LMFDB - Embedded newform 289.4.b.f.288.10

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			288.10
Character	$\chi$	$=$	289.288
Dual form			289.4.b.f.288.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.26162 q^{2} -6.09538i q^{3} -6.40830 q^{4} +13.2627i q^{5} +7.69007i q^{6} +27.2593i q^{7} +18.1779 q^{8} -10.1536 q^{9} +O(q^{10})$	\(q-1.26162 q^{2} -6.09538i q^{3} -6.40830 q^{4} +13.2627i q^{5} +7.69007i q^{6} +27.2593i q^{7} +18.1779 q^{8} -10.1536 q^{9} -16.7325i q^{10} -45.5149i q^{11} +39.0610i q^{12} -52.5379 q^{13} -34.3910i q^{14} +80.8409 q^{15} +28.3328 q^{16} +12.8100 q^{18} -3.08418 q^{19} -84.9911i q^{20} +166.156 q^{21} +57.4227i q^{22} -112.369i q^{23} -110.801i q^{24} -50.8981 q^{25} +66.2831 q^{26} -102.685i q^{27} -174.686i q^{28} -18.6294i q^{29} -101.991 q^{30} +238.763i q^{31} -181.168 q^{32} -277.430 q^{33} -361.531 q^{35} +65.0674 q^{36} -162.717i q^{37} +3.89108 q^{38} +320.238i q^{39} +241.087i q^{40} -383.816i q^{41} -209.626 q^{42} -468.761 q^{43} +291.673i q^{44} -134.664i q^{45} +141.767i q^{46} -199.727 q^{47} -172.699i q^{48} -400.071 q^{49} +64.2142 q^{50} +336.679 q^{52} +105.679 q^{53} +129.550i q^{54} +603.649 q^{55} +495.516i q^{56} +18.7993i q^{57} +23.5033i q^{58} -207.142 q^{59} -518.053 q^{60} -586.101i q^{61} -301.230i q^{62} -276.780i q^{63} +1.90375 q^{64} -696.792i q^{65} +350.013 q^{66} +401.953 q^{67} -684.930 q^{69} +456.116 q^{70} -481.559i q^{71} -184.571 q^{72} -725.281i q^{73} +205.287i q^{74} +310.243i q^{75} +19.7644 q^{76} +1240.71 q^{77} -404.020i q^{78} +382.621i q^{79} +375.768i q^{80} -900.052 q^{81} +484.231i q^{82} +182.391 q^{83} -1064.78 q^{84} +591.401 q^{86} -113.553 q^{87} -827.364i q^{88} -623.009 q^{89} +169.895i q^{90} -1432.15i q^{91} +720.094i q^{92} +1455.35 q^{93} +251.981 q^{94} -40.9045i q^{95} +1104.29i q^{96} -369.544i q^{97} +504.739 q^{98} +462.140i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.26162	−0.446051	−0.223026	−	0.974813i	$-0.571593\pi$
$2$	−1.26162	−0.446051	−0.223026	+	0.974813i	$0.571593\pi$
$3$	− 6.09538i	− 1.17306i	−0.809929	−	0.586528i	$-0.800494\pi$
$3$	− 6.09538i	− 1.17306i	0.809929	−	0.586528i	$-0.199506\pi$
$4$	−6.40830	−0.801038
$5$	13.2627i	1.18625i	0.805111	+	0.593124i	$0.202106\pi$
$5$	13.2627i	1.18625i	−0.805111	+	0.593124i	$0.797894\pi$
$6$	7.69007i	0.523243i
$7$	27.2593i	1.47186i	0.677055	+	0.735932i	$0.263256\pi$
$7$	27.2593i	1.47186i	−0.677055	+	0.735932i	$0.736744\pi$
$8$	18.1779	0.803356
$9$	−10.1536	−0.376059
$10$	− 16.7325i	− 0.529128i
$11$	− 45.5149i	− 1.24757i	−0.781596	−	0.623785i	$-0.785594\pi$
$11$	− 45.5149i	− 1.24757i	0.781596	−	0.623785i	$-0.214406\pi$
$12$	39.0610i	0.939662i
$13$	−52.5379	−1.12088	−0.560438	−	0.828196i	$-0.689367\pi$
$13$	−52.5379	−1.12088	−0.560438	+	0.828196i	$0.689367\pi$
$14$	− 34.3910i	− 0.656527i
$15$	80.8409	1.39153
$16$	28.3328	0.442700
$17$	0	0
$17$	0	0
$18$	12.8100	0.167742
$19$	−3.08418	−0.0372400	−0.0186200	−	0.999827i	$-0.505927\pi$
$19$	−3.08418	−0.0372400	−0.0186200	+	0.999827i	$0.505927\pi$
$20$	− 84.9911i	− 0.950230i
$21$	166.156	1.72658
$22$	57.4227i	0.556480i
$23$	− 112.369i	− 1.01872i	−0.860554	−	0.509359i	$-0.829882\pi$
$23$	− 112.369i	− 1.01872i	0.860554	−	0.509359i	$-0.170118\pi$
$24$	− 110.801i	− 0.942381i
$25$	−50.8981	−0.407185
$26$	66.2831	0.499968
$27$	− 102.685i	− 0.731917i
$28$	− 174.686i	− 1.17902i
$29$	− 18.6294i	− 0.119290i	−0.998220	−	0.0596448i	$-0.981003\pi$
$29$	− 18.6294i	− 0.119290i	0.998220	−	0.0596448i	$-0.0189968\pi$
$30$	−101.991	−0.620696
$31$	238.763i	1.38333i	0.722219	+	0.691664i	$0.243122\pi$
$31$	238.763i	1.38333i	−0.722219	+	0.691664i	$0.756878\pi$
$32$	−181.168	−1.00082
$33$	−277.430	−1.46347
$34$	0	0
$35$	−361.531	−1.74600
$36$	65.0674	0.301238
$37$	− 162.717i	− 0.722986i	−0.932375	−	0.361493i	$-0.882267\pi$
$37$	− 162.717i	− 0.722986i	0.932375	−	0.361493i	$-0.117733\pi$
$38$	3.89108	0.0166110
$39$	320.238i	1.31485i
$40$	241.087i	0.952979i
$41$	− 383.816i	− 1.46200i	−0.682378	−	0.730999i	$-0.739054\pi$
$41$	− 383.816i	− 1.46200i	0.682378	−	0.730999i	$-0.260946\pi$
$42$	−209.626	−0.770143
$43$	−468.761	−1.66245	−0.831226	−	0.555935i	$-0.812360\pi$
$43$	−468.761	−1.66245	−0.831226	+	0.555935i	$0.812360\pi$
$44$	291.673i	0.999350i
$45$	− 134.664i	− 0.446100i
$46$	141.767i	0.454401i
$47$	−199.727	−0.619856	−0.309928	−	0.950760i	$-0.600305\pi$
$47$	−199.727	−0.619856	−0.309928	+	0.950760i	$0.600305\pi$
$48$	− 172.699i	− 0.519312i
$49$	−400.071	−1.16639
$50$	64.2142	0.181625
$51$	0	0
$52$	336.679	0.897864
$53$	105.679	0.273889	0.136944	−	0.990579i	$-0.456272\pi$
$53$	105.679	0.273889	0.136944	+	0.990579i	$0.456272\pi$
$54$	129.550i	0.326473i
$55$	603.649	1.47993
$56$	495.516i	1.18243i
$57$	18.7993i	0.0436846i
$58$	23.5033i	0.0532093i
$59$	−207.142	−0.457078	−0.228539	−	0.973535i	$-0.573395\pi$
$59$	−207.142	−0.457078	−0.228539	+	0.973535i	$0.573395\pi$
$60$	−518.053	−1.11467
$61$	− 586.101i	− 1.23021i	−0.788447	−	0.615103i	$-0.789114\pi$
$61$	− 586.101i	− 1.23021i	0.788447	−	0.615103i	$-0.210886\pi$
$62$	− 301.230i	− 0.617036i
$63$	− 276.780i	− 0.553508i
$64$	1.90375	0.00371826
$65$	− 696.792i	− 1.32964i
$66$	350.013	0.652782
$67$	401.953	0.732931	0.366466	−	0.930432i	$-0.380568\pi$
$67$	401.953	0.732931	0.366466	+	0.930432i	$0.380568\pi$
$68$	0	0
$69$	−684.930	−1.19501
$70$	456.116	0.778805
$71$	− 481.559i	− 0.804937i	−0.915434	−	0.402468i	$-0.868152\pi$
$71$	− 481.559i	− 0.804937i	0.915434	−	0.402468i	$-0.131848\pi$
$72$	−184.571	−0.302109
$73$	− 725.281i	− 1.16285i	−0.813602	−	0.581423i	$-0.802496\pi$
$73$	− 725.281i	− 1.16285i	0.813602	−	0.581423i	$-0.197504\pi$
$74$	205.287i	0.322489i
$75$	310.243i	0.477650i
$76$	19.7644	0.0298307
$77$	1240.71	1.83625
$78$	− 404.020i	− 0.586491i
$79$	382.621i	0.544914i	0.962168	+	0.272457i	$0.0878363\pi$
$79$	382.621i	0.544914i	−0.962168	+	0.272457i	$0.912164\pi$
$80$	375.768i	0.525152i
$81$	−900.052	−1.23464
$82$	484.231i	0.652127i
$83$	182.391	0.241204	0.120602	−	0.992701i	$-0.461517\pi$
$83$	182.391	0.241204	0.120602	+	0.992701i	$0.461517\pi$
$84$	−1064.78	−1.38306
$85$	0	0
$86$	591.401	0.741539
$87$	−113.553	−0.139933
$88$	− 827.364i	− 1.00224i
$89$	−623.009	−0.742010	−0.371005	−	0.928631i	$-0.620987\pi$
$89$	−623.009	−0.742010	−0.371005	+	0.928631i	$0.620987\pi$
$90$	169.895i	0.198983i
$91$	− 1432.15i	− 1.64978i
$92$	720.094i	0.816032i
$93$	1455.35	1.62272
$94$	251.981	0.276488
$95$	− 40.9045i	− 0.0441759i
$96$	1104.29i	1.17402i
$97$	− 369.544i	− 0.386820i	−0.981118	−	0.193410i	$-0.938045\pi$
$97$	− 369.544i	− 0.386820i	0.981118	−	0.193410i	$-0.0619547\pi$
$98$	504.739	0.520268
$99$	462.140i	0.469160i
$100$	326.170	0.326170
$101$	810.381	0.798375	0.399188	−	0.916869i	$-0.369292\pi$
$101$	810.381	0.798375	0.399188	+	0.916869i	$0.369292\pi$
$102$	0	0
$103$	772.683	0.739172	0.369586	−	0.929197i	$-0.379500\pi$
$103$	772.683	0.739172	0.369586	+	0.929197i	$0.379500\pi$
$104$	−955.026	−0.900462
$105$	2203.67i	2.04815i
$106$	−133.327	−0.122169
$107$	− 996.966i	− 0.900751i	−0.892839	−	0.450375i	$-0.851290\pi$
$107$	− 996.966i	− 0.900751i	0.892839	−	0.450375i	$-0.148710\pi$
$108$	658.038i	0.586293i
$109$	− 700.996i	− 0.615993i	−0.951388	−	0.307996i	$-0.900342\pi$
$109$	− 700.996i	− 0.615993i	0.951388	−	0.307996i	$-0.0996585\pi$
$110$	−761.578	−0.660123
$111$	−991.820	−0.848102
$112$	772.333i	0.651595i
$113$	− 1459.26i	− 1.21483i	−0.794384	−	0.607415i	$-0.792206\pi$
$113$	− 1459.26i	− 1.21483i	0.794384	−	0.607415i	$-0.207794\pi$
$114$	− 23.7176i	− 0.0194856i
$115$	1490.31	1.20845
$116$	119.383i	0.0955554i
$117$	533.449	0.421516
$118$	261.335	0.203880
$119$	0	0
$120$	1469.51	1.11790
$121$	−740.607	−0.556429
$122$	739.439i	0.548735i
$123$	−2339.50	−1.71501
$124$	− 1530.07i	− 1.10810i
$125$	982.788i	0.703226i
$126$	349.193i	0.246893i
$127$	−426.418	−0.297941	−0.148970	−	0.988842i	$-0.547596\pi$
$127$	−426.418	−0.297941	−0.148970	+	0.988842i	$0.547596\pi$
$128$	1446.94	0.999164
$129$	2857.28i	1.95015i
$130$	879.089i	0.593086i
$131$	− 19.9037i	− 0.0132747i	−0.999978	−	0.00663737i	$-0.997887\pi$
$131$	− 19.9037i	− 0.0132747i	0.999978	−	0.00663737i	$-0.00211276\pi$
$132$	1777.86	1.17229
$133$	− 84.0728i	− 0.0548123i
$134$	−507.114	−0.326925
$135$	1361.88	0.868235
$136$	0	0
$137$	−1824.89	−1.13804	−0.569019	−	0.822324i	$-0.692677\pi$
$137$	−1824.89	−1.13804	−0.569019	+	0.822324i	$0.692677\pi$
$138$	864.125	0.533037
$139$	1803.87i	1.10074i	0.834922	+	0.550368i	$0.185512\pi$
$139$	1803.87i	1.10074i	−0.834922	+	0.550368i	$0.814488\pi$
$140$	2316.80	1.39861
$141$	1217.41i	0.727126i
$142$	607.546i	0.359043i
$143$	2391.26i	1.39837i
$144$	−287.680	−0.166482
$145$	247.076	0.141507
$146$	915.032i	0.518689i
$147$	2438.58i	1.36824i
$148$	1042.74i	0.579139i
$149$	2742.46	1.50786	0.753929	−	0.656955i	$-0.228156\pi$
$149$	2742.46	1.50786	0.753929	+	0.656955i	$0.228156\pi$
$150$	− 391.410i	− 0.213057i
$151$	−2085.22	−1.12379	−0.561896	−	0.827208i	$-0.689928\pi$
$151$	−2085.22	−1.12379	−0.561896	+	0.827208i	$0.689928\pi$
$152$	−56.0639	−0.0299170
$153$	0	0
$154$	−1565.30	−0.819063
$155$	−3166.64	−1.64097
$156$	− 2052.18i	− 1.05324i
$157$	−2703.53	−1.37430	−0.687149	−	0.726516i	$-0.741138\pi$
$157$	−2703.53	−1.37430	−0.687149	+	0.726516i	$0.741138\pi$
$158$	− 482.724i	− 0.243060i
$159$	− 644.152i	− 0.321287i
$160$	− 2402.77i	− 1.18722i
$161$	3063.10	1.49942
$162$	1135.53	0.550712
$163$	− 2923.57i	− 1.40486i	−0.711755	−	0.702428i	$-0.752099\pi$
$163$	− 2923.57i	− 1.40486i	0.711755	−	0.702428i	$-0.247901\pi$
$164$	2459.61i	1.17112i
$165$	− 3679.46i	− 1.73604i
$166$	−230.108	−0.107590
$167$	2419.31i	1.12103i	0.828145	+	0.560514i	$0.189397\pi$
$167$	2419.31i	1.12103i	−0.828145	+	0.560514i	$0.810603\pi$
$168$	3020.36	1.38706
$169$	563.229	0.256363
$170$	0	0
$171$	31.3156	0.0140045
$172$	3003.97	1.33169
$173$	393.655i	0.173000i	0.996252	+	0.0865000i	$0.0275683\pi$
$173$	393.655i	0.173000i	−0.996252	+	0.0865000i	$0.972432\pi$
$174$	143.262	0.0624174
$175$	− 1387.45i	− 0.599321i
$176$	− 1289.57i	− 0.552299i
$177$	1262.61i	0.536178i
$178$	786.004	0.330975
$179$	−1918.60	−0.801133	−0.400567	−	0.916268i	$-0.631187\pi$
$179$	−1918.60	−0.801133	−0.400567	+	0.916268i	$0.631187\pi$
$180$	862.966i	0.357343i
$181$	− 1677.03i	− 0.688688i	−0.938844	−	0.344344i	$-0.888101\pi$
$181$	− 1677.03i	− 0.688688i	0.938844	−	0.344344i	$-0.111899\pi$
$182$	1806.83i	0.735886i
$183$	−3572.51	−1.44310
$184$	− 2042.63i	− 0.818393i
$185$	2158.06	0.857640
$186$	−1836.11	−0.723817
$187$	0	0
$188$	1279.91	0.496529
$189$	2799.13	1.07728
$190$	51.6061i	0.0197047i
$191$	2347.19	0.889196	0.444598	−	0.895730i	$-0.353347\pi$
$191$	2347.19	0.889196	0.444598	+	0.895730i	$0.353347\pi$
$192$	− 11.6041i	− 0.00436173i
$193$	650.982i	0.242791i	0.992604	+	0.121396i	$0.0387369\pi$
$193$	650.982i	0.242791i	−0.992604	+	0.121396i	$0.961263\pi$
$194$	466.226i	0.172542i
$195$	−4247.21	−1.55974
$196$	2563.77	0.934320
$197$	− 2681.01i	− 0.969614i	−0.874621	−	0.484807i	$-0.838890\pi$
$197$	− 2681.01i	− 0.969614i	0.874621	−	0.484807i	$-0.161110\pi$
$198$	− 583.047i	− 0.209269i
$199$	1784.74i	0.635762i	0.948131	+	0.317881i	$0.102971\pi$
$199$	1784.74i	0.635762i	−0.948131	+	0.317881i	$0.897029\pi$
$200$	−925.218	−0.327114
$201$	− 2450.06i	− 0.859769i
$202$	−1022.40	−0.356116
$203$	507.825	0.175578
$204$	0	0
$205$	5090.42	1.73429
$206$	−974.835	−0.329709
$207$	1140.95i	0.383099i
$208$	−1488.55	−0.496212
$209$	140.376i	0.0464595i
$210$	− 2780.20i	− 0.913581i
$211$	68.5651i	0.0223707i	0.999937	+	0.0111853i	$0.00356048\pi$
$211$	68.5651i	0.0223707i	−0.999937	+	0.0111853i	$0.996440\pi$
$212$	−677.222	−0.219395
$213$	−2935.28	−0.944235
$214$	1257.80i	0.401781i
$215$	− 6217.02i	− 1.97208i
$216$	− 1866.60i	− 0.587990i
$217$	−6508.53	−2.03607
$218$	884.393i	0.274765i
$219$	−4420.86	−1.36408
$220$	−3868.36	−1.18548
$221$	0	0
$222$	1251.30	0.378297
$223$	−4734.22	−1.42164	−0.710822	−	0.703372i	$-0.751677\pi$
$223$	−4734.22	−1.42164	−0.710822	+	0.703372i	$0.751677\pi$
$224$	− 4938.52i	− 1.47308i
$225$	516.799	0.153126
$226$	1841.04i	0.541877i
$227$	1722.86i	0.503745i	0.967760	+	0.251873i	$0.0810464\pi$
$227$	1722.86i	0.503745i	−0.967760	+	0.251873i	$0.918954\pi$
$228$	− 120.471i	− 0.0349930i
$229$	313.375	0.0904297	0.0452149	−	0.998977i	$-0.485603\pi$
$229$	313.375	0.0904297	0.0452149	+	0.998977i	$0.485603\pi$
$230$	−1880.21	−0.539032
$231$	− 7562.57i	− 2.15403i
$232$	− 338.643i	− 0.0958319i
$233$	2101.61i	0.590905i	0.955357	+	0.295453i	$0.0954705\pi$
$233$	2101.61i	0.590905i	−0.955357	+	0.295453i	$0.904530\pi$
$234$	−673.012	−0.188018
$235$	− 2648.92i	− 0.735303i
$236$	1327.43	0.366137
$237$	2332.22	0.639215
$238$	0	0
$239$	−5525.44	−1.49544	−0.747721	−	0.664013i	$-0.768852\pi$
$239$	−5525.44	−1.49544	−0.747721	+	0.664013i	$0.768852\pi$
$240$	2290.45	0.616033
$241$	− 2005.92i	− 0.536153i	−0.963398	−	0.268076i	$-0.913612\pi$
$241$	− 2005.92i	− 0.536153i	0.963398	−	0.268076i	$-0.0863880\pi$
$242$	934.367	0.248196
$243$	2713.65i	0.716383i
$244$	3755.91i	0.985442i
$245$	− 5306.00i	− 1.38362i
$246$	2951.57	0.764981
$247$	162.037	0.0417414
$248$	4340.21i	1.11130i
$249$	− 1111.74i	− 0.282946i
$250$	− 1239.91i	− 0.313675i
$251$	4427.81	1.11347	0.556735	−	0.830690i	$-0.312054\pi$
$251$	4427.81	1.11347	0.556735	+	0.830690i	$0.312054\pi$
$252$	1773.69i	0.443381i
$253$	−5114.46	−1.27092
$254$	537.979	0.132897
$255$	0	0
$256$	−1840.73	−0.449397
$257$	6198.01	1.50436	0.752182	−	0.658956i	$-0.229002\pi$
$257$	6198.01	1.50436	0.752182	+	0.658956i	$0.229002\pi$
$258$	− 3604.81i	− 0.869867i
$259$	4435.55	1.06414
$260$	4465.26i	1.06509i
$261$	189.156i	0.0448599i
$262$	25.1109i	0.00592122i
$263$	−477.072	−0.111854	−0.0559269	−	0.998435i	$-0.517811\pi$
$263$	−477.072	−0.111854	−0.0559269	+	0.998435i	$0.517811\pi$
$264$	−5043.09	−1.17569
$265$	1401.58i	0.324900i
$266$	106.068i	0.0244491i
$267$	3797.48i	0.870419i
$268$	−2575.84	−0.587106
$269$	7583.93i	1.71896i	0.511168	+	0.859481i	$0.329213\pi$
$269$	7583.93i	1.71896i	−0.511168	+	0.859481i	$0.670787\pi$
$270$	−1718.18	−0.387278
$271$	−5431.92	−1.21759	−0.608793	−	0.793329i	$-0.708346\pi$
$271$	−5431.92	−1.21759	−0.608793	+	0.793329i	$0.708346\pi$
$272$	0	0
$273$	−8729.47	−1.93528
$274$	2302.33	0.507624
$275$	2316.62i	0.507991i
$276$	4389.24	0.957251
$277$	4091.98i	0.887594i	0.896127	+	0.443797i	$0.146369\pi$
$277$	4091.98i	0.887594i	−0.896127	+	0.443797i	$0.853631\pi$
$278$	− 2275.81i	− 0.490985i
$279$	− 2424.31i	− 0.520213i
$280$	−6571.86	−1.40266
$281$	3410.26	0.723983	0.361991	−	0.932181i	$-0.382097\pi$
$281$	3410.26	0.723983	0.361991	+	0.932181i	$0.382097\pi$
$282$	− 1535.92i	− 0.324336i
$283$	4322.86i	0.908011i	0.890999	+	0.454006i	$0.150005\pi$
$283$	4322.86i	0.908011i	−0.890999	+	0.454006i	$0.849995\pi$
$284$	3085.97i	0.644785i
$285$	−249.328	−0.0518208
$286$	− 3016.87i	− 0.623745i
$287$	10462.6	2.15186
$288$	1839.51	0.376369
$289$	0	0
$290$	−311.716	−0.0631194
$291$	−2252.51	−0.453761
$292$	4647.82i	0.931484i
$293$	3073.56	0.612831	0.306416	−	0.951898i	$-0.400870\pi$
$293$	3073.56	0.612831	0.306416	+	0.951898i	$0.400870\pi$
$294$	− 3076.57i	− 0.610304i
$295$	− 2747.26i	− 0.542208i
$296$	− 2957.84i	− 0.580815i
$297$	−4673.70	−0.913117
$298$	−3459.95	−0.672583
$299$	5903.62i	1.14186i
$300$	− 1988.13i	− 0.382616i
$301$	− 12778.1i	− 2.44690i
$302$	2630.76	0.501269
$303$	− 4939.57i	− 0.936538i
$304$	−87.3836	−0.0164862
$305$	7773.26	1.45933
$306$	0	0
$307$	−3954.45	−0.735154	−0.367577	−	0.929993i	$-0.619813\pi$
$307$	−3954.45	−0.735154	−0.367577	+	0.929993i	$0.619813\pi$
$308$	−7950.82	−1.47091
$309$	− 4709.79i	− 0.867089i
$310$	3995.11	0.731957
$311$	8984.40i	1.63813i	0.573701	+	0.819065i	$0.305507\pi$
$311$	8984.40i	1.63813i	−0.573701	+	0.819065i	$0.694493\pi$
$312$	5821.24i	1.05629i
$313$	− 1737.01i	− 0.313680i	−0.987624	−	0.156840i	$-0.949869\pi$
$313$	− 1737.01i	− 0.313680i	0.987624	−	0.156840i	$-0.0501306\pi$
$314$	3410.83	0.613008
$315$	3670.84	0.656598
$316$	− 2451.95i	− 0.436497i
$317$	− 8232.55i	− 1.45863i	−0.684177	−	0.729316i	$-0.739838\pi$
$317$	− 8232.55i	− 1.45863i	0.684177	−	0.729316i	$-0.260162\pi$
$318$	812.678i	0.143310i
$319$	−847.916	−0.148822
$320$	25.2488i	0.00441078i
$321$	−6076.88	−1.05663
$322$	−3864.48	−0.668817
$323$	0	0
$324$	5767.81	0.988993
$325$	2674.08	0.456403
$326$	3688.44i	0.626638i
$327$	−4272.83	−0.722594
$328$	− 6976.95i	− 1.17451i
$329$	− 5444.44i	− 0.912345i
$330$	4642.10i	0.774361i
$331$	4153.12	0.689655	0.344828	−	0.938666i	$-0.387937\pi$
$331$	4153.12	0.689655	0.344828	+	0.938666i	$0.387937\pi$
$332$	−1168.81	−0.193214
$333$	1652.16i	0.271885i
$334$	− 3052.26i	− 0.500037i
$335$	5330.97i	0.869439i
$336$	4707.66	0.764357
$337$	6569.49i	1.06191i	0.847401	+	0.530954i	$0.178166\pi$
$337$	6569.49i	1.06191i	−0.847401	+	0.530954i	$0.821834\pi$
$338$	−710.584	−0.114351
$339$	−8894.75	−1.42506
$340$	0	0
$341$	10867.3	1.72580
$342$	−39.5085	−0.00624671
$343$	− 1555.70i	− 0.244898i
$344$	−8521.08	−1.33554
$345$	− 9084.00i	− 1.41758i
$346$	− 496.644i	− 0.0771669i
$347$	2038.58i	0.315380i	0.987489	+	0.157690i	$0.0504047\pi$
$347$	2038.58i	0.315380i	−0.987489	+	0.157690i	$0.949595\pi$
$348$	727.684	0.112092
$349$	1791.21	0.274731	0.137365	−	0.990520i	$-0.456137\pi$
$349$	1791.21	0.274731	0.137365	+	0.990520i	$0.456137\pi$
$350$	1750.44i	0.267328i
$351$	5394.86i	0.820388i
$352$	8245.86i	1.24860i
$353$	−5928.71	−0.893919	−0.446959	−	0.894554i	$-0.647493\pi$
$353$	−5928.71	−0.893919	−0.446959	+	0.894554i	$0.647493\pi$
$354$	− 1592.94i	− 0.239163i
$355$	6386.75	0.954854
$356$	3992.43	0.594378
$357$	0	0
$358$	2420.55	0.357347
$359$	−413.998	−0.0608634	−0.0304317	−	0.999537i	$-0.509688\pi$
$359$	−413.998	−0.0608634	−0.0304317	+	0.999537i	$0.509688\pi$
$360$	− 2447.90i	− 0.358377i
$361$	−6849.49	−0.998613
$362$	2115.78i	0.307190i
$363$	4514.28i	0.652722i
$364$	9177.63i	1.32154i
$365$	9619.15	1.37942
$366$	4507.16	0.643697
$367$	− 6979.15i	− 0.992667i	−0.868132	−	0.496334i	$-0.834679\pi$
$367$	− 6979.15i	− 0.992667i	0.868132	−	0.496334i	$-0.165321\pi$
$368$	− 3183.73i	− 0.450987i
$369$	3897.11i	0.549798i
$370$	−2722.66	−0.382552
$371$	2880.73i	0.403127i
$372$	−9326.34	−1.29986
$373$	6855.64	0.951667	0.475833	−	0.879535i	$-0.342147\pi$
$373$	6855.64	0.951667	0.475833	+	0.879535i	$0.342147\pi$
$374$	0	0
$375$	5990.46	0.824923
$376$	−3630.62	−0.497965
$377$	978.750i	0.133709i
$378$	−3531.45	−0.480524
$379$	− 8206.78i	− 1.11228i	−0.831089	−	0.556140i	$-0.812282\pi$
$379$	− 8206.78i	− 1.11228i	0.831089	−	0.556140i	$-0.187718\pi$
$380$	262.128i	0.0353866i
$381$	2599.18i	0.349501i
$382$	−2961.27	−0.396627
$383$	889.401	0.118659	0.0593293	−	0.998238i	$-0.481104\pi$
$383$	889.401	0.118659	0.0593293	+	0.998238i	$0.481104\pi$
$384$	− 8819.67i	− 1.17208i
$385$	16455.1i	2.17825i
$386$	− 821.294i	− 0.108297i
$387$	4759.62	0.625180
$388$	2368.15i	0.309857i
$389$	12992.8	1.69347	0.846736	−	0.532013i	$-0.178564\pi$
$389$	12992.8	1.69347	0.846736	+	0.532013i	$0.178564\pi$
$390$	5358.38	0.695723
$391$	0	0
$392$	−7272.43	−0.937023
$393$	−121.320	−0.0155720
$394$	3382.42i	0.432498i
$395$	−5074.57	−0.646403
$396$	− 2961.54i	− 0.375815i
$397$	− 4325.29i	− 0.546801i	−0.961900	−	0.273400i	$-0.911852\pi$
$397$	− 4325.29i	− 0.546801i	0.961900	−	0.273400i	$-0.0881484\pi$
$398$	− 2251.67i	− 0.283583i
$399$	−512.455	−0.0642979
$400$	−1442.09	−0.180261
$401$	11849.1i	1.47560i	0.675022	+	0.737798i	$0.264134\pi$
$401$	11849.1i	1.47560i	−0.675022	+	0.737798i	$0.735866\pi$
$402$	3091.05i	0.383501i
$403$	− 12544.1i	− 1.55054i
$404$	−5193.17	−0.639529
$405$	− 11937.1i	− 1.46459i
$406$	−640.685	−0.0783168
$407$	−7406.04	−0.901975
$408$	0	0
$409$	7677.24	0.928154	0.464077	−	0.885795i	$-0.346386\pi$
$409$	7677.24	0.928154	0.464077	+	0.885795i	$0.346386\pi$
$410$	−6422.19	−0.773584
$411$	11123.4i	1.33498i
$412$	−4951.59	−0.592105
$413$	− 5646.55i	− 0.672757i
$414$	− 1439.45i	− 0.170882i
$415$	2418.98i	0.286128i
$416$	9518.20	1.12180
$417$	10995.3	1.29122
$418$	− 177.102i	− 0.0207233i
$419$	3235.83i	0.377280i	0.982046	+	0.188640i	$0.0604080\pi$
$419$	3235.83i	0.377280i	−0.982046	+	0.188640i	$0.939592\pi$
$420$	− 14121.8i	− 1.64065i
$421$	14496.6	1.67820	0.839100	−	0.543977i	$-0.183082\pi$
$421$	14496.6	1.67820	0.839100	+	0.543977i	$0.183082\pi$
$422$	− 86.5033i	− 0.00997848i
$423$	2027.95	0.233103
$424$	1921.02	0.220030
$425$	0	0
$426$	3703.22	0.421178
$427$	15976.7	1.81070
$428$	6388.86i	0.721536i
$429$	14575.6	1.64037
$430$	7843.54i	0.879649i
$431$	− 7136.25i	− 0.797543i	−0.917050	−	0.398771i	$-0.869437\pi$
$431$	− 7136.25i	− 0.797543i	0.917050	−	0.398771i	$-0.130563\pi$
$432$	− 2909.36i	− 0.324020i
$433$	−11434.3	−1.26904	−0.634521	−	0.772905i	$-0.718803\pi$
$433$	−11434.3	−1.26904	−0.634521	+	0.772905i	$0.718803\pi$
$434$	8211.32	0.908193
$435$	− 1506.02i	− 0.165996i
$436$	4492.20i	0.493434i
$437$	346.566i	0.0379371i
$438$	5577.46	0.608451
$439$	− 13308.0i	− 1.44682i	−0.690417	−	0.723412i	$-0.742573\pi$
$439$	− 13308.0i	− 1.44682i	0.690417	−	0.723412i	$-0.257427\pi$
$440$	10973.0	1.18891
$441$	4062.16	0.438630
$442$	0	0
$443$	−10897.9	−1.16880	−0.584398	−	0.811467i	$-0.698669\pi$
$443$	−10897.9	−1.16880	−0.584398	+	0.811467i	$0.698669\pi$
$444$	6355.88	0.679362
$445$	− 8262.76i	− 0.880208i
$446$	5972.80	0.634126
$447$	− 16716.3i	− 1.76880i
$448$	51.8949i	0.00547278i
$449$	16756.4i	1.76121i	0.473855	+	0.880603i	$0.342862\pi$
$449$	16756.4i	1.76121i	−0.473855	+	0.880603i	$0.657138\pi$
$450$	−652.006	−0.0683019
$451$	−17469.3	−1.82394
$452$	9351.40i	0.973126i
$453$	12710.2i	1.31827i
$454$	− 2173.60i	− 0.224696i
$455$	18994.1	1.95705
$456$	341.730i	0.0350943i
$457$	−13561.8	−1.38817	−0.694087	−	0.719891i	$-0.744192\pi$
$457$	−13561.8	−1.38817	−0.694087	+	0.719891i	$0.744192\pi$
$458$	−395.361	−0.0403363
$459$	0	0
$460$	−9550.36	−0.968017
$461$	−15516.4	−1.56762	−0.783810	−	0.621000i	$-0.786727\pi$
$461$	−15516.4	−1.56762	−0.783810	+	0.621000i	$0.786727\pi$
$462$	9541.11i	0.960807i
$463$	−15186.4	−1.52435	−0.762174	−	0.647372i	$-0.775868\pi$
$463$	−15186.4	−1.52435	−0.762174	+	0.647372i	$0.775868\pi$
$464$	− 527.824i	− 0.0528095i
$465$	19301.8i	1.92495i
$466$	− 2651.44i	− 0.263574i
$467$	−10544.3	−1.04482	−0.522412	−	0.852693i	$-0.674968\pi$
$467$	−10544.3	−1.04482	−0.522412	+	0.852693i	$0.674968\pi$
$468$	−3418.50	−0.337650
$469$	10957.0i	1.07878i
$470$	3341.94i	0.327983i
$471$	16479.0i	1.61213i
$472$	−3765.40	−0.367196
$473$	21335.6i	2.07402i
$474$	−2942.38	−0.285123
$475$	156.979	0.0151636
$476$	0	0
$477$	−1073.02	−0.102998
$478$	6971.02	0.667044
$479$	− 5762.22i	− 0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$479$	− 5762.22i	− 0.549650i	0.961494	−	0.274825i	$-0.0886200\pi$
$480$	−14645.8	−1.39268
$481$	8548.79i	0.810377i
$482$	2530.72i	0.239152i
$483$	− 18670.7i	− 1.75890i
$484$	4746.03	0.445721
$485$	4901.14	0.458864
$486$	− 3423.61i	− 0.319544i
$487$	− 5369.36i	− 0.499608i	−0.968296	−	0.249804i	$-0.919634\pi$
$487$	− 5369.36i	− 0.499608i	0.968296	−	0.249804i	$-0.0803661\pi$
$488$	− 10654.1i	− 0.988293i
$489$	−17820.2	−1.64797
$490$	6694.18i	0.617167i
$491$	−17878.0	−1.64322	−0.821611	−	0.570049i	$-0.806924\pi$
$491$	−17878.0	−1.64322	−0.821611	+	0.570049i	$0.806924\pi$
$492$	14992.2	1.37378
$493$	0	0
$494$	−204.429	−0.0186188
$495$	−6129.21	−0.556540
$496$	6764.84i	0.612400i
$497$	13127.0	1.18476
$498$	1402.60i	0.126209i
$499$	3507.03i	0.314622i	0.987549	+	0.157311i	$0.0502825\pi$
$499$	3507.03i	0.314622i	−0.987549	+	0.157311i	$0.949718\pi$
$500$	− 6298.01i	− 0.563311i
$501$	14746.6	1.31503
$502$	−5586.23	−0.496665
$503$	− 1207.56i	− 0.107043i	−0.998567	−	0.0535214i	$-0.982955\pi$
$503$	− 1207.56i	− 0.107043i	0.998567	−	0.0535214i	$-0.0170445\pi$
$504$	− 5031.27i	− 0.444664i
$505$	10747.8i	0.947071i
$506$	6452.52	0.566897
$507$	− 3433.09i	− 0.300728i
$508$	2732.62	0.238662
$509$	−14092.1	−1.22716	−0.613578	−	0.789634i	$-0.710270\pi$
$509$	−14092.1	−1.22716	−0.613578	+	0.789634i	$0.710270\pi$
$510$	0	0
$511$	19770.7	1.71155
$512$	−9253.25	−0.798710
$513$	316.700i	0.0272566i
$514$	−7819.56	−0.671023
$515$	10247.8i	0.876841i
$516$	− 18310.3i	− 1.56214i
$517$	9090.58i	0.773314i
$518$	−5595.99	−0.474660
$519$	2399.47	0.202939
$520$	− 12666.2i	− 1.06817i
$521$	− 10080.1i	− 0.847635i	−0.905748	−	0.423818i	$-0.860690\pi$
$521$	− 10080.1i	− 0.847635i	0.905748	−	0.423818i	$-0.139310\pi$
$522$	− 238.643i	− 0.0200098i
$523$	12330.2	1.03090	0.515452	−	0.856919i	$-0.327624\pi$
$523$	12330.2	1.03090	0.515452	+	0.856919i	$0.327624\pi$
$524$	127.549i	0.0106336i
$525$	−8457.01	−0.703036
$526$	601.886	0.0498925
$527$	0	0
$528$	−7860.38	−0.647877
$529$	−459.760	−0.0377875
$530$	− 1768.27i	− 0.144922i
$531$	2103.24	0.171888
$532$	538.764i	0.0439067i
$533$	20164.9i	1.63872i
$534$	− 4790.99i	− 0.388252i
$535$	13222.4	1.06851
$536$	7306.65	0.588805
$537$	11694.6i	0.939774i
$538$	− 9568.07i	− 0.766745i
$539$	18209.2i	1.45515i
$540$	−8727.33	−0.695490
$541$	− 7122.72i	− 0.566044i	−0.959113	−	0.283022i	$-0.908663\pi$
$541$	− 7122.72i	− 0.566044i	0.959113	−	0.283022i	$-0.0913369\pi$
$542$	6853.04	0.543106
$543$	−10222.1	−0.807869
$544$	0	0
$545$	9297.07	0.730720
$546$	11013.3	0.863235
$547$	− 14919.7i	− 1.16622i	−0.812394	−	0.583109i	$-0.801836\pi$
$547$	− 14919.7i	− 1.16622i	0.812394	−	0.583109i	$-0.198164\pi$
$548$	11694.5	0.911612
$549$	5951.04i	0.462630i
$550$	− 2922.70i	− 0.226590i
$551$	57.4566i	0.00444234i
$552$	−12450.6	−0.960021
$553$	−10430.0	−0.802040
$554$	− 5162.55i	− 0.395913i
$555$	− 13154.2i	− 1.00606i
$556$	− 11559.7i	− 0.881731i
$557$	−18935.0	−1.44039	−0.720197	−	0.693769i	$-0.755949\pi$
$557$	−18935.0	−1.44039	−0.720197	+	0.693769i	$0.755949\pi$
$558$	3058.57i	0.232042i
$559$	24627.7	1.86340
$560$	−10243.2	−0.772953
$561$	0	0
$562$	−4302.47	−0.322934
$563$	−8150.42	−0.610123	−0.305062	−	0.952333i	$-0.598677\pi$
$563$	−8150.42	−0.610123	−0.305062	+	0.952333i	$0.598677\pi$
$564$	− 7801.56i	− 0.582456i
$565$	19353.7	1.44109
$566$	− 5453.82i	− 0.405020i
$567$	− 24534.8i	− 1.81722i
$568$	− 8753.71i	− 0.646650i
$569$	−18218.2	−1.34226	−0.671131	−	0.741338i	$-0.734191\pi$
$569$	−18218.2	−1.34226	−0.671131	+	0.741338i	$0.734191\pi$
$570$	314.558	0.0231147
$571$	2443.35i	0.179073i	0.995984	+	0.0895367i	$0.0285386\pi$
$571$	2443.35i	0.179073i	−0.995984	+	0.0895367i	$0.971461\pi$
$572$	− 15323.9i	− 1.12015i
$573$	− 14307.0i	− 1.04308i
$574$	−13199.8	−0.959842
$575$	5719.36i	0.414806i
$576$	−19.3299	−0.00139829
$577$	−852.752	−0.0615261	−0.0307630	−	0.999527i	$-0.509794\pi$
$577$	−852.752	−0.0615261	−0.0307630	+	0.999527i	$0.509794\pi$
$578$	0	0
$579$	3967.98	0.284807
$580$	−1583.34	−0.113352
$581$	4971.84i	0.355020i
$582$	2841.82	0.202401
$583$	− 4809.96i	− 0.341695i
$584$	− 13184.1i	− 0.934179i
$585$	7074.95i	0.500022i
$586$	−3877.68	−0.273354
$587$	−900.272	−0.0633019	−0.0316509	−	0.999499i	$-0.510076\pi$
$587$	−900.272	−0.0633019	−0.0316509	+	0.999499i	$0.510076\pi$
$588$	− 15627.2i	− 1.09601i
$589$	− 736.390i	− 0.0515152i
$590$	3466.00i	0.241853i
$591$	−16341.7	−1.13741
$592$	− 4610.22i	− 0.320066i
$593$	25007.2	1.73174	0.865870	−	0.500268i	$-0.166765\pi$
$593$	25007.2	1.73174	0.865870	+	0.500268i	$0.166765\pi$
$594$	5896.46	0.407297
$595$	0	0
$596$	−17574.5	−1.20785
$597$	10878.6	0.745784
$598$	− 7448.15i	− 0.509327i
$599$	−3448.82	−0.235251	−0.117625	−	0.993058i	$-0.537528\pi$
$599$	−3448.82	−0.235251	−0.117625	+	0.993058i	$0.537528\pi$
$600$	5639.55i	0.383723i
$601$	942.873i	0.0639943i	0.999488	+	0.0319972i	$0.0101868\pi$
$601$	942.873i	0.0639943i	−0.999488	+	0.0319972i	$0.989813\pi$
$602$	16121.2i	1.09145i
$603$	−4081.27	−0.275626
$604$	13362.7	0.900201
$605$	− 9822.41i	− 0.660063i
$606$	6231.89i	0.417744i
$607$	10173.9i	0.680306i	0.940370	+	0.340153i	$0.110479\pi$
$607$	10173.9i	0.680306i	−0.940370	+	0.340153i	$0.889521\pi$
$608$	558.756	0.0372707
$609$	− 3095.39i	− 0.205963i
$610$	−9806.93	−0.650936
$611$	10493.3	0.694782
$612$	0	0
$613$	−17327.0	−1.14165	−0.570824	−	0.821073i	$-0.693376\pi$
$613$	−17327.0	−1.14165	−0.570824	+	0.821073i	$0.693376\pi$
$614$	4989.03	0.327916
$615$	− 31028.0i	− 2.03442i
$616$	22553.4	1.47516
$617$	19782.8i	1.29080i	0.763844	+	0.645401i	$0.223310\pi$
$617$	19782.8i	1.29080i	−0.763844	+	0.645401i	$0.776690\pi$
$618$	5941.98i	0.386766i
$619$	− 15033.2i	− 0.976147i	−0.872802	−	0.488074i	$-0.837700\pi$
$619$	− 15033.2i	− 0.976147i	0.872802	−	0.488074i	$-0.162300\pi$
$620$	20292.8	1.31448
$621$	−11538.6	−0.745618
$622$	− 11334.9i	− 0.730690i
$623$	− 16982.8i	− 1.09214i
$624$	9073.25i	0.582084i
$625$	−19396.6	−1.24139
$626$	2191.46i	0.139917i
$627$	855.647	0.0544996
$628$	17325.0	1.10087
$629$	0	0
$630$	−4631.22	−0.292877
$631$	24060.5	1.51796	0.758979	−	0.651115i	$-0.225698\pi$
$631$	24060.5	1.51796	0.758979	+	0.651115i	$0.225698\pi$
$632$	6955.23i	0.437760i
$633$	417.930	0.0262421
$634$	10386.4i	0.650625i
$635$	− 5655.44i	− 0.353432i
$636$	4127.92i	0.257363i
$637$	21018.9	1.30737
$638$	1069.75	0.0663822
$639$	4889.55i	0.302704i
$640$	19190.3i	1.18526i
$641$	− 4886.07i	− 0.301074i	−0.988604	−	0.150537i	$-0.951900\pi$
$641$	− 4886.07i	− 0.301074i	0.988604	−	0.150537i	$-0.0481002\pi$
$642$	7666.74	0.471312
$643$	− 24508.4i	− 1.50314i	−0.659655	−	0.751568i	$-0.729298\pi$
$643$	− 24508.4i	− 1.50314i	0.659655	−	0.751568i	$-0.270702\pi$
$644$	−19629.3	−1.20109
$645$	−37895.1	−2.31336
$646$	0	0
$647$	2932.50	0.178189	0.0890947	−	0.996023i	$-0.471603\pi$
$647$	2932.50	0.178189	0.0890947	+	0.996023i	$0.471603\pi$
$648$	−16361.0	−0.991854
$649$	9428.06i	0.570237i
$650$	−3373.68	−0.203579
$651$	39671.9i	2.38843i
$652$	18735.1i	1.12534i
$653$	− 19977.8i	− 1.19723i	−0.801035	−	0.598617i	$-0.795717\pi$
$653$	− 19977.8i	− 1.19723i	0.801035	−	0.598617i	$-0.204283\pi$
$654$	5390.71	0.322314
$655$	263.975	0.0157471
$656$	− 10874.6i	− 0.647227i
$657$	7364.21i	0.437299i
$658$	6868.83i	0.406953i
$659$	28222.7	1.66829	0.834143	−	0.551548i	$-0.185963\pi$
$659$	28222.7	1.66829	0.834143	+	0.551548i	$0.185963\pi$
$660$	23579.1i	1.39063i
$661$	−20712.8	−1.21881	−0.609405	−	0.792859i	$-0.708592\pi$
$661$	−20712.8	−1.21881	−0.609405	+	0.792859i	$0.708592\pi$
$662$	−5239.67	−0.307622
$663$	0	0
$664$	3315.47	0.193773
$665$	1115.03	0.0650210
$666$	− 2084.41i	− 0.121275i
$667$	−2093.37	−0.121522
$668$	− 15503.7i	− 0.897987i
$669$	28856.8i	1.66767i
$670$	− 6725.68i	− 0.387814i
$671$	−26676.3	−1.53477
$672$	−30102.2	−1.72800
$673$	− 395.468i	− 0.0226511i	−0.999936	−	0.0113255i	$-0.996395\pi$
$673$	− 395.468i	− 0.0226511i	0.999936	−	0.0113255i	$-0.00360511\pi$
$674$	− 8288.23i	− 0.473666i
$675$	5226.48i	0.298025i
$676$	−3609.34	−0.205356
$677$	− 2297.47i	− 0.130427i	−0.997871	−	0.0652135i	$-0.979227\pi$
$677$	− 2297.47i	− 0.130427i	0.997871	−	0.0652135i	$-0.0207729\pi$
$678$	11221.8	0.635652
$679$	10073.5	0.569347
$680$	0	0
$681$	10501.5	0.590921
$682$	−13710.4	−0.769794
$683$	− 28611.7i	− 1.60292i	−0.598048	−	0.801461i	$-0.704057\pi$
$683$	− 28611.7i	− 1.60292i	0.598048	−	0.801461i	$-0.295943\pi$
$684$	−200.680	−0.0112181
$685$	− 24202.9i	− 1.35000i
$686$	1962.71i	0.109237i
$687$	− 1910.14i	− 0.106079i
$688$	−13281.3	−0.735968
$689$	−5552.14	−0.306995
$690$	11460.6i	0.632315i
$691$	− 4902.87i	− 0.269919i	−0.990851	−	0.134959i	$-0.956910\pi$
$691$	− 4902.87i	− 0.269919i	0.990851	−	0.134959i	$-0.0430904\pi$
$692$	− 2522.66i	− 0.138580i
$693$	−12597.6	−0.690540
$694$	− 2571.93i	− 0.140676i
$695$	−23924.1	−1.30575
$696$	−2064.16	−0.112416
$697$	0	0
$698$	−2259.83	−0.122544
$699$	12810.1	0.693165
$700$	8891.18i	0.480079i
$701$	1714.99	0.0924027	0.0462014	−	0.998932i	$-0.485288\pi$
$701$	1714.99	0.0924027	0.0462014	+	0.998932i	$0.485288\pi$
$702$	− 6806.28i	− 0.365935i
$703$	501.849i	0.0269240i
$704$	− 86.6490i	− 0.00463879i
$705$	−16146.1	−0.862552
$706$	7479.80	0.398734
$707$	22090.4i	1.17510i
$708$	− 8091.18i	− 0.429499i
$709$	− 13939.8i	− 0.738393i	−0.929351	−	0.369196i	$-0.879633\pi$
$709$	− 13939.8i	− 0.738393i	0.929351	−	0.369196i	$-0.120367\pi$
$710$	−8057.67	−0.425914
$711$	− 3884.98i	− 0.204920i
$712$	−11325.0	−0.596098
$713$	26829.6	1.40922
$714$	0	0
$715$	−31714.4	−1.65881
$716$	12295.0	0.641738
$717$	33679.6i	1.75424i
$718$	522.310	0.0271482
$719$	− 1779.18i	− 0.0922839i	−0.998935	−	0.0461419i	$-0.985307\pi$
$719$	− 1779.18i	− 0.0922839i	0.998935	−	0.0461419i	$-0.0146926\pi$
$720$	− 3815.40i	− 0.197488i
$721$	21062.8i	1.08796i
$722$	8641.48	0.445433
$723$	−12226.9	−0.628937
$724$	10746.9i	0.551665i
$725$	948.201i	0.0485729i
$726$	− 5695.32i	− 0.291148i
$727$	9141.02	0.466330	0.233165	−	0.972437i	$-0.425092\pi$
$727$	9141.02	0.466330	0.233165	+	0.972437i	$0.425092\pi$
$728$	− 26033.4i	− 1.32536i
$729$	−7760.65	−0.394282
$730$	−12135.8	−0.615294
$731$	0	0
$732$	22893.7	1.15598
$733$	−12176.0	−0.613547	−0.306773	−	0.951783i	$-0.599249\pi$
$733$	−12176.0	−0.613547	−0.306773	+	0.951783i	$0.599249\pi$
$734$	8805.07i	0.442781i
$735$	−32342.1	−1.62307
$736$	20357.7i	1.01956i
$737$	− 18294.9i	− 0.914383i
$738$	− 4916.69i	− 0.245238i
$739$	−25025.4	−1.24570	−0.622852	−	0.782339i	$-0.714026\pi$
$739$	−25025.4	−1.24570	−0.622852	+	0.782339i	$0.714026\pi$
$740$	−13829.5	−0.687003
$741$	− 987.673i	− 0.0489650i
$742$	− 3634.40i	− 0.179816i
$743$	− 10781.3i	− 0.532339i	−0.963926	−	0.266170i	$-0.914242\pi$
$743$	− 10781.3i	− 0.532339i	0.963926	−	0.266170i	$-0.0857581\pi$
$744$	26455.2	1.30362
$745$	36372.3i	1.78869i
$746$	−8649.24	−0.424492
$747$	−1851.92	−0.0907072
$748$	0	0
$749$	27176.6	1.32578
$750$	−7557.71	−0.367958
$751$	33910.7i	1.64769i	0.566812	+	0.823847i	$0.308177\pi$
$751$	33910.7i	1.64769i	−0.566812	+	0.823847i	$0.691823\pi$
$752$	−5658.84	−0.274410
$753$	− 26989.2i	− 1.30616i
$754$	− 1234.81i	− 0.0596410i
$755$	− 27655.5i	− 1.33310i
$756$	−17937.7	−0.862945
$757$	31545.0	1.51456	0.757281	−	0.653090i	$-0.226528\pi$
$757$	31545.0	1.51456	0.757281	+	0.653090i	$0.226528\pi$
$758$	10353.9i	0.496134i
$759$	31174.5i	1.49086i
$760$	− 743.556i	− 0.0354890i
$761$	9150.51	0.435881	0.217941	−	0.975962i	$-0.430066\pi$
$761$	9150.51	0.435881	0.217941	+	0.975962i	$0.430066\pi$
$762$	− 3279.19i	− 0.155895i
$763$	19108.7	0.906658
$764$	−15041.5	−0.712280
$765$	0	0
$766$	−1122.09	−0.0529279
$767$	10882.8	0.512328
$768$	11219.9i	0.527168i
$769$	33797.5	1.58488	0.792438	−	0.609952i	$-0.208811\pi$
$769$	33797.5	1.58488	0.792438	+	0.609952i	$0.208811\pi$
$770$	− 20760.1i	− 0.971612i
$771$	− 37779.2i	− 1.76470i
$772$	− 4171.69i	− 0.194485i
$773$	33969.4	1.58059	0.790295	−	0.612727i	$-0.209927\pi$
$773$	33969.4	1.58059	0.790295	+	0.612727i	$0.209927\pi$
$774$	−6004.85	−0.278863
$775$	− 12152.6i	− 0.563270i
$776$	− 6717.52i	− 0.310754i
$777$	− 27036.3i	− 1.24829i
$778$	−16392.0	−0.755376
$779$	1183.76i	0.0544449i
$780$	27217.4	1.24941
$781$	−21918.1	−1.00421
$782$	0	0
$783$	−1912.96	−0.0873100
$784$	−11335.1	−0.516359
$785$	− 35856.0i	− 1.63026i
$786$	153.061	0.00694592
$787$	− 10962.7i	− 0.496543i	−0.968691	−	0.248271i	$-0.920138\pi$
$787$	− 10962.7i	− 0.496543i	0.968691	−	0.248271i	$-0.0798624\pi$
$788$	17180.7i	0.776698i
$789$	2907.93i	0.131211i
$790$	6402.20	0.288329
$791$	39778.5	1.78807
$792$	8400.72i	0.376902i
$793$	30792.5i	1.37891i
$794$	5456.89i	0.243901i
$795$	8543.17	0.381126
$796$	− 11437.1i	− 0.509270i
$797$	8041.29	0.357386	0.178693	−	0.983905i	$-0.442813\pi$
$797$	8041.29	0.357386	0.178693	+	0.983905i	$0.442813\pi$
$798$	646.526	0.0286802
$799$	0	0
$800$	9221.12	0.407520
$801$	6325.79	0.279040
$802$	− 14949.1i	− 0.658192i
$803$	−33011.1	−1.45073
$804$	15700.7i	0.688708i
$805$	40624.8i	1.77868i
$806$	15826.0i	0.691620i
$807$	46226.9	2.01644
$808$	14731.0	0.641379
$809$	− 36568.0i	− 1.58920i	−0.607134	−	0.794599i	$-0.707681\pi$
$809$	− 36568.0i	− 1.58920i	0.607134	−	0.794599i	$-0.292319\pi$
$810$	15060.1i	0.653282i
$811$	37315.7i	1.61570i	0.589390	+	0.807849i	$0.299368\pi$
$811$	37315.7i	1.61570i	−0.589390	+	0.807849i	$0.700632\pi$
$812$	−3254.30	−0.140645
$813$	33109.6i	1.42830i
$814$	9343.64	0.402327
$815$	38774.3	1.66651
$816$	0	0
$817$	1445.75	0.0619097
$818$	−9685.79	−0.414005
$819$	14541.4i	0.620414i
$820$	−32620.9	−1.38924
$821$	− 27365.1i	− 1.16328i	−0.813448	−	0.581638i	$-0.802412\pi$
$821$	− 27365.1i	− 1.16328i	0.813448	−	0.581638i	$-0.197588\pi$
$822$	− 14033.6i	− 0.595471i
$823$	21138.4i	0.895308i	0.894207	+	0.447654i	$0.147740\pi$
$823$	21138.4i	0.895308i	−0.894207	+	0.447654i	$0.852260\pi$
$824$	14045.7	0.593818
$825$	14120.7	0.595902
$826$	7123.83i	0.300084i
$827$	43453.3i	1.82711i	0.406717	+	0.913554i	$0.366673\pi$
$827$	43453.3i	1.82711i	−0.406717	+	0.913554i	$0.633327\pi$
$828$	− 7311.55i	− 0.306877i
$829$	36460.2	1.52752	0.763760	−	0.645501i	$-0.223351\pi$
$829$	36460.2	1.52752	0.763760	+	0.645501i	$0.223351\pi$
$830$	− 3051.85i	− 0.127628i
$831$	24942.2	1.04120
$832$	−100.019	−0.00416771
$833$	0	0
$834$	−13871.9	−0.575952
$835$	−32086.5	−1.32982
$836$	− 899.575i	− 0.0372158i
$837$	24517.4	1.01248
$838$	− 4082.39i	− 0.168286i
$839$	− 20236.0i	− 0.832687i	−0.909207	−	0.416344i	$-0.863311\pi$
$839$	− 20236.0i	− 0.832687i	0.909207	−	0.416344i	$-0.136689\pi$
$840$	40058.0i	1.64539i
$841$	24041.9	0.985770
$842$	−18289.3	−0.748564
$843$	− 20786.8i	− 0.849272i
$844$	− 439.386i	− 0.0179198i
$845$	7469.92i	0.304110i
$846$	−2558.51	−0.103976
$847$	− 20188.4i	− 0.818988i
$848$	2994.18	0.121251
$849$	26349.4	1.06515
$850$	0	0
$851$	−18284.3	−0.736519
$852$	18810.2	0.756368
$853$	− 6934.19i	− 0.278338i	−0.990269	−	0.139169i	$-0.955557\pi$
$853$	− 6934.19i	− 0.278338i	0.990269	−	0.139169i	$-0.0444431\pi$
$854$	−20156.6	−0.807664
$855$	415.328i	0.0166128i
$856$	− 18122.7i	− 0.723623i
$857$	19207.8i	0.765607i	0.923830	+	0.382803i	$0.125041\pi$
$857$	19207.8i	0.765607i	−0.923830	+	0.382803i	$0.874959\pi$
$858$	−18388.9	−0.731688
$859$	9954.55	0.395395	0.197698	−	0.980263i	$-0.436654\pi$
$859$	9954.55	0.395395	0.197698	+	0.980263i	$0.436654\pi$
$860$	39840.6i	1.57971i
$861$	− 63773.2i	− 2.52426i
$862$	9003.26i	0.355745i
$863$	−33698.0	−1.32919	−0.664596	−	0.747203i	$-0.731396\pi$
$863$	−33698.0	−1.32919	−0.664596	+	0.747203i	$0.731396\pi$
$864$	18603.3i	0.732519i
$865$	−5220.91	−0.205221
$866$	14425.7	0.566058
$867$	0	0
$868$	41708.6	1.63097
$869$	17415.0	0.679818
$870$	1900.03i	0.0740425i
$871$	−21117.8	−0.821525
$872$	− 12742.6i	− 0.494861i
$873$	3752.20i	0.145467i
$874$	− 437.236i	− 0.0169219i
$875$	−26790.1	−1.03505
$876$	28330.2	1.09268
$877$	4469.65i	0.172097i	0.996291	+	0.0860487i	$0.0274241\pi$
$877$	4469.65i	0.172097i	−0.996291	+	0.0860487i	$0.972576\pi$
$878$	16789.7i	0.645358i
$879$	− 18734.5i	− 0.718885i
$880$	17103.1	0.655164
$881$	− 2081.32i	− 0.0795930i	−0.999208	−	0.0397965i	$-0.987329\pi$
$881$	− 2081.32i	− 0.0795930i	0.999208	−	0.0397965i	$-0.0126710\pi$
$882$	−5124.91	−0.195652
$883$	10124.4	0.385857	0.192929	−	0.981213i	$-0.438201\pi$
$883$	10124.4	0.385857	0.192929	+	0.981213i	$0.438201\pi$
$884$	0	0
$885$	−16745.6	−0.636040
$886$	13749.1	0.521343
$887$	42725.3i	1.61733i	0.588268	+	0.808666i	$0.299810\pi$
$887$	42725.3i	1.61733i	−0.588268	+	0.808666i	$0.700190\pi$
$888$	−18029.2	−0.681328
$889$	− 11623.9i	− 0.438529i
$890$	10424.5i	0.392618i
$891$	40965.8i	1.54030i
$892$	30338.3	1.13879
$893$	615.996	0.0230835
$894$	21089.7i	0.788977i
$895$	− 25445.7i	− 0.950343i
$896$	39442.7i	1.47063i
$897$	35984.8	1.33946
$898$	− 21140.2i	− 0.785588i
$899$	4448.02	0.165017
$900$	−3311.80	−0.122659
$901$	0	0
$902$	22039.7	0.813573
$903$	−77887.4	−2.87036
$904$	− 26526.3i	− 0.975941i
$905$	22241.8	0.816954
$906$	− 16035.5i	− 0.588017i
$907$	− 21767.6i	− 0.796893i	−0.917192	−	0.398446i	$-0.869550\pi$
$907$	− 21767.6i	− 0.796893i	0.917192	−	0.398446i	$-0.130450\pi$
$908$	− 11040.6i	− 0.403519i
$909$	−8228.28	−0.300236
$910$	−23963.4	−0.872943
$911$	− 21654.2i	− 0.787524i	−0.919212	−	0.393762i	$-0.871173\pi$
$911$	− 21654.2i	− 0.787524i	0.919212	−	0.393762i	$-0.128827\pi$
$912$	532.636i	0.0193392i
$913$	− 8301.49i	− 0.300919i
$914$	17109.9	0.619197
$915$	− 47380.9i	− 1.71187i
$916$	−2008.20	−0.0724376
$917$	542.560	0.0195386
$918$	0	0
$919$	21445.1	0.769760	0.384880	−	0.922967i	$-0.374243\pi$
$919$	21445.1	0.769760	0.384880	+	0.922967i	$0.374243\pi$
$920$	27090.6	0.970818
$921$	24103.8i	0.862376i
$922$	19575.9	0.699239
$923$	25300.1i	0.902234i
$924$	48463.2i	1.72546i
$925$	8281.97i	0.294389i
$926$	19159.6	0.679938
$927$	−7845.51	−0.277972
$928$	3375.06i	0.119388i
$929$	24271.4i	0.857177i	0.903500	+	0.428589i	$0.140989\pi$
$929$	24271.4i	0.857177i	−0.903500	+	0.428589i	$0.859011\pi$
$930$	− 24351.7i	− 0.858626i
$931$	1233.89	0.0434363
$932$	− 13467.7i	− 0.473338i
$933$	54763.3	1.92162
$934$	13303.0	0.466046
$935$	0	0
$936$	9696.96	0.338627
$937$	−38491.5	−1.34201	−0.671005	−	0.741453i	$-0.734137\pi$
$937$	−38491.5	−1.34201	−0.671005	+	0.741453i	$0.734137\pi$
$938$	− 13823.6i	− 0.481190i
$939$	−10587.7	−0.367964
$940$	16975.1i	0.589006i
$941$	31323.4i	1.08514i	0.840012	+	0.542568i	$0.182548\pi$
$941$	31323.4i	1.08514i	−0.840012	+	0.542568i	$0.817452\pi$
$942$	− 20790.3i	− 0.719092i
$943$	−43128.9	−1.48937
$944$	−5868.92	−0.202349
$945$	37123.9i	1.27793i
$946$	− 26917.5i	− 0.925121i
$947$	− 32096.4i	− 1.10136i	−0.834715	−	0.550682i	$-0.814368\pi$
$947$	− 32096.4i	− 1.10136i	0.834715	−	0.550682i	$-0.185632\pi$
$948$	−14945.6	−0.512035
$949$	38104.7i	1.30341i
$950$	−198.049	−0.00676373
$951$	−50180.5	−1.71106
$952$	0	0
$953$	10493.3	0.356675	0.178338	−	0.983969i	$-0.442928\pi$
$953$	10493.3	0.356675	0.178338	+	0.983969i	$0.442928\pi$
$954$	1353.75	0.0459426
$955$	31129.9i	1.05481i
$956$	35408.7	1.19791
$957$	5168.37i	0.174576i
$958$	7269.75i	0.245172i
$959$	− 49745.4i	− 1.67504i
$960$	153.901	0.00517409
$961$	−27217.0	−0.913597
$962$	− 10785.4i	− 0.361470i
$963$	10122.8i	0.338736i
$964$	12854.6i	0.429479i
$965$	−8633.74	−0.288010
$966$	23555.4i	0.784559i
$967$	−31468.2	−1.04648	−0.523242	−	0.852184i	$-0.675278\pi$
$967$	−31468.2	−1.04648	−0.523242	+	0.852184i	$0.675278\pi$
$968$	−13462.6	−0.447010
$969$	0	0
$970$	−6183.39	−0.204677
$971$	46923.8	1.55083	0.775415	−	0.631452i	$-0.217541\pi$
$971$	46923.8	1.55083	0.775415	+	0.631452i	$0.217541\pi$
$972$	− 17389.9i	− 0.573850i
$973$	−49172.3	−1.62013
$974$	6774.11i	0.222851i
$975$	− 16299.5i	− 0.535387i
$976$	− 16605.9i	− 0.544612i
$977$	−8666.25	−0.283785	−0.141893	−	0.989882i	$-0.545319\pi$
$977$	−8666.25	−0.283785	−0.141893	+	0.989882i	$0.545319\pi$
$978$	22482.4	0.735081
$979$	28356.2i	0.925709i
$980$	34002.5i	1.10834i
$981$	7117.63i	0.231650i
$982$	22555.3	0.732961
$983$	35155.3i	1.14067i	0.821412	+	0.570336i	$0.193187\pi$
$983$	35155.3i	1.14067i	−0.821412	+	0.570336i	$0.806813\pi$
$984$	−42527.1	−1.37776
$985$	35557.3	1.15020
$986$	0	0
$987$	−33185.9	−1.07023
$988$	−1038.38	−0.0334365
$989$	52674.2i	1.69357i
$990$	7732.75	0.248246
$991$	− 11236.2i	− 0.360170i	−0.983651	−	0.180085i	$-0.942363\pi$
$991$	− 11236.2i	− 0.360170i	0.983651	−	0.180085i	$-0.0576374\pi$
$992$	− 43256.3i	− 1.38447i
$993$	− 25314.8i	− 0.809004i
$994$	−16561.3	−0.528463
$995$	−23670.4	−0.754172
$996$	7124.36i	0.226651i
$997$	− 18825.0i	− 0.597987i	−0.954255	−	0.298994i	$-0.903349\pi$
$997$	− 18825.0i	− 0.597987i	0.954255	−	0.298994i	$-0.0966509\pi$
$998$	− 4424.56i	− 0.140338i
$999$	−16708.6	−0.529166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.b.f.288.10		24
17.4	even	4		289.4.a.i.1.8	yes	12
17.13	even	4		289.4.a.h.1.8	✓	12
17.16	even	2	inner	289.4.b.f.288.9		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.4.a.h.1.8	✓	12	17.13	even	4
289.4.a.i.1.8	yes	12	17.4	even	4
289.4.b.f.288.9		24	17.16	even	2	inner
289.4.b.f.288.10		24	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.26162 q^{2} -6.09538i q^{3} -6.40830 q^{4} +13.2627i q^{5} +7.69007i q^{6} +27.2593i q^{7} +18.1779 q^{8} -10.1536 q^{9} +O(q^{10})\)	\(q-1.26162 q^{2} -6.09538i q^{3} -6.40830 q^{4} +13.2627i q^{5} +7.69007i q^{6} +27.2593i q^{7} +18.1779 q^{8} -10.1536 q^{9} -16.7325i q^{10} -45.5149i q^{11} +39.0610i q^{12} -52.5379 q^{13} -34.3910i q^{14} +80.8409 q^{15} +28.3328 q^{16} +12.8100 q^{18} -3.08418 q^{19} -84.9911i q^{20} +166.156 q^{21} +57.4227i q^{22} -112.369i q^{23} -110.801i q^{24} -50.8981 q^{25} +66.2831 q^{26} -102.685i q^{27} -174.686i q^{28} -18.6294i q^{29} -101.991 q^{30} +238.763i q^{31} -181.168 q^{32} -277.430 q^{33} -361.531 q^{35} +65.0674 q^{36} -162.717i q^{37} +3.89108 q^{38} +320.238i q^{39} +241.087i q^{40} -383.816i q^{41} -209.626 q^{42} -468.761 q^{43} +291.673i q^{44} -134.664i q^{45} +141.767i q^{46} -199.727 q^{47} -172.699i q^{48} -400.071 q^{49} +64.2142 q^{50} +336.679 q^{52} +105.679 q^{53} +129.550i q^{54} +603.649 q^{55} +495.516i q^{56} +18.7993i q^{57} +23.5033i q^{58} -207.142 q^{59} -518.053 q^{60} -586.101i q^{61} -301.230i q^{62} -276.780i q^{63} +1.90375 q^{64} -696.792i q^{65} +350.013 q^{66} +401.953 q^{67} -684.930 q^{69} +456.116 q^{70} -481.559i q^{71} -184.571 q^{72} -725.281i q^{73} +205.287i q^{74} +310.243i q^{75} +19.7644 q^{76} +1240.71 q^{77} -404.020i q^{78} +382.621i q^{79} +375.768i q^{80} -900.052 q^{81} +484.231i q^{82} +182.391 q^{83} -1064.78 q^{84} +591.401 q^{86} -113.553 q^{87} -827.364i q^{88} -623.009 q^{89} +169.895i q^{90} -1432.15i q^{91} +720.094i q^{92} +1455.35 q^{93} +251.981 q^{94} -40.9045i q^{95} +1104.29i q^{96} -369.544i q^{97} +504.739 q^{98} +462.140i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) \) 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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