LMFDB - Embedded newform 575.4.a.o.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,4,Mod(1,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	575.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$33.9260982533$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 3 x^{12} - 85 x^{11} + 222 x^{10} + 2763 x^{9} - 6071 x^{8} - 43370 x^{7} + 78313 x^{6} + \cdots - 1836192 $ x^13 - 3x^12 - 85x^11 + 222x^10 + 2763x^9 - 6071x^8 - 43370x^7 + 78313x^6 + 340247x^5 - 505261x^4 - 1252464x^3 + 1561412x^2 + 1655688x - 1836192
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.74638$ of defining polynomial
Character	$\chi$	$=$	575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.74638 q^{2} +7.96785 q^{3} -4.95017 q^{4} +13.9149 q^{6} -27.5580 q^{7} -22.6159 q^{8} +36.4866 q^{9} +60.9612 q^{11} -39.4422 q^{12} +70.6157 q^{13} -48.1266 q^{14} +0.105454 q^{16} -9.53767 q^{17} +63.7193 q^{18} +137.393 q^{19} -219.578 q^{21} +106.461 q^{22} -23.0000 q^{23} -180.200 q^{24} +123.322 q^{26} +75.5876 q^{27} +136.417 q^{28} -193.189 q^{29} +170.657 q^{31} +181.111 q^{32} +485.729 q^{33} -16.6564 q^{34} -180.615 q^{36} +216.022 q^{37} +239.939 q^{38} +562.655 q^{39} +94.8947 q^{41} -383.466 q^{42} +495.951 q^{43} -301.768 q^{44} -40.1667 q^{46} -36.6486 q^{47} +0.840243 q^{48} +416.442 q^{49} -75.9947 q^{51} -349.559 q^{52} -691.376 q^{53} +132.004 q^{54} +623.248 q^{56} +1094.72 q^{57} -337.381 q^{58} +872.536 q^{59} -382.989 q^{61} +298.032 q^{62} -1005.50 q^{63} +315.445 q^{64} +848.267 q^{66} -325.119 q^{67} +47.2130 q^{68} -183.260 q^{69} +145.392 q^{71} -825.176 q^{72} -65.3155 q^{73} +377.256 q^{74} -680.116 q^{76} -1679.97 q^{77} +982.608 q^{78} +699.567 q^{79} -382.867 q^{81} +165.722 q^{82} +544.415 q^{83} +1086.95 q^{84} +866.117 q^{86} -1539.30 q^{87} -1378.69 q^{88} -837.249 q^{89} -1946.03 q^{91} +113.854 q^{92} +1359.77 q^{93} -64.0023 q^{94} +1443.07 q^{96} -253.821 q^{97} +727.266 q^{98} +2224.26 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 3 q^{2} + 3 q^{3} + 75 q^{4} + 62 q^{6} + q^{7} - 78 q^{8} + 108 q^{9} + 118 q^{11} - 27 q^{12} + 83 q^{13} + 40 q^{14} + 411 q^{16} + 71 q^{17} - 217 q^{18} + 360 q^{19} + 434 q^{21} + 169 q^{22}+ \cdots + 4599 q^{99}+O(q^{100}) $ 13 * q - 3 * q^2 + 3 * q^3 + 75 * q^4 + 62 * q^6 + q^7 - 78 * q^8 + 108 * q^9 + 118 * q^11 - 27 * q^12 + 83 * q^13 + 40 * q^14 + 411 * q^16 + 71 * q^17 - 217 * q^18 + 360 * q^19 + 434 * q^21 + 169 * q^22 - 299 * q^23 + 621 * q^24 + 269 * q^26 - 18 * q^27 + 626 * q^28 + 505 * q^29 + 130 * q^31 - 1207 * q^32 + 883 * q^33 + 487 * q^34 + 1441 * q^36 + 152 * q^37 + 661 * q^38 + 295 * q^39 + 330 * q^41 - 1056 * q^42 + 941 * q^43 - 253 * q^44 + 69 * q^46 - 874 * q^47 + 442 * q^48 + 1708 * q^49 - 1047 * q^51 - 1685 * q^52 + 1320 * q^53 + 3338 * q^54 + 220 * q^56 - 1763 * q^57 + 1376 * q^58 + 1177 * q^59 + 822 * q^61 - 3066 * q^62 + 887 * q^63 + 6328 * q^64 - 1423 * q^66 - 2453 * q^67 + 3293 * q^68 - 69 * q^69 + 550 * q^71 - 4917 * q^72 + 1582 * q^73 + 2504 * q^74 + 1265 * q^76 - 733 * q^77 + 6167 * q^78 + 3403 * q^79 - 439 * q^81 - 5232 * q^82 - 76 * q^83 + 5964 * q^84 - 544 * q^86 - 1523 * q^87 + 7321 * q^88 + 2403 * q^89 + 2451 * q^91 - 1725 * q^92 + 2251 * q^93 + 6595 * q^94 + 4550 * q^96 - 2208 * q^97 + 4895 * q^98 + 4599 * 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 $

$2$	1.74638	0.617438	0.308719	−	0.951153i	$-0.400100\pi$
$2$	1.74638	0.617438	0.308719	+	0.951153i	$0.400100\pi$
$3$	7.96785	1.53341	0.766706	−	0.641998i	$-0.221894\pi$
$3$	7.96785	1.53341	0.766706	+	0.641998i	$0.221894\pi$
$4$	−4.95017	−0.618771
$5$	0	0
$5$	0	0
$6$	13.9149	0.946787
$7$	−27.5580	−1.48799	−0.743996	−	0.668185i	$-0.767072\pi$
$7$	−27.5580	−1.48799	−0.743996	+	0.668185i	$0.767072\pi$
$8$	−22.6159	−0.999490
$9$	36.4866	1.35135
$10$	0	0
$11$	60.9612	1.67095	0.835477	−	0.549526i	$-0.185192\pi$
$11$	60.9612	1.67095	0.835477	+	0.549526i	$0.185192\pi$
$12$	−39.4422	−0.948831
$13$	70.6157	1.50656	0.753280	−	0.657700i	$-0.228471\pi$
$13$	70.6157	1.50656	0.753280	+	0.657700i	$0.228471\pi$
$14$	−48.1266	−0.918742
$15$	0	0
$16$	0.105454	0.00164772
$17$	−9.53767	−0.136072	−0.0680360	−	0.997683i	$-0.521673\pi$
$17$	−9.53767	−0.136072	−0.0680360	+	0.997683i	$0.521673\pi$
$18$	63.7193	0.834377
$19$	137.393	1.65895	0.829474	−	0.558545i	$-0.188640\pi$
$19$	137.393	1.65895	0.829474	+	0.558545i	$0.188640\pi$
$20$	0	0
$21$	−219.578	−2.28170
$22$	106.461	1.03171
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−180.200	−1.53263
$25$	0	0
$26$	123.322	0.930207
$27$	75.5876	0.538772
$28$	136.417	0.920725
$29$	−193.189	−1.23704	−0.618522	−	0.785768i	$-0.712268\pi$
$29$	−193.189	−1.23704	−0.618522	+	0.785768i	$0.712268\pi$
$30$	0	0
$31$	170.657	0.988742	0.494371	−	0.869251i	$-0.335399\pi$
$31$	170.657	0.988742	0.494371	+	0.869251i	$0.335399\pi$
$32$	181.111	1.00051
$33$	485.729	2.56226
$34$	−16.6564	−0.0840160
$35$	0	0
$36$	−180.615	−0.836179
$37$	216.022	0.959832	0.479916	−	0.877314i	$-0.340667\pi$
$37$	216.022	0.959832	0.479916	+	0.877314i	$0.340667\pi$
$38$	239.939	1.02430
$39$	562.655	2.31018
$40$	0	0
$41$	94.8947	0.361465	0.180733	−	0.983532i	$-0.442153\pi$
$41$	94.8947	0.361465	0.180733	+	0.983532i	$0.442153\pi$
$42$	−383.466	−1.40881
$43$	495.951	1.75888	0.879439	−	0.476011i	$-0.157918\pi$
$43$	495.951	1.75888	0.879439	+	0.476011i	$0.157918\pi$
$44$	−301.768	−1.03394
$45$	0	0
$46$	−40.1667	−0.128745
$47$	−36.6486	−0.113739	−0.0568697	−	0.998382i	$-0.518112\pi$
$47$	−36.6486	−0.113739	−0.0568697	+	0.998382i	$0.518112\pi$
$48$	0.840243	0.00252664
$49$	416.442	1.21412
$50$	0	0
$51$	−75.9947	−0.208655
$52$	−349.559	−0.932215
$53$	−691.376	−1.79184	−0.895922	−	0.444211i	$-0.853484\pi$
$53$	−691.376	−1.79184	−0.895922	+	0.444211i	$0.853484\pi$
$54$	132.004	0.332658
$55$	0	0
$56$	623.248	1.48723
$57$	1094.72	2.54385
$58$	−337.381	−0.763797
$59$	872.536	1.92533	0.962665	−	0.270696i	$-0.0872537\pi$
$59$	872.536	1.92533	0.962665	+	0.270696i	$0.0872537\pi$
$60$	0	0
$61$	−382.989	−0.803880	−0.401940	−	0.915666i	$-0.631664\pi$
$61$	−382.989	−0.803880	−0.401940	+	0.915666i	$0.631664\pi$
$62$	298.032	0.610486
$63$	−1005.50	−2.01080
$64$	315.445	0.616103
$65$	0	0
$66$	848.267	1.58204
$67$	−325.119	−0.592831	−0.296415	−	0.955059i	$-0.595791\pi$
$67$	−325.119	−0.592831	−0.296415	+	0.955059i	$0.595791\pi$
$68$	47.2130	0.0841974
$69$	−183.260	−0.319739
$70$	0	0
$71$	145.392	0.243027	0.121513	−	0.992590i	$-0.461225\pi$
$71$	145.392	0.243027	0.121513	+	0.992590i	$0.461225\pi$
$72$	−825.176	−1.35067
$73$	−65.3155	−0.104721	−0.0523603	−	0.998628i	$-0.516674\pi$
$73$	−65.3155	−0.104721	−0.0523603	+	0.998628i	$0.516674\pi$
$74$	377.256	0.592637
$75$	0	0
$76$	−680.116	−1.02651
$77$	−1679.97	−2.48636
$78$	982.608	1.42639
$79$	699.567	0.996296	0.498148	−	0.867092i	$-0.334014\pi$
$79$	699.567	0.996296	0.498148	+	0.867092i	$0.334014\pi$
$80$	0	0
$81$	−382.867	−0.525195
$82$	165.722	0.223182
$83$	544.415	0.719968	0.359984	−	0.932958i	$-0.382782\pi$
$83$	544.415	0.719968	0.359984	+	0.932958i	$0.382782\pi$
$84$	1086.95	1.41185
$85$	0	0
$86$	866.117	1.08600
$87$	−1539.30	−1.89690
$88$	−1378.69	−1.67010
$89$	−837.249	−0.997171	−0.498585	−	0.866841i	$-0.666147\pi$
$89$	−837.249	−0.997171	−0.498585	+	0.866841i	$0.666147\pi$
$90$	0	0
$91$	−1946.03	−2.24175
$92$	113.854	0.129023
$93$	1359.77	1.51615
$94$	−64.0023	−0.0702270
$95$	0	0
$96$	1443.07	1.53419
$97$	−253.821	−0.265687	−0.132843	−	0.991137i	$-0.542411\pi$
$97$	−253.821	−0.265687	−0.132843	+	0.991137i	$0.542411\pi$
$98$	727.266	0.749642
$99$	2224.26	2.25805
$100$	0	0
$101$	−785.461	−0.773824	−0.386912	−	0.922117i	$-0.626458\pi$
$101$	−785.461	−0.773824	−0.386912	+	0.922117i	$0.626458\pi$
$102$	−132.715	−0.128831
$103$	−143.585	−0.137358	−0.0686790	−	0.997639i	$-0.521878\pi$
$103$	−143.585	−0.137358	−0.0686790	+	0.997639i	$0.521878\pi$
$104$	−1597.04	−1.50579
$105$	0	0
$106$	−1207.40	−1.10635
$107$	−746.981	−0.674891	−0.337446	−	0.941345i	$-0.609563\pi$
$107$	−746.981	−0.674891	−0.337446	+	0.941345i	$0.609563\pi$
$108$	−374.171	−0.333376
$109$	−184.744	−0.162342	−0.0811709	−	0.996700i	$-0.525866\pi$
$109$	−184.744	−0.162342	−0.0811709	+	0.996700i	$0.525866\pi$
$110$	0	0
$111$	1721.23	1.47182
$112$	−2.90610	−0.00245180
$113$	−41.9387	−0.0349138	−0.0174569	−	0.999848i	$-0.505557\pi$
$113$	−41.9387	−0.0349138	−0.0174569	+	0.999848i	$0.505557\pi$
$114$	1911.80	1.57067
$115$	0	0
$116$	956.316	0.765446
$117$	2576.53	2.03590
$118$	1523.78	1.18877
$119$	262.839	0.202474
$120$	0	0
$121$	2385.26	1.79208
$122$	−668.843	−0.496346
$123$	756.107	0.554275
$124$	−844.783	−0.611804
$125$	0	0
$126$	−1755.98	−1.24155
$127$	−1493.28	−1.04336	−0.521682	−	0.853140i	$-0.674695\pi$
$127$	−1493.28	−1.04336	−0.521682	+	0.853140i	$0.674695\pi$
$128$	−898.004	−0.620102
$129$	3951.66	2.69709
$130$	0	0
$131$	−114.667	−0.0764770	−0.0382385	−	0.999269i	$-0.512175\pi$
$131$	−114.667	−0.0764770	−0.0382385	+	0.999269i	$0.512175\pi$
$132$	−2404.44	−1.58545
$133$	−3786.26	−2.46850
$134$	−567.781	−0.366036
$135$	0	0
$136$	215.703	0.136003
$137$	2262.22	1.41076	0.705380	−	0.708829i	$-0.250776\pi$
$137$	2262.22	1.41076	0.705380	+	0.708829i	$0.250776\pi$
$138$	−320.042	−0.197419
$139$	−2804.79	−1.71150	−0.855751	−	0.517387i	$-0.826905\pi$
$139$	−2804.79	−1.71150	−0.855751	+	0.517387i	$0.826905\pi$
$140$	0	0
$141$	−292.011	−0.174409
$142$	253.910	0.150054
$143$	4304.82	2.51739
$144$	3.84766	0.00222666
$145$	0	0
$146$	−114.065	−0.0646584
$147$	3318.15	1.86174
$148$	−1069.34	−0.593916
$149$	−232.626	−0.127902	−0.0639511	−	0.997953i	$-0.520370\pi$
$149$	−232.626	−0.127902	−0.0639511	+	0.997953i	$0.520370\pi$
$150$	0	0
$151$	−1426.30	−0.768680	−0.384340	−	0.923192i	$-0.625571\pi$
$151$	−1426.30	−0.768680	−0.384340	+	0.923192i	$0.625571\pi$
$152$	−3107.25	−1.65810
$153$	−347.997	−0.183882
$154$	−2933.86	−1.53517
$155$	0	0
$156$	−2785.24	−1.42947
$157$	962.502	0.489274	0.244637	−	0.969615i	$-0.421331\pi$
$157$	962.502	0.489274	0.244637	+	0.969615i	$0.421331\pi$
$158$	1221.71	0.615151
$159$	−5508.77	−2.74764
$160$	0	0
$161$	633.834	0.310268
$162$	−668.631	−0.324275
$163$	−1761.39	−0.846396	−0.423198	−	0.906037i	$-0.639093\pi$
$163$	−1761.39	−0.846396	−0.423198	+	0.906037i	$0.639093\pi$
$164$	−469.745	−0.223664
$165$	0	0
$166$	950.755	0.444535
$167$	−2012.99	−0.932756	−0.466378	−	0.884586i	$-0.654441\pi$
$167$	−2012.99	−0.932756	−0.466378	+	0.884586i	$0.654441\pi$
$168$	4965.94	2.28054
$169$	2789.58	1.26972
$170$	0	0
$171$	5012.98	2.24183
$172$	−2455.04	−1.08834
$173$	−540.225	−0.237414	−0.118707	−	0.992929i	$-0.537875\pi$
$173$	−540.225	−0.237414	−0.118707	+	0.992929i	$0.537875\pi$
$174$	−2688.20	−1.17122
$175$	0	0
$176$	6.42861	0.00275327
$177$	6952.23	2.95233
$178$	−1462.15	−0.615691
$179$	−918.831	−0.383668	−0.191834	−	0.981427i	$-0.561444\pi$
$179$	−918.831	−0.383668	−0.191834	+	0.981427i	$0.561444\pi$
$180$	0	0
$181$	3173.59	1.30327	0.651633	−	0.758534i	$-0.274084\pi$
$181$	3173.59	1.30327	0.651633	+	0.758534i	$0.274084\pi$
$182$	−3398.50	−1.38414
$183$	−3051.59	−1.23268
$184$	520.165	0.208408
$185$	0	0
$186$	2374.68	0.936128
$187$	−581.427	−0.227370
$188$	181.417	0.0703786
$189$	−2083.04	−0.801688
$190$	0	0
$191$	2829.53	1.07192	0.535962	−	0.844242i	$-0.319949\pi$
$191$	2829.53	1.07192	0.535962	+	0.844242i	$0.319949\pi$
$192$	2513.42	0.944741
$193$	1987.90	0.741410	0.370705	−	0.928751i	$-0.379116\pi$
$193$	1987.90	0.741410	0.370705	+	0.928751i	$0.379116\pi$
$194$	−443.267	−0.164045
$195$	0	0
$196$	−2061.46	−0.751260
$197$	−1644.24	−0.594658	−0.297329	−	0.954775i	$-0.596096\pi$
$197$	−1644.24	−0.594658	−0.297329	+	0.954775i	$0.596096\pi$
$198$	3884.41	1.39421
$199$	1995.20	0.710734	0.355367	−	0.934727i	$-0.384356\pi$
$199$	1995.20	0.710734	0.355367	+	0.934727i	$0.384356\pi$
$200$	0	0
$201$	−2590.50	−0.909054
$202$	−1371.71	−0.477788
$203$	5323.89	1.84071
$204$	376.186	0.129109
$205$	0	0
$206$	−250.754	−0.0848101
$207$	−839.191	−0.281777
$208$	7.44672	0.00248239
$209$	8375.61	2.77202
$210$	0	0
$211$	3654.46	1.19234	0.596170	−	0.802858i	$-0.296689\pi$
$211$	3654.46	1.19234	0.596170	+	0.802858i	$0.296689\pi$
$212$	3422.42	1.10874
$213$	1158.46	0.372661
$214$	−1304.51	−0.416703
$215$	0	0
$216$	−1709.48	−0.538497
$217$	−4702.98	−1.47124
$218$	−322.633	−0.100236
$219$	−520.424	−0.160580
$220$	0	0
$221$	−673.509	−0.205001
$222$	3005.92	0.908757
$223$	−3845.98	−1.15491	−0.577457	−	0.816421i	$-0.695955\pi$
$223$	−3845.98	−1.15491	−0.577457	+	0.816421i	$0.695955\pi$
$224$	−4991.06	−1.48875
$225$	0	0
$226$	−73.2408	−0.0215571
$227$	3520.54	1.02937	0.514683	−	0.857380i	$-0.327910\pi$
$227$	3520.54	1.02937	0.514683	+	0.857380i	$0.327910\pi$
$228$	−5419.06	−1.57406
$229$	−731.753	−0.211160	−0.105580	−	0.994411i	$-0.533670\pi$
$229$	−731.753	−0.211160	−0.105580	+	0.994411i	$0.533670\pi$
$230$	0	0
$231$	−13385.7	−3.81262
$232$	4369.13	1.23641
$233$	−5819.25	−1.63619	−0.818094	−	0.575085i	$-0.804969\pi$
$233$	−5819.25	−1.63619	−0.818094	+	0.575085i	$0.804969\pi$
$234$	4499.59	1.25704
$235$	0	0
$236$	−4319.20	−1.19134
$237$	5574.04	1.52773
$238$	459.016	0.125015
$239$	−2213.55	−0.599091	−0.299545	−	0.954082i	$-0.596835\pi$
$239$	−2213.55	−0.599091	−0.299545	+	0.954082i	$0.596835\pi$
$240$	0	0
$241$	5765.08	1.54092	0.770460	−	0.637489i	$-0.220027\pi$
$241$	5765.08	1.54092	0.770460	+	0.637489i	$0.220027\pi$
$242$	4165.57	1.10650
$243$	−5091.49	−1.34411
$244$	1895.86	0.497417
$245$	0	0
$246$	1320.45	0.342230
$247$	9702.07	2.49930
$248$	−3859.57	−0.988237
$249$	4337.82	1.10401
$250$	0	0
$251$	−1076.57	−0.270726	−0.135363	−	0.990796i	$-0.543220\pi$
$251$	−1076.57	−0.270726	−0.135363	+	0.990796i	$0.543220\pi$
$252$	4977.37	1.24423
$253$	−1402.11	−0.348418
$254$	−2607.83	−0.644213
$255$	0	0
$256$	−4091.81	−0.998978
$257$	−3475.19	−0.843488	−0.421744	−	0.906715i	$-0.638582\pi$
$257$	−3475.19	−0.843488	−0.421744	+	0.906715i	$0.638582\pi$
$258$	6901.09	1.66528
$259$	−5953.13	−1.42822
$260$	0	0
$261$	−7048.80	−1.67168
$262$	−200.252	−0.0472198
$263$	576.620	0.135194	0.0675968	−	0.997713i	$-0.478467\pi$
$263$	576.620	0.135194	0.0675968	+	0.997713i	$0.478467\pi$
$264$	−10985.2	−2.56095
$265$	0	0
$266$	−6612.24	−1.52415
$267$	−6671.07	−1.52907
$268$	1609.39	0.366826
$269$	−1857.96	−0.421123	−0.210561	−	0.977581i	$-0.567529\pi$
$269$	−1857.96	−0.421123	−0.210561	+	0.977581i	$0.567529\pi$
$270$	0	0
$271$	8634.76	1.93551	0.967757	−	0.251887i	$-0.0810511\pi$
$271$	8634.76	1.93551	0.967757	+	0.251887i	$0.0810511\pi$
$272$	−1.00579	−0.000224209 0
$273$	−15505.6	−3.43752
$274$	3950.68	0.871057
$275$	0	0
$276$	907.170	0.197845
$277$	−1614.11	−0.350117	−0.175059	−	0.984558i	$-0.556011\pi$
$277$	−1614.11	−0.350117	−0.175059	+	0.984558i	$0.556011\pi$
$278$	−4898.22	−1.05675
$279$	6226.71	1.33614
$280$	0	0
$281$	1372.23	0.291319	0.145659	−	0.989335i	$-0.453470\pi$
$281$	1372.23	0.291319	0.145659	+	0.989335i	$0.453470\pi$
$282$	−509.961	−0.107687
$283$	−3394.23	−0.712953	−0.356477	−	0.934304i	$-0.616022\pi$
$283$	−3394.23	−0.712953	−0.356477	+	0.934304i	$0.616022\pi$
$284$	−719.717	−0.150378
$285$	0	0
$286$	7517.83	1.55433
$287$	−2615.11	−0.537857
$288$	6608.13	1.35204
$289$	−4822.03	−0.981484
$290$	0	0
$291$	−2022.41	−0.407407
$292$	323.322	0.0647980
$293$	3622.97	0.722376	0.361188	−	0.932493i	$-0.382371\pi$
$293$	3622.97	0.722376	0.361188	+	0.932493i	$0.382371\pi$
$294$	5794.74	1.14951
$295$	0	0
$296$	−4885.53	−0.959343
$297$	4607.91	0.900262
$298$	−406.252	−0.0789717
$299$	−1624.16	−0.314139
$300$	0	0
$301$	−13667.4	−2.61720
$302$	−2490.86	−0.474612
$303$	−6258.43	−1.18659
$304$	14.4886	0.00273348
$305$	0	0
$306$	−607.734	−0.113535
$307$	−4972.83	−0.924477	−0.462239	−	0.886756i	$-0.652954\pi$
$307$	−4972.83	−0.924477	−0.462239	+	0.886756i	$0.652954\pi$
$308$	8316.11	1.53849
$309$	−1144.07	−0.210627
$310$	0	0
$311$	−3408.78	−0.621525	−0.310763	−	0.950488i	$-0.600584\pi$
$311$	−3408.78	−0.621525	−0.310763	+	0.950488i	$0.600584\pi$
$312$	−12724.9	−2.30900
$313$	−5053.61	−0.912610	−0.456305	−	0.889823i	$-0.650827\pi$
$313$	−5053.61	−0.912610	−0.456305	+	0.889823i	$0.650827\pi$
$314$	1680.89	0.302096
$315$	0	0
$316$	−3462.97	−0.616479
$317$	2191.69	0.388321	0.194161	−	0.980970i	$-0.437802\pi$
$317$	2191.69	0.388321	0.194161	+	0.980970i	$0.437802\pi$
$318$	−9620.40	−1.69649
$319$	−11777.0	−2.06704
$320$	0	0
$321$	−5951.83	−1.03489
$322$	1106.91	0.191571
$323$	−1310.40	−0.225736
$324$	1895.26	0.324975
$325$	0	0
$326$	−3076.05	−0.522597
$327$	−1472.01	−0.248937
$328$	−2146.13	−0.361281
$329$	1009.96	0.169243
$330$	0	0
$331$	6443.90	1.07006	0.535028	−	0.844834i	$-0.320301\pi$
$331$	6443.90	1.07006	0.535028	+	0.844834i	$0.320301\pi$
$332$	−2694.95	−0.445495
$333$	7881.90	1.29707
$334$	−3515.45	−0.575919
$335$	0	0
$336$	−23.1554	−0.00375961
$337$	−11361.5	−1.83649	−0.918246	−	0.396010i	$-0.870394\pi$
$337$	−11361.5	−1.83649	−0.918246	+	0.396010i	$0.870394\pi$
$338$	4871.66	0.783974
$339$	−334.161	−0.0535373
$340$	0	0
$341$	10403.5	1.65214
$342$	8754.56	1.38419
$343$	−2023.92	−0.318605
$344$	−11216.4	−1.75798
$345$	0	0
$346$	−943.437	−0.146588
$347$	5654.15	0.874729	0.437364	−	0.899284i	$-0.355912\pi$
$347$	5654.15	0.874729	0.437364	+	0.899284i	$0.355912\pi$
$348$	7619.78	1.17374
$349$	3458.22	0.530414	0.265207	−	0.964192i	$-0.414560\pi$
$349$	3458.22	0.530414	0.265207	+	0.964192i	$0.414560\pi$
$350$	0	0
$351$	5337.67	0.811692
$352$	11040.8	1.67180
$353$	−4298.91	−0.648181	−0.324090	−	0.946026i	$-0.605058\pi$
$353$	−4298.91	−0.648181	−0.324090	+	0.946026i	$0.605058\pi$
$354$	12141.2	1.82288
$355$	0	0
$356$	4144.52	0.617020
$357$	2094.26	0.310476
$358$	−1604.63	−0.236891
$359$	−4720.65	−0.694001	−0.347001	−	0.937865i	$-0.612800\pi$
$359$	−4720.65	−0.694001	−0.347001	+	0.937865i	$0.612800\pi$
$360$	0	0
$361$	12017.7	1.75211
$362$	5542.29	0.804686
$363$	19005.4	2.74800
$364$	9633.15	1.38713
$365$	0	0
$366$	−5329.24	−0.761103
$367$	−2253.88	−0.320576	−0.160288	−	0.987070i	$-0.551242\pi$
$367$	−2253.88	−0.320576	−0.160288	+	0.987070i	$0.551242\pi$
$368$	−2.42545	−0.000343574 0
$369$	3462.38	0.488468
$370$	0	0
$371$	19052.9	2.66625
$372$	−6731.10	−0.938148
$373$	971.159	0.134812	0.0674058	−	0.997726i	$-0.478528\pi$
$373$	971.159	0.134812	0.0674058	+	0.997726i	$0.478528\pi$
$374$	−1015.39	−0.140387
$375$	0	0
$376$	828.841	0.113681
$377$	−13642.2	−1.86368
$378$	−3637.78	−0.494992
$379$	4577.70	0.620424	0.310212	−	0.950667i	$-0.399600\pi$
$379$	4577.70	0.620424	0.310212	+	0.950667i	$0.399600\pi$
$380$	0	0
$381$	−11898.2	−1.59991
$382$	4941.42	0.661846
$383$	7852.16	1.04759	0.523795	−	0.851844i	$-0.324516\pi$
$383$	7852.16	1.04759	0.523795	+	0.851844i	$0.324516\pi$
$384$	−7155.15	−0.950872
$385$	0	0
$386$	3471.62	0.457775
$387$	18095.5	2.37687
$388$	1256.46	0.164399
$389$	5290.91	0.689613	0.344807	−	0.938674i	$-0.387944\pi$
$389$	5290.91	0.689613	0.344807	+	0.938674i	$0.387944\pi$
$390$	0	0
$391$	219.366	0.0283730
$392$	−9418.21	−1.21350
$393$	−913.648	−0.117271
$394$	−2871.47	−0.367164
$395$	0	0
$396$	−11010.5	−1.39722
$397$	10062.3	1.27207	0.636037	−	0.771658i	$-0.280572\pi$
$397$	10062.3	1.27207	0.636037	+	0.771658i	$0.280572\pi$
$398$	3484.37	0.438834
$399$	−30168.4	−3.78523
$400$	0	0
$401$	−3378.97	−0.420792	−0.210396	−	0.977616i	$-0.567475\pi$
$401$	−3378.97	−0.420792	−0.210396	+	0.977616i	$0.567475\pi$
$402$	−4523.99	−0.561284
$403$	12051.1	1.48960
$404$	3888.16	0.478820
$405$	0	0
$406$	9297.53	1.13652
$407$	13169.0	1.60383
$408$	1718.69	0.208548
$409$	−9626.70	−1.16384	−0.581919	−	0.813247i	$-0.697698\pi$
$409$	−9626.70	−1.16384	−0.581919	+	0.813247i	$0.697698\pi$
$410$	0	0
$411$	18025.0	2.16328
$412$	710.771	0.0849932
$413$	−24045.3	−2.86487
$414$	−1465.54	−0.173980
$415$	0	0
$416$	12789.3	1.50732
$417$	−22348.1	−2.62444
$418$	14627.0	1.71155
$419$	652.487	0.0760765	0.0380383	−	0.999276i	$-0.487889\pi$
$419$	652.487	0.0760765	0.0380383	+	0.999276i	$0.487889\pi$
$420$	0	0
$421$	−3349.84	−0.387794	−0.193897	−	0.981022i	$-0.562113\pi$
$421$	−3349.84	−0.387794	−0.193897	+	0.981022i	$0.562113\pi$
$422$	6382.07	0.736195
$423$	−1337.18	−0.153702
$424$	15636.1	1.79093
$425$	0	0
$426$	2023.12	0.230095
$427$	10554.4	1.19617
$428$	3697.68	0.417603
$429$	34300.1	3.86020
$430$	0	0
$431$	−2617.96	−0.292582	−0.146291	−	0.989242i	$-0.546734\pi$
$431$	−2617.96	−0.292582	−0.146291	+	0.989242i	$0.546734\pi$
$432$	7.97103	0.000887746 0
$433$	−303.016	−0.0336306	−0.0168153	−	0.999859i	$-0.505353\pi$
$433$	−303.016	−0.0336306	−0.0168153	+	0.999859i	$0.505353\pi$
$434$	−8213.17	−0.908398
$435$	0	0
$436$	914.513	0.100452
$437$	−3160.03	−0.345915
$438$	−908.856	−0.0991480
$439$	4887.75	0.531389	0.265694	−	0.964057i	$-0.414399\pi$
$439$	4887.75	0.531389	0.265694	+	0.964057i	$0.414399\pi$
$440$	0	0
$441$	15194.6	1.64070
$442$	−1176.20	−0.126575
$443$	2134.04	0.228875	0.114437	−	0.993430i	$-0.463494\pi$
$443$	2134.04	0.228875	0.114437	+	0.993430i	$0.463494\pi$
$444$	−8520.37	−0.910718
$445$	0	0
$446$	−6716.53	−0.713087
$447$	−1853.53	−0.196127
$448$	−8693.03	−0.916756
$449$	−2412.13	−0.253531	−0.126765	−	0.991933i	$-0.540460\pi$
$449$	−2412.13	−0.253531	−0.126765	+	0.991933i	$0.540460\pi$
$450$	0	0
$451$	5784.89	0.603991
$452$	207.604	0.0216036
$453$	−11364.5	−1.17870
$454$	6148.18	0.635570
$455$	0	0
$456$	−24758.1	−2.54256
$457$	5233.87	0.535733	0.267867	−	0.963456i	$-0.413681\pi$
$457$	5233.87	0.535733	0.267867	+	0.963456i	$0.413681\pi$
$458$	−1277.92	−0.130378
$459$	−720.929	−0.0733118
$460$	0	0
$461$	1170.36	0.118241	0.0591203	−	0.998251i	$-0.481170\pi$
$461$	1170.36	0.118241	0.0591203	+	0.998251i	$0.481170\pi$
$462$	−23376.5	−2.35406
$463$	−1030.98	−0.103486	−0.0517428	−	0.998660i	$-0.516478\pi$
$463$	−1030.98	−0.103486	−0.0517428	+	0.998660i	$0.516478\pi$
$464$	−20.3726	−0.00203830
$465$	0	0
$466$	−10162.6	−1.01024
$467$	−9611.60	−0.952402	−0.476201	−	0.879336i	$-0.657987\pi$
$467$	−9611.60	−0.952402	−0.476201	+	0.879336i	$0.657987\pi$
$468$	−12754.2	−1.25975
$469$	8959.64	0.882127
$470$	0	0
$471$	7669.07	0.750259
$472$	−19733.2	−1.92435
$473$	30233.7	2.93900
$474$	9734.38	0.943280
$475$	0	0
$476$	−1301.10	−0.125285
$477$	−25225.9	−2.42142
$478$	−3865.69	−0.369901
$479$	2740.84	0.261445	0.130722	−	0.991419i	$-0.458270\pi$
$479$	2740.84	0.261445	0.130722	+	0.991419i	$0.458270\pi$
$480$	0	0
$481$	15254.5	1.44604
$482$	10068.0	0.951422
$483$	5050.29	0.475768
$484$	−11807.5	−1.10889
$485$	0	0
$486$	−8891.67	−0.829906
$487$	599.429	0.0557756	0.0278878	−	0.999611i	$-0.491122\pi$
$487$	599.429	0.0557756	0.0278878	+	0.999611i	$0.491122\pi$
$488$	8661.63	0.803470
$489$	−14034.5	−1.29787
$490$	0	0
$491$	8417.89	0.773715	0.386858	−	0.922139i	$-0.373561\pi$
$491$	8417.89	0.773715	0.386858	+	0.922139i	$0.373561\pi$
$492$	−3742.85	−0.342969
$493$	1842.57	0.168327
$494$	16943.5	1.54316
$495$	0	0
$496$	17.9965	0.00162917
$497$	−4006.72	−0.361622
$498$	7575.47	0.681656
$499$	−4002.81	−0.359099	−0.179549	−	0.983749i	$-0.557464\pi$
$499$	−4002.81	−0.359099	−0.179549	+	0.983749i	$0.557464\pi$
$500$	0	0
$501$	−16039.2	−1.43030
$502$	−1880.09	−0.167156
$503$	−5847.99	−0.518387	−0.259194	−	0.965825i	$-0.583457\pi$
$503$	−5847.99	−0.518387	−0.259194	+	0.965825i	$0.583457\pi$
$504$	22740.2	2.00978
$505$	0	0
$506$	−2448.61	−0.215126
$507$	22226.9	1.94701
$508$	7391.99	0.645603
$509$	−12474.1	−1.08625	−0.543127	−	0.839650i	$-0.682760\pi$
$509$	−12474.1	−1.08625	−0.543127	+	0.839650i	$0.682760\pi$
$510$	0	0
$511$	1799.96	0.155823
$512$	38.1784	0.00329544
$513$	10385.2	0.893794
$514$	−6068.99	−0.520801
$515$	0	0
$516$	−19561.4	−1.66888
$517$	−2234.14	−0.190053
$518$	−10396.4	−0.881838
$519$	−4304.43	−0.364053
$520$	0	0
$521$	13917.3	1.17031	0.585154	−	0.810923i	$-0.301034\pi$
$521$	13917.3	1.17031	0.585154	+	0.810923i	$0.301034\pi$
$522$	−12309.9	−1.03216
$523$	−12279.6	−1.02667	−0.513335	−	0.858188i	$-0.671590\pi$
$523$	−12279.6	−1.02667	−0.513335	+	0.858188i	$0.671590\pi$
$524$	567.620	0.0473217
$525$	0	0
$526$	1007.00	0.0834736
$527$	−1627.67	−0.134540
$528$	51.2222	0.00422189
$529$	529.000	0.0434783
$530$	0	0
$531$	31835.8	2.60180
$532$	18742.6	1.52744
$533$	6701.06	0.544569
$534$	−11650.2	−0.944108
$535$	0	0
$536$	7352.86	0.592528
$537$	−7321.10	−0.588322
$538$	−3244.70	−0.260017
$539$	25386.8	2.02873
$540$	0	0
$541$	10350.6	0.822562	0.411281	−	0.911509i	$-0.365081\pi$
$541$	10350.6	0.822562	0.411281	+	0.911509i	$0.365081\pi$
$542$	15079.5	1.19506
$543$	25286.7	1.99845
$544$	−1727.38	−0.136141
$545$	0	0
$546$	−27078.7	−2.12246
$547$	−806.728	−0.0630589	−0.0315294	−	0.999503i	$-0.510038\pi$
$547$	−806.728	−0.0630589	−0.0315294	+	0.999503i	$0.510038\pi$
$548$	−11198.3	−0.872937
$549$	−13973.9	−1.08633
$550$	0	0
$551$	−26542.7	−2.05219
$552$	4144.60	0.319576
$553$	−19278.6	−1.48248
$554$	−2818.84	−0.216175
$555$	0	0
$556$	13884.2	1.05903
$557$	2775.50	0.211134	0.105567	−	0.994412i	$-0.466334\pi$
$557$	2775.50	0.211134	0.105567	+	0.994412i	$0.466334\pi$
$558$	10874.2	0.824984
$559$	35021.9	2.64986
$560$	0	0
$561$	−4632.72	−0.348652
$562$	2396.44	0.179871
$563$	−14230.1	−1.06524	−0.532619	−	0.846355i	$-0.678792\pi$
$563$	−14230.1	−1.06524	−0.532619	+	0.846355i	$0.678792\pi$
$564$	1445.50	0.107919
$565$	0	0
$566$	−5927.60	−0.440204
$567$	10551.0	0.781486
$568$	−3288.18	−0.242903
$569$	19128.1	1.40930	0.704650	−	0.709555i	$-0.251104\pi$
$569$	19128.1	1.40930	0.704650	+	0.709555i	$0.251104\pi$
$570$	0	0
$571$	−3025.84	−0.221765	−0.110882	−	0.993834i	$-0.535368\pi$
$571$	−3025.84	−0.221765	−0.110882	+	0.993834i	$0.535368\pi$
$572$	−21309.5	−1.55769
$573$	22545.2	1.64370
$574$	−4566.97	−0.332093
$575$	0	0
$576$	11509.5	0.832574
$577$	−26089.9	−1.88239	−0.941195	−	0.337865i	$-0.890295\pi$
$577$	−26089.9	−1.88239	−0.941195	+	0.337865i	$0.890295\pi$
$578$	−8421.09	−0.606006
$579$	15839.3	1.13689
$580$	0	0
$581$	−15003.0	−1.07131
$582$	−3531.89	−0.251549
$583$	−42147.1	−2.99409
$584$	1477.17	0.104667
$585$	0	0
$586$	6327.08	0.446022
$587$	−8017.17	−0.563720	−0.281860	−	0.959455i	$-0.590951\pi$
$587$	−8017.17	−0.563720	−0.281860	+	0.959455i	$0.590951\pi$
$588$	−16425.4	−1.15199
$589$	23447.1	1.64027
$590$	0	0
$591$	−13101.1	−0.911856
$592$	22.7804	0.00158154
$593$	18589.1	1.28729	0.643645	−	0.765324i	$-0.277421\pi$
$593$	18589.1	1.28729	0.643645	+	0.765324i	$0.277421\pi$
$594$	8047.15	0.555856
$595$	0	0
$596$	1151.54	0.0791422
$597$	15897.5	1.08985
$598$	−2836.40	−0.193961
$599$	−26260.4	−1.79127	−0.895636	−	0.444787i	$-0.853279\pi$
$599$	−26260.4	−1.79127	−0.895636	+	0.444787i	$0.853279\pi$
$600$	0	0
$601$	−9005.84	−0.611241	−0.305621	−	0.952153i	$-0.598864\pi$
$601$	−9005.84	−0.611241	−0.305621	+	0.952153i	$0.598864\pi$
$602$	−23868.4	−1.61596
$603$	−11862.5	−0.801125
$604$	7060.43	0.475637
$605$	0	0
$606$	−10929.6	−0.732647
$607$	−22583.2	−1.51009	−0.755043	−	0.655676i	$-0.772384\pi$
$607$	−22583.2	−1.51009	−0.755043	+	0.655676i	$0.772384\pi$
$608$	24883.3	1.65979
$609$	42420.0	2.82257
$610$	0	0
$611$	−2587.97	−0.171355
$612$	1722.64	0.113780
$613$	1316.87	0.0867667	0.0433833	−	0.999058i	$-0.486186\pi$
$613$	1316.87	0.0867667	0.0433833	+	0.999058i	$0.486186\pi$
$614$	−8684.44	−0.570807
$615$	0	0
$616$	37993.9	2.48510
$617$	25475.8	1.66226	0.831132	−	0.556076i	$-0.187694\pi$
$617$	25475.8	1.66226	0.831132	+	0.556076i	$0.187694\pi$
$618$	−1997.97	−0.130049
$619$	22782.0	1.47930	0.739650	−	0.672992i	$-0.234991\pi$
$619$	22782.0	1.47930	0.739650	+	0.672992i	$0.234991\pi$
$620$	0	0
$621$	−1738.51	−0.112342
$622$	−5953.02	−0.383753
$623$	23072.9	1.48378
$624$	59.3343	0.00380653
$625$	0	0
$626$	−8825.52	−0.563480
$627$	66735.6	4.25066
$628$	−4764.54	−0.302748
$629$	−2060.35	−0.130606
$630$	0	0
$631$	−17322.3	−1.09285	−0.546425	−	0.837508i	$-0.684012\pi$
$631$	−17322.3	−1.09285	−0.546425	+	0.837508i	$0.684012\pi$
$632$	−15821.3	−0.995788
$633$	29118.2	1.82835
$634$	3827.52	0.239764
$635$	0	0
$636$	27269.3	1.70016
$637$	29407.4	1.82914
$638$	−20567.1	−1.27627
$639$	5304.87	0.328416
$640$	0	0
$641$	−25514.3	−1.57216	−0.786080	−	0.618125i	$-0.787893\pi$
$641$	−25514.3	−1.57216	−0.786080	+	0.618125i	$0.787893\pi$
$642$	−10394.1	−0.638978
$643$	13167.6	0.807589	0.403794	−	0.914850i	$-0.367691\pi$
$643$	13167.6	0.807589	0.403794	+	0.914850i	$0.367691\pi$
$644$	−3137.58	−0.191984
$645$	0	0
$646$	−2288.46	−0.139378
$647$	−14419.8	−0.876197	−0.438098	−	0.898927i	$-0.644348\pi$
$647$	−14419.8	−0.876197	−0.438098	+	0.898927i	$0.644348\pi$
$648$	8658.88	0.524927
$649$	53190.8	3.21714
$650$	0	0
$651$	−37472.6	−2.25602
$652$	8719.16	0.523725
$653$	18605.4	1.11499	0.557493	−	0.830182i	$-0.311763\pi$
$653$	18605.4	1.11499	0.557493	+	0.830182i	$0.311763\pi$
$654$	−2570.69	−0.153703
$655$	0	0
$656$	10.0070	0.000595594 0
$657$	−2383.14	−0.141515
$658$	1763.78	0.104497
$659$	−27497.4	−1.62541	−0.812705	−	0.582675i	$-0.802006\pi$
$659$	−27497.4	−1.62541	−0.812705	+	0.582675i	$0.802006\pi$
$660$	0	0
$661$	5841.00	0.343704	0.171852	−	0.985123i	$-0.445025\pi$
$661$	5841.00	0.343704	0.171852	+	0.985123i	$0.445025\pi$
$662$	11253.5	0.660693
$663$	−5366.42	−0.314350
$664$	−12312.4	−0.719601
$665$	0	0
$666$	13764.8	0.800862
$667$	4443.34	0.257941
$668$	9964.66	0.577162
$669$	−30644.2	−1.77096
$670$	0	0
$671$	−23347.4	−1.34325
$672$	−39768.0	−2.28286
$673$	−31184.0	−1.78611	−0.893057	−	0.449944i	$-0.851444\pi$
$673$	−31184.0	−1.78611	−0.893057	+	0.449944i	$0.851444\pi$
$674$	−19841.4	−1.13392
$675$	0	0
$676$	−13808.9	−0.785666
$677$	−628.810	−0.0356974	−0.0178487	−	0.999841i	$-0.505682\pi$
$677$	−628.810	−0.0356974	−0.0178487	+	0.999841i	$0.505682\pi$
$678$	−583.572	−0.0330559
$679$	6994.79	0.395339
$680$	0	0
$681$	28051.1	1.57844
$682$	18168.4	1.02009
$683$	13252.5	0.742451	0.371225	−	0.928543i	$-0.378938\pi$
$683$	13252.5	0.742451	0.371225	+	0.928543i	$0.378938\pi$
$684$	−24815.1	−1.38718
$685$	0	0
$686$	−3534.53	−0.196719
$687$	−5830.50	−0.323795
$688$	52.3001	0.00289814
$689$	−48822.0	−2.69952
$690$	0	0
$691$	−20653.5	−1.13704	−0.568521	−	0.822669i	$-0.692484\pi$
$691$	−20653.5	−1.13704	−0.568521	+	0.822669i	$0.692484\pi$
$692$	2674.20	0.146905
$693$	−61296.2	−3.35996
$694$	9874.29	0.540091
$695$	0	0
$696$	34812.6	1.89593
$697$	−905.075	−0.0491853
$698$	6039.36	0.327498
$699$	−46366.9	−2.50895
$700$	0	0
$701$	−6385.56	−0.344050	−0.172025	−	0.985093i	$-0.555031\pi$
$701$	−6385.56	−0.344050	−0.172025	+	0.985093i	$0.555031\pi$
$702$	9321.59	0.501169
$703$	29679.8	1.59231
$704$	19229.9	1.02948
$705$	0	0
$706$	−7507.52	−0.400211
$707$	21645.7	1.15144
$708$	−34414.7	−1.82681
$709$	−23481.3	−1.24381	−0.621904	−	0.783094i	$-0.713641\pi$
$709$	−23481.3	−1.24381	−0.621904	+	0.783094i	$0.713641\pi$
$710$	0	0
$711$	25524.8	1.34635
$712$	18935.1	0.996662
$713$	−3925.12	−0.206167
$714$	3657.37	0.191700
$715$	0	0
$716$	4548.36	0.237403
$717$	−17637.2	−0.918653
$718$	−8244.04	−0.428503
$719$	−26590.6	−1.37923	−0.689613	−	0.724178i	$-0.742219\pi$
$719$	−26590.6	−1.37923	−0.689613	+	0.724178i	$0.742219\pi$
$720$	0	0
$721$	3956.92	0.204388
$722$	20987.5	1.08182
$723$	45935.3	2.36287
$724$	−15709.8	−0.806423
$725$	0	0
$726$	33190.6	1.69672
$727$	−30288.8	−1.54518	−0.772592	−	0.634903i	$-0.781040\pi$
$727$	−30288.8	−1.54518	−0.772592	+	0.634903i	$0.781040\pi$
$728$	44011.1	2.24060
$729$	−30230.8	−1.53588
$730$	0	0
$731$	−4730.21	−0.239334
$732$	15105.9	0.762746
$733$	−10849.1	−0.546685	−0.273342	−	0.961917i	$-0.588129\pi$
$733$	−10849.1	−0.546685	−0.273342	+	0.961917i	$0.588129\pi$
$734$	−3936.12	−0.197936
$735$	0	0
$736$	−4165.56	−0.208620
$737$	−19819.7	−0.990592
$738$	6046.63	0.301598
$739$	19017.8	0.946660	0.473330	−	0.880885i	$-0.343052\pi$
$739$	19017.8	0.946660	0.473330	+	0.880885i	$0.343052\pi$
$740$	0	0
$741$	77304.6	3.83246
$742$	33273.6	1.64624
$743$	−13686.8	−0.675801	−0.337901	−	0.941182i	$-0.609717\pi$
$743$	−13686.8	−0.675801	−0.337901	+	0.941182i	$0.609717\pi$
$744$	−30752.5	−1.51538
$745$	0	0
$746$	1696.01	0.0832377
$747$	19863.9	0.972932
$748$	2878.16	0.140690
$749$	20585.3	1.00423
$750$	0	0
$751$	19702.9	0.957347	0.478674	−	0.877993i	$-0.341118\pi$
$751$	19702.9	0.957347	0.478674	+	0.877993i	$0.341118\pi$
$752$	−3.86475	−0.000187411 0
$753$	−8577.91	−0.415135
$754$	−23824.4	−1.15071
$755$	0	0
$756$	10311.4	0.496061
$757$	41079.0	1.97231	0.986157	−	0.165813i	$-0.0530247\pi$
$757$	41079.0	1.97231	0.986157	+	0.165813i	$0.0530247\pi$
$758$	7994.39	0.383073
$759$	−11171.8	−0.534268
$760$	0	0
$761$	−11409.9	−0.543506	−0.271753	−	0.962367i	$-0.587603\pi$
$761$	−11409.9	−0.543506	−0.271753	+	0.962367i	$0.587603\pi$
$762$	−20778.8	−0.987844
$763$	5091.17	0.241563
$764$	−14006.6	−0.663275
$765$	0	0
$766$	13712.8	0.646821
$767$	61614.7	2.90062
$768$	−32602.9	−1.53185
$769$	−2253.98	−0.105697	−0.0528483	−	0.998603i	$-0.516830\pi$
$769$	−2253.98	−0.105697	−0.0528483	+	0.998603i	$0.516830\pi$
$770$	0	0
$771$	−27689.8	−1.29342
$772$	−9840.43	−0.458763
$773$	−30875.1	−1.43661	−0.718306	−	0.695728i	$-0.755082\pi$
$773$	−30875.1	−1.43661	−0.718306	+	0.695728i	$0.755082\pi$
$774$	31601.7	1.46757
$775$	0	0
$776$	5740.38	0.265551
$777$	−47433.6	−2.19005
$778$	9239.92	0.425793
$779$	13037.8	0.599652
$780$	0	0
$781$	8863.29	0.406087
$782$	383.096	0.0175185
$783$	−14602.7	−0.666484
$784$	43.9156	0.00200053
$785$	0	0
$786$	−1595.57	−0.0724075
$787$	13216.7	0.598634	0.299317	−	0.954154i	$-0.403241\pi$
$787$	13216.7	0.598634	0.299317	+	0.954154i	$0.403241\pi$
$788$	8139.28	0.367957
$789$	4594.42	0.207308
$790$	0	0
$791$	1155.75	0.0519515
$792$	−50303.7	−2.25690
$793$	−27045.0	−1.21109
$794$	17572.6	0.785427
$795$	0	0
$796$	−9876.57	−0.439781
$797$	−40132.2	−1.78363	−0.891817	−	0.452395i	$-0.850570\pi$
$797$	−40132.2	−1.78363	−0.891817	+	0.452395i	$0.850570\pi$
$798$	−52685.3	−2.33714
$799$	349.542	0.0154767
$800$	0	0
$801$	−30548.3	−1.34753
$802$	−5900.96	−0.259813
$803$	−3981.71	−0.174983
$804$	12823.4	0.562496
$805$	0	0
$806$	21045.8	0.919734
$807$	−14804.0	−0.645755
$808$	17763.9	0.773430
$809$	14870.1	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$809$	14870.1	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$810$	0	0
$811$	282.093	0.0122141	0.00610704	−	0.999981i	$-0.498056\pi$
$811$	282.093	0.0122141	0.00610704	+	0.999981i	$0.498056\pi$
$812$	−26354.1	−1.13898
$813$	68800.4	2.96794
$814$	22998.0	0.990268
$815$	0	0
$816$	−8.01396	−0.000343805 0
$817$	68139.9	2.91789
$818$	−16811.9	−0.718598
$819$	−71003.8	−3.02940
$820$	0	0
$821$	−31977.5	−1.35934	−0.679672	−	0.733516i	$-0.737878\pi$
$821$	−31977.5	−1.35934	−0.679672	+	0.733516i	$0.737878\pi$
$822$	31478.4	1.33569
$823$	35152.5	1.48887	0.744434	−	0.667696i	$-0.232719\pi$
$823$	35152.5	1.48887	0.744434	+	0.667696i	$0.232719\pi$
$824$	3247.31	0.137288
$825$	0	0
$826$	−41992.2	−1.76888
$827$	23945.0	1.00683	0.503416	−	0.864044i	$-0.332076\pi$
$827$	23945.0	1.00683	0.503416	+	0.864044i	$0.332076\pi$
$828$	4154.14	0.174355
$829$	−24120.6	−1.01055	−0.505274	−	0.862959i	$-0.668608\pi$
$829$	−24120.6	−1.01055	−0.505274	+	0.862959i	$0.668608\pi$
$830$	0	0
$831$	−12861.0	−0.536874
$832$	22275.4	0.928196
$833$	−3971.89	−0.165207
$834$	−39028.2	−1.62043
$835$	0	0
$836$	−41460.7	−1.71525
$837$	12899.6	0.532706
$838$	1139.49	0.0469725
$839$	22867.8	0.940981	0.470490	−	0.882405i	$-0.344077\pi$
$839$	22867.8	0.940981	0.470490	+	0.882405i	$0.344077\pi$
$840$	0	0
$841$	12932.9	0.530276
$842$	−5850.09	−0.239439
$843$	10933.7	0.446712
$844$	−18090.2	−0.737785
$845$	0	0
$846$	−2335.23	−0.0949016
$847$	−65733.1	−2.66661
$848$	−72.9085	−0.00295246
$849$	−27044.7	−1.09325
$850$	0	0
$851$	−4968.51	−0.200139
$852$	−5734.59	−0.230591
$853$	−39570.7	−1.58836	−0.794182	−	0.607680i	$-0.792100\pi$
$853$	−39570.7	−1.58836	−0.794182	+	0.607680i	$0.792100\pi$
$854$	18432.0	0.738558
$855$	0	0
$856$	16893.6	0.674547
$857$	−34117.0	−1.35988	−0.679938	−	0.733270i	$-0.737993\pi$
$857$	−34117.0	−1.35988	−0.679938	+	0.733270i	$0.737993\pi$
$858$	59901.0	2.38343
$859$	−25440.7	−1.01051	−0.505254	−	0.862971i	$-0.668601\pi$
$859$	−25440.7	−1.01051	−0.505254	+	0.862971i	$0.668601\pi$
$860$	0	0
$861$	−20836.8	−0.824757
$862$	−4571.95	−0.180651
$863$	6323.33	0.249419	0.124710	−	0.992193i	$-0.460200\pi$
$863$	6323.33	0.249419	0.124710	+	0.992193i	$0.460200\pi$
$864$	13689.8	0.539045
$865$	0	0
$866$	−529.181	−0.0207648
$867$	−38421.2	−1.50502
$868$	23280.5	0.910359
$869$	42646.4	1.66476
$870$	0	0
$871$	−22958.5	−0.893135
$872$	4178.15	0.162259
$873$	−9261.06	−0.359037
$874$	−5518.60	−0.213581
$875$	0	0
$876$	2576.18	0.0993621
$877$	44457.7	1.71178	0.855889	−	0.517160i	$-0.173011\pi$
$877$	44457.7	1.71178	0.855889	+	0.517160i	$0.173011\pi$
$878$	8535.86	0.328099
$879$	28867.3	1.10770
$880$	0	0
$881$	−17088.2	−0.653480	−0.326740	−	0.945114i	$-0.605950\pi$
$881$	−17088.2	−0.653480	−0.326740	+	0.945114i	$0.605950\pi$
$882$	26535.4	1.01303
$883$	−37260.9	−1.42008	−0.710040	−	0.704162i	$-0.751323\pi$
$883$	−37260.9	−1.42008	−0.710040	+	0.704162i	$0.751323\pi$
$884$	3333.98	0.126848
$885$	0	0
$886$	3726.84	0.141316
$887$	−14770.5	−0.559126	−0.279563	−	0.960127i	$-0.590190\pi$
$887$	−14770.5	−0.559126	−0.279563	+	0.960127i	$0.590190\pi$
$888$	−38927.1	−1.47107
$889$	41151.8	1.55252
$890$	0	0
$891$	−23340.0	−0.877576
$892$	19038.2	0.714627
$893$	−5035.25	−0.188688
$894$	−3236.96	−0.121096
$895$	0	0
$896$	24747.2	0.922706
$897$	−12941.1	−0.481705
$898$	−4212.49	−0.156540
$899$	−32969.1	−1.22312
$900$	0	0
$901$	6594.11	0.243820
$902$	10102.6	0.372927
$903$	−108900.	−4.01324
$904$	948.481	0.0348960
$905$	0	0
$906$	−19846.8	−0.727777
$907$	−24238.3	−0.887342	−0.443671	−	0.896190i	$-0.646324\pi$
$907$	−24238.3	−0.887342	−0.443671	+	0.896190i	$0.646324\pi$
$908$	−17427.2	−0.636942
$909$	−28658.8	−1.04571
$910$	0	0
$911$	−18340.4	−0.667009	−0.333505	−	0.942748i	$-0.608231\pi$
$911$	−18340.4	−0.667009	−0.333505	+	0.942748i	$0.608231\pi$
$912$	115.443	0.00419156
$913$	33188.2	1.20303
$914$	9140.31	0.330782
$915$	0	0
$916$	3622.30	0.130660
$917$	3159.99	0.113797
$918$	−1259.01	−0.0452654
$919$	29898.4	1.07318	0.536592	−	0.843842i	$-0.319711\pi$
$919$	29898.4	1.07318	0.536592	+	0.843842i	$0.319711\pi$
$920$	0	0
$921$	−39622.8	−1.41760
$922$	2043.88	0.0730062
$923$	10267.0	0.366134
$924$	66261.5	2.35914
$925$	0	0
$926$	−1800.49	−0.0638959
$927$	−5238.94	−0.185620
$928$	−34988.6	−1.23767
$929$	3164.90	0.111773	0.0558865	−	0.998437i	$-0.482202\pi$
$929$	3164.90	0.111773	0.0558865	+	0.998437i	$0.482202\pi$
$930$	0	0
$931$	57216.1	2.01416
$932$	28806.2	1.01242
$933$	−27160.7	−0.953055
$934$	−16785.5	−0.588049
$935$	0	0
$936$	−58270.4	−2.03486
$937$	−7829.27	−0.272968	−0.136484	−	0.990642i	$-0.543580\pi$
$937$	−7829.27	−0.272968	−0.136484	+	0.990642i	$0.543580\pi$
$938$	15646.9	0.544658
$939$	−40266.4	−1.39941
$940$	0	0
$941$	12422.9	0.430367	0.215183	−	0.976574i	$-0.430965\pi$
$941$	12422.9	0.430367	0.215183	+	0.976574i	$0.430965\pi$
$942$	13393.1	0.463238
$943$	−2182.58	−0.0753707
$944$	92.0125	0.00317241
$945$	0	0
$946$	52799.5	1.81465
$947$	29718.7	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$947$	29718.7	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$948$	−27592.4	−0.945316
$949$	−4612.30	−0.157768
$950$	0	0
$951$	17463.1	0.595456
$952$	−5944.33	−0.202371
$953$	20281.7	0.689389	0.344695	−	0.938715i	$-0.387982\pi$
$953$	20281.7	0.689389	0.344695	+	0.938715i	$0.387982\pi$
$954$	−44054.0	−1.49507
$955$	0	0
$956$	10957.4	0.370700
$957$	−93837.4	−3.16963
$958$	4786.53	0.161426
$959$	−62342.1	−2.09920
$960$	0	0
$961$	−667.021	−0.0223900
$962$	26640.2	0.892842
$963$	−27254.8	−0.912018
$964$	−28538.1	−0.953476
$965$	0	0
$966$	8819.71	0.293757
$967$	−36677.2	−1.21971	−0.609855	−	0.792513i	$-0.708772\pi$
$967$	−36677.2	−1.21971	−0.609855	+	0.792513i	$0.708772\pi$
$968$	−53944.8	−1.79117
$969$	−10441.1	−0.346147
$970$	0	0
$971$	28329.9	0.936302	0.468151	−	0.883648i	$-0.344920\pi$
$971$	28329.9	0.936302	0.468151	+	0.883648i	$0.344920\pi$
$972$	25203.7	0.831697
$973$	77294.3	2.54670
$974$	1046.83	0.0344380
$975$	0	0
$976$	−40.3878	−0.00132457
$977$	3627.47	0.118785	0.0593926	−	0.998235i	$-0.481084\pi$
$977$	3627.47	0.118785	0.0593926	+	0.998235i	$0.481084\pi$
$978$	−24509.5	−0.801357
$979$	−51039.7	−1.66623
$980$	0	0
$981$	−6740.67	−0.219381
$982$	14700.8	0.477721
$983$	−32607.9	−1.05802	−0.529008	−	0.848617i	$-0.677436\pi$
$983$	−32607.9	−1.05802	−0.529008	+	0.848617i	$0.677436\pi$
$984$	−17100.0	−0.553993
$985$	0	0
$986$	3217.82	0.103931
$987$	8047.22	0.259520
$988$	−48026.9	−1.54650
$989$	−11406.9	−0.366752
$990$	0	0
$991$	11086.4	0.355369	0.177685	−	0.984087i	$-0.443139\pi$
$991$	11086.4	0.355369	0.177685	+	0.984087i	$0.443139\pi$
$992$	30908.0	0.989243
$993$	51344.0	1.64084
$994$	−6997.25	−0.223279
$995$	0	0
$996$	−21472.9	−0.683128
$997$	26663.5	0.846982	0.423491	−	0.905900i	$-0.360805\pi$
$997$	26663.5	0.846982	0.423491	+	0.905900i	$0.360805\pi$
$998$	−6990.42	−0.221721
$999$	16328.6	0.517131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.4.a.o.1.8	✓	13
5.2	odd	4		575.4.b.l.24.16		26
5.3	odd	4		575.4.b.l.24.11		26
5.4	even	2		575.4.a.p.1.6	yes	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
575.4.a.o.1.8	✓	13	1.1	even	1	trivial
575.4.a.p.1.6	yes	13	5.4	even	2
575.4.b.l.24.11		26	5.3	odd	4
575.4.b.l.24.16		26	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	575.a (trivial)

\(f(q)\)	\(=\)	\(q+1.74638 q^{2} +7.96785 q^{3} -4.95017 q^{4} +13.9149 q^{6} -27.5580 q^{7} -22.6159 q^{8} +36.4866 q^{9} +60.9612 q^{11} -39.4422 q^{12} +70.6157 q^{13} -48.1266 q^{14} +0.105454 q^{16} -9.53767 q^{17} +63.7193 q^{18} +137.393 q^{19} -219.578 q^{21} +106.461 q^{22} -23.0000 q^{23} -180.200 q^{24} +123.322 q^{26} +75.5876 q^{27} +136.417 q^{28} -193.189 q^{29} +170.657 q^{31} +181.111 q^{32} +485.729 q^{33} -16.6564 q^{34} -180.615 q^{36} +216.022 q^{37} +239.939 q^{38} +562.655 q^{39} +94.8947 q^{41} -383.466 q^{42} +495.951 q^{43} -301.768 q^{44} -40.1667 q^{46} -36.6486 q^{47} +0.840243 q^{48} +416.442 q^{49} -75.9947 q^{51} -349.559 q^{52} -691.376 q^{53} +132.004 q^{54} +623.248 q^{56} +1094.72 q^{57} -337.381 q^{58} +872.536 q^{59} -382.989 q^{61} +298.032 q^{62} -1005.50 q^{63} +315.445 q^{64} +848.267 q^{66} -325.119 q^{67} +47.2130 q^{68} -183.260 q^{69} +145.392 q^{71} -825.176 q^{72} -65.3155 q^{73} +377.256 q^{74} -680.116 q^{76} -1679.97 q^{77} +982.608 q^{78} +699.567 q^{79} -382.867 q^{81} +165.722 q^{82} +544.415 q^{83} +1086.95 q^{84} +866.117 q^{86} -1539.30 q^{87} -1378.69 q^{88} -837.249 q^{89} -1946.03 q^{91} +113.854 q^{92} +1359.77 q^{93} -64.0023 q^{94} +1443.07 q^{96} -253.821 q^{97} +727.266 q^{98} +2224.26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 3 q^{2} + 3 q^{3} + 75 q^{4} + 62 q^{6} + q^{7} - 78 q^{8} + 108 q^{9} + 118 q^{11} - 27 q^{12} + 83 q^{13} + 40 q^{14} + 411 q^{16} + 71 q^{17} - 217 q^{18} + 360 q^{19} + 434 q^{21} + 169 q^{22}+ \cdots + 4599 q^{99}+O(q^{100}) \) 13 * q - 3 * q^2 + 3 * q^3 + 75 * q^4 + 62 * q^6 + q^7 - 78 * q^8 + 108 * q^9 + 118 * q^11 - 27 * q^12 + 83 * q^13 + 40 * q^14 + 411 * q^16 + 71 * q^17 - 217 * q^18 + 360 * q^19 + 434 * q^21 + 169 * q^22 - 299 * q^23 + 621 * q^24 + 269 * q^26 - 18 * q^27 + 626 * q^28 + 505 * q^29 + 130 * q^31 - 1207 * q^32 + 883 * q^33 + 487 * q^34 + 1441 * q^36 + 152 * q^37 + 661 * q^38 + 295 * q^39 + 330 * q^41 - 1056 * q^42 + 941 * q^43 - 253 * q^44 + 69 * q^46 - 874 * q^47 + 442 * q^48 + 1708 * q^49 - 1047 * q^51 - 1685 * q^52 + 1320 * q^53 + 3338 * q^54 + 220 * q^56 - 1763 * q^57 + 1376 * q^58 + 1177 * q^59 + 822 * q^61 - 3066 * q^62 + 887 * q^63 + 6328 * q^64 - 1423 * q^66 - 2453 * q^67 + 3293 * q^68 - 69 * q^69 + 550 * q^71 - 4917 * q^72 + 1582 * q^73 + 2504 * q^74 + 1265 * q^76 - 733 * q^77 + 6167 * q^78 + 3403 * q^79 - 439 * q^81 - 5232 * q^82 - 76 * q^83 + 5964 * q^84 - 544 * q^86 - 1523 * q^87 + 7321 * q^88 + 2403 * q^89 + 2451 * q^91 - 1725 * q^92 + 2251 * q^93 + 6595 * q^94 + 4550 * q^96 - 2208 * q^97 + 4895 * q^98 + 4599 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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