LMFDB - Embedded newform 289.4.b.f.288.3

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			288.3
Character	$\chi$	$=$	289.288
Dual form			289.4.b.f.288.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.42326 q^{2} +9.44971i q^{3} +11.5652 q^{4} -8.63042i q^{5} -41.7985i q^{6} -12.6513i q^{7} -15.7698 q^{8} -62.2970 q^{9} +O(q^{10})$	\(q-4.42326 q^{2} +9.44971i q^{3} +11.5652 q^{4} -8.63042i q^{5} -41.7985i q^{6} -12.6513i q^{7} -15.7698 q^{8} -62.2970 q^{9} +38.1746i q^{10} -14.5171i q^{11} +109.288i q^{12} -26.6619 q^{13} +55.9602i q^{14} +81.5550 q^{15} -22.7676 q^{16} +275.556 q^{18} +125.285 q^{19} -99.8126i q^{20} +119.552 q^{21} +64.2131i q^{22} +2.31970i q^{23} -149.020i q^{24} +50.5159 q^{25} +117.932 q^{26} -333.547i q^{27} -146.315i q^{28} -21.8015i q^{29} -360.739 q^{30} +323.318i q^{31} +226.866 q^{32} +137.183 q^{33} -109.186 q^{35} -720.478 q^{36} +73.2182i q^{37} -554.167 q^{38} -251.947i q^{39} +136.100i q^{40} +179.916i q^{41} -528.807 q^{42} -186.872 q^{43} -167.894i q^{44} +537.649i q^{45} -10.2606i q^{46} +235.952 q^{47} -215.147i q^{48} +182.944 q^{49} -223.445 q^{50} -308.350 q^{52} -200.645 q^{53} +1475.36i q^{54} -125.289 q^{55} +199.510i q^{56} +1183.90i q^{57} +96.4337i q^{58} +718.203 q^{59} +943.200 q^{60} +727.874i q^{61} -1430.12i q^{62} +788.141i q^{63} -821.345 q^{64} +230.103i q^{65} -606.795 q^{66} +76.9332 q^{67} -21.9205 q^{69} +482.959 q^{70} -923.747i q^{71} +982.414 q^{72} +820.959i q^{73} -323.863i q^{74} +477.361i q^{75} +1448.94 q^{76} -183.661 q^{77} +1114.43i q^{78} -51.0071i q^{79} +196.494i q^{80} +1469.90 q^{81} -795.815i q^{82} -1183.28 q^{83} +1382.64 q^{84} +826.585 q^{86} +206.018 q^{87} +228.933i q^{88} -86.8644 q^{89} -2378.16i q^{90} +337.309i q^{91} +26.8278i q^{92} -3055.26 q^{93} -1043.68 q^{94} -1081.26i q^{95} +2143.82i q^{96} +758.448i q^{97} -809.206 q^{98} +904.375i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.42326	−1.56386	−0.781929	−	0.623368i	$-0.785764\pi$
$2$	−4.42326	−1.56386	−0.781929	+	0.623368i	$0.785764\pi$
$3$	9.44971i	1.81860i	0.416144	+	0.909299i	$0.363381\pi$
$3$	9.44971i	1.81860i	−0.416144	+	0.909299i	$0.636619\pi$
$4$	11.5652	1.44565
$5$	− 8.63042i	− 0.771928i	−0.922514	−	0.385964i	$-0.873869\pi$
$5$	− 8.63042i	− 0.771928i	0.922514	−	0.385964i	$-0.126131\pi$
$6$	− 41.7985i	− 2.84403i
$7$	− 12.6513i	− 0.683108i	−0.939862	−	0.341554i	$-0.889047\pi$
$7$	− 12.6513i	− 0.683108i	0.939862	−	0.341554i	$-0.110953\pi$
$8$	−15.7698	−0.696935
$9$	−62.2970	−2.30730
$10$	38.1746i	1.20719i
$11$	− 14.5171i	− 0.397917i	−0.980008	−	0.198958i	$-0.936244\pi$
$11$	− 14.5171i	− 0.397917i	0.980008	−	0.198958i	$-0.0637559\pi$
$12$	109.288i	2.62906i
$13$	−26.6619	−0.568822	−0.284411	−	0.958702i	$-0.591798\pi$
$13$	−26.6619	−0.568822	−0.284411	+	0.958702i	$0.591798\pi$
$14$	55.9602i	1.06828i
$15$	81.5550	1.40383
$16$	−22.7676	−0.355744
$17$	0	0
$17$	0	0
$18$	275.556	3.60829
$19$	125.285	1.51275	0.756376	−	0.654137i	$-0.226968\pi$
$19$	125.285	1.51275	0.756376	+	0.654137i	$0.226968\pi$
$20$	− 99.8126i	− 1.11594i
$21$	119.552	1.24230
$22$	64.2131i	0.622285i
$23$	2.31970i	0.0210301i	0.999945	+	0.0105150i	$0.00334710\pi$
$23$	2.31970i	0.0210301i	−0.999945	+	0.0105150i	$0.996653\pi$
$24$	− 149.020i	− 1.26744i
$25$	50.5159	0.404127
$26$	117.932	0.889556
$27$	− 333.547i	− 2.37745i
$28$	− 146.315i	− 0.987536i
$29$	− 21.8015i	− 0.139601i	−0.997561	−	0.0698007i	$-0.977764\pi$
$29$	− 21.8015i	− 0.139601i	0.997561	−	0.0698007i	$-0.0222363\pi$
$30$	−360.739	−2.19538
$31$	323.318i	1.87321i	0.350383	+	0.936606i	$0.386051\pi$
$31$	323.318i	1.87321i	−0.350383	+	0.936606i	$0.613949\pi$
$32$	226.866	1.25327
$33$	137.183	0.723650
$34$	0	0
$35$	−109.186	−0.527310
$36$	−720.478	−3.33555
$37$	73.2182i	0.325324i	0.986682	+	0.162662i	$0.0520081\pi$
$37$	73.2182i	0.325324i	−0.986682	+	0.162662i	$0.947992\pi$
$38$	−554.167	−2.36573
$39$	− 251.947i	− 1.03446i
$40$	136.100i	0.537983i
$41$	179.916i	0.685321i	0.939459	+	0.342661i	$0.111328\pi$
$41$	179.916i	0.685321i	−0.939459	+	0.342661i	$0.888672\pi$
$42$	−528.807	−1.94278
$43$	−186.872	−0.662739	−0.331370	−	0.943501i	$-0.607511\pi$
$43$	−186.872	−0.662739	−0.331370	+	0.943501i	$0.607511\pi$
$44$	− 167.894i	− 0.575249i
$45$	537.649i	1.78107i
$46$	− 10.2606i	− 0.0328880i
$47$	235.952	0.732280	0.366140	−	0.930560i	$-0.380679\pi$
$47$	235.952	0.732280	0.366140	+	0.930560i	$0.380679\pi$
$48$	− 215.147i	− 0.646956i
$49$	182.944	0.533363
$50$	−223.445	−0.631997
$51$	0	0
$52$	−308.350	−0.822318
$53$	−200.645	−0.520014	−0.260007	−	0.965607i	$-0.583725\pi$
$53$	−200.645	−0.520014	−0.260007	+	0.965607i	$0.583725\pi$
$54$	1475.36i	3.71799i
$55$	−125.289	−0.307163
$56$	199.510i	0.476082i
$57$	1183.90i	2.75109i
$58$	96.4337i	0.218317i
$59$	718.203	1.58478	0.792391	−	0.610014i	$-0.208836\pi$
$59$	718.203	1.58478	0.792391	+	0.610014i	$0.208836\pi$
$60$	943.200	2.02944
$61$	727.874i	1.52778i	0.645345	+	0.763891i	$0.276714\pi$
$61$	727.874i	1.52778i	−0.645345	+	0.763891i	$0.723286\pi$
$62$	− 1430.12i	− 2.92944i
$63$	788.141i	1.57613i
$64$	−821.345	−1.60419
$65$	230.103i	0.439089i
$66$	−606.795	−1.13169
$67$	76.9332	0.140282	0.0701410	−	0.997537i	$-0.477655\pi$
$67$	76.9332	0.140282	0.0701410	+	0.997537i	$0.477655\pi$
$68$	0	0
$69$	−21.9205	−0.0382452
$70$	482.959	0.824639
$71$	− 923.747i	− 1.54406i	−0.635583	−	0.772032i	$-0.719240\pi$
$71$	− 923.747i	− 1.54406i	0.635583	−	0.772032i	$-0.280760\pi$
$72$	982.414	1.60804
$73$	820.959i	1.31625i	0.752910	+	0.658123i	$0.228650\pi$
$73$	820.959i	1.31625i	−0.752910	+	0.658123i	$0.771350\pi$
$74$	− 323.863i	− 0.508761i
$75$	477.361i	0.734945i
$76$	1448.94	2.18691
$77$	−183.661	−0.271820
$78$	1114.43i	1.61774i
$79$	− 51.0071i	− 0.0726423i	−0.999340	−	0.0363212i	$-0.988436\pi$
$79$	− 51.0071i	− 0.0726423i	0.999340	−	0.0363212i	$-0.0115639\pi$
$80$	196.494i	0.274609i
$81$	1469.90	2.01632
$82$	− 795.815i	− 1.07174i
$83$	−1183.28	−1.56485	−0.782424	−	0.622746i	$-0.786017\pi$
$83$	−1183.28	−1.56485	−0.782424	+	0.622746i	$0.786017\pi$
$84$	1382.64	1.79593
$85$	0	0
$86$	826.585	1.03643
$87$	206.018	0.253879
$88$	228.933i	0.277322i
$89$	−86.8644	−0.103456	−0.0517281	−	0.998661i	$-0.516473\pi$
$89$	−86.8644	−0.103456	−0.0517281	+	0.998661i	$0.516473\pi$
$90$	− 2378.16i	− 2.78534i
$91$	337.309i	0.388567i
$92$	26.8278i	0.0304021i
$93$	−3055.26	−3.40662
$94$	−1043.68	−1.14518
$95$	− 1081.26i	− 1.16774i
$96$	2143.82i	2.27919i
$97$	758.448i	0.793904i	0.917839	+	0.396952i	$0.129932\pi$
$97$	758.448i	0.793904i	−0.917839	+	0.396952i	$0.870068\pi$
$98$	−809.206	−0.834104
$99$	904.375i	0.918112i
$100$	584.227	0.584227
$101$	991.783	0.977090	0.488545	−	0.872539i	$-0.337528\pi$
$101$	991.783	0.977090	0.488545	+	0.872539i	$0.337528\pi$
$102$	0	0
$103$	−596.460	−0.570592	−0.285296	−	0.958439i	$-0.592092\pi$
$103$	−596.460	−0.570592	−0.285296	+	0.958439i	$0.592092\pi$
$104$	420.454	0.396432
$105$	− 1031.78i	− 0.958966i
$106$	887.506	0.813228
$107$	− 786.005i	− 0.710149i	−0.934838	−	0.355075i	$-0.884455\pi$
$107$	− 786.005i	− 0.710149i	0.934838	−	0.355075i	$-0.115545\pi$
$108$	− 3857.54i	− 3.43696i
$109$	489.932i	0.430523i	0.976556	+	0.215261i	$0.0690604\pi$
$109$	489.932i	0.430523i	−0.976556	+	0.215261i	$0.930940\pi$
$110$	554.186	0.480359
$111$	−691.891	−0.591634
$112$	288.041i	0.243012i
$113$	− 162.976i	− 0.135677i	−0.997696	−	0.0678385i	$-0.978390\pi$
$113$	− 162.976i	− 0.135677i	0.997696	−	0.0678385i	$-0.0216103\pi$
$114$	− 5236.72i	− 4.30231i
$115$	20.0200	0.0162337
$116$	− 252.139i	− 0.201815i
$117$	1660.96	1.31244
$118$	−3176.80	−2.47837
$119$	0	0
$120$	−1286.11	−0.978376
$121$	1120.25	0.841662
$122$	− 3219.57i	− 2.38923i
$123$	−1700.15	−1.24632
$124$	3739.24i	2.70801i
$125$	− 1514.78i	− 1.08389i
$126$	− 3486.15i	− 2.46485i
$127$	334.191	0.233501	0.116751	−	0.993161i	$-0.462752\pi$
$127$	334.191	0.233501	0.116751	+	0.993161i	$0.462752\pi$
$128$	1818.09	1.25545
$129$	− 1765.89i	− 1.20526i
$130$	− 1017.81i	− 0.686673i
$131$	697.659i	0.465304i	0.972560	+	0.232652i	$0.0747403\pi$
$131$	697.659i	0.465304i	−0.972560	+	0.232652i	$0.925260\pi$
$132$	1586.55	1.04615
$133$	− 1585.02i	− 1.03337i
$134$	−340.295	−0.219381
$135$	−2878.65	−1.83522
$136$	0	0
$137$	2065.04	1.28780	0.643898	−	0.765111i	$-0.277316\pi$
$137$	2065.04	1.28780	0.643898	+	0.765111i	$0.277316\pi$
$138$	96.9601	0.0598101
$139$	1815.36i	1.10775i	0.832601	+	0.553873i	$0.186851\pi$
$139$	1815.36i	1.10775i	−0.832601	+	0.553873i	$0.813149\pi$
$140$	−1262.76	−0.762307
$141$	2229.68i	1.33172i
$142$	4085.97i	2.41470i
$143$	387.055i	0.226344i
$144$	1418.36	0.820808
$145$	−188.156	−0.107762
$146$	− 3631.31i	− 2.05842i
$147$	1728.76i	0.969973i
$148$	846.784i	0.470305i
$149$	860.952	0.473368	0.236684	−	0.971587i	$-0.423939\pi$
$149$	860.952	0.473368	0.236684	+	0.971587i	$0.423939\pi$
$150$	− 2111.49i	− 1.14935i
$151$	1008.79	0.543668	0.271834	−	0.962344i	$-0.412370\pi$
$151$	1008.79	0.543668	0.271834	+	0.962344i	$0.412370\pi$
$152$	−1975.72	−1.05429
$153$	0	0
$154$	812.382	0.425088
$155$	2790.37	1.44599
$156$	− 2913.82i	− 1.49546i
$157$	2083.02	1.05887	0.529436	−	0.848350i	$-0.322404\pi$
$157$	2083.02	1.05887	0.529436	+	0.848350i	$0.322404\pi$
$158$	225.617i	0.113602i
$159$	− 1896.04i	− 0.945697i
$160$	− 1957.95i	− 0.967433i
$161$	29.3473	0.0143658
$162$	−6501.75	−3.15325
$163$	2555.18i	1.22784i	0.789370	+	0.613918i	$0.210408\pi$
$163$	2555.18i	1.22784i	−0.789370	+	0.613918i	$0.789592\pi$
$164$	2080.77i	0.990735i
$165$	− 1183.95i	− 0.558606i
$166$	5233.97	2.44720
$167$	2473.04i	1.14592i	0.819582	+	0.572962i	$0.194206\pi$
$167$	2473.04i	1.14592i	−0.819582	+	0.572962i	$0.805794\pi$
$168$	−1885.31	−0.865802
$169$	−1486.14	−0.676442
$170$	0	0
$171$	−7804.87	−3.49037
$172$	−2161.22	−0.958089
$173$	1895.93i	0.833206i	0.909089	+	0.416603i	$0.136779\pi$
$173$	1895.93i	0.833206i	−0.909089	+	0.416603i	$0.863221\pi$
$174$	−911.271	−0.397030
$175$	− 639.094i	− 0.276063i
$176$	330.521i	0.141557i
$177$	6786.81i	2.88208i
$178$	384.224	0.161791
$179$	2662.80	1.11188	0.555940	−	0.831222i	$-0.312358\pi$
$179$	2662.80	1.11188	0.555940	+	0.831222i	$0.312358\pi$
$180$	6218.03i	2.57480i
$181$	− 1410.66i	− 0.579303i	−0.957132	−	0.289651i	$-0.906461\pi$
$181$	− 1410.66i	− 0.579303i	0.957132	−	0.289651i	$-0.0935394\pi$
$182$	− 1492.00i	− 0.607663i
$183$	−6878.20	−2.77842
$184$	− 36.5813i	− 0.0146566i
$185$	631.904	0.251127
$186$	13514.2	5.32747
$187$	0	0
$188$	2728.84	1.05862
$189$	−4219.81	−1.62405
$190$	4782.69i	1.82617i
$191$	4025.80	1.52511	0.762557	−	0.646921i	$-0.223944\pi$
$191$	4025.80	1.52511	0.762557	+	0.646921i	$0.223944\pi$
$192$	− 7761.47i	− 2.91737i
$193$	− 3819.08i	− 1.42437i	−0.701991	−	0.712186i	$-0.747705\pi$
$193$	− 3819.08i	− 1.42437i	0.701991	−	0.712186i	$-0.252295\pi$
$194$	− 3354.81i	− 1.24155i
$195$	−2174.41	−0.798527
$196$	2115.78	0.771057
$197$	− 657.014i	− 0.237616i	−0.992917	−	0.118808i	$-0.962093\pi$
$197$	− 657.014i	− 0.237616i	0.992917	−	0.118808i	$-0.0379073\pi$
$198$	− 4000.28i	− 1.43580i
$199$	3078.87i	1.09676i	0.836230	+	0.548379i	$0.184755\pi$
$199$	3078.87i	1.09676i	−0.836230	+	0.548379i	$0.815245\pi$
$200$	−796.627	−0.281650
$201$	726.997i	0.255116i
$202$	−4386.91	−1.52803
$203$	−275.818	−0.0953628
$204$	0	0
$205$	1552.75	0.529019
$206$	2638.30	0.892325
$207$	− 144.511i	− 0.0485226i
$208$	607.028	0.202355
$209$	− 1818.78i	− 0.601949i
$210$	4563.83i	1.49969i
$211$	− 1405.99i	− 0.458732i	−0.973340	−	0.229366i	$-0.926335\pi$
$211$	− 1405.99i	− 0.458732i	0.973340	−	0.229366i	$-0.0736652\pi$
$212$	−2320.50	−0.751759
$213$	8729.14	2.80803
$214$	3476.70i	1.11057i
$215$	1612.79i	0.511587i
$216$	5259.98i	1.65693i
$217$	4090.40	1.27961
$218$	− 2167.10i	− 0.673277i
$219$	−7757.83	−2.39372
$220$	−1448.99	−0.444051
$221$	0	0
$222$	3060.41	0.925232
$223$	3601.84	1.08160	0.540800	−	0.841151i	$-0.318121\pi$
$223$	3601.84	1.08160	0.540800	+	0.841151i	$0.318121\pi$
$224$	− 2870.16i	− 0.856118i
$225$	−3146.99	−0.932442
$226$	720.886i	0.212180i
$227$	3977.19i	1.16289i	0.813587	+	0.581443i	$0.197512\pi$
$227$	3977.19i	1.16289i	−0.813587	+	0.581443i	$0.802488\pi$
$228$	13692.1i	3.97711i
$229$	3018.77	0.871117	0.435559	−	0.900160i	$-0.356551\pi$
$229$	3018.77	0.871117	0.435559	+	0.900160i	$0.356551\pi$
$230$	−88.5536	−0.0253872
$231$	− 1735.55i	− 0.494332i
$232$	343.806i	0.0972930i
$233$	− 1131.49i	− 0.318138i	−0.987267	−	0.159069i	$-0.949151\pi$
$233$	− 1131.49i	− 0.318138i	0.987267	−	0.159069i	$-0.0508493\pi$
$234$	−7346.84	−2.05247
$235$	− 2036.37i	− 0.565267i
$236$	8306.17	2.29104
$237$	482.002	0.132107
$238$	0	0
$239$	2866.47	0.775802	0.387901	−	0.921701i	$-0.373200\pi$
$239$	2866.47	0.775802	0.387901	+	0.921701i	$0.373200\pi$
$240$	−1856.81	−0.499403
$241$	718.184i	0.191960i	0.995383	+	0.0959799i	$0.0305985\pi$
$241$	718.184i	0.191960i	−0.995383	+	0.0959799i	$0.969402\pi$
$242$	−4955.17	−1.31624
$243$	4884.37i	1.28944i
$244$	8418.01i	2.20864i
$245$	− 1578.88i	− 0.411718i
$246$	7520.22	1.94907
$247$	−3340.33	−0.860486
$248$	− 5098.67i	− 1.30551i
$249$	− 11181.7i	− 2.84583i
$250$	6700.24i	1.69504i
$251$	−6234.39	−1.56777	−0.783887	−	0.620903i	$-0.786766\pi$
$251$	−6234.39	−1.56777	−0.783887	+	0.620903i	$0.786766\pi$
$252$	9115.02i	2.27854i
$253$	33.6754	0.00836821
$254$	−1478.21	−0.365163
$255$	0	0
$256$	−1471.14	−0.359164
$257$	−3681.61	−0.893590	−0.446795	−	0.894636i	$-0.647435\pi$
$257$	−3681.61	−0.893590	−0.446795	+	0.894636i	$0.647435\pi$
$258$	7810.99i	1.88485i
$259$	926.309	0.222232
$260$	2661.19i	0.634770i
$261$	1358.17i	0.322102i
$262$	− 3085.93i	− 0.727669i
$263$	−7086.65	−1.66153	−0.830764	−	0.556625i	$-0.812096\pi$
$263$	−7086.65	−1.66153	−0.830764	+	0.556625i	$0.812096\pi$
$264$	−2163.35	−0.504337
$265$	1731.65i	0.401414i
$266$	7010.95i	1.61605i
$267$	− 820.843i	− 0.188145i
$268$	889.749	0.202799
$269$	− 584.965i	− 0.132587i	−0.997800	−	0.0662935i	$-0.978883\pi$
$269$	− 584.965i	− 0.132587i	0.997800	−	0.0662935i	$-0.0211174\pi$
$270$	12733.0	2.87002
$271$	−255.939	−0.0573697	−0.0286849	−	0.999589i	$-0.509132\pi$
$271$	−255.939	−0.0573697	−0.0286849	+	0.999589i	$0.509132\pi$
$272$	0	0
$273$	−3187.47	−0.706647
$274$	−9134.19	−2.01393
$275$	− 733.347i	− 0.160809i
$276$	−253.515	−0.0552892
$277$	5483.90i	1.18951i	0.803905	+	0.594757i	$0.202752\pi$
$277$	5483.90i	1.18951i	−0.803905	+	0.594757i	$0.797248\pi$
$278$	− 8029.79i	− 1.73236i
$279$	− 20141.7i	− 4.32206i
$280$	1721.85	0.367501
$281$	−3612.64	−0.766946	−0.383473	−	0.923552i	$-0.625272\pi$
$281$	−3612.64	−0.766946	−0.383473	+	0.923552i	$0.625272\pi$
$282$	− 9862.45i	− 2.08262i
$283$	3778.61i	0.793693i	0.917885	+	0.396846i	$0.129895\pi$
$283$	3778.61i	0.793693i	−0.917885	+	0.396846i	$0.870105\pi$
$284$	− 10683.3i	− 2.23218i
$285$	10217.6	2.12364
$286$	− 1712.04i	− 0.353969i
$287$	2276.18	0.468149
$288$	−14133.1	−2.89166
$289$	0	0
$290$	832.263	0.168525
$291$	−7167.11	−1.44379
$292$	9494.57i	1.90283i
$293$	8361.99	1.66728	0.833639	−	0.552309i	$-0.186253\pi$
$293$	8361.99	1.66728	0.833639	+	0.552309i	$0.186253\pi$
$294$	− 7646.77i	− 1.51690i
$295$	− 6198.39i	− 1.22334i
$296$	− 1154.64i	− 0.226730i
$297$	−4842.15	−0.946027
$298$	−3808.21	−0.740281
$299$	− 61.8477i	− 0.0119624i
$300$	5520.77i	1.06247i
$301$	2364.19i	0.452723i
$302$	−4462.12	−0.850219
$303$	9372.06i	1.77693i
$304$	−2852.44	−0.538153
$305$	6281.86	1.17934
$306$	0	0
$307$	−4070.96	−0.756814	−0.378407	−	0.925639i	$-0.623528\pi$
$307$	−4070.96	−0.756814	−0.378407	+	0.925639i	$0.623528\pi$
$308$	−2124.08	−0.392957
$309$	− 5636.38i	− 1.03768i
$310$	−12342.5	−2.26132
$311$	2434.77i	0.443933i	0.975054	+	0.221967i	$0.0712476\pi$
$311$	2434.77i	0.443933i	−0.975054	+	0.221967i	$0.928752\pi$
$312$	3973.17i	0.720950i
$313$	4352.54i	0.786006i	0.919537	+	0.393003i	$0.128564\pi$
$313$	4352.54i	0.786006i	−0.919537	+	0.393003i	$0.871436\pi$
$314$	−9213.72	−1.65593
$315$	6801.99	1.21666
$316$	− 589.907i	− 0.105015i
$317$	− 3770.58i	− 0.668065i	−0.942561	−	0.334033i	$-0.891590\pi$
$317$	− 3770.58i	− 0.668065i	0.942561	−	0.334033i	$-0.108410\pi$
$318$	8386.67i	1.47893i
$319$	−316.496	−0.0555497
$320$	7088.55i	1.23832i
$321$	7427.52	1.29148
$322$	−129.811	−0.0224661
$323$	0	0
$324$	16999.7	2.91490
$325$	−1346.85	−0.229876
$326$	− 11302.2i	− 1.92016i
$327$	−4629.72	−0.782948
$328$	− 2837.25i	− 0.477624i
$329$	− 2985.11i	− 0.500227i
$330$	5236.89i	0.873580i
$331$	9325.54	1.54857	0.774287	−	0.632835i	$-0.218109\pi$
$331$	9325.54	1.54857	0.774287	+	0.632835i	$0.218109\pi$
$332$	−13684.9	−2.26222
$333$	− 4561.28i	− 0.750620i
$334$	− 10938.9i	− 1.79206i
$335$	− 663.966i	− 0.108288i
$336$	−2721.90	−0.441941
$337$	1183.38i	0.191284i	0.995416	+	0.0956421i	$0.0304904\pi$
$337$	1183.38i	0.191284i	−0.995416	+	0.0956421i	$0.969510\pi$
$338$	6573.59	1.05786
$339$	1540.08	0.246742
$340$	0	0
$341$	4693.65	0.745383
$342$	34522.9	5.45844
$343$	− 6653.89i	− 1.04745i
$344$	2946.95	0.461886
$345$	189.183i	0.0295225i
$346$	− 8386.17i	− 1.30302i
$347$	4879.95i	0.754956i	0.926019	+	0.377478i	$0.123209\pi$
$347$	4879.95i	0.754956i	−0.926019	+	0.377478i	$0.876791\pi$
$348$	2382.64	0.367020
$349$	−1965.33	−0.301438	−0.150719	−	0.988577i	$-0.548159\pi$
$349$	−1965.33	−0.301438	−0.150719	+	0.988577i	$0.548159\pi$
$350$	2826.88i	0.431723i
$351$	8892.99i	1.35234i
$352$	− 3293.44i	− 0.498696i
$353$	−7433.23	−1.12077	−0.560384	−	0.828233i	$-0.689346\pi$
$353$	−7433.23	−1.12077	−0.560384	+	0.828233i	$0.689346\pi$
$354$	− 30019.8i	− 4.50716i
$355$	−7972.32	−1.19191
$356$	−1004.60	−0.149562
$357$	0	0
$358$	−11778.2	−1.73882
$359$	5316.51	0.781601	0.390800	−	0.920475i	$-0.372198\pi$
$359$	5316.51	0.781601	0.390800	+	0.920475i	$0.372198\pi$
$360$	− 8478.64i	− 1.24129i
$361$	8837.27	1.28842
$362$	6239.73i	0.905947i
$363$	10586.1i	1.53065i
$364$	3901.05i	0.561732i
$365$	7085.22	1.01605
$366$	30424.0	4.34506
$367$	− 413.067i	− 0.0587518i	−0.999568	−	0.0293759i	$-0.990648\pi$
$367$	− 413.067i	− 0.0587518i	0.999568	−	0.0293759i	$-0.00935199\pi$
$368$	− 52.8141i	− 0.00748132i
$369$	− 11208.2i	− 1.58124i
$370$	−2795.07	−0.392727
$371$	2538.43i	0.355226i
$372$	−35334.7	−4.92478
$373$	5546.88	0.769990	0.384995	−	0.922919i	$-0.374203\pi$
$373$	5546.88	0.769990	0.384995	+	0.922919i	$0.374203\pi$
$374$	0	0
$375$	14314.2	1.97115
$376$	−3720.93	−0.510351
$377$	581.270i	0.0794083i
$378$	18665.3	2.53979
$379$	− 13601.4i	− 1.84342i	−0.387883	−	0.921708i	$-0.626794\pi$
$379$	− 13601.4i	− 1.84342i	0.387883	−	0.921708i	$-0.373206\pi$
$380$	− 12505.0i	− 1.68814i
$381$	3158.01i	0.424645i
$382$	−17807.2	−2.38506
$383$	−2571.35	−0.343055	−0.171527	−	0.985179i	$-0.554870\pi$
$383$	−2571.35	−0.343055	−0.171527	+	0.985179i	$0.554870\pi$
$384$	17180.5i	2.28317i
$385$	1585.07i	0.209826i
$386$	16892.8i	2.22751i
$387$	11641.6	1.52914
$388$	8771.60i	1.14771i
$389$	9976.83	1.30037	0.650187	−	0.759775i	$-0.274691\pi$
$389$	9976.83	1.30037	0.650187	+	0.759775i	$0.274691\pi$
$390$	9617.98	1.24878
$391$	0	0
$392$	−2884.99	−0.371719
$393$	−6592.68	−0.846200
$394$	2906.14i	0.371597i
$395$	−440.212	−0.0560747
$396$	10459.3i	1.32727i
$397$	3477.77i	0.439658i	0.975538	+	0.219829i	$0.0705500\pi$
$397$	3477.77i	0.439658i	−0.975538	+	0.219829i	$0.929450\pi$
$398$	− 13618.6i	− 1.71517i
$399$	14978.0	1.87929
$400$	−1150.13	−0.143766
$401$	10435.4i	1.29955i	0.760128	+	0.649773i	$0.225136\pi$
$401$	10435.4i	1.29955i	−0.760128	+	0.649773i	$0.774864\pi$
$402$	− 3215.69i	− 0.398966i
$403$	− 8620.27i	− 1.06552i
$404$	11470.2	1.41253
$405$	− 12685.9i	− 1.55646i
$406$	1220.02	0.149134
$407$	1062.92	0.129452
$408$	0	0
$409$	−14641.1	−1.77006	−0.885029	−	0.465536i	$-0.845861\pi$
$409$	−14641.1	−1.77006	−0.885029	+	0.465536i	$0.845861\pi$
$410$	−6868.22	−0.827310
$411$	19514.0i	2.34198i
$412$	−6898.19	−0.824877
$413$	− 9086.24i	− 1.08258i
$414$	639.207i	0.0758824i
$415$	10212.2i	1.20795i
$416$	−6048.67	−0.712886
$417$	−17154.6	−2.01454
$418$	8044.92i	0.941363i
$419$	− 331.177i	− 0.0386134i	−0.999814	−	0.0193067i	$-0.993854\pi$
$419$	− 331.177i	− 0.0386134i	0.999814	−	0.0193067i	$-0.00614590\pi$
$420$	− 11932.7i	− 1.38633i
$421$	601.612	0.0696455	0.0348227	−	0.999394i	$-0.488913\pi$
$421$	601.612	0.0696455	0.0348227	+	0.999394i	$0.488913\pi$
$422$	6219.06i	0.717391i
$423$	−14699.1	−1.68959
$424$	3164.14	0.362416
$425$	0	0
$426$	−38611.2	−4.39136
$427$	9208.58	1.04364
$428$	− 9090.31i	− 1.02663i
$429$	−3657.56	−0.411628
$430$	− 7133.77i	− 0.800049i
$431$	− 10219.6i	− 1.14214i	−0.820903	−	0.571068i	$-0.806529\pi$
$431$	− 10219.6i	− 1.14214i	0.820903	−	0.571068i	$-0.193471\pi$
$432$	7594.07i	0.845763i
$433$	1098.15	0.121879	0.0609396	−	0.998141i	$-0.480590\pi$
$433$	1098.15	0.121879	0.0609396	+	0.998141i	$0.480590\pi$
$434$	−18092.9	−2.00112
$435$	− 1778.02i	− 0.195976i
$436$	5666.17i	0.622386i
$437$	290.623i	0.0318133i
$438$	34314.9	3.74344
$439$	− 1804.38i	− 0.196170i	−0.995178	−	0.0980850i	$-0.968728\pi$
$439$	− 1804.38i	− 0.196170i	0.995178	−	0.0980850i	$-0.0312717\pi$
$440$	1975.79	0.214073
$441$	−11396.8	−1.23063
$442$	0	0
$443$	−8570.23	−0.919151	−0.459575	−	0.888139i	$-0.651998\pi$
$443$	−8570.23	−0.919151	−0.459575	+	0.888139i	$0.651998\pi$
$444$	−8001.86	−0.855297
$445$	749.676i	0.0798608i
$446$	−15931.9	−1.69147
$447$	8135.74i	0.860867i
$448$	10391.1i	1.09583i
$449$	− 4142.84i	− 0.435440i	−0.976011	−	0.217720i	$-0.930138\pi$
$449$	− 4142.84i	− 0.435440i	0.976011	−	0.217720i	$-0.0698619\pi$
$450$	13919.9	1.45821
$451$	2611.87	0.272701
$452$	− 1884.85i	− 0.196142i
$453$	9532.73i	0.988713i
$454$	− 17592.1i	− 1.81859i
$455$	2911.12	0.299946
$456$	− 18670.0i	− 1.91733i
$457$	10190.2	1.04306	0.521530	−	0.853233i	$-0.325361\pi$
$457$	10190.2	1.04306	0.521530	+	0.853233i	$0.325361\pi$
$458$	−13352.8	−1.36230
$459$	0	0
$460$	231.535	0.0234682
$461$	10235.7	1.03411	0.517055	−	0.855952i	$-0.327028\pi$
$461$	10235.7	1.03411	0.517055	+	0.855952i	$0.327028\pi$
$462$	7676.77i	0.773064i
$463$	−4240.64	−0.425658	−0.212829	−	0.977090i	$-0.568268\pi$
$463$	−4240.64	−0.425658	−0.212829	+	0.977090i	$0.568268\pi$
$464$	496.369i	0.0496624i
$465$	26368.2i	2.62967i
$466$	5004.86i	0.497523i
$467$	−59.5197	−0.00589774	−0.00294887	−	0.999996i	$-0.500939\pi$
$467$	−59.5197	−0.00589774	−0.00294887	+	0.999996i	$0.500939\pi$
$468$	19209.3	1.89733
$469$	− 973.309i	− 0.0958278i
$470$	9007.37i	0.883998i
$471$	19683.9i	1.92566i
$472$	−11325.9	−1.10449
$473$	2712.85i	0.263715i
$474$	−2132.02	−0.206597
$475$	6328.87	0.611344
$476$	0	0
$477$	12499.6	1.19983
$478$	−12679.2	−1.21324
$479$	14778.8i	1.40973i	0.709341	+	0.704865i	$0.248993\pi$
$479$	14778.8i	1.40973i	−0.709341	+	0.704865i	$0.751007\pi$
$480$	18502.0	1.75937
$481$	− 1952.14i	− 0.185052i
$482$	− 3176.71i	− 0.300198i
$483$	277.324i	0.0261256i
$484$	12956.0	1.21675
$485$	6545.72	0.612837
$486$	− 21604.8i	− 2.01649i
$487$	− 3920.93i	− 0.364834i	−0.983221	−	0.182417i	$-0.941608\pi$
$487$	− 3920.93i	− 0.364834i	0.983221	−	0.182417i	$-0.0583921\pi$
$488$	− 11478.5i	− 1.06476i
$489$	−24145.7	−2.23294
$490$	6983.79i	0.643868i
$491$	−1593.61	−0.146473	−0.0732367	−	0.997315i	$-0.523333\pi$
$491$	−1593.61	−0.146473	−0.0732367	+	0.997315i	$0.523333\pi$
$492$	−19662.6	−1.80175
$493$	0	0
$494$	14775.1	1.34568
$495$	7805.13	0.708717
$496$	− 7361.18i	− 0.666385i
$497$	−11686.6	−1.05476
$498$	49459.5i	4.45047i
$499$	− 297.390i	− 0.0266794i	−0.999911	−	0.0133397i	$-0.995754\pi$
$499$	− 297.390i	− 0.0266794i	0.999911	−	0.0133397i	$-0.00424629\pi$
$500$	− 17518.7i	− 1.56692i
$501$	−23369.5	−2.08398
$502$	27576.3	2.45178
$503$	− 15738.5i	− 1.39512i	−0.716525	−	0.697561i	$-0.754268\pi$
$503$	− 15738.5i	− 1.39512i	0.716525	−	0.697561i	$-0.245732\pi$
$504$	− 12428.9i	− 1.09846i
$505$	− 8559.50i	− 0.754243i
$506$	−148.955	−0.0130867
$507$	− 14043.6i	− 1.23018i
$508$	3864.99	0.337561
$509$	−19131.6	−1.66599	−0.832997	−	0.553277i	$-0.813377\pi$
$509$	−19131.6	−1.66599	−0.832997	+	0.553277i	$0.813377\pi$
$510$	0	0
$511$	10386.2	0.899139
$512$	−8037.53	−0.693773
$513$	− 41788.3i	− 3.59649i
$514$	16284.7	1.39745
$515$	5147.70i	0.440456i
$516$	− 20422.9i	− 1.74238i
$517$	− 3425.35i	− 0.291386i
$518$	−4097.30	−0.347539
$519$	−17916.0	−1.51527
$520$	− 3628.69i	− 0.306017i
$521$	− 16429.7i	− 1.38157i	−0.723058	−	0.690787i	$-0.757264\pi$
$521$	− 16429.7i	− 1.38157i	0.723058	−	0.690787i	$-0.242736\pi$
$522$	− 6007.53i	− 0.503722i
$523$	3606.66	0.301546	0.150773	−	0.988568i	$-0.451824\pi$
$523$	3606.66	0.301546	0.150773	+	0.988568i	$0.451824\pi$
$524$	8068.58i	0.672667i
$525$	6039.25	0.502047
$526$	31346.1	2.59839
$527$	0	0
$528$	−3123.33	−0.257434
$529$	12161.6	0.999558
$530$	− 7659.55i	− 0.627754i
$531$	−44741.9	−3.65656
$532$	− 18331.1i	− 1.49390i
$533$	− 4796.90i	− 0.389826i
$534$	3630.80i	0.294232i
$535$	−6783.55	−0.548184
$536$	−1213.22	−0.0977674
$537$	25162.6i	2.02206i
$538$	2587.45i	0.207347i
$539$	− 2655.82i	− 0.212234i
$540$	−33292.2	−2.65309
$541$	− 24648.2i	− 1.95880i	−0.201931	−	0.979400i	$-0.564722\pi$
$541$	− 24648.2i	− 1.95880i	0.201931	−	0.979400i	$-0.435278\pi$
$542$	1132.08	0.0897181
$543$	13330.4	1.05352
$544$	0	0
$545$	4228.32	0.332333
$546$	14099.0	1.10509
$547$	− 6616.10i	− 0.517156i	−0.965990	−	0.258578i	$-0.916746\pi$
$547$	− 6616.10i	− 0.517156i	0.965990	−	0.258578i	$-0.0832539\pi$
$548$	23882.6	1.86170
$549$	− 45344.4i	− 3.52505i
$550$	3243.78i	0.251482i
$551$	− 2731.40i	− 0.211182i
$552$	345.683	0.0266544
$553$	−645.308	−0.0496226
$554$	− 24256.7i	− 1.86023i
$555$	5971.31i	0.456699i
$556$	20995.0i	1.60141i
$557$	6856.99	0.521616	0.260808	−	0.965391i	$-0.416011\pi$
$557$	6856.99	0.521616	0.260808	+	0.965391i	$0.416011\pi$
$558$	89092.1i	6.75909i
$559$	4982.38	0.376980
$560$	2485.91	0.187588
$561$	0	0
$562$	15979.6	1.19939
$563$	−20537.2	−1.53737	−0.768684	−	0.639629i	$-0.779088\pi$
$563$	−20537.2	−1.53737	−0.768684	+	0.639629i	$0.779088\pi$
$564$	25786.7i	1.92521i
$565$	−1406.55	−0.104733
$566$	− 16713.8i	− 1.24122i
$567$	− 18596.2i	− 1.37737i
$568$	14567.3i	1.07611i
$569$	12241.2	0.901891	0.450945	−	0.892552i	$-0.351087\pi$
$569$	12241.2	0.901891	0.450945	+	0.892552i	$0.351087\pi$
$570$	−45195.0	−3.32107
$571$	− 5037.80i	− 0.369221i	−0.982812	−	0.184611i	$-0.940898\pi$
$571$	− 5037.80i	− 0.369221i	0.982812	−	0.184611i	$-0.0591024\pi$
$572$	4476.37i	0.327214i
$573$	38042.7i	2.77357i
$574$	−10068.1	−0.732118
$575$	117.182i	0.00849882i
$576$	51167.3	3.70134
$577$	−6963.10	−0.502388	−0.251194	−	0.967937i	$-0.580823\pi$
$577$	−6963.10	−0.502388	−0.251194	+	0.967937i	$0.580823\pi$
$578$	0	0
$579$	36089.2	2.59036
$580$	−2176.07	−0.155787
$581$	14970.1i	1.06896i
$582$	31702.0	2.25789
$583$	2912.80i	0.206922i
$584$	− 12946.4i	− 0.917338i
$585$	− 14334.8i	− 1.01311i
$586$	−36987.2	−2.60739
$587$	−4148.82	−0.291721	−0.145860	−	0.989305i	$-0.546595\pi$
$587$	−4148.82	−0.291721	−0.145860	+	0.989305i	$0.546595\pi$
$588$	19993.5i	1.40224i
$589$	40506.8i	2.83371i
$590$	27417.1i	1.91313i
$591$	6208.59	0.432127
$592$	− 1667.01i	− 0.115732i
$593$	−11579.9	−0.801903	−0.400951	−	0.916099i	$-0.631320\pi$
$593$	−11579.9	−0.801903	−0.400951	+	0.916099i	$0.631320\pi$
$594$	21418.1	1.47945
$595$	0	0
$596$	9957.08	0.684326
$597$	−29094.4	−1.99456
$598$	273.568i	0.0187074i
$599$	−4270.75	−0.291316	−0.145658	−	0.989335i	$-0.546530\pi$
$599$	−4270.75	−0.291316	−0.145658	+	0.989335i	$0.546530\pi$
$600$	− 7527.90i	− 0.512208i
$601$	− 8584.45i	− 0.582640i	−0.956626	−	0.291320i	$-0.905906\pi$
$601$	− 8584.45i	− 0.582640i	0.956626	−	0.291320i	$-0.0940945\pi$
$602$	− 10457.4i	− 0.707994i
$603$	−4792.71	−0.323672
$604$	11666.8	0.785954
$605$	− 9668.25i	− 0.649703i
$606$	− 41455.0i	− 2.77887i
$607$	− 27576.2i	− 1.84396i	−0.387241	−	0.921979i	$-0.626572\pi$
$607$	− 27576.2i	− 1.84396i	0.387241	−	0.921979i	$-0.373428\pi$
$608$	28422.8	1.89588
$609$	− 2606.40i	− 0.173427i
$610$	−27786.3	−1.84432
$611$	−6290.93	−0.416537
$612$	0	0
$613$	−4036.96	−0.265989	−0.132995	−	0.991117i	$-0.542459\pi$
$613$	−4036.96	−0.265989	−0.132995	+	0.991117i	$0.542459\pi$
$614$	18006.9	1.18355
$615$	14673.0i	0.962072i
$616$	2896.31	0.189441
$617$	9008.21i	0.587775i	0.955840	+	0.293887i	$0.0949490\pi$
$617$	9008.21i	0.587775i	−0.955840	+	0.293887i	$0.905051\pi$
$618$	24931.1i	1.62278i
$619$	5131.94i	0.333231i	0.986022	+	0.166616i	$0.0532839\pi$
$619$	5131.94i	0.333231i	−0.986022	+	0.166616i	$0.946716\pi$
$620$	32271.2	2.09039
$621$	773.729	0.0499979
$622$	− 10769.6i	− 0.694249i
$623$	1098.95i	0.0706718i
$624$	5736.24i	0.368002i
$625$	−6758.66	−0.432554
$626$	− 19252.4i	− 1.22920i
$627$	17186.9	1.09470
$628$	24090.5	1.53076
$629$	0	0
$630$	−30086.9	−1.90269
$631$	14562.4	0.918734	0.459367	−	0.888247i	$-0.348076\pi$
$631$	14562.4	0.918734	0.459367	+	0.888247i	$0.348076\pi$
$632$	804.373i	0.0506270i
$633$	13286.2	0.834248
$634$	16678.2i	1.04476i
$635$	− 2884.21i	− 0.180246i
$636$	− 21928.1i	− 1.36715i
$637$	−4877.62	−0.303388
$638$	1399.94	0.0868718
$639$	57546.7i	3.56262i
$640$	− 15690.9i	− 0.969121i
$641$	19333.1i	1.19128i	0.803252	+	0.595640i	$0.203101\pi$
$641$	19333.1i	1.19128i	−0.803252	+	0.595640i	$0.796899\pi$
$642$	−32853.8	−2.01968
$643$	19912.0i	1.22123i	0.791928	+	0.610615i	$0.209078\pi$
$643$	19912.0i	1.22123i	−0.791928	+	0.610615i	$0.790922\pi$
$644$	339.408	0.0207679
$645$	−15240.4	−0.930371
$646$	0	0
$647$	7742.24	0.470447	0.235223	−	0.971941i	$-0.424418\pi$
$647$	7742.24	0.470447	0.235223	+	0.971941i	$0.424418\pi$
$648$	−23180.1	−1.40525
$649$	− 10426.3i	− 0.630611i
$650$	5957.46	0.359494
$651$	38653.1i	2.32709i
$652$	29551.2i	1.77502i
$653$	10554.3i	0.632498i	0.948676	+	0.316249i	$0.102423\pi$
$653$	10554.3i	0.632498i	−0.948676	+	0.316249i	$0.897577\pi$
$654$	20478.4	1.22442
$655$	6021.09	0.359181
$656$	− 4096.26i	− 0.243799i
$657$	− 51143.3i	− 3.03697i
$658$	13203.9i	0.782283i
$659$	−21754.7	−1.28595	−0.642977	−	0.765885i	$-0.722301\pi$
$659$	−21754.7	−1.28595	−0.642977	+	0.765885i	$0.722301\pi$
$660$	− 13692.6i	− 0.807549i
$661$	−24822.0	−1.46061	−0.730306	−	0.683120i	$-0.760623\pi$
$661$	−24822.0	−1.46061	−0.730306	+	0.683120i	$0.760623\pi$
$662$	−41249.2	−2.42175
$663$	0	0
$664$	18660.2	1.09060
$665$	−13679.4	−0.797690
$666$	20175.7i	1.17386i
$667$	50.5730	0.00293582
$668$	28601.2i	1.65661i
$669$	34036.3i	1.96700i
$670$	2936.89i	0.169346i
$671$	10566.7	0.607930
$672$	27122.1	1.55693
$673$	− 23576.4i	− 1.35038i	−0.737645	−	0.675189i	$-0.764062\pi$
$673$	− 23576.4i	− 1.35038i	0.737645	−	0.675189i	$-0.235938\pi$
$674$	− 5234.39i	− 0.299141i
$675$	− 16849.4i	− 0.960791i
$676$	−17187.6	−0.977899
$677$	− 5217.93i	− 0.296221i	−0.988971	−	0.148110i	$-0.952681\pi$
$677$	− 5217.93i	− 0.296221i	0.988971	−	0.148110i	$-0.0473191\pi$
$678$	−6812.16	−0.385869
$679$	9595.38	0.542322
$680$	0	0
$681$	−37583.3	−2.11482
$682$	−20761.2	−1.16567
$683$	13939.9i	0.780958i	0.920612	+	0.390479i	$0.127691\pi$
$683$	13939.9i	0.780958i	−0.920612	+	0.390479i	$0.872309\pi$
$684$	−90264.9	−5.04586
$685$	− 17822.1i	− 0.994086i
$686$	29431.9i	1.63807i
$687$	28526.5i	1.58421i
$688$	4254.64	0.235766
$689$	5349.59	0.295795
$690$	− 836.806i	− 0.0461691i
$691$	14140.0i	0.778455i	0.921142	+	0.389228i	$0.127258\pi$
$691$	14140.0i	0.778455i	−0.921142	+	0.389228i	$0.872742\pi$
$692$	21926.8i	1.20452i
$693$	11441.6	0.627170
$694$	− 21585.3i	− 1.18064i
$695$	15667.3	0.855100
$696$	−3248.87	−0.176937
$697$	0	0
$698$	8693.18	0.471406
$699$	10692.2	0.578565
$700$	− 7391.25i	− 0.399090i
$701$	23850.7	1.28506	0.642532	−	0.766259i	$-0.277884\pi$
$701$	23850.7	1.28506	0.642532	+	0.766259i	$0.277884\pi$
$702$	− 39336.0i	− 2.11487i
$703$	9173.13i	0.492135i
$704$	11923.6i	0.638334i
$705$	19243.1	1.02799
$706$	32879.1	1.75272
$707$	− 12547.4i	− 0.667458i
$708$	78490.9i	4.16648i
$709$	− 28474.5i	− 1.50830i	−0.656703	−	0.754149i	$-0.728049\pi$
$709$	− 28474.5i	− 1.50830i	0.656703	−	0.754149i	$-0.271951\pi$
$710$	35263.6	1.86397
$711$	3177.59i	0.167607i
$712$	1369.84	0.0721023
$713$	−750.001	−0.0393938
$714$	0	0
$715$	3340.44	0.174721
$716$	30795.8	1.60739
$717$	27087.4i	1.41087i
$718$	−23516.3	−1.22231
$719$	5053.95i	0.262143i	0.991373	+	0.131071i	$0.0418417\pi$
$719$	5053.95i	0.262143i	−0.991373	+	0.131071i	$0.958158\pi$
$720$	− 12241.0i	− 0.633604i
$721$	7546.02i	0.389776i
$722$	−39089.5	−2.01490
$723$	−6786.63	−0.349098
$724$	− 16314.6i	− 0.837470i
$725$	− 1101.32i	− 0.0564167i
$726$	− 46824.9i	− 2.39371i
$727$	20929.6	1.06773	0.533863	−	0.845571i	$-0.320740\pi$
$727$	20929.6	1.06773	0.533863	+	0.845571i	$0.320740\pi$
$728$	− 5319.31i	− 0.270806i
$729$	−6468.60	−0.328639
$730$	−31339.8	−1.58895
$731$	0	0
$732$	−79547.8	−4.01663
$733$	20140.2	1.01487	0.507433	−	0.861691i	$-0.330594\pi$
$733$	20140.2	1.01487	0.507433	+	0.861691i	$0.330594\pi$
$734$	1827.10i	0.0918795i
$735$	14919.9	0.748749
$736$	526.261i	0.0263563i
$737$	− 1116.85i	− 0.0558205i
$738$	49576.9i	2.47283i
$739$	−5481.57	−0.272859	−0.136430	−	0.990650i	$-0.543563\pi$
$739$	−5481.57	−0.272859	−0.136430	+	0.990650i	$0.543563\pi$
$740$	7308.10	0.363042
$741$	− 31565.2i	− 1.56488i
$742$	− 11228.1i	− 0.555523i
$743$	− 32587.9i	− 1.60906i	−0.593911	−	0.804531i	$-0.702417\pi$
$743$	− 32587.9i	− 1.60906i	0.593911	−	0.804531i	$-0.297583\pi$
$744$	48180.9	2.37419
$745$	− 7430.37i	− 0.365406i
$746$	−24535.3	−1.20415
$747$	73715.1	3.61057
$748$	0	0
$749$	−9944.02	−0.485109
$750$	−63315.4	−3.08260
$751$	3824.32i	0.185821i	0.995674	+	0.0929104i	$0.0296170\pi$
$751$	3824.32i	0.185821i	−0.995674	+	0.0929104i	$0.970383\pi$
$752$	−5372.07	−0.260504
$753$	− 58913.2i	− 2.85115i
$754$	− 2571.11i	− 0.124183i
$755$	− 8706.24i	− 0.419672i
$756$	−48803.0	−2.34782
$757$	16870.1	0.809980	0.404990	−	0.914321i	$-0.367275\pi$
$757$	16870.1	0.809980	0.404990	+	0.914321i	$0.367275\pi$
$758$	60162.3i	2.88284i
$759$	318.223i	0.0152184i
$760$	17051.3i	0.813836i
$761$	−19951.4	−0.950378	−0.475189	−	0.879884i	$-0.657620\pi$
$761$	−19951.4	−0.950378	−0.475189	+	0.879884i	$0.657620\pi$
$762$	− 13968.7i	− 0.664084i
$763$	6198.30	0.294094
$764$	46559.2	2.20478
$765$	0	0
$766$	11373.8	0.536489
$767$	−19148.7	−0.901458
$768$	− 13901.8i	− 0.653175i
$769$	−19560.4	−0.917249	−0.458625	−	0.888630i	$-0.651658\pi$
$769$	−19560.4	−0.917249	−0.458625	+	0.888630i	$0.651658\pi$
$770$	− 7011.19i	− 0.328137i
$771$	− 34790.2i	− 1.62508i
$772$	− 44168.5i	− 2.05914i
$773$	29989.8	1.39542	0.697710	−	0.716381i	$-0.254203\pi$
$773$	29989.8	1.39542	0.697710	+	0.716381i	$0.254203\pi$
$774$	−51493.8	−2.39135
$775$	16332.7i	0.757016i
$776$	− 11960.6i	− 0.553299i
$777$	8753.35i	0.404150i
$778$	−44130.1	−2.03360
$779$	22540.7i	1.03672i
$780$	−25147.5	−1.15439
$781$	−13410.2	−0.614409
$782$	0	0
$783$	−7271.82	−0.331895
$784$	−4165.19	−0.189741
$785$	− 17977.3i	− 0.817373i
$786$	29161.1	1.32334
$787$	36288.4i	1.64364i	0.569749	+	0.821819i	$0.307040\pi$
$787$	36288.4i	1.64364i	−0.569749	+	0.821819i	$0.692960\pi$
$788$	− 7598.50i	− 0.343509i
$789$	− 66966.8i	− 3.02165i
$790$	1947.17	0.0876928
$791$	−2061.87	−0.0926822
$792$	− 14261.8i	− 0.639864i
$793$	− 19406.5i	− 0.869036i
$794$	− 15383.1i	− 0.687563i
$795$	−16363.6	−0.730010
$796$	35607.7i	1.58553i
$797$	29090.7	1.29291	0.646453	−	0.762954i	$-0.276252\pi$
$797$	29090.7	1.29291	0.646453	+	0.762954i	$0.276252\pi$
$798$	−66251.5	−2.93894
$799$	0	0
$800$	11460.3	0.506480
$801$	5411.39	0.238704
$802$	− 46158.4i	− 2.03231i
$803$	11918.0	0.523757
$804$	8407.87i	0.368809i
$805$	− 253.280i	− 0.0110894i
$806$	38129.7i	1.66633i
$807$	5527.75	0.241123
$808$	−15640.3	−0.680968
$809$	3569.56i	0.155129i	0.996987	+	0.0775644i	$0.0247143\pi$
$809$	3569.56i	0.155129i	−0.996987	+	0.0775644i	$0.975286\pi$
$810$	56112.8i	2.43408i
$811$	− 24700.0i	− 1.06946i	−0.845022	−	0.534732i	$-0.820413\pi$
$811$	− 24700.0i	− 1.06946i	0.845022	−	0.534732i	$-0.179587\pi$
$812$	−3189.90	−0.137861
$813$	− 2418.55i	− 0.104332i
$814$	−4701.57	−0.202445
$815$	22052.3	0.947801
$816$	0	0
$817$	−23412.3	−1.00256
$818$	64761.2	2.76812
$819$	− 21013.3i	− 0.896539i
$820$	17957.9	0.764776
$821$	6655.21i	0.282909i	0.989945	+	0.141455i	$0.0451779\pi$
$821$	6655.21i	0.282909i	−0.989945	+	0.141455i	$0.954822\pi$
$822$	− 86315.5i	− 3.66253i
$823$	4887.46i	0.207006i	0.994629	+	0.103503i	$0.0330052\pi$
$823$	4887.46i	0.207006i	−0.994629	+	0.103503i	$0.966995\pi$
$824$	9406.08	0.397665
$825$	6929.91	0.292447
$826$	40190.8i	1.69300i
$827$	− 21935.8i	− 0.922347i	−0.887310	−	0.461174i	$-0.847429\pi$
$827$	− 21935.8i	− 0.922347i	0.887310	−	0.461174i	$-0.152571\pi$
$828$	− 1671.29i	− 0.0701467i
$829$	41942.6	1.75721	0.878605	−	0.477550i	$-0.158475\pi$
$829$	41942.6	1.75721	0.878605	+	0.477550i	$0.158475\pi$
$830$	− 45171.4i	− 1.88906i
$831$	−51821.3	−2.16325
$832$	21898.6	0.912497
$833$	0	0
$834$	75879.2	3.15046
$835$	21343.3	0.884571
$836$	− 21034.5i	− 0.870209i
$837$	107842.	4.45347
$838$	1464.88i	0.0603859i
$839$	42161.2i	1.73488i	0.497540	+	0.867441i	$0.334237\pi$
$839$	42161.2i	1.73488i	−0.497540	+	0.867441i	$0.665763\pi$
$840$	16271.0i	0.668336i
$841$	23913.7	0.980511
$842$	−2661.08	−0.108916
$843$	− 34138.4i	− 1.39477i
$844$	− 16260.6i	− 0.663166i
$845$	12826.0i	0.522164i
$846$	65018.0	2.64227
$847$	− 14172.7i	− 0.574947i
$848$	4568.22	0.184992
$849$	−35706.8	−1.44341
$850$	0	0
$851$	−169.844	−0.00684159
$852$	100954.	4.05943
$853$	− 8674.96i	− 0.348212i	−0.984727	−	0.174106i	$-0.944296\pi$
$853$	− 8674.96i	− 0.348212i	0.984727	−	0.174106i	$-0.0557035\pi$
$854$	−40731.9	−1.63211
$855$	67359.3i	2.69431i
$856$	12395.2i	0.494928i
$857$	− 26939.7i	− 1.07380i	−0.843647	−	0.536898i	$-0.819596\pi$
$857$	− 26939.7i	− 1.07380i	0.843647	−	0.536898i	$-0.180404\pi$
$858$	16178.3	0.643728
$859$	−3089.68	−0.122722	−0.0613611	−	0.998116i	$-0.519544\pi$
$859$	−3089.68	−0.122722	−0.0613611	+	0.998116i	$0.519544\pi$
$860$	18652.2i	0.739576i
$861$	21509.2i	0.851374i
$862$	45203.9i	1.78614i
$863$	−10012.6	−0.394939	−0.197469	−	0.980309i	$-0.563272\pi$
$863$	−10012.6	−0.394939	−0.197469	+	0.980309i	$0.563272\pi$
$864$	− 75670.3i	− 2.97958i
$865$	16362.6	0.643175
$866$	−4857.40	−0.190602
$867$	0	0
$868$	47306.4	1.84987
$869$	−740.477	−0.0289056
$870$	7864.65i	0.306479i
$871$	−2051.19	−0.0797954
$872$	− 7726.15i	− 0.300046i
$873$	− 47249.0i	− 1.83177i
$874$	− 1285.50i	− 0.0497514i
$875$	−19163.9	−0.740411
$876$	−89720.9	−3.46049
$877$	− 17171.8i	− 0.661177i	−0.943775	−	0.330588i	$-0.892753\pi$
$877$	− 17171.8i	− 0.661177i	0.943775	−	0.330588i	$-0.107247\pi$
$878$	7981.26i	0.306782i
$879$	79018.4i	3.03211i
$880$	2852.53	0.109271
$881$	46614.5i	1.78261i	0.453400	+	0.891307i	$0.350211\pi$
$881$	46614.5i	1.78261i	−0.453400	+	0.891307i	$0.649789\pi$
$882$	50411.2	1.92453
$883$	−48461.9	−1.84697	−0.923484	−	0.383637i	$-0.874671\pi$
$883$	−48461.9	−1.84697	−0.923484	+	0.383637i	$0.874671\pi$
$884$	0	0
$885$	58573.0	2.22476
$886$	37908.3	1.43742
$887$	7302.52i	0.276431i	0.990402	+	0.138216i	$0.0441367\pi$
$887$	7302.52i	0.276431i	−0.990402	+	0.138216i	$0.955863\pi$
$888$	10911.0	0.412330
$889$	− 4227.96i	− 0.159507i
$890$	− 3316.01i	− 0.124891i
$891$	− 21338.8i	− 0.802329i
$892$	41656.0	1.56362
$893$	29561.2	1.10776
$894$	− 35986.5i	− 1.34627i
$895$	− 22981.0i	− 0.858292i
$896$	− 23001.3i	− 0.857612i
$897$	584.443	0.0217547
$898$	18324.8i	0.680966i
$899$	7048.82	0.261503
$900$	−36395.6	−1.34799
$901$	0	0
$902$	−11553.0	−0.426465
$903$	−22340.9	−0.823320
$904$	2570.11i	0.0945581i
$905$	−12174.6	−0.447180
$906$	− 42165.7i	− 1.54621i
$907$	− 30995.0i	− 1.13470i	−0.823478	−	0.567349i	$-0.807969\pi$
$907$	− 30995.0i	− 1.13470i	0.823478	−	0.567349i	$-0.192031\pi$
$908$	45997.0i	1.68113i
$909$	−61785.1	−2.25444
$910$	−12876.6	−0.469072
$911$	29080.3i	1.05760i	0.848746	+	0.528801i	$0.177358\pi$
$911$	29080.3i	1.05760i	−0.848746	+	0.528801i	$0.822642\pi$
$912$	− 26954.7i	− 0.978683i
$913$	17177.9i	0.622679i
$914$	−45074.0	−1.63120
$915$	59361.7i	2.14474i
$916$	34912.7	1.25933
$917$	8826.33	0.317853
$918$	0	0
$919$	−52815.9	−1.89579	−0.947897	−	0.318577i	$-0.896795\pi$
$919$	−52815.9	−1.89579	−0.947897	+	0.318577i	$0.896795\pi$
$920$	−315.712	−0.0113138
$921$	− 38469.4i	− 1.37634i
$922$	−45275.2	−1.61720
$923$	24628.8i	0.878297i
$924$	− 20072.0i	− 0.714631i
$925$	3698.68i	0.131472i
$926$	18757.5	0.665668
$927$	37157.7	1.31653
$928$	− 4946.02i	− 0.174958i
$929$	37249.7i	1.31553i	0.753225	+	0.657763i	$0.228497\pi$
$929$	37249.7i	1.31553i	−0.753225	+	0.657763i	$0.771503\pi$
$930$	− 116633.i	− 4.11242i
$931$	22920.0	0.806846
$932$	− 13085.9i	− 0.459917i
$933$	−23007.9	−0.807336
$934$	263.271	0.00922323
$935$	0	0
$936$	−26193.0	−0.914686
$937$	−48587.8	−1.69402	−0.847009	−	0.531579i	$-0.821599\pi$
$937$	−48587.8	−1.69402	−0.847009	+	0.531579i	$0.821599\pi$
$938$	4305.19i	0.149861i
$939$	−41130.2	−1.42943
$940$	− 23551.0i	− 0.817179i
$941$	17402.8i	0.602886i	0.953484	+	0.301443i	$0.0974684\pi$
$941$	17402.8i	0.602886i	−0.953484	+	0.301443i	$0.902532\pi$
$942$	− 87067.0i	− 3.01146i
$943$	−417.352	−0.0144123
$944$	−16351.8	−0.563777
$945$	36418.8i	1.25365i
$946$	− 11999.7i	− 0.412413i
$947$	− 40669.1i	− 1.39553i	−0.716325	−	0.697766i	$-0.754177\pi$
$947$	− 40669.1i	− 1.39553i	0.716325	−	0.697766i	$-0.245823\pi$
$948$	5574.46	0.190981
$949$	− 21888.3i	− 0.748710i
$950$	−27994.2	−0.956055
$951$	35630.9	1.21494
$952$	0	0
$953$	−46549.9	−1.58227	−0.791133	−	0.611645i	$-0.790508\pi$
$953$	−46549.9	−1.58227	−0.791133	+	0.611645i	$0.790508\pi$
$954$	−55289.0	−1.87636
$955$	− 34744.3i	− 1.17728i
$956$	33151.4	1.12154
$957$	− 2990.79i	− 0.101023i
$958$	− 65370.5i	− 2.20462i
$959$	− 26125.5i	− 0.879704i
$960$	−66984.7	−2.25200
$961$	−74743.4	−2.50893
$962$	8634.81i	0.289394i
$963$	48965.8i	1.63853i
$964$	8305.95i	0.277507i
$965$	−32960.3	−1.09951
$966$	− 1226.68i	− 0.0408568i
$967$	−41120.0	−1.36746	−0.683728	−	0.729737i	$-0.739643\pi$
$967$	−41120.0	−1.36746	−0.683728	+	0.729737i	$0.739643\pi$
$968$	−17666.2	−0.586584
$969$	0	0
$970$	−28953.4	−0.958389
$971$	−25706.4	−0.849595	−0.424798	−	0.905288i	$-0.639655\pi$
$971$	−25706.4	−0.849595	−0.424798	+	0.905288i	$0.639655\pi$
$972$	56488.8i	1.86407i
$973$	22966.7	0.756710
$974$	17343.3i	0.570549i
$975$	− 12727.3i	− 0.418052i
$976$	− 16572.0i	− 0.543500i
$977$	−32403.0	−1.06107	−0.530533	−	0.847664i	$-0.678008\pi$
$977$	−32403.0	−1.06107	−0.530533	+	0.847664i	$0.678008\pi$
$978$	106803.	3.49200
$979$	1261.02i	0.0411670i
$980$	− 18260.1i	− 0.595200i
$981$	− 30521.3i	− 0.993345i
$982$	7048.93	0.229063
$983$	− 29111.3i	− 0.944563i	−0.881448	−	0.472282i	$-0.843431\pi$
$983$	− 29111.3i	− 0.944563i	0.881448	−	0.472282i	$-0.156569\pi$
$984$	26811.2	0.868606
$985$	−5670.30	−0.183422
$986$	0	0
$987$	28208.4	0.909711
$988$	−38631.6	−1.24396
$989$	− 433.488i	− 0.0139374i
$990$	−34524.1	−1.10833
$991$	− 31298.2i	− 1.00325i	−0.865086	−	0.501624i	$-0.832736\pi$
$991$	− 31298.2i	− 1.00325i	0.865086	−	0.501624i	$-0.167264\pi$
$992$	73349.7i	2.34764i
$993$	88123.6i	2.81623i
$994$	51693.0	1.64950
$995$	26571.9	0.846619
$996$	− 129319.i	− 4.11408i
$997$	11418.1i	0.362702i	0.983418	+	0.181351i	$0.0580470\pi$
$997$	11418.1i	0.362702i	−0.983418	+	0.181351i	$0.941953\pi$
$998$	1315.43i	0.0417228i
$999$	24421.7	0.773442

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.b.f.288.3		24
17.4	even	4		289.4.a.i.1.11	yes	12
17.13	even	4		289.4.a.h.1.11	✓	12
17.16	even	2	inner	289.4.b.f.288.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.4.a.h.1.11	✓	12	17.13	even	4
289.4.a.i.1.11	yes	12	17.4	even	4
289.4.b.f.288.3		24	1.1	even	1	trivial
289.4.b.f.288.4		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)

\(f(q)\)	\(=\)	\(q-4.42326 q^{2} +9.44971i q^{3} +11.5652 q^{4} -8.63042i q^{5} -41.7985i q^{6} -12.6513i q^{7} -15.7698 q^{8} -62.2970 q^{9} +O(q^{10})\)	\(q-4.42326 q^{2} +9.44971i q^{3} +11.5652 q^{4} -8.63042i q^{5} -41.7985i q^{6} -12.6513i q^{7} -15.7698 q^{8} -62.2970 q^{9} +38.1746i q^{10} -14.5171i q^{11} +109.288i q^{12} -26.6619 q^{13} +55.9602i q^{14} +81.5550 q^{15} -22.7676 q^{16} +275.556 q^{18} +125.285 q^{19} -99.8126i q^{20} +119.552 q^{21} +64.2131i q^{22} +2.31970i q^{23} -149.020i q^{24} +50.5159 q^{25} +117.932 q^{26} -333.547i q^{27} -146.315i q^{28} -21.8015i q^{29} -360.739 q^{30} +323.318i q^{31} +226.866 q^{32} +137.183 q^{33} -109.186 q^{35} -720.478 q^{36} +73.2182i q^{37} -554.167 q^{38} -251.947i q^{39} +136.100i q^{40} +179.916i q^{41} -528.807 q^{42} -186.872 q^{43} -167.894i q^{44} +537.649i q^{45} -10.2606i q^{46} +235.952 q^{47} -215.147i q^{48} +182.944 q^{49} -223.445 q^{50} -308.350 q^{52} -200.645 q^{53} +1475.36i q^{54} -125.289 q^{55} +199.510i q^{56} +1183.90i q^{57} +96.4337i q^{58} +718.203 q^{59} +943.200 q^{60} +727.874i q^{61} -1430.12i q^{62} +788.141i q^{63} -821.345 q^{64} +230.103i q^{65} -606.795 q^{66} +76.9332 q^{67} -21.9205 q^{69} +482.959 q^{70} -923.747i q^{71} +982.414 q^{72} +820.959i q^{73} -323.863i q^{74} +477.361i q^{75} +1448.94 q^{76} -183.661 q^{77} +1114.43i q^{78} -51.0071i q^{79} +196.494i q^{80} +1469.90 q^{81} -795.815i q^{82} -1183.28 q^{83} +1382.64 q^{84} +826.585 q^{86} +206.018 q^{87} +228.933i q^{88} -86.8644 q^{89} -2378.16i q^{90} +337.309i q^{91} +26.8278i q^{92} -3055.26 q^{93} -1043.68 q^{94} -1081.26i q^{95} +2143.82i q^{96} +758.448i q^{97} -809.206 q^{98} +904.375i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) \) 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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