6873.66	0.347073
$733$	−20860.7	−1.05117	−0.525584	−	0.850742i	$-0.676153\pi$
$733$	−20860.7	−1.05117	−0.525584	+	0.850742i	$0.676153\pi$
$734$	−1933.80	−0.0972449
$735$	0	0
$736$	−669.065	−0.0335082
$737$	−23108.0	−1.15495
$738$	−33.1469	−0.00165333
$739$	12402.1	0.617346	0.308673	−	0.951168i	$-0.400115\pi$
$739$	12402.1	0.617346	0.308673	+	0.951168i	$0.400115\pi$
$740$	44322.1	2.20178
$741$	19915.9	0.987352
$742$	0	0
$743$	11254.3	0.555695	0.277847	−	0.960625i	$-0.410379\pi$
$743$	11254.3	0.555695	0.277847	+	0.960625i	$0.410379\pi$
$744$	−471.744	−0.0232459
$745$	−24153.8	−1.18782
$746$	−2215.23	−0.108720
$747$	684.993	0.0335510
$748$	−22552.2	−1.10239
$749$	0	0
$750$	1542.51	0.0750994
$751$	1281.14	0.0622498	0.0311249	−	0.999516i	$-0.490091\pi$
$751$	1281.14	0.0622498	0.0311249	+	0.999516i	$0.490091\pi$
$752$	−3386.41	−0.164215
$753$	3507.34	0.169741
$754$	−392.719	−0.0189682
$755$	45567.0	2.19649
$756$	0	0
$757$	26012.4	1.24893	0.624463	−	0.781054i	$-0.285318\pi$
$757$	26012.4	1.24893	0.624463	+	0.781054i	$0.285318\pi$
$758$	1076.64	0.0515901
$759$	−2424.48	−0.115946
$760$	3704.57	0.176814
$761$	11446.5	0.545248	0.272624	−	0.962121i	$-0.412108\pi$
$761$	11446.5	0.545248	0.272624	+	0.962121i	$0.412108\pi$
$762$	966.358	0.0459415
$763$	0	0
$764$	19246.6	0.911412
$765$	−3568.56	−0.168655
$766$	556.007	0.0262263
$767$	28395.8	1.33678
$768$	−19644.4	−0.922989
$769$	−5414.70	−0.253913	−0.126956	−	0.991908i	$-0.540521\pi$
$769$	−5414.70	−0.253913	−0.126956	+	0.991908i	$0.540521\pi$
$770$	0	0
$771$	−3753.61	−0.175335
$772$	−35192.8	−1.64070
$773$	−13222.0	−0.615218	−0.307609	−	0.951513i	$-0.599529\pi$
$773$	−13222.0	−0.615218	−0.307609	+	0.951513i	$0.599529\pi$
$774$	80.9901	0.00376115
$775$	−6603.46	−0.306069
$776$	−2675.40	−0.123765
$777$	0	0
$778$	811.513	0.0373961
$779$	6474.63	0.297789
$780$	43863.8	2.01356
$781$	−24448.8	−1.12016
$782$	379.602	0.0173588
$783$	5011.19	0.228717
$784$	0	0
$785$	40878.2	1.85861
$786$	1359.31	0.0616858
$787$	−6413.52	−0.290492	−0.145246	−	0.989396i	$-0.546397\pi$
$787$	−6413.52	−0.290492	−0.145246	+	0.989396i	$0.546397\pi$
$788$	−17761.8	−0.802968
$789$	−15472.0	−0.698123
$790$	−512.987	−0.0231028
$791$	0	0
$792$	−142.092	−0.00637504
$793$	−10257.5	−0.459336
$794$	994.110	0.0444328
$795$	−18426.1	−0.822020
$796$	29408.3	1.30948
$797$	−9137.86	−0.406122	−0.203061	−	0.979166i	$-0.565089\pi$
$797$	−9137.86	−0.406122	−0.203061	+	0.979166i	$0.565089\pi$
$798$	0	0
$799$	5807.77	0.257152
$800$	7718.58	0.341116
$801$	−713.978	−0.0314946
$802$	−491.426	−0.0216370
$803$	16399.0	0.720681
$804$	35329.9	1.54974
$805$	0	0
$806$	351.195	0.0153478
$807$	9003.16	0.392721
$808$	4137.97	0.180165
$809$	−14991.7	−0.651522	−0.325761	−	0.945452i	$-0.605620\pi$
$809$	−14991.7	−0.651522	−0.325761	+	0.945452i	$0.605620\pi$
$810$	2364.53	0.102569
$811$	26765.1	1.15888	0.579439	−	0.815016i	$-0.303272\pi$
$811$	26765.1	1.15888	0.579439	+	0.815016i	$0.303272\pi$
$812$	0	0
$813$	−9043.38	−0.390117
$814$	−1503.06	−0.0647203
$815$	−6822.25	−0.293219
$816$	34323.4	1.47250
$817$	−15819.9	−0.677439
$818$	3045.70	0.130184
$819$	0	0
$820$	14260.1	0.607297
$821$	−39156.5	−1.66452	−0.832259	−	0.554387i	$-0.812953\pi$
$821$	−39156.5	−1.66452	−0.832259	+	0.554387i	$0.812953\pi$
$822$	2292.96	0.0972947
$823$	−20156.1	−0.853701	−0.426850	−	0.904322i	$-0.640377\pi$
$823$	−20156.1	−0.853701	−0.426850	+	0.904322i	$0.640377\pi$
$824$	−5571.82	−0.235563
$825$	27969.6	1.18034
$826$	0	0
$827$	−31882.8	−1.34060	−0.670299	−	0.742091i	$-0.733834\pi$
$827$	−31882.8	−1.34060	−0.670299	+	0.742091i	$0.733834\pi$
$828$	−263.601	−0.0110637
$829$	−21380.8	−0.895762	−0.447881	−	0.894093i	$-0.647821\pi$
$829$	−21380.8	−0.895762	−0.447881	+	0.894093i	$0.647821\pi$
$830$	1333.88	0.0557827
$831$	−44260.1	−1.84761
$832$	29727.5	1.23872
$833$	0	0
$834$	−71.5521	−0.00297080
$835$	33889.3	1.40454
$836$	13846.2	0.572824
$837$	−4481.33	−0.185062
$838$	882.694	0.0363868
$839$	30576.8	1.25820	0.629100	−	0.777325i	$-0.283424\pi$
$839$	30576.8	1.25820	0.629100	+	0.777325i	$0.283424\pi$
$840$	0	0
$841$	−23187.3	−0.950728
$842$	560.034	0.0229217
$843$	−22296.3	−0.910943
$844$	−13287.6	−0.541916
$845$	−25065.2	−1.02044
$846$	18.2549	0.000741864 0
$847$	0	0
$848$	−12603.1	−0.510370
$849$	−7424.28	−0.300119
$850$	−4379.23	−0.176713
$851$	−5589.40	−0.225149
$852$	37379.9	1.50307
$853$	−3297.10	−0.132345	−0.0661726	−	0.997808i	$-0.521079\pi$
$853$	−3297.10	−0.132345	−0.0661726	+	0.997808i	$0.521079\pi$
$854$	0	0
$855$	2190.96	0.0876365
$856$	301.975	0.0120576
$857$	−21502.3	−0.857064	−0.428532	−	0.903527i	$-0.640969\pi$
$857$	−21502.3	−0.857064	−0.428532	+	0.903527i	$0.640969\pi$
$858$	−1487.52	−0.0591878
$859$	36005.6	1.43014	0.715072	−	0.699051i	$-0.246394\pi$
$859$	36005.6	1.43014	0.715072	+	0.699051i	$0.246394\pi$
$860$	−34842.6	−1.38154
$861$	0	0
$862$	−2594.87	−0.102531
$863$	4007.03	0.158055	0.0790273	−	0.996872i	$-0.474819\pi$
$863$	4007.03	0.158055	0.0790273	+	0.996872i	$0.474819\pi$
$864$	5238.08	0.206254
$865$	5599.22	0.220092
$866$	−1739.26	−0.0682478
$867$	−34198.6	−1.33961
$868$	0	0
$869$	−3843.36	−0.150031
$870$	607.531	0.0236750
$871$	−52722.4	−2.05101
$872$	−5459.54	−0.212022
$873$	−1582.29	−0.0613430
$874$	−233.061	−0.00901993
$875$	0	0
$876$	−25072.4	−0.967030
$877$	−1336.61	−0.0514643	−0.0257321	−	0.999669i	$-0.508192\pi$
$877$	−1336.61	−0.0514643	−0.0257321	+	0.999669i	$0.508192\pi$
$878$	−697.669	−0.0268168
$879$	−37804.3	−1.45063
$880$	30357.0	1.16288
$881$	41849.4	1.60039	0.800195	−	0.599740i	$-0.204729\pi$
$881$	41849.4	1.60039	0.800195	+	0.599740i	$0.204729\pi$
$882$	0	0
$883$	−20647.8	−0.786923	−0.393462	−	0.919341i	$-0.628723\pi$
$883$	−20647.8	−0.786923	−0.393462	+	0.919341i	$0.628723\pi$
$884$	−51454.3	−1.95769
$885$	−43927.9	−1.66850
$886$	−3451.64	−0.130880
$887$	44612.5	1.68877	0.844385	−	0.535736i	$-0.179966\pi$
$887$	44612.5	1.68877	0.844385	+	0.535736i	$0.179966\pi$
$888$	4606.48	0.174080
$889$	0	0
$890$	−1390.32	−0.0523637
$891$	17715.4	0.666091
$892$	46192.7	1.73391
$893$	−3565.75	−0.133621
$894$	−1252.34	−0.0468508
$895$	34590.8	1.29189
$896$	0	0
$897$	−5531.60	−0.205903
$898$	−761.009	−0.0282797
$899$	−1074.63	−0.0398674
$900$	3041.00	0.112629
$901$	21614.7	0.799210
$902$	−483.592	−0.0178513
$903$	0	0
$904$	−5320.68	−0.195756
$905$	35514.3	1.30446
$906$	2362.58	0.0866353
$907$	−31664.3	−1.15920	−0.579600	−	0.814901i	$-0.696791\pi$
$907$	−31664.3	−1.15920	−0.579600	+	0.814901i	$0.696791\pi$
$908$	−39307.2	−1.43662
$909$	2447.28	0.0892972
$910$	0	0
$911$	16484.8	0.599524	0.299762	−	0.954014i	$-0.403093\pi$
$911$	16484.8	0.599524	0.299762	+	0.954014i	$0.403093\pi$
$912$	−21073.2	−0.765137
$913$	9993.59	0.362256
$914$	−340.499	−0.0123224
$915$	15868.2	0.573318
$916$	−4669.41	−0.168430
$917$	0	0
$918$	−2971.89	−0.106849
$919$	18662.2	0.669870	0.334935	−	0.942241i	$-0.391286\pi$
$919$	18662.2	0.669870	0.334935	+	0.942241i	$0.391286\pi$
$920$	−1028.94	−0.0368729
$921$	−46928.3	−1.67898
$922$	3232.89	0.115477
$923$	−55781.6	−1.98925
$924$	0	0
$925$	64481.3	2.29204
$926$	158.296	0.00561763
$927$	−3295.29	−0.116755
$928$	1256.10	0.0444326
$929$	−33832.7	−1.19485	−0.597424	−	0.801926i	$-0.703809\pi$
$929$	−33832.7	−1.19485	−0.597424	+	0.801926i	$0.703809\pi$
$930$	−543.293	−0.0191562
$931$	0	0
$932$	26690.8	0.938074
$933$	−35471.1	−1.24467
$934$	−1236.19	−0.0433078
$935$	−52062.9	−1.82100
$936$	−324.193	−0.0113211
$937$	3652.05	0.127329	0.0636645	−	0.997971i	$-0.479721\pi$
$937$	3652.05	0.127329	0.0636645	+	0.997971i	$0.479721\pi$
$938$	0	0
$939$	−44104.7	−1.53280
$940$	−7853.42	−0.272500
$941$	30066.8	1.04160	0.520802	−	0.853678i	$-0.325633\pi$
$941$	30066.8	1.04160	0.520802	+	0.853678i	$0.325633\pi$
$942$	2119.48	0.0733081
$943$	−1798.32	−0.0621010
$944$	−30046.0	−1.03593
$945$	0	0
$946$	1181.59	0.0406098
$947$	−57360.1	−1.96827	−0.984135	−	0.177421i	$-0.943225\pi$
$947$	−57360.1	−1.96827	−0.984135	+	0.177421i	$0.943225\pi$
$948$	5876.12	0.201316
$949$	37415.3	1.27982
$950$	2688.68	0.0918235
$951$	−45796.5	−1.56157
$952$	0	0
$953$	37618.1	1.27867	0.639333	−	0.768930i	$-0.279211\pi$
$953$	37618.1	1.27867	0.639333	+	0.768930i	$0.279211\pi$
$954$	67.9389	0.00230566
$955$	44431.8	1.50553
$956$	40937.8	1.38496
$957$	4551.70	0.153747
$958$	3094.18	0.104351
$959$	0	0
$960$	−45988.0	−1.54610
$961$	961.000	0.0322581
$962$	−3429.34	−0.114934
$963$	178.594	0.00597624
$964$	−13390.7	−0.447391
$965$	−81244.3	−2.71020
$966$	0	0
$967$	−6302.73	−0.209599	−0.104799	−	0.994493i	$-0.533420\pi$
$967$	−6302.73	−0.209599	−0.104799	+	0.994493i	$0.533420\pi$
$968$	1961.18	0.0651184
$969$	36141.1	1.19816
$970$	−3081.18	−0.101990
$971$	−48114.9	−1.59020	−0.795098	−	0.606481i	$-0.792581\pi$
$971$	−48114.9	−1.59020	−0.795098	+	0.606481i	$0.792581\pi$
$972$	3998.96	0.131961
$973$	0	0
$974$	1262.67	0.0415385
$975$	63814.6	2.09610
$976$	10853.6	0.355958
$977$	33496.5	1.09688	0.548438	−	0.836191i	$-0.315223\pi$
$977$	33496.5	1.09688	0.548438	+	0.836191i	$0.315223\pi$
$978$	−353.724	−0.0115653
$979$	−10416.5	−0.340053
$980$	0	0
$981$	−3228.88	−0.105087
$982$	−438.534	−0.0142507
$983$	14695.7	0.476825	0.238412	−	0.971164i	$-0.423373\pi$
$983$	14695.7	0.476825	0.238412	+	0.971164i	$0.423373\pi$
$984$	1482.08	0.0480151
$985$	−41004.0	−1.32639
$986$	−712.663	−0.0230181
$987$	0	0
$988$	31591.0	1.01725
$989$	4393.95	0.141273
$990$	−163.643	−0.00525346
$991$	−46952.3	−1.50503	−0.752517	−	0.658573i	$-0.771160\pi$
$991$	−46952.3	−1.50503	−0.752517	+	0.658573i	$0.771160\pi$
$992$	−1123.28	−0.0359519
$993$	7223.61	0.230850
$994$	0	0
$995$	67890.5	2.16309
$996$	−15279.2	−0.486085
$997$	−38016.7	−1.20762	−0.603812	−	0.797127i	$-0.706352\pi$
$997$	−38016.7	−1.20762	−0.603812	+	0.797127i	$0.706352\pi$
$998$	1631.65	0.0517525
$999$	43759.2	1.38586

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1519.4.a.h.1.11	✓	23
7.6	odd	2		1519.4.a.i.1.11	yes	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1519.4.a.h.1.11	✓	23	1.1	even	1	trivial
1519.4.a.i.1.11	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)

\(f(q)\)	\(=\)	\(q+0.189863 q^{2} -5.02070 q^{3} -7.96395 q^{4} -18.3852 q^{5} -0.953245 q^{6} -3.03096 q^{8} -1.79257 q^{9} -3.49067 q^{10} -26.1525 q^{11} +39.9846 q^{12} -59.6686 q^{13} +92.3065 q^{15} +63.1361 q^{16} -108.280 q^{17} -0.340344 q^{18} +66.4797 q^{19} +146.419 q^{20} -4.96539 q^{22} -18.4646 q^{23} +15.2176 q^{24} +213.015 q^{25} -11.3289 q^{26} +144.559 q^{27} +34.6654 q^{29} +17.5256 q^{30} -31.0000 q^{31} +36.2349 q^{32} +131.304 q^{33} -20.5583 q^{34} +14.2760 q^{36} +302.708 q^{37} +12.6220 q^{38} +299.578 q^{39} +55.7248 q^{40} +97.3925 q^{41} -237.966 q^{43} +208.277 q^{44} +32.9568 q^{45} -3.50575 q^{46} -53.6367 q^{47} -316.988 q^{48} +40.4436 q^{50} +543.640 q^{51} +475.198 q^{52} -199.618 q^{53} +27.4464 q^{54} +480.818 q^{55} -333.775 q^{57} +6.58168 q^{58} -475.892 q^{59} -735.124 q^{60} +171.908 q^{61} -5.88575 q^{62} -498.210 q^{64} +1097.02 q^{65} +24.9297 q^{66} +883.588 q^{67} +862.335 q^{68} +92.7054 q^{69} +934.857 q^{71} +5.43323 q^{72} -627.052 q^{73} +57.4731 q^{74} -1069.48 q^{75} -529.441 q^{76} +56.8788 q^{78} +146.960 q^{79} -1160.77 q^{80} -677.387 q^{81} +18.4912 q^{82} -382.128 q^{83} +1990.74 q^{85} -45.1809 q^{86} -174.045 q^{87} +79.2672 q^{88} +398.297 q^{89} +6.25728 q^{90} +147.051 q^{92} +155.642 q^{93} -10.1836 q^{94} -1222.24 q^{95} -181.925 q^{96} +882.691 q^{97} +46.8803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 5 q^{2} - 6 q^{3} + 91 q^{4} - 40 q^{5} - 36 q^{6} + 39 q^{8} + 211 q^{9} - 40 q^{10} + 44 q^{11} - 414 q^{12} + 20 q^{13} + 523 q^{16} - 306 q^{17} + 51 q^{18} - 296 q^{19} - 400 q^{20} - 326 q^{22}+ \cdots - 3456 q^{99}+O(q^{100}) \) 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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