LMFDB - Embedded newform 1521.4.a.bg.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 $ x^9 - 4x^8 - 46x^7 + 145x^6 + 680x^5 - 1501x^4 - 3203x^3 + 4784x^2 + 3584x - 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 169)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$5.84018$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.84018 q^{2} +15.4273 q^{4} -15.2399 q^{5} +4.31620 q^{7} +35.9495 q^{8} +O(q^{10})$	$q+4.84018 q^{2} +15.4273 q^{4} -15.2399 q^{5} +4.31620 q^{7} +35.9495 q^{8} -73.7636 q^{10} -24.5691 q^{11} +20.8912 q^{14} +50.5834 q^{16} +127.160 q^{17} +51.7462 q^{19} -235.110 q^{20} -118.919 q^{22} -87.3684 q^{23} +107.253 q^{25} +66.5874 q^{28} -225.726 q^{29} -108.720 q^{31} -42.7634 q^{32} +615.479 q^{34} -65.7783 q^{35} -115.756 q^{37} +250.461 q^{38} -547.865 q^{40} -191.885 q^{41} -123.301 q^{43} -379.036 q^{44} -422.878 q^{46} -36.7339 q^{47} -324.370 q^{49} +519.124 q^{50} -119.162 q^{53} +374.430 q^{55} +155.165 q^{56} -1092.55 q^{58} -804.553 q^{59} +678.886 q^{61} -526.226 q^{62} -611.650 q^{64} +87.4806 q^{67} +1961.74 q^{68} -318.379 q^{70} -981.049 q^{71} -263.862 q^{73} -560.281 q^{74} +798.304 q^{76} -106.045 q^{77} +321.051 q^{79} -770.883 q^{80} -928.757 q^{82} -1042.54 q^{83} -1937.91 q^{85} -596.798 q^{86} -883.248 q^{88} -344.746 q^{89} -1347.86 q^{92} -177.799 q^{94} -788.604 q^{95} +482.521 q^{97} -1570.01 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + 37 q^{4} - 30 q^{5} + 38 q^{7} - 60 q^{8}+O(q^{10}) $ 9 * q - 5 * q^2 + 37 * q^4 - 30 * q^5 + 38 * q^7 - 60 * q^8	$ 9 q - 5 q^{2} + 37 q^{4} - 30 q^{5} + 38 q^{7} - 60 q^{8} - 147 q^{10} - 181 q^{11} + 147 q^{14} + 269 q^{16} + 55 q^{17} + 161 q^{19} - 370 q^{20} + 340 q^{22} + 204 q^{23} + 307 q^{25} + 344 q^{28} - 280 q^{29} + 706 q^{31} - 680 q^{32} + 216 q^{34} - 20 q^{35} + 298 q^{37} + 739 q^{38} + 13 q^{40} - 1201 q^{41} - 533 q^{43} - 355 q^{44} - 840 q^{46} - 956 q^{47} + 403 q^{49} + 1156 q^{50} + 278 q^{53} - 250 q^{55} - 250 q^{56} - 2877 q^{58} - 1377 q^{59} - 136 q^{61} - 2035 q^{62} + 284 q^{64} - 931 q^{67} + 1536 q^{68} - 4854 q^{70} - 2046 q^{71} - 45 q^{73} + 1990 q^{74} - 3608 q^{76} + 718 q^{77} + 412 q^{79} + 787 q^{80} + 2757 q^{82} - 3709 q^{83} - 2106 q^{85} - 125 q^{86} - 636 q^{88} - 1663 q^{89} - 4010 q^{92} - 2531 q^{94} + 1614 q^{95} - 1087 q^{97} + 282 q^{98}+O(q^{100}) $ 9 * q - 5 * q^2 + 37 * q^4 - 30 * q^5 + 38 * q^7 - 60 * q^8 - 147 * q^10 - 181 * q^11 + 147 * q^14 + 269 * q^16 + 55 * q^17 + 161 * q^19 - 370 * q^20 + 340 * q^22 + 204 * q^23 + 307 * q^25 + 344 * q^28 - 280 * q^29 + 706 * q^31 - 680 * q^32 + 216 * q^34 - 20 * q^35 + 298 * q^37 + 739 * q^38 + 13 * q^40 - 1201 * q^41 - 533 * q^43 - 355 * q^44 - 840 * q^46 - 956 * q^47 + 403 * q^49 + 1156 * q^50 + 278 * q^53 - 250 * q^55 - 250 * q^56 - 2877 * q^58 - 1377 * q^59 - 136 * q^61 - 2035 * q^62 + 284 * q^64 - 931 * q^67 + 1536 * q^68 - 4854 * q^70 - 2046 * q^71 - 45 * q^73 + 1990 * q^74 - 3608 * q^76 + 718 * q^77 + 412 * q^79 + 787 * q^80 + 2757 * q^82 - 3709 * q^83 - 2106 * q^85 - 125 * q^86 - 636 * q^88 - 1663 * q^89 - 4010 * q^92 - 2531 * q^94 + 1614 * q^95 - 1087 * q^97 + 282 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	4.84018	1.71126	0.855630	−	0.517587i	$-0.173170\pi$
$2$	4.84018	1.71126	0.855630	+	0.517587i	$0.173170\pi$
$3$	0	0
$3$	0	0
$4$	15.4273	1.92841
$5$	−15.2399	−1.36309	−0.681547	−	0.731774i	$-0.738692\pi$
$5$	−15.2399	−1.36309	−0.681547	+	0.731774i	$0.738692\pi$
$6$	0	0
$7$	4.31620	0.233053	0.116527	−	0.993188i	$-0.462824\pi$
$7$	4.31620	0.233053	0.116527	+	0.993188i	$0.462824\pi$
$8$	35.9495	1.58876
$9$	0	0
$10$	−73.7636	−2.33261
$11$	−24.5691	−0.673443	−0.336722	−	0.941604i	$-0.609318\pi$
$11$	−24.5691	−0.673443	−0.336722	+	0.941604i	$0.609318\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	20.8912	0.398815
$15$	0	0
$16$	50.5834	0.790365
$17$	127.160	1.81417	0.907086	−	0.420944i	$-0.138301\pi$
$17$	127.160	1.81417	0.907086	+	0.420944i	$0.138301\pi$
$18$	0	0
$19$	51.7462	0.624810	0.312405	−	0.949949i	$-0.398865\pi$
$19$	51.7462	0.624810	0.312405	+	0.949949i	$0.398865\pi$
$20$	−235.110	−2.62861
$21$	0	0
$22$	−118.919	−1.15244
$23$	−87.3684	−0.792068	−0.396034	−	0.918236i	$-0.629614\pi$
$23$	−87.3684	−0.792068	−0.396034	+	0.918236i	$0.629614\pi$
$24$	0	0
$25$	107.253	0.858024
$26$	0	0
$27$	0	0
$28$	66.5874	0.449423
$29$	−225.726	−1.44539	−0.722693	−	0.691169i	$-0.757096\pi$
$29$	−225.726	−1.44539	−0.722693	+	0.691169i	$0.757096\pi$
$30$	0	0
$31$	−108.720	−0.629895	−0.314948	−	0.949109i	$-0.601987\pi$
$31$	−108.720	−0.629895	−0.314948	+	0.949109i	$0.601987\pi$
$32$	−42.7634	−0.236237
$33$	0	0
$34$	615.479	3.10452
$35$	−65.7783	−0.317673
$36$	0	0
$37$	−115.756	−0.514330	−0.257165	−	0.966367i	$-0.582788\pi$
$37$	−115.756	−0.514330	−0.257165	+	0.966367i	$0.582788\pi$
$38$	250.461	1.06921
$39$	0	0
$40$	−547.865	−2.16563
$41$	−191.885	−0.730912	−0.365456	−	0.930829i	$-0.619087\pi$
$41$	−191.885	−0.730912	−0.365456	+	0.930829i	$0.619087\pi$
$42$	0	0
$43$	−123.301	−0.437284	−0.218642	−	0.975805i	$-0.570163\pi$
$43$	−123.301	−0.437284	−0.218642	+	0.975805i	$0.570163\pi$
$44$	−379.036	−1.29868
$45$	0	0
$46$	−422.878	−1.35543
$47$	−36.7339	−0.114004	−0.0570020	−	0.998374i	$-0.518154\pi$
$47$	−36.7339	−0.114004	−0.0570020	+	0.998374i	$0.518154\pi$
$48$	0	0
$49$	−324.370	−0.945686
$50$	519.124	1.46830
$51$	0	0
$52$	0	0
$53$	−119.162	−0.308833	−0.154416	−	0.988006i	$-0.549350\pi$
$53$	−119.162	−0.308833	−0.154416	+	0.988006i	$0.549350\pi$
$54$	0	0
$55$	374.430	0.917966
$56$	155.165	0.370265
$57$	0	0
$58$	−1092.55	−2.47343
$59$	−804.553	−1.77532	−0.887660	−	0.460500i	$-0.847670\pi$
$59$	−804.553	−1.77532	−0.887660	+	0.460500i	$0.847670\pi$
$60$	0	0
$61$	678.886	1.42496	0.712479	−	0.701693i	$-0.247572\pi$
$61$	678.886	1.42496	0.712479	+	0.701693i	$0.247572\pi$
$62$	−526.226	−1.07791
$63$	0	0
$64$	−611.650	−1.19463
$65$	0	0
$66$	0	0
$67$	87.4806	0.159514	0.0797571	−	0.996814i	$-0.474586\pi$
$67$	87.4806	0.159514	0.0797571	+	0.996814i	$0.474586\pi$
$68$	1961.74	3.49848
$69$	0	0
$70$	−318.379	−0.543622
$71$	−981.049	−1.63985	−0.819923	−	0.572473i	$-0.805984\pi$
$71$	−981.049	−1.63985	−0.819923	+	0.572473i	$0.805984\pi$
$72$	0	0
$73$	−263.862	−0.423051	−0.211526	−	0.977372i	$-0.567843\pi$
$73$	−263.862	−0.423051	−0.211526	+	0.977372i	$0.567843\pi$
$74$	−560.281	−0.880153
$75$	0	0
$76$	798.304	1.20489
$77$	−106.045	−0.156948
$78$	0	0
$79$	321.051	0.457228	0.228614	−	0.973517i	$-0.426581\pi$
$79$	321.051	0.457228	0.228614	+	0.973517i	$0.426581\pi$
$80$	−770.883	−1.07734
$81$	0	0
$82$	−928.757	−1.25078
$83$	−1042.54	−1.37872	−0.689362	−	0.724417i	$-0.742109\pi$
$83$	−1042.54	−1.37872	−0.689362	+	0.724417i	$0.742109\pi$
$84$	0	0
$85$	−1937.91	−2.47289
$86$	−596.798	−0.748307
$87$	0	0
$88$	−883.248	−1.06994
$89$	−344.746	−0.410595	−0.205298	−	0.978700i	$-0.565816\pi$
$89$	−344.746	−0.410595	−0.205298	+	0.978700i	$0.565816\pi$
$90$	0	0
$91$	0	0
$92$	−1347.86	−1.52743
$93$	0	0
$94$	−177.799	−0.195091
$95$	−788.604	−0.851674
$96$	0	0
$97$	482.521	0.505079	0.252539	−	0.967587i	$-0.418734\pi$
$97$	482.521	0.505079	0.252539	+	0.967587i	$0.418734\pi$
$98$	−1570.01	−1.61832
$99$	0	0
$100$	1654.63	1.65463
$101$	−839.779	−0.827338	−0.413669	−	0.910427i	$-0.635753\pi$
$101$	−839.779	−0.827338	−0.413669	+	0.910427i	$0.635753\pi$
$102$	0	0
$103$	159.823	0.152892	0.0764459	−	0.997074i	$-0.475643\pi$
$103$	159.823	0.152892	0.0764459	+	0.997074i	$0.475643\pi$
$104$	0	0
$105$	0	0
$106$	−576.764	−0.528494
$107$	−1075.62	−0.971818	−0.485909	−	0.874009i	$-0.661511\pi$
$107$	−1075.62	−0.971818	−0.485909	+	0.874009i	$0.661511\pi$
$108$	0	0
$109$	1161.67	1.02080	0.510401	−	0.859936i	$-0.329497\pi$
$109$	1161.67	1.02080	0.510401	+	0.859936i	$0.329497\pi$
$110$	1812.31	1.57088
$111$	0	0
$112$	218.328	0.184197
$113$	1463.48	1.21834	0.609172	−	0.793038i	$-0.291502\pi$
$113$	1463.48	1.21834	0.609172	+	0.793038i	$0.291502\pi$
$114$	0	0
$115$	1331.48	1.07966
$116$	−3482.34	−2.78730
$117$	0	0
$118$	−3894.18	−3.03803
$119$	548.851	0.422799
$120$	0	0
$121$	−727.357	−0.546474
$122$	3285.93	2.43848
$123$	0	0
$124$	−1677.26	−1.21470
$125$	270.461	0.193526
$126$	0	0
$127$	289.923	0.202571	0.101285	−	0.994857i	$-0.467704\pi$
$127$	289.923	0.202571	0.101285	+	0.994857i	$0.467704\pi$
$128$	−2618.38	−1.80808
$129$	0	0
$130$	0	0
$131$	1201.97	0.801653	0.400826	−	0.916154i	$-0.368723\pi$
$131$	1201.97	0.801653	0.400826	+	0.916154i	$0.368723\pi$
$132$	0	0
$133$	223.347	0.145614
$134$	423.422	0.272971
$135$	0	0
$136$	4571.35	2.88228
$137$	779.620	0.486186	0.243093	−	0.970003i	$-0.421838\pi$
$137$	779.620	0.486186	0.243093	+	0.970003i	$0.421838\pi$
$138$	0	0
$139$	2031.48	1.23963	0.619814	−	0.784749i	$-0.287208\pi$
$139$	2031.48	1.23963	0.619814	+	0.784749i	$0.287208\pi$
$140$	−1014.78	−0.612605
$141$	0	0
$142$	−4748.45	−2.80621
$143$	0	0
$144$	0	0
$145$	3440.03	1.97020
$146$	−1277.14	−0.723951
$147$	0	0
$148$	−1785.81	−0.991841
$149$	2603.60	1.43151	0.715757	−	0.698350i	$-0.246082\pi$
$149$	2603.60	1.43151	0.715757	+	0.698350i	$0.246082\pi$
$150$	0	0
$151$	−206.776	−0.111438	−0.0557192	−	0.998446i	$-0.517745\pi$
$151$	−206.776	−0.111438	−0.0557192	+	0.998446i	$0.517745\pi$
$152$	1860.25	0.992671
$153$	0	0
$154$	−513.279	−0.268579
$155$	1656.88	0.858606
$156$	0	0
$157$	699.208	0.355433	0.177716	−	0.984082i	$-0.443129\pi$
$157$	699.208	0.355433	0.177716	+	0.984082i	$0.443129\pi$
$158$	1553.94	0.782437
$159$	0	0
$160$	651.708	0.322013
$161$	−377.100	−0.184594
$162$	0	0
$163$	2944.14	1.41474	0.707371	−	0.706842i	$-0.249881\pi$
$163$	2944.14	1.41474	0.707371	+	0.706842i	$0.249881\pi$
$164$	−2960.27	−1.40950
$165$	0	0
$166$	−5046.10	−2.35936
$167$	−2740.82	−1.27000	−0.635002	−	0.772510i	$-0.719001\pi$
$167$	−2740.82	−1.27000	−0.635002	+	0.772510i	$0.719001\pi$
$168$	0	0
$169$	0	0
$170$	−9379.81	−4.23176
$171$	0	0
$172$	−1902.20	−0.843264
$173$	−338.812	−0.148898	−0.0744491	−	0.997225i	$-0.523720\pi$
$173$	−338.812	−0.148898	−0.0744491	+	0.997225i	$0.523720\pi$
$174$	0	0
$175$	462.926	0.199965
$176$	−1242.79	−0.532266
$177$	0	0
$178$	−1668.63	−0.702635
$179$	1094.40	0.456979	0.228489	−	0.973546i	$-0.426621\pi$
$179$	1094.40	0.456979	0.228489	+	0.973546i	$0.426621\pi$
$180$	0	0
$181$	−1420.26	−0.583243	−0.291622	−	0.956534i	$-0.594195\pi$
$181$	−1420.26	−0.583243	−0.291622	+	0.956534i	$0.594195\pi$
$182$	0	0
$183$	0	0
$184$	−3140.85	−1.25840
$185$	1764.11	0.701080
$186$	0	0
$187$	−3124.22	−1.22174
$188$	−566.705	−0.219847
$189$	0	0
$190$	−3816.98	−1.45744
$191$	896.779	0.339731	0.169866	−	0.985467i	$-0.445667\pi$
$191$	896.779	0.339731	0.169866	+	0.985467i	$0.445667\pi$
$192$	0	0
$193$	−2589.97	−0.965959	−0.482980	−	0.875632i	$-0.660445\pi$
$193$	−2589.97	−0.965959	−0.482980	+	0.875632i	$0.660445\pi$
$194$	2335.49	0.864321
$195$	0	0
$196$	−5004.16	−1.82367
$197$	1481.87	0.535932	0.267966	−	0.963428i	$-0.413648\pi$
$197$	1481.87	0.535932	0.267966	+	0.963428i	$0.413648\pi$
$198$	0	0
$199$	−3599.45	−1.28220	−0.641101	−	0.767456i	$-0.721522\pi$
$199$	−3599.45	−1.28220	−0.641101	+	0.767456i	$0.721522\pi$
$200$	3855.69	1.36319
$201$	0	0
$202$	−4064.68	−1.41579
$203$	−974.278	−0.336852
$204$	0	0
$205$	2924.30	0.996301
$206$	773.572	0.261638
$207$	0	0
$208$	0	0
$209$	−1271.36	−0.420774
$210$	0	0
$211$	3051.22	0.995518	0.497759	−	0.867315i	$-0.334156\pi$
$211$	3051.22	0.995518	0.497759	+	0.867315i	$0.334156\pi$
$212$	−1838.35	−0.595557
$213$	0	0
$214$	−5206.21	−1.66303
$215$	1879.09	0.596059
$216$	0	0
$217$	−469.259	−0.146799
$218$	5622.67	1.74686
$219$	0	0
$220$	5776.45	1.77022
$221$	0	0
$222$	0	0
$223$	−2599.87	−0.780718	−0.390359	−	0.920663i	$-0.627649\pi$
$223$	−2599.87	−0.780718	−0.390359	+	0.920663i	$0.627649\pi$
$224$	−184.576	−0.0550558
$225$	0	0
$226$	7083.52	2.08491
$227$	993.208	0.290403	0.145202	−	0.989402i	$-0.453617\pi$
$227$	993.208	0.290403	0.145202	+	0.989402i	$0.453617\pi$
$228$	0	0
$229$	437.772	0.126327	0.0631633	−	0.998003i	$-0.479881\pi$
$229$	437.772	0.126327	0.0631633	+	0.998003i	$0.479881\pi$
$230$	6444.60	1.84758
$231$	0	0
$232$	−8114.72	−2.29637
$233$	−2933.08	−0.824689	−0.412344	−	0.911028i	$-0.635290\pi$
$233$	−2933.08	−0.824689	−0.412344	+	0.911028i	$0.635290\pi$
$234$	0	0
$235$	559.819	0.155398
$236$	−12412.1	−3.42355
$237$	0	0
$238$	2656.53	0.723519
$239$	−5813.48	−1.57340	−0.786700	−	0.617335i	$-0.788212\pi$
$239$	−5813.48	−1.57340	−0.786700	+	0.617335i	$0.788212\pi$
$240$	0	0
$241$	1148.25	0.306910	0.153455	−	0.988156i	$-0.450960\pi$
$241$	1148.25	0.306910	0.153455	+	0.988156i	$0.450960\pi$
$242$	−3520.54	−0.935160
$243$	0	0
$244$	10473.4	2.74791
$245$	4943.36	1.28906
$246$	0	0
$247$	0	0
$248$	−3908.44	−1.00075
$249$	0	0
$250$	1309.08	0.331174
$251$	3255.80	0.818741	0.409371	−	0.912368i	$-0.365748\pi$
$251$	3255.80	0.818741	0.409371	+	0.912368i	$0.365748\pi$
$252$	0	0
$253$	2146.57	0.533413
$254$	1403.28	0.346652
$255$	0	0
$256$	−7780.24	−1.89947
$257$	−6320.90	−1.53419	−0.767095	−	0.641534i	$-0.778298\pi$
$257$	−6320.90	−1.53419	−0.767095	+	0.641534i	$0.778298\pi$
$258$	0	0
$259$	−499.628	−0.119866
$260$	0	0
$261$	0	0
$262$	5817.74	1.37184
$263$	6168.99	1.44637	0.723187	−	0.690653i	$-0.242677\pi$
$263$	6168.99	1.44637	0.723187	+	0.690653i	$0.242677\pi$
$264$	0	0
$265$	1816.01	0.420968
$266$	1081.04	0.249183
$267$	0	0
$268$	1349.59	0.307609
$269$	−843.598	−0.191209	−0.0956043	−	0.995419i	$-0.530478\pi$
$269$	−843.598	−0.191209	−0.0956043	+	0.995419i	$0.530478\pi$
$270$	0	0
$271$	−2063.06	−0.462442	−0.231221	−	0.972901i	$-0.574272\pi$
$271$	−2063.06	−0.462442	−0.231221	+	0.972901i	$0.574272\pi$
$272$	6432.20	1.43386
$273$	0	0
$274$	3773.50	0.831990
$275$	−2635.12	−0.577831
$276$	0	0
$277$	6582.49	1.42781	0.713905	−	0.700242i	$-0.246925\pi$
$277$	6582.49	1.42781	0.713905	+	0.700242i	$0.246925\pi$
$278$	9832.74	2.12133
$279$	0	0
$280$	−2364.70	−0.504706
$281$	935.025	0.198502	0.0992508	−	0.995062i	$-0.468355\pi$
$281$	935.025	0.198502	0.0992508	+	0.995062i	$0.468355\pi$
$282$	0	0
$283$	−5338.77	−1.12140	−0.560701	−	0.828018i	$-0.689468\pi$
$283$	−5338.77	−1.12140	−0.560701	+	0.828018i	$0.689468\pi$
$284$	−15134.9	−3.16230
$285$	0	0
$286$	0	0
$287$	−828.215	−0.170341
$288$	0	0
$289$	11256.8	2.29122
$290$	16650.3	3.37152
$291$	0	0
$292$	−4070.69	−0.815818
$293$	1763.76	0.351673	0.175837	−	0.984419i	$-0.443737\pi$
$293$	1763.76	0.351673	0.175837	+	0.984419i	$0.443737\pi$
$294$	0	0
$295$	12261.3	2.41993
$296$	−4161.38	−0.817146
$297$	0	0
$298$	12601.9	2.44969
$299$	0	0
$300$	0	0
$301$	−532.192	−0.101910
$302$	−1000.83	−0.190700
$303$	0	0
$304$	2617.50	0.493828
$305$	−10346.1	−1.94235
$306$	0	0
$307$	−4736.59	−0.880558	−0.440279	−	0.897861i	$-0.645121\pi$
$307$	−4736.59	−0.880558	−0.440279	+	0.897861i	$0.645121\pi$
$308$	−1636.00	−0.302661
$309$	0	0
$310$	8019.60	1.46930
$311$	4746.90	0.865505	0.432753	−	0.901513i	$-0.357542\pi$
$311$	4746.90	0.865505	0.432753	+	0.901513i	$0.357542\pi$
$312$	0	0
$313$	10115.9	1.82679	0.913393	−	0.407078i	$-0.133452\pi$
$313$	10115.9	1.82679	0.913393	+	0.407078i	$0.133452\pi$
$314$	3384.29	0.608238
$315$	0	0
$316$	4952.95	0.881725
$317$	5906.13	1.04644	0.523219	−	0.852198i	$-0.324731\pi$
$317$	5906.13	1.04644	0.523219	+	0.852198i	$0.324731\pi$
$318$	0	0
$319$	5545.89	0.973386
$320$	9321.45	1.62839
$321$	0	0
$322$	−1825.23	−0.315888
$323$	6580.07	1.13351
$324$	0	0
$325$	0	0
$326$	14250.2	2.42099
$327$	0	0
$328$	−6898.16	−1.16124
$329$	−158.551	−0.0265690
$330$	0	0
$331$	−2802.82	−0.465429	−0.232715	−	0.972545i	$-0.574761\pi$
$331$	−2802.82	−0.465429	−0.232715	+	0.972545i	$0.574761\pi$
$332$	−16083.6	−2.65875
$333$	0	0
$334$	−13266.0	−2.17331
$335$	−1333.19	−0.217433
$336$	0	0
$337$	−244.919	−0.0395892	−0.0197946	−	0.999804i	$-0.506301\pi$
$337$	−244.919	−0.0395892	−0.0197946	+	0.999804i	$0.506301\pi$
$338$	0	0
$339$	0	0
$340$	−29896.7	−4.76875
$341$	2671.17	0.424199
$342$	0	0
$343$	−2880.51	−0.453448
$344$	−4432.60	−0.694738
$345$	0	0
$346$	−1639.91	−0.254804
$347$	−10356.8	−1.60226	−0.801129	−	0.598492i	$-0.795767\pi$
$347$	−10356.8	−1.60226	−0.801129	+	0.598492i	$0.795767\pi$
$348$	0	0
$349$	−8433.58	−1.29352	−0.646761	−	0.762693i	$-0.723877\pi$
$349$	−8433.58	−1.29352	−0.646761	+	0.762693i	$0.723877\pi$
$350$	2240.64	0.342193
$351$	0	0
$352$	1050.66	0.159092
$353$	−4733.72	−0.713741	−0.356871	−	0.934154i	$-0.616156\pi$
$353$	−4733.72	−0.713741	−0.356871	+	0.934154i	$0.616156\pi$
$354$	0	0
$355$	14951.0	2.23526
$356$	−5318.50	−0.791797
$357$	0	0
$358$	5297.08	0.782010
$359$	−7561.34	−1.11162	−0.555811	−	0.831309i	$-0.687592\pi$
$359$	−7561.34	−1.11162	−0.555811	+	0.831309i	$0.687592\pi$
$360$	0	0
$361$	−4181.33	−0.609613
$362$	−6874.30	−0.998081
$363$	0	0
$364$	0	0
$365$	4021.22	0.576659
$366$	0	0
$367$	−11940.0	−1.69826	−0.849130	−	0.528184i	$-0.822873\pi$
$367$	−11940.0	−1.69826	−0.849130	+	0.528184i	$0.822873\pi$
$368$	−4419.39	−0.626023
$369$	0	0
$370$	8538.60	1.19973
$371$	−514.327	−0.0719745
$372$	0	0
$373$	−5797.12	−0.804728	−0.402364	−	0.915480i	$-0.631811\pi$
$373$	−5797.12	−0.804728	−0.402364	+	0.915480i	$0.631811\pi$
$374$	−15121.8	−2.09072
$375$	0	0
$376$	−1320.56	−0.181125
$377$	0	0
$378$	0	0
$379$	8846.31	1.19896	0.599478	−	0.800391i	$-0.295375\pi$
$379$	8846.31	1.19896	0.599478	+	0.800391i	$0.295375\pi$
$380$	−12166.0	−1.64238
$381$	0	0
$382$	4340.57	0.581369
$383$	−1047.61	−0.139766	−0.0698830	−	0.997555i	$-0.522263\pi$
$383$	−1047.61	−0.139766	−0.0698830	+	0.997555i	$0.522263\pi$
$384$	0	0
$385$	1616.12	0.213935
$386$	−12535.9	−1.65301
$387$	0	0
$388$	7444.01	0.974000
$389$	11858.4	1.54562	0.772808	−	0.634640i	$-0.218852\pi$
$389$	11858.4	1.54562	0.772808	+	0.634640i	$0.218852\pi$
$390$	0	0
$391$	−11109.8	−1.43695
$392$	−11660.9	−1.50247
$393$	0	0
$394$	7172.50	0.917120
$395$	−4892.76	−0.623245
$396$	0	0
$397$	10480.3	1.32492	0.662460	−	0.749097i	$-0.269512\pi$
$397$	10480.3	1.32492	0.662460	+	0.749097i	$0.269512\pi$
$398$	−17422.0	−2.19418
$399$	0	0
$400$	5425.22	0.678152
$401$	−7936.40	−0.988341	−0.494171	−	0.869365i	$-0.664528\pi$
$401$	−7936.40	−0.988341	−0.494171	+	0.869365i	$0.664528\pi$
$402$	0	0
$403$	0	0
$404$	−12955.5	−1.59545
$405$	0	0
$406$	−4715.68	−0.576441
$407$	2844.03	0.346372
$408$	0	0
$409$	−5835.68	−0.705516	−0.352758	−	0.935715i	$-0.614756\pi$
$409$	−5835.68	−0.705516	−0.352758	+	0.935715i	$0.614756\pi$
$410$	14154.1	1.70493
$411$	0	0
$412$	2465.64	0.294838
$413$	−3472.61	−0.413744
$414$	0	0
$415$	15888.2	1.87933
$416$	0	0
$417$	0	0
$418$	−6153.60	−0.720054
$419$	8544.29	0.996219	0.498109	−	0.867114i	$-0.334028\pi$
$419$	8544.29	0.996219	0.498109	+	0.867114i	$0.334028\pi$
$420$	0	0
$421$	−16524.6	−1.91297	−0.956484	−	0.291786i	$-0.905750\pi$
$421$	−16524.6	−1.91297	−0.956484	+	0.291786i	$0.905750\pi$
$422$	14768.4	1.70359
$423$	0	0
$424$	−4283.81	−0.490661
$425$	13638.3	1.55660
$426$	0	0
$427$	2930.21	0.332091
$428$	−16594.0	−1.87407
$429$	0	0
$430$	9095.11	1.02001
$431$	6836.58	0.764052	0.382026	−	0.924152i	$-0.375226\pi$
$431$	6836.58	0.764052	0.382026	+	0.924152i	$0.375226\pi$
$432$	0	0
$433$	−6290.65	−0.698174	−0.349087	−	0.937090i	$-0.613508\pi$
$433$	−6290.65	−0.698174	−0.349087	+	0.937090i	$0.613508\pi$
$434$	−2271.30	−0.251211
$435$	0	0
$436$	17921.4	1.96853
$437$	−4520.98	−0.494892
$438$	0	0
$439$	8868.59	0.964180	0.482090	−	0.876122i	$-0.339878\pi$
$439$	8868.59	0.964180	0.482090	+	0.876122i	$0.339878\pi$
$440$	13460.6	1.45843
$441$	0	0
$442$	0	0
$443$	4310.67	0.462316	0.231158	−	0.972916i	$-0.425749\pi$
$443$	4310.67	0.462316	0.231158	+	0.972916i	$0.425749\pi$
$444$	0	0
$445$	5253.87	0.559680
$446$	−12583.8	−1.33601
$447$	0	0
$448$	−2640.00	−0.278412
$449$	4604.48	0.483962	0.241981	−	0.970281i	$-0.422203\pi$
$449$	4604.48	0.483962	0.241981	+	0.970281i	$0.422203\pi$
$450$	0	0
$451$	4714.45	0.492228
$452$	22577.6	2.34947
$453$	0	0
$454$	4807.30	0.496956
$455$	0	0
$456$	0	0
$457$	16379.2	1.67656	0.838281	−	0.545238i	$-0.183561\pi$
$457$	16379.2	1.67656	0.838281	+	0.545238i	$0.183561\pi$
$458$	2118.89	0.216178
$459$	0	0
$460$	20541.2	2.08204
$461$	−9631.51	−0.973068	−0.486534	−	0.873662i	$-0.661739\pi$
$461$	−9631.51	−0.973068	−0.486534	+	0.873662i	$0.661739\pi$
$462$	0	0
$463$	−17855.0	−1.79220	−0.896102	−	0.443848i	$-0.853613\pi$
$463$	−17855.0	−1.79220	−0.896102	+	0.443848i	$0.853613\pi$
$464$	−11418.0	−1.14238
$465$	0	0
$466$	−14196.6	−1.41126
$467$	−8220.28	−0.814538	−0.407269	−	0.913308i	$-0.633519\pi$
$467$	−8220.28	−0.814538	−0.407269	+	0.913308i	$0.633519\pi$
$468$	0	0
$469$	377.584	0.0371753
$470$	2709.62	0.265927
$471$	0	0
$472$	−28923.3	−2.82055
$473$	3029.40	0.294486
$474$	0	0
$475$	5549.93	0.536102
$476$	8467.29	0.815331
$477$	0	0
$478$	−28138.3	−2.69250
$479$	3164.10	0.301819	0.150909	−	0.988548i	$-0.451780\pi$
$479$	3164.10	0.301819	0.150909	+	0.988548i	$0.451780\pi$
$480$	0	0
$481$	0	0
$482$	5557.73	0.525203
$483$	0	0
$484$	−11221.2	−1.05383
$485$	−7353.55	−0.688469
$486$	0	0
$487$	−7487.99	−0.696741	−0.348371	−	0.937357i	$-0.613265\pi$
$487$	−7487.99	−0.696741	−0.348371	+	0.937357i	$0.613265\pi$
$488$	24405.6	2.26391
$489$	0	0
$490$	23926.7	2.20592
$491$	16013.7	1.47187	0.735933	−	0.677054i	$-0.236744\pi$
$491$	16013.7	1.47187	0.735933	+	0.677054i	$0.236744\pi$
$492$	0	0
$493$	−28703.4	−2.62218
$494$	0	0
$495$	0	0
$496$	−5499.44	−0.497847
$497$	−4234.41	−0.382171
$498$	0	0
$499$	10343.2	0.927904	0.463952	−	0.885860i	$-0.346431\pi$
$499$	10343.2	0.927904	0.463952	+	0.885860i	$0.346431\pi$
$500$	4172.48	0.373198
$501$	0	0
$502$	15758.6	1.40108
$503$	18615.3	1.65013	0.825063	−	0.565040i	$-0.191139\pi$
$503$	18615.3	1.65013	0.825063	+	0.565040i	$0.191139\pi$
$504$	0	0
$505$	12798.1	1.12774
$506$	10389.8	0.912809
$507$	0	0
$508$	4472.73	0.390641
$509$	3628.41	0.315965	0.157983	−	0.987442i	$-0.449501\pi$
$509$	3628.41	0.315965	0.157983	+	0.987442i	$0.449501\pi$
$510$	0	0
$511$	−1138.88	−0.0985935
$512$	−16710.7	−1.44241
$513$	0	0
$514$	−30594.3	−2.62540
$515$	−2435.68	−0.208406
$516$	0	0
$517$	902.521	0.0767753
$518$	−2418.29	−0.205122
$519$	0	0
$520$	0	0
$521$	3105.46	0.261137	0.130569	−	0.991439i	$-0.458320\pi$
$521$	3105.46	0.261137	0.130569	+	0.991439i	$0.458320\pi$
$522$	0	0
$523$	1911.23	0.159794	0.0798969	−	0.996803i	$-0.474541\pi$
$523$	1911.23	0.159794	0.0798969	+	0.996803i	$0.474541\pi$
$524$	18543.1	1.54592
$525$	0	0
$526$	29859.0	2.47512
$527$	−13824.9	−1.14274
$528$	0	0
$529$	−4533.77	−0.372628
$530$	8789.80	0.720386
$531$	0	0
$532$	3445.64	0.280804
$533$	0	0
$534$	0	0
$535$	16392.4	1.32468
$536$	3144.88	0.253430
$537$	0	0
$538$	−4083.16	−0.327208
$539$	7969.50	0.636866
$540$	0	0
$541$	−15251.2	−1.21202	−0.606008	−	0.795459i	$-0.707230\pi$
$541$	−15251.2	−1.21202	−0.606008	+	0.795459i	$0.707230\pi$
$542$	−9985.55	−0.791358
$543$	0	0
$544$	−5437.82	−0.428575
$545$	−17703.6	−1.39145
$546$	0	0
$547$	1838.85	0.143736	0.0718681	−	0.997414i	$-0.477104\pi$
$547$	1838.85	0.143736	0.0718681	+	0.997414i	$0.477104\pi$
$548$	12027.4	0.937567
$549$	0	0
$550$	−12754.4	−0.988819
$551$	−11680.4	−0.903091
$552$	0	0
$553$	1385.72	0.106558
$554$	31860.4	2.44336
$555$	0	0
$556$	31340.3	2.39051
$557$	−6301.29	−0.479343	−0.239672	−	0.970854i	$-0.577040\pi$
$557$	−6301.29	−0.479343	−0.239672	+	0.970854i	$0.577040\pi$
$558$	0	0
$559$	0	0
$560$	−3327.29	−0.251078
$561$	0	0
$562$	4525.69	0.339688
$563$	−11759.8	−0.880316	−0.440158	−	0.897920i	$-0.645078\pi$
$563$	−11759.8	−0.880316	−0.440158	+	0.897920i	$0.645078\pi$
$564$	0	0
$565$	−22303.3	−1.66072
$566$	−25840.6	−1.91901
$567$	0	0
$568$	−35268.2	−2.60532
$569$	17219.4	1.26867	0.634337	−	0.773056i	$-0.281273\pi$
$569$	17219.4	1.26867	0.634337	+	0.773056i	$0.281273\pi$
$570$	0	0
$571$	825.501	0.0605011	0.0302506	−	0.999542i	$-0.490369\pi$
$571$	825.501	0.0605011	0.0302506	+	0.999542i	$0.490369\pi$
$572$	0	0
$573$	0	0
$574$	−4008.71	−0.291498
$575$	−9370.52	−0.679614
$576$	0	0
$577$	1073.19	0.0774308	0.0387154	−	0.999250i	$-0.487673\pi$
$577$	1073.19	0.0774308	0.0387154	+	0.999250i	$0.487673\pi$
$578$	54484.8	3.92088
$579$	0	0
$580$	53070.3	3.79935
$581$	−4499.83	−0.321316
$582$	0	0
$583$	2927.71	0.207981
$584$	−9485.71	−0.672126
$585$	0	0
$586$	8536.93	0.601804
$587$	14939.4	1.05045	0.525225	−	0.850963i	$-0.323981\pi$
$587$	14939.4	1.05045	0.525225	+	0.850963i	$0.323981\pi$
$588$	0	0
$589$	−5625.86	−0.393565
$590$	59346.7	4.14113
$591$	0	0
$592$	−5855.34	−0.406509
$593$	−6296.13	−0.436006	−0.218003	−	0.975948i	$-0.569954\pi$
$593$	−6296.13	−0.436006	−0.218003	+	0.975948i	$0.569954\pi$
$594$	0	0
$595$	−8364.40	−0.576314
$596$	40166.6	2.76055
$597$	0	0
$598$	0	0
$599$	3518.96	0.240035	0.120017	−	0.992772i	$-0.461705\pi$
$599$	3518.96	0.240035	0.120017	+	0.992772i	$0.461705\pi$
$600$	0	0
$601$	5380.65	0.365194	0.182597	−	0.983188i	$-0.441550\pi$
$601$	5380.65	0.365194	0.182597	+	0.983188i	$0.441550\pi$
$602$	−2575.90	−0.174395
$603$	0	0
$604$	−3190.00	−0.214899
$605$	11084.8	0.744895
$606$	0	0
$607$	20708.6	1.38474	0.692369	−	0.721543i	$-0.256567\pi$
$607$	20708.6	1.38474	0.692369	+	0.721543i	$0.256567\pi$
$608$	−2212.84	−0.147603
$609$	0	0
$610$	−50077.1	−3.32387
$611$	0	0
$612$	0	0
$613$	8563.66	0.564246	0.282123	−	0.959378i	$-0.408961\pi$
$613$	8563.66	0.564246	0.282123	+	0.959378i	$0.408961\pi$
$614$	−22925.9	−1.50687
$615$	0	0
$616$	−3812.28	−0.249352
$617$	−30449.6	−1.98680	−0.993400	−	0.114703i	$-0.963409\pi$
$617$	−30449.6	−1.98680	−0.993400	+	0.114703i	$0.963409\pi$
$618$	0	0
$619$	26233.1	1.70339	0.851693	−	0.524041i	$-0.175576\pi$
$619$	26233.1	1.70339	0.851693	+	0.524041i	$0.175576\pi$
$620$	25561.2	1.65575
$621$	0	0
$622$	22975.8	1.48111
$623$	−1487.99	−0.0956905
$624$	0	0
$625$	−17528.4	−1.12182
$626$	48962.7	3.12611
$627$	0	0
$628$	10786.9	0.685421
$629$	−14719.6	−0.933084
$630$	0	0
$631$	−20639.5	−1.30213	−0.651066	−	0.759021i	$-0.725678\pi$
$631$	−20639.5	−1.30213	−0.651066	+	0.759021i	$0.725678\pi$
$632$	11541.6	0.726425
$633$	0	0
$634$	28586.7	1.79073
$635$	−4418.38	−0.276123
$636$	0	0
$637$	0	0
$638$	26843.1	1.66572
$639$	0	0
$640$	39903.8	2.46459
$641$	4431.77	0.273080	0.136540	−	0.990635i	$-0.456402\pi$
$641$	4431.77	0.273080	0.136540	+	0.990635i	$0.456402\pi$
$642$	0	0
$643$	14176.1	0.869439	0.434719	−	0.900566i	$-0.356848\pi$
$643$	14176.1	0.869439	0.434719	+	0.900566i	$0.356848\pi$
$644$	−5817.63	−0.355973
$645$	0	0
$646$	31848.7	1.93974
$647$	−5650.79	−0.343362	−0.171681	−	0.985153i	$-0.554920\pi$
$647$	−5650.79	−0.343362	−0.171681	+	0.985153i	$0.554920\pi$
$648$	0	0
$649$	19767.2	1.19558
$650$	0	0
$651$	0	0
$652$	45420.2	2.72821
$653$	18436.3	1.10485	0.552425	−	0.833562i	$-0.313702\pi$
$653$	18436.3	1.10485	0.552425	+	0.833562i	$0.313702\pi$
$654$	0	0
$655$	−18317.8	−1.09273
$656$	−9706.18	−0.577687
$657$	0	0
$658$	−767.415	−0.0454665
$659$	15331.9	0.906294	0.453147	−	0.891436i	$-0.350301\pi$
$659$	15331.9	0.906294	0.453147	+	0.891436i	$0.350301\pi$
$660$	0	0
$661$	−22295.1	−1.31192	−0.655961	−	0.754795i	$-0.727736\pi$
$661$	−22295.1	−1.31192	−0.655961	+	0.754795i	$0.727736\pi$
$662$	−13566.2	−0.796471
$663$	0	0
$664$	−37478.9	−2.19046
$665$	−3403.78	−0.198485
$666$	0	0
$667$	19721.3	1.14484
$668$	−42283.4	−2.44909
$669$	0	0
$670$	−6452.88	−0.372084
$671$	−16679.7	−0.959629
$672$	0	0
$673$	−16171.7	−0.926259	−0.463129	−	0.886291i	$-0.653274\pi$
$673$	−16171.7	−0.926259	−0.463129	+	0.886291i	$0.653274\pi$
$674$	−1185.45	−0.0677475
$675$	0	0
$676$	0	0
$677$	−33614.4	−1.90828	−0.954141	−	0.299358i	$-0.903227\pi$
$677$	−33614.4	−1.90828	−0.954141	+	0.299358i	$0.903227\pi$
$678$	0	0
$679$	2082.66	0.117710
$680$	−69666.7	−3.92882
$681$	0	0
$682$	12928.9	0.725915
$683$	22477.5	1.25926	0.629632	−	0.776893i	$-0.283206\pi$
$683$	22477.5	1.25926	0.629632	+	0.776893i	$0.283206\pi$
$684$	0	0
$685$	−11881.3	−0.662717
$686$	−13942.2	−0.775968
$687$	0	0
$688$	−6236.97	−0.345614
$689$	0	0
$690$	0	0
$691$	−7618.93	−0.419447	−0.209723	−	0.977761i	$-0.567256\pi$
$691$	−7618.93	−0.419447	−0.209723	+	0.977761i	$0.567256\pi$
$692$	−5226.95	−0.287137
$693$	0	0
$694$	−50128.9	−2.74188
$695$	−30959.5	−1.68973
$696$	0	0
$697$	−24400.2	−1.32600
$698$	−40820.0	−2.21355
$699$	0	0
$700$	7141.70	0.385616
$701$	18164.3	0.978684	0.489342	−	0.872092i	$-0.337237\pi$
$701$	18164.3	0.978684	0.489342	+	0.872092i	$0.337237\pi$
$702$	0	0
$703$	−5989.95	−0.321359
$704$	15027.7	0.804514
$705$	0	0
$706$	−22912.0	−1.22140
$707$	−3624.66	−0.192814
$708$	0	0
$709$	16706.5	0.884945	0.442472	−	0.896782i	$-0.354101\pi$
$709$	16706.5	0.884945	0.442472	+	0.896782i	$0.354101\pi$
$710$	72365.7	3.82512
$711$	0	0
$712$	−12393.4	−0.652336
$713$	9498.72	0.498920
$714$	0	0
$715$	0	0
$716$	16883.6	0.881244
$717$	0	0
$718$	−36598.2	−1.90227
$719$	−4549.39	−0.235972	−0.117986	−	0.993015i	$-0.537644\pi$
$719$	−4549.39	−0.235972	−0.117986	+	0.993015i	$0.537644\pi$
$720$	0	0
$721$	689.830	0.0356319
$722$	−20238.4	−1.04321
$723$	0	0
$724$	−21910.8	−1.12473
$725$	−24209.8	−1.24018
$726$	0	0
$727$	−6246.70	−0.318676	−0.159338	−	0.987224i	$-0.550936\pi$
$727$	−6246.70	−0.318676	−0.159338	+	0.987224i	$0.550936\pi$
$728$	0	0
$729$	0	0
$730$	19463.4	0.986813
$731$	−15679.0	−0.793309
$732$	0	0
$733$	−3447.16	−0.173702	−0.0868511	−	0.996221i	$-0.527680\pi$
$733$	−3447.16	−0.173702	−0.0868511	+	0.996221i	$0.527680\pi$
$734$	−57791.6	−2.90617
$735$	0	0
$736$	3736.17	0.187116
$737$	−2149.32	−0.107424
$738$	0	0
$739$	37032.9	1.84341	0.921703	−	0.387895i	$-0.126798\pi$
$739$	37032.9	1.84341	0.921703	+	0.387895i	$0.126798\pi$
$740$	27215.4	1.35197
$741$	0	0
$742$	−2489.43	−0.123167
$743$	−15298.0	−0.755356	−0.377678	−	0.925937i	$-0.623277\pi$
$743$	−15298.0	−0.755356	−0.377678	+	0.925937i	$0.623277\pi$
$744$	0	0
$745$	−39678.5	−1.95129
$746$	−28059.1	−1.37710
$747$	0	0
$748$	−48198.4	−2.35603
$749$	−4642.62	−0.226485
$750$	0	0
$751$	−9844.54	−0.478339	−0.239169	−	0.970978i	$-0.576875\pi$
$751$	−9844.54	−0.478339	−0.239169	+	0.970978i	$0.576875\pi$
$752$	−1858.12	−0.0901048
$753$	0	0
$754$	0	0
$755$	3151.24	0.151901
$756$	0	0
$757$	5921.31	0.284298	0.142149	−	0.989845i	$-0.454599\pi$
$757$	5921.31	0.284298	0.142149	+	0.989845i	$0.454599\pi$
$758$	42817.7	2.05173
$759$	0	0
$760$	−28349.9	−1.35310
$761$	17872.5	0.851348	0.425674	−	0.904877i	$-0.360037\pi$
$761$	17872.5	0.851348	0.425674	+	0.904877i	$0.360037\pi$
$762$	0	0
$763$	5013.99	0.237901
$764$	13834.9	0.655142
$765$	0	0
$766$	−5070.61	−0.239176
$767$	0	0
$768$	0	0
$769$	−11265.2	−0.528262	−0.264131	−	0.964487i	$-0.585085\pi$
$769$	−11265.2	−0.528262	−0.264131	+	0.964487i	$0.585085\pi$
$770$	7822.29	0.366099
$771$	0	0
$772$	−39956.3	−1.86277
$773$	27603.7	1.28439	0.642196	−	0.766540i	$-0.278023\pi$
$773$	27603.7	1.28439	0.642196	+	0.766540i	$0.278023\pi$
$774$	0	0
$775$	−11660.6	−0.540465
$776$	17346.4	0.802447
$777$	0	0
$778$	57396.7	2.64495
$779$	−9929.31	−0.456681
$780$	0	0
$781$	24103.5	1.10434
$782$	−53773.4	−2.45899
$783$	0	0
$784$	−16407.7	−0.747437
$785$	−10655.8	−0.484488
$786$	0	0
$787$	−24163.1	−1.09444	−0.547218	−	0.836990i	$-0.684313\pi$
$787$	−24163.1	−1.09444	−0.547218	+	0.836990i	$0.684313\pi$
$788$	22861.2	1.03350
$789$	0	0
$790$	−23681.8	−1.06653
$791$	6316.70	0.283939
$792$	0	0
$793$	0	0
$794$	50726.7	2.26728
$795$	0	0
$796$	−55529.8	−2.47262
$797$	−30385.6	−1.35046	−0.675228	−	0.737609i	$-0.735955\pi$
$797$	−30385.6	−1.35046	−0.675228	+	0.737609i	$0.735955\pi$
$798$	0	0
$799$	−4671.10	−0.206823
$800$	−4586.51	−0.202697
$801$	0	0
$802$	−38413.6	−1.69131
$803$	6482.87	0.284901
$804$	0	0
$805$	5746.94	0.251619
$806$	0	0
$807$	0	0
$808$	−30189.6	−1.31444
$809$	41976.9	1.82426	0.912131	−	0.409899i	$-0.134436\pi$
$809$	41976.9	1.82426	0.912131	+	0.409899i	$0.134436\pi$
$810$	0	0
$811$	−9674.05	−0.418868	−0.209434	−	0.977823i	$-0.567162\pi$
$811$	−9674.05	−0.418868	−0.209434	+	0.977823i	$0.567162\pi$
$812$	−15030.5	−0.649590
$813$	0	0
$814$	13765.6	0.592733
$815$	−44868.3	−1.92843
$816$	0	0
$817$	−6380.35	−0.273219
$818$	−28245.7	−1.20732
$819$	0	0
$820$	45114.0	1.92128
$821$	−25426.0	−1.08085	−0.540423	−	0.841394i	$-0.681736\pi$
$821$	−25426.0	−1.08085	−0.540423	+	0.841394i	$0.681736\pi$
$822$	0	0
$823$	14388.3	0.609408	0.304704	−	0.952447i	$-0.401442\pi$
$823$	14388.3	0.609408	0.304704	+	0.952447i	$0.401442\pi$
$824$	5745.56	0.242908
$825$	0	0
$826$	−16808.1	−0.708024
$827$	−3850.25	−0.161894	−0.0809469	−	0.996718i	$-0.525794\pi$
$827$	−3850.25	−0.161894	−0.0809469	+	0.996718i	$0.525794\pi$
$828$	0	0
$829$	−1918.58	−0.0803802	−0.0401901	−	0.999192i	$-0.512796\pi$
$829$	−1918.58	−0.0803802	−0.0401901	+	0.999192i	$0.512796\pi$
$830$	76901.7	3.21602
$831$	0	0
$832$	0	0
$833$	−41247.1	−1.71564
$834$	0	0
$835$	41769.6	1.73113
$836$	−19613.7	−0.811426
$837$	0	0
$838$	41355.9	1.70479
$839$	−31412.1	−1.29257	−0.646285	−	0.763096i	$-0.723678\pi$
$839$	−31412.1	−1.29257	−0.646285	+	0.763096i	$0.723678\pi$
$840$	0	0
$841$	26563.1	1.08914
$842$	−79981.9	−3.27359
$843$	0	0
$844$	47072.0	1.91977
$845$	0	0
$846$	0	0
$847$	−3139.42	−0.127357
$848$	−6027.61	−0.244091
$849$	0	0
$850$	66012.0	2.66376
$851$	10113.4	0.407384
$852$	0	0
$853$	18315.5	0.735184	0.367592	−	0.929987i	$-0.380182\pi$
$853$	18315.5	0.735184	0.367592	+	0.929987i	$0.380182\pi$
$854$	14182.7	0.568295
$855$	0	0
$856$	−38668.1	−1.54398
$857$	−9579.31	−0.381824	−0.190912	−	0.981607i	$-0.561144\pi$
$857$	−9579.31	−0.381824	−0.190912	+	0.981607i	$0.561144\pi$
$858$	0	0
$859$	−7136.55	−0.283465	−0.141732	−	0.989905i	$-0.545267\pi$
$859$	−7136.55	−0.283465	−0.141732	+	0.989905i	$0.545267\pi$
$860$	28989.3	1.14945
$861$	0	0
$862$	33090.3	1.30749
$863$	−17239.2	−0.679986	−0.339993	−	0.940428i	$-0.610425\pi$
$863$	−17239.2	−0.679986	−0.339993	+	0.940428i	$0.610425\pi$
$864$	0	0
$865$	5163.44	0.202962
$866$	−30447.8	−1.19476
$867$	0	0
$868$	−7239.41	−0.283089
$869$	−7887.94	−0.307917
$870$	0	0
$871$	0	0
$872$	41761.3	1.62181
$873$	0	0
$874$	−21882.3	−0.846889
$875$	1167.36	0.0451019
$876$	0	0
$877$	−33048.7	−1.27249	−0.636247	−	0.771486i	$-0.719514\pi$
$877$	−33048.7	−1.27249	−0.636247	+	0.771486i	$0.719514\pi$
$878$	42925.6	1.64996
$879$	0	0
$880$	18939.9	0.725529
$881$	−19527.8	−0.746775	−0.373387	−	0.927676i	$-0.621804\pi$
$881$	−19527.8	−0.746775	−0.373387	+	0.927676i	$0.621804\pi$
$882$	0	0
$883$	−26361.3	−1.00467	−0.502337	−	0.864672i	$-0.667526\pi$
$883$	−26361.3	−1.00467	−0.502337	+	0.864672i	$0.667526\pi$
$884$	0	0
$885$	0	0
$886$	20864.4	0.791144
$887$	−21405.8	−0.810301	−0.405151	−	0.914250i	$-0.632781\pi$
$887$	−21405.8	−0.810301	−0.405151	+	0.914250i	$0.632781\pi$
$888$	0	0
$889$	1251.37	0.0472098
$890$	25429.7	0.957758
$891$	0	0
$892$	−40109.0	−1.50555
$893$	−1900.84	−0.0712308
$894$	0	0
$895$	−16678.5	−0.622905
$896$	−11301.5	−0.421379
$897$	0	0
$898$	22286.5	0.828184
$899$	24541.0	0.910442
$900$	0	0
$901$	−15152.7	−0.560276
$902$	22818.8	0.842330
$903$	0	0
$904$	52611.5	1.93565
$905$	21644.5	0.795015
$906$	0	0
$907$	28887.5	1.05754	0.528772	−	0.848764i	$-0.322653\pi$
$907$	28887.5	1.05754	0.528772	+	0.848764i	$0.322653\pi$
$908$	15322.5	0.560018
$909$	0	0
$910$	0	0
$911$	31360.9	1.14054	0.570271	−	0.821456i	$-0.306838\pi$
$911$	31360.9	1.14054	0.570271	+	0.821456i	$0.306838\pi$
$912$	0	0
$913$	25614.4	0.928492
$914$	79278.4	2.86903
$915$	0	0
$916$	6753.64	0.243610
$917$	5187.94	0.186828
$918$	0	0
$919$	−1104.61	−0.0396493	−0.0198246	−	0.999803i	$-0.506311\pi$
$919$	−1104.61	−0.0396493	−0.0198246	+	0.999803i	$0.506311\pi$
$920$	47866.0	1.71532
$921$	0	0
$922$	−46618.2	−1.66517
$923$	0	0
$924$	0	0
$925$	−12415.2	−0.441308
$926$	−86421.2	−3.06693
$927$	0	0
$928$	9652.81	0.341454
$929$	38435.8	1.35742	0.678708	−	0.734409i	$-0.262540\pi$
$929$	38435.8	1.35742	0.678708	+	0.734409i	$0.262540\pi$
$930$	0	0
$931$	−16784.9	−0.590874
$932$	−45249.5	−1.59034
$933$	0	0
$934$	−39787.6	−1.39389
$935$	47612.7	1.66535
$936$	0	0
$937$	21008.7	0.732469	0.366235	−	0.930522i	$-0.380647\pi$
$937$	21008.7	0.732469	0.366235	+	0.930522i	$0.380647\pi$
$938$	1827.57	0.0636166
$939$	0	0
$940$	8636.50	0.299672
$941$	12695.6	0.439814	0.219907	−	0.975521i	$-0.429425\pi$
$941$	12695.6	0.439814	0.219907	+	0.975521i	$0.429425\pi$
$942$	0	0
$943$	16764.7	0.578932
$944$	−40697.0	−1.40315
$945$	0	0
$946$	14662.8	0.503942
$947$	48719.4	1.67177	0.835886	−	0.548902i	$-0.184954\pi$
$947$	48719.4	1.67177	0.835886	+	0.548902i	$0.184954\pi$
$948$	0	0
$949$	0	0
$950$	26862.7	0.917410
$951$	0	0
$952$	19730.9	0.671725
$953$	11347.3	0.385704	0.192852	−	0.981228i	$-0.438226\pi$
$953$	11347.3	0.385704	0.192852	+	0.981228i	$0.438226\pi$
$954$	0	0
$955$	−13666.8	−0.463085
$956$	−89686.3	−3.03417
$957$	0	0
$958$	15314.8	0.516491
$959$	3365.00	0.113307
$960$	0	0
$961$	−17970.9	−0.603232
$962$	0	0
$963$	0	0
$964$	17714.4	0.591849
$965$	39470.8	1.31669
$966$	0	0
$967$	10585.2	0.352014	0.176007	−	0.984389i	$-0.443682\pi$
$967$	10585.2	0.352014	0.176007	+	0.984389i	$0.443682\pi$
$968$	−26148.1	−0.868215
$969$	0	0
$970$	−35592.5	−1.17815
$971$	53730.8	1.77580	0.887900	−	0.460036i	$-0.152164\pi$
$971$	53730.8	1.77580	0.887900	+	0.460036i	$0.152164\pi$
$972$	0	0
$973$	8768.30	0.288899
$974$	−36243.2	−1.19231
$975$	0	0
$976$	34340.4	1.12624
$977$	9534.31	0.312210	0.156105	−	0.987740i	$-0.450106\pi$
$977$	9534.31	0.312210	0.156105	+	0.987740i	$0.450106\pi$
$978$	0	0
$979$	8470.11	0.276513
$980$	76262.7	2.48584
$981$	0	0
$982$	77508.9	2.51875
$983$	8972.94	0.291142	0.145571	−	0.989348i	$-0.453498\pi$
$983$	8972.94	0.291142	0.145571	+	0.989348i	$0.453498\pi$
$984$	0	0
$985$	−22583.4	−0.730526
$986$	−138929.	−4.48723
$987$	0	0
$988$	0	0
$989$	10772.6	0.346359
$990$	0	0
$991$	−34900.6	−1.11872	−0.559362	−	0.828924i	$-0.688954\pi$
$991$	−34900.6	−1.11872	−0.559362	+	0.828924i	$0.688954\pi$
$992$	4649.26	0.148804
$993$	0	0
$994$	−20495.3	−0.653995
$995$	54855.1	1.74776
$996$	0	0
$997$	−48154.9	−1.52967	−0.764836	−	0.644225i	$-0.777180\pi$
$997$	−48154.9	−1.52967	−0.764836	+	0.644225i	$0.777180\pi$
$998$	50062.8	1.58788
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bg.1.9		9
3.2	odd	2		169.4.a.l.1.1	yes	9
13.12	even	2		1521.4.a.bh.1.1		9
39.2	even	12		169.4.e.h.147.2		36
39.5	even	4		169.4.b.g.168.17		18
39.8	even	4		169.4.b.g.168.2		18
39.11	even	12		169.4.e.h.147.17		36
39.17	odd	6		169.4.c.l.146.1		18
39.20	even	12		169.4.e.h.23.2		36
39.23	odd	6		169.4.c.l.22.1		18
39.29	odd	6		169.4.c.k.22.9		18
39.32	even	12		169.4.e.h.23.17		36
39.35	odd	6		169.4.c.k.146.9		18
39.38	odd	2		169.4.a.k.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.9	✓	9	39.38	odd	2
169.4.a.l.1.1	yes	9	3.2	odd	2
169.4.b.g.168.2		18	39.8	even	4
169.4.b.g.168.17		18	39.5	even	4
169.4.c.k.22.9		18	39.29	odd	6
169.4.c.k.146.9		18	39.35	odd	6
169.4.c.l.22.1		18	39.23	odd	6
169.4.c.l.146.1		18	39.17	odd	6
169.4.e.h.23.2		36	39.20	even	12
169.4.e.h.23.17		36	39.32	even	12
169.4.e.h.147.2		36	39.2	even	12
169.4.e.h.147.17		36	39.11	even	12
1521.4.a.bg.1.9		9	1.1	even	1	trivial
1521.4.a.bh.1.1		9	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+4.84018 q^{2} +15.4273 q^{4} -15.2399 q^{5} +4.31620 q^{7} +35.9495 q^{8} +O(q^{10})\)	\(q+4.84018 q^{2} +15.4273 q^{4} -15.2399 q^{5} +4.31620 q^{7} +35.9495 q^{8} -73.7636 q^{10} -24.5691 q^{11} +20.8912 q^{14} +50.5834 q^{16} +127.160 q^{17} +51.7462 q^{19} -235.110 q^{20} -118.919 q^{22} -87.3684 q^{23} +107.253 q^{25} +66.5874 q^{28} -225.726 q^{29} -108.720 q^{31} -42.7634 q^{32} +615.479 q^{34} -65.7783 q^{35} -115.756 q^{37} +250.461 q^{38} -547.865 q^{40} -191.885 q^{41} -123.301 q^{43} -379.036 q^{44} -422.878 q^{46} -36.7339 q^{47} -324.370 q^{49} +519.124 q^{50} -119.162 q^{53} +374.430 q^{55} +155.165 q^{56} -1092.55 q^{58} -804.553 q^{59} +678.886 q^{61} -526.226 q^{62} -611.650 q^{64} +87.4806 q^{67} +1961.74 q^{68} -318.379 q^{70} -981.049 q^{71} -263.862 q^{73} -560.281 q^{74} +798.304 q^{76} -106.045 q^{77} +321.051 q^{79} -770.883 q^{80} -928.757 q^{82} -1042.54 q^{83} -1937.91 q^{85} -596.798 q^{86} -883.248 q^{88} -344.746 q^{89} -1347.86 q^{92} -177.799 q^{94} -788.604 q^{95} +482.521 q^{97} -1570.01 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + 37 q^{4} - 30 q^{5} + 38 q^{7} - 60 q^{8}+O(q^{10}) \) 9 * q - 5 * q^2 + 37 * q^4 - 30 * q^5 + 38 * q^7 - 60 * q^8	\( 9 q - 5 q^{2} + 37 q^{4} - 30 q^{5} + 38 q^{7} - 60 q^{8} - 147 q^{10} - 181 q^{11} + 147 q^{14} + 269 q^{16} + 55 q^{17} + 161 q^{19} - 370 q^{20} + 340 q^{22} + 204 q^{23} + 307 q^{25} + 344 q^{28} - 280 q^{29} + 706 q^{31} - 680 q^{32} + 216 q^{34} - 20 q^{35} + 298 q^{37} + 739 q^{38} + 13 q^{40} - 1201 q^{41} - 533 q^{43} - 355 q^{44} - 840 q^{46} - 956 q^{47} + 403 q^{49} + 1156 q^{50} + 278 q^{53} - 250 q^{55} - 250 q^{56} - 2877 q^{58} - 1377 q^{59} - 136 q^{61} - 2035 q^{62} + 284 q^{64} - 931 q^{67} + 1536 q^{68} - 4854 q^{70} - 2046 q^{71} - 45 q^{73} + 1990 q^{74} - 3608 q^{76} + 718 q^{77} + 412 q^{79} + 787 q^{80} + 2757 q^{82} - 3709 q^{83} - 2106 q^{85} - 125 q^{86} - 636 q^{88} - 1663 q^{89} - 4010 q^{92} - 2531 q^{94} + 1614 q^{95} - 1087 q^{97} + 282 q^{98}+O(q^{100}) \) 9 * q - 5 * q^2 + 37 * q^4 - 30 * q^5 + 38 * q^7 - 60 * q^8 - 147 * q^10 - 181 * q^11 + 147 * q^14 + 269 * q^16 + 55 * q^17 + 161 * q^19 - 370 * q^20 + 340 * q^22 + 204 * q^23 + 307 * q^25 + 344 * q^28 - 280 * q^29 + 706 * q^31 - 680 * q^32 + 216 * q^34 - 20 * q^35 + 298 * q^37 + 739 * q^38 + 13 * q^40 - 1201 * q^41 - 533 * q^43 - 355 * q^44 - 840 * q^46 - 956 * q^47 + 403 * q^49 + 1156 * q^50 + 278 * q^53 - 250 * q^55 - 250 * q^56 - 2877 * q^58 - 1377 * q^59 - 136 * q^61 - 2035 * q^62 + 284 * q^64 - 931 * q^67 + 1536 * q^68 - 4854 * q^70 - 2046 * q^71 - 45 * q^73 + 1990 * q^74 - 3608 * q^76 + 718 * q^77 + 412 * q^79 + 787 * q^80 + 2757 * q^82 - 3709 * q^83 - 2106 * q^85 - 125 * q^86 - 636 * q^88 - 1663 * q^89 - 4010 * q^92 - 2531 * q^94 + 1614 * q^95 - 1087 * q^97 + 282 * q^98

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