LMFDB - Embedded newform 289.4.b.f.288.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [289,4,Mod(288,289)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(289, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("289.288");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 289 = 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	289.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$17.0515519917$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			288.11
Character	$\chi$	$=$	289.288
Dual form			289.4.b.f.288.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.447590 q^{2} +9.51519i q^{3} -7.79966 q^{4} -11.7425i q^{5} -4.25890i q^{6} -2.27797i q^{7} +7.07177 q^{8} -63.5388 q^{9} +O(q^{10})$	\(q-0.447590 q^{2} +9.51519i q^{3} -7.79966 q^{4} -11.7425i q^{5} -4.25890i q^{6} -2.27797i q^{7} +7.07177 q^{8} -63.5388 q^{9} +5.25584i q^{10} -22.6868i q^{11} -74.2152i q^{12} +67.4022 q^{13} +1.01960i q^{14} +111.732 q^{15} +59.2321 q^{16} +28.4393 q^{18} -42.9965 q^{19} +91.5878i q^{20} +21.6753 q^{21} +10.1544i q^{22} -117.444i q^{23} +67.2892i q^{24} -12.8870 q^{25} -30.1686 q^{26} -347.673i q^{27} +17.7674i q^{28} +226.336i q^{29} -50.0103 q^{30} -0.673139i q^{31} -83.0858 q^{32} +215.869 q^{33} -26.7491 q^{35} +495.581 q^{36} -99.9003i q^{37} +19.2448 q^{38} +641.345i q^{39} -83.0405i q^{40} +154.092i q^{41} -9.70165 q^{42} +321.792 q^{43} +176.949i q^{44} +746.106i q^{45} +52.5667i q^{46} -30.4214 q^{47} +563.604i q^{48} +337.811 q^{49} +5.76808 q^{50} -525.715 q^{52} +361.595 q^{53} +155.615i q^{54} -266.400 q^{55} -16.1093i q^{56} -409.119i q^{57} -101.306i q^{58} -147.417 q^{59} -871.475 q^{60} -321.075i q^{61} +0.301290i q^{62} +144.739i q^{63} -436.668 q^{64} -791.473i q^{65} -96.6208 q^{66} +612.509 q^{67} +1117.50 q^{69} +11.9726 q^{70} +248.823i q^{71} -449.331 q^{72} -701.031i q^{73} +44.7144i q^{74} -122.622i q^{75} +335.358 q^{76} -51.6798 q^{77} -287.059i q^{78} -773.040i q^{79} -695.534i q^{80} +1592.63 q^{81} -68.9702i q^{82} +1005.91 q^{83} -169.060 q^{84} -144.031 q^{86} -2153.63 q^{87} -160.436i q^{88} +1641.08 q^{89} -333.949i q^{90} -153.540i q^{91} +916.023i q^{92} +6.40505 q^{93} +13.6163 q^{94} +504.887i q^{95} -790.577i q^{96} -479.663i q^{97} -151.201 q^{98} +1441.49i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) $ 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9	$ 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9} - 144 q^{13} + 276 q^{15} + 384 q^{16} + 378 q^{18} + 132 q^{19} + 492 q^{21} - 888 q^{25} - 1056 q^{26} - 3780 q^{30} + 2706 q^{32} + 1932 q^{33} - 132 q^{35} + 1326 q^{36} - 120 q^{38} - 1608 q^{42} - 972 q^{43} - 1776 q^{47} + 1140 q^{49} + 870 q^{50} + 450 q^{52} - 2052 q^{53} + 1944 q^{55} + 1584 q^{59} - 2916 q^{60} - 3714 q^{64} + 1188 q^{66} + 1248 q^{67} - 3012 q^{69} + 3300 q^{70} + 2910 q^{72} - 1350 q^{76} + 2016 q^{77} + 5040 q^{81} - 1344 q^{83} - 1554 q^{84} + 5556 q^{86} - 1452 q^{87} - 1812 q^{89} - 264 q^{93} - 1470 q^{94} + 3822 q^{98}+O(q^{100}) $ 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9 - 144 * q^13 + 276 * q^15 + 384 * q^16 + 378 * q^18 + 132 * q^19 + 492 * q^21 - 888 * q^25 - 1056 * q^26 - 3780 * q^30 + 2706 * q^32 + 1932 * q^33 - 132 * q^35 + 1326 * q^36 - 120 * q^38 - 1608 * q^42 - 972 * q^43 - 1776 * q^47 + 1140 * q^49 + 870 * q^50 + 450 * q^52 - 2052 * q^53 + 1944 * q^55 + 1584 * q^59 - 2916 * q^60 - 3714 * q^64 + 1188 * q^66 + 1248 * q^67 - 3012 * q^69 + 3300 * q^70 + 2910 * q^72 - 1350 * q^76 + 2016 * q^77 + 5040 * q^81 - 1344 * q^83 - 1554 * q^84 + 5556 * q^86 - 1452 * q^87 - 1812 * q^89 - 264 * q^93 - 1470 * q^94 + 3822 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/289\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.447590	−0.158247	−0.0791235	−	0.996865i	$-0.525212\pi$
$2$	−0.447590	−0.158247	−0.0791235	+	0.996865i	$0.525212\pi$
$3$	9.51519i	1.83120i	0.402092	+	0.915599i	$0.368283\pi$
$3$	9.51519i	1.83120i	−0.402092	+	0.915599i	$0.631717\pi$
$4$	−7.79966	−0.974958
$5$	− 11.7425i	− 1.05028i	−0.851015	−	0.525142i	$-0.824012\pi$
$5$	− 11.7425i	− 1.05028i	0.851015	−	0.525142i	$-0.175988\pi$
$6$	− 4.25890i	− 0.289782i
$7$	− 2.27797i	− 0.122999i	−0.998107	−	0.0614994i	$-0.980412\pi$
$7$	− 2.27797i	− 0.122999i	0.998107	−	0.0614994i	$-0.0195882\pi$
$8$	7.07177	0.312531
$9$	−63.5388	−2.35329
$10$	5.25584i	0.166204i
$11$	− 22.6868i	− 0.621848i	−0.950435	−	0.310924i	$-0.899362\pi$
$11$	− 22.6868i	− 0.621848i	0.950435	−	0.310924i	$-0.100638\pi$
$12$	− 74.2152i	− 1.78534i
$13$	67.4022	1.43800	0.719001	−	0.695009i	$-0.244600\pi$
$13$	67.4022	1.43800	0.719001	+	0.695009i	$0.244600\pi$
$14$	1.01960i	0.0194642i
$15$	111.732	1.92328
$16$	59.2321	0.925501
$17$	0	0
$17$	0	0
$18$	28.4393	0.372400
$19$	−42.9965	−0.519161	−0.259581	−	0.965721i	$-0.583584\pi$
$19$	−42.9965	−0.519161	−0.259581	+	0.965721i	$0.583584\pi$
$20$	91.5878i	1.02398i
$21$	21.6753	0.225235
$22$	10.1544i	0.0984055i
$23$	− 117.444i	− 1.06473i	−0.846516	−	0.532364i	$-0.821304\pi$
$23$	− 117.444i	− 1.06473i	0.846516	−	0.532364i	$-0.178696\pi$
$24$	67.2892i	0.572306i
$25$	−12.8870	−0.103096
$26$	−30.1686	−0.227559
$27$	− 347.673i	− 2.47814i
$28$	17.7674i	0.119919i
$29$	226.336i	1.44930i	0.689120	+	0.724648i	$0.257997\pi$
$29$	226.336i	1.44930i	−0.689120	+	0.724648i	$0.742003\pi$
$30$	−50.0103	−0.304353
$31$	− 0.673139i	− 0.00389998i	−0.999998	−	0.00194999i	$-0.999379\pi$
$31$	− 0.673139i	− 0.00389998i	0.999998	−	0.00194999i	$-0.000620702\pi$
$32$	−83.0858	−0.458989
$33$	215.869	1.13873
$34$	0	0
$35$	−26.7491	−0.129184
$36$	495.581	2.29436
$37$	− 99.9003i	− 0.443878i	−0.975060	−	0.221939i	$-0.928761\pi$
$37$	− 99.9003i	− 0.443878i	0.975060	−	0.221939i	$-0.0712387\pi$
$38$	19.2448	0.0821557
$39$	641.345i	2.63327i
$40$	− 83.0405i	− 0.328246i
$41$	154.092i	0.586956i	0.955966	+	0.293478i	$0.0948128\pi$
$41$	154.092i	0.586956i	−0.955966	+	0.293478i	$0.905187\pi$
$42$	−9.70165	−0.0356428
$43$	321.792	1.14123	0.570615	−	0.821218i	$-0.306705\pi$
$43$	321.792	1.14123	0.570615	+	0.821218i	$0.306705\pi$
$44$	176.949i	0.606275i
$45$	746.106i	2.47162i
$46$	52.5667i	0.168490i
$47$	−30.4214	−0.0944132	−0.0472066	−	0.998885i	$-0.515032\pi$
$47$	−30.4214	−0.0944132	−0.0472066	+	0.998885i	$0.515032\pi$
$48$	563.604i	1.69478i
$49$	337.811	0.984871
$50$	5.76808	0.0163146
$51$	0	0
$52$	−525.715	−1.40199
$53$	361.595	0.937148	0.468574	−	0.883424i	$-0.344768\pi$
$53$	361.595	0.937148	0.468574	+	0.883424i	$0.344768\pi$
$54$	155.615i	0.392158i
$55$	−266.400	−0.653116
$56$	− 16.1093i	− 0.0384410i
$57$	− 409.119i	− 0.950687i
$58$	− 101.306i	− 0.229347i
$59$	−147.417	−0.325289	−0.162645	−	0.986685i	$-0.552002\pi$
$59$	−147.417	−0.325289	−0.162645	+	0.986685i	$0.552002\pi$
$60$	−871.475	−1.87511
$61$	− 321.075i	− 0.673925i	−0.941518	−	0.336963i	$-0.890600\pi$
$61$	− 321.075i	− 0.673925i	0.941518	−	0.336963i	$-0.109400\pi$
$62$	0.301290i	0 0.000617160i
$63$	144.739i	0.289452i
$64$	−436.668	−0.852867
$65$	− 791.473i	− 1.51031i
$66$	−96.6208	−0.180200
$67$	612.509	1.11686	0.558432	−	0.829550i	$-0.311403\pi$
$67$	612.509	1.11686	0.558432	+	0.829550i	$0.311403\pi$
$68$	0	0
$69$	1117.50	1.94973
$70$	11.9726	0.0204429
$71$	248.823i	0.415913i	0.978138	+	0.207956i	$0.0666812\pi$
$71$	248.823i	0.415913i	−0.978138	+	0.207956i	$0.933319\pi$
$72$	−449.331	−0.735475
$73$	− 701.031i	− 1.12397i	−0.827149	−	0.561983i	$-0.810039\pi$
$73$	− 701.031i	− 1.12397i	0.827149	−	0.561983i	$-0.189961\pi$
$74$	44.7144i	0.0702424i
$75$	− 122.622i	− 0.188789i
$76$	335.358	0.506160
$77$	−51.6798	−0.0764865
$78$	− 287.059i	− 0.416706i
$79$	− 773.040i	− 1.10093i	−0.834857	−	0.550467i	$-0.814450\pi$
$79$	− 773.040i	− 1.10093i	0.834857	−	0.550467i	$-0.185550\pi$
$80$	− 695.534i	− 0.972038i
$81$	1592.63	2.18467
$82$	− 68.9702i	− 0.0928840i
$83$	1005.91	1.33028	0.665139	−	0.746720i	$-0.268372\pi$
$83$	1005.91	1.33028	0.665139	+	0.746720i	$0.268372\pi$
$84$	−169.060	−0.219595
$85$	0	0
$86$	−144.031	−0.180596
$87$	−2153.63	−2.65395
$88$	− 160.436i	− 0.194347i
$89$	1641.08	1.95454	0.977271	−	0.211994i	$-0.0679958\pi$
$89$	1641.08	1.95454	0.977271	+	0.211994i	$0.0679958\pi$
$90$	− 333.949i	− 0.391126i
$91$	− 153.540i	− 0.176872i
$92$	916.023i	1.03806i
$93$	6.40505	0.00714164
$94$	13.6163	0.0149406
$95$	504.887i	0.545267i
$96$	− 790.577i	− 0.840499i
$97$	− 479.663i	− 0.502087i	−0.967976	−	0.251043i	$-0.919226\pi$
$97$	− 479.663i	− 0.502087i	0.967976	−	0.251043i	$-0.0807737\pi$
$98$	−151.201	−0.155853
$99$	1441.49i	1.46339i
$100$	100.514	0.100514
$101$	−814.043	−0.801984	−0.400992	−	0.916082i	$-0.631334\pi$
$101$	−814.043	−0.801984	−0.400992	+	0.916082i	$0.631334\pi$
$102$	0	0
$103$	−543.806	−0.520222	−0.260111	−	0.965579i	$-0.583759\pi$
$103$	−543.806	−0.520222	−0.260111	+	0.965579i	$0.583759\pi$
$104$	476.653	0.449420
$105$	− 254.523i	− 0.236561i
$106$	−161.846	−0.148301
$107$	− 171.706i	− 0.155135i	−0.996987	−	0.0775673i	$-0.975285\pi$
$107$	− 171.706i	− 0.155135i	0.996987	−	0.0775673i	$-0.0247153\pi$
$108$	2711.73i	2.41608i
$109$	− 1310.16i	− 1.15129i	−0.817700	−	0.575644i	$-0.804751\pi$
$109$	− 1310.16i	− 1.15129i	0.817700	−	0.575644i	$-0.195249\pi$
$110$	119.238	0.103354
$111$	950.570	0.812830
$112$	− 134.929i	− 0.113836i
$113$	− 879.939i	− 0.732546i	−0.930507	−	0.366273i	$-0.880634\pi$
$113$	− 879.939i	− 0.732546i	0.930507	−	0.366273i	$-0.119366\pi$
$114$	183.118i	0.150443i
$115$	−1379.09	−1.11827
$116$	− 1765.35i	− 1.41300i
$117$	−4282.65	−3.38403
$118$	65.9824	0.0514760
$119$	0	0
$120$	790.145	0.601084
$121$	816.310	0.613306
$122$	143.710i	0.106647i
$123$	−1466.22	−1.07483
$124$	5.25026i	0.00380232i
$125$	− 1316.49i	− 0.942004i
$126$	− 64.7839i	− 0.0458048i
$127$	1303.39	0.910684	0.455342	−	0.890317i	$-0.349517\pi$
$127$	1303.39	0.910684	0.455342	+	0.890317i	$0.349517\pi$
$128$	860.135	0.593952
$129$	3061.91i	2.08982i
$130$	354.255i	0.239002i
$131$	1523.87i	1.01634i	0.861256	+	0.508171i	$0.169678\pi$
$131$	1523.87i	1.01634i	−0.861256	+	0.508171i	$0.830322\pi$
$132$	−1683.71	−1.11021
$133$	97.9447i	0.0638562i
$134$	−274.153	−0.176740
$135$	−4082.56	−2.60275
$136$	0	0
$137$	−2617.59	−1.63238	−0.816189	−	0.577785i	$-0.803917\pi$
$137$	−2617.59	−1.63238	−0.816189	+	0.577785i	$0.803917\pi$
$138$	−500.182	−0.308538
$139$	− 232.950i	− 0.142148i	−0.997471	−	0.0710741i	$-0.977357\pi$
$139$	− 232.950i	− 0.142148i	0.997471	−	0.0710741i	$-0.0226427\pi$
$140$	208.634	0.125949
$141$	− 289.466i	− 0.172889i
$142$	− 111.370i	− 0.0658169i
$143$	− 1529.14i	− 0.894218i
$144$	−3763.53	−2.17797
$145$	2657.76	1.52217
$146$	313.774i	0.177864i
$147$	3214.33i	1.80349i
$148$	779.189i	0.432763i
$149$	429.754	0.236287	0.118144	−	0.992997i	$-0.462306\pi$
$149$	429.754	0.236287	0.118144	+	0.992997i	$0.462306\pi$
$150$	54.8844i	0.0298753i
$151$	1373.00	0.739953	0.369976	−	0.929041i	$-0.379366\pi$
$151$	1373.00	0.739953	0.369976	+	0.929041i	$0.379366\pi$
$152$	−304.061	−0.162254
$153$	0	0
$154$	23.1314	0.0121038
$155$	−7.90436	−0.00409609
$156$	− 5002.27i	− 2.56732i
$157$	616.059	0.313165	0.156582	−	0.987665i	$-0.449952\pi$
$157$	616.059	0.313165	0.156582	+	0.987665i	$0.449952\pi$
$158$	346.005i	0.174220i
$159$	3440.64i	1.71610i
$160$	975.638i	0.482068i
$161$	−267.534	−0.130960
$162$	−712.844	−0.345718
$163$	3460.81i	1.66301i	0.555515	+	0.831507i	$0.312521\pi$
$163$	3460.81i	1.66301i	−0.555515	+	0.831507i	$0.687479\pi$
$164$	− 1201.87i	− 0.572257i
$165$	− 2534.85i	− 1.19599i
$166$	−450.235	−0.210512
$167$	− 2424.76i	− 1.12355i	−0.827288	−	0.561777i	$-0.810118\pi$
$167$	− 2424.76i	− 1.12355i	0.827288	−	0.561777i	$-0.189882\pi$
$168$	153.283	0.0703930
$169$	2346.06	1.06785
$170$	0	0
$171$	2731.94	1.22174
$172$	−2509.87	−1.11265
$173$	2853.14i	1.25388i	0.779069	+	0.626938i	$0.215692\pi$
$173$	2853.14i	1.25388i	−0.779069	+	0.626938i	$0.784308\pi$
$174$	963.943	0.419979
$175$	29.3562i	0.0126807i
$176$	− 1343.79i	− 0.575521i
$177$	− 1402.70i	− 0.595669i
$178$	−734.531	−0.309300
$179$	−1195.13	−0.499040	−0.249520	−	0.968370i	$-0.580273\pi$
$179$	−1195.13	−0.499040	−0.249520	+	0.968370i	$0.580273\pi$
$180$	− 5819.37i	− 2.40972i
$181$	− 2669.80i	− 1.09638i	−0.836354	−	0.548190i	$-0.815317\pi$
$181$	− 2669.80i	− 1.09638i	0.836354	−	0.548190i	$-0.184683\pi$
$182$	68.7231i	0.0279895i
$183$	3055.09	1.23409
$184$	− 830.536i	− 0.332760i
$185$	−1173.08	−0.466198
$186$	−2.86683	−0.00113014
$187$	0	0
$188$	237.277	0.0920489
$189$	−791.989	−0.304808
$190$	− 225.982i	− 0.0862868i
$191$	−3394.56	−1.28598	−0.642988	−	0.765876i	$-0.722306\pi$
$191$	−3394.56	−1.28598	−0.642988	+	0.765876i	$0.722306\pi$
$192$	− 4154.98i	− 1.56177i
$193$	− 3003.88i	− 1.12033i	−0.828381	−	0.560165i	$-0.810738\pi$
$193$	− 3003.88i	− 1.12033i	0.828381	−	0.560165i	$-0.189262\pi$
$194$	214.692i	0.0794537i
$195$	7531.01	2.76568
$196$	−2634.81	−0.960208
$197$	− 1636.63i	− 0.591902i	−0.955203	−	0.295951i	$-0.904363\pi$
$197$	− 1636.63i	− 0.591902i	0.955203	−	0.295951i	$-0.0956365\pi$
$198$	− 645.197i	− 0.231576i
$199$	1600.54i	0.570146i	0.958506	+	0.285073i	$0.0920179\pi$
$199$	1600.54i	0.570146i	−0.958506	+	0.285073i	$0.907982\pi$
$200$	−91.1337	−0.0322206
$201$	5828.14i	2.04520i
$202$	364.358	0.126911
$203$	515.587	0.178262
$204$	0	0
$205$	1809.44	0.616470
$206$	243.402	0.0823235
$207$	7462.24i	2.50561i
$208$	3992.37	1.33087
$209$	975.452i	0.322839i
$210$	113.922i	0.0374350i
$211$	4282.36i	1.39720i	0.715510	+	0.698602i	$0.246194\pi$
$211$	4282.36i	1.39720i	−0.715510	+	0.698602i	$0.753806\pi$
$212$	−2820.32	−0.913680
$213$	−2367.59	−0.761619
$214$	76.8537i	0.0245496i
$215$	− 3778.65i	− 1.19861i
$216$	− 2458.66i	− 0.774495i
$217$	−1.53339	−0.000479693 0
$218$	586.414i	0.182188i
$219$	6670.44	2.05820
$220$	2077.83	0.636761
$221$	0	0
$222$	−425.465	−0.128628
$223$	1396.84	0.419460	0.209730	−	0.977759i	$-0.432742\pi$
$223$	1396.84	0.419460	0.209730	+	0.977759i	$0.432742\pi$
$224$	189.267i	0.0564551i
$225$	818.822	0.242614
$226$	393.852i	0.115923i
$227$	1612.43i	0.471458i	0.971819	+	0.235729i	$0.0757477\pi$
$227$	1612.43i	0.471458i	−0.971819	+	0.235729i	$0.924252\pi$
$228$	3190.99i	0.926880i
$229$	2530.45	0.730204	0.365102	−	0.930968i	$-0.381034\pi$
$229$	2530.45	0.730204	0.365102	+	0.930968i	$0.381034\pi$
$230$	617.266	0.176962
$231$	− 491.743i	− 0.140062i
$232$	1600.60i	0.452950i
$233$	− 4689.59i	− 1.31856i	−0.751896	−	0.659282i	$-0.770860\pi$
$233$	− 4689.59i	− 1.31856i	0.751896	−	0.659282i	$-0.229140\pi$
$234$	1916.87	0.535512
$235$	357.225i	0.0991607i
$236$	1149.80	0.317143
$237$	7355.62	2.01603
$238$	0	0
$239$	−6332.33	−1.71383	−0.856913	−	0.515462i	$-0.827620\pi$
$239$	−6332.33	−1.71383	−0.856913	+	0.515462i	$0.827620\pi$
$240$	6618.14	1.78000
$241$	− 4438.39i	− 1.18631i	−0.805087	−	0.593157i	$-0.797881\pi$
$241$	− 4438.39i	− 1.18631i	0.805087	−	0.593157i	$-0.202119\pi$
$242$	−365.372	−0.0970537
$243$	5766.97i	1.52243i
$244$	2504.28i	0.657049i
$245$	− 3966.75i	− 1.03439i
$246$	656.265	0.170089
$247$	−2898.06	−0.746555
$248$	− 4.76029i	− 0.00121886i
$249$	9571.43i	2.43600i
$250$	589.248i	0.149069i
$251$	2085.26	0.524385	0.262193	−	0.965016i	$-0.415554\pi$
$251$	2085.26	0.524385	0.262193	+	0.965016i	$0.415554\pi$
$252$	− 1128.92i	− 0.282203i
$253$	−2664.42	−0.662098
$254$	−583.383	−0.144113
$255$	0	0
$256$	3108.36	0.758876
$257$	537.124	0.130369	0.0651846	−	0.997873i	$-0.479236\pi$
$257$	537.124	0.130369	0.0651846	+	0.997873i	$0.479236\pi$
$258$	− 1370.48i	− 0.330707i
$259$	−227.570	−0.0545965
$260$	6173.22i	1.47249i
$261$	− 14381.1i	− 3.41061i
$262$	− 682.068i	− 0.160833i
$263$	−6617.18	−1.55146	−0.775728	−	0.631068i	$-0.782617\pi$
$263$	−6617.18	−1.55146	−0.775728	+	0.631068i	$0.782617\pi$
$264$	1526.58	0.355887
$265$	− 4246.03i	− 0.984271i
$266$	− 43.8390i	− 0.0101051i
$267$	15615.2i	3.57915i
$268$	−4777.36	−1.08890
$269$	− 617.429i	− 0.139945i	−0.997549	−	0.0699726i	$-0.977709\pi$
$269$	− 617.429i	− 0.139945i	0.997549	−	0.0699726i	$-0.0222912\pi$
$270$	1827.31	0.411877
$271$	−5332.00	−1.19519	−0.597594	−	0.801799i	$-0.703877\pi$
$271$	−5332.00	−1.19519	−0.597594	+	0.801799i	$0.703877\pi$
$272$	0	0
$273$	1460.96	0.323889
$274$	1171.61	0.258319
$275$	292.364i	0.0641099i
$276$	−8716.12	−1.90090
$277$	− 2486.63i	− 0.539376i	−0.962948	−	0.269688i	$-0.913079\pi$
$277$	− 2486.63i	− 0.539376i	0.962948	−	0.269688i	$-0.0869205\pi$
$278$	104.266i	0.0224945i
$279$	42.7704i	0.00917777i
$280$	−189.164	−0.0403739
$281$	4951.96	1.05128	0.525639	−	0.850708i	$-0.323826\pi$
$281$	4951.96	1.05128	0.525639	+	0.850708i	$0.323826\pi$
$282$	129.562i	0.0273592i
$283$	8216.86i	1.72594i	0.505253	+	0.862971i	$0.331399\pi$
$283$	8216.86i	1.72594i	−0.505253	+	0.862971i	$0.668601\pi$
$284$	− 1940.73i	− 0.405498i
$285$	−4804.10	−0.998491
$286$	684.428i	0.141507i
$287$	351.018	0.0721949
$288$	5279.17	1.08013
$289$	0	0
$290$	−1189.59	−0.240879
$291$	4564.08	0.919420
$292$	5467.81i	1.09582i
$293$	−737.775	−0.147103	−0.0735516	−	0.997291i	$-0.523433\pi$
$293$	−737.775	−0.147103	−0.0735516	+	0.997291i	$0.523433\pi$
$294$	− 1438.70i	− 0.285397i
$295$	1731.05i	0.341646i
$296$	− 706.472i	− 0.138726i
$297$	−7887.59	−1.54102
$298$	−192.354	−0.0373917
$299$	− 7915.98i	− 1.53108i
$300$	956.410i	0.184061i
$301$	− 733.033i	− 0.140370i
$302$	−614.539	−0.117095
$303$	− 7745.77i	− 1.46859i
$304$	−2546.77	−0.480484
$305$	−3770.23	−0.707813
$306$	0	0
$307$	5994.47	1.11441	0.557203	−	0.830377i	$-0.311875\pi$
$307$	5994.47	1.11441	0.557203	+	0.830377i	$0.311875\pi$
$308$	403.085	0.0745712
$309$	− 5174.42i	− 0.952629i
$310$	3.53791	0.000648193 0
$311$	9460.76i	1.72498i	0.506070	+	0.862492i	$0.331098\pi$
$311$	9460.76i	1.72498i	−0.506070	+	0.862492i	$0.668902\pi$
$312$	4535.44i	0.822977i
$313$	4674.92i	0.844225i	0.906544	+	0.422112i	$0.138711\pi$
$313$	4674.92i	0.844225i	−0.906544	+	0.422112i	$0.861289\pi$
$314$	−275.742	−0.0495574
$315$	1699.61	0.304006
$316$	6029.45i	1.07336i
$317$	− 692.929i	− 0.122772i	−0.998114	−	0.0613861i	$-0.980448\pi$
$317$	− 692.929i	− 0.122772i	0.998114	−	0.0613861i	$-0.0195521\pi$
$318$	− 1540.00i	− 0.271568i
$319$	5134.84	0.901241
$320$	5127.59i	0.895753i
$321$	1633.81	0.284082
$322$	119.745	0.0207241
$323$	0	0
$324$	−12422.0	−2.12997
$325$	−868.611	−0.148252
$326$	− 1549.02i	− 0.263167i
$327$	12466.4	2.10824
$328$	1089.71i	0.183442i
$329$	69.2991i	0.0116127i
$330$	1134.57i	0.189261i
$331$	7067.34	1.17358	0.586792	−	0.809738i	$-0.300391\pi$
$331$	7067.34	1.17358	0.586792	+	0.809738i	$0.300391\pi$
$332$	−7845.76	−1.29696
$333$	6347.54i	1.04457i
$334$	1085.30i	0.177799i
$335$	− 7192.40i	− 1.17302i
$336$	1283.87	0.208455
$337$	5897.11i	0.953223i	0.879114	+	0.476611i	$0.158135\pi$
$337$	5897.11i	0.953223i	−0.879114	+	0.476611i	$0.841865\pi$
$338$	−1050.07	−0.168984
$339$	8372.78	1.34144
$340$	0	0
$341$	−15.2714	−0.00242519
$342$	−1222.79	−0.193336
$343$	− 1550.87i	− 0.244137i
$344$	2275.64	0.356669
$345$	− 13122.3i	− 2.04777i
$346$	− 1277.04i	− 0.198422i
$347$	− 9529.27i	− 1.47423i	−0.675767	−	0.737115i	$-0.736188\pi$
$347$	− 9529.27i	− 1.47423i	0.675767	−	0.737115i	$-0.263812\pi$
$348$	16797.6	2.58749
$349$	2516.87	0.386031	0.193016	−	0.981196i	$-0.438173\pi$
$349$	2516.87	0.386031	0.193016	+	0.981196i	$0.438173\pi$
$350$	− 13.1395i	− 0.00200668i
$351$	− 23433.9i	− 3.56356i
$352$	1884.95i	0.285421i
$353$	9296.21	1.40166	0.700832	−	0.713327i	$-0.252812\pi$
$353$	9296.21	1.40166	0.700832	+	0.713327i	$0.252812\pi$
$354$	627.834i	0.0942628i
$355$	2921.81	0.436826
$356$	−12799.9	−1.90560
$357$	0	0
$358$	534.928	0.0789716
$359$	−10477.8	−1.54038	−0.770192	−	0.637812i	$-0.779840\pi$
$359$	−10477.8	−1.54038	−0.770192	+	0.637812i	$0.779840\pi$
$360$	5276.29i	0.772458i
$361$	−5010.30	−0.730472
$362$	1194.98i	0.173499i
$363$	7767.34i	1.12308i
$364$	1197.56i	0.172443i
$365$	−8231.88	−1.18048
$366$	−1367.43	−0.195291
$367$	− 2779.47i	− 0.395332i	−0.980269	−	0.197666i	$-0.936664\pi$
$367$	− 2779.47i	− 0.395332i	0.980269	−	0.197666i	$-0.0633362\pi$
$368$	− 6956.44i	− 0.985406i
$369$	− 9790.84i	− 1.38128i
$370$	525.060	0.0737745
$371$	− 823.702i	− 0.115268i
$372$	−49.9572	−0.00696280
$373$	2548.58	0.353782	0.176891	−	0.984230i	$-0.443396\pi$
$373$	2548.58	0.353782	0.176891	+	0.984230i	$0.443396\pi$
$374$	0	0
$375$	12526.7	1.72500
$376$	−215.133	−0.0295071
$377$	15255.6i	2.08409i
$378$	354.486	0.0482349
$379$	− 11378.4i	− 1.54213i	−0.636754	−	0.771067i	$-0.719723\pi$
$379$	− 11378.4i	− 1.54213i	0.636754	−	0.771067i	$-0.280277\pi$
$380$	− 3937.95i	− 0.531612i
$381$	12402.0i	1.66764i
$382$	1519.37	0.203502
$383$	633.266	0.0844867	0.0422433	−	0.999107i	$-0.486550\pi$
$383$	633.266	0.0844867	0.0422433	+	0.999107i	$0.486550\pi$
$384$	8184.34i	1.08764i
$385$	606.852i	0.0803326i
$386$	1344.50i	0.177289i
$387$	−20446.3	−2.68564
$388$	3741.21i	0.489513i
$389$	−12163.4	−1.58537	−0.792685	−	0.609632i	$-0.791317\pi$
$389$	−12163.4	−1.58537	−0.792685	+	0.609632i	$0.791317\pi$
$390$	−3370.80	−0.437660
$391$	0	0
$392$	2388.92	0.307803
$393$	−14499.9	−1.86113
$394$	732.537i	0.0936667i
$395$	−9077.45	−1.15629
$396$	− 11243.1i	− 1.42674i
$397$	11911.8i	1.50589i	0.658084	+	0.752944i	$0.271367\pi$
$397$	11911.8i	1.50589i	−0.658084	+	0.752944i	$0.728633\pi$
$398$	− 716.384i	− 0.0902239i
$399$	−931.962	−0.116933
$400$	−763.322	−0.0954153
$401$	− 7875.79i	− 0.980793i	−0.871499	−	0.490396i	$-0.836852\pi$
$401$	− 7875.79i	− 0.980793i	0.871499	−	0.490396i	$-0.163148\pi$
$402$	− 2608.62i	− 0.323647i
$403$	− 45.3711i	− 0.00560818i
$404$	6349.26	0.781900
$405$	− 18701.5i	− 2.29453i
$406$	−230.772	−0.0282094
$407$	−2266.42	−0.276025
$408$	0	0
$409$	−4351.34	−0.526064	−0.263032	−	0.964787i	$-0.584722\pi$
$409$	−4351.34	−0.526064	−0.263032	+	0.964787i	$0.584722\pi$
$410$	−809.885	−0.0975545
$411$	− 24906.9i	− 2.98921i
$412$	4241.51	0.507194
$413$	335.812i	0.0400102i
$414$	− 3340.02i	− 0.396505i
$415$	− 11811.9i	− 1.39717i
$416$	−5600.17	−0.660026
$417$	2216.57	0.260301
$418$	− 436.602i	− 0.0510883i
$419$	2066.20i	0.240908i	0.992719	+	0.120454i	$0.0384351\pi$
$419$	2066.20i	0.240908i	−0.992719	+	0.120454i	$0.961565\pi$
$420$	1985.19i	0.230637i
$421$	10563.5	1.22289	0.611443	−	0.791289i	$-0.290589\pi$
$421$	10563.5	1.22289	0.611443	+	0.791289i	$0.290589\pi$
$422$	− 1916.74i	− 0.221103i
$423$	1932.94	0.222181
$424$	2557.11	0.292888
$425$	0	0
$426$	1059.71	0.120524
$427$	−731.399	−0.0828920
$428$	1339.25i	0.151250i
$429$	14550.1	1.63749
$430$	1691.29i	0.189677i
$431$	4347.98i	0.485928i	0.970035	+	0.242964i	$0.0781197\pi$
$431$	4347.98i	0.485928i	−0.970035	+	0.242964i	$0.921880\pi$
$432$	− 20593.4i	− 2.29352i
$433$	−949.003	−0.105326	−0.0526630	−	0.998612i	$-0.516771\pi$
$433$	−949.003	−0.105326	−0.0526630	+	0.998612i	$0.516771\pi$
$434$	0.686331	7.59100e−5 0
$435$	25289.1i	2.78740i
$436$	10218.8i	1.12246i
$437$	5049.67i	0.552765i
$438$	−2985.62	−0.325704
$439$	7256.78i	0.788946i	0.918908	+	0.394473i	$0.129073\pi$
$439$	7256.78i	0.788946i	−0.918908	+	0.394473i	$0.870927\pi$
$440$	−1883.92	−0.204119
$441$	−21464.1	−2.31769
$442$	0	0
$443$	−3674.08	−0.394043	−0.197021	−	0.980399i	$-0.563127\pi$
$443$	−3674.08	−0.394043	−0.197021	+	0.980399i	$0.563127\pi$
$444$	−7414.12	−0.792475
$445$	− 19270.4i	− 2.05282i
$446$	−625.213	−0.0663782
$447$	4089.19i	0.432689i
$448$	994.717i	0.104902i
$449$	13012.3i	1.36769i	0.729630	+	0.683843i	$0.239693\pi$
$449$	13012.3i	1.36769i	−0.729630	+	0.683843i	$0.760307\pi$
$450$	−366.497	−0.0383929
$451$	3495.86	0.364997
$452$	6863.23i	0.714201i
$453$	13064.3i	1.35500i
$454$	− 721.708i	− 0.0746067i
$455$	−1802.95	−0.185766
$456$	− 2893.20i	− 0.297119i
$457$	13.1283	0.00134380	0.000671900	−	1.00000i	$-0.499786\pi$
$457$	13.1283	0.00134380	0.000671900		1.00000i	$0.499786\pi$
$458$	−1132.60	−0.115553
$459$	0	0
$460$	10756.4	1.09026
$461$	14382.6	1.45307	0.726537	−	0.687128i	$-0.241129\pi$
$461$	14382.6	1.45307	0.726537	+	0.687128i	$0.241129\pi$
$462$	220.099i	0.0221644i
$463$	−13264.3	−1.33141	−0.665705	−	0.746215i	$-0.731869\pi$
$463$	−13264.3	−1.33141	−0.665705	+	0.746215i	$0.731869\pi$
$464$	13406.4i	1.34132i
$465$	− 75.2114i	− 0.00750075i
$466$	2099.01i	0.208659i
$467$	−12992.0	−1.28736	−0.643681	−	0.765294i	$-0.722594\pi$
$467$	−12992.0	−1.28736	−0.643681	+	0.765294i	$0.722594\pi$
$468$	33403.3	3.29929
$469$	− 1395.28i	− 0.137373i
$470$	− 159.890i	− 0.0156919i
$471$	5861.92i	0.573467i
$472$	−1042.50	−0.101663
$473$	− 7300.43i	− 0.709671i
$474$	−3292.30	−0.319030
$475$	554.094	0.0535234
$476$	0	0
$477$	−22975.3	−2.20538
$478$	2834.29	0.271208
$479$	8697.74i	0.829665i	0.909898	+	0.414833i	$0.136160\pi$
$479$	8697.74i	0.829665i	−0.909898	+	0.414833i	$0.863840\pi$
$480$	−9283.37	−0.882763
$481$	− 6733.50i	− 0.638298i
$482$	1986.58i	0.187730i
$483$	− 2545.63i	− 0.239814i
$484$	−6366.94	−0.597947
$485$	−5632.46	−0.527333
$486$	− 2581.24i	− 0.240920i
$487$	− 1219.66i	− 0.113487i	−0.998389	−	0.0567435i	$-0.981928\pi$
$487$	− 1219.66i	− 0.113487i	0.998389	−	0.0567435i	$-0.0180717\pi$
$488$	− 2270.57i	− 0.210623i
$489$	−32930.2	−3.04531
$490$	1775.48i	0.163690i
$491$	−11340.1	−1.04231	−0.521154	−	0.853463i	$-0.674498\pi$
$491$	−11340.1	−1.04231	−0.521154	+	0.853463i	$0.674498\pi$
$492$	11436.0	1.04792
$493$	0	0
$494$	1297.14	0.118140
$495$	16926.7	1.53697
$496$	− 39.8714i	− 0.00360944i
$497$	566.811	0.0511568
$498$	− 4284.07i	− 0.385490i
$499$	7357.73i	0.660074i	0.943968	+	0.330037i	$0.107061\pi$
$499$	7357.73i	0.660074i	−0.943968	+	0.330037i	$0.892939\pi$
$500$	10268.2i	0.918414i
$501$	23072.1	2.05745
$502$	−933.343	−0.0829824
$503$	− 4549.92i	− 0.403321i	−0.979455	−	0.201661i	$-0.935366\pi$
$503$	− 4549.92i	− 0.403321i	0.979455	−	0.201661i	$-0.0646338\pi$
$504$	1023.56i	0.0904626i
$505$	9558.93i	0.842310i
$506$	1192.57	0.104775
$507$	22323.2i	1.95544i
$508$	−10166.0	−0.887879
$509$	−13630.1	−1.18692	−0.593461	−	0.804863i	$-0.702239\pi$
$509$	−13630.1	−1.18692	−0.593461	+	0.804863i	$0.702239\pi$
$510$	0	0
$511$	−1596.93	−0.138246
$512$	−8272.35	−0.714042
$513$	14948.7i	1.28655i
$514$	−240.411	−0.0206305
$515$	6385.66i	0.546380i
$516$	− 23881.9i	− 2.03748i
$517$	690.165i	0.0587106i
$518$	101.858	0.00863974
$519$	−27148.2	−2.29609
$520$	− 5597.11i	− 0.472018i
$521$	− 14004.2i	− 1.17761i	−0.808275	−	0.588805i	$-0.799598\pi$
$521$	− 14004.2i	− 1.17761i	0.808275	−	0.588805i	$-0.200402\pi$
$522$	6436.84i	0.539718i
$523$	−5652.21	−0.472570	−0.236285	−	0.971684i	$-0.575930\pi$
$523$	−5652.21	−0.472570	−0.236285	+	0.971684i	$0.575930\pi$
$524$	− 11885.7i	− 0.990891i
$525$	−279.329	−0.0232208
$526$	2961.78	0.245513
$527$	0	0
$528$	12786.4	1.05389
$529$	−1626.06	−0.133645
$530$	1900.48i	0.155758i
$531$	9366.69	0.765499
$532$	− 763.935i	− 0.0622571i
$533$	10386.2i	0.844044i
$534$	− 6989.20i	− 0.566390i
$535$	−2016.26	−0.162935
$536$	4331.52	0.349055
$537$	− 11371.9i	− 0.913842i
$538$	276.355i	0.0221459i
$539$	− 7663.84i	− 0.612440i
$540$	31842.6	2.53757
$541$	8809.75i	0.700112i	0.936729	+	0.350056i	$0.113837\pi$
$541$	8809.75i	0.700112i	−0.936729	+	0.350056i	$0.886163\pi$
$542$	2386.55	0.189135
$543$	25403.7	2.00769
$544$	0	0
$545$	−15384.6	−1.20918
$546$	−653.913	−0.0512544
$547$	2198.83i	0.171874i	0.996301	+	0.0859372i	$0.0273884\pi$
$547$	2198.83i	0.171874i	−0.996301	+	0.0859372i	$0.972612\pi$
$548$	20416.3	1.59150
$549$	20400.7i	1.58594i
$550$	− 130.859i	− 0.0101452i
$551$	− 9731.65i	− 0.752418i
$552$	7902.70	0.609350
$553$	−1760.96	−0.135414
$554$	1112.99i	0.0853546i
$555$	− 11162.1i	− 0.853702i
$556$	1816.93i	0.138588i
$557$	19799.3	1.50614	0.753072	−	0.657938i	$-0.228571\pi$
$557$	19799.3	1.50614	0.753072	+	0.657938i	$0.228571\pi$
$558$	− 19.1436i	− 0.00145235i
$559$	21689.5	1.64109
$560$	−1584.41	−0.119560
$561$	0	0
$562$	−2216.45	−0.166362
$563$	20671.9	1.54745	0.773726	−	0.633521i	$-0.218391\pi$
$563$	20671.9	1.54745	0.773726	+	0.633521i	$0.218391\pi$
$564$	2257.73i	0.168560i
$565$	−10332.7	−0.769381
$566$	− 3677.78i	− 0.273125i
$567$	− 3627.96i	− 0.268712i
$568$	1759.62i	0.129986i
$569$	10735.1	0.790928	0.395464	−	0.918481i	$-0.370584\pi$
$569$	10735.1	0.790928	0.395464	+	0.918481i	$0.370584\pi$
$570$	2150.26	0.158008
$571$	− 1413.36i	− 0.103586i	−0.998658	−	0.0517929i	$-0.983506\pi$
$571$	− 1413.36i	− 0.103586i	0.998658	−	0.0517929i	$-0.0164936\pi$
$572$	11926.8i	0.871825i
$573$	− 32299.8i	− 2.35488i
$574$	−157.112	−0.0114246
$575$	1513.50i	0.109769i
$576$	27745.3	2.00704
$577$	13903.6	1.00314	0.501572	−	0.865116i	$-0.332755\pi$
$577$	13903.6	1.00314	0.501572	+	0.865116i	$0.332755\pi$
$578$	0	0
$579$	28582.4	2.05155
$580$	−20729.6	−1.48405
$581$	− 2291.43i	− 0.163623i
$582$	−2042.84	−0.145495
$583$	− 8203.42i	− 0.582763i
$584$	− 4957.53i	− 0.351274i
$585$	50289.2i	3.55419i
$586$	330.220	0.0232786
$587$	−7153.64	−0.503002	−0.251501	−	0.967857i	$-0.580924\pi$
$587$	−7153.64	−0.503002	−0.251501	+	0.967857i	$0.580924\pi$
$588$	− 25070.7i	− 1.75833i
$589$	28.9426i	0.00202472i
$590$	− 774.800i	− 0.0540644i
$591$	15572.8	1.08389
$592$	− 5917.30i	− 0.410810i
$593$	22686.9	1.57106	0.785530	−	0.618824i	$-0.212390\pi$
$593$	22686.9	1.57106	0.785530	+	0.618824i	$0.212390\pi$
$594$	3530.40	0.243862
$595$	0	0
$596$	−3351.94	−0.230370
$597$	−15229.4	−1.04405
$598$	3543.11i	0.242289i
$599$	9190.22	0.626882	0.313441	−	0.949608i	$-0.398518\pi$
$599$	9190.22	0.626882	0.313441	+	0.949608i	$0.398518\pi$
$600$	− 867.154i	− 0.0590024i
$601$	− 12350.7i	− 0.838260i	−0.907926	−	0.419130i	$-0.862335\pi$
$601$	− 12350.7i	− 0.838260i	0.907926	−	0.419130i	$-0.137665\pi$
$602$	328.098i	0.0222131i
$603$	−38918.1	−2.62830
$604$	−10708.9	−0.721423
$605$	− 9585.54i	− 0.644145i
$606$	3466.93i	0.232400i
$607$	− 22662.6i	− 1.51540i	−0.652606	−	0.757698i	$-0.726324\pi$
$607$	− 22662.6i	− 1.51540i	0.652606	−	0.757698i	$-0.273676\pi$
$608$	3572.40	0.238289
$609$	4905.91i	0.326432i
$610$	1687.52	0.112009
$611$	−2050.47	−0.135766
$612$	0	0
$613$	5314.94	0.350193	0.175097	−	0.984551i	$-0.443976\pi$
$613$	5314.94	0.350193	0.175097	+	0.984551i	$0.443976\pi$
$614$	−2683.06	−0.176351
$615$	17217.1i	1.12888i
$616$	−365.468	−0.0239044
$617$	− 7956.05i	− 0.519122i	−0.965727	−	0.259561i	$-0.916422\pi$
$617$	− 7956.05i	− 0.519122i	0.965727	−	0.259561i	$-0.0835779\pi$
$618$	2316.02i	0.150751i
$619$	− 13874.9i	− 0.900933i	−0.892793	−	0.450467i	$-0.851258\pi$
$619$	− 13874.9i	− 0.900933i	0.892793	−	0.450467i	$-0.148742\pi$
$620$	61.6513	0.00399351
$621$	−40832.1	−2.63854
$622$	− 4234.54i	− 0.272974i
$623$	− 3738.33i	− 0.240406i
$624$	37988.2i	2.43709i
$625$	−17069.8	−1.09247
$626$	− 2092.45i	− 0.133596i
$627$	−9281.60	−0.591183
$628$	−4805.06	−0.305323
$629$	0	0
$630$	−760.727	−0.0481081
$631$	23668.0	1.49320	0.746598	−	0.665276i	$-0.231686\pi$
$631$	23668.0	1.49320	0.746598	+	0.665276i	$0.231686\pi$
$632$	− 5466.76i	− 0.344076i
$633$	−40747.5	−2.55856
$634$	310.148i	0.0194283i
$635$	− 15305.1i	− 0.956477i
$636$	− 26835.8i	− 1.67313i
$637$	22769.2	1.41625
$638$	−2298.30	−0.142619
$639$	− 15809.9i	− 0.978762i
$640$	− 10100.2i	− 0.623818i
$641$	18274.3i	1.12604i	0.826442	+	0.563021i	$0.190361\pi$
$641$	18274.3i	1.12604i	−0.826442	+	0.563021i	$0.809639\pi$
$642$	−731.277	−0.0449552
$643$	− 19527.2i	− 1.19763i	−0.800887	−	0.598815i	$-0.795639\pi$
$643$	− 19527.2i	− 1.19763i	0.800887	−	0.598815i	$-0.204361\pi$
$644$	2086.67	0.127681
$645$	35954.6	2.19490
$646$	0	0
$647$	−4471.27	−0.271690	−0.135845	−	0.990730i	$-0.543375\pi$
$647$	−4471.27	−0.271690	−0.135845	+	0.990730i	$0.543375\pi$
$648$	11262.7	0.682778
$649$	3344.42i	0.202280i
$650$	388.781	0.0234604
$651$	− 14.5905i	0 0.000878413i
$652$	− 26993.1i	− 1.62137i
$653$	− 10869.7i	− 0.651399i	−0.945473	−	0.325699i	$-0.894400\pi$
$653$	− 10869.7i	− 0.651399i	0.945473	−	0.325699i	$-0.105600\pi$
$654$	−5579.84	−0.333622
$655$	17894.1	1.06745
$656$	9127.21i	0.543228i
$657$	44542.6i	2.64501i
$658$	− 31.0176i	− 0.00183768i
$659$	−3827.54	−0.226252	−0.113126	−	0.993581i	$-0.536086\pi$
$659$	−3827.54	−0.226252	−0.113126	+	0.993581i	$0.536086\pi$
$660$	19771.0i	1.16604i
$661$	−15347.3	−0.903087	−0.451543	−	0.892249i	$-0.649126\pi$
$661$	−15347.3	−0.903087	−0.451543	+	0.892249i	$0.649126\pi$
$662$	−3163.27	−0.185716
$663$	0	0
$664$	7113.57	0.415753
$665$	1150.12	0.0670672
$666$	− 2841.09i	− 0.165301i
$667$	26581.8	1.54310
$668$	18912.3i	1.09542i
$669$	13291.2i	0.768114i
$670$	3219.25i	0.185627i
$671$	−7284.16	−0.419079
$672$	−1800.91	−0.103380
$673$	26133.5i	1.49684i	0.663227	+	0.748418i	$0.269186\pi$
$673$	26133.5i	1.49684i	−0.663227	+	0.748418i	$0.730814\pi$
$674$	− 2639.49i	− 0.150845i
$675$	4480.45i	0.255486i
$676$	−18298.5	−1.04111
$677$	− 19539.0i	− 1.10922i	−0.832109	−	0.554612i	$-0.812867\pi$
$677$	− 19539.0i	− 1.10922i	0.832109	−	0.554612i	$-0.187133\pi$
$678$	−3747.57	−0.212278
$679$	−1092.66	−0.0617561
$680$	0	0
$681$	−15342.6	−0.863332
$682$	6.83531	0.000383779 0
$683$	16057.8i	0.899612i	0.893126	+	0.449806i	$0.148507\pi$
$683$	16057.8i	0.899612i	−0.893126	+	0.449806i	$0.851493\pi$
$684$	−21308.2	−1.19114
$685$	30737.1i	1.71446i
$686$	694.152i	0.0386339i
$687$	24077.7i	1.33715i
$688$	19060.4	1.05621
$689$	24372.3	1.34762
$690$	5873.40i	0.324053i
$691$	14698.6i	0.809206i	0.914493	+	0.404603i	$0.132590\pi$
$691$	14698.6i	0.809206i	−0.914493	+	0.404603i	$0.867410\pi$
$692$	− 22253.6i	− 1.22248i
$693$	3283.67	0.179995
$694$	4265.20i	0.233292i
$695$	−2735.43	−0.149296
$696$	−15230.0	−0.829441
$697$	0	0
$698$	−1126.53	−0.0610883
$699$	44622.3	2.41455
$700$	− 228.968i	− 0.0123631i
$701$	−7042.75	−0.379459	−0.189730	−	0.981836i	$-0.560761\pi$
$701$	−7042.75	−0.379459	−0.189730	+	0.981836i	$0.560761\pi$
$702$	10488.8i	0.563923i
$703$	4295.36i	0.230445i
$704$	9906.60i	0.530354i
$705$	−3399.06	−0.181583
$706$	−4160.89	−0.221809
$707$	1854.37i	0.0986431i
$708$	10940.6i	0.580752i
$709$	5951.66i	0.315260i	0.987498	+	0.157630i	$0.0503853\pi$
$709$	5951.66i	0.315260i	−0.987498	+	0.157630i	$0.949615\pi$
$710$	−1307.77	−0.0691265
$711$	49118.0i	2.59082i
$712$	11605.3	0.610855
$713$	−79.0561	−0.00415242
$714$	0	0
$715$	−17956.0	−0.939182
$716$	9321.61	0.486543
$717$	− 60253.3i	− 3.13835i
$718$	4689.76	0.243761
$719$	− 34823.3i	− 1.80625i	−0.429381	−	0.903123i	$-0.641268\pi$
$719$	− 34823.3i	− 1.80625i	0.429381	−	0.903123i	$-0.358732\pi$
$720$	44193.4i	2.28749i
$721$	1238.77i	0.0639866i
$722$	2242.56	0.115595
$723$	42232.1	2.17238
$724$	20823.6i	1.06892i
$725$	− 2916.79i	− 0.149416i
$726$	− 3476.58i	− 0.177725i
$727$	−3241.15	−0.165348	−0.0826738	−	0.996577i	$-0.526346\pi$
$727$	−3241.15	−0.165348	−0.0826738	+	0.996577i	$0.526346\pi$
$728$	− 1085.80i	− 0.0552781i
$729$	−11872.9	−0.603204
$730$	3684.51	0.186808
$731$	0	0
$732$	−23828.7	−1.20319
$733$	−18578.8	−0.936185	−0.468093	−	0.883679i	$-0.655059\pi$
$733$	−18578.8	−0.936185	−0.468093	+	0.883679i	$0.655059\pi$
$734$	1244.06i	0.0625601i
$735$	37744.4	1.89418
$736$	9757.92i	0.488698i
$737$	− 13895.9i	− 0.694519i
$738$	4382.28i	0.218583i
$739$	−30393.0	−1.51289	−0.756444	−	0.654058i	$-0.773065\pi$
$739$	−30393.0	−1.51289	−0.756444	+	0.654058i	$0.773065\pi$
$740$	9149.64	0.454524
$741$	− 27575.6i	− 1.36709i
$742$	368.681i	0.0182408i
$743$	37449.0i	1.84908i	0.381081	+	0.924542i	$0.375552\pi$
$743$	37449.0i	1.84908i	−0.381081	+	0.924542i	$0.624448\pi$
$744$	45.2950	0.00223198
$745$	− 5046.40i	− 0.248169i
$746$	−1140.72	−0.0559849
$747$	−63914.3	−3.13052
$748$	0	0
$749$	−391.140	−0.0190814
$750$	−5606.80	−0.272975
$751$	− 5541.28i	− 0.269247i	−0.990897	−	0.134623i	$-0.957018\pi$
$751$	− 5541.28i	− 0.269247i	0.990897	−	0.134623i	$-0.0429825\pi$
$752$	−1801.92	−0.0873795
$753$	19841.7i	0.960253i
$754$	− 6828.23i	− 0.329801i
$755$	− 16122.5i	− 0.777160i
$756$	6177.25	0.297175
$757$	−13531.4	−0.649680	−0.324840	−	0.945769i	$-0.605310\pi$
$757$	−13531.4	−0.649680	−0.324840	+	0.945769i	$0.605310\pi$
$758$	5092.85i	0.244038i
$759$	− 25352.5i	− 1.21243i
$760$	3570.45i	0.170413i
$761$	−37110.8	−1.76776	−0.883879	−	0.467715i	$-0.845077\pi$
$761$	−37110.8	−1.76776	−0.883879	+	0.467715i	$0.845077\pi$
$762$	− 5551.00i	− 0.263899i
$763$	−2984.50	−0.141607
$764$	26476.4	1.25377
$765$	0	0
$766$	−283.444	−0.0133698
$767$	−9936.24	−0.467766
$768$	29576.6i	1.38965i
$769$	−820.351	−0.0384689	−0.0192345	−	0.999815i	$-0.506123\pi$
$769$	−820.351	−0.0384689	−0.0192345	+	0.999815i	$0.506123\pi$
$770$	− 271.621i	− 0.0127124i
$771$	5110.84i	0.238732i
$772$	23429.2i	1.09227i
$773$	25113.2	1.16851	0.584255	−	0.811570i	$-0.301387\pi$
$773$	25113.2	1.16851	0.584255	+	0.811570i	$0.301387\pi$
$774$	9151.55	0.424994
$775$	8.67473i	0 0.000402072i
$776$	− 3392.07i	− 0.156918i
$777$	− 2165.37i	− 0.0999771i
$778$	5444.21	0.250880
$779$	− 6625.43i	− 0.304725i
$780$	−58739.3	−2.69642
$781$	5644.99	0.258634
$782$	0	0
$783$	78691.0	3.59155
$784$	20009.2	0.911499
$785$	− 7234.10i	− 0.328912i
$786$	6490.00	0.294517
$787$	− 20784.0i	− 0.941384i	−0.882298	−	0.470692i	$-0.844004\pi$
$787$	− 20784.0i	− 0.941384i	0.882298	−	0.470692i	$-0.155996\pi$
$788$	12765.1i	0.577080i
$789$	− 62963.7i	− 2.84102i
$790$	4062.97	0.182980
$791$	−2004.47	−0.0901023
$792$	10193.9i	0.457354i
$793$	− 21641.2i	− 0.969105i
$794$	− 5331.62i	− 0.238302i
$795$	40401.8	1.80240
$796$	− 12483.7i	− 0.555868i
$797$	−4974.23	−0.221074	−0.110537	−	0.993872i	$-0.535257\pi$
$797$	−4974.23	−0.221074	−0.110537	+	0.993872i	$0.535257\pi$
$798$	417.137	0.0185044
$799$	0	0
$800$	1070.73	0.0473198
$801$	−104272.	−4.59960
$802$	3525.12i	0.155207i
$803$	−15904.1	−0.698935
$804$	− 45457.5i	− 1.99398i
$805$	3141.52i	0.137545i
$806$	20.3076i	0 0.000887477i
$807$	5874.95	0.256268
$808$	−5756.73	−0.250645
$809$	− 27460.1i	− 1.19338i	−0.802471	−	0.596692i	$-0.796481\pi$
$809$	− 27460.1i	− 1.19338i	0.802471	−	0.596692i	$-0.203519\pi$
$810$	8370.59i	0.363102i
$811$	14356.6i	0.621612i	0.950473	+	0.310806i	$0.100599\pi$
$811$	14356.6i	0.621612i	−0.950473	+	0.310806i	$0.899401\pi$
$812$	−4021.40	−0.173798
$813$	− 50735.0i	− 2.18863i
$814$	1014.43	0.0436801
$815$	40638.6	1.74664
$816$	0	0
$817$	−13835.9	−0.592482
$818$	1947.62	0.0832480
$819$	9755.76i	0.416232i
$820$	−14113.0	−0.601033
$821$	− 15953.0i	− 0.678155i	−0.940758	−	0.339077i	$-0.889885\pi$
$821$	− 15953.0i	− 0.678155i	0.940758	−	0.339077i	$-0.110115\pi$
$822$	11148.1i	0.473033i
$823$	17581.1i	0.744641i	0.928104	+	0.372321i	$0.121438\pi$
$823$	17581.1i	0.744641i	−0.928104	+	0.372321i	$0.878562\pi$
$824$	−3845.67	−0.162585
$825$	−2781.90	−0.117398
$826$	− 150.306i	− 0.00633149i
$827$	− 39492.2i	− 1.66056i	−0.557349	−	0.830278i	$-0.688182\pi$
$827$	− 39492.2i	− 1.66056i	0.557349	−	0.830278i	$-0.311818\pi$
$828$	− 58202.9i	− 2.44286i
$829$	15789.1	0.661494	0.330747	−	0.943720i	$-0.392699\pi$
$829$	15789.1	0.661494	0.330747	+	0.943720i	$0.392699\pi$
$830$	5286.90i	0.221098i
$831$	23660.7	0.987704
$832$	−29432.4	−1.22642
$833$	0	0
$834$	−992.112	−0.0411919
$835$	−28472.8	−1.18005
$836$	− 7608.19i	− 0.314755i
$837$	−234.032	−0.00966469
$838$	− 924.812i	− 0.0381230i
$839$	− 9439.37i	− 0.388419i	−0.980960	−	0.194209i	$-0.937786\pi$
$839$	− 9439.37i	− 0.388419i	0.980960	−	0.194209i	$-0.0622141\pi$
$840$	− 1799.93i	− 0.0739326i
$841$	−26839.0	−1.10046
$842$	−4728.13	−0.193518
$843$	47118.8i	1.92510i
$844$	− 33401.0i	− 1.36222i
$845$	− 27548.7i	− 1.12154i
$846$	−865.164	−0.0351595
$847$	− 1859.53i	− 0.0754359i
$848$	21418.0	0.867331
$849$	−78185.0	−3.16054
$850$	0	0
$851$	−11732.7	−0.472610
$852$	18466.4	0.742546
$853$	2324.12i	0.0932898i	0.998912	+	0.0466449i	$0.0148529\pi$
$853$	2324.12i	0.0932898i	−0.998912	+	0.0466449i	$0.985147\pi$
$854$	327.367	0.0131174
$855$	− 32079.9i	− 1.28317i
$856$	− 1214.26i	− 0.0484844i
$857$	22464.0i	0.895399i	0.894184	+	0.447699i	$0.147756\pi$
$857$	22464.0i	0.895399i	−0.894184	+	0.447699i	$0.852244\pi$
$858$	−6512.46	−0.259128
$859$	1314.36	0.0522064	0.0261032	−	0.999659i	$-0.491690\pi$
$859$	1314.36	0.0522064	0.0261032	+	0.999659i	$0.491690\pi$
$860$	29472.2i	1.16860i
$861$	3340.00i	0.132203i
$862$	− 1946.11i	− 0.0768966i
$863$	−15380.5	−0.606674	−0.303337	−	0.952883i	$-0.598101\pi$
$863$	−15380.5	−0.606674	−0.303337	+	0.952883i	$0.598101\pi$
$864$	28886.7i	1.13744i
$865$	33503.1	1.31693
$866$	424.764	0.0166675
$867$	0	0
$868$	11.9599	0.000467681 0
$869$	−17537.8	−0.684614
$870$	− 11319.1i	− 0.441097i
$871$	41284.5	1.60605
$872$	− 9265.14i	− 0.359813i
$873$	30477.2i	1.18155i
$874$	− 2260.18i	− 0.0874734i
$875$	−2998.93	−0.115865
$876$	−52027.2	−2.00666
$877$	14688.9i	0.565574i	0.959183	+	0.282787i	$0.0912590\pi$
$877$	14688.9i	0.565574i	−0.959183	+	0.282787i	$0.908741\pi$
$878$	− 3248.06i	− 0.124848i
$879$	− 7020.06i	− 0.269375i
$880$	−15779.4	−0.604460
$881$	9512.25i	0.363764i	0.983320	+	0.181882i	$0.0582189\pi$
$881$	9512.25i	0.363764i	−0.983320	+	0.181882i	$0.941781\pi$
$882$	9607.11	0.366767
$883$	14662.1	0.558798	0.279399	−	0.960175i	$-0.409865\pi$
$883$	14662.1	0.558798	0.279399	+	0.960175i	$0.409865\pi$
$884$	0	0
$885$	−16471.2	−0.625621
$886$	1644.48	0.0623561
$887$	− 27847.1i	− 1.05413i	−0.849825	−	0.527065i	$-0.823292\pi$
$887$	− 27847.1i	− 1.05413i	0.849825	−	0.527065i	$-0.176708\pi$
$888$	6722.21	0.254034
$889$	− 2969.08i	− 0.112013i
$890$	8625.25i	0.324853i
$891$	− 36131.6i	− 1.35853i
$892$	−10894.9	−0.408956
$893$	1308.01	0.0490157
$894$	− 1830.28i	− 0.0684717i
$895$	14033.8i	0.524134i
$896$	− 1959.36i	− 0.0730554i
$897$	75322.0	2.80371
$898$	− 5824.20i	− 0.216432i
$899$	152.356	0.00565222
$900$	−6386.54	−0.236538
$901$	0	0
$902$	−1564.71	−0.0577597
$903$	6974.95	0.257045
$904$	− 6222.72i	− 0.228943i
$905$	−31350.2	−1.15151
$906$	− 5847.46i	− 0.214425i
$907$	141.986i	0.00519799i	0.999997	+	0.00259899i	$0.000827286\pi$
$907$	141.986i	0.00519799i	−0.999997	+	0.00259899i	$0.999173\pi$
$908$	− 12576.4i	− 0.459651i
$909$	51723.3	1.88730
$910$	806.983	0.0293969
$911$	32439.3i	1.17976i	0.807490	+	0.589881i	$0.200825\pi$
$911$	32439.3i	1.17976i	−0.807490	+	0.589881i	$0.799175\pi$
$912$	− 24233.0i	− 0.879862i
$913$	− 22820.9i	− 0.827230i
$914$	−5.87610	−0.000212652 0
$915$	− 35874.5i	− 1.29615i
$916$	−19736.6	−0.711918
$917$	3471.32	0.125009
$918$	0	0
$919$	11802.3	0.423638	0.211819	−	0.977309i	$-0.432061\pi$
$919$	11802.3	0.423638	0.211819	+	0.977309i	$0.432061\pi$
$920$	−9752.59	−0.349493
$921$	57038.5i	2.04070i
$922$	−6437.53	−0.229944
$923$	16771.2i	0.598083i
$924$	3835.43i	0.136555i
$925$	1287.41i	0.0457620i
$926$	5936.95	0.210691
$927$	34552.8	1.22423
$928$	− 18805.3i	− 0.665210i
$929$	30089.9i	1.06267i	0.847163	+	0.531334i	$0.178309\pi$
$929$	30089.9i	1.06267i	−0.847163	+	0.531334i	$0.821691\pi$
$930$	33.6639i	0.00118697i
$931$	−14524.7	−0.511307
$932$	36577.2i	1.28554i
$933$	−90020.9	−3.15879
$934$	5815.09	0.203721
$935$	0	0
$936$	−30285.9	−1.05761
$937$	34961.5	1.21894	0.609468	−	0.792810i	$-0.291383\pi$
$937$	34961.5	1.21894	0.609468	+	0.792810i	$0.291383\pi$
$938$	624.512i	0.0217389i
$939$	−44482.8	−1.54594
$940$	− 2786.23i	− 0.0966775i
$941$	− 8933.80i	− 0.309494i	−0.987954	−	0.154747i	$-0.950544\pi$
$941$	− 8933.80i	− 0.309494i	0.987954	−	0.154747i	$-0.0494562\pi$
$942$	− 2623.74i	− 0.0907494i
$943$	18097.2	0.624948
$944$	−8731.81	−0.301055
$945$	9299.95i	0.320135i
$946$	3267.60i	0.112303i
$947$	17816.2i	0.611350i	0.952136	+	0.305675i	$0.0988822\pi$
$947$	17816.2i	0.611350i	−0.952136	+	0.305675i	$0.901118\pi$
$948$	−57371.4	−1.96554
$949$	− 47251.1i	− 1.61626i
$950$	−248.007	−0.00846991
$951$	6593.35	0.224820
$952$	0	0
$953$	36249.3	1.23214	0.616070	−	0.787691i	$-0.288724\pi$
$953$	36249.3	1.23214	0.616070	+	0.787691i	$0.288724\pi$
$954$	10283.5	0.348994
$955$	39860.7i	1.35064i
$956$	49390.0	1.67091
$957$	48858.9i	1.65035i
$958$	− 3893.02i	− 0.131292i
$959$	5962.79i	0.200781i
$960$	−48789.9	−1.64030
$961$	29790.5	0.999985
$962$	3013.85i	0.101009i
$963$	10910.0i	0.365076i
$964$	34617.9i	1.15661i
$965$	−35273.1	−1.17666
$966$	1139.40i	0.0379499i
$967$	−10020.3	−0.333227	−0.166613	−	0.986022i	$-0.553283\pi$
$967$	−10020.3	−0.333227	−0.166613	+	0.986022i	$0.553283\pi$
$968$	5772.75	0.191677
$969$	0	0
$970$	2521.03	0.0834489
$971$	−24562.5	−0.811789	−0.405894	−	0.913920i	$-0.633040\pi$
$971$	−24562.5	−0.811789	−0.405894	+	0.913920i	$0.633040\pi$
$972$	− 44980.4i	− 1.48431i
$973$	−530.654	−0.0174841
$974$	545.908i	0.0179590i
$975$	− 8264.99i	− 0.271479i
$976$	− 19017.9i	− 0.623718i
$977$	17102.8	0.560048	0.280024	−	0.959993i	$-0.409658\pi$
$977$	17102.8	0.560048	0.280024	+	0.959993i	$0.409658\pi$
$978$	14739.2	0.481911
$979$	− 37230.9i	− 1.21543i
$980$	30939.3i	1.00849i
$981$	83245.8i	2.70931i
$982$	5075.73	0.164942
$983$	51204.2i	1.66140i	0.556718	+	0.830702i	$0.312061\pi$
$983$	51204.2i	1.66140i	−0.556718	+	0.830702i	$0.687939\pi$
$984$	−10368.8	−0.335919
$985$	−19218.1	−0.621665
$986$	0	0
$987$	−659.394	−0.0212652
$988$	22603.9	0.727859
$989$	− 37792.5i	− 1.21510i
$990$	−7576.24	−0.243221
$991$	19176.3i	0.614686i	0.951599	+	0.307343i	$0.0994399\pi$
$991$	19176.3i	0.614686i	−0.951599	+	0.307343i	$0.900560\pi$
$992$	55.9283i	0.00179005i
$993$	67247.0i	2.14906i
$994$	−253.699	−0.00809541
$995$	18794.4	0.598815
$996$	− 74653.9i	− 2.37500i
$997$	− 18251.7i	− 0.579775i	−0.957061	−	0.289888i	$-0.906382\pi$
$997$	− 18251.7i	− 0.579775i	0.957061	−	0.289888i	$-0.0936179\pi$
$998$	− 3293.24i	− 0.104455i
$999$	−34732.6	−1.09999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.b.f.288.11		24
17.4	even	4		289.4.a.i.1.7	yes	12
17.13	even	4		289.4.a.h.1.7	✓	12
17.16	even	2	inner	289.4.b.f.288.12		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.4.a.h.1.7	✓	12	17.13	even	4
289.4.a.i.1.7	yes	12	17.4	even	4
289.4.b.f.288.11		24	1.1	even	1	trivial
289.4.b.f.288.12		24	17.16	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:	\( N \)	\(=\)	\( 289 = 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	289.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.447590 q^{2} +9.51519i q^{3} -7.79966 q^{4} -11.7425i q^{5} -4.25890i q^{6} -2.27797i q^{7} +7.07177 q^{8} -63.5388 q^{9} +O(q^{10})\)	\(q-0.447590 q^{2} +9.51519i q^{3} -7.79966 q^{4} -11.7425i q^{5} -4.25890i q^{6} -2.27797i q^{7} +7.07177 q^{8} -63.5388 q^{9} +5.25584i q^{10} -22.6868i q^{11} -74.2152i q^{12} +67.4022 q^{13} +1.01960i q^{14} +111.732 q^{15} +59.2321 q^{16} +28.4393 q^{18} -42.9965 q^{19} +91.5878i q^{20} +21.6753 q^{21} +10.1544i q^{22} -117.444i q^{23} +67.2892i q^{24} -12.8870 q^{25} -30.1686 q^{26} -347.673i q^{27} +17.7674i q^{28} +226.336i q^{29} -50.0103 q^{30} -0.673139i q^{31} -83.0858 q^{32} +215.869 q^{33} -26.7491 q^{35} +495.581 q^{36} -99.9003i q^{37} +19.2448 q^{38} +641.345i q^{39} -83.0405i q^{40} +154.092i q^{41} -9.70165 q^{42} +321.792 q^{43} +176.949i q^{44} +746.106i q^{45} +52.5667i q^{46} -30.4214 q^{47} +563.604i q^{48} +337.811 q^{49} +5.76808 q^{50} -525.715 q^{52} +361.595 q^{53} +155.615i q^{54} -266.400 q^{55} -16.1093i q^{56} -409.119i q^{57} -101.306i q^{58} -147.417 q^{59} -871.475 q^{60} -321.075i q^{61} +0.301290i q^{62} +144.739i q^{63} -436.668 q^{64} -791.473i q^{65} -96.6208 q^{66} +612.509 q^{67} +1117.50 q^{69} +11.9726 q^{70} +248.823i q^{71} -449.331 q^{72} -701.031i q^{73} +44.7144i q^{74} -122.622i q^{75} +335.358 q^{76} -51.6798 q^{77} -287.059i q^{78} -773.040i q^{79} -695.534i q^{80} +1592.63 q^{81} -68.9702i q^{82} +1005.91 q^{83} -169.060 q^{84} -144.031 q^{86} -2153.63 q^{87} -160.436i q^{88} +1641.08 q^{89} -333.949i q^{90} -153.540i q^{91} +916.023i q^{92} +6.40505 q^{93} +13.6163 q^{94} +504.887i q^{95} -790.577i q^{96} -479.663i q^{97} -151.201 q^{98} +1441.49i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) \) 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9	\( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9} - 144 q^{13} + 276 q^{15} + 384 q^{16} + 378 q^{18} + 132 q^{19} + 492 q^{21} - 888 q^{25} - 1056 q^{26} - 3780 q^{30} + 2706 q^{32} + 1932 q^{33} - 132 q^{35} + 1326 q^{36} - 120 q^{38} - 1608 q^{42} - 972 q^{43} - 1776 q^{47} + 1140 q^{49} + 870 q^{50} + 450 q^{52} - 2052 q^{53} + 1944 q^{55} + 1584 q^{59} - 2916 q^{60} - 3714 q^{64} + 1188 q^{66} + 1248 q^{67} - 3012 q^{69} + 3300 q^{70} + 2910 q^{72} - 1350 q^{76} + 2016 q^{77} + 5040 q^{81} - 1344 q^{83} - 1554 q^{84} + 5556 q^{86} - 1452 q^{87} - 1812 q^{89} - 264 q^{93} - 1470 q^{94} + 3822 q^{98}+O(q^{100}) \) 24 * q + 96 * q^4 + 102 * q^8 - 216 * q^9 - 144 * q^13 + 276 * q^15 + 384 * q^16 + 378 * q^18 + 132 * q^19 + 492 * q^21 - 888 * q^25 - 1056 * q^26 - 3780 * q^30 + 2706 * q^32 + 1932 * q^33 - 132 * q^35 + 1326 * q^36 - 120 * q^38 - 1608 * q^42 - 972 * q^43 - 1776 * q^47 + 1140 * q^49 + 870 * q^50 + 450 * q^52 - 2052 * q^53 + 1944 * q^55 + 1584 * q^59 - 2916 * q^60 - 3714 * q^64 + 1188 * q^66 + 1248 * q^67 - 3012 * q^69 + 3300 * q^70 + 2910 * q^72 - 1350 * q^76 + 2016 * q^77 + 5040 * q^81 - 1344 * q^83 - 1554 * q^84 + 5556 * q^86 - 1452 * q^87 - 1812 * q^89 - 264 * q^93 - 1470 * q^94 + 3822 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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