LMFDB - Embedded newform 289.4.b.f.288.5

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			288.5
Character	$\chi$	$=$	289.288
Dual form			289.4.b.f.288.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.28857 q^{2} -2.85039i q^{3} +10.3918 q^{4} +6.36629i q^{5} +12.2241i q^{6} +29.4308i q^{7} -10.2575 q^{8} +18.8753 q^{9} +O(q^{10})$	\(q-4.28857 q^{2} -2.85039i q^{3} +10.3918 q^{4} +6.36629i q^{5} +12.2241i q^{6} +29.4308i q^{7} -10.2575 q^{8} +18.8753 q^{9} -27.3023i q^{10} +61.3595i q^{11} -29.6207i q^{12} +18.5582 q^{13} -126.216i q^{14} +18.1464 q^{15} -39.1446 q^{16} -80.9480 q^{18} -115.793 q^{19} +66.1574i q^{20} +83.8892 q^{21} -263.144i q^{22} -7.38352i q^{23} +29.2379i q^{24} +84.4703 q^{25} -79.5883 q^{26} -130.762i q^{27} +305.840i q^{28} -164.974i q^{29} -77.8221 q^{30} +127.132i q^{31} +249.934 q^{32} +174.898 q^{33} -187.365 q^{35} +196.149 q^{36} +158.120i q^{37} +496.588 q^{38} -52.8982i q^{39} -65.3022i q^{40} -31.3736i q^{41} -359.764 q^{42} -157.708 q^{43} +637.637i q^{44} +120.166i q^{45} +31.6647i q^{46} -460.704 q^{47} +111.577i q^{48} -523.171 q^{49} -362.257 q^{50} +192.854 q^{52} +166.846 q^{53} +560.783i q^{54} -390.632 q^{55} -301.886i q^{56} +330.056i q^{57} +707.502i q^{58} +343.136 q^{59} +188.574 q^{60} +112.359i q^{61} -545.216i q^{62} +555.515i q^{63} -758.704 q^{64} +118.147i q^{65} -750.063 q^{66} -984.209 q^{67} -21.0459 q^{69} +803.528 q^{70} -524.337i q^{71} -193.613 q^{72} +852.636i q^{73} -678.106i q^{74} -240.773i q^{75} -1203.30 q^{76} -1805.86 q^{77} +226.858i q^{78} +201.021i q^{79} -249.206i q^{80} +136.909 q^{81} +134.548i q^{82} +22.5351 q^{83} +871.761 q^{84} +676.342 q^{86} -470.240 q^{87} -629.395i q^{88} +502.230 q^{89} -515.338i q^{90} +546.184i q^{91} -76.7282i q^{92} +362.377 q^{93} +1975.76 q^{94} -737.175i q^{95} -712.410i q^{96} +680.691i q^{97} +2243.65 q^{98} +1158.18i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.28857	−1.51624	−0.758119	−	0.652116i	$-0.773881\pi$
$2$	−4.28857	−1.51624	−0.758119	+	0.652116i	$0.773881\pi$
$3$	− 2.85039i	− 0.548557i	−0.961650	−	0.274279i	$-0.911561\pi$
$3$	− 2.85039i	− 0.548557i	0.961650	−	0.274279i	$-0.0884391\pi$
$4$	10.3918	1.29898
$5$	6.36629i	0.569418i	0.958614	+	0.284709i	$0.0918971\pi$
$5$	6.36629i	0.569418i	−0.958614	+	0.284709i	$0.908103\pi$
$6$	12.2241i	0.831744i
$7$	29.4308i	1.58911i	0.607190	+	0.794556i	$0.292297\pi$
$7$	29.4308i	1.58911i	−0.607190	+	0.794556i	$0.707703\pi$
$8$	−10.2575	−0.453322
$9$	18.8753	0.699085
$10$	− 27.3023i	− 0.863374i
$11$	61.3595i	1.68187i	0.541136	+	0.840935i	$0.317995\pi$
$11$	61.3595i	1.68187i	−0.541136	+	0.840935i	$0.682005\pi$
$12$	− 29.6207i	− 0.712564i
$13$	18.5582	0.395933	0.197967	−	0.980209i	$-0.436566\pi$
$13$	18.5582	0.395933	0.197967	+	0.980209i	$0.436566\pi$
$14$	− 126.216i	− 2.40947i
$15$	18.1464	0.312359
$16$	−39.1446	−0.611634
$17$	0	0
$17$	0	0
$18$	−80.9480	−1.05998
$19$	−115.793	−1.39815	−0.699075	−	0.715049i	$-0.746404\pi$
$19$	−115.793	−1.39815	−0.699075	+	0.715049i	$0.746404\pi$
$20$	66.1574i	0.739662i
$21$	83.8892	0.871720
$22$	− 263.144i	− 2.55012i
$23$	− 7.38352i	− 0.0669378i	−0.999440	−	0.0334689i	$-0.989345\pi$
$23$	− 7.38352i	− 0.0669378i	0.999440	−	0.0334689i	$-0.0106555\pi$
$24$	29.2379i	0.248673i
$25$	84.4703	0.675763
$26$	−79.5883	−0.600329
$27$	− 130.762i	− 0.932046i
$28$	305.840i	2.06422i
$29$	− 164.974i	− 1.05638i	−0.849127	−	0.528188i	$-0.822872\pi$
$29$	− 164.974i	− 1.05638i	0.849127	−	0.528188i	$-0.177128\pi$
$30$	−77.8221	−0.473610
$31$	127.132i	0.736570i	0.929713	+	0.368285i	$0.120055\pi$
$31$	127.132i	0.736570i	−0.929713	+	0.368285i	$0.879945\pi$
$32$	249.934	1.38070
$33$	174.898	0.922603
$34$	0	0
$35$	−187.365	−0.904870
$36$	196.149	0.908096
$37$	158.120i	0.702559i	0.936271	+	0.351280i	$0.114253\pi$
$37$	158.120i	0.702559i	−0.936271	+	0.351280i	$0.885747\pi$
$38$	496.588	2.11993
$39$	− 52.8982i	− 0.217192i
$40$	− 65.3022i	− 0.258130i
$41$	− 31.3736i	− 0.119506i	−0.998213	−	0.0597528i	$-0.980969\pi$
$41$	− 31.3736i	− 0.119506i	0.998213	−	0.0597528i	$-0.0190313\pi$
$42$	−359.764	−1.32173
$43$	−157.708	−0.559309	−0.279654	−	0.960101i	$-0.590220\pi$
$43$	−157.708	−0.559309	−0.279654	+	0.960101i	$0.590220\pi$
$44$	637.637i	2.18471i
$45$	120.166i	0.398072i
$46$	31.6647i	0.101494i
$47$	−460.704	−1.42980	−0.714900	−	0.699227i	$-0.753528\pi$
$47$	−460.704	−1.42980	−0.714900	+	0.699227i	$0.753528\pi$
$48$	111.577i	0.335517i
$49$	−523.171	−1.52528
$50$	−362.257	−1.02462
$51$	0	0
$52$	192.854	0.514308
$53$	166.846	0.432417	0.216208	−	0.976347i	$-0.430631\pi$
$53$	166.846	0.432417	0.216208	+	0.976347i	$0.430631\pi$
$54$	560.783i	1.41320i
$55$	−390.632	−0.957688
$56$	− 301.886i	− 0.720379i
$57$	330.056i	0.766965i
$58$	707.502i	1.60172i
$59$	343.136	0.757161	0.378581	−	0.925568i	$-0.376412\pi$
$59$	343.136	0.757161	0.378581	+	0.925568i	$0.376412\pi$
$60$	188.574	0.405747
$61$	112.359i	0.235837i	0.993023	+	0.117918i	$0.0376221\pi$
$61$	112.359i	0.235837i	−0.993023	+	0.117918i	$0.962378\pi$
$62$	− 545.216i	− 1.11682i
$63$	555.515i	1.11092i
$64$	−758.704	−1.48184
$65$	118.147i	0.225452i
$66$	−750.063	−1.39889
$67$	−984.209	−1.79463	−0.897316	−	0.441390i	$-0.854486\pi$
$67$	−984.209	−1.79463	−0.897316	+	0.441390i	$0.854486\pi$
$68$	0	0
$69$	−21.0459	−0.0367192
$70$	803.528	1.37200
$71$	− 524.337i	− 0.876442i	−0.898867	−	0.438221i	$-0.855609\pi$
$71$	− 524.337i	− 0.876442i	0.898867	−	0.438221i	$-0.144391\pi$
$72$	−193.613	−0.316910
$73$	852.636i	1.36703i	0.729934	+	0.683517i	$0.239551\pi$
$73$	852.636i	1.36703i	−0.729934	+	0.683517i	$0.760449\pi$
$74$	− 678.106i	− 1.06525i
$75$	− 240.773i	− 0.370695i
$76$	−1203.30	−1.81616
$77$	−1805.86	−2.67268
$78$	226.858i	0.329315i
$79$	201.021i	0.286286i	0.989702	+	0.143143i	$0.0457209\pi$
$79$	201.021i	0.286286i	−0.989702	+	0.143143i	$0.954279\pi$
$80$	− 249.206i	− 0.348276i
$81$	136.909	0.187804
$82$	134.548i	0.181199i
$83$	22.5351	0.0298017	0.0149009	−	0.999889i	$-0.495257\pi$
$83$	22.5351	0.0298017	0.0149009	+	0.999889i	$0.495257\pi$
$84$	871.761	1.13234
$85$	0	0
$86$	676.342	0.848045
$87$	−470.240	−0.579483
$88$	− 629.395i	− 0.762428i
$89$	502.230	0.598161	0.299080	−	0.954228i	$-0.403320\pi$
$89$	502.230	0.598161	0.299080	+	0.954228i	$0.403320\pi$
$90$	− 515.338i	− 0.603572i
$91$	546.184i	0.629182i
$92$	− 76.7282i	− 0.0869507i
$93$	362.377	0.404051
$94$	1975.76	2.16792
$95$	− 737.175i	− 0.796132i
$96$	− 712.410i	− 0.757396i
$97$	680.691i	0.712512i	0.934388	+	0.356256i	$0.115947\pi$
$97$	680.691i	0.712512i	−0.934388	+	0.356256i	$0.884053\pi$
$98$	2243.65	2.31269
$99$	1158.18i	1.17577i
$100$	877.801	0.877801
$101$	1921.92	1.89345	0.946724	−	0.322045i	$-0.104370\pi$
$101$	1921.92	1.89345	0.946724	+	0.322045i	$0.104370\pi$
$102$	0	0
$103$	−1123.80	−1.07506	−0.537529	−	0.843245i	$-0.680642\pi$
$103$	−1123.80	−1.07506	−0.537529	+	0.843245i	$0.680642\pi$
$104$	−190.361	−0.179485
$105$	534.063i	0.496373i
$106$	−715.532	−0.655647
$107$	− 122.786i	− 0.110937i	−0.998460	−	0.0554683i	$-0.982335\pi$
$107$	− 122.786i	− 0.110937i	0.998460	−	0.0554683i	$-0.0176652\pi$
$108$	− 1358.86i	− 1.21071i
$109$	− 791.497i	− 0.695520i	−0.937584	−	0.347760i	$-0.886942\pi$
$109$	− 791.497i	− 0.695520i	0.937584	−	0.347760i	$-0.113058\pi$
$110$	1675.25	1.45208
$111$	450.702	0.385394
$112$	− 1152.06i	− 0.971956i
$113$	73.7910i	0.0614307i	0.999528	+	0.0307154i	$0.00977854\pi$
$113$	73.7910i	0.0614307i	−0.999528	+	0.0307154i	$0.990221\pi$
$114$	− 1415.47i	− 1.16290i
$115$	47.0056	0.0381156
$116$	− 1714.38i	− 1.37221i
$117$	350.292	0.276791
$118$	−1471.56	−1.14804
$119$	0	0
$120$	−186.137	−0.141599
$121$	−2433.98	−1.82869
$122$	− 481.857i	− 0.357585i
$123$	−89.4269	−0.0655557
$124$	1321.14i	0.956788i
$125$	1333.55i	0.954210i
$126$	− 2382.36i	− 1.68443i
$127$	1475.81	1.03115	0.515577	−	0.856843i	$-0.327577\pi$
$127$	1475.81	1.03115	0.515577	+	0.856843i	$0.327577\pi$
$128$	1254.28	0.866122
$129$	449.529i	0.306813i
$130$	− 506.682i	− 0.341838i
$131$	− 1029.79i	− 0.686821i	−0.939185	−	0.343410i	$-0.888418\pi$
$131$	− 1029.79i	− 0.686821i	0.939185	−	0.343410i	$-0.111582\pi$
$132$	1817.51	1.19844
$133$	− 3407.89i	− 2.22182i
$134$	4220.85	2.72109
$135$	832.471	0.530724
$136$	0	0
$137$	−1018.26	−0.635008	−0.317504	−	0.948257i	$-0.602845\pi$
$137$	−1018.26	−0.635008	−0.317504	+	0.948257i	$0.602845\pi$
$138$	90.2567	0.0556751
$139$	− 1506.59i	− 0.919332i	−0.888092	−	0.459666i	$-0.847969\pi$
$139$	− 1506.59i	− 0.919332i	0.888092	−	0.459666i	$-0.152031\pi$
$140$	−1947.06	−1.17541
$141$	1313.18i	0.784327i
$142$	2248.66i	1.32890i
$143$	1138.72i	0.665908i
$144$	−738.865	−0.427584
$145$	1050.27	0.601520
$146$	− 3656.59i	− 2.07275i
$147$	1491.24i	0.836704i
$148$	1643.15i	0.912609i
$149$	−424.614	−0.233462	−0.116731	−	0.993164i	$-0.537241\pi$
$149$	−424.614	−0.233462	−0.116731	+	0.993164i	$0.537241\pi$
$150$	1032.57i	0.562061i
$151$	−1566.83	−0.844418	−0.422209	−	0.906499i	$-0.638745\pi$
$151$	−1566.83	−0.844418	−0.422209	+	0.906499i	$0.638745\pi$
$152$	1187.75	0.633811
$153$	0	0
$154$	7744.54	4.05242
$155$	−809.362	−0.419416
$156$	− 549.709i	− 0.282128i
$157$	−762.291	−0.387500	−0.193750	−	0.981051i	$-0.562065\pi$
$157$	−762.291	−0.387500	−0.193750	+	0.981051i	$0.562065\pi$
$158$	− 862.092i	− 0.434078i
$159$	− 475.577i	− 0.237206i
$160$	1591.15i	0.786199i
$161$	217.303	0.106372
$162$	−587.145	−0.284756
$163$	2253.74i	1.08299i	0.840705	+	0.541493i	$0.182141\pi$
$163$	2253.74i	1.08299i	−0.840705	+	0.541493i	$0.817859\pi$
$164$	− 326.029i	− 0.155235i
$165$	1113.45i	0.525347i
$166$	−96.6432	−0.0451865
$167$	1995.81i	0.924792i	0.886673	+	0.462396i	$0.153010\pi$
$167$	1995.81i	0.924792i	−0.886673	+	0.462396i	$0.846990\pi$
$168$	−860.493	−0.395169
$169$	−1852.59	−0.843237
$170$	0	0
$171$	−2185.63	−0.977425
$172$	−1638.88	−0.726529
$173$	− 1853.93i	− 0.814749i	−0.913261	−	0.407375i	$-0.866444\pi$
$173$	− 1853.93i	− 0.814749i	0.913261	−	0.407375i	$-0.133556\pi$
$174$	2016.66	0.878634
$175$	2486.03i	1.07386i
$176$	− 2401.89i	− 1.02869i
$177$	− 978.071i	− 0.415346i
$178$	−2153.85	−0.906954
$179$	1623.39	0.677865	0.338933	−	0.940811i	$-0.389934\pi$
$179$	1623.39	0.677865	0.338933	+	0.940811i	$0.389934\pi$
$180$	1248.74i	0.517086i
$181$	− 759.107i	− 0.311735i	−0.987778	−	0.155867i	$-0.950183\pi$
$181$	− 759.107i	− 0.311735i	0.987778	−	0.155867i	$-0.0498173\pi$
$182$	− 2342.35i	− 0.953990i
$183$	320.266	0.129370
$184$	75.7364i	0.0303444i
$185$	−1006.64	−0.400050
$186$	−1554.08	−0.612637
$187$	0	0
$188$	−4787.55	−1.85728
$189$	3848.44	1.48113
$190$	3161.42i	1.20713i
$191$	−1878.32	−0.711572	−0.355786	−	0.934567i	$-0.615787\pi$
$191$	−1878.32	−0.711572	−0.355786	+	0.934567i	$0.615787\pi$
$192$	2162.60i	0.812876i
$193$	− 497.364i	− 0.185498i	−0.995690	−	0.0927489i	$-0.970435\pi$
$193$	− 497.364i	− 0.185498i	0.995690	−	0.0927489i	$-0.0295654\pi$
$194$	− 2919.19i	− 1.08034i
$195$	336.765	0.123673
$196$	−5436.70	−1.98130
$197$	1052.05i	0.380486i	0.981737	+	0.190243i	$0.0609276\pi$
$197$	1052.05i	0.380486i	−0.981737	+	0.190243i	$0.939072\pi$
$198$	− 4966.92i	− 1.78275i
$199$	1783.55i	0.635338i	0.948202	+	0.317669i	$0.102900\pi$
$199$	1783.55i	0.635338i	−0.948202	+	0.317669i	$0.897100\pi$
$200$	−866.454	−0.306338
$201$	2805.38i	0.984458i
$202$	−8242.29	−2.87092
$203$	4855.31	1.67870
$204$	0	0
$205$	199.733	0.0680487
$206$	4819.48	1.63004
$207$	− 139.366i	− 0.0467952i
$208$	−726.455	−0.242166
$209$	− 7105.02i	− 2.35151i
$210$	− 2290.37i	− 0.752620i
$211$	3756.09i	1.22550i	0.790278	+	0.612748i	$0.209936\pi$
$211$	3756.09i	1.22550i	−0.790278	+	0.612748i	$0.790064\pi$
$212$	1733.84	0.561700
$213$	−1494.57	−0.480779
$214$	526.578i	0.168206i
$215$	− 1004.02i	− 0.318481i
$216$	1341.30i	0.422517i
$217$	−3741.61	−1.17049
$218$	3394.39i	1.05457i
$219$	2430.34	0.749897
$220$	−4059.38	−1.24402
$221$	0	0
$222$	−1932.87	−0.584349
$223$	3356.40	1.00790	0.503949	−	0.863733i	$-0.331880\pi$
$223$	3356.40	1.00790	0.503949	+	0.863733i	$0.331880\pi$
$224$	7355.76i	2.19410i
$225$	1594.40	0.472415
$226$	− 316.458i	− 0.0931436i
$227$	1988.64i	0.581457i	0.956806	+	0.290728i	$0.0938976\pi$
$227$	1988.64i	0.581457i	−0.956806	+	0.290728i	$0.906102\pi$
$228$	3429.89i	0.996271i
$229$	−5203.50	−1.50156	−0.750779	−	0.660554i	$-0.770322\pi$
$229$	−5203.50	−1.50156	−0.750779	+	0.660554i	$0.770322\pi$
$230$	−201.587	−0.0577924
$231$	5147.39i	1.46612i
$232$	1692.22i	0.478878i
$233$	5702.70i	1.60342i	0.597714	+	0.801709i	$0.296076\pi$
$233$	5702.70i	1.60342i	−0.597714	+	0.801709i	$0.703924\pi$
$234$	−1502.25	−0.419681
$235$	− 2932.98i	− 0.814154i
$236$	3565.81	0.983536
$237$	572.988	0.157045
$238$	0	0
$239$	1418.04	0.383787	0.191894	−	0.981416i	$-0.438537\pi$
$239$	1418.04	0.383787	0.191894	+	0.981416i	$0.438537\pi$
$240$	−710.334	−0.191049
$241$	− 6215.71i	− 1.66137i	−0.556746	−	0.830683i	$-0.687950\pi$
$241$	− 6215.71i	− 1.66137i	0.556746	−	0.830683i	$-0.312050\pi$
$242$	10438.3	2.77273
$243$	− 3920.83i	− 1.03507i
$244$	1167.61i	0.306347i
$245$	− 3330.66i	− 0.868523i
$246$	383.514	0.0993981
$247$	−2148.92	−0.553573
$248$	− 1304.06i	− 0.333903i
$249$	− 64.2337i	− 0.0163480i
$250$	− 5719.02i	− 1.44681i
$251$	1547.45	0.389139	0.194570	−	0.980889i	$-0.437669\pi$
$251$	1547.45	0.389139	0.194570	+	0.980889i	$0.437669\pi$
$252$	5772.81i	1.44307i
$253$	453.049	0.112581
$254$	−6329.10	−1.56348
$255$	0	0
$256$	690.569	0.168596
$257$	−7251.45	−1.76005	−0.880025	−	0.474928i	$-0.842474\pi$
$257$	−7251.45	−1.76005	−0.880025	+	0.474928i	$0.842474\pi$
$258$	− 1927.84i	− 0.465201i
$259$	−4653.58	−1.11645
$260$	1227.76i	0.292857i
$261$	− 3113.93i	− 0.738496i
$262$	4416.34i	1.04138i
$263$	−4635.13	−1.08675	−0.543373	−	0.839491i	$-0.682853\pi$
$263$	−4635.13	−1.08675	−0.543373	+	0.839491i	$0.682853\pi$
$264$	−1794.02	−0.418236
$265$	1062.19i	0.246226i
$266$	14615.0i	3.36880i
$267$	− 1431.55i	− 0.328126i
$268$	−10227.7	−2.33119
$269$	− 7116.95i	− 1.61311i	−0.591156	−	0.806557i	$-0.701328\pi$
$269$	− 7116.95i	− 1.61311i	0.591156	−	0.806557i	$-0.298672\pi$
$270$	−3570.11	−0.804704
$271$	163.713	0.0366970	0.0183485	−	0.999832i	$-0.494159\pi$
$271$	163.713	0.0366970	0.0183485	+	0.999832i	$0.494159\pi$
$272$	0	0
$273$	1556.83	0.345143
$274$	4366.89	0.962823
$275$	5183.05i	1.13655i
$276$	−218.705	−0.0476975
$277$	4174.84i	0.905567i	0.891620	+	0.452784i	$0.149569\pi$
$277$	4174.84i	0.905567i	−0.891620	+	0.452784i	$0.850431\pi$
$278$	6461.11i	1.39393i
$279$	2399.66i	0.514925i
$280$	1921.90	0.410197
$281$	6920.09	1.46910	0.734551	−	0.678553i	$-0.237393\pi$
$281$	6920.09	1.46910	0.734551	+	0.678553i	$0.237393\pi$
$282$	− 5631.68i	− 1.18923i
$283$	6292.91i	1.32182i	0.750466	+	0.660909i	$0.229829\pi$
$283$	6292.91i	1.32182i	−0.750466	+	0.660909i	$0.770171\pi$
$284$	− 5448.82i	− 1.13848i
$285$	−2101.23	−0.436724
$286$	− 4883.50i	− 1.00968i
$287$	923.350	0.189908
$288$	4717.58	0.965230
$289$	0	0
$290$	−4504.17	−0.912048
$291$	1940.23	0.390854
$292$	8860.45i	1.77575i
$293$	2292.96	0.457189	0.228595	−	0.973522i	$-0.426587\pi$
$293$	2292.96	0.457189	0.228595	+	0.973522i	$0.426587\pi$
$294$	− 6395.29i	− 1.26864i
$295$	2184.50i	0.431142i
$296$	− 1621.91i	− 0.318485i
$297$	8023.51	1.56758
$298$	1820.99	0.353983
$299$	− 137.025i	− 0.0265029i
$300$	− 2502.07i	− 0.481524i
$301$	− 4641.47i	− 0.888804i
$302$	6719.47	1.28034
$303$	− 5478.22i	− 1.03867i
$304$	4532.69	0.855156
$305$	−715.308	−0.134290
$306$	0	0
$307$	2300.11	0.427604	0.213802	−	0.976877i	$-0.431415\pi$
$307$	2300.11	0.427604	0.213802	+	0.976877i	$0.431415\pi$
$308$	−18766.1	−3.47176
$309$	3203.26i	0.589731i
$310$	3471.01	0.635935
$311$	− 5775.22i	− 1.05300i	−0.850175	−	0.526499i	$-0.823504\pi$
$311$	− 5775.22i	− 1.05300i	0.850175	−	0.526499i	$-0.176496\pi$
$312$	542.603i	0.0984579i
$313$	2799.69i	0.505585i	0.967521	+	0.252792i	$0.0813490\pi$
$313$	2799.69i	0.505585i	−0.967521	+	0.252792i	$0.918651\pi$
$314$	3269.14	0.587542
$315$	−3536.57	−0.632581
$316$	2088.97i	0.371880i
$317$	5140.47i	0.910781i	0.890292	+	0.455390i	$0.150500\pi$
$317$	5140.47i	0.910781i	−0.890292	+	0.455390i	$0.849500\pi$
$318$	2039.54i	0.359660i
$319$	10122.7	1.77669
$320$	− 4830.13i	− 0.843789i
$321$	−349.989	−0.0608550
$322$	−931.917	−0.161285
$323$	0	0
$324$	1422.74	0.243954
$325$	1567.62	0.267557
$326$	− 9665.33i	− 1.64206i
$327$	−2256.07	−0.381533
$328$	321.815i	0.0541745i
$329$	− 13558.9i	− 2.27211i
$330$	− 4775.12i	− 0.796551i
$331$	−459.510	−0.0763050	−0.0381525	−	0.999272i	$-0.512147\pi$
$331$	−459.510	−0.0763050	−0.0381525	+	0.999272i	$0.512147\pi$
$332$	234.180	0.0387118
$333$	2984.55i	0.491148i
$334$	− 8559.16i	− 1.40221i
$335$	− 6265.76i	− 1.02190i
$336$	−3283.81	−0.533174
$337$	6233.46i	1.00759i	0.863823	+	0.503796i	$0.168063\pi$
$337$	6233.46i	1.00759i	−0.863823	+	0.503796i	$0.831937\pi$
$338$	7944.97	1.27855
$339$	210.333	0.0336983
$340$	0	0
$341$	−7800.78	−1.23882
$342$	9373.24	1.48201
$343$	− 5302.57i	− 0.834729i
$344$	1617.69	0.253547
$345$	− 133.984i	− 0.0209086i
$346$	7950.70i	1.23535i
$347$	− 6370.01i	− 0.985476i	−0.870178	−	0.492738i	$-0.835996\pi$
$347$	− 6370.01i	− 0.985476i	0.870178	−	0.492738i	$-0.164004\pi$
$348$	−4886.65	−0.752735
$349$	6494.79	0.996155	0.498078	−	0.867132i	$-0.334039\pi$
$349$	6494.79	0.996155	0.498078	+	0.867132i	$0.334039\pi$
$350$	− 10661.5i	− 1.62823i
$351$	− 2426.72i	− 0.369028i
$352$	15335.8i	2.32217i
$353$	502.630	0.0757856	0.0378928	−	0.999282i	$-0.487935\pi$
$353$	502.630	0.0757856	0.0378928	+	0.999282i	$0.487935\pi$
$354$	4194.53i	0.629764i
$355$	3338.09	0.499062
$356$	5219.09	0.776998
$357$	0	0
$358$	−6962.02	−1.02781
$359$	974.431	0.143255	0.0716275	−	0.997431i	$-0.477181\pi$
$359$	974.431	0.143255	0.0716275	+	0.997431i	$0.477181\pi$
$360$	− 1232.60i	− 0.180455i
$361$	6549.11	0.954821
$362$	3255.48i	0.472664i
$363$	6937.80i	1.00314i
$364$	5675.84i	0.817294i
$365$	−5428.13	−0.778415
$366$	−1373.48	−0.196156
$367$	− 4693.11i	− 0.667515i	−0.942659	−	0.333758i	$-0.891683\pi$
$367$	− 4693.11i	− 0.667515i	0.942659	−	0.333758i	$-0.108317\pi$
$368$	289.025i	0.0409415i
$369$	− 592.186i	− 0.0835446i
$370$	4317.02	0.606571
$371$	4910.42i	0.687159i
$372$	3765.76	0.524853
$373$	8651.16	1.20091	0.600456	−	0.799658i	$-0.294986\pi$
$373$	8651.16	1.20091	0.600456	+	0.799658i	$0.294986\pi$
$374$	0	0
$375$	3801.13	0.523439
$376$	4725.67	0.648159
$377$	− 3061.63i	− 0.418254i
$378$	−16504.3	−2.24574
$379$	3490.01i	0.473007i	0.971631	+	0.236503i	$0.0760014\pi$
$379$	3490.01i	0.473007i	−0.971631	+	0.236503i	$0.923999\pi$
$380$	− 7660.59i	− 1.03416i
$381$	− 4206.62i	− 0.565648i
$382$	8055.29	1.07891
$383$	−3928.41	−0.524105	−0.262052	−	0.965054i	$-0.584399\pi$
$383$	−3928.41	−0.524105	−0.262052	+	0.965054i	$0.584399\pi$
$384$	− 3575.18i	− 0.475118i
$385$	− 11496.6i	− 1.52187i
$386$	2132.98i	0.281259i
$387$	−2976.79	−0.391004
$388$	7073.62i	0.925537i
$389$	−6961.15	−0.907312	−0.453656	−	0.891177i	$-0.649880\pi$
$389$	−6961.15	−0.907312	−0.453656	+	0.891177i	$0.649880\pi$
$390$	−1444.24	−0.187518
$391$	0	0
$392$	5366.43	0.691443
$393$	−2935.31	−0.376761
$394$	− 4511.81i	− 0.576908i
$395$	−1279.76	−0.163017
$396$	12035.6i	1.52730i
$397$	− 5331.94i	− 0.674062i	−0.941494	−	0.337031i	$-0.890577\pi$
$397$	− 5331.94i	− 0.674062i	0.941494	−	0.337031i	$-0.109423\pi$
$398$	− 7648.86i	− 0.963323i
$399$	−9713.81	−1.21879
$400$	−3306.56	−0.413320
$401$	− 546.021i	− 0.0679975i	−0.999422	−	0.0339988i	$-0.989176\pi$
$401$	− 546.021i	− 0.0679975i	0.999422	−	0.0339988i	$-0.0108242\pi$
$402$	− 12031.1i	− 1.49267i
$403$	2359.35i	0.291632i
$404$	19972.3	2.45955
$405$	871.604i	0.106939i
$406$	−20822.3	−2.54531
$407$	−9702.13	−1.18161
$408$	0	0
$409$	15964.6	1.93007	0.965037	−	0.262114i	$-0.0844196\pi$
$409$	15964.6	1.93007	0.965037	+	0.262114i	$0.0844196\pi$
$410$	−856.571	−0.103178
$411$	2902.44i	0.348338i
$412$	−11678.3	−1.39648
$413$	10098.8i	1.20321i
$414$	597.681i	0.0709527i
$415$	143.465i	0.0169697i
$416$	4638.34	0.546667
$417$	−4294.36	−0.504306
$418$	30470.4i	3.56544i
$419$	− 7643.91i	− 0.891240i	−0.895222	−	0.445620i	$-0.852983\pi$
$419$	− 7643.91i	− 0.891240i	0.895222	−	0.445620i	$-0.147017\pi$
$420$	5549.89i	0.644778i
$421$	−7524.14	−0.871031	−0.435516	−	0.900181i	$-0.643434\pi$
$421$	−7524.14	−0.871031	−0.435516	+	0.900181i	$0.643434\pi$
$422$	− 16108.2i	− 1.85814i
$423$	−8695.92	−0.999551
$424$	−1711.43	−0.196024
$425$	0	0
$426$	6409.55	0.728975
$427$	−3306.80	−0.374771
$428$	− 1275.97i	− 0.144104i
$429$	3245.80	0.365289
$430$	4305.79i	0.482892i
$431$	12222.4i	1.36597i	0.730431	+	0.682986i	$0.239319\pi$
$431$	12222.4i	1.36597i	−0.730431	+	0.682986i	$0.760681\pi$
$432$	5118.64i	0.570071i
$433$	15420.3	1.71144	0.855719	−	0.517440i	$-0.173115\pi$
$433$	15420.3	1.71144	0.855719	+	0.517440i	$0.173115\pi$
$434$	16046.1	1.77475
$435$	− 2993.68i	− 0.329968i
$436$	− 8225.10i	− 0.903465i
$437$	854.963i	0.0935890i
$438$	−10422.7	−1.13702
$439$	17832.7i	1.93874i	0.245607	+	0.969369i	$0.421013\pi$
$439$	17832.7i	1.93874i	−0.245607	+	0.969369i	$0.578987\pi$
$440$	4006.91	0.434141
$441$	−9875.00	−1.06630
$442$	0	0
$443$	241.629	0.0259145	0.0129573	−	0.999916i	$-0.495875\pi$
$443$	241.629	0.0259145	0.0129573	+	0.999916i	$0.495875\pi$
$444$	4683.62	0.500618
$445$	3197.35i	0.340604i
$446$	−14394.2	−1.52821
$447$	1210.32i	0.128067i
$448$	− 22329.2i	− 2.35482i
$449$	14481.4i	1.52209i	0.648698	+	0.761046i	$0.275314\pi$
$449$	14481.4i	1.52209i	−0.648698	+	0.761046i	$0.724686\pi$
$450$	−6837.70	−0.716294
$451$	1925.07	0.200993
$452$	766.823i	0.0797971i
$453$	4466.08i	0.463211i
$454$	− 8528.42i	− 0.881627i
$455$	−3477.16	−0.358268
$456$	− 3385.55i	− 0.347682i
$457$	1814.51	0.185731	0.0928654	−	0.995679i	$-0.470397\pi$
$457$	1814.51	0.185731	0.0928654	+	0.995679i	$0.470397\pi$
$458$	22315.6	2.27672
$459$	0	0
$460$	488.474	0.0495114
$461$	13434.1	1.35724	0.678621	−	0.734489i	$-0.262578\pi$
$461$	13434.1	1.35724	0.678621	+	0.734489i	$0.262578\pi$
$462$	− 22075.0i	− 2.22299i
$463$	−16593.9	−1.66562	−0.832810	−	0.553559i	$-0.813269\pi$
$463$	−16593.9	−1.66562	−0.832810	+	0.553559i	$0.813269\pi$
$464$	6457.84i	0.646116i
$465$	2307.00i	0.230074i
$466$	− 24456.4i	− 2.43116i
$467$	3944.12	0.390818	0.195409	−	0.980722i	$-0.437397\pi$
$467$	3944.12	0.390818	0.195409	+	0.980722i	$0.437397\pi$
$468$	3640.17	0.359545
$469$	− 28966.0i	− 2.85187i
$470$	12578.3i	1.23445i
$471$	2172.83i	0.212566i
$472$	−3519.72	−0.343238
$473$	− 9676.89i	− 0.940685i
$474$	−2457.30	−0.238117
$475$	−9781.11	−0.944817
$476$	0	0
$477$	3149.27	0.302296
$478$	−6081.35	−0.581913
$479$	11162.8i	1.06480i	0.846492	+	0.532401i	$0.178710\pi$
$479$	11162.8i	1.06480i	−0.846492	+	0.532401i	$0.821290\pi$
$480$	4535.41	0.431275
$481$	2934.42i	0.278166i
$482$	26656.5i	2.51903i
$483$	− 619.397i	− 0.0583510i
$484$	−25293.5	−2.37543
$485$	−4333.48	−0.405717
$486$	16814.7i	1.56941i
$487$	− 9984.01i	− 0.928991i	−0.885575	−	0.464496i	$-0.846236\pi$
$487$	− 9984.01i	− 0.928991i	0.885575	−	0.464496i	$-0.153764\pi$
$488$	− 1152.52i	− 0.106910i
$489$	6424.04	0.594080
$490$	14283.8i	1.31689i
$491$	19639.7	1.80514	0.902572	−	0.430540i	$-0.141677\pi$
$491$	19639.7	1.80514	0.902572	+	0.430540i	$0.141677\pi$
$492$	−929.309	−0.0851554
$493$	0	0
$494$	9215.80	0.839349
$495$	−7373.30	−0.669505
$496$	− 4976.55i	− 0.450511i
$497$	15431.7	1.39277
$498$	275.470i	0.0247874i
$499$	− 5764.32i	− 0.517127i	−0.965994	−	0.258564i	$-0.916751\pi$
$499$	− 5764.32i	− 0.517127i	0.965994	−	0.258564i	$-0.0832491\pi$
$500$	13858.0i	1.23950i
$501$	5688.83	0.507302
$502$	−6636.33	−0.590027
$503$	1747.34i	0.154890i	0.996997	+	0.0774452i	$0.0246763\pi$
$503$	1747.34i	0.154890i	−0.996997	+	0.0774452i	$0.975324\pi$
$504$	− 5698.19i	− 0.503606i
$505$	12235.5i	1.07816i
$506$	−1942.93	−0.170699
$507$	5280.61i	0.462564i
$508$	15336.3	1.33945
$509$	−9110.89	−0.793385	−0.396693	−	0.917952i	$-0.629842\pi$
$509$	−9110.89	−0.793385	−0.396693	+	0.917952i	$0.629842\pi$
$510$	0	0
$511$	−25093.8	−2.17237
$512$	−12995.8	−1.12175
$513$	15141.4i	1.30314i
$514$	31098.3	2.66865
$515$	− 7154.42i	− 0.612158i
$516$	4671.43i	0.398543i
$517$	− 28268.5i	− 2.40474i
$518$	19957.2	1.69280
$519$	−5284.42	−0.446937
$520$	− 1211.89i	− 0.102202i
$521$	9231.61i	0.776285i	0.921599	+	0.388142i	$0.126883\pi$
$521$	9231.61i	0.776285i	−0.921599	+	0.388142i	$0.873117\pi$
$522$	13354.3i	1.11974i
$523$	14927.5	1.24806	0.624028	−	0.781402i	$-0.285495\pi$
$523$	14927.5	1.24806	0.624028	+	0.781402i	$0.285495\pi$
$524$	− 10701.4i	− 0.892165i
$525$	7086.14	0.589076
$526$	19878.1	1.64777
$527$	0	0
$528$	−6846.32	−0.564295
$529$	12112.5	0.995519
$530$	− 4555.28i	− 0.373338i
$531$	6476.79	0.529320
$532$	− 35414.2i	− 2.88609i
$533$	− 582.239i	− 0.0473162i
$534$	6139.31i	0.497516i
$535$	781.694	0.0631693
$536$	10095.5	0.813545
$537$	− 4627.29i	− 0.371848i
$538$	30521.5i	2.44587i
$539$	− 32101.5i	− 2.56532i
$540$	8650.90	0.689399
$541$	− 9005.10i	− 0.715637i	−0.933791	−	0.357818i	$-0.883521\pi$
$541$	− 9005.10i	− 0.715637i	0.933791	−	0.357818i	$-0.116479\pi$
$542$	−702.096	−0.0556413
$543$	−2163.75	−0.171004
$544$	0	0
$545$	5038.90	0.396042
$546$	−6676.59	−0.523318
$547$	6774.33i	0.529524i	0.964314	+	0.264762i	$0.0852933\pi$
$547$	6774.33i	0.529524i	−0.964314	+	0.264762i	$0.914707\pi$
$548$	−10581.6	−0.824861
$549$	2120.80i	0.164870i
$550$	− 22227.9i	− 1.72327i
$551$	19102.9i	1.47697i
$552$	215.878	0.0166456
$553$	−5916.20	−0.454941
$554$	− 17904.1i	− 1.37306i
$555$	2869.30i	0.219450i
$556$	− 15656.2i	− 1.19419i
$557$	−8558.56	−0.651055	−0.325528	−	0.945533i	$-0.605542\pi$
$557$	−8558.56	−0.651055	−0.325528	+	0.945533i	$0.605542\pi$
$558$	− 10291.1i	− 0.780748i
$559$	−2926.79	−0.221449
$560$	7334.32	0.553450
$561$	0	0
$562$	−29677.3	−2.22751
$563$	−13915.1	−1.04165	−0.520827	−	0.853662i	$-0.674376\pi$
$563$	−13915.1	−1.04165	−0.520827	+	0.853662i	$0.674376\pi$
$564$	13646.4i	1.01882i
$565$	−469.775	−0.0349798
$566$	− 26987.6i	− 2.00419i
$567$	4029.35i	0.298442i
$568$	5378.39i	0.397310i
$569$	12378.8	0.912031	0.456015	−	0.889972i	$-0.349276\pi$
$569$	12378.8	0.912031	0.456015	+	0.889972i	$0.349276\pi$
$570$	9011.29	0.662178
$571$	− 785.845i	− 0.0575947i	−0.999585	−	0.0287974i	$-0.990832\pi$
$571$	− 785.845i	− 0.0575947i	0.999585	−	0.0287974i	$-0.00916775\pi$
$572$	11833.4i	0.865000i
$573$	5353.93i	0.390338i
$574$	−3959.85	−0.287946
$575$	− 623.688i	− 0.0452341i
$576$	−14320.8	−1.03593
$577$	19662.8	1.41867	0.709334	−	0.704873i	$-0.248996\pi$
$577$	19662.8	1.41867	0.709334	+	0.704873i	$0.248996\pi$
$578$	0	0
$579$	−1417.68	−0.101756
$580$	10914.2	0.781361
$581$	663.224i	0.0473583i
$582$	−8320.82	−0.592627
$583$	10237.6i	0.727269i
$584$	− 8745.92i	− 0.619707i
$585$	2230.06i	0.157610i
$586$	−9833.54	−0.693208
$587$	10024.7	0.704875	0.352438	−	0.935835i	$-0.385353\pi$
$587$	10024.7	0.704875	0.352438	+	0.935835i	$0.385353\pi$
$588$	15496.7i	1.08686i
$589$	− 14721.1i	− 1.02983i
$590$	− 9368.40i	− 0.653713i
$591$	2998.76	0.208719
$592$	− 6189.52i	− 0.429709i
$593$	−770.054	−0.0533260	−0.0266630	−	0.999644i	$-0.508488\pi$
$593$	−770.054	−0.0533260	−0.0266630	+	0.999644i	$0.508488\pi$
$594$	−34409.4	−2.37682
$595$	0	0
$596$	−4412.52	−0.303261
$597$	5083.80	0.348519
$598$	587.641i	0.0401847i
$599$	20011.7	1.36503	0.682517	−	0.730870i	$-0.260885\pi$
$599$	20011.7	1.36503	0.682517	+	0.730870i	$0.260885\pi$
$600$	2469.73i	0.168044i
$601$	16303.2i	1.10652i	0.833008	+	0.553261i	$0.186617\pi$
$601$	16303.2i	1.10652i	−0.833008	+	0.553261i	$0.813383\pi$
$602$	19905.3i	1.34764i
$603$	−18577.2	−1.25460
$604$	−16282.2	−1.09688
$605$	− 15495.5i	− 1.04129i
$606$	23493.7i	1.57486i
$607$	− 26777.6i	− 1.79056i	−0.445506	−	0.895279i	$-0.646976\pi$
$607$	− 26777.6i	− 1.79056i	0.445506	−	0.895279i	$-0.353024\pi$
$608$	−28940.7	−1.93043
$609$	− 13839.5i	− 0.920864i
$610$	3067.65	0.203615
$611$	−8549.85	−0.566105
$612$	0	0
$613$	1385.13	0.0912639	0.0456320	−	0.998958i	$-0.485470\pi$
$613$	1385.13	0.0912639	0.0456320	+	0.998958i	$0.485470\pi$
$614$	−9864.19	−0.648349
$615$	− 569.318i	− 0.0373286i
$616$	18523.6	1.21158
$617$	818.646i	0.0534156i	0.999643	+	0.0267078i	$0.00850237\pi$
$617$	818.646i	0.0534156i	−0.999643	+	0.0267078i	$0.991498\pi$
$618$	− 13737.4i	− 0.894173i
$619$	− 9468.86i	− 0.614839i	−0.951574	−	0.307420i	$-0.900534\pi$
$619$	− 9468.86i	− 0.614839i	0.951574	−	0.307420i	$-0.0994655\pi$
$620$	−8410.75	−0.544813
$621$	−965.486	−0.0623891
$622$	24767.4i	1.59660i
$623$	14781.0i	0.950545i
$624$	2070.68i	0.132842i
$625$	2069.03	0.132418
$626$	− 12006.7i	− 0.766587i
$627$	−20252.1	−1.28994
$628$	−7921.60	−0.503354
$629$	0	0
$630$	15166.8	0.959143
$631$	−11291.5	−0.712374	−0.356187	−	0.934415i	$-0.615923\pi$
$631$	−11291.5	−0.712374	−0.356187	+	0.934415i	$0.615923\pi$
$632$	− 2061.97i	− 0.129780i
$633$	10706.3	0.672255
$634$	− 22045.3i	− 1.38096i
$635$	9395.41i	0.587158i
$636$	− 4942.11i	− 0.308125i
$637$	−9709.13	−0.603909
$638$	−43412.0	−2.69388
$639$	− 9897.02i	− 0.612707i
$640$	7985.10i	0.493186i
$641$	27604.9i	1.70098i	0.525993	+	0.850489i	$0.323694\pi$
$641$	27604.9i	1.70098i	−0.525993	+	0.850489i	$0.676306\pi$
$642$	1500.95	0.0922707
$643$	30899.1i	1.89509i	0.319625	+	0.947544i	$0.396443\pi$
$643$	30899.1i	1.89509i	−0.319625	+	0.947544i	$0.603557\pi$
$644$	2258.17	0.138175
$645$	−2861.84	−0.174705
$646$	0	0
$647$	26023.7	1.58129	0.790647	−	0.612272i	$-0.209744\pi$
$647$	26023.7	1.58129	0.790647	+	0.612272i	$0.209744\pi$
$648$	−1404.35	−0.0851357
$649$	21054.7i	1.27345i
$650$	−6722.85	−0.405680
$651$	10665.0i	0.642082i
$652$	23420.5i	1.40677i
$653$	18446.4i	1.10546i	0.833362	+	0.552728i	$0.186413\pi$
$653$	18446.4i	1.10546i	−0.833362	+	0.552728i	$0.813587\pi$
$654$	9675.33	0.578494
$655$	6555.97	0.391088
$656$	1228.11i	0.0730938i
$657$	16093.8i	0.955673i
$658$	58148.2i	3.44506i
$659$	−987.581	−0.0583774	−0.0291887	−	0.999574i	$-0.509292\pi$
$659$	−987.581	−0.0583774	−0.0291887	+	0.999574i	$0.509292\pi$
$660$	11570.8i	0.682414i
$661$	−4126.78	−0.242834	−0.121417	−	0.992602i	$-0.538744\pi$
$661$	−4126.78	−0.242834	−0.121417	+	0.992602i	$0.538744\pi$
$662$	1970.64	0.115697
$663$	0	0
$664$	−231.153	−0.0135098
$665$	21695.6	1.26514
$666$	− 12799.5i	− 0.744698i
$667$	−1218.09	−0.0707115
$668$	20740.1i	1.20128i
$669$	− 9567.05i	− 0.552890i
$670$	26871.2i	1.54944i
$671$	−6894.26	−0.396647
$672$	20966.8	1.20359
$673$	7788.28i	0.446086i	0.974809	+	0.223043i	$0.0715991\pi$
$673$	7788.28i	0.446086i	−0.974809	+	0.223043i	$0.928401\pi$
$674$	− 26732.6i	− 1.52775i
$675$	− 11045.5i	− 0.629842i
$676$	−19251.8	−1.09535
$677$	− 849.049i	− 0.0482003i	−0.999710	−	0.0241001i	$-0.992328\pi$
$677$	− 849.049i	− 0.0482003i	0.999710	−	0.0241001i	$-0.00767206\pi$
$678$	−902.027	−0.0510946
$679$	−20033.3	−1.13226
$680$	0	0
$681$	5668.39	0.318962
$682$	33454.2	1.87834
$683$	− 18449.2i	− 1.03359i	−0.856110	−	0.516793i	$-0.827126\pi$
$683$	− 18449.2i	− 1.03359i	0.856110	−	0.516793i	$-0.172874\pi$
$684$	−22712.7	−1.26965
$685$	− 6482.56i	− 0.361585i
$686$	22740.4i	1.26565i
$687$	14832.0i	0.823691i
$688$	6173.42	0.342092
$689$	3096.37	0.171208
$690$	574.601i	0.0317024i
$691$	− 33194.1i	− 1.82744i	−0.406339	−	0.913722i	$-0.633195\pi$
$691$	− 33194.1i	− 1.82744i	0.406339	−	0.913722i	$-0.366805\pi$
$692$	− 19265.7i	− 1.05834i
$693$	−34086.1	−1.86843
$694$	27318.2i	1.49422i
$695$	9591.38	0.523484
$696$	4823.49	0.262692
$697$	0	0
$698$	−27853.4	−1.51041
$699$	16254.9	0.879567
$700$	25834.4i	1.39492i
$701$	−1102.73	−0.0594146	−0.0297073	−	0.999559i	$-0.509458\pi$
$701$	−1102.73	−0.0594146	−0.0297073	+	0.999559i	$0.509458\pi$
$702$	10407.2i	0.559534i
$703$	− 18309.2i	− 0.982282i
$704$	− 46553.7i	− 2.49227i
$705$	−8360.12	−0.446610
$706$	−2155.57	−0.114909
$707$	56563.7i	3.00890i
$708$	− 10163.9i	− 0.539526i
$709$	5881.49i	0.311543i	0.987793	+	0.155771i	$0.0497863\pi$
$709$	5881.49i	0.311543i	−0.987793	+	0.155771i	$0.950214\pi$
$710$	−14315.6	−0.756697
$711$	3794.33i	0.200138i
$712$	−5151.63	−0.271159
$713$	938.685	0.0493044
$714$	0	0
$715$	−7249.45	−0.379180
$716$	16870.0	0.880532
$717$	− 4041.95i	− 0.210529i
$718$	−4178.92	−0.217209
$719$	− 13392.6i	− 0.694661i	−0.937743	−	0.347330i	$-0.887088\pi$
$719$	− 13392.6i	− 0.694661i	0.937743	−	0.347330i	$-0.112912\pi$
$720$	− 4703.83i	− 0.243474i
$721$	− 33074.2i	− 1.70839i
$722$	−28086.3	−1.44774
$723$	−17717.2	−0.911355
$724$	− 7888.51i	− 0.404937i
$725$	− 13935.4i	− 0.713859i
$726$	− 29753.2i	− 1.52100i
$727$	30483.7	1.55513	0.777564	−	0.628804i	$-0.216455\pi$
$727$	30483.7	1.55513	0.777564	+	0.628804i	$0.216455\pi$
$728$	− 5602.48i	− 0.285222i
$729$	−7479.33	−0.379989
$730$	23278.9	1.18026
$731$	0	0
$732$	3328.14	0.168049
$733$	−23656.0	−1.19203	−0.596013	−	0.802975i	$-0.703249\pi$
$733$	−23656.0	−1.19203	−0.596013	+	0.802975i	$0.703249\pi$
$734$	20126.7i	1.01211i
$735$	−9493.67	−0.476434
$736$	− 1845.39i	− 0.0924214i
$737$	− 60390.5i	− 3.01834i
$738$	2539.63i	0.126674i
$739$	27884.4	1.38801	0.694007	−	0.719968i	$-0.255843\pi$
$739$	27884.4	1.38801	0.694007	+	0.719968i	$0.255843\pi$
$740$	−10460.8	−0.519656
$741$	6125.26i	0.303667i
$742$	− 21058.7i	− 1.04190i
$743$	3391.10i	0.167439i	0.996489	+	0.0837196i	$0.0266800\pi$
$743$	3391.10i	0.167439i	−0.996489	+	0.0837196i	$0.973320\pi$
$744$	−3717.08	−0.183165
$745$	− 2703.22i	− 0.132937i
$746$	−37101.1	−1.82087
$747$	425.356	0.0208339
$748$	0	0
$749$	3613.70	0.176291
$750$	−16301.4	−0.793658
$751$	− 19843.5i	− 0.964181i	−0.876121	−	0.482091i	$-0.839878\pi$
$751$	− 19843.5i	− 0.964181i	0.876121	−	0.482091i	$-0.160122\pi$
$752$	18034.1	0.874514
$753$	− 4410.82i	− 0.213465i
$754$	13130.0i	0.634173i
$755$	− 9974.92i	− 0.480827i
$756$	39992.3	1.92395
$757$	23915.1	1.14823	0.574114	−	0.818775i	$-0.305347\pi$
$757$	23915.1	1.14823	0.574114	+	0.818775i	$0.305347\pi$
$758$	− 14967.1i	− 0.717191i
$759$	− 1291.36i	− 0.0617570i
$760$	7561.57i	0.360904i
$761$	17162.3	0.817520	0.408760	−	0.912642i	$-0.365961\pi$
$761$	17162.3	0.817520	0.408760	+	0.912642i	$0.365961\pi$
$762$	18040.4i	0.857656i
$763$	23294.4	1.10526
$764$	−19519.1	−0.924317
$765$	0	0
$766$	16847.2	0.794668
$767$	6368.00	0.299785
$768$	− 1968.39i	− 0.0924845i
$769$	−18387.3	−0.862240	−0.431120	−	0.902295i	$-0.641881\pi$
$769$	−18387.3	−0.862240	−0.431120	+	0.902295i	$0.641881\pi$
$770$	49304.0i	2.30752i
$771$	20669.4i	0.965488i
$772$	− 5168.52i	− 0.240957i
$773$	−4428.78	−0.206070	−0.103035	−	0.994678i	$-0.532855\pi$
$773$	−4428.78	−0.206070	−0.103035	+	0.994678i	$0.532855\pi$
$774$	12766.2	0.592855
$775$	10738.9i	0.497746i
$776$	− 6982.18i	− 0.322997i
$777$	13264.5i	0.612435i
$778$	29853.4	1.37570
$779$	3632.86i	0.167087i
$780$	3499.61	0.160649
$781$	32173.1	1.47406
$782$	0	0
$783$	−21572.4	−0.984591
$784$	20479.3	0.932913
$785$	− 4852.97i	− 0.220650i
$786$	12588.3	0.571259
$787$	− 17665.8i	− 0.800151i	−0.916482	−	0.400075i	$-0.868984\pi$
$787$	− 17665.8i	− 0.800151i	0.916482	−	0.400075i	$-0.131016\pi$
$788$	10932.8i	0.494243i
$789$	13211.9i	0.596143i
$790$	5488.33	0.247172
$791$	−2171.73	−0.0976203
$792$	− 11880.0i	− 0.533002i
$793$	2085.18i	0.0933756i
$794$	22866.4i	1.02204i
$795$	3027.66	0.135069
$796$	18534.3i	0.825290i
$797$	−12688.4	−0.563921	−0.281960	−	0.959426i	$-0.590985\pi$
$797$	−12688.4	−0.563921	−0.281960	+	0.959426i	$0.590985\pi$
$798$	41658.3	1.84798
$799$	0	0
$800$	21112.0	0.933029
$801$	9479.74	0.418165
$802$	2341.65i	0.103100i
$803$	−52317.3	−2.29918
$804$	29153.0i	1.27879i
$805$	1383.41i	0.0605700i
$806$	− 10118.3i	− 0.442184i
$807$	−20286.1	−0.884886
$808$	−19714.1	−0.858342
$809$	− 28633.8i	− 1.24439i	−0.782863	−	0.622194i	$-0.786242\pi$
$809$	− 28633.8i	− 1.24439i	0.782863	−	0.622194i	$-0.213758\pi$
$810$	− 3737.94i	− 0.162145i
$811$	2305.93i	0.0998425i	0.998753	+	0.0499213i	$0.0158970\pi$
$811$	2305.93i	0.0998425i	−0.998753	+	0.0499213i	$0.984103\pi$
$812$	50455.6	2.18060
$813$	− 466.646i	− 0.0201304i
$814$	41608.3	1.79161
$815$	−14348.0	−0.616672
$816$	0	0
$817$	18261.6	0.781997
$818$	−68465.4	−2.92645
$819$	10309.4i	0.439852i
$820$	2075.60	0.0883938
$821$	− 28803.5i	− 1.22442i	−0.790695	−	0.612210i	$-0.790281\pi$
$821$	− 28803.5i	− 1.22442i	0.790695	−	0.612210i	$-0.209719\pi$
$822$	− 12447.3i	− 0.528164i
$823$	14597.0i	0.618248i	0.951022	+	0.309124i	$0.100036\pi$
$823$	14597.0i	0.618248i	−0.951022	+	0.309124i	$0.899964\pi$
$824$	11527.3	0.487347
$825$	14773.7	0.623460
$826$	− 43309.3i	− 1.82436i
$827$	32943.4i	1.38519i	0.721326	+	0.692596i	$0.243533\pi$
$827$	32943.4i	1.38519i	−0.721326	+	0.692596i	$0.756467\pi$
$828$	− 1448.27i	− 0.0607859i
$829$	−35091.9	−1.47020	−0.735098	−	0.677961i	$-0.762864\pi$
$829$	−35091.9	−1.47020	−0.735098	+	0.677961i	$0.762864\pi$
$830$	− 615.259i	− 0.0257300i
$831$	11899.9	0.496756
$832$	−14080.2	−0.586711
$833$	0	0
$834$	18416.7	0.764648
$835$	−12705.9	−0.526594
$836$	− 73834.1i	− 3.05455i
$837$	16624.1	0.686517
$838$	32781.5i	1.35133i
$839$	19365.1i	0.796848i	0.917201	+	0.398424i	$0.130443\pi$
$839$	19365.1i	0.796848i	−0.917201	+	0.398424i	$0.869557\pi$
$840$	− 5478.15i	− 0.225017i
$841$	−2827.42	−0.115930
$842$	32267.8	1.32069
$843$	− 19724.9i	− 0.805887i
$844$	39032.6i	1.59189i
$845$	− 11794.1i	− 0.480155i
$846$	37293.0	1.51556
$847$	− 71634.1i	− 2.90599i
$848$	−6531.13	−0.264481
$849$	17937.2	0.725094
$850$	0	0
$851$	1167.48	0.0470278
$852$	−15531.3	−0.624521
$853$	− 37350.6i	− 1.49925i	−0.661863	−	0.749625i	$-0.730234\pi$
$853$	− 37350.6i	− 1.49925i	0.661863	−	0.749625i	$-0.269766\pi$
$854$	14181.4	0.568242
$855$	− 13914.4i	− 0.556564i
$856$	1259.48i	0.0502899i
$857$	− 14189.2i	− 0.565569i	−0.959183	−	0.282785i	$-0.908742\pi$
$857$	− 14189.2i	− 0.565569i	0.959183	−	0.282785i	$-0.0912582\pi$
$858$	−13919.9	−0.553865
$859$	−24014.5	−0.953859	−0.476930	−	0.878942i	$-0.658250\pi$
$859$	−24014.5	−0.953859	−0.476930	+	0.878942i	$0.658250\pi$
$860$	− 10433.6i	− 0.413699i
$861$	− 2631.90i	− 0.104175i
$862$	− 52416.8i	− 2.07114i
$863$	−8498.07	−0.335200	−0.167600	−	0.985855i	$-0.553602\pi$
$863$	−8498.07	−0.335200	−0.167600	+	0.985855i	$0.553602\pi$
$864$	− 32682.0i	− 1.28688i
$865$	11802.7	0.463933
$866$	−66131.1	−2.59495
$867$	0	0
$868$	−38882.1	−1.52044
$869$	−12334.5	−0.481497
$870$	12838.6i	0.500310i
$871$	−18265.2	−0.710554
$872$	8118.78i	0.315294i
$873$	12848.2i	0.498106i
$874$	− 3666.57i	− 0.141903i
$875$	−39247.4	−1.51635
$876$	25255.7	0.974100
$877$	− 16013.2i	− 0.616566i	−0.951295	−	0.308283i	$-0.900246\pi$
$877$	− 16013.2i	− 0.616566i	0.951295	−	0.308283i	$-0.0997544\pi$
$878$	− 76476.6i	− 2.93959i
$879$	− 6535.84i	− 0.250794i
$880$	15291.1	0.585755
$881$	− 16824.6i	− 0.643399i	−0.946842	−	0.321699i	$-0.895746\pi$
$881$	− 16824.6i	− 0.643399i	0.946842	−	0.321699i	$-0.104254\pi$
$882$	42349.6	1.61676
$883$	11327.3	0.431703	0.215851	−	0.976426i	$-0.430747\pi$
$883$	11327.3	0.431703	0.215851	+	0.976426i	$0.430747\pi$
$884$	0	0
$885$	6226.69	0.236506
$886$	−1036.24	−0.0392926
$887$	− 11077.4i	− 0.419327i	−0.977774	−	0.209663i	$-0.932763\pi$
$887$	− 11077.4i	− 0.419327i	0.977774	−	0.209663i	$-0.0672368\pi$
$888$	−4623.08	−0.174708
$889$	43434.1i	1.63862i
$890$	− 13712.0i	− 0.516436i
$891$	8400.68i	0.315862i
$892$	34879.2	1.30924
$893$	53346.5	1.99907
$894$	− 5190.52i	− 0.194180i
$895$	10335.0i	0.385989i
$896$	36914.4i	1.37637i
$897$	−390.575	−0.0145384
$898$	− 62104.5i	− 2.30785i
$899$	20973.5	0.778095
$900$	16568.7	0.613657
$901$	0	0
$902$	−8255.78	−0.304753
$903$	−13230.0	−0.487560
$904$	− 756.911i	− 0.0278479i
$905$	4832.70	0.177508
$906$	− 19153.1i	− 0.702339i
$907$	− 1761.65i	− 0.0644924i	−0.999480	−	0.0322462i	$-0.989734\pi$
$907$	− 1761.65i	− 0.0644924i	0.999480	−	0.0322462i	$-0.0102661\pi$
$908$	20665.6i	0.755299i
$909$	36276.8	1.32368
$910$	14912.1	0.543220
$911$	44996.2i	1.63643i	0.574909	+	0.818217i	$0.305037\pi$
$911$	44996.2i	1.63643i	−0.574909	+	0.818217i	$0.694963\pi$
$912$	− 12919.9i	− 0.469102i
$913$	1382.74i	0.0501227i
$914$	−7781.63	−0.281612
$915$	2038.90i	0.0736657i
$916$	−54073.8	−1.95049
$917$	30307.6	1.09144
$918$	0	0
$919$	34461.8	1.23699	0.618493	−	0.785790i	$-0.287743\pi$
$919$	34461.8	1.23699	0.618493	+	0.785790i	$0.287743\pi$
$920$	−482.160	−0.0172786
$921$	− 6556.21i	− 0.234565i
$922$	−57613.1	−2.05790
$923$	− 9730.78i	− 0.347012i
$924$	53490.8i	1.90446i
$925$	13356.4i	0.474763i
$926$	71163.9	2.52548
$927$	−21212.0	−0.751557
$928$	− 41232.7i	− 1.45854i
$929$	37878.7i	1.33774i	0.743379	+	0.668870i	$0.233222\pi$
$929$	37878.7i	1.33774i	−0.743379	+	0.668870i	$0.766778\pi$
$930$	− 9893.71i	− 0.348847i
$931$	60579.8	2.13257
$932$	59261.5i	2.08281i
$933$	−16461.6	−0.577630
$934$	−16914.6	−0.592573
$935$	0	0
$936$	−3593.12	−0.125475
$937$	599.505	0.0209018	0.0104509	−	0.999945i	$-0.496673\pi$
$937$	599.505	0.0209018	0.0104509	+	0.999945i	$0.496673\pi$
$938$	124223.i	4.32412i
$939$	7980.22	0.277342
$940$	− 30479.0i	− 1.05757i
$941$	− 10118.0i	− 0.350519i	−0.984522	−	0.175260i	$-0.943924\pi$
$941$	− 10118.0i	− 0.350519i	0.984522	−	0.175260i	$-0.0560765\pi$
$942$	− 9318.32i	− 0.322301i
$943$	−231.647	−0.00799945
$944$	−13431.9	−0.463106
$945$	24500.3i	0.843380i
$946$	41500.0i	1.42630i
$947$	39320.2i	1.34924i	0.738163	+	0.674622i	$0.235693\pi$
$947$	39320.2i	1.34924i	−0.738163	+	0.674622i	$0.764307\pi$
$948$	5954.39	0.203997
$949$	15823.4i	0.541254i
$950$	41947.0	1.43257
$951$	14652.3	0.499616
$952$	0	0
$953$	−41203.8	−1.40055	−0.700274	−	0.713874i	$-0.746939\pi$
$953$	−41203.8	−1.40055	−0.700274	+	0.713874i	$0.746939\pi$
$954$	−13505.9	−0.458353
$955$	− 11957.9i	− 0.405182i
$956$	14736.0	0.498531
$957$	− 28853.7i	− 0.974615i
$958$	− 47872.3i	− 1.61449i
$959$	− 29968.3i	− 1.00910i
$960$	−13767.7	−0.462867
$961$	13628.3	0.457465
$962$	− 12584.5i	− 0.421766i
$963$	− 2317.63i	− 0.0775540i
$964$	− 64592.6i	− 2.15808i
$965$	3166.37	0.105626
$966$	2656.33i	0.0884740i
$967$	−9403.83	−0.312727	−0.156363	−	0.987700i	$-0.549977\pi$
$967$	−9403.83	−0.312727	−0.156363	+	0.987700i	$0.549977\pi$
$968$	24966.6	0.828984
$969$	0	0
$970$	18584.4	0.615164
$971$	44819.8	1.48129	0.740646	−	0.671895i	$-0.234519\pi$
$971$	44819.8	1.48129	0.740646	+	0.671895i	$0.234519\pi$
$972$	− 40744.6i	− 1.34453i
$973$	44340.1	1.46092
$974$	42817.1i	1.40857i
$975$	− 4468.33i	− 0.146770i
$976$	− 4398.23i	− 0.144246i
$977$	−13700.4	−0.448632	−0.224316	−	0.974516i	$-0.572015\pi$
$977$	−13700.4	−0.448632	−0.224316	+	0.974516i	$0.572015\pi$
$978$	−27549.9	−0.900766
$979$	30816.6i	1.00603i
$980$	− 34611.6i	− 1.12819i
$981$	− 14939.7i	− 0.486227i
$982$	−84226.0	−2.73703
$983$	34126.9i	1.10730i	0.832748	+	0.553652i	$0.186766\pi$
$983$	34126.9i	1.10730i	−0.832748	+	0.553652i	$0.813234\pi$
$984$	917.297	0.0297178
$985$	−6697.69	−0.216656
$986$	0	0
$987$	−38648.1	−1.24638
$988$	−22331.2	−0.719080
$989$	1164.44i	0.0374389i
$990$	31620.9	1.01513
$991$	38645.7i	1.23877i	0.785087	+	0.619386i	$0.212618\pi$
$991$	38645.7i	1.23877i	−0.785087	+	0.619386i	$0.787382\pi$
$992$	31774.8i	1.01699i
$993$	1309.78i	0.0418577i
$994$	−66179.7	−2.11176
$995$	−11354.6	−0.361773
$996$	− 667.505i	− 0.0212356i
$997$	− 13993.3i	− 0.444505i	−0.974989	−	0.222252i	$-0.928659\pi$
$997$	− 13993.3i	− 0.444505i	0.974989	−	0.222252i	$-0.0713409\pi$
$998$	24720.7i	0.784088i
$999$	20676.1	0.654817

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.b.f.288.5		24
17.4	even	4		289.4.a.h.1.10	✓	12
17.13	even	4		289.4.a.i.1.10	yes	12
17.16	even	2	inner	289.4.b.f.288.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.4.a.h.1.10	✓	12	17.4	even	4
289.4.a.i.1.10	yes	12	17.13	even	4
289.4.b.f.288.5		24	1.1	even	1	trivial
289.4.b.f.288.6		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)

\(f(q)\)	\(=\)	\(q-4.28857 q^{2} -2.85039i q^{3} +10.3918 q^{4} +6.36629i q^{5} +12.2241i q^{6} +29.4308i q^{7} -10.2575 q^{8} +18.8753 q^{9} +O(q^{10})\)	\(q-4.28857 q^{2} -2.85039i q^{3} +10.3918 q^{4} +6.36629i q^{5} +12.2241i q^{6} +29.4308i q^{7} -10.2575 q^{8} +18.8753 q^{9} -27.3023i q^{10} +61.3595i q^{11} -29.6207i q^{12} +18.5582 q^{13} -126.216i q^{14} +18.1464 q^{15} -39.1446 q^{16} -80.9480 q^{18} -115.793 q^{19} +66.1574i q^{20} +83.8892 q^{21} -263.144i q^{22} -7.38352i q^{23} +29.2379i q^{24} +84.4703 q^{25} -79.5883 q^{26} -130.762i q^{27} +305.840i q^{28} -164.974i q^{29} -77.8221 q^{30} +127.132i q^{31} +249.934 q^{32} +174.898 q^{33} -187.365 q^{35} +196.149 q^{36} +158.120i q^{37} +496.588 q^{38} -52.8982i q^{39} -65.3022i q^{40} -31.3736i q^{41} -359.764 q^{42} -157.708 q^{43} +637.637i q^{44} +120.166i q^{45} +31.6647i q^{46} -460.704 q^{47} +111.577i q^{48} -523.171 q^{49} -362.257 q^{50} +192.854 q^{52} +166.846 q^{53} +560.783i q^{54} -390.632 q^{55} -301.886i q^{56} +330.056i q^{57} +707.502i q^{58} +343.136 q^{59} +188.574 q^{60} +112.359i q^{61} -545.216i q^{62} +555.515i q^{63} -758.704 q^{64} +118.147i q^{65} -750.063 q^{66} -984.209 q^{67} -21.0459 q^{69} +803.528 q^{70} -524.337i q^{71} -193.613 q^{72} +852.636i q^{73} -678.106i q^{74} -240.773i q^{75} -1203.30 q^{76} -1805.86 q^{77} +226.858i q^{78} +201.021i q^{79} -249.206i q^{80} +136.909 q^{81} +134.548i q^{82} +22.5351 q^{83} +871.761 q^{84} +676.342 q^{86} -470.240 q^{87} -629.395i q^{88} +502.230 q^{89} -515.338i q^{90} +546.184i q^{91} -76.7282i q^{92} +362.377 q^{93} +1975.76 q^{94} -737.175i q^{95} -712.410i q^{96} +680.691i q^{97} +2243.65 q^{98} +1158.18i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more