LMFDB - Embedded newform 1521.4.a.q.1.1

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

$f(q)$	$=$	$q-3.46410 q^{2} +4.00000 q^{4} -13.8564 q^{5} -22.5167 q^{7} +13.8564 q^{8} +O(q^{10})$	$q-3.46410 q^{2} +4.00000 q^{4} -13.8564 q^{5} -22.5167 q^{7} +13.8564 q^{8} +48.0000 q^{10} +22.5167 q^{11} +78.0000 q^{14} -80.0000 q^{16} +27.0000 q^{17} -88.3346 q^{19} -55.4256 q^{20} -78.0000 q^{22} +57.0000 q^{23} +67.0000 q^{25} -90.0666 q^{28} +69.0000 q^{29} -72.7461 q^{31} +166.277 q^{32} -93.5307 q^{34} +312.000 q^{35} -39.8372 q^{37} +306.000 q^{38} -192.000 q^{40} -393.176 q^{41} +85.0000 q^{43} +90.0666 q^{44} -197.454 q^{46} -342.946 q^{47} +164.000 q^{49} -232.095 q^{50} -426.000 q^{53} -312.000 q^{55} -312.000 q^{56} -239.023 q^{58} -19.0526 q^{59} -17.0000 q^{61} +252.000 q^{62} +64.0000 q^{64} -164.545 q^{67} +108.000 q^{68} -1080.80 q^{70} -583.701 q^{71} -1004.59 q^{73} +138.000 q^{74} -353.338 q^{76} -507.000 q^{77} -1244.00 q^{79} +1108.51 q^{80} +1362.00 q^{82} +426.084 q^{83} -374.123 q^{85} -294.449 q^{86} +312.000 q^{88} -306.573 q^{89} +228.000 q^{92} +1188.00 q^{94} +1224.00 q^{95} +1234.95 q^{97} -568.113 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{4}+O(q^{10}) $ 2 * q + 8 * q^4	$ 2 q + 8 q^{4} + 96 q^{10} + 156 q^{14} - 160 q^{16} + 54 q^{17} - 156 q^{22} + 114 q^{23} + 134 q^{25} + 138 q^{29} + 624 q^{35} + 612 q^{38} - 384 q^{40} + 170 q^{43} + 328 q^{49} - 852 q^{53} - 624 q^{55} - 624 q^{56} - 34 q^{61} + 504 q^{62} + 128 q^{64} + 216 q^{68} + 276 q^{74} - 1014 q^{77} - 2488 q^{79} + 2724 q^{82} + 624 q^{88} + 456 q^{92} + 2376 q^{94} + 2448 q^{95}+O(q^{100}) $ 2 * q + 8 * q^4 + 96 * q^10 + 156 * q^14 - 160 * q^16 + 54 * q^17 - 156 * q^22 + 114 * q^23 + 134 * q^25 + 138 * q^29 + 624 * q^35 + 612 * q^38 - 384 * q^40 + 170 * q^43 + 328 * q^49 - 852 * q^53 - 624 * q^55 - 624 * q^56 - 34 * q^61 + 504 * q^62 + 128 * q^64 + 216 * q^68 + 276 * q^74 - 1014 * q^77 - 2488 * q^79 + 2724 * q^82 + 624 * q^88 + 456 * q^92 + 2376 * q^94 + 2448 * q^95

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 $

$2$	−3.46410	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−3.46410	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−13.8564	−1.23935	−0.619677	−	0.784857i	$-0.712737\pi$
$5$	−13.8564	−1.23935	−0.619677	+	0.784857i	$0.712737\pi$
$6$	0	0
$7$	−22.5167	−1.21579	−0.607893	−	0.794019i	$-0.707985\pi$
$7$	−22.5167	−1.21579	−0.607893	+	0.794019i	$0.707985\pi$
$8$	13.8564	0.612372
$9$	0	0
$10$	48.0000	1.51789
$11$	22.5167	0.617184	0.308592	−	0.951194i	$-0.400142\pi$
$11$	22.5167	0.617184	0.308592	+	0.951194i	$0.400142\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	78.0000	1.48903
$15$	0	0
$16$	−80.0000	−1.25000
$17$	27.0000	0.385204	0.192602	−	0.981277i	$-0.438307\pi$
$17$	27.0000	0.385204	0.192602	+	0.981277i	$0.438307\pi$
$18$	0	0
$19$	−88.3346	−1.06660	−0.533299	−	0.845927i	$-0.679048\pi$
$19$	−88.3346	−1.06660	−0.533299	+	0.845927i	$0.679048\pi$
$20$	−55.4256	−0.619677
$21$	0	0
$22$	−78.0000	−0.755893
$23$	57.0000	0.516753	0.258377	−	0.966044i	$-0.416812\pi$
$23$	57.0000	0.516753	0.258377	+	0.966044i	$0.416812\pi$
$24$	0	0
$25$	67.0000	0.536000
$26$	0	0
$27$	0	0
$28$	−90.0666	−0.607893
$29$	69.0000	0.441827	0.220913	−	0.975293i	$-0.429096\pi$
$29$	69.0000	0.441827	0.220913	+	0.975293i	$0.429096\pi$
$30$	0	0
$31$	−72.7461	−0.421471	−0.210735	−	0.977543i	$-0.567586\pi$
$31$	−72.7461	−0.421471	−0.210735	+	0.977543i	$0.567586\pi$
$32$	166.277	0.918559
$33$	0	0
$34$	−93.5307	−0.471776
$35$	312.000	1.50679
$36$	0	0
$37$	−39.8372	−0.177005	−0.0885026	−	0.996076i	$-0.528208\pi$
$37$	−39.8372	−0.177005	−0.0885026	+	0.996076i	$0.528208\pi$
$38$	306.000	1.30631
$39$	0	0
$40$	−192.000	−0.758947
$41$	−393.176	−1.49765	−0.748826	−	0.662767i	$-0.769382\pi$
$41$	−393.176	−1.49765	−0.748826	+	0.662767i	$0.769382\pi$
$42$	0	0
$43$	85.0000	0.301451	0.150725	−	0.988576i	$-0.451839\pi$
$43$	85.0000	0.301451	0.150725	+	0.988576i	$0.451839\pi$
$44$	90.0666	0.308592
$45$	0	0
$46$	−197.454	−0.632891
$47$	−342.946	−1.06434	−0.532168	−	0.846639i	$-0.678623\pi$
$47$	−342.946	−1.06434	−0.532168	+	0.846639i	$0.678623\pi$
$48$	0	0
$49$	164.000	0.478134
$50$	−232.095	−0.656463
$51$	0	0
$52$	0	0
$53$	−426.000	−1.10407	−0.552034	−	0.833822i	$-0.686148\pi$
$53$	−426.000	−1.10407	−0.552034	+	0.833822i	$0.686148\pi$
$54$	0	0
$55$	−312.000	−0.764910
$56$	−312.000	−0.744513
$57$	0	0
$58$	−239.023	−0.541125
$59$	−19.0526	−0.0420412	−0.0210206	−	0.999779i	$-0.506692\pi$
$59$	−19.0526	−0.0420412	−0.0210206	+	0.999779i	$0.506692\pi$
$60$	0	0
$61$	−17.0000	−0.0356824	−0.0178412	−	0.999841i	$-0.505679\pi$
$61$	−17.0000	−0.0356824	−0.0178412	+	0.999841i	$0.505679\pi$
$62$	252.000	0.516194
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	−164.545	−0.300035	−0.150018	−	0.988683i	$-0.547933\pi$
$67$	−164.545	−0.300035	−0.150018	+	0.988683i	$0.547933\pi$
$68$	108.000	0.192602
$69$	0	0
$70$	−1080.80	−1.84543
$71$	−583.701	−0.975670	−0.487835	−	0.872936i	$-0.662213\pi$
$71$	−583.701	−0.975670	−0.487835	+	0.872936i	$0.662213\pi$
$72$	0	0
$73$	−1004.59	−1.61066	−0.805331	−	0.592826i	$-0.798012\pi$
$73$	−1004.59	−1.61066	−0.805331	+	0.592826i	$0.798012\pi$
$74$	138.000	0.216786
$75$	0	0
$76$	−353.338	−0.533299
$77$	−507.000	−0.750364
$78$	0	0
$79$	−1244.00	−1.77166	−0.885829	−	0.464012i	$-0.846409\pi$
$79$	−1244.00	−1.77166	−0.885829	+	0.464012i	$0.846409\pi$
$80$	1108.51	1.54919
$81$	0	0
$82$	1362.00	1.83424
$83$	426.084	0.563480	0.281740	−	0.959491i	$-0.409088\pi$
$83$	426.084	0.563480	0.281740	+	0.959491i	$0.409088\pi$
$84$	0	0
$85$	−374.123	−0.477404
$86$	−294.449	−0.369200
$87$	0	0
$88$	312.000	0.377947
$89$	−306.573	−0.365131	−0.182566	−	0.983194i	$-0.558440\pi$
$89$	−306.573	−0.365131	−0.182566	+	0.983194i	$0.558440\pi$
$90$	0	0
$91$	0	0
$92$	228.000	0.258377
$93$	0	0
$94$	1188.00	1.30354
$95$	1224.00	1.32189
$96$	0	0
$97$	1234.95	1.29268	0.646342	−	0.763048i	$-0.276298\pi$
$97$	1234.95	1.29268	0.646342	+	0.763048i	$0.276298\pi$
$98$	−568.113	−0.585592
$99$	0	0
$100$	268.000	0.268000
$101$	1959.00	1.92998	0.964989	−	0.262290i	$-0.0844778\pi$
$101$	1959.00	1.92998	0.964989	+	0.262290i	$0.0844778\pi$
$102$	0	0
$103$	−1856.00	−1.77551	−0.887753	−	0.460320i	$-0.847735\pi$
$103$	−1856.00	−1.77551	−0.887753	+	0.460320i	$0.847735\pi$
$104$	0	0
$105$	0	0
$106$	1475.71	1.35220
$107$	255.000	0.230390	0.115195	−	0.993343i	$-0.463251\pi$
$107$	255.000	0.230390	0.115195	+	0.993343i	$0.463251\pi$
$108$	0	0
$109$	−609.682	−0.535752	−0.267876	−	0.963453i	$-0.586322\pi$
$109$	−609.682	−0.535752	−0.267876	+	0.963453i	$0.586322\pi$
$110$	1080.80	0.936820
$111$	0	0
$112$	1801.33	1.51973
$113$	−411.000	−0.342156	−0.171078	−	0.985257i	$-0.554725\pi$
$113$	−411.000	−0.342156	−0.171078	+	0.985257i	$0.554725\pi$
$114$	0	0
$115$	−789.815	−0.640440
$116$	276.000	0.220913
$117$	0	0
$118$	66.0000	0.0514898
$119$	−607.950	−0.468325
$120$	0	0
$121$	−824.000	−0.619083
$122$	58.8897	0.0437018
$123$	0	0
$124$	−290.985	−0.210735
$125$	803.672	0.575061
$126$	0	0
$127$	2243.00	1.56720	0.783599	−	0.621267i	$-0.213382\pi$
$127$	2243.00	1.56720	0.783599	+	0.621267i	$0.213382\pi$
$128$	−1551.92	−1.07165
$129$	0	0
$130$	0	0
$131$	372.000	0.248105	0.124053	−	0.992276i	$-0.460411\pi$
$131$	372.000	0.248105	0.124053	+	0.992276i	$0.460411\pi$
$132$	0	0
$133$	1989.00	1.29675
$134$	570.000	0.367466
$135$	0	0
$136$	374.123	0.235888
$137$	−1189.92	−0.742056	−0.371028	−	0.928622i	$-0.620995\pi$
$137$	−1189.92	−0.742056	−0.371028	+	0.928622i	$0.620995\pi$
$138$	0	0
$139$	−2545.00	−1.55298	−0.776490	−	0.630130i	$-0.783002\pi$
$139$	−2545.00	−1.55298	−0.776490	+	0.630130i	$0.783002\pi$
$140$	1248.00	0.753395
$141$	0	0
$142$	2022.00	1.19495
$143$	0	0
$144$	0	0
$145$	−956.092	−0.547580
$146$	3480.00	1.97265
$147$	0	0
$148$	−159.349	−0.0885026
$149$	−1304.23	−0.717094	−0.358547	−	0.933512i	$-0.616728\pi$
$149$	−1304.23	−0.717094	−0.358547	+	0.933512i	$0.616728\pi$
$150$	0	0
$151$	86.6025	0.0466729	0.0233365	−	0.999728i	$-0.492571\pi$
$151$	86.6025	0.0466729	0.0233365	+	0.999728i	$0.492571\pi$
$152$	−1224.00	−0.653155
$153$	0	0
$154$	1756.30	0.919004
$155$	1008.00	0.522352
$156$	0	0
$157$	−1534.00	−0.779787	−0.389893	−	0.920860i	$-0.627488\pi$
$157$	−1534.00	−0.779787	−0.389893	+	0.920860i	$0.627488\pi$
$158$	4309.34	2.16983
$159$	0	0
$160$	−2304.00	−1.13842
$161$	−1283.45	−0.628261
$162$	0	0
$163$	1633.32	0.784858	0.392429	−	0.919782i	$-0.371635\pi$
$163$	1633.32	0.784858	0.392429	+	0.919782i	$0.371635\pi$
$164$	−1572.70	−0.748826
$165$	0	0
$166$	−1476.00	−0.690119
$167$	−1626.40	−0.753618	−0.376809	−	0.926291i	$-0.622979\pi$
$167$	−1626.40	−0.753618	−0.376809	+	0.926291i	$0.622979\pi$
$168$	0	0
$169$	0	0
$170$	1296.00	0.584698
$171$	0	0
$172$	340.000	0.150725
$173$	−873.000	−0.383659	−0.191829	−	0.981428i	$-0.561442\pi$
$173$	−873.000	−0.383659	−0.191829	+	0.981428i	$0.561442\pi$
$174$	0	0
$175$	−1508.62	−0.651661
$176$	−1801.33	−0.771481
$177$	0	0
$178$	1062.00	0.447193
$179$	−1287.00	−0.537402	−0.268701	−	0.963224i	$-0.586594\pi$
$179$	−1287.00	−0.537402	−0.268701	+	0.963224i	$0.586594\pi$
$180$	0	0
$181$	−2.00000	−0.000821319 0	−0.000410660	−	1.00000i	$-0.500131\pi$
$181$	−2.00000	−0.000821319 0	−0.000410660		1.00000i	$0.500131\pi$
$182$	0	0
$183$	0	0
$184$	789.815	0.316445
$185$	552.000	0.219372
$186$	0	0
$187$	607.950	0.237742
$188$	−1371.78	−0.532168
$189$	0	0
$190$	−4240.06	−1.61898
$191$	2841.00	1.07627	0.538135	−	0.842859i	$-0.319129\pi$
$191$	2841.00	1.07627	0.538135	+	0.842859i	$0.319129\pi$
$192$	0	0
$193$	−4245.26	−1.58332	−0.791659	−	0.610964i	$-0.790782\pi$
$193$	−4245.26	−1.58332	−0.791659	+	0.610964i	$0.790782\pi$
$194$	−4278.00	−1.58321
$195$	0	0
$196$	656.000	0.239067
$197$	2752.23	0.995371	0.497686	−	0.867357i	$-0.334183\pi$
$197$	2752.23	0.995371	0.497686	+	0.867357i	$0.334183\pi$
$198$	0	0
$199$	1685.00	0.600234	0.300117	−	0.953902i	$-0.402974\pi$
$199$	1685.00	0.600234	0.300117	+	0.953902i	$0.402974\pi$
$200$	928.379	0.328232
$201$	0	0
$202$	−6786.18	−2.36373
$203$	−1553.65	−0.537167
$204$	0	0
$205$	5448.00	1.85612
$206$	6429.37	2.17454
$207$	0	0
$208$	0	0
$209$	−1989.00	−0.658287
$210$	0	0
$211$	1681.00	0.548459	0.274229	−	0.961664i	$-0.411577\pi$
$211$	1681.00	0.548459	0.274229	+	0.961664i	$0.411577\pi$
$212$	−1704.00	−0.552034
$213$	0	0
$214$	−883.346	−0.282170
$215$	−1177.79	−0.373604
$216$	0	0
$217$	1638.00	0.512418
$218$	2112.00	0.656159
$219$	0	0
$220$	−1248.00	−0.382455
$221$	0	0
$222$	0	0
$223$	4096.30	1.23008	0.615042	−	0.788495i	$-0.289139\pi$
$223$	4096.30	1.23008	0.615042	+	0.788495i	$0.289139\pi$
$224$	−3744.00	−1.11677
$225$	0	0
$226$	1423.75	0.419054
$227$	−438.209	−0.128128	−0.0640638	−	0.997946i	$-0.520406\pi$
$227$	−438.209	−0.128128	−0.0640638	+	0.997946i	$0.520406\pi$
$228$	0	0
$229$	−180.133	−0.0519805	−0.0259903	−	0.999662i	$-0.508274\pi$
$229$	−180.133	−0.0519805	−0.0259903	+	0.999662i	$0.508274\pi$
$230$	2736.00	0.784376
$231$	0	0
$232$	956.092	0.270563
$233$	−5778.00	−1.62459	−0.812295	−	0.583247i	$-0.801782\pi$
$233$	−5778.00	−1.62459	−0.812295	+	0.583247i	$0.801782\pi$
$234$	0	0
$235$	4752.00	1.31909
$236$	−76.2102	−0.0210206
$237$	0	0
$238$	2106.00	0.573579
$239$	1860.22	0.503464	0.251732	−	0.967797i	$-0.419000\pi$
$239$	1860.22	0.503464	0.251732	+	0.967797i	$0.419000\pi$
$240$	0	0
$241$	−2059.41	−0.550449	−0.275224	−	0.961380i	$-0.588752\pi$
$241$	−2059.41	−0.550449	−0.275224	+	0.961380i	$0.588752\pi$
$242$	2854.42	0.758219
$243$	0	0
$244$	−68.0000	−0.0178412
$245$	−2272.45	−0.592578
$246$	0	0
$247$	0	0
$248$	−1008.00	−0.258097
$249$	0	0
$250$	−2784.00	−0.704302
$251$	4491.00	1.12936	0.564680	−	0.825310i	$-0.309000\pi$
$251$	4491.00	1.12936	0.564680	+	0.825310i	$0.309000\pi$
$252$	0	0
$253$	1283.45	0.318932
$254$	−7769.98	−1.91942
$255$	0	0
$256$	4864.00	1.18750
$257$	5451.00	1.32305	0.661525	−	0.749923i	$-0.269909\pi$
$257$	5451.00	1.32305	0.661525	+	0.749923i	$0.269909\pi$
$258$	0	0
$259$	897.000	0.215200
$260$	0	0
$261$	0	0
$262$	−1288.65	−0.303866
$263$	783.000	0.183581	0.0917906	−	0.995778i	$-0.470741\pi$
$263$	783.000	0.183581	0.0917906	+	0.995778i	$0.470741\pi$
$264$	0	0
$265$	5902.83	1.36833
$266$	−6890.10	−1.58819
$267$	0	0
$268$	−658.179	−0.150018
$269$	5085.00	1.15256	0.576279	−	0.817253i	$-0.304504\pi$
$269$	5085.00	1.15256	0.576279	+	0.817253i	$0.304504\pi$
$270$	0	0
$271$	−1325.02	−0.297008	−0.148504	−	0.988912i	$-0.547446\pi$
$271$	−1325.02	−0.297008	−0.148504	+	0.988912i	$0.547446\pi$
$272$	−2160.00	−0.481505
$273$	0	0
$274$	4122.00	0.908829
$275$	1508.62	0.330811
$276$	0	0
$277$	3421.00	0.742050	0.371025	−	0.928623i	$-0.379006\pi$
$277$	3421.00	0.742050	0.371025	+	0.928623i	$0.379006\pi$
$278$	8816.14	1.90200
$279$	0	0
$280$	4323.20	0.922716
$281$	810.600	0.172087	0.0860433	−	0.996291i	$-0.472578\pi$
$281$	810.600	0.172087	0.0860433	+	0.996291i	$0.472578\pi$
$282$	0	0
$283$	−7177.00	−1.50752	−0.753760	−	0.657149i	$-0.771762\pi$
$283$	−7177.00	−1.50752	−0.753760	+	0.657149i	$0.771762\pi$
$284$	−2334.80	−0.487835
$285$	0	0
$286$	0	0
$287$	8853.00	1.82082
$288$	0	0
$289$	−4184.00	−0.851618
$290$	3312.00	0.670646
$291$	0	0
$292$	−4018.36	−0.805331
$293$	9313.24	1.85695	0.928473	−	0.371400i	$-0.121122\pi$
$293$	9313.24	1.85695	0.928473	+	0.371400i	$0.121122\pi$
$294$	0	0
$295$	264.000	0.0521040
$296$	−552.000	−0.108393
$297$	0	0
$298$	4518.00	0.878257
$299$	0	0
$300$	0	0
$301$	−1913.92	−0.366499
$302$	−300.000	−0.0571625
$303$	0	0
$304$	7066.77	1.33325
$305$	235.559	0.0442232
$306$	0	0
$307$	4777.00	0.888070	0.444035	−	0.896009i	$-0.353547\pi$
$307$	4777.00	0.888070	0.444035	+	0.896009i	$0.353547\pi$
$308$	−2028.00	−0.375182
$309$	0	0
$310$	−3491.81	−0.639748
$311$	6192.00	1.12899	0.564495	−	0.825436i	$-0.309071\pi$
$311$	6192.00	1.12899	0.564495	+	0.825436i	$0.309071\pi$
$312$	0	0
$313$	−770.000	−0.139051	−0.0695255	−	0.997580i	$-0.522149\pi$
$313$	−770.000	−0.139051	−0.0695255	+	0.997580i	$0.522149\pi$
$314$	5313.93	0.955040
$315$	0	0
$316$	−4976.00	−0.885829
$317$	−8057.50	−1.42762	−0.713808	−	0.700341i	$-0.753031\pi$
$317$	−8057.50	−1.42762	−0.713808	+	0.700341i	$0.753031\pi$
$318$	0	0
$319$	1553.65	0.272689
$320$	−886.810	−0.154919
$321$	0	0
$322$	4446.00	0.769459
$323$	−2385.03	−0.410857
$324$	0	0
$325$	0	0
$326$	−5658.00	−0.961250
$327$	0	0
$328$	−5448.00	−0.917120
$329$	7722.00	1.29400
$330$	0	0
$331$	−5277.56	−0.876377	−0.438189	−	0.898883i	$-0.644380\pi$
$331$	−5277.56	−0.876377	−0.438189	+	0.898883i	$0.644380\pi$
$332$	1704.34	0.281740
$333$	0	0
$334$	5634.00	0.922990
$335$	2280.00	0.371850
$336$	0	0
$337$	−8278.00	−1.33808	−0.669038	−	0.743228i	$-0.733294\pi$
$337$	−8278.00	−1.33808	−0.669038	+	0.743228i	$0.733294\pi$
$338$	0	0
$339$	0	0
$340$	−1496.49	−0.238702
$341$	−1638.00	−0.260125
$342$	0	0
$343$	4030.48	0.634477
$344$	1177.79	0.184600
$345$	0	0
$346$	3024.16	0.469884
$347$	−6867.00	−1.06236	−0.531181	−	0.847258i	$-0.678252\pi$
$347$	−6867.00	−1.06236	−0.531181	+	0.847258i	$0.678252\pi$
$348$	0	0
$349$	12153.8	1.86412	0.932060	−	0.362303i	$-0.118010\pi$
$349$	12153.8	1.86412	0.932060	+	0.362303i	$0.118010\pi$
$350$	5226.00	0.798118
$351$	0	0
$352$	3744.00	0.566920
$353$	5807.57	0.875653	0.437827	−	0.899059i	$-0.355748\pi$
$353$	5807.57	0.875653	0.437827	+	0.899059i	$0.355748\pi$
$354$	0	0
$355$	8088.00	1.20920
$356$	−1226.29	−0.182566
$357$	0	0
$358$	4458.30	0.658180
$359$	1340.61	0.197088	0.0985439	−	0.995133i	$-0.468581\pi$
$359$	1340.61	0.197088	0.0985439	+	0.995133i	$0.468581\pi$
$360$	0	0
$361$	944.000	0.137629
$362$	6.92820	0.00100591
$363$	0	0
$364$	0	0
$365$	13920.0	1.99618
$366$	0	0
$367$	3665.00	0.521285	0.260642	−	0.965435i	$-0.416066\pi$
$367$	3665.00	0.521285	0.260642	+	0.965435i	$0.416066\pi$
$368$	−4560.00	−0.645941
$369$	0	0
$370$	−1912.18	−0.268675
$371$	9592.10	1.34231
$372$	0	0
$373$	5371.00	0.745576	0.372788	−	0.927917i	$-0.378402\pi$
$373$	5371.00	0.745576	0.372788	+	0.927917i	$0.378402\pi$
$374$	−2106.00	−0.291173
$375$	0	0
$376$	−4752.00	−0.651770
$377$	0	0
$378$	0	0
$379$	11509.5	1.55990	0.779950	−	0.625842i	$-0.215244\pi$
$379$	11509.5	1.55990	0.779950	+	0.625842i	$0.215244\pi$
$380$	4896.00	0.660946
$381$	0	0
$382$	−9841.51	−1.31816
$383$	−2419.67	−0.322819	−0.161409	−	0.986888i	$-0.551604\pi$
$383$	−2419.67	−0.322819	−0.161409	+	0.986888i	$0.551604\pi$
$384$	0	0
$385$	7025.20	0.929967
$386$	14706.0	1.93916
$387$	0	0
$388$	4939.81	0.646342
$389$	−9858.00	−1.28489	−0.642443	−	0.766334i	$-0.722079\pi$
$389$	−9858.00	−1.28489	−0.642443	+	0.766334i	$0.722079\pi$
$390$	0	0
$391$	1539.00	0.199055
$392$	2272.45	0.292796
$393$	0	0
$394$	−9534.00	−1.21908
$395$	17237.4	2.19571
$396$	0	0
$397$	8720.88	1.10249	0.551245	−	0.834344i	$-0.314153\pi$
$397$	8720.88	1.10249	0.551245	+	0.834344i	$0.314153\pi$
$398$	−5837.01	−0.735133
$399$	0	0
$400$	−5360.00	−0.670000
$401$	7584.65	0.944537	0.472269	−	0.881455i	$-0.343435\pi$
$401$	7584.65	0.944537	0.472269	+	0.881455i	$0.343435\pi$
$402$	0	0
$403$	0	0
$404$	7836.00	0.964989
$405$	0	0
$406$	5382.00	0.657892
$407$	−897.000	−0.109245
$408$	0	0
$409$	4304.15	0.520358	0.260179	−	0.965560i	$-0.416218\pi$
$409$	4304.15	0.520358	0.260179	+	0.965560i	$0.416218\pi$
$410$	−18872.4	−2.27327
$411$	0	0
$412$	−7424.00	−0.887753
$413$	429.000	0.0511131
$414$	0	0
$415$	−5904.00	−0.698352
$416$	0	0
$417$	0	0
$418$	6890.10	0.806234
$419$	−5397.00	−0.629262	−0.314631	−	0.949214i	$-0.601881\pi$
$419$	−5397.00	−0.629262	−0.314631	+	0.949214i	$0.601881\pi$
$420$	0	0
$421$	7260.76	0.840541	0.420270	−	0.907399i	$-0.361935\pi$
$421$	7260.76	0.840541	0.420270	+	0.907399i	$0.361935\pi$
$422$	−5823.15	−0.671722
$423$	0	0
$424$	−5902.83	−0.676101
$425$	1809.00	0.206469
$426$	0	0
$427$	382.783	0.0433822
$428$	1020.00	0.115195
$429$	0	0
$430$	4080.00	0.457570
$431$	−486.706	−0.0543940	−0.0271970	−	0.999630i	$-0.508658\pi$
$431$	−486.706	−0.0543940	−0.0271970	+	0.999630i	$0.508658\pi$
$432$	0	0
$433$	−12139.0	−1.34726	−0.673629	−	0.739069i	$-0.735266\pi$
$433$	−12139.0	−1.34726	−0.673629	+	0.739069i	$0.735266\pi$
$434$	−5674.20	−0.627581
$435$	0	0
$436$	−2438.73	−0.267876
$437$	−5035.07	−0.551167
$438$	0	0
$439$	−461.000	−0.0501192	−0.0250596	−	0.999686i	$-0.507978\pi$
$439$	−461.000	−0.0501192	−0.0250596	+	0.999686i	$0.507978\pi$
$440$	−4323.20	−0.468410
$441$	0	0
$442$	0	0
$443$	−12156.0	−1.30372	−0.651861	−	0.758338i	$-0.726012\pi$
$443$	−12156.0	−1.30372	−0.651861	+	0.758338i	$0.726012\pi$
$444$	0	0
$445$	4248.00	0.452527
$446$	−14190.0	−1.50654
$447$	0	0
$448$	−1441.07	−0.151973
$449$	−296.181	−0.0311306	−0.0155653	−	0.999879i	$-0.504955\pi$
$449$	−296.181	−0.0311306	−0.0155653	+	0.999879i	$0.504955\pi$
$450$	0	0
$451$	−8853.00	−0.924327
$452$	−1644.00	−0.171078
$453$	0	0
$454$	1518.00	0.156924
$455$	0	0
$456$	0	0
$457$	−611.414	−0.0625837	−0.0312918	−	0.999510i	$-0.509962\pi$
$457$	−611.414	−0.0625837	−0.0312918	+	0.999510i	$0.509962\pi$
$458$	624.000	0.0636629
$459$	0	0
$460$	−3159.26	−0.320220
$461$	13127.2	1.32624	0.663119	−	0.748514i	$-0.269233\pi$
$461$	13127.2	1.32624	0.663119	+	0.748514i	$0.269233\pi$
$462$	0	0
$463$	834.848	0.0837985	0.0418992	−	0.999122i	$-0.486659\pi$
$463$	834.848	0.0837985	0.0418992	+	0.999122i	$0.486659\pi$
$464$	−5520.00	−0.552284
$465$	0	0
$466$	20015.6	1.98971
$467$	14496.0	1.43639	0.718196	−	0.695841i	$-0.244968\pi$
$467$	14496.0	1.43639	0.718196	+	0.695841i	$0.244968\pi$
$468$	0	0
$469$	3705.00	0.364778
$470$	−16461.4	−1.61555
$471$	0	0
$472$	−264.000	−0.0257449
$473$	1913.92	0.186051
$474$	0	0
$475$	−5918.42	−0.571696
$476$	−2431.80	−0.234162
$477$	0	0
$478$	−6444.00	−0.616614
$479$	8897.54	0.848725	0.424362	−	0.905492i	$-0.360498\pi$
$479$	8897.54	0.848725	0.424362	+	0.905492i	$0.360498\pi$
$480$	0	0
$481$	0	0
$482$	7134.00	0.674159
$483$	0	0
$484$	−3296.00	−0.309542
$485$	−17112.0	−1.60209
$486$	0	0
$487$	4754.48	0.442394	0.221197	−	0.975229i	$-0.429004\pi$
$487$	4754.48	0.442394	0.221197	+	0.975229i	$0.429004\pi$
$488$	−235.559	−0.0218509
$489$	0	0
$490$	7872.00	0.725757
$491$	−1635.00	−0.150278	−0.0751390	−	0.997173i	$-0.523940\pi$
$491$	−1635.00	−0.150278	−0.0751390	+	0.997173i	$0.523940\pi$
$492$	0	0
$493$	1863.00	0.170193
$494$	0	0
$495$	0	0
$496$	5819.69	0.526838
$497$	13143.0	1.18621
$498$	0	0
$499$	−14434.9	−1.29498	−0.647490	−	0.762074i	$-0.724181\pi$
$499$	−14434.9	−1.29498	−0.647490	+	0.762074i	$0.724181\pi$
$500$	3214.69	0.287530
$501$	0	0
$502$	−15557.3	−1.38318
$503$	−12687.0	−1.12462	−0.562312	−	0.826925i	$-0.690088\pi$
$503$	−12687.0	−1.12462	−0.562312	+	0.826925i	$0.690088\pi$
$504$	0	0
$505$	−27144.7	−2.39193
$506$	−4446.00	−0.390610
$507$	0	0
$508$	8972.00	0.783599
$509$	−5748.68	−0.500600	−0.250300	−	0.968168i	$-0.580529\pi$
$509$	−5748.68	−0.500600	−0.250300	+	0.968168i	$0.580529\pi$
$510$	0	0
$511$	22620.0	1.95822
$512$	−4434.05	−0.382733
$513$	0	0
$514$	−18882.8	−1.62040
$515$	25717.5	2.20048
$516$	0	0
$517$	−7722.00	−0.656892
$518$	−3107.30	−0.263565
$519$	0	0
$520$	0	0
$521$	−6054.00	−0.509080	−0.254540	−	0.967062i	$-0.581924\pi$
$521$	−6054.00	−0.509080	−0.254540	+	0.967062i	$0.581924\pi$
$522$	0	0
$523$	−14803.0	−1.23765	−0.618824	−	0.785530i	$-0.712391\pi$
$523$	−14803.0	−1.23765	−0.618824	+	0.785530i	$0.712391\pi$
$524$	1488.00	0.124053
$525$	0	0
$526$	−2712.39	−0.224840
$527$	−1964.15	−0.162352
$528$	0	0
$529$	−8918.00	−0.732966
$530$	−20448.0	−1.67586
$531$	0	0
$532$	7956.00	0.648377
$533$	0	0
$534$	0	0
$535$	−3533.38	−0.285536
$536$	−2280.00	−0.183733
$537$	0	0
$538$	−17615.0	−1.41159
$539$	3692.73	0.295097
$540$	0	0
$541$	21470.5	1.70626	0.853132	−	0.521695i	$-0.174700\pi$
$541$	21470.5	1.70626	0.853132	+	0.521695i	$0.174700\pi$
$542$	4590.00	0.363759
$543$	0	0
$544$	4489.48	0.353832
$545$	8448.00	0.663986
$546$	0	0
$547$	−13516.0	−1.05649	−0.528247	−	0.849091i	$-0.677151\pi$
$547$	−13516.0	−1.05649	−0.528247	+	0.849091i	$0.677151\pi$
$548$	−4759.68	−0.371028
$549$	0	0
$550$	−5226.00	−0.405159
$551$	−6095.09	−0.471251
$552$	0	0
$553$	28010.7	2.15396
$554$	−11850.7	−0.908822
$555$	0	0
$556$	−10180.0	−0.776490
$557$	−2890.79	−0.219905	−0.109952	−	0.993937i	$-0.535070\pi$
$557$	−2890.79	−0.219905	−0.109952	+	0.993937i	$0.535070\pi$
$558$	0	0
$559$	0	0
$560$	−24960.0	−1.88349
$561$	0	0
$562$	−2808.00	−0.210762
$563$	11583.0	0.867079	0.433539	−	0.901135i	$-0.357265\pi$
$563$	11583.0	0.867079	0.433539	+	0.901135i	$0.357265\pi$
$564$	0	0
$565$	5694.98	0.424053
$566$	24861.9	1.84633
$567$	0	0
$568$	−8088.00	−0.597473
$569$	12879.0	0.948885	0.474443	−	0.880286i	$-0.342650\pi$
$569$	12879.0	0.948885	0.474443	+	0.880286i	$0.342650\pi$
$570$	0	0
$571$	11636.0	0.852805	0.426402	−	0.904534i	$-0.359781\pi$
$571$	11636.0	0.852805	0.426402	+	0.904534i	$0.359781\pi$
$572$	0	0
$573$	0	0
$574$	−30667.7	−2.23004
$575$	3819.00	0.276980
$576$	0	0
$577$	12311.4	0.888269	0.444134	−	0.895960i	$-0.353511\pi$
$577$	12311.4	0.888269	0.444134	+	0.895960i	$0.353511\pi$
$578$	14493.8	1.04301
$579$	0	0
$580$	−3824.37	−0.273790
$581$	−9594.00	−0.685071
$582$	0	0
$583$	−9592.10	−0.681414
$584$	−13920.0	−0.986325
$585$	0	0
$586$	−32262.0	−2.27428
$587$	−15645.6	−1.10011	−0.550054	−	0.835129i	$-0.685393\pi$
$587$	−15645.6	−1.10011	−0.550054	+	0.835129i	$0.685393\pi$
$588$	0	0
$589$	6426.00	0.449539
$590$	−914.523	−0.0638141
$591$	0	0
$592$	3186.97	0.221256
$593$	25821.4	1.78813	0.894063	−	0.447942i	$-0.147843\pi$
$593$	25821.4	1.78813	0.894063	+	0.447942i	$0.147843\pi$
$594$	0	0
$595$	8424.00	0.580421
$596$	−5216.94	−0.358547
$597$	0	0
$598$	0	0
$599$	−1668.00	−0.113777	−0.0568887	−	0.998381i	$-0.518118\pi$
$599$	−1668.00	−0.113777	−0.0568887	+	0.998381i	$0.518118\pi$
$600$	0	0
$601$	13699.0	0.929773	0.464887	−	0.885370i	$-0.346095\pi$
$601$	13699.0	0.929773	0.464887	+	0.885370i	$0.346095\pi$
$602$	6630.00	0.448868
$603$	0	0
$604$	346.410	0.0233365
$605$	11417.7	0.767264
$606$	0	0
$607$	−23173.0	−1.54953	−0.774764	−	0.632251i	$-0.782131\pi$
$607$	−23173.0	−1.54953	−0.774764	+	0.632251i	$0.782131\pi$
$608$	−14688.0	−0.979732
$609$	0	0
$610$	−816.000	−0.0541621
$611$	0	0
$612$	0	0
$613$	−16615.6	−1.09477	−0.547387	−	0.836880i	$-0.684377\pi$
$613$	−16615.6	−1.09477	−0.547387	+	0.836880i	$0.684377\pi$
$614$	−16548.0	−1.08766
$615$	0	0
$616$	−7025.20	−0.459502
$617$	−28393.5	−1.85264	−0.926321	−	0.376736i	$-0.877046\pi$
$617$	−28393.5	−1.85264	−0.926321	+	0.376736i	$0.877046\pi$
$618$	0	0
$619$	6245.78	0.405556	0.202778	−	0.979225i	$-0.435003\pi$
$619$	6245.78	0.405556	0.202778	+	0.979225i	$0.435003\pi$
$620$	4032.00	0.261176
$621$	0	0
$622$	−21449.7	−1.38273
$623$	6903.00	0.443921
$624$	0	0
$625$	−19511.0	−1.24870
$626$	2667.36	0.170302
$627$	0	0
$628$	−6136.00	−0.389893
$629$	−1075.60	−0.0681830
$630$	0	0
$631$	−22379.8	−1.41193	−0.705964	−	0.708247i	$-0.749486\pi$
$631$	−22379.8	−1.41193	−0.705964	+	0.708247i	$0.749486\pi$
$632$	−17237.4	−1.08491
$633$	0	0
$634$	27912.0	1.74847
$635$	−31079.9	−1.94231
$636$	0	0
$637$	0	0
$638$	−5382.00	−0.333974
$639$	0	0
$640$	21504.0	1.32816
$641$	19827.0	1.22172	0.610858	−	0.791740i	$-0.290825\pi$
$641$	19827.0	1.22172	0.610858	+	0.791740i	$0.290825\pi$
$642$	0	0
$643$	−8450.68	−0.518293	−0.259146	−	0.965838i	$-0.583441\pi$
$643$	−8450.68	−0.518293	−0.259146	+	0.965838i	$0.583441\pi$
$644$	−5133.80	−0.314130
$645$	0	0
$646$	8262.00	0.503195
$647$	2949.00	0.179192	0.0895959	−	0.995978i	$-0.471442\pi$
$647$	2949.00	0.179192	0.0895959	+	0.995978i	$0.471442\pi$
$648$	0	0
$649$	−429.000	−0.0259472
$650$	0	0
$651$	0	0
$652$	6533.30	0.392429
$653$	−12039.0	−0.721474	−0.360737	−	0.932668i	$-0.617475\pi$
$653$	−12039.0	−0.721474	−0.360737	+	0.932668i	$0.617475\pi$
$654$	0	0
$655$	−5154.58	−0.307490
$656$	31454.0	1.87206
$657$	0	0
$658$	−26749.8	−1.58483
$659$	−3363.00	−0.198792	−0.0993960	−	0.995048i	$-0.531691\pi$
$659$	−3363.00	−0.198792	−0.0993960	+	0.995048i	$0.531691\pi$
$660$	0	0
$661$	10158.5	0.597759	0.298880	−	0.954291i	$-0.403387\pi$
$661$	10158.5	0.597759	0.298880	+	0.954291i	$0.403387\pi$
$662$	18282.0	1.07334
$663$	0	0
$664$	5904.00	0.345060
$665$	−27560.4	−1.60714
$666$	0	0
$667$	3933.00	0.228315
$668$	−6505.58	−0.376809
$669$	0	0
$670$	−7898.15	−0.455421
$671$	−382.783	−0.0220226
$672$	0	0
$673$	18169.0	1.04066	0.520329	−	0.853966i	$-0.325809\pi$
$673$	18169.0	1.04066	0.520329	+	0.853966i	$0.325809\pi$
$674$	28675.8	1.63880
$675$	0	0
$676$	0	0
$677$	−9042.00	−0.513312	−0.256656	−	0.966503i	$-0.582621\pi$
$677$	−9042.00	−0.513312	−0.256656	+	0.966503i	$0.582621\pi$
$678$	0	0
$679$	−27807.0	−1.57163
$680$	−5184.00	−0.292349
$681$	0	0
$682$	5674.20	0.318587
$683$	−12462.1	−0.698169	−0.349084	−	0.937091i	$-0.613507\pi$
$683$	−12462.1	−0.698169	−0.349084	+	0.937091i	$0.613507\pi$
$684$	0	0
$685$	16488.0	0.919670
$686$	−13962.0	−0.777072
$687$	0	0
$688$	−6800.00	−0.376813
$689$	0	0
$690$	0	0
$691$	4318.00	0.237720	0.118860	−	0.992911i	$-0.462076\pi$
$691$	4318.00	0.237720	0.118860	+	0.992911i	$0.462076\pi$
$692$	−3492.00	−0.191829
$693$	0	0
$694$	23788.0	1.30112
$695$	35264.6	1.92469
$696$	0	0
$697$	−10615.7	−0.576901
$698$	−42102.0	−2.28307
$699$	0	0
$700$	−6034.47	−0.325830
$701$	−18270.0	−0.984377	−0.492189	−	0.870489i	$-0.663803\pi$
$701$	−18270.0	−0.984377	−0.492189	+	0.870489i	$0.663803\pi$
$702$	0	0
$703$	3519.00	0.188793
$704$	1441.07	0.0771481
$705$	0	0
$706$	−20118.0	−1.07245
$707$	−44110.1	−2.34644
$708$	0	0
$709$	−1629.86	−0.0863338	−0.0431669	−	0.999068i	$-0.513745\pi$
$709$	−1629.86	−0.0863338	−0.0431669	+	0.999068i	$0.513745\pi$
$710$	−28017.7	−1.48096
$711$	0	0
$712$	−4248.00	−0.223596
$713$	−4146.53	−0.217796
$714$	0	0
$715$	0	0
$716$	−5148.00	−0.268701
$717$	0	0
$718$	−4644.00	−0.241382
$719$	9831.00	0.509923	0.254961	−	0.966951i	$-0.417937\pi$
$719$	9831.00	0.509923	0.254961	+	0.966951i	$0.417937\pi$
$720$	0	0
$721$	41790.9	2.15863
$722$	−3270.11	−0.168561
$723$	0	0
$724$	−8.00000	−0.000410660 0
$725$	4623.00	0.236819
$726$	0	0
$727$	15464.0	0.788897	0.394448	−	0.918918i	$-0.370936\pi$
$727$	15464.0	0.788897	0.394448	+	0.918918i	$0.370936\pi$
$728$	0	0
$729$	0	0
$730$	−48220.3	−2.44481
$731$	2295.00	0.116120
$732$	0	0
$733$	−12616.3	−0.635733	−0.317866	−	0.948136i	$-0.602966\pi$
$733$	−12616.3	−0.635733	−0.317866	+	0.948136i	$0.602966\pi$
$734$	−12695.9	−0.638441
$735$	0	0
$736$	9477.78	0.474668
$737$	−3705.00	−0.185177
$738$	0	0
$739$	−16283.0	−0.810528	−0.405264	−	0.914200i	$-0.632820\pi$
$739$	−16283.0	−0.810528	−0.405264	+	0.914200i	$0.632820\pi$
$740$	2208.00	0.109686
$741$	0	0
$742$	−33228.0	−1.64399
$743$	10806.3	0.533571	0.266786	−	0.963756i	$-0.414038\pi$
$743$	10806.3	0.533571	0.266786	+	0.963756i	$0.414038\pi$
$744$	0	0
$745$	18072.0	0.888734
$746$	−18605.7	−0.913140
$747$	0	0
$748$	2431.80	0.118871
$749$	−5741.75	−0.280105
$750$	0	0
$751$	13615.0	0.661542	0.330771	−	0.943711i	$-0.392691\pi$
$751$	13615.0	0.661542	0.330771	+	0.943711i	$0.392691\pi$
$752$	27435.7	1.33042
$753$	0	0
$754$	0	0
$755$	−1200.00	−0.0578443
$756$	0	0
$757$	5551.00	0.266519	0.133259	−	0.991081i	$-0.457456\pi$
$757$	5551.00	0.266519	0.133259	+	0.991081i	$0.457456\pi$
$758$	−39870.0	−1.91048
$759$	0	0
$760$	16960.2	0.809490
$761$	10082.3	0.480265	0.240133	−	0.970740i	$-0.422809\pi$
$761$	10082.3	0.480265	0.240133	+	0.970740i	$0.422809\pi$
$762$	0	0
$763$	13728.0	0.651359
$764$	11364.0	0.538135
$765$	0	0
$766$	8382.00	0.395371
$767$	0	0
$768$	0	0
$769$	−29758.4	−1.39547	−0.697733	−	0.716357i	$-0.745808\pi$
$769$	−29758.4	−1.39547	−0.697733	+	0.716357i	$0.745808\pi$
$770$	−24336.0	−1.13897
$771$	0	0
$772$	−16981.0	−0.791659
$773$	−27735.3	−1.29052	−0.645259	−	0.763964i	$-0.723251\pi$
$773$	−27735.3	−1.29052	−0.645259	+	0.763964i	$0.723251\pi$
$774$	0	0
$775$	−4873.99	−0.225908
$776$	17112.0	0.791604
$777$	0	0
$778$	34149.1	1.57366
$779$	34731.0	1.59739
$780$	0	0
$781$	−13143.0	−0.602168
$782$	−5331.25	−0.243792
$783$	0	0
$784$	−13120.0	−0.597668
$785$	21255.7	0.966432
$786$	0	0
$787$	31549.3	1.42899	0.714493	−	0.699643i	$-0.246658\pi$
$787$	31549.3	1.42899	0.714493	+	0.699643i	$0.246658\pi$
$788$	11008.9	0.497686
$789$	0	0
$790$	−59712.0	−2.68919
$791$	9254.35	0.415988
$792$	0	0
$793$	0	0
$794$	−30210.0	−1.35027
$795$	0	0
$796$	6740.00	0.300117
$797$	1455.00	0.0646659	0.0323330	−	0.999477i	$-0.489706\pi$
$797$	1455.00	0.0646659	0.0323330	+	0.999477i	$0.489706\pi$
$798$	0	0
$799$	−9259.54	−0.409986
$800$	11140.6	0.492347
$801$	0	0
$802$	−26274.0	−1.15682
$803$	−22620.0	−0.994075
$804$	0	0
$805$	17784.0	0.778638
$806$	0	0
$807$	0	0
$808$	27144.7	1.18187
$809$	−1659.00	−0.0720981	−0.0360490	−	0.999350i	$-0.511477\pi$
$809$	−1659.00	−0.0720981	−0.0360490	+	0.999350i	$0.511477\pi$
$810$	0	0
$811$	4402.87	0.190636	0.0953180	−	0.995447i	$-0.469613\pi$
$811$	4402.87	0.190636	0.0953180	+	0.995447i	$0.469613\pi$
$812$	−6214.60	−0.268583
$813$	0	0
$814$	3107.30	0.133797
$815$	−22632.0	−0.972717
$816$	0	0
$817$	−7508.44	−0.321526
$818$	−14910.0	−0.637306
$819$	0	0
$820$	21792.0	0.928061
$821$	−28701.8	−1.22010	−0.610049	−	0.792364i	$-0.708850\pi$
$821$	−28701.8	−1.22010	−0.610049	+	0.792364i	$0.708850\pi$
$822$	0	0
$823$	−15779.0	−0.668313	−0.334156	−	0.942518i	$-0.608451\pi$
$823$	−15779.0	−0.668313	−0.334156	+	0.942518i	$0.608451\pi$
$824$	−25717.5	−1.08727
$825$	0	0
$826$	−1486.10	−0.0626005
$827$	−7354.29	−0.309231	−0.154615	−	0.987975i	$-0.549414\pi$
$827$	−7354.29	−0.309231	−0.154615	+	0.987975i	$0.549414\pi$
$828$	0	0
$829$	−17371.0	−0.727768	−0.363884	−	0.931444i	$-0.618550\pi$
$829$	−17371.0	−0.727768	−0.363884	+	0.931444i	$0.618550\pi$
$830$	20452.1	0.855303
$831$	0	0
$832$	0	0
$833$	4428.00	0.184179
$834$	0	0
$835$	22536.0	0.934001
$836$	−7956.00	−0.329144
$837$	0	0
$838$	18695.8	0.770685
$839$	29474.3	1.21283	0.606416	−	0.795148i	$-0.292607\pi$
$839$	29474.3	1.21283	0.606416	+	0.795148i	$0.292607\pi$
$840$	0	0
$841$	−19628.0	−0.804789
$842$	−25152.0	−1.02945
$843$	0	0
$844$	6724.00	0.274229
$845$	0	0
$846$	0	0
$847$	18553.7	0.752673
$848$	34080.0	1.38008
$849$	0	0
$850$	−6266.56	−0.252872
$851$	−2270.72	−0.0914680
$852$	0	0
$853$	−2909.85	−0.116801	−0.0584005	−	0.998293i	$-0.518600\pi$
$853$	−2909.85	−0.116801	−0.0584005	+	0.998293i	$0.518600\pi$
$854$	−1326.00	−0.0531321
$855$	0	0
$856$	3533.38	0.141085
$857$	−5346.00	−0.213087	−0.106544	−	0.994308i	$-0.533978\pi$
$857$	−5346.00	−0.213087	−0.106544	+	0.994308i	$0.533978\pi$
$858$	0	0
$859$	24244.0	0.962974	0.481487	−	0.876453i	$-0.340097\pi$
$859$	24244.0	0.962974	0.481487	+	0.876453i	$0.340097\pi$
$860$	−4711.18	−0.186802
$861$	0	0
$862$	1686.00	0.0666188
$863$	−32780.8	−1.29301	−0.646507	−	0.762908i	$-0.723771\pi$
$863$	−32780.8	−1.29301	−0.646507	+	0.762908i	$0.723771\pi$
$864$	0	0
$865$	12096.6	0.475489
$866$	42050.7	1.65005
$867$	0	0
$868$	6552.00	0.256209
$869$	−28010.7	−1.09344
$870$	0	0
$871$	0	0
$872$	−8448.00	−0.328080
$873$	0	0
$874$	17442.0	0.675039
$875$	−18096.0	−0.699150
$876$	0	0
$877$	−4543.17	−0.174928	−0.0874640	−	0.996168i	$-0.527876\pi$
$877$	−4543.17	−0.174928	−0.0874640	+	0.996168i	$0.527876\pi$
$878$	1596.95	0.0613832
$879$	0	0
$880$	24960.0	0.956138
$881$	−20517.0	−0.784603	−0.392302	−	0.919837i	$-0.628321\pi$
$881$	−20517.0	−0.784603	−0.392302	+	0.919837i	$0.628321\pi$
$882$	0	0
$883$	23852.0	0.909042	0.454521	−	0.890736i	$-0.349811\pi$
$883$	23852.0	0.909042	0.454521	+	0.890736i	$0.349811\pi$
$884$	0	0
$885$	0	0
$886$	42109.6	1.59673
$887$	−38757.0	−1.46712	−0.733558	−	0.679626i	$-0.762142\pi$
$887$	−38757.0	−1.46712	−0.733558	+	0.679626i	$0.762142\pi$
$888$	0	0
$889$	−50504.9	−1.90538
$890$	−14715.5	−0.554230
$891$	0	0
$892$	16385.2	0.615042
$893$	30294.0	1.13522
$894$	0	0
$895$	17833.2	0.666031
$896$	34944.0	1.30290
$897$	0	0
$898$	1026.00	0.0381270
$899$	−5019.48	−0.186217
$900$	0	0
$901$	−11502.0	−0.425291
$902$	30667.7	1.13206
$903$	0	0
$904$	−5694.98	−0.209527
$905$	27.7128	0.00101791
$906$	0	0
$907$	39071.0	1.43035	0.715177	−	0.698943i	$-0.246346\pi$
$907$	39071.0	1.43035	0.715177	+	0.698943i	$0.246346\pi$
$908$	−1752.84	−0.0640638
$909$	0	0
$910$	0	0
$911$	53040.0	1.92897	0.964486	−	0.264134i	$-0.0850860\pi$
$911$	53040.0	1.92897	0.964486	+	0.264134i	$0.0850860\pi$
$912$	0	0
$913$	9594.00	0.347771
$914$	2118.00	0.0766490
$915$	0	0
$916$	−720.533	−0.0259903
$917$	−8376.20	−0.301643
$918$	0	0
$919$	367.000	0.0131732	0.00658662	−	0.999978i	$-0.497903\pi$
$919$	367.000	0.0131732	0.00658662	+	0.999978i	$0.497903\pi$
$920$	−10944.0	−0.392188
$921$	0	0
$922$	−45474.0	−1.62430
$923$	0	0
$924$	0	0
$925$	−2669.09	−0.0948748
$926$	−2892.00	−0.102632
$927$	0	0
$928$	11473.1	0.405844
$929$	29935.0	1.05720	0.528599	−	0.848872i	$-0.322718\pi$
$929$	29935.0	1.05720	0.528599	+	0.848872i	$0.322718\pi$
$930$	0	0
$931$	−14486.9	−0.509976
$932$	−23112.0	−0.812295
$933$	0	0
$934$	−50215.6	−1.75921
$935$	−8424.00	−0.294646
$936$	0	0
$937$	42166.0	1.47012	0.735060	−	0.678002i	$-0.237154\pi$
$937$	42166.0	1.47012	0.735060	+	0.678002i	$0.237154\pi$
$938$	−12834.5	−0.446760
$939$	0	0
$940$	19008.0	0.659545
$941$	35022.1	1.21327	0.606635	−	0.794981i	$-0.292519\pi$
$941$	35022.1	1.21327	0.606635	+	0.794981i	$0.292519\pi$
$942$	0	0
$943$	−22411.0	−0.773916
$944$	1524.20	0.0525515
$945$	0	0
$946$	−6630.00	−0.227865
$947$	2599.81	0.0892106	0.0446053	−	0.999005i	$-0.485797\pi$
$947$	2599.81	0.0892106	0.0446053	+	0.999005i	$0.485797\pi$
$948$	0	0
$949$	0	0
$950$	20502.0	0.700182
$951$	0	0
$952$	−8424.00	−0.286789
$953$	10623.0	0.361084	0.180542	−	0.983567i	$-0.442215\pi$
$953$	10623.0	0.361084	0.180542	+	0.983567i	$0.442215\pi$
$954$	0	0
$955$	−39366.1	−1.33388
$956$	7440.89	0.251732
$957$	0	0
$958$	−30822.0	−1.03947
$959$	26793.0	0.902180
$960$	0	0
$961$	−24499.0	−0.822362
$962$	0	0
$963$	0	0
$964$	−8237.63	−0.275224
$965$	58824.0	1.96229
$966$	0	0
$967$	20199.2	0.671729	0.335864	−	0.941910i	$-0.390972\pi$
$967$	20199.2	0.671729	0.335864	+	0.941910i	$0.390972\pi$
$968$	−11417.7	−0.379110
$969$	0	0
$970$	59277.7	1.96216
$971$	2325.00	0.0768412	0.0384206	−	0.999262i	$-0.487767\pi$
$971$	2325.00	0.0768412	0.0384206	+	0.999262i	$0.487767\pi$
$972$	0	0
$973$	57304.9	1.88809
$974$	−16470.0	−0.541820
$975$	0	0
$976$	1360.00	0.0446030
$977$	32938.4	1.07860	0.539300	−	0.842113i	$-0.318689\pi$
$977$	32938.4	1.07860	0.539300	+	0.842113i	$0.318689\pi$
$978$	0	0
$979$	−6903.00	−0.225353
$980$	−9089.80	−0.296289
$981$	0	0
$982$	5663.81	0.184052
$983$	42702.0	1.38554	0.692768	−	0.721161i	$-0.256391\pi$
$983$	42702.0	1.38554	0.692768	+	0.721161i	$0.256391\pi$
$984$	0	0
$985$	−38136.0	−1.23362
$986$	−6453.62	−0.208443
$987$	0	0
$988$	0	0
$989$	4845.00	0.155776
$990$	0	0
$991$	−4843.00	−0.155240	−0.0776201	−	0.996983i	$-0.524732\pi$
$991$	−4843.00	−0.155240	−0.0776201	+	0.996983i	$0.524732\pi$
$992$	−12096.0	−0.387146
$993$	0	0
$994$	−45528.7	−1.45280
$995$	−23348.0	−0.743902
$996$	0	0
$997$	10943.0	0.347611	0.173806	−	0.984780i	$-0.444394\pi$
$997$	10943.0	0.347611	0.173806	+	0.984780i	$0.444394\pi$
$998$	50004.0	1.58602
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.q.1.1		2
3.2	odd	2		169.4.a.h.1.2		2
13.6	odd	12		117.4.q.c.10.1		2
13.11	odd	12		117.4.q.c.82.1		2
13.12	even	2	inner	1521.4.a.q.1.2		2
39.2	even	12		169.4.e.b.147.1		2
39.5	even	4		169.4.b.b.168.1		2
39.8	even	4		169.4.b.b.168.2		2
39.11	even	12		13.4.e.a.4.1	✓	2
39.17	odd	6		169.4.c.i.146.2		4
39.20	even	12		169.4.e.b.23.1		2
39.23	odd	6		169.4.c.i.22.2		4
39.29	odd	6		169.4.c.i.22.1		4
39.32	even	12		13.4.e.a.10.1	yes	2
39.35	odd	6		169.4.c.i.146.1		4
39.38	odd	2		169.4.a.h.1.1		2
156.11	odd	12		208.4.w.a.17.1		2
156.71	odd	12		208.4.w.a.49.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.e.a.4.1	✓	2	39.11	even	12
13.4.e.a.10.1	yes	2	39.32	even	12
117.4.q.c.10.1		2	13.6	odd	12
117.4.q.c.82.1		2	13.11	odd	12
169.4.a.h.1.1		2	39.38	odd	2
169.4.a.h.1.2		2	3.2	odd	2
169.4.b.b.168.1		2	39.5	even	4
169.4.b.b.168.2		2	39.8	even	4
169.4.c.i.22.1		4	39.29	odd	6
169.4.c.i.22.2		4	39.23	odd	6
169.4.c.i.146.1		4	39.35	odd	6
169.4.c.i.146.2		4	39.17	odd	6
169.4.e.b.23.1		2	39.20	even	12
169.4.e.b.147.1		2	39.2	even	12
208.4.w.a.17.1		2	156.11	odd	12
208.4.w.a.49.1		2	156.71	odd	12
1521.4.a.q.1.1		2	1.1	even	1	trivial
1521.4.a.q.1.2		2	13.12	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-3.46410 q^{2} +4.00000 q^{4} -13.8564 q^{5} -22.5167 q^{7} +13.8564 q^{8} +O(q^{10})\)	\(q-3.46410 q^{2} +4.00000 q^{4} -13.8564 q^{5} -22.5167 q^{7} +13.8564 q^{8} +48.0000 q^{10} +22.5167 q^{11} +78.0000 q^{14} -80.0000 q^{16} +27.0000 q^{17} -88.3346 q^{19} -55.4256 q^{20} -78.0000 q^{22} +57.0000 q^{23} +67.0000 q^{25} -90.0666 q^{28} +69.0000 q^{29} -72.7461 q^{31} +166.277 q^{32} -93.5307 q^{34} +312.000 q^{35} -39.8372 q^{37} +306.000 q^{38} -192.000 q^{40} -393.176 q^{41} +85.0000 q^{43} +90.0666 q^{44} -197.454 q^{46} -342.946 q^{47} +164.000 q^{49} -232.095 q^{50} -426.000 q^{53} -312.000 q^{55} -312.000 q^{56} -239.023 q^{58} -19.0526 q^{59} -17.0000 q^{61} +252.000 q^{62} +64.0000 q^{64} -164.545 q^{67} +108.000 q^{68} -1080.80 q^{70} -583.701 q^{71} -1004.59 q^{73} +138.000 q^{74} -353.338 q^{76} -507.000 q^{77} -1244.00 q^{79} +1108.51 q^{80} +1362.00 q^{82} +426.084 q^{83} -374.123 q^{85} -294.449 q^{86} +312.000 q^{88} -306.573 q^{89} +228.000 q^{92} +1188.00 q^{94} +1224.00 q^{95} +1234.95 q^{97} -568.113 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{4}+O(q^{10}) \) 2 * q + 8 * q^4	\( 2 q + 8 q^{4} + 96 q^{10} + 156 q^{14} - 160 q^{16} + 54 q^{17} - 156 q^{22} + 114 q^{23} + 134 q^{25} + 138 q^{29} + 624 q^{35} + 612 q^{38} - 384 q^{40} + 170 q^{43} + 328 q^{49} - 852 q^{53} - 624 q^{55} - 624 q^{56} - 34 q^{61} + 504 q^{62} + 128 q^{64} + 216 q^{68} + 276 q^{74} - 1014 q^{77} - 2488 q^{79} + 2724 q^{82} + 624 q^{88} + 456 q^{92} + 2376 q^{94} + 2448 q^{95}+O(q^{100}) \) 2 * q + 8 * q^4 + 96 * q^10 + 156 * q^14 - 160 * q^16 + 54 * q^17 - 156 * q^22 + 114 * q^23 + 134 * q^25 + 138 * q^29 + 624 * q^35 + 612 * q^38 - 384 * q^40 + 170 * q^43 + 328 * q^49 - 852 * q^53 - 624 * q^55 - 624 * q^56 - 34 * q^61 + 504 * q^62 + 128 * q^64 + 216 * q^68 + 276 * q^74 - 1014 * q^77 - 2488 * q^79 + 2724 * q^82 + 624 * q^88 + 456 * q^92 + 2376 * q^94 + 2448 * q^95

Introduction

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