LMFDB - Embedded newform 289.4.b.f.288.17

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			288.17
Character	$\chi$	$=$	289.288
Dual form			289.4.b.f.288.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.16473 q^{2} -9.14133i q^{3} -3.31395 q^{4} -16.2298i q^{5} -19.7885i q^{6} +12.2246i q^{7} -24.4916 q^{8} -56.5639 q^{9} +O(q^{10})$	\(q+2.16473 q^{2} -9.14133i q^{3} -3.31395 q^{4} -16.2298i q^{5} -19.7885i q^{6} +12.2246i q^{7} -24.4916 q^{8} -56.5639 q^{9} -35.1331i q^{10} -11.1761i q^{11} +30.2939i q^{12} +28.9000 q^{13} +26.4629i q^{14} -148.362 q^{15} -26.5061 q^{16} -122.445 q^{18} +79.5850 q^{19} +53.7848i q^{20} +111.749 q^{21} -24.1932i q^{22} -45.0447i q^{23} +223.886i q^{24} -138.406 q^{25} +62.5606 q^{26} +270.253i q^{27} -40.5117i q^{28} -20.3774i q^{29} -321.163 q^{30} -1.03839i q^{31} +138.555 q^{32} -102.164 q^{33} +198.402 q^{35} +187.450 q^{36} +219.921i q^{37} +172.280 q^{38} -264.184i q^{39} +397.494i q^{40} -310.992i q^{41} +241.906 q^{42} -483.760 q^{43} +37.0371i q^{44} +918.020i q^{45} -97.5095i q^{46} -632.422 q^{47} +242.301i q^{48} +193.560 q^{49} -299.612 q^{50} -95.7732 q^{52} -490.172 q^{53} +585.024i q^{54} -181.386 q^{55} -299.400i q^{56} -727.512i q^{57} -44.1115i q^{58} -147.956 q^{59} +491.665 q^{60} +176.395i q^{61} -2.24783i q^{62} -691.469i q^{63} +511.982 q^{64} -469.041i q^{65} -221.158 q^{66} +809.312 q^{67} -411.768 q^{69} +429.487 q^{70} -714.189i q^{71} +1385.34 q^{72} -780.184i q^{73} +476.070i q^{74} +1265.22i q^{75} -263.741 q^{76} +136.623 q^{77} -571.887i q^{78} -230.265i q^{79} +430.188i q^{80} +943.248 q^{81} -673.213i q^{82} -236.046 q^{83} -370.330 q^{84} -1047.21 q^{86} -186.276 q^{87} +273.721i q^{88} -688.557 q^{89} +1987.26i q^{90} +353.290i q^{91} +149.276i q^{92} -9.49224 q^{93} -1369.02 q^{94} -1291.65i q^{95} -1266.57i q^{96} +1846.11i q^{97} +419.004 q^{98} +632.164i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.16473	0.765347	0.382673	−	0.923884i	$-0.375003\pi$
$2$	2.16473	0.765347	0.382673	+	0.923884i	$0.375003\pi$
$3$	− 9.14133i	− 1.75925i	−0.475668	−	0.879625i	$-0.657794\pi$
$3$	− 9.14133i	− 1.75925i	0.475668	−	0.879625i	$-0.342206\pi$
$4$	−3.31395	−0.414244
$5$	− 16.2298i	− 1.45164i	−0.687886	−	0.725819i	$-0.741461\pi$
$5$	− 16.2298i	− 1.45164i	0.687886	−	0.725819i	$-0.258539\pi$
$6$	− 19.7885i	− 1.34644i
$7$	12.2246i	0.660065i	0.943970	+	0.330032i	$0.107060\pi$
$7$	12.2246i	0.660065i	−0.943970	+	0.330032i	$0.892940\pi$
$8$	−24.4916	−1.08239
$9$	−56.5639	−2.09496
$10$	− 35.1331i	− 1.11101i
$11$	− 11.1761i	− 0.306338i	−0.988200	−	0.153169i	$-0.951052\pi$
$11$	− 11.1761i	− 0.306338i	0.988200	−	0.153169i	$-0.0489480\pi$
$12$	30.2939i	0.728759i
$13$	28.9000	0.616570	0.308285	−	0.951294i	$-0.400245\pi$
$13$	28.9000	0.616570	0.308285	+	0.951294i	$0.400245\pi$
$14$	26.4629i	0.505178i
$15$	−148.362	−2.55379
$16$	−26.5061	−0.414158
$17$	0	0
$17$	0	0
$18$	−122.445	−1.60337
$19$	79.5850	0.960950	0.480475	−	0.877009i	$-0.340464\pi$
$19$	79.5850	0.960950	0.480475	+	0.877009i	$0.340464\pi$
$20$	53.7848i	0.601332i
$21$	111.749	1.16122
$22$	− 24.1932i	− 0.234455i
$23$	− 45.0447i	− 0.408368i	−0.978933	−	0.204184i	$-0.934546\pi$
$23$	− 45.0447i	− 0.408368i	0.978933	−	0.204184i	$-0.0654541\pi$
$24$	223.886i	1.90419i
$25$	−138.406	−1.10725
$26$	62.5606	0.471890
$27$	270.253i	1.92631i
$28$	− 40.5117i	− 0.273428i
$29$	− 20.3774i	− 0.130482i	−0.997870	−	0.0652411i	$-0.979218\pi$
$29$	− 20.3774i	− 0.130482i	0.997870	−	0.0652411i	$-0.0207817\pi$
$30$	−321.163	−1.95454
$31$	− 1.03839i	− 0.00601613i	−0.999995	−	0.00300806i	$-0.999043\pi$
$31$	− 1.03839i	− 0.00601613i	0.999995	−	0.00300806i	$-0.000957498\pi$
$32$	138.555	0.765413
$33$	−102.164	−0.538926
$34$	0	0
$35$	198.402	0.958175
$36$	187.450	0.867824
$37$	219.921i	0.977159i	0.872520	+	0.488579i	$0.162485\pi$
$37$	219.921i	0.977159i	−0.872520	+	0.488579i	$0.837515\pi$
$38$	172.280	0.735460
$39$	− 264.184i	− 1.08470i
$40$	397.494i	1.57123i
$41$	− 310.992i	− 1.18460i	−0.805716	−	0.592302i	$-0.798219\pi$
$41$	− 310.992i	− 1.18460i	0.805716	−	0.592302i	$-0.201781\pi$
$42$	241.906	0.888735
$43$	−483.760	−1.71564	−0.857822	−	0.513947i	$-0.828183\pi$
$43$	−483.760	−1.71564	−0.857822	+	0.513947i	$0.828183\pi$
$44$	37.0371i	0.126899i
$45$	918.020i	3.04112i
$46$	− 97.5095i	− 0.312543i
$47$	−632.422	−1.96273	−0.981364	−	0.192156i	$-0.938452\pi$
$47$	−632.422	−1.96273	−0.981364	+	0.192156i	$0.938452\pi$
$48$	242.301i	0.728606i
$49$	193.560	0.564315
$50$	−299.612	−0.847431
$51$	0	0
$52$	−95.7732	−0.255411
$53$	−490.172	−1.27038	−0.635192	−	0.772354i	$-0.719079\pi$
$53$	−490.172	−1.27038	−0.635192	+	0.772354i	$0.719079\pi$
$54$	585.024i	1.47429i
$55$	−181.386	−0.444692
$56$	− 299.400i	− 0.714446i
$57$	− 727.512i	− 1.69055i
$58$	− 44.1115i	− 0.0998642i
$59$	−147.956	−0.326478	−0.163239	−	0.986587i	$-0.552194\pi$
$59$	−147.956	−0.326478	−0.163239	+	0.986587i	$0.552194\pi$
$60$	491.665	1.05789
$61$	176.395i	0.370246i	0.982715	+	0.185123i	$0.0592684\pi$
$61$	176.395i	0.370246i	−0.982715	+	0.185123i	$0.940732\pi$
$62$	− 2.24783i	− 0.00460442i
$63$	− 691.469i	− 1.38281i
$64$	511.982	0.999964
$65$	− 469.041i	− 0.895037i
$66$	−221.158	−0.412465
$67$	809.312	1.47572	0.737860	−	0.674954i	$-0.235836\pi$
$67$	809.312	1.47572	0.737860	+	0.674954i	$0.235836\pi$
$68$	0	0
$69$	−411.768	−0.718421
$70$	429.487	0.733336
$71$	− 714.189i	− 1.19378i	−0.802322	−	0.596892i	$-0.796402\pi$
$71$	− 714.189i	− 1.19378i	0.802322	−	0.596892i	$-0.203598\pi$
$72$	1385.34	2.26756
$73$	− 780.184i	− 1.25087i	−0.780276	−	0.625436i	$-0.784921\pi$
$73$	− 780.184i	− 1.25087i	0.780276	−	0.625436i	$-0.215079\pi$
$74$	476.070i	0.747865i
$75$	1265.22i	1.94793i
$76$	−263.741	−0.398068
$77$	136.623	0.202203
$78$	− 571.887i	− 0.830172i
$79$	− 230.265i	− 0.327935i	−0.986466	−	0.163968i	$-0.947571\pi$
$79$	− 230.265i	− 0.327935i	0.986466	−	0.163968i	$-0.0524293\pi$
$80$	430.188i	0.601207i
$81$	943.248	1.29389
$82$	− 673.213i	− 0.906633i
$83$	−236.046	−0.312161	−0.156080	−	0.987744i	$-0.549886\pi$
$83$	−236.046	−0.312161	−0.156080	+	0.987744i	$0.549886\pi$
$84$	−370.330	−0.481028
$85$	0	0
$86$	−1047.21	−1.31306
$87$	−186.276	−0.229551
$88$	273.721i	0.331577i
$89$	−688.557	−0.820078	−0.410039	−	0.912068i	$-0.634485\pi$
$89$	−688.557	−0.820078	−0.410039	+	0.912068i	$0.634485\pi$
$90$	1987.26i	2.32751i
$91$	353.290i	0.406976i
$92$	149.276i	0.169164i
$93$	−9.49224	−0.0105839
$94$	−1369.02	−1.50217
$95$	− 1291.65i	− 1.39495i
$96$	− 1266.57i	− 1.34655i
$97$	1846.11i	1.93241i	0.257775	+	0.966205i	$0.417011\pi$
$97$	1846.11i	1.93241i	−0.257775	+	0.966205i	$0.582989\pi$
$98$	419.004	0.431896
$99$	632.164i	0.641766i
$100$	458.672	0.458672
$101$	−735.748	−0.724848	−0.362424	−	0.932013i	$-0.618051\pi$
$101$	−735.748	−0.724848	−0.362424	+	0.932013i	$0.618051\pi$
$102$	0	0
$103$	387.334	0.370535	0.185268	−	0.982688i	$-0.440685\pi$
$103$	387.334	0.370535	0.185268	+	0.982688i	$0.440685\pi$
$104$	−707.808	−0.667368
$105$	− 1813.66i	− 1.68567i
$106$	−1061.09	−0.972284
$107$	− 793.424i	− 0.716852i	−0.933558	−	0.358426i	$-0.883314\pi$
$107$	− 793.424i	− 0.716852i	0.933558	−	0.358426i	$-0.116686\pi$
$108$	− 895.606i	− 0.797961i
$109$	− 1709.67i	− 1.50236i	−0.660100	−	0.751178i	$-0.729486\pi$
$109$	− 1709.67i	− 1.50236i	0.660100	−	0.751178i	$-0.270514\pi$
$110$	−392.651	−0.340344
$111$	2010.37	1.71907
$112$	− 324.025i	− 0.273371i
$113$	42.1279i	0.0350713i	0.999846	+	0.0175357i	$0.00558206\pi$
$113$	42.1279i	0.0350713i	−0.999846	+	0.0175357i	$0.994418\pi$
$114$	− 1574.87i	− 1.29386i
$115$	−731.066	−0.592802
$116$	67.5297i	0.0540515i
$117$	−1634.70	−1.29169
$118$	−320.284	−0.249869
$119$	0	0
$120$	3633.63	2.76419
$121$	1206.09	0.906157
$122$	381.846i	0.283367i
$123$	−2842.88	−2.08401
$124$	3.44117i	0.00249215i
$125$	217.583i	0.155690i
$126$	− 1496.84i	− 1.05833i
$127$	1536.78	1.07376	0.536878	−	0.843660i	$-0.319603\pi$
$127$	1536.78	1.07376	0.536878	+	0.843660i	$0.319603\pi$
$128$	−0.135867	−9.38207e−5 0
$129$	4422.21i	3.01825i
$130$	− 1015.35i	− 0.685013i
$131$	264.718i	0.176554i	0.996096	+	0.0882768i	$0.0281360\pi$
$131$	264.718i	0.176554i	−0.996096	+	0.0882768i	$0.971864\pi$
$132$	338.568	0.223247
$133$	972.892i	0.634289i
$134$	1751.94	1.12944
$135$	4386.15	2.79630
$136$	0	0
$137$	686.450	0.428083	0.214042	−	0.976825i	$-0.431337\pi$
$137$	686.450	0.428083	0.214042	+	0.976825i	$0.431337\pi$
$138$	−891.366	−0.549841
$139$	43.2364i	0.0263832i	0.999913	+	0.0131916i	$0.00419914\pi$
$139$	43.2364i	0.0263832i	−0.999913	+	0.0131916i	$0.995801\pi$
$140$	−657.496	−0.396918
$141$	5781.18i	3.45293i
$142$	− 1546.02i	− 0.913659i
$143$	− 322.989i	− 0.188879i
$144$	1499.29	0.867643
$145$	−330.721	−0.189413
$146$	− 1688.89i	− 0.957350i
$147$	− 1769.39i	− 0.992770i
$148$	− 728.810i	− 0.404782i
$149$	335.407	0.184413	0.0922067	−	0.995740i	$-0.470608\pi$
$149$	335.407	0.184413	0.0922067	+	0.995740i	$0.470608\pi$
$150$	2738.85i	1.49084i
$151$	2432.29	1.31084	0.655421	−	0.755264i	$-0.272491\pi$
$151$	2432.29	1.31084	0.655421	+	0.755264i	$0.272491\pi$
$152$	−1949.17	−1.04012
$153$	0	0
$154$	295.752	0.154756
$155$	−16.8528	−0.00873324
$156$	875.494i	0.449331i
$157$	2706.98	1.37605	0.688026	−	0.725686i	$-0.258477\pi$
$157$	2706.98	1.37605	0.688026	+	0.725686i	$0.258477\pi$
$158$	− 498.462i	− 0.250984i
$159$	4480.83i	2.23492i
$160$	− 2248.71i	− 1.11110i
$161$	550.652	0.269549
$162$	2041.87	0.990277
$163$	− 190.765i	− 0.0916678i	−0.998949	−	0.0458339i	$-0.985405\pi$
$163$	− 190.765i	− 0.0916678i	0.998949	−	0.0458339i	$-0.0145945\pi$
$164$	1030.61i	0.490715i
$165$	1658.11i	0.782325i
$166$	−510.974	−0.238911
$167$	− 2757.18i	− 1.27759i	−0.769379	−	0.638793i	$-0.779434\pi$
$167$	− 2757.18i	− 1.27759i	0.769379	−	0.638793i	$-0.220566\pi$
$168$	−2736.91	−1.25689
$169$	−1361.79	−0.619841
$170$	0	0
$171$	−4501.63	−2.01315
$172$	1603.16	0.710696
$173$	− 1453.00i	− 0.638554i	−0.947661	−	0.319277i	$-0.896560\pi$
$173$	− 1453.00i	− 0.638554i	0.947661	−	0.319277i	$-0.103440\pi$
$174$	−403.238	−0.175686
$175$	− 1691.96i	− 0.730858i
$176$	296.235i	0.126872i
$177$	1352.51i	0.574356i
$178$	−1490.54	−0.627644
$179$	1171.47	0.489160	0.244580	−	0.969629i	$-0.421350\pi$
$179$	1171.47	0.489160	0.244580	+	0.969629i	$0.421350\pi$
$180$	− 3042.28i	− 1.25977i
$181$	− 2737.41i	− 1.12414i	−0.827089	−	0.562072i	$-0.810005\pi$
$181$	− 2737.41i	− 1.12414i	0.827089	−	0.562072i	$-0.189995\pi$
$182$	764.776i	0.311478i
$183$	1612.48	0.651355
$184$	1103.22i	0.442012i
$185$	3569.28	1.41848
$186$	−20.5481	−0.00810033
$187$	0	0
$188$	2095.82	0.813049
$189$	−3303.73	−1.27149
$190$	− 2796.07i	− 1.06762i
$191$	−1345.56	−0.509744	−0.254872	−	0.966975i	$-0.582033\pi$
$191$	−1345.56	−0.509744	−0.254872	+	0.966975i	$0.582033\pi$
$192$	− 4680.19i	− 1.75919i
$193$	− 3787.61i	− 1.41263i	−0.707896	−	0.706317i	$-0.750355\pi$
$193$	− 3787.61i	− 1.41263i	0.707896	−	0.706317i	$-0.249645\pi$
$194$	3996.32i	1.47896i
$195$	−4287.66	−1.57459
$196$	−641.448	−0.233764
$197$	− 72.4095i	− 0.0261876i	−0.999914	−	0.0130938i	$-0.995832\pi$
$197$	− 72.4095i	− 0.0261876i	0.999914	−	0.0130938i	$-0.00416801\pi$
$198$	1368.46i	0.491174i
$199$	2703.59i	0.963077i	0.876425	+	0.481539i	$0.159922\pi$
$199$	2703.59i	0.963077i	−0.876425	+	0.481539i	$0.840078\pi$
$200$	3389.80	1.19847
$201$	− 7398.19i	− 2.59616i
$202$	−1592.69	−0.554760
$203$	249.105	0.0861267
$204$	0	0
$205$	−5047.33	−1.71962
$206$	838.472	0.283588
$207$	2547.90i	0.855514i
$208$	−766.025	−0.255357
$209$	− 889.450i	− 0.294376i
$210$	− 3926.08i	− 1.29012i
$211$	2338.29i	0.762911i	0.924387	+	0.381456i	$0.124577\pi$
$211$	2338.29i	0.762911i	−0.924387	+	0.381456i	$0.875423\pi$
$212$	1624.41	0.526249
$213$	−6528.64	−2.10016
$214$	− 1717.55i	− 0.548640i
$215$	7851.33i	2.49049i
$216$	− 6618.94i	− 2.08501i
$217$	12.6938	0.00397103
$218$	− 3700.97i	− 1.14982i
$219$	−7131.91	−2.20059
$220$	601.105	0.184211
$221$	0	0
$222$	4351.91	1.31568
$223$	−1737.70	−0.521815	−0.260908	−	0.965364i	$-0.584022\pi$
$223$	−1737.70	−0.521815	−0.260908	+	0.965364i	$0.584022\pi$
$224$	1693.77i	0.505222i
$225$	7828.80	2.31965
$226$	91.1954i	0.0268417i
$227$	4480.83i	1.31015i	0.755565	+	0.655073i	$0.227362\pi$
$227$	4480.83i	1.31015i	−0.755565	+	0.655073i	$0.772638\pi$
$228$	2410.94i	0.700301i
$229$	3917.73	1.13053	0.565264	−	0.824910i	$-0.308774\pi$
$229$	3917.73	1.13053	0.565264	+	0.824910i	$0.308774\pi$
$230$	−1582.56	−0.453699
$231$	− 1248.92i	− 0.355726i
$232$	499.075i	0.141232i
$233$	− 4166.02i	− 1.17135i	−0.810546	−	0.585675i	$-0.800829\pi$
$233$	− 4166.02i	− 1.17135i	0.810546	−	0.585675i	$-0.199171\pi$
$234$	−3538.67	−0.988590
$235$	10264.1i	2.84917i
$236$	490.318	0.135242
$237$	−2104.93	−0.576920
$238$	0	0
$239$	−1780.06	−0.481768	−0.240884	−	0.970554i	$-0.577437\pi$
$239$	−1780.06	−0.481768	−0.240884	+	0.970554i	$0.577437\pi$
$240$	3932.49	1.05767
$241$	− 2469.12i	− 0.659958i	−0.943988	−	0.329979i	$-0.892958\pi$
$241$	− 2469.12i	− 0.659958i	0.943988	−	0.329979i	$-0.107042\pi$
$242$	2610.87	0.693524
$243$	− 1325.70i	− 0.349975i
$244$	− 584.563i	− 0.153372i
$245$	− 3141.44i	− 0.819180i
$246$	−6154.06	−1.59499
$247$	2300.00	0.592493
$248$	25.4318i	0.00651178i
$249$	2157.77i	0.549169i
$250$	471.009i	0.119157i
$251$	−4824.05	−1.21311	−0.606557	−	0.795040i	$-0.707450\pi$
$251$	−4824.05	−1.21311	−0.606557	+	0.795040i	$0.707450\pi$
$252$	2291.50i	0.572820i
$253$	−503.424	−0.125099
$254$	3326.71	0.821796
$255$	0	0
$256$	−4096.15	−1.00004
$257$	−5558.77	−1.34921	−0.674604	−	0.738180i	$-0.735686\pi$
$257$	−5558.77	−1.34921	−0.674604	+	0.738180i	$0.735686\pi$
$258$	9572.88i	2.31000i
$259$	−2688.45	−0.644988
$260$	1554.38i	0.370764i
$261$	1152.62i	0.273355i
$262$	573.043i	0.135125i
$263$	3172.23	0.743757	0.371878	−	0.928281i	$-0.378714\pi$
$263$	3172.23	0.743757	0.371878	+	0.928281i	$0.378714\pi$
$264$	2502.17	0.583326
$265$	7955.40i	1.84414i
$266$	2106.05i	0.485451i
$267$	6294.33i	1.44272i
$268$	−2682.02	−0.611309
$269$	− 2849.92i	− 0.645959i	−0.946406	−	0.322979i	$-0.895316\pi$
$269$	− 2849.92i	− 0.645959i	0.946406	−	0.322979i	$-0.104684\pi$
$270$	9494.83	2.14014
$271$	−2565.11	−0.574980	−0.287490	−	0.957784i	$-0.592821\pi$
$271$	−2565.11	−0.574980	−0.287490	+	0.957784i	$0.592821\pi$
$272$	0	0
$273$	3229.54	0.715973
$274$	1485.98	0.327632
$275$	1546.84i	0.339194i
$276$	1364.58	0.297602
$277$	− 6260.95i	− 1.35806i	−0.734108	−	0.679032i	$-0.762400\pi$
$277$	− 6260.95i	− 1.35806i	0.734108	−	0.679032i	$-0.237600\pi$
$278$	93.5951i	0.0201923i
$279$	58.7352i	0.0126035i
$280$	−4859.20	−1.03712
$281$	1807.59	0.383744	0.191872	−	0.981420i	$-0.438544\pi$
$281$	1807.59	0.383744	0.191872	+	0.981420i	$0.438544\pi$
$282$	12514.7i	2.64269i
$283$	− 1264.09i	− 0.265521i	−0.991148	−	0.132761i	$-0.957616\pi$
$283$	− 1264.09i	− 0.265521i	0.991148	−	0.132761i	$-0.0423841\pi$
$284$	2366.79i	0.494518i
$285$	−11807.4	−2.45407
$286$	− 699.184i	− 0.144558i
$287$	3801.74	0.781915
$288$	−7837.18	−1.60351
$289$	0	0
$290$	−715.921	−0.144967
$291$	16875.9	3.39959
$292$	2585.49i	0.518166i
$293$	3708.40	0.739410	0.369705	−	0.929149i	$-0.379459\pi$
$293$	3708.40	0.739410	0.369705	+	0.929149i	$0.379459\pi$
$294$	− 3830.26i	− 0.759813i
$295$	2401.29i	0.473927i
$296$	− 5386.24i	− 1.05766i
$297$	3020.38	0.590101
$298$	726.065	0.141140
$299$	− 1301.79i	− 0.251788i
$300$	− 4192.88i	− 0.806919i
$301$	− 5913.76i	− 1.13244i
$302$	5265.25	1.00325
$303$	6725.71i	1.27519i
$304$	−2109.49	−0.397985
$305$	2862.85	0.537463
$306$	0	0
$307$	4689.47	0.871799	0.435899	−	0.899995i	$-0.356430\pi$
$307$	4689.47	0.871799	0.435899	+	0.899995i	$0.356430\pi$
$308$	−452.763	−0.0837615
$309$	− 3540.75i	− 0.651864i
$310$	−36.4818	−0.00668396
$311$	2725.79i	0.496995i	0.968633	+	0.248497i	$0.0799368\pi$
$311$	2725.79i	0.496995i	−0.968633	+	0.248497i	$0.920063\pi$
$312$	6470.30i	1.17407i
$313$	− 1406.42i	− 0.253980i	−0.991904	−	0.126990i	$-0.959468\pi$
$313$	− 1406.42i	− 0.253980i	0.991904	−	0.126990i	$-0.0405316\pi$
$314$	5859.87	1.05316
$315$	−11222.4	−2.00734
$316$	763.089i	0.135845i
$317$	− 836.717i	− 0.148248i	−0.997249	−	0.0741241i	$-0.976384\pi$
$317$	− 836.717i	− 0.148248i	0.997249	−	0.0741241i	$-0.0236161\pi$
$318$	9699.77i	1.71049i
$319$	−227.740	−0.0399717
$320$	− 8309.36i	− 1.45159i
$321$	−7252.95	−1.26112
$322$	1192.01	0.206299
$323$	0	0
$324$	−3125.88	−0.535987
$325$	−3999.94	−0.682698
$326$	− 412.954i	− 0.0701577i
$327$	−15628.7	−2.64302
$328$	7616.70i	1.28220i
$329$	− 7731.09i	− 1.29553i
$330$	3589.35i	0.598750i
$331$	−2544.99	−0.422614	−0.211307	−	0.977420i	$-0.567772\pi$
$331$	−2544.99	−0.422614	−0.211307	+	0.977420i	$0.567772\pi$
$332$	782.244	0.129311
$333$	− 12439.6i	− 2.04711i
$334$	− 5968.54i	− 0.977796i
$335$	− 13135.0i	− 2.14221i
$336$	−2962.02	−0.480927
$337$	9258.04i	1.49649i	0.663422	+	0.748246i	$0.269104\pi$
$337$	9258.04i	1.49649i	−0.663422	+	0.748246i	$0.730896\pi$
$338$	−2947.91	−0.474393
$339$	385.105	0.0616992
$340$	0	0
$341$	−11.6051	−0.00184297
$342$	−9744.81	−1.54076
$343$	6559.21i	1.03255i
$344$	11848.1	1.85699
$345$	6682.92i	1.04289i
$346$	− 3145.36i	− 0.488716i
$347$	11972.3i	1.85217i	0.377310	+	0.926087i	$0.376849\pi$
$347$	11972.3i	1.85217i	−0.377310	+	0.926087i	$0.623151\pi$
$348$	617.311	0.0950901
$349$	−1887.57	−0.289511	−0.144755	−	0.989467i	$-0.546240\pi$
$349$	−1887.57	−0.289511	−0.144755	+	0.989467i	$0.546240\pi$
$350$	− 3662.63i	− 0.559360i
$351$	7810.31i	1.18770i
$352$	− 1548.50i	− 0.234475i
$353$	7441.02	1.12194	0.560971	−	0.827836i	$-0.310428\pi$
$353$	7441.02	1.12194	0.560971	+	0.827836i	$0.310428\pi$
$354$	2927.82i	0.439581i
$355$	−11591.1	−1.73294
$356$	2281.85	0.339712
$357$	0	0
$358$	2535.91	0.374377
$359$	−11518.8	−1.69342	−0.846711	−	0.532052i	$-0.821421\pi$
$359$	−11518.8	−1.69342	−0.846711	+	0.532052i	$0.821421\pi$
$360$	− 22483.8i	− 3.29167i
$361$	−525.233	−0.0765757
$362$	− 5925.74i	− 0.860359i
$363$	− 11025.3i	− 1.59416i
$364$	− 1170.79i	− 0.168588i
$365$	−12662.2	−1.81581
$366$	3490.58	0.498513
$367$	− 702.580i	− 0.0999302i	−0.998751	−	0.0499651i	$-0.984089\pi$
$367$	− 702.580i	− 0.0999302i	0.998751	−	0.0499651i	$-0.0159110\pi$
$368$	1193.96i	0.169129i
$369$	17590.9i	2.48170i
$370$	7726.52	1.08563
$371$	− 5992.15i	− 0.838536i
$372$	31.4569	0.00438431
$373$	8797.32	1.22120	0.610600	−	0.791939i	$-0.290928\pi$
$373$	8797.32	1.22120	0.610600	+	0.791939i	$0.290928\pi$
$374$	0	0
$375$	1989.00	0.273897
$376$	15489.0	2.12443
$377$	− 588.906i	− 0.0804515i
$378$	−7151.67	−0.973128
$379$	3113.75i	0.422012i	0.977485	+	0.211006i	$0.0676740\pi$
$379$	3113.75i	0.422012i	−0.977485	+	0.211006i	$0.932326\pi$
$380$	4280.46i	0.577850i
$381$	− 14048.2i	− 1.88901i
$382$	−2912.77	−0.390131
$383$	9735.60	1.29887	0.649434	−	0.760418i	$-0.275006\pi$
$383$	9735.60	1.29887	0.649434	+	0.760418i	$0.275006\pi$
$384$	1.24200i	0 0.000165054i
$385$	− 2217.37i	− 0.293526i
$386$	− 8199.15i	− 1.08115i
$387$	27363.3	3.59420
$388$	− 6117.91i	− 0.800490i
$389$	1772.22	0.230990	0.115495	−	0.993308i	$-0.463155\pi$
$389$	1772.22	0.230990	0.115495	+	0.993308i	$0.463155\pi$
$390$	−9281.61	−1.20511
$391$	0	0
$392$	−4740.60	−0.610807
$393$	2419.88	0.310602
$394$	− 156.747i	− 0.0200426i
$395$	−3737.16	−0.476043
$396$	− 2094.96i	− 0.265848i
$397$	1201.80i	0.151931i	0.997110	+	0.0759653i	$0.0242038\pi$
$397$	1201.80i	0.151931i	−0.997110	+	0.0759653i	$0.975796\pi$
$398$	5852.54i	0.737088i
$399$	8893.53	1.11587
$400$	3668.61	0.458576
$401$	− 13633.2i	− 1.69778i	−0.528566	−	0.848892i	$-0.677270\pi$
$401$	− 13633.2i	− 1.69778i	0.528566	−	0.848892i	$-0.322730\pi$
$402$	− 16015.1i	− 1.98696i
$403$	− 30.0094i	− 0.00370937i
$404$	2438.23	0.300264
$405$	− 15308.7i	− 1.87826i
$406$	539.244	0.0659168
$407$	2457.87	0.299341
$408$	0	0
$409$	7345.95	0.888103	0.444051	−	0.896001i	$-0.353541\pi$
$409$	7345.95	0.888103	0.444051	+	0.896001i	$0.353541\pi$
$410$	−10926.1	−1.31610
$411$	− 6275.07i	− 0.753105i
$412$	−1283.61	−0.153492
$413$	− 1808.69i	− 0.215496i
$414$	5515.51i	0.654765i
$415$	3830.97i	0.453145i
$416$	4004.23	0.471931
$417$	395.238	0.0464146
$418$	− 1925.42i	− 0.225300i
$419$	− 3293.55i	− 0.384011i	−0.981394	−	0.192005i	$-0.938501\pi$
$419$	− 3293.55i	− 0.384011i	0.981394	−	0.192005i	$-0.0614991\pi$
$420$	6010.39i	0.698278i
$421$	−9775.57	−1.13167	−0.565834	−	0.824519i	$-0.691446\pi$
$421$	−9775.57	−1.13167	−0.565834	+	0.824519i	$0.691446\pi$
$422$	5061.75i	0.583892i
$423$	35772.2	4.11184
$424$	12005.1	1.37505
$425$	0	0
$426$	−14132.7	−1.60735
$427$	−2156.35	−0.244386
$428$	2629.37i	0.296952i
$429$	−2952.55	−0.332286
$430$	16996.0i	1.90609i
$431$	7842.98i	0.876527i	0.898847	+	0.438263i	$0.144406\pi$
$431$	7842.98i	0.876527i	−0.898847	+	0.438263i	$0.855594\pi$
$432$	− 7163.35i	− 0.797794i
$433$	5178.49	0.574740	0.287370	−	0.957820i	$-0.407219\pi$
$433$	5178.49	0.574740	0.287370	+	0.957820i	$0.407219\pi$
$434$	27.4787	0.00303922
$435$	3023.23i	0.333225i
$436$	5665.77i	0.622342i
$437$	− 3584.88i	− 0.392421i
$438$	−15438.7	−1.68422
$439$	− 5661.88i	− 0.615550i	−0.951459	−	0.307775i	$-0.900416\pi$
$439$	− 5661.88i	− 0.615550i	0.951459	−	0.307775i	$-0.0995845\pi$
$440$	4442.44	0.481329
$441$	−10948.5	−1.18222
$442$	0	0
$443$	−3020.29	−0.323924	−0.161962	−	0.986797i	$-0.551782\pi$
$443$	−3020.29	−0.323924	−0.161962	+	0.986797i	$0.551782\pi$
$444$	−6662.29	−0.712113
$445$	11175.1i	1.19046i
$446$	−3761.64	−0.399370
$447$	− 3066.06i	− 0.324429i
$448$	6258.76i	0.660041i
$449$	− 13308.6i	− 1.39883i	−0.714718	−	0.699413i	$-0.753445\pi$
$449$	− 13308.6i	− 1.39883i	0.714718	−	0.699413i	$-0.246555\pi$
$450$	16947.2	1.77533
$451$	−3475.68	−0.362890
$452$	− 139.610i	− 0.0145281i
$453$	− 22234.4i	− 2.30610i
$454$	9699.78i	1.00272i
$455$	5733.82	0.590782
$456$	17818.0i	1.82983i
$457$	−7260.23	−0.743150	−0.371575	−	0.928403i	$-0.621182\pi$
$457$	−7260.23	−0.743150	−0.371575	+	0.928403i	$0.621182\pi$
$458$	8480.83	0.865247
$459$	0	0
$460$	2422.72	0.245565
$461$	2580.20	0.260677	0.130338	−	0.991470i	$-0.458394\pi$
$461$	2580.20	0.260677	0.130338	+	0.991470i	$0.458394\pi$
$462$	− 2703.56i	− 0.272254i
$463$	14392.8	1.44469	0.722345	−	0.691533i	$-0.243064\pi$
$463$	14392.8	1.44469	0.722345	+	0.691533i	$0.243064\pi$
$464$	540.125i	0.0540402i
$465$	154.057i	0.0153639i
$466$	− 9018.29i	− 0.896490i
$467$	11696.0	1.15894	0.579471	−	0.814993i	$-0.303259\pi$
$467$	11696.0	1.15894	0.579471	+	0.814993i	$0.303259\pi$
$468$	5417.30	0.535075
$469$	9893.50i	0.974071i
$470$	22218.9i	2.18060i
$471$	− 24745.4i	− 2.42082i
$472$	3623.68	0.353375
$473$	5406.55i	0.525568i
$474$	−4556.61	−0.441544
$475$	−11015.1	−1.06401
$476$	0	0
$477$	27726.0	2.66140
$478$	−3853.35	−0.368720
$479$	− 13997.5i	− 1.33520i	−0.744521	−	0.667599i	$-0.767322\pi$
$479$	− 13997.5i	− 1.33520i	0.744521	−	0.667599i	$-0.232678\pi$
$480$	−20556.2	−1.95471
$481$	6355.73i	0.602487i
$482$	− 5344.97i	− 0.505097i
$483$	− 5033.69i	− 0.474205i
$484$	−3996.94	−0.375370
$485$	29961.9	2.80516
$486$	− 2869.78i	− 0.267852i
$487$	2888.42i	0.268761i	0.990930	+	0.134381i	$0.0429045\pi$
$487$	2888.42i	0.268761i	−0.990930	+	0.134381i	$0.957095\pi$
$488$	− 4320.19i	− 0.400750i
$489$	−1743.84	−0.161267
$490$	− 6800.36i	− 0.626957i
$491$	9420.11	0.865832	0.432916	−	0.901434i	$-0.357485\pi$
$491$	9420.11	0.865832	0.432916	+	0.901434i	$0.357485\pi$
$492$	9421.17	0.863291
$493$	0	0
$494$	4978.88	0.453463
$495$	10259.9	0.931612
$496$	27.5236i	0.00249162i
$497$	8730.65	0.787975
$498$	4670.98i	0.420305i
$499$	713.936i	0.0640484i	0.999487	+	0.0320242i	$0.0101954\pi$
$499$	713.936i	0.0640484i	−0.999487	+	0.0320242i	$0.989805\pi$
$500$	− 721.061i	− 0.0644937i
$501$	−25204.3	−2.24759
$502$	−10442.8	−0.928453
$503$	17604.4i	1.56052i	0.625456	+	0.780260i	$0.284913\pi$
$503$	17604.4i	1.56052i	−0.625456	+	0.780260i	$0.715087\pi$
$504$	16935.2i	1.49673i
$505$	11941.0i	1.05222i
$506$	−1089.78	−0.0957440
$507$	12448.6i	1.09046i
$508$	−5092.81	−0.444797
$509$	12291.2	1.07033	0.535165	−	0.844748i	$-0.320249\pi$
$509$	12291.2	1.07033	0.535165	+	0.844748i	$0.320249\pi$
$510$	0	0
$511$	9537.41	0.825656
$512$	−8865.96	−0.765280
$513$	21508.1i	1.85108i
$514$	−12033.2	−1.03261
$515$	− 6286.35i	− 0.537883i
$516$	− 14655.0i	− 1.25029i
$517$	7068.02i	0.601259i
$518$	−5819.75	−0.493639
$519$	−13282.4	−1.12338
$520$	11487.6i	0.968776i
$521$	8736.73i	0.734670i	0.930089	+	0.367335i	$0.119730\pi$
$521$	8736.73i	0.734670i	−0.930089	+	0.367335i	$0.880270\pi$
$522$	2495.12i	0.209211i
$523$	−2513.00	−0.210107	−0.105053	−	0.994467i	$-0.533501\pi$
$523$	−2513.00	−0.210107	−0.105053	+	0.994467i	$0.533501\pi$
$524$	− 877.264i	− 0.0731363i
$525$	−15466.8	−1.28576
$526$	6867.01	0.569232
$527$	0	0
$528$	2707.98	0.223200
$529$	10138.0	0.833236
$530$	17221.3i	1.41140i
$531$	8368.95	0.683957
$532$	− 3224.12i	− 0.262751i
$533$	− 8987.66i	− 0.730392i
$534$	13625.5i	1.10418i
$535$	−12877.1	−1.04061
$536$	−19821.4	−1.59730
$537$	− 10708.8i	− 0.860554i
$538$	− 6169.31i	− 0.494383i
$539$	− 2163.25i	− 0.172871i
$540$	−14535.5	−1.15835
$541$	− 9702.15i	− 0.771031i	−0.922701	−	0.385516i	$-0.874024\pi$
$541$	− 9702.15i	− 0.771031i	0.922701	−	0.385516i	$-0.125976\pi$
$542$	−5552.77	−0.440059
$543$	−25023.5	−1.97765
$544$	0	0
$545$	−27747.6	−2.18088
$546$	6991.07	0.547968
$547$	− 16198.6i	− 1.26618i	−0.774077	−	0.633092i	$-0.781785\pi$
$547$	− 16198.6i	− 1.26618i	0.774077	−	0.633092i	$-0.218215\pi$
$548$	−2274.86	−0.177331
$549$	− 9977.56i	− 0.775650i
$550$	3348.50i	0.259601i
$551$	− 1621.73i	− 0.125387i
$552$	10084.9	0.777610
$553$	2814.90	0.216459
$554$	− 13553.2i	− 1.03939i
$555$	− 32628.0i	− 2.49546i
$556$	− 143.283i	− 0.0109291i
$557$	−9873.27	−0.751066	−0.375533	−	0.926809i	$-0.622540\pi$
$557$	−9873.27	−0.751066	−0.375533	+	0.926809i	$0.622540\pi$
$558$	127.146i	0.00964608i
$559$	−13980.7	−1.05782
$560$	−5258.87	−0.396835
$561$	0	0
$562$	3912.95	0.293697
$563$	−3283.56	−0.245801	−0.122900	−	0.992419i	$-0.539220\pi$
$563$	−3283.56	−0.245801	−0.122900	+	0.992419i	$0.539220\pi$
$564$	− 19158.6i	− 1.43036i
$565$	683.727	0.0509108
$566$	− 2736.42i	− 0.203216i
$567$	11530.8i	0.854053i
$568$	17491.7i	1.29214i
$569$	15065.3	1.10996	0.554982	−	0.831862i	$-0.312725\pi$
$569$	15065.3	1.10996	0.554982	+	0.831862i	$0.312725\pi$
$570$	−25559.8	−1.87821
$571$	− 2567.87i	− 0.188200i	−0.995563	−	0.0941000i	$-0.970003\pi$
$571$	− 2567.87i	− 0.188200i	0.995563	−	0.0941000i	$-0.0299973\pi$
$572$	1070.37i	0.0782421i
$573$	12300.2i	0.896767i
$574$	8229.73	0.598436
$575$	6234.47i	0.452166i
$576$	−28959.7	−2.09488
$577$	−23042.3	−1.66250	−0.831249	−	0.555900i	$-0.812374\pi$
$577$	−23042.3	−1.66250	−0.831249	+	0.555900i	$0.812374\pi$
$578$	0	0
$579$	−34623.8	−2.48517
$580$	1095.99	0.0784632
$581$	− 2885.56i	− 0.206046i
$582$	36531.7	2.60187
$583$	5478.22i	0.389167i
$584$	19108.0i	1.35393i
$585$	26530.8i	1.87506i
$586$	8027.68	0.565905
$587$	4094.68	0.287914	0.143957	−	0.989584i	$-0.454017\pi$
$587$	4094.68	0.287914	0.143957	+	0.989584i	$0.454017\pi$
$588$	5863.69i	0.411249i
$589$	− 82.6401i	− 0.00578120i
$590$	5198.14i	0.362719i
$591$	−661.919	−0.0460706
$592$	− 5829.26i	− 0.404698i
$593$	25698.4	1.77961	0.889803	−	0.456344i	$-0.150841\pi$
$593$	25698.4	1.77961	0.889803	+	0.456344i	$0.150841\pi$
$594$	6538.30	0.451632
$595$	0	0
$596$	−1111.52	−0.0763922
$597$	24714.4	1.69429
$598$	− 2818.02i	− 0.192705i
$599$	1085.55	0.0740472	0.0370236	−	0.999314i	$-0.488212\pi$
$599$	1085.55	0.0740472	0.0370236	+	0.999314i	$0.488212\pi$
$600$	− 30987.3i	− 2.10842i
$601$	− 11163.9i	− 0.757713i	−0.925455	−	0.378857i	$-0.876317\pi$
$601$	− 11163.9i	− 0.757713i	0.925455	−	0.378857i	$-0.123683\pi$
$602$	− 12801.7i	− 0.866706i
$603$	−45777.8	−3.09157
$604$	−8060.50	−0.543008
$605$	− 19574.7i	− 1.31541i
$606$	14559.3i	0.975961i
$607$	167.486i	0.0111994i	0.999984	+	0.00559971i	$0.00178245\pi$
$607$	167.486i	0.0111994i	−0.999984	+	0.00559971i	$0.998218\pi$
$608$	11026.9	0.735524
$609$	− 2277.15i	− 0.151518i
$610$	6197.29	0.411346
$611$	−18277.0	−1.21016
$612$	0	0
$613$	−6533.58	−0.430487	−0.215244	−	0.976560i	$-0.569055\pi$
$613$	−6533.58	−0.430487	−0.215244	+	0.976560i	$0.569055\pi$
$614$	10151.4	0.667228
$615$	46139.3i	3.02523i
$616$	−3346.12	−0.218862
$617$	− 9073.82i	− 0.592056i	−0.955179	−	0.296028i	$-0.904338\pi$
$617$	− 9073.82i	− 0.592056i	0.955179	−	0.296028i	$-0.0956621\pi$
$618$	− 7664.75i	− 0.498902i
$619$	26617.2i	1.72833i	0.503212	+	0.864163i	$0.332151\pi$
$619$	26617.2i	1.72833i	−0.503212	+	0.864163i	$0.667849\pi$
$620$	55.8495	0.00361769
$621$	12173.5	0.786641
$622$	5900.60i	0.380374i
$623$	− 8417.32i	− 0.541304i
$624$	7002.49i	0.449237i
$625$	−13769.5	−0.881246
$626$	− 3044.52i	− 0.194383i
$627$	−8130.76	−0.517881
$628$	−8970.79	−0.570022
$629$	0	0
$630$	−24293.5	−1.53631
$631$	−5977.98	−0.377147	−0.188573	−	0.982059i	$-0.560386\pi$
$631$	−5977.98	−0.377147	−0.188573	+	0.982059i	$0.560386\pi$
$632$	5639.58i	0.354953i
$633$	21375.0	1.34215
$634$	− 1811.26i	− 0.113461i
$635$	− 24941.6i	− 1.55870i
$636$	− 14849.2i	− 0.925803i
$637$	5593.88	0.347940
$638$	−492.995	−0.0305922
$639$	40397.3i	2.50093i
$640$	2.20509i	0 0.000136194i
$641$	7927.25i	0.488467i	0.969716	+	0.244234i	$0.0785363\pi$
$641$	7927.25i	0.488467i	−0.969716	+	0.244234i	$0.921464\pi$
$642$	−15700.7	−0.965195
$643$	9197.33i	0.564086i	0.959402	+	0.282043i	$0.0910121\pi$
$643$	9197.33i	0.564086i	−0.959402	+	0.282043i	$0.908988\pi$
$644$	−1824.83	−0.111659
$645$	71771.5	4.38140
$646$	0	0
$647$	19035.8	1.15669	0.578343	−	0.815794i	$-0.303700\pi$
$647$	19035.8	1.15669	0.578343	+	0.815794i	$0.303700\pi$
$648$	−23101.7	−1.40049
$649$	1653.57i	0.100013i
$650$	−8658.79	−0.522501
$651$	− 116.039i	− 0.00698604i
$652$	632.186i	0.0379729i
$653$	19349.6i	1.15958i	0.814764	+	0.579792i	$0.196866\pi$
$653$	19349.6i	1.15958i	−0.814764	+	0.579792i	$0.803134\pi$
$654$	−33831.8	−2.02283
$655$	4296.32	0.256292
$656$	8243.17i	0.490613i
$657$	44130.2i	2.62052i
$658$	− 16735.7i	− 0.991528i
$659$	16666.8	0.985201	0.492600	−	0.870256i	$-0.336046\pi$
$659$	16666.8	0.985201	0.492600	+	0.870256i	$0.336046\pi$
$660$	− 5494.90i	− 0.324074i
$661$	6024.30	0.354490	0.177245	−	0.984167i	$-0.443281\pi$
$661$	6024.30	0.354490	0.177245	+	0.984167i	$0.443281\pi$
$662$	−5509.21	−0.323446
$663$	0	0
$664$	5781.14	0.337879
$665$	15789.8	0.920758
$666$	− 26928.4i	− 1.56675i
$667$	−917.893	−0.0532848
$668$	9137.16i	0.529232i
$669$	15884.9i	0.918003i
$670$	− 28433.7i	− 1.63953i
$671$	1971.40	0.113421
$672$	15483.3	0.888812
$673$	− 27296.1i	− 1.56343i	−0.623637	−	0.781714i	$-0.714346\pi$
$673$	− 27296.1i	− 1.56343i	0.623637	−	0.781714i	$-0.285654\pi$
$674$	20041.1i	1.14534i
$675$	− 37404.8i	− 2.13290i
$676$	4512.91	0.256766
$677$	− 13081.5i	− 0.742632i	−0.928507	−	0.371316i	$-0.878907\pi$
$677$	− 13081.5i	− 0.742632i	0.928507	−	0.371316i	$-0.121093\pi$
$678$	833.647	0.0472213
$679$	−22567.9	−1.27552
$680$	0	0
$681$	40960.8	2.30487
$682$	−25.1220	−0.00141051
$683$	15698.0i	0.879452i	0.898132	+	0.439726i	$0.144925\pi$
$683$	15698.0i	0.879452i	−0.898132	+	0.439726i	$0.855075\pi$
$684$	14918.2	0.833936
$685$	− 11140.9i	− 0.621422i
$686$	14198.9i	0.790258i
$687$	− 35813.3i	− 1.98888i
$688$	12822.6	0.710547
$689$	−14166.0	−0.783281
$690$	14466.7i	0.798170i
$691$	13621.7i	0.749919i	0.927041	+	0.374959i	$0.122343\pi$
$691$	13621.7i	0.749919i	−0.927041	+	0.374959i	$0.877657\pi$
$692$	4815.19i	0.264517i
$693$	−7727.93	−0.423607
$694$	25916.7i	1.41756i
$695$	701.718	0.0382988
$696$	4562.21	0.248463
$697$	0	0
$698$	−4086.08	−0.221576
$699$	−38082.9	−2.06070
$700$	5607.07i	0.302753i
$701$	5733.01	0.308892	0.154446	−	0.988001i	$-0.450641\pi$
$701$	5733.01	0.308892	0.154446	+	0.988001i	$0.450641\pi$
$702$	16907.2i	0.909004i
$703$	17502.4i	0.939000i
$704$	− 5721.96i	− 0.306327i
$705$	93827.4	5.01240
$706$	16107.8	0.858674
$707$	− 8994.20i	− 0.478447i
$708$	− 4482.16i	− 0.237924i
$709$	− 25474.5i	− 1.34939i	−0.738097	−	0.674694i	$-0.764275\pi$
$709$	− 25474.5i	− 1.34939i	0.738097	−	0.674694i	$-0.235725\pi$
$710$	−25091.7	−1.32630
$711$	13024.7i	0.687011i
$712$	16863.9	0.887642
$713$	−46.7739	−0.00245679
$714$	0	0
$715$	−5242.05	−0.274184
$716$	−3882.19	−0.202632
$717$	16272.1i	0.847551i
$718$	−24935.1	−1.29606
$719$	− 15508.0i	− 0.804380i	−0.915556	−	0.402190i	$-0.868249\pi$
$719$	− 15508.0i	− 0.804380i	0.915556	−	0.402190i	$-0.131751\pi$
$720$	− 24333.1i	− 1.25950i
$721$	4734.99i	0.244577i
$722$	−1136.99	−0.0586070
$723$	−22571.0	−1.16103
$724$	9071.64i	0.465670i
$725$	2820.36i	0.144477i
$726$	− 23866.8i	− 1.22008i
$727$	−32127.0	−1.63896	−0.819481	−	0.573106i	$-0.805738\pi$
$727$	−32127.0	−1.63896	−0.819481	+	0.573106i	$0.805738\pi$
$728$	− 8652.65i	− 0.440506i
$729$	13349.0	0.678200
$730$	−27410.3	−1.38973
$731$	0	0
$732$	−5343.68	−0.269820
$733$	−1723.27	−0.0868353	−0.0434177	−	0.999057i	$-0.513825\pi$
$733$	−1723.27	−0.0868353	−0.0434177	+	0.999057i	$0.513825\pi$
$734$	− 1520.89i	− 0.0764812i
$735$	−28716.9	−1.44114
$736$	− 6241.15i	− 0.312570i
$737$	− 9044.96i	− 0.452070i
$738$	38079.5i	1.89936i
$739$	−22411.5	−1.11559	−0.557796	−	0.829978i	$-0.688353\pi$
$739$	−22411.5	−1.11559	−0.557796	+	0.829978i	$0.688353\pi$
$740$	−11828.4	−0.587597
$741$	− 21025.1i	− 1.04234i
$742$	− 12971.4i	− 0.641771i
$743$	23971.3i	1.18361i	0.806082	+	0.591804i	$0.201584\pi$
$743$	23971.3i	1.18361i	−0.806082	+	0.591804i	$0.798416\pi$
$744$	232.481	0.0114558
$745$	− 5443.59i	− 0.267701i
$746$	19043.8	0.934642
$747$	13351.7	0.653964
$748$	0	0
$749$	9699.26	0.473169
$750$	4305.64	0.209627
$751$	− 26268.1i	− 1.27635i	−0.769892	−	0.638175i	$-0.779690\pi$
$751$	− 26268.1i	− 1.27635i	0.769892	−	0.638175i	$-0.220310\pi$
$752$	16763.0	0.812879
$753$	44098.3i	2.13417i
$754$	− 1274.82i	− 0.0615733i
$755$	− 39475.6i	− 1.90287i
$756$	10948.4	0.526706
$757$	345.842	0.0166048	0.00830239	−	0.999966i	$-0.497357\pi$
$757$	345.842	0.0166048	0.00830239	+	0.999966i	$0.497357\pi$
$758$	6740.42i	0.322986i
$759$	4601.97i	0.220080i
$760$	31634.6i	1.50988i
$761$	28929.2	1.37803	0.689015	−	0.724747i	$-0.258043\pi$
$761$	28929.2	1.37803	0.689015	+	0.724747i	$0.258043\pi$
$762$	− 30410.5i	− 1.44574i
$763$	20900.0	0.991652
$764$	4459.11	0.211159
$765$	0	0
$766$	21074.9	0.994084
$767$	−4275.92	−0.201297
$768$	37444.2i	1.75931i
$769$	−29320.3	−1.37492	−0.687462	−	0.726221i	$-0.741275\pi$
$769$	−29320.3	−1.37492	−0.687462	+	0.726221i	$0.741275\pi$
$770$	− 4799.99i	− 0.224649i
$771$	50814.6i	2.37359i
$772$	12552.0i	0.585175i
$773$	−25830.3	−1.20188	−0.600938	−	0.799296i	$-0.705206\pi$
$773$	−25830.3	−1.20188	−0.600938	+	0.799296i	$0.705206\pi$
$774$	59234.2	2.75081
$775$	143.720i	0.00666137i
$776$	− 45214.2i	− 2.09162i
$777$	24576.0i	1.13469i
$778$	3836.37	0.176787
$779$	− 24750.3i	− 1.13834i
$780$	14209.1	0.652266
$781$	−7981.85	−0.365702
$782$	0	0
$783$	5507.05	0.251349
$784$	−5130.51	−0.233715
$785$	− 43933.7i	− 1.99753i
$786$	5238.37	0.237718
$787$	− 13639.9i	− 0.617801i	−0.951094	−	0.308900i	$-0.900039\pi$
$787$	− 13639.9i	− 0.617801i	0.951094	−	0.308900i	$-0.0999610\pi$
$788$	239.962i	0.0108481i
$789$	− 28998.4i	− 1.30845i
$790$	−8089.94	−0.364338
$791$	−514.995	−0.0231493
$792$	− 15482.7i	− 0.694640i
$793$	5097.80i	0.228283i
$794$	2601.56i	0.116280i
$795$	72722.9	3.24430
$796$	− 8959.57i	− 0.398949i
$797$	−35932.1	−1.59697	−0.798483	−	0.602017i	$-0.794364\pi$
$797$	−35932.1	−1.59697	−0.798483	+	0.602017i	$0.794364\pi$
$798$	19252.1	0.854030
$799$	0	0
$800$	−19176.8	−0.847505
$801$	38947.5	1.71803
$802$	− 29512.3i	− 1.29939i
$803$	−8719.42	−0.383190
$804$	24517.3i	1.07544i
$805$	− 8936.97i	− 0.391288i
$806$	− 64.9622i	− 0.00283895i
$807$	−26052.1	−1.13640
$808$	18019.7	0.784566
$809$	19296.4i	0.838598i	0.907848	+	0.419299i	$0.137724\pi$
$809$	19296.4i	0.838598i	−0.907848	+	0.419299i	$0.862276\pi$
$810$	− 33139.2i	− 1.43752i
$811$	5464.45i	0.236600i	0.992978	+	0.118300i	$0.0377445\pi$
$811$	5464.45i	0.236600i	−0.992978	+	0.118300i	$0.962256\pi$
$812$	−825.522	−0.0356775
$813$	23448.6i	1.01153i
$814$	5320.61	0.229100
$815$	−3096.08	−0.133068
$816$	0	0
$817$	−38500.0	−1.64865
$818$	15902.0	0.679707
$819$	− 19983.4i	− 0.852598i
$820$	16726.6	0.712341
$821$	− 16693.5i	− 0.709631i	−0.934936	−	0.354816i	$-0.884544\pi$
$821$	− 16693.5i	− 0.709631i	0.934936	−	0.354816i	$-0.115456\pi$
$822$	− 13583.8i	− 0.576387i
$823$	− 5833.41i	− 0.247072i	−0.992340	−	0.123536i	$-0.960577\pi$
$823$	− 5833.41i	− 0.247072i	0.992340	−	0.123536i	$-0.0394234\pi$
$824$	−9486.44	−0.401063
$825$	14140.2	0.596726
$826$	− 3915.33i	− 0.164930i
$827$	− 11048.3i	− 0.464556i	−0.972649	−	0.232278i	$-0.925382\pi$
$827$	− 11048.3i	− 0.464556i	0.972649	−	0.232278i	$-0.0746180\pi$
$828$	− 8443.63i	− 0.354392i
$829$	−26287.1	−1.10131	−0.550657	−	0.834732i	$-0.685623\pi$
$829$	−26287.1	−1.10131	−0.550657	+	0.834732i	$0.685623\pi$
$830$	8293.01i	0.346813i
$831$	−57233.4	−2.38917
$832$	14796.3	0.616548
$833$	0	0
$834$	855.583	0.0355233
$835$	−44748.4	−1.85459
$836$	2947.60i	0.121943i
$837$	280.628	0.0115889
$838$	− 7129.64i	− 0.293901i
$839$	− 8311.95i	− 0.342027i	−0.985269	−	0.171013i	$-0.945296\pi$
$839$	− 8311.95i	− 0.342027i	0.985269	−	0.171013i	$-0.0547041\pi$
$840$	44419.5i	1.82455i
$841$	23973.8	0.982974
$842$	−21161.5	−0.866119
$843$	− 16523.8i	− 0.675101i
$844$	− 7748.97i	− 0.316032i
$845$	22101.6i	0.899785i
$846$	77437.2	3.14698
$847$	14744.0i	0.598122i
$848$	12992.5	0.526139
$849$	−11555.5	−0.467118
$850$	0	0
$851$	9906.29	0.399040
$852$	21635.6	0.869981
$853$	3875.16i	0.155549i	0.996971	+	0.0777743i	$0.0247814\pi$
$853$	3875.16i	0.155549i	−0.996971	+	0.0777743i	$0.975219\pi$
$854$	−4667.91	−0.187040
$855$	73060.6i	2.92236i
$856$	19432.2i	0.775912i
$857$	31317.9i	1.24831i	0.781301	+	0.624154i	$0.214556\pi$
$857$	31317.9i	1.24831i	−0.781301	+	0.624154i	$0.785444\pi$
$858$	−6391.47	−0.254314
$859$	−35801.6	−1.42204	−0.711021	−	0.703171i	$-0.751767\pi$
$859$	−35801.6	−1.42204	−0.711021	+	0.703171i	$0.751767\pi$
$860$	− 26018.9i	− 1.03167i
$861$	− 34753.0i	− 1.37558i
$862$	16977.9i	0.670847i
$863$	−5935.67	−0.234128	−0.117064	−	0.993124i	$-0.537348\pi$
$863$	−5935.67	−0.234128	−0.117064	+	0.993124i	$0.537348\pi$
$864$	37444.8i	1.47442i
$865$	−23582.0	−0.926950
$866$	11210.0	0.439875
$867$	0	0
$868$	−42.0668	−0.00164498
$869$	−2573.47	−0.100459
$870$	6544.46i	0.255032i
$871$	23389.1	0.909885
$872$	41872.6i	1.62613i
$873$	− 104423.i	− 4.04832i
$874$	− 7760.29i	− 0.300338i
$875$	−2659.86	−0.102765
$876$	23634.8	0.911583
$877$	− 43699.3i	− 1.68258i	−0.540585	−	0.841290i	$-0.681797\pi$
$877$	− 43699.3i	− 1.68258i	0.540585	−	0.841290i	$-0.318203\pi$
$878$	− 12256.4i	− 0.471110i
$879$	− 33899.7i	− 1.30081i
$880$	4807.83	0.184173
$881$	30329.5i	1.15985i	0.814671	+	0.579924i	$0.196918\pi$
$881$	30329.5i	1.15985i	−0.814671	+	0.579924i	$0.803082\pi$
$882$	−23700.5	−0.904805
$883$	35558.7	1.35521	0.677603	−	0.735428i	$-0.263019\pi$
$883$	35558.7	1.35521	0.677603	+	0.735428i	$0.263019\pi$
$884$	0	0
$885$	21951.0	0.833757
$886$	−6538.11	−0.247914
$887$	− 23689.6i	− 0.896752i	−0.893845	−	0.448376i	$-0.852003\pi$
$887$	− 23689.6i	− 0.896752i	0.893845	−	0.448376i	$-0.147997\pi$
$888$	−49237.3	−1.86069
$889$	18786.5i	0.708749i
$890$	24191.1i	0.911111i
$891$	− 10541.8i	− 0.396369i
$892$	5758.65	0.216159
$893$	−50331.3	−1.88608
$894$	− 6637.19i	− 0.248301i
$895$	− 19012.7i	− 0.710083i
$896$	− 1.66091i	0 6.19277e-5i
$897$	−11900.1	−0.442957
$898$	− 28809.6i	− 1.07059i
$899$	−21.1596	−0.000784998 0
$900$	−25944.3	−0.960900
$901$	0	0
$902$	−7523.90	−0.277737
$903$	−54059.6	−1.99224
$904$	− 1031.78i	− 0.0379608i
$905$	−44427.6	−1.63185
$906$	− 48131.4i	− 1.76496i
$907$	12176.0i	0.445752i	0.974847	+	0.222876i	$0.0715445\pi$
$907$	12176.0i	0.445752i	−0.974847	+	0.222876i	$0.928456\pi$
$908$	− 14849.3i	− 0.542721i
$909$	41616.8	1.51853
$910$	12412.2	0.452153
$911$	34830.5i	1.26673i	0.773855	+	0.633363i	$0.218326\pi$
$911$	34830.5i	1.26673i	−0.773855	+	0.633363i	$0.781674\pi$
$912$	19283.5i	0.700154i
$913$	2638.07i	0.0956269i
$914$	−15716.4	−0.568767
$915$	− 26170.2i	− 0.945531i
$916$	−12983.2	−0.468315
$917$	−3236.07	−0.116537
$918$	0	0
$919$	−15886.7	−0.570244	−0.285122	−	0.958491i	$-0.592034\pi$
$919$	−15886.7	−0.570244	−0.285122	+	0.958491i	$0.592034\pi$
$920$	17905.0	0.641642
$921$	− 42868.0i	− 1.53371i
$922$	5585.44	0.199508
$923$	− 20640.1i	− 0.736052i
$924$	4138.85i	0.147357i
$925$	− 30438.5i	− 1.08196i
$926$	31156.6	1.10569
$927$	−21909.1	−0.776256
$928$	− 2823.38i	− 0.0998728i
$929$	− 14115.7i	− 0.498514i	−0.968437	−	0.249257i	$-0.919814\pi$
$929$	− 14115.7i	− 0.498514i	0.968437	−	0.249257i	$-0.0801864\pi$
$930$	333.492i	0.0117587i
$931$	15404.5	0.542278
$932$	13806.0i	0.485225i
$933$	24917.4	0.874338
$934$	25318.6	0.886993
$935$	0	0
$936$	40036.4	1.39811
$937$	39312.3	1.37063	0.685314	−	0.728248i	$-0.259665\pi$
$937$	39312.3	1.37063	0.685314	+	0.728248i	$0.259665\pi$
$938$	21416.7i	0.745502i
$939$	−12856.6	−0.446814
$940$	− 34014.7i	− 1.18025i
$941$	30478.0i	1.05585i	0.849291	+	0.527924i	$0.177030\pi$
$941$	30478.0i	1.05585i	−0.849291	+	0.527924i	$0.822970\pi$
$942$	− 53567.0i	− 1.85277i
$943$	−14008.5	−0.483754
$944$	3921.73	0.135213
$945$	53618.8i	1.84574i
$946$	11703.7i	0.402242i
$947$	− 45331.1i	− 1.55551i	−0.628570	−	0.777753i	$-0.716360\pi$
$947$	− 45331.1i	− 1.55551i	0.628570	−	0.777753i	$-0.283640\pi$
$948$	6975.65	0.238986
$949$	− 22547.3i	− 0.771250i
$950$	−23844.6	−0.814339
$951$	−7648.70	−0.260806
$952$	0	0
$953$	33782.9	1.14830	0.574152	−	0.818748i	$-0.305332\pi$
$953$	33782.9	1.14830	0.574152	+	0.818748i	$0.305332\pi$
$954$	60019.3	2.03690
$955$	21838.1i	0.739964i
$956$	5899.04	0.199570
$957$	2081.84i	0.0703202i
$958$	− 30300.7i	− 1.02189i
$959$	8391.56i	0.282563i
$960$	−75958.6	−2.55370
$961$	29789.9	0.999964
$962$	13758.4i	0.461111i
$963$	44879.1i	1.50178i
$964$	8182.55i	0.273384i
$965$	−61472.2	−2.05063
$966$	− 10896.6i	− 0.362931i
$967$	45742.7	1.52119	0.760593	−	0.649229i	$-0.224908\pi$
$967$	45742.7	1.52119	0.760593	+	0.649229i	$0.224908\pi$
$968$	−29539.2	−0.980813
$969$	0	0
$970$	64859.5	2.14692
$971$	−7001.98	−0.231415	−0.115708	−	0.993283i	$-0.536914\pi$
$971$	−7001.98	−0.231415	−0.115708	+	0.993283i	$0.536914\pi$
$972$	4393.31i	0.144975i
$973$	−528.547	−0.0174146
$974$	6252.64i	0.205696i
$975$	36564.8i	1.20104i
$976$	− 4675.53i	− 0.153340i
$977$	−36142.8	−1.18353	−0.591767	−	0.806109i	$-0.701569\pi$
$977$	−36142.8	−1.18353	−0.591767	+	0.806109i	$0.701569\pi$
$978$	−3774.95	−0.123425
$979$	7695.39i	0.251221i
$980$	10410.6i	0.339341i
$981$	96705.6i	3.14737i
$982$	20392.0	0.662662
$983$	21919.3i	0.711207i	0.934637	+	0.355603i	$0.115725\pi$
$983$	21919.3i	0.711207i	−0.934637	+	0.355603i	$0.884275\pi$
$984$	69626.7	2.25571
$985$	−1175.19	−0.0380150
$986$	0	0
$987$	−70672.4	−2.27916
$988$	−7622.11	−0.245437
$989$	21790.8i	0.700614i
$990$	22209.9	0.713006
$991$	− 19931.0i	− 0.638880i	−0.947607	−	0.319440i	$-0.896505\pi$
$991$	− 19931.0i	− 0.638880i	0.947607	−	0.319440i	$-0.103495\pi$
$992$	− 143.873i	− 0.00460482i
$993$	23264.6i	0.743483i
$994$	18899.5	0.603074
$995$	43878.7	1.39804
$996$	− 7150.75i	− 0.227490i
$997$	18298.2i	0.581253i	0.956837	+	0.290626i	$0.0938637\pi$
$997$	18298.2i	0.581253i	−0.956837	+	0.290626i	$0.906136\pi$
$998$	1545.48i	0.0490193i
$999$	−59434.5	−1.88231

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	289.4.b.f.288.17		24
17.4	even	4		289.4.a.h.1.4	✓	12
17.13	even	4		289.4.a.i.1.4	yes	12
17.16	even	2	inner	289.4.b.f.288.18		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.4.a.h.1.4	✓	12	17.4	even	4
289.4.a.i.1.4	yes	12	17.13	even	4
289.4.b.f.288.17		24	1.1	even	1	trivial
289.4.b.f.288.18		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)

\(f(q)\)	\(=\)	\(q+2.16473 q^{2} -9.14133i q^{3} -3.31395 q^{4} -16.2298i q^{5} -19.7885i q^{6} +12.2246i q^{7} -24.4916 q^{8} -56.5639 q^{9} +O(q^{10})\)	\(q+2.16473 q^{2} -9.14133i q^{3} -3.31395 q^{4} -16.2298i q^{5} -19.7885i q^{6} +12.2246i q^{7} -24.4916 q^{8} -56.5639 q^{9} -35.1331i q^{10} -11.1761i q^{11} +30.2939i q^{12} +28.9000 q^{13} +26.4629i q^{14} -148.362 q^{15} -26.5061 q^{16} -122.445 q^{18} +79.5850 q^{19} +53.7848i q^{20} +111.749 q^{21} -24.1932i q^{22} -45.0447i q^{23} +223.886i q^{24} -138.406 q^{25} +62.5606 q^{26} +270.253i q^{27} -40.5117i q^{28} -20.3774i q^{29} -321.163 q^{30} -1.03839i q^{31} +138.555 q^{32} -102.164 q^{33} +198.402 q^{35} +187.450 q^{36} +219.921i q^{37} +172.280 q^{38} -264.184i q^{39} +397.494i q^{40} -310.992i q^{41} +241.906 q^{42} -483.760 q^{43} +37.0371i q^{44} +918.020i q^{45} -97.5095i q^{46} -632.422 q^{47} +242.301i q^{48} +193.560 q^{49} -299.612 q^{50} -95.7732 q^{52} -490.172 q^{53} +585.024i q^{54} -181.386 q^{55} -299.400i q^{56} -727.512i q^{57} -44.1115i q^{58} -147.956 q^{59} +491.665 q^{60} +176.395i q^{61} -2.24783i q^{62} -691.469i q^{63} +511.982 q^{64} -469.041i q^{65} -221.158 q^{66} +809.312 q^{67} -411.768 q^{69} +429.487 q^{70} -714.189i q^{71} +1385.34 q^{72} -780.184i q^{73} +476.070i q^{74} +1265.22i q^{75} -263.741 q^{76} +136.623 q^{77} -571.887i q^{78} -230.265i q^{79} +430.188i q^{80} +943.248 q^{81} -673.213i q^{82} -236.046 q^{83} -370.330 q^{84} -1047.21 q^{86} -186.276 q^{87} +273.721i q^{88} -688.557 q^{89} +1987.26i q^{90} +353.290i q^{91} +149.276i q^{92} -9.49224 q^{93} -1369.02 q^{94} -1291.65i q^{95} -1266.57i q^{96} +1846.11i q^{97} +419.004 q^{98} +632.164i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 96 q^{4} + 102 q^{8} - 216 q^{9}+O(q^{10}) \) 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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