LMFDB - Embedded newform 119.4.a.e.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [119,4,Mod(1,119)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("119.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(119, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 119 = 7 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	119.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.02122729068$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 53x^{7} + 90x^{6} + 880x^{5} - 1087x^{4} - 4674x^{3} + 2515x^{2} + 1814x + 36 $ x^9 - 2x^8 - 53x^7 + 90x^6 + 880x^5 - 1087x^4 - 4674x^3 + 2515x^2 + 1814x + 36
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.0204472$ of defining polynomial
Character	$\chi$	$=$	119.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.0204472 q^{2} -8.34282 q^{3} -7.99958 q^{4} -11.9677 q^{5} +0.170588 q^{6} +7.00000 q^{7} +0.327147 q^{8} +42.6027 q^{9} +0.244705 q^{10} -69.2380 q^{11} +66.7391 q^{12} +75.3431 q^{13} -0.143131 q^{14} +99.8441 q^{15} +63.9900 q^{16} +17.0000 q^{17} -0.871107 q^{18} -44.5275 q^{19} +95.7363 q^{20} -58.3998 q^{21} +1.41572 q^{22} -51.3581 q^{23} -2.72933 q^{24} +18.2250 q^{25} -1.54056 q^{26} -130.171 q^{27} -55.9971 q^{28} -78.5173 q^{29} -2.04153 q^{30} +191.447 q^{31} -3.92559 q^{32} +577.640 q^{33} -0.347603 q^{34} -83.7736 q^{35} -340.804 q^{36} -283.110 q^{37} +0.910463 q^{38} -628.574 q^{39} -3.91518 q^{40} +194.140 q^{41} +1.19411 q^{42} +113.892 q^{43} +553.875 q^{44} -509.855 q^{45} +1.05013 q^{46} +185.276 q^{47} -533.857 q^{48} +49.0000 q^{49} -0.372650 q^{50} -141.828 q^{51} -602.713 q^{52} +693.163 q^{53} +2.66163 q^{54} +828.617 q^{55} +2.29003 q^{56} +371.485 q^{57} +1.60546 q^{58} -611.213 q^{59} -798.711 q^{60} +339.563 q^{61} -3.91455 q^{62} +298.219 q^{63} -511.839 q^{64} -901.681 q^{65} -11.8111 q^{66} -56.5036 q^{67} -135.993 q^{68} +428.471 q^{69} +1.71294 q^{70} +161.004 q^{71} +13.9373 q^{72} +224.490 q^{73} +5.78881 q^{74} -152.048 q^{75} +356.201 q^{76} -484.666 q^{77} +12.8526 q^{78} +1243.28 q^{79} -765.810 q^{80} -64.2819 q^{81} -3.96962 q^{82} -445.110 q^{83} +467.174 q^{84} -203.450 q^{85} -2.32878 q^{86} +655.056 q^{87} -22.6510 q^{88} +220.783 q^{89} +10.4251 q^{90} +527.402 q^{91} +410.843 q^{92} -1597.21 q^{93} -3.78837 q^{94} +532.890 q^{95} +32.7505 q^{96} -148.526 q^{97} -1.00191 q^{98} -2949.73 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 2 q^{2} + 11 q^{3} + 38 q^{4} - 3 q^{5} + 9 q^{6} + 63 q^{7} + 24 q^{8} + 74 q^{9} + 134 q^{10} - 8 q^{11} + 56 q^{12} + 164 q^{13} + 14 q^{14} + 34 q^{15} + 178 q^{16} + 153 q^{17} + 98 q^{18} + 244 q^{19}+ \cdots - 2396 q^{99}+O(q^{100}) $ 9 * q + 2 * q^2 + 11 * q^3 + 38 * q^4 - 3 * q^5 + 9 * q^6 + 63 * q^7 + 24 * q^8 + 74 * q^9 + 134 * q^10 - 8 * q^11 + 56 * q^12 + 164 * q^13 + 14 * q^14 + 34 * q^15 + 178 * q^16 + 153 * q^17 + 98 * q^18 + 244 * q^19 - 41 * q^20 + 77 * q^21 - 80 * q^22 - 14 * q^23 + 298 * q^24 + 684 * q^25 + 326 * q^26 + 218 * q^27 + 266 * q^28 - 234 * q^29 - 335 * q^30 + 555 * q^31 - 181 * q^32 + 458 * q^33 + 34 * q^34 - 21 * q^35 - 1221 * q^36 - 364 * q^37 - 714 * q^38 - 52 * q^39 + 123 * q^40 - 45 * q^41 + 63 * q^42 - 135 * q^43 - 748 * q^44 - 844 * q^45 - 1576 * q^46 - 172 * q^47 - 949 * q^48 + 441 * q^49 - 2901 * q^50 + 187 * q^51 - 1596 * q^52 + 101 * q^53 - 1163 * q^54 + 1260 * q^55 + 168 * q^56 - 602 * q^57 + 1062 * q^58 + 280 * q^59 - 1727 * q^60 + 639 * q^61 - 1708 * q^62 + 518 * q^63 - 2390 * q^64 + 638 * q^65 - 2476 * q^66 + 35 * q^67 + 646 * q^68 + 1288 * q^69 + 938 * q^70 - 1616 * q^71 + 1335 * q^72 + 1049 * q^73 - 370 * q^74 + 1260 * q^75 + 4964 * q^76 - 56 * q^77 - 4714 * q^78 + 2304 * q^79 - 3996 * q^80 - 791 * q^81 - 215 * q^82 + 2508 * q^83 + 392 * q^84 - 51 * q^85 + 623 * q^86 + 166 * q^87 - 416 * q^88 + 2762 * q^89 + 2935 * q^90 + 1148 * q^91 - 2392 * q^92 + 2784 * q^93 - 862 * q^94 - 3462 * q^95 + 2928 * q^96 + 3107 * q^97 + 98 * q^98 - 2396 * q^99

Coefficient data

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

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

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

$2$	−0.0204472	−0.00722918	−0.00361459	−	0.999993i	$-0.501151\pi$
$2$	−0.0204472	−0.00722918	−0.00361459	+	0.999993i	$0.501151\pi$
$3$	−8.34282	−1.60558	−0.802789	−	0.596264i	$-0.796651\pi$
$3$	−8.34282	−1.60558	−0.802789	+	0.596264i	$0.796651\pi$
$4$	−7.99958	−0.999948
$5$	−11.9677	−1.07042	−0.535210	−	0.844719i	$-0.679768\pi$
$5$	−11.9677	−1.07042	−0.535210	+	0.844719i	$0.679768\pi$
$6$	0.170588	0.0116070
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0.327147	0.0144580
$9$	42.6027	1.57788
$10$	0.244705	0.00773827
$11$	−69.2380	−1.89782	−0.948911	−	0.315545i	$-0.897813\pi$
$11$	−69.2380	−1.89782	−0.948911	+	0.315545i	$0.897813\pi$
$12$	66.7391	1.60549
$13$	75.3431	1.60742	0.803709	−	0.595023i	$-0.202857\pi$
$13$	75.3431	1.60742	0.803709	+	0.595023i	$0.202857\pi$
$14$	−0.143131	−0.00273237
$15$	99.8441	1.71864
$16$	63.9900	0.999843
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	−0.871107	−0.0114068
$19$	−44.5275	−0.537648	−0.268824	−	0.963189i	$-0.586635\pi$
$19$	−44.5275	−0.537648	−0.268824	+	0.963189i	$0.586635\pi$
$20$	95.7363	1.07036
$21$	−58.3998	−0.606851
$22$	1.41572	0.0137197
$23$	−51.3581	−0.465604	−0.232802	−	0.972524i	$-0.574789\pi$
$23$	−51.3581	−0.465604	−0.232802	+	0.972524i	$0.574789\pi$
$24$	−2.72933	−0.0232134
$25$	18.2250	0.145800
$26$	−1.54056	−0.0116203
$27$	−130.171	−0.927828
$28$	−55.9971	−0.377945
$29$	−78.5173	−0.502769	−0.251384	−	0.967887i	$-0.580886\pi$
$29$	−78.5173	−0.502769	−0.251384	+	0.967887i	$0.580886\pi$
$30$	−2.04153	−0.0124244
$31$	191.447	1.10919	0.554594	−	0.832121i	$-0.312873\pi$
$31$	191.447	1.10919	0.554594	+	0.832121i	$0.312873\pi$
$32$	−3.92559	−0.0216860
$33$	577.640	3.04710
$34$	−0.347603	−0.00175333
$35$	−83.7736	−0.404581
$36$	−340.804	−1.57780
$37$	−283.110	−1.25792	−0.628959	−	0.777438i	$-0.716519\pi$
$37$	−283.110	−1.25792	−0.628959	+	0.777438i	$0.716519\pi$
$38$	0.910463	0.00388675
$39$	−628.574	−2.58083
$40$	−3.91518	−0.0154761
$41$	194.140	0.739502	0.369751	−	0.929131i	$-0.379443\pi$
$41$	194.140	0.739502	0.369751	+	0.929131i	$0.379443\pi$
$42$	1.19411	0.00438704
$43$	113.892	0.403916	0.201958	−	0.979394i	$-0.435270\pi$
$43$	113.892	0.403916	0.201958	+	0.979394i	$0.435270\pi$
$44$	553.875	1.89772
$45$	−509.855	−1.68899
$46$	1.05013	0.00336594
$47$	185.276	0.575005	0.287503	−	0.957780i	$-0.407175\pi$
$47$	185.276	0.575005	0.287503	+	0.957780i	$0.407175\pi$
$48$	−533.857	−1.60533
$49$	49.0000	0.142857
$50$	−0.372650	−0.00105401
$51$	−141.828	−0.389410
$52$	−602.713	−1.60733
$53$	693.163	1.79648	0.898238	−	0.439508i	$-0.144847\pi$
$53$	693.163	1.79648	0.898238	+	0.439508i	$0.144847\pi$
$54$	2.66163	0.00670744
$55$	828.617	2.03147
$56$	2.29003	0.00546461
$57$	371.485	0.863235
$58$	1.60546	0.00363461
$59$	−611.213	−1.34870	−0.674349	−	0.738413i	$-0.735576\pi$
$59$	−611.213	−1.34870	−0.674349	+	0.738413i	$0.735576\pi$
$60$	−798.711	−1.71855
$61$	339.563	0.712730	0.356365	−	0.934347i	$-0.384016\pi$
$61$	339.563	0.712730	0.356365	+	0.934347i	$0.384016\pi$
$62$	−3.91455	−0.00801853
$63$	298.219	0.596382
$64$	−511.839	−0.999686
$65$	−901.681	−1.72061
$66$	−11.8111	−0.0220280
$67$	−56.5036	−0.103030	−0.0515150	−	0.998672i	$-0.516405\pi$
$67$	−56.5036	−0.103030	−0.0515150	+	0.998672i	$0.516405\pi$
$68$	−135.993	−0.242523
$69$	428.471	0.747564
$70$	1.71294	0.00292479
$71$	161.004	0.269122	0.134561	−	0.990905i	$-0.457038\pi$
$71$	161.004	0.269122	0.134561	+	0.990905i	$0.457038\pi$
$72$	13.9373	0.0228130
$73$	224.490	0.359925	0.179963	−	0.983673i	$-0.442402\pi$
$73$	224.490	0.359925	0.179963	+	0.983673i	$0.442402\pi$
$74$	5.78881	0.00909372
$75$	−152.048	−0.234093
$76$	356.201	0.537620
$77$	−484.666	−0.717309
$78$	12.8526	0.0186573
$79$	1243.28	1.77064	0.885319	−	0.464984i	$-0.153940\pi$
$79$	1243.28	1.77064	0.885319	+	0.464984i	$0.153940\pi$
$80$	−765.810	−1.07025
$81$	−64.2819	−0.0881782
$82$	−3.96962	−0.00534600
$83$	−445.110	−0.588640	−0.294320	−	0.955707i	$-0.595093\pi$
$83$	−445.110	−0.588640	−0.294320	+	0.955707i	$0.595093\pi$
$84$	467.174	0.606819
$85$	−203.450	−0.259615
$86$	−2.32878	−0.00291998
$87$	655.056	0.807234
$88$	−22.6510	−0.0274387
$89$	220.783	0.262955	0.131477	−	0.991319i	$-0.458028\pi$
$89$	220.783	0.262955	0.131477	+	0.991319i	$0.458028\pi$
$90$	10.4251	0.0122100
$91$	527.402	0.607547
$92$	410.843	0.465580
$93$	−1597.21	−1.78089
$94$	−3.78837	−0.00415682
$95$	532.890	0.575509
$96$	32.7505	0.0348186
$97$	−148.526	−0.155470	−0.0777348	−	0.996974i	$-0.524769\pi$
$97$	−148.526	−0.155470	−0.0777348	+	0.996974i	$0.524769\pi$
$98$	−1.00191	−0.00103274
$99$	−2949.73	−2.99453
$100$	−145.792	−0.145792
$101$	1777.60	1.75127	0.875634	−	0.482976i	$-0.160444\pi$
$101$	1777.60	1.75127	0.875634	+	0.482976i	$0.160444\pi$
$102$	2.89999	0.00281511
$103$	−67.5774	−0.0646466	−0.0323233	−	0.999477i	$-0.510291\pi$
$103$	−67.5774	−0.0646466	−0.0323233	+	0.999477i	$0.510291\pi$
$104$	24.6483	0.0232400
$105$	698.909	0.649586
$106$	−14.1733	−0.0129871
$107$	174.399	0.157568	0.0787839	−	0.996892i	$-0.474896\pi$
$107$	174.399	0.157568	0.0787839	+	0.996892i	$0.474896\pi$
$108$	1041.31	0.927780
$109$	1700.63	1.49442	0.747208	−	0.664591i	$-0.231394\pi$
$109$	1700.63	1.49442	0.747208	+	0.664591i	$0.231394\pi$
$110$	−16.9429	−0.0146858
$111$	2361.94	2.01968
$112$	447.930	0.377905
$113$	−1584.74	−1.31929	−0.659643	−	0.751579i	$-0.729293\pi$
$113$	−1584.74	−1.31929	−0.659643	+	0.751579i	$0.729293\pi$
$114$	−7.59584	−0.00624048
$115$	614.636	0.498392
$116$	628.105	0.502742
$117$	3209.82	2.53631
$118$	12.4976	0.00974998
$119$	119.000	0.0916698
$120$	32.6637	0.0248481
$121$	3462.90	2.60173
$122$	−6.94311	−0.00515246
$123$	−1619.68	−1.18733
$124$	−1531.49	−1.10913
$125$	1277.85	0.914353
$126$	−6.09775	−0.00431135
$127$	−2620.42	−1.83090	−0.915451	−	0.402430i	$-0.868166\pi$
$127$	−2620.42	−1.83090	−0.915451	+	0.402430i	$0.868166\pi$
$128$	41.8704	0.0289130
$129$	−950.182	−0.648518
$130$	18.4369	0.0124386
$131$	−318.971	−0.212738	−0.106369	−	0.994327i	$-0.533922\pi$
$131$	−318.971	−0.212738	−0.106369	+	0.994327i	$0.533922\pi$
$132$	−4620.88	−3.04694
$133$	−311.692	−0.203212
$134$	1.15534	0.000744823 0
$135$	1557.84	0.993166
$136$	5.56150	0.00350658
$137$	373.544	0.232949	0.116474	−	0.993194i	$-0.462841\pi$
$137$	373.544	0.232949	0.116474	+	0.993194i	$0.462841\pi$
$138$	−8.76105	−0.00540428
$139$	−1176.24	−0.717749	−0.358874	−	0.933386i	$-0.616839\pi$
$139$	−1176.24	−0.717749	−0.358874	+	0.933386i	$0.616839\pi$
$140$	670.154	0.404560
$141$	−1545.72	−0.923215
$142$	−3.29208	−0.00194553
$143$	−5216.60	−3.05059
$144$	2726.15	1.57763
$145$	939.668	0.538174
$146$	−4.59019	−0.00260197
$147$	−408.798	−0.229368
$148$	2264.76	1.25785
$149$	−1174.07	−0.645529	−0.322764	−	0.946479i	$-0.604612\pi$
$149$	−1174.07	−0.645529	−0.322764	+	0.946479i	$0.604612\pi$
$150$	3.10895	0.00169230
$151$	1771.81	0.954886	0.477443	−	0.878663i	$-0.341564\pi$
$151$	1771.81	0.954886	0.477443	+	0.878663i	$0.341564\pi$
$152$	−14.5670	−0.00777331
$153$	724.246	0.382692
$154$	9.91007	0.00518556
$155$	−2291.17	−1.18730
$156$	5028.33	2.58070
$157$	2045.48	1.03979	0.519895	−	0.854230i	$-0.325971\pi$
$157$	2045.48	1.03979	0.519895	+	0.854230i	$0.325971\pi$
$158$	−25.4217	−0.0128003
$159$	−5782.94	−2.88438
$160$	46.9802	0.0232132
$161$	−359.507	−0.175982
$162$	1.31439	0.000637456 0
$163$	−3368.05	−1.61844	−0.809222	−	0.587503i	$-0.800111\pi$
$163$	−3368.05	−1.61844	−0.809222	+	0.587503i	$0.800111\pi$
$164$	−1553.04	−0.739463
$165$	−6913.00	−3.26168
$166$	9.10125	0.00425539
$167$	3306.72	1.53222	0.766112	−	0.642707i	$-0.222189\pi$
$167$	3306.72	1.53222	0.766112	+	0.642707i	$0.222189\pi$
$168$	−19.1053	−0.00877385
$169$	3479.59	1.58379
$170$	4.15999	0.00187681
$171$	−1896.99	−0.848343
$172$	−911.090	−0.403895
$173$	−2134.49	−0.938049	−0.469025	−	0.883185i	$-0.655394\pi$
$173$	−2134.49	−0.938049	−0.469025	+	0.883185i	$0.655394\pi$
$174$	−13.3941	−0.00583564
$175$	127.575	0.0551071
$176$	−4430.54	−1.89752
$177$	5099.24	2.16544
$178$	−4.51440	−0.00190095
$179$	−3885.16	−1.62229	−0.811147	−	0.584843i	$-0.801156\pi$
$179$	−3885.16	−1.62229	−0.811147	+	0.584843i	$0.801156\pi$
$180$	4078.63	1.68890
$181$	−755.818	−0.310384	−0.155192	−	0.987884i	$-0.549600\pi$
$181$	−755.818	−0.310384	−0.155192	+	0.987884i	$0.549600\pi$
$182$	−10.7839	−0.00439207
$183$	−2832.91	−1.14434
$184$	−16.8016	−0.00673170
$185$	3388.16	1.34650
$186$	32.6584	0.0128744
$187$	−1177.05	−0.460289
$188$	−1482.13	−0.574975
$189$	−911.195	−0.350686
$190$	−10.8961	−0.00416046
$191$	−915.498	−0.346823	−0.173411	−	0.984849i	$-0.555479\pi$
$191$	−915.498	−0.346823	−0.173411	+	0.984849i	$0.555479\pi$
$192$	4270.19	1.60507
$193$	2760.87	1.02970	0.514849	−	0.857281i	$-0.327848\pi$
$193$	2760.87	1.02970	0.514849	+	0.857281i	$0.327848\pi$
$194$	3.03695	0.00112392
$195$	7522.57	2.76258
$196$	−391.980	−0.142850
$197$	210.484	0.0761235	0.0380618	−	0.999275i	$-0.487882\pi$
$197$	210.484	0.0761235	0.0380618	+	0.999275i	$0.487882\pi$
$198$	60.3137	0.0216480
$199$	−1629.97	−0.580631	−0.290316	−	0.956931i	$-0.593760\pi$
$199$	−1629.97	−0.580631	−0.290316	+	0.956931i	$0.593760\pi$
$200$	5.96224	0.00210797
$201$	471.399	0.165423
$202$	−36.3470	−0.0126602
$203$	−549.621	−0.190029
$204$	1134.56	0.389389
$205$	−2323.40	−0.791578
$206$	1.38177	0.000467342 0
$207$	−2187.99	−0.734667
$208$	4821.20	1.60717
$209$	3082.99	1.02036
$210$	−14.2907	−0.00469598
$211$	1567.79	0.511522	0.255761	−	0.966740i	$-0.417674\pi$
$211$	1567.79	0.511522	0.255761	+	0.966740i	$0.417674\pi$
$212$	−5545.02	−1.79638
$213$	−1343.23	−0.432096
$214$	−3.56597	−0.00113909
$215$	−1363.02	−0.432360
$216$	−42.5850	−0.0134145
$217$	1340.13	0.419234
$218$	−34.7732	−0.0108034
$219$	−1872.88	−0.577888
$220$	−6628.59	−2.03136
$221$	1280.83	0.389856
$222$	−48.2950	−0.0146007
$223$	3382.37	1.01570	0.507848	−	0.861447i	$-0.330441\pi$
$223$	3382.37	1.01570	0.507848	+	0.861447i	$0.330441\pi$
$224$	−27.4791	−0.00819655
$225$	776.433	0.230054
$226$	32.4034	0.00953736
$227$	2470.11	0.722233	0.361117	−	0.932521i	$-0.382396\pi$
$227$	2470.11	0.722233	0.361117	+	0.932521i	$0.382396\pi$
$228$	−2971.73	−0.863190
$229$	5184.13	1.49597	0.747985	−	0.663716i	$-0.231022\pi$
$229$	5184.13	1.49597	0.747985	+	0.663716i	$0.231022\pi$
$230$	−12.5676	−0.00360297
$231$	4043.48	1.15169
$232$	−25.6867	−0.00726902
$233$	−1037.90	−0.291823	−0.145912	−	0.989298i	$-0.546611\pi$
$233$	−1037.90	−0.291823	−0.145912	+	0.989298i	$0.546611\pi$
$234$	−65.6319	−0.0183354
$235$	−2217.32	−0.615497
$236$	4889.45	1.34863
$237$	−10372.5	−2.84290
$238$	−2.43322	−0.000662698 0
$239$	4604.37	1.24616	0.623080	−	0.782158i	$-0.285881\pi$
$239$	4604.37	1.24616	0.623080	+	0.782158i	$0.285881\pi$
$240$	6389.02	1.71837
$241$	−5261.84	−1.40641	−0.703206	−	0.710986i	$-0.748249\pi$
$241$	−5261.84	−1.40641	−0.703206	+	0.710986i	$0.748249\pi$
$242$	−70.8066	−0.0188083
$243$	4050.90	1.06941
$244$	−2716.36	−0.712693
$245$	−586.415	−0.152917
$246$	33.1179	0.00858341
$247$	−3354.84	−0.864224
$248$	62.6312	0.0160366
$249$	3713.47	0.945107
$250$	−26.1284	−0.00661003
$251$	1950.12	0.490401	0.245200	−	0.969472i	$-0.421146\pi$
$251$	1950.12	0.490401	0.245200	+	0.969472i	$0.421146\pi$
$252$	−2385.63	−0.596351
$253$	3555.93	0.883634
$254$	53.5803	0.0132359
$255$	1697.35	0.416832
$256$	4093.86	0.999477
$257$	−6002.11	−1.45681	−0.728407	−	0.685145i	$-0.759739\pi$
$257$	−6002.11	−1.45681	−0.728407	+	0.685145i	$0.759739\pi$
$258$	19.4286	0.00468826
$259$	−1981.77	−0.475448
$260$	7213.07	1.72052
$261$	−3345.05	−0.793308
$262$	6.52207	0.00153792
$263$	2614.58	0.613012	0.306506	−	0.951869i	$-0.400840\pi$
$263$	2614.58	0.613012	0.306506	+	0.951869i	$0.400840\pi$
$264$	188.973	0.0440549
$265$	−8295.54	−1.92299
$266$	6.37324	0.00146906
$267$	−1841.96	−0.422194
$268$	452.005	0.103025
$269$	−3359.11	−0.761371	−0.380685	−	0.924705i	$-0.624312\pi$
$269$	−3359.11	−0.761371	−0.380685	+	0.924705i	$0.624312\pi$
$270$	−31.8535	−0.00717978
$271$	7741.51	1.73529	0.867644	−	0.497186i	$-0.165633\pi$
$271$	7741.51	1.73529	0.867644	+	0.497186i	$0.165633\pi$
$272$	1087.83	0.242498
$273$	−4400.02	−0.975463
$274$	−7.63793	−0.00168403
$275$	−1261.86	−0.276702
$276$	−3427.59	−0.747525
$277$	4799.95	1.04116	0.520580	−	0.853813i	$-0.325716\pi$
$277$	4799.95	1.04116	0.520580	+	0.853813i	$0.325716\pi$
$278$	24.0508	0.00518874
$279$	8156.15	1.75017
$280$	−27.4063	−0.00584943
$281$	4713.91	1.00074	0.500370	−	0.865811i	$-0.333197\pi$
$281$	4713.91	1.00074	0.500370	+	0.865811i	$0.333197\pi$
$282$	31.6057	0.00667409
$283$	1762.06	0.370119	0.185060	−	0.982727i	$-0.440752\pi$
$283$	1762.06	0.370119	0.185060	+	0.982727i	$0.440752\pi$
$284$	−1287.96	−0.269108
$285$	−4445.81	−0.924024
$286$	106.665	0.0220533
$287$	1358.98	0.279506
$288$	−167.241	−0.0342179
$289$	289.000	0.0588235
$290$	−19.2136	−0.00389056
$291$	1239.13	0.249619
$292$	−1795.82	−0.359906
$293$	5712.12	1.13893	0.569464	−	0.822017i	$-0.307151\pi$
$293$	5712.12	1.13893	0.569464	+	0.822017i	$0.307151\pi$
$294$	8.35879	0.00165814
$295$	7314.79	1.44367
$296$	−92.6185	−0.0181870
$297$	9012.76	1.76085
$298$	24.0065	0.00466665
$299$	−3869.48	−0.748420
$300$	1216.32	0.234080
$301$	797.245	0.152666
$302$	−36.2286	−0.00690305
$303$	−14830.2	−2.81179
$304$	−2849.31	−0.537563
$305$	−4063.77	−0.762921
$306$	−14.8088	−0.00276655
$307$	−7791.05	−1.44840	−0.724199	−	0.689591i	$-0.757790\pi$
$307$	−7791.05	−1.44840	−0.724199	+	0.689591i	$0.757790\pi$
$308$	3877.12	0.717271
$309$	563.786	0.103795
$310$	46.8481	0.00858320
$311$	4492.19	0.819064	0.409532	−	0.912296i	$-0.365692\pi$
$311$	4492.19	0.819064	0.409532	+	0.912296i	$0.365692\pi$
$312$	−205.636	−0.0373136
$313$	6630.29	1.19734	0.598668	−	0.800998i	$-0.295697\pi$
$313$	6630.29	1.19734	0.598668	+	0.800998i	$0.295697\pi$
$314$	−41.8244	−0.00751684
$315$	−3568.98	−0.638379
$316$	−9945.75	−1.77055
$317$	7059.01	1.25071	0.625353	−	0.780342i	$-0.284955\pi$
$317$	7059.01	1.25071	0.625353	+	0.780342i	$0.284955\pi$
$318$	118.245	0.0208517
$319$	5436.38	0.954165
$320$	6125.52	1.07008
$321$	−1454.98	−0.252987
$322$	7.35091	0.00127221
$323$	−756.967	−0.130399
$324$	514.228	0.0881736
$325$	1373.13	0.234361
$326$	68.8673	0.0117000
$327$	−14188.1	−2.39940
$328$	63.5123	0.0106917
$329$	1296.93	0.217332
$330$	141.352	0.0235793
$331$	6836.44	1.13524	0.567621	−	0.823290i	$-0.307864\pi$
$331$	6836.44	1.13524	0.567621	+	0.823290i	$0.307864\pi$
$332$	3560.69	0.588609
$333$	−12061.2	−1.98484
$334$	−67.6131	−0.0110767
$335$	676.215	0.110285
$336$	−3737.00	−0.606756
$337$	−5137.61	−0.830456	−0.415228	−	0.909717i	$-0.636298\pi$
$337$	−5137.61	−0.830456	−0.415228	+	0.909717i	$0.636298\pi$
$338$	−71.1479	−0.0114495
$339$	13221.2	2.11822
$340$	1627.52	0.259601
$341$	−13255.4	−2.10504
$342$	38.7882	0.00613283
$343$	343.000	0.0539949
$344$	37.2595	0.00583981
$345$	−5127.80	−0.800207
$346$	43.6445	0.00678133
$347$	16.9107	0.00261618	0.00130809	−	0.999999i	$-0.499584\pi$
$347$	16.9107	0.00261618	0.00130809	+	0.999999i	$0.499584\pi$
$348$	−5240.17	−0.807192
$349$	565.027	0.0866624	0.0433312	−	0.999061i	$-0.486203\pi$
$349$	565.027	0.0866624	0.0433312	+	0.999061i	$0.486203\pi$
$350$	−2.60855	−0.000398379 0
$351$	−9807.47	−1.49141
$352$	271.800	0.0411562
$353$	1908.19	0.287714	0.143857	−	0.989599i	$-0.454050\pi$
$353$	1908.19	0.287714	0.143857	+	0.989599i	$0.454050\pi$
$354$	−104.265	−0.0156543
$355$	−1926.84	−0.288073
$356$	−1766.17	−0.262941
$357$	−992.796	−0.147183
$358$	79.4407	0.0117279
$359$	−2618.85	−0.385008	−0.192504	−	0.981296i	$-0.561661\pi$
$359$	−2618.85	−0.385008	−0.192504	+	0.981296i	$0.561661\pi$
$360$	−166.798	−0.0244194
$361$	−4876.30	−0.710935
$362$	15.4544	0.00224382
$363$	−28890.3	−4.17727
$364$	−4218.99	−0.607515
$365$	−2686.62	−0.385271
$366$	57.9252	0.00827267
$367$	1611.70	0.229237	0.114618	−	0.993410i	$-0.463436\pi$
$367$	1611.70	0.229237	0.114618	+	0.993410i	$0.463436\pi$
$368$	−3286.40	−0.465531
$369$	8270.89	1.16684
$370$	−69.2785	−0.00973410
$371$	4852.14	0.679004
$372$	12777.0	1.78080
$373$	−11845.1	−1.64428	−0.822141	−	0.569285i	$-0.807220\pi$
$373$	−11845.1	−1.64428	−0.822141	+	0.569285i	$0.807220\pi$
$374$	24.0673	0.00332752
$375$	−10660.9	−1.46807
$376$	60.6124	0.00831342
$377$	−5915.74	−0.808159
$378$	18.6314	0.00253517
$379$	3158.08	0.428020	0.214010	−	0.976831i	$-0.431348\pi$
$379$	3158.08	0.428020	0.214010	+	0.976831i	$0.431348\pi$
$380$	−4262.90	−0.575479
$381$	21861.7	2.93965
$382$	18.7194	0.00250724
$383$	−6718.35	−0.896323	−0.448161	−	0.893953i	$-0.647921\pi$
$383$	−6718.35	−0.896323	−0.448161	+	0.893953i	$0.647921\pi$
$384$	−349.318	−0.0464220
$385$	5800.32	0.767822
$386$	−56.4521	−0.00744387
$387$	4852.12	0.637330
$388$	1188.15	0.155462
$389$	1221.71	0.159237	0.0796184	−	0.996825i	$-0.474630\pi$
$389$	1221.71	0.159237	0.0796184	+	0.996825i	$0.474630\pi$
$390$	−153.816	−0.0199712
$391$	−873.087	−0.112926
$392$	16.0302	0.00206543
$393$	2661.12	0.341567
$394$	−4.30380	−0.000550311 0
$395$	−14879.2	−1.89533
$396$	23596.6	2.99437
$397$	−5381.44	−0.680320	−0.340160	−	0.940368i	$-0.610481\pi$
$397$	−5381.44	−0.680320	−0.340160	+	0.940368i	$0.610481\pi$
$398$	33.3284	0.00419749
$399$	2600.40	0.326272
$400$	1166.21	0.145777
$401$	12397.7	1.54391	0.771956	−	0.635675i	$-0.219278\pi$
$401$	12397.7	1.54391	0.771956	+	0.635675i	$0.219278\pi$
$402$	−9.63880	−0.00119587
$403$	14424.2	1.78293
$404$	−14220.1	−1.75118
$405$	769.304	0.0943877
$406$	11.2382	0.00137375
$407$	19602.0	2.38730
$408$	−46.3986	−0.00563008
$409$	−5664.85	−0.684862	−0.342431	−	0.939543i	$-0.611250\pi$
$409$	−5664.85	−0.684862	−0.342431	+	0.939543i	$0.611250\pi$
$410$	47.5071	0.00572246
$411$	−3116.41	−0.374017
$412$	540.591	0.0646432
$413$	−4278.49	−0.509760
$414$	44.7384	0.00531104
$415$	5326.92	0.630092
$416$	−295.766	−0.0348585
$417$	9813.14	1.15240
$418$	−63.0386	−0.00737636
$419$	14028.0	1.63559	0.817794	−	0.575510i	$-0.195197\pi$
$419$	14028.0	1.63559	0.817794	+	0.575510i	$0.195197\pi$
$420$	−5590.98	−0.649552
$421$	14096.9	1.63192	0.815962	−	0.578106i	$-0.196208\pi$
$421$	14096.9	1.63192	0.815962	+	0.578106i	$0.196208\pi$
$422$	−32.0569	−0.00369788
$423$	7893.25	0.907288
$424$	226.766	0.0259734
$425$	309.824	0.0353616
$426$	27.4652	0.00312370
$427$	2376.94	0.269387
$428$	−1395.12	−0.157560
$429$	43521.2	4.89796
$430$	27.8700	0.00312561
$431$	4248.58	0.474818	0.237409	−	0.971410i	$-0.423702\pi$
$431$	4248.58	0.474818	0.237409	+	0.971410i	$0.423702\pi$
$432$	−8329.62	−0.927683
$433$	937.201	0.104016	0.0520081	−	0.998647i	$-0.483438\pi$
$433$	937.201	0.104016	0.0520081	+	0.998647i	$0.483438\pi$
$434$	−27.4019	−0.00303072
$435$	−7839.49	−0.864079
$436$	−13604.4	−1.49434
$437$	2286.85	0.250331
$438$	38.2952	0.00417766
$439$	−4569.76	−0.496817	−0.248409	−	0.968655i	$-0.579908\pi$
$439$	−4569.76	−0.496817	−0.248409	+	0.968655i	$0.579908\pi$
$440$	271.079	0.0293709
$441$	2087.53	0.225411
$442$	−26.1895	−0.00281834
$443$	−5024.96	−0.538923	−0.269462	−	0.963011i	$-0.586846\pi$
$443$	−5024.96	−0.538923	−0.269462	+	0.963011i	$0.586846\pi$
$444$	−18894.5	−2.01958
$445$	−2642.26	−0.281472
$446$	−69.1601	−0.00734265
$447$	9795.08	1.03645
$448$	−3582.88	−0.377846
$449$	−4591.17	−0.482563	−0.241281	−	0.970455i	$-0.577568\pi$
$449$	−4591.17	−0.482563	−0.241281	+	0.970455i	$0.577568\pi$
$450$	−15.8759	−0.00166310
$451$	−13441.9	−1.40344
$452$	12677.2	1.31922
$453$	−14781.9	−1.53314
$454$	−50.5069	−0.00522116
$455$	−6311.77	−0.650330
$456$	121.530	0.0124806
$457$	3061.00	0.313320	0.156660	−	0.987653i	$-0.449927\pi$
$457$	3061.00	0.313320	0.156660	+	0.987653i	$0.449927\pi$
$458$	−106.001	−0.0108146
$459$	−2212.90	−0.225031
$460$	−4916.83	−0.498366
$461$	1438.10	0.145291	0.0726453	−	0.997358i	$-0.476856\pi$
$461$	1438.10	0.145291	0.0726453	+	0.997358i	$0.476856\pi$
$462$	−82.6780	−0.00832581
$463$	5152.77	0.517213	0.258606	−	0.965983i	$-0.416737\pi$
$463$	5152.77	0.517213	0.258606	+	0.965983i	$0.416737\pi$
$464$	−5024.32	−0.502690
$465$	19114.8	1.90630
$466$	21.2221	0.00210964
$467$	17831.0	1.76685	0.883425	−	0.468573i	$-0.155232\pi$
$467$	17831.0	1.76685	0.883425	+	0.468573i	$0.155232\pi$
$468$	−25677.2	−2.53618
$469$	−395.525	−0.0389417
$470$	45.3380	0.00444954
$471$	−17065.1	−1.66946
$472$	−199.956	−0.0194995
$473$	−7885.66	−0.766560
$474$	212.089	0.0205518
$475$	−811.512	−0.0783889
$476$	−951.950	−0.0916651
$477$	29530.6	2.83462
$478$	−94.1466	−0.00900871
$479$	−5902.23	−0.563005	−0.281503	−	0.959560i	$-0.590833\pi$
$479$	−5902.23	−0.563005	−0.281503	+	0.959560i	$0.590833\pi$
$480$	−391.947	−0.0372706
$481$	−21330.4	−2.02200
$482$	107.590	0.0101672
$483$	2999.30	0.282553
$484$	−27701.7	−2.60159
$485$	1777.51	0.166418
$486$	−82.8297	−0.00773093
$487$	−195.193	−0.0181623	−0.00908115	−	0.999959i	$-0.502891\pi$
$487$	−195.193	−0.0181623	−0.00908115	+	0.999959i	$0.502891\pi$
$488$	111.087	0.0103046
$489$	28099.1	2.59854
$490$	11.9906	0.00110547
$491$	1176.02	0.108092	0.0540458	−	0.998538i	$-0.482788\pi$
$491$	1176.02	0.108092	0.0540458	+	0.998538i	$0.482788\pi$
$492$	12956.7	1.18727
$493$	−1334.79	−0.121939
$494$	68.5972	0.00624764
$495$	35301.3	3.20541
$496$	12250.7	1.10902
$497$	1127.03	0.101718
$498$	−75.9302	−0.00683235
$499$	−9379.34	−0.841437	−0.420718	−	0.907191i	$-0.638222\pi$
$499$	−9379.34	−0.841437	−0.420718	+	0.907191i	$0.638222\pi$
$500$	−10222.2	−0.914306
$501$	−27587.3	−2.46010
$502$	−39.8746	−0.00354520
$503$	17417.2	1.54393	0.771965	−	0.635665i	$-0.219274\pi$
$503$	17417.2	1.54393	0.771965	+	0.635665i	$0.219274\pi$
$504$	97.5614	0.00862248
$505$	−21273.7	−1.87459
$506$	−72.7089	−0.00638795
$507$	−29029.6	−2.54290
$508$	20962.3	1.83081
$509$	−6927.46	−0.603250	−0.301625	−	0.953427i	$-0.597529\pi$
$509$	−6927.46	−0.603250	−0.301625	+	0.953427i	$0.597529\pi$
$510$	−34.7061	−0.00301336
$511$	1571.43	0.136039
$512$	−418.672	−0.0361384
$513$	5796.18	0.498845
$514$	122.726	0.0105316
$515$	808.743	0.0691990
$516$	7601.06	0.648485
$517$	−12828.1	−1.09126
$518$	40.5217	0.00343710
$519$	17807.7	1.50611
$520$	−294.982	−0.0248766
$521$	2456.06	0.206529	0.103265	−	0.994654i	$-0.467071\pi$
$521$	2456.06	0.206529	0.103265	+	0.994654i	$0.467071\pi$
$522$	68.3969	0.00573497
$523$	−13939.6	−1.16546	−0.582729	−	0.812666i	$-0.698015\pi$
$523$	−13939.6	−1.16546	−0.582729	+	0.812666i	$0.698015\pi$
$524$	2551.63	0.212726
$525$	−1064.33	−0.0884787
$526$	−53.4610	−0.00443158
$527$	3254.60	0.269018
$528$	36963.2	3.04662
$529$	−9529.35	−0.783213
$530$	169.621	0.0139016
$531$	−26039.3	−2.12808
$532$	2493.41	0.203201
$533$	14627.1	1.18869
$534$	37.6629	0.00305212
$535$	−2087.14	−0.168664
$536$	−18.4850	−0.00148961
$537$	32413.2	2.60472
$538$	68.6845	0.00550409
$539$	−3392.66	−0.271117
$540$	−12462.1	−0.993114
$541$	7127.43	0.566418	0.283209	−	0.959058i	$-0.408601\pi$
$541$	7127.43	0.566418	0.283209	+	0.959058i	$0.408601\pi$
$542$	−158.292	−0.0125447
$543$	6305.65	0.498345
$544$	−66.7351	−0.00525964
$545$	−20352.6	−1.59965
$546$	89.9682	0.00705180
$547$	8358.71	0.653369	0.326684	−	0.945134i	$-0.394069\pi$
$547$	8358.71	0.653369	0.326684	+	0.945134i	$0.394069\pi$
$548$	−2988.19	−0.232937
$549$	14466.3	1.12460
$550$	25.8015	0.00200033
$551$	3496.18	0.270312
$552$	140.173	0.0108083
$553$	8702.99	0.669238
$554$	−98.1456	−0.00752673
$555$	−28266.9	−2.16191
$556$	9409.40	0.717711
$557$	−21226.6	−1.61472	−0.807362	−	0.590057i	$-0.799105\pi$
$557$	−21226.6	−1.61472	−0.807362	+	0.590057i	$0.799105\pi$
$558$	−166.771	−0.0126523
$559$	8580.99	0.649262
$560$	−5360.67	−0.404517
$561$	9819.88	0.739030
$562$	−96.3863	−0.00723454
$563$	−2736.05	−0.204815	−0.102407	−	0.994743i	$-0.532655\pi$
$563$	−2736.05	−0.204815	−0.102407	+	0.994743i	$0.532655\pi$
$564$	12365.1	0.923167
$565$	18965.6	1.41219
$566$	−36.0293	−0.00267566
$567$	−449.973	−0.0333282
$568$	52.6719	0.00389096
$569$	−2731.50	−0.201249	−0.100624	−	0.994924i	$-0.532084\pi$
$569$	−2731.50	−0.201249	−0.100624	+	0.994924i	$0.532084\pi$
$570$	90.9044	0.00667994
$571$	−679.490	−0.0497999	−0.0249000	−	0.999690i	$-0.507927\pi$
$571$	−679.490	−0.0497999	−0.0249000	+	0.999690i	$0.507927\pi$
$572$	41730.7	3.05043
$573$	7637.84	0.556851
$574$	−27.7874	−0.00202060
$575$	−935.999	−0.0678850
$576$	−21805.8	−1.57738
$577$	6896.56	0.497586	0.248793	−	0.968557i	$-0.419966\pi$
$577$	6896.56	0.497586	0.248793	+	0.968557i	$0.419966\pi$
$578$	−5.90925	−0.000425246 0
$579$	−23033.4	−1.65326
$580$	−7516.95	−0.538146
$581$	−3115.77	−0.222485
$582$	−25.3367	−0.00180454
$583$	−47993.2	−3.40939
$584$	73.4411	0.00520380
$585$	−38414.1	−2.71492
$586$	−116.797	−0.00823351
$587$	−6876.39	−0.483507	−0.241754	−	0.970338i	$-0.577723\pi$
$587$	−6876.39	−0.483507	−0.241754	+	0.970338i	$0.577723\pi$
$588$	3270.22	0.229356
$589$	−8524.64	−0.596353
$590$	−149.567	−0.0104366
$591$	−1756.03	−0.122222
$592$	−18116.2	−1.25772
$593$	−13903.3	−0.962802	−0.481401	−	0.876500i	$-0.659872\pi$
$593$	−13903.3	−0.962802	−0.481401	+	0.876500i	$0.659872\pi$
$594$	−184.286	−0.0127295
$595$	−1424.15	−0.0981253
$596$	9392.09	0.645495
$597$	13598.6	0.932248
$598$	79.1201	0.00541047
$599$	57.5719	0.00392709	0.00196354	−	0.999998i	$-0.499375\pi$
$599$	57.5719	0.00392709	0.00196354	+	0.999998i	$0.499375\pi$
$600$	−49.7419	−0.00338451
$601$	−13428.9	−0.911439	−0.455719	−	0.890123i	$-0.650618\pi$
$601$	−13428.9	−0.911439	−0.455719	+	0.890123i	$0.650618\pi$
$602$	−16.3014	−0.00110365
$603$	−2407.20	−0.162569
$604$	−14173.7	−0.954836
$605$	−41442.8	−2.78494
$606$	303.237	0.0203270
$607$	−13233.3	−0.884879	−0.442440	−	0.896798i	$-0.645887\pi$
$607$	−13233.3	−0.884879	−0.442440	+	0.896798i	$0.645887\pi$
$608$	174.797	0.0116595
$609$	4585.39	0.305106
$610$	83.0928	0.00551530
$611$	13959.3	0.924273
$612$	−5793.67	−0.382672
$613$	5047.07	0.332544	0.166272	−	0.986080i	$-0.446827\pi$
$613$	5047.07	0.332544	0.166272	+	0.986080i	$0.446827\pi$
$614$	159.305	0.0104707
$615$	19383.7	1.27094
$616$	−158.557	−0.0103708
$617$	−8218.89	−0.536272	−0.268136	−	0.963381i	$-0.586408\pi$
$617$	−8218.89	−0.536272	−0.268136	+	0.963381i	$0.586408\pi$
$618$	−11.5279	−0.000750354 0
$619$	−2840.18	−0.184421	−0.0922104	−	0.995740i	$-0.529393\pi$
$619$	−2840.18	−0.184421	−0.0922104	+	0.995740i	$0.529393\pi$
$620$	18328.4	1.18724
$621$	6685.32	0.432001
$622$	−91.8528	−0.00592116
$623$	1545.48	0.0993875
$624$	−40222.5	−2.58043
$625$	−17571.0	−1.12454
$626$	−135.571	−0.00865576
$627$	−25720.9	−1.63827
$628$	−16363.0	−1.03974
$629$	−4812.87	−0.305090
$630$	72.9758	0.00461496
$631$	−12817.6	−0.808652	−0.404326	−	0.914615i	$-0.632494\pi$
$631$	−12817.6	−0.808652	−0.404326	+	0.914615i	$0.632494\pi$
$632$	406.737	0.0255999
$633$	−13079.8	−0.821288
$634$	−144.337	−0.00904158
$635$	31360.3	1.95983
$636$	46261.1	2.88423
$637$	3691.81	0.229631
$638$	−111.159	−0.00689783
$639$	6859.20	0.424641
$640$	−501.091	−0.0309490
$641$	31419.6	1.93604	0.968019	−	0.250876i	$-0.0807185\pi$
$641$	31419.6	1.93604	0.968019	+	0.250876i	$0.0807185\pi$
$642$	29.7502	0.00182889
$643$	9523.47	0.584089	0.292044	−	0.956405i	$-0.405665\pi$
$643$	9523.47	0.584089	0.292044	+	0.956405i	$0.405665\pi$
$644$	2875.90	0.175973
$645$	11371.5	0.694187
$646$	15.4779	0.000942676 0
$647$	−4184.16	−0.254245	−0.127122	−	0.991887i	$-0.540574\pi$
$647$	−4184.16	−0.254245	−0.127122	+	0.991887i	$0.540574\pi$
$648$	−21.0296	−0.00127488
$649$	42319.1	2.55959
$650$	−28.0766	−0.00169424
$651$	−11180.4	−0.673113
$652$	26943.0	1.61836
$653$	17737.5	1.06297	0.531486	−	0.847067i	$-0.321634\pi$
$653$	17737.5	1.06297	0.531486	+	0.847067i	$0.321634\pi$
$654$	290.107	0.0173457
$655$	3817.34	0.227719
$656$	12423.0	0.739386
$657$	9563.87	0.567918
$658$	−26.5186	−0.00157113
$659$	−13168.9	−0.778434	−0.389217	−	0.921146i	$-0.627254\pi$
$659$	−13168.9	−0.778434	−0.389217	+	0.921146i	$0.627254\pi$
$660$	55301.1	3.26151
$661$	27406.7	1.61270	0.806352	−	0.591435i	$-0.201438\pi$
$661$	27406.7	1.61270	0.806352	+	0.591435i	$0.201438\pi$
$662$	−139.786	−0.00820687
$663$	−10685.8	−0.625944
$664$	−145.616	−0.00851055
$665$	3730.23	0.217522
$666$	246.619	0.0143488
$667$	4032.50	0.234091
$668$	−26452.3	−1.53214
$669$	−28218.5	−1.63078
$670$	−13.8267	−0.000797273 0
$671$	−23510.6	−1.35263
$672$	229.254	0.0131602
$673$	−28992.2	−1.66057	−0.830287	−	0.557337i	$-0.811823\pi$
$673$	−28992.2	−1.66057	−0.830287	+	0.557337i	$0.811823\pi$
$674$	105.050	0.00600352
$675$	−2372.36	−0.135277
$676$	−27835.2	−1.58371
$677$	−12159.5	−0.690292	−0.345146	−	0.938549i	$-0.612171\pi$
$677$	−12159.5	−0.690292	−0.345146	+	0.938549i	$0.612171\pi$
$678$	−270.336	−0.0153130
$679$	−1039.68	−0.0587620
$680$	−66.5581	−0.00375351
$681$	−20607.7	−1.15960
$682$	271.036	0.0152177
$683$	−8523.47	−0.477513	−0.238757	−	0.971079i	$-0.576740\pi$
$683$	−8523.47	−0.477513	−0.238757	+	0.971079i	$0.576740\pi$
$684$	15175.1	0.848298
$685$	−4470.44	−0.249353
$686$	−7.01340	−0.000390339 0
$687$	−43250.3	−2.40189
$688$	7287.96	0.403853
$689$	52225.1	2.88769
$690$	104.849	0.00578485
$691$	16880.0	0.929299	0.464650	−	0.885495i	$-0.346180\pi$
$691$	16880.0	0.929299	0.464650	+	0.885495i	$0.346180\pi$
$692$	17075.1	0.938000
$693$	−20648.1	−1.13183
$694$	−0.345777	−1.89128e−5 0
$695$	14076.8	0.768293
$696$	214.299	0.0116710
$697$	3300.38	0.179356
$698$	−11.5532	−0.000626498 0
$699$	8658.98	0.468545
$700$	−1020.54	−0.0551042
$701$	−17669.7	−0.952031	−0.476015	−	0.879437i	$-0.657919\pi$
$701$	−17669.7	−0.952031	−0.476015	+	0.879437i	$0.657919\pi$
$702$	200.535	0.0107817
$703$	12606.2	0.676317
$704$	35438.7	1.89723
$705$	18498.7	0.988229
$706$	−39.0173	−0.00207993
$707$	12443.2	0.661917
$708$	−40791.8	−2.16532
$709$	−2054.13	−0.108807	−0.0544037	−	0.998519i	$-0.517326\pi$
$709$	−2054.13	−0.108807	−0.0544037	+	0.998519i	$0.517326\pi$
$710$	39.3985	0.00208253
$711$	52967.3	2.79385
$712$	72.2285	0.00380180
$713$	−9832.34	−0.516443
$714$	20.2999	0.00106401
$715$	62430.6	3.26541
$716$	31079.7	1.62221
$717$	−38413.5	−2.00080
$718$	53.5482	0.00278329
$719$	−409.281	−0.0212289	−0.0106145	−	0.999944i	$-0.503379\pi$
$719$	−409.281	−0.0212289	−0.0106145	+	0.999944i	$0.503379\pi$
$720$	−32625.6	−1.68873
$721$	−473.042	−0.0244341
$722$	99.7068	0.00513948
$723$	43898.6	2.25810
$724$	6046.23	0.310368
$725$	−1430.97	−0.0733035
$726$	590.727	0.0301983
$727$	10961.5	0.559200	0.279600	−	0.960117i	$-0.409798\pi$
$727$	10961.5	0.559200	0.279600	+	0.960117i	$0.409798\pi$
$728$	172.538	0.00878390
$729$	−32060.4	−1.62883
$730$	54.9339	0.00278520
$731$	1936.17	0.0979640
$732$	22662.1	1.14428
$733$	8325.43	0.419518	0.209759	−	0.977753i	$-0.432732\pi$
$733$	8325.43	0.419518	0.209759	+	0.977753i	$0.432732\pi$
$734$	−32.9547	−0.00165719
$735$	4892.36	0.245520
$736$	201.611	0.0100971
$737$	3912.19	0.195532
$738$	−169.117	−0.00843533
$739$	181.394	0.00902934	0.00451467	−	0.999990i	$-0.498563\pi$
$739$	181.394	0.00902934	0.00451467	+	0.999990i	$0.498563\pi$
$740$	−27103.9	−1.34643
$741$	27988.8	1.38758
$742$	−99.2128	−0.00490865
$743$	12938.2	0.638839	0.319420	−	0.947613i	$-0.396512\pi$
$743$	12938.2	0.638839	0.319420	+	0.947613i	$0.396512\pi$
$744$	−522.521	−0.0257481
$745$	14050.9	0.690987
$746$	242.200	0.0118868
$747$	−18962.9	−0.928802
$748$	9415.87	0.460265
$749$	1220.79	0.0595550
$750$	217.985	0.0106129
$751$	−21345.4	−1.03715	−0.518577	−	0.855031i	$-0.673538\pi$
$751$	−21345.4	−1.03715	−0.518577	+	0.855031i	$0.673538\pi$
$752$	11855.8	0.574915
$753$	−16269.5	−0.787376
$754$	120.960	0.00584233
$755$	−21204.4	−1.02213
$756$	7289.18	0.350668
$757$	−16229.7	−0.779231	−0.389616	−	0.920978i	$-0.627392\pi$
$757$	−16229.7	−0.779231	−0.389616	+	0.920978i	$0.627392\pi$
$758$	−64.5739	−0.00309423
$759$	−29666.5	−1.41874
$760$	174.333	0.00832070
$761$	4693.56	0.223576	0.111788	−	0.993732i	$-0.464342\pi$
$761$	4693.56	0.223576	0.111788	+	0.993732i	$0.464342\pi$
$762$	−447.011	−0.0212513
$763$	11904.4	0.564836
$764$	7323.60	0.346804
$765$	−8667.53	−0.409641
$766$	137.372	0.00647968
$767$	−46050.7	−2.16792
$768$	−34154.4	−1.60474
$769$	34006.6	1.59468	0.797340	−	0.603531i	$-0.206240\pi$
$769$	34006.6	1.59468	0.797340	+	0.603531i	$0.206240\pi$
$770$	−118.600	−0.00555073
$771$	50074.5	2.33903
$772$	−22085.8	−1.02964
$773$	−22113.6	−1.02894	−0.514470	−	0.857509i	$-0.672011\pi$
$773$	−22113.6	−1.02894	−0.514470	+	0.857509i	$0.672011\pi$
$774$	−99.2123	−0.00460738
$775$	3489.11	0.161719
$776$	−48.5899	−0.00224778
$777$	16533.5	0.763369
$778$	−24.9806	−0.00115115
$779$	−8644.57	−0.397592
$780$	−60177.4	−2.76243
$781$	−11147.6	−0.510745
$782$	17.8522	0.000816360 0
$783$	10220.6	0.466483
$784$	3135.51	0.142835
$785$	−24479.6	−1.11301
$786$	−54.4125	−0.00246925
$787$	26230.0	1.18806	0.594028	−	0.804445i	$-0.297537\pi$
$787$	26230.0	1.18806	0.594028	+	0.804445i	$0.297537\pi$
$788$	−1683.78	−0.0761195
$789$	−21813.0	−0.984238
$790$	304.238	0.0137017
$791$	−11093.1	−0.498643
$792$	−964.994	−0.0432949
$793$	25583.7	1.14565
$794$	110.036	0.00491816
$795$	69208.3	3.08750
$796$	13039.1	0.580601
$797$	−29226.6	−1.29894	−0.649472	−	0.760386i	$-0.725010\pi$
$797$	−29226.6	−1.29894	−0.649472	+	0.760386i	$0.725010\pi$
$798$	−53.1709	−0.00235868
$799$	3149.69	0.139459
$800$	−71.5438	−0.00316182
$801$	9405.96	0.414911
$802$	−253.497	−0.0111612
$803$	−15543.2	−0.683074
$804$	−3771.00	−0.165414
$805$	4302.45	0.188375
$806$	−294.935	−0.0128891
$807$	28024.5	1.22244
$808$	581.537	0.0253198
$809$	−34364.6	−1.49344	−0.746722	−	0.665136i	$-0.768374\pi$
$809$	−34364.6	−1.49344	−0.746722	+	0.665136i	$0.768374\pi$
$810$	−15.7301	−0.000682346 0
$811$	10275.8	0.444923	0.222462	−	0.974941i	$-0.428591\pi$
$811$	10275.8	0.444923	0.222462	+	0.974941i	$0.428591\pi$
$812$	4396.74	0.190019
$813$	−64586.0	−2.78614
$814$	−400.805	−0.0172583
$815$	40307.7	1.73242
$816$	−9075.57	−0.389349
$817$	−5071.33	−0.217165
$818$	115.830	0.00495100
$819$	22468.8	0.958635
$820$	18586.3	0.791537
$821$	7440.09	0.316274	0.158137	−	0.987417i	$-0.449451\pi$
$821$	7440.09	0.316274	0.158137	+	0.987417i	$0.449451\pi$
$822$	63.7219	0.00270384
$823$	22397.6	0.948640	0.474320	−	0.880352i	$-0.342694\pi$
$823$	22397.6	0.948640	0.474320	+	0.880352i	$0.342694\pi$
$824$	−22.1077	−0.000934660 0
$825$	10527.5	0.444266
$826$	87.4832	0.00368515
$827$	12458.0	0.523831	0.261915	−	0.965091i	$-0.415646\pi$
$827$	12458.0	0.523831	0.261915	+	0.965091i	$0.415646\pi$
$828$	17503.0	0.734629
$829$	39954.6	1.67392	0.836960	−	0.547264i	$-0.184331\pi$
$829$	39954.6	1.67392	0.836960	+	0.547264i	$0.184331\pi$
$830$	−108.921	−0.00455505
$831$	−40045.1	−1.67166
$832$	−38563.6	−1.60691
$833$	833.000	0.0346479
$834$	−200.651	−0.00833092
$835$	−39573.7	−1.64012
$836$	−24662.7	−1.02031
$837$	−24920.8	−1.02914
$838$	−286.833	−0.0118240
$839$	24065.9	0.990283	0.495142	−	0.868812i	$-0.335116\pi$
$839$	24065.9	0.990283	0.495142	+	0.868812i	$0.335116\pi$
$840$	228.646	0.00939171
$841$	−18224.0	−0.747224
$842$	−288.242	−0.0117975
$843$	−39327.3	−1.60677
$844$	−12541.7	−0.511495
$845$	−41642.5	−1.69532
$846$	−161.395	−0.00655895
$847$	24240.3	0.983360
$848$	44355.5	1.79620
$849$	−14700.6	−0.594255
$850$	−6.33505	−0.000255636 0
$851$	14540.0	0.585692
$852$	10745.2	0.432073
$853$	12546.3	0.503607	0.251804	−	0.967778i	$-0.418976\pi$
$853$	12546.3	0.503607	0.251804	+	0.967778i	$0.418976\pi$
$854$	−48.6018	−0.00194745
$855$	22702.6	0.908083
$856$	57.0540	0.00227811
$857$	−28426.3	−1.13305	−0.566525	−	0.824045i	$-0.691712\pi$
$857$	−28426.3	−1.13305	−0.566525	+	0.824045i	$0.691712\pi$
$858$	−889.888	−0.0354082
$859$	−14817.1	−0.588538	−0.294269	−	0.955723i	$-0.595076\pi$
$859$	−14817.1	−0.588538	−0.294269	+	0.955723i	$0.595076\pi$
$860$	10903.6	0.432337
$861$	−11337.7	−0.448768
$862$	−86.8716	−0.00343255
$863$	24385.9	0.961883	0.480942	−	0.876753i	$-0.340295\pi$
$863$	24385.9	0.961883	0.480942	+	0.876753i	$0.340295\pi$
$864$	510.997	0.0201209
$865$	25544.9	1.00411
$866$	−19.1632	−0.000751952 0
$867$	−2411.08	−0.0944457
$868$	−10720.5	−0.419212
$869$	−86082.5	−3.36035
$870$	160.296	0.00624659
$871$	−4257.15	−0.165612
$872$	556.357	0.0216062
$873$	−6327.62	−0.245312
$874$	−46.7597	−0.00180969
$875$	8944.93	0.345593
$876$	14982.2	0.577858
$877$	21344.7	0.821847	0.410924	−	0.911670i	$-0.365206\pi$
$877$	21344.7	0.821847	0.410924	+	0.911670i	$0.365206\pi$
$878$	93.4389	0.00359158
$879$	−47655.2	−1.82864
$880$	53023.2	2.03115
$881$	10102.1	0.386322	0.193161	−	0.981167i	$-0.438126\pi$
$881$	10102.1	0.386322	0.193161	+	0.981167i	$0.438126\pi$
$882$	−42.6842	−0.00162954
$883$	−44942.3	−1.71283	−0.856415	−	0.516288i	$-0.827314\pi$
$883$	−44942.3	−1.71283	−0.856415	+	0.516288i	$0.827314\pi$
$884$	−10246.1	−0.389836
$885$	−61026.0	−2.31793
$886$	102.746	0.00389597
$887$	−18492.5	−0.700018	−0.350009	−	0.936746i	$-0.613821\pi$
$887$	−18492.5	−0.700018	−0.350009	+	0.936746i	$0.613821\pi$
$888$	772.700	0.0292006
$889$	−18342.9	−0.692016
$890$	54.0268	0.00203481
$891$	4450.75	0.167346
$892$	−27057.5	−1.01564
$893$	−8249.87	−0.309150
$894$	−200.282	−0.00749266
$895$	46496.3	1.73654
$896$	293.093	0.0109281
$897$	32282.4	1.20165
$898$	93.8766	0.00348853
$899$	−15031.9	−0.557665
$900$	−6211.14	−0.230042
$901$	11783.8	0.435710
$902$	274.849	0.0101457
$903$	−6651.28	−0.245117
$904$	−518.441	−0.0190742
$905$	9045.37	0.332241
$906$	302.249	0.0110834
$907$	19894.2	0.728307	0.364154	−	0.931339i	$-0.381358\pi$
$907$	19894.2	0.728307	0.364154	+	0.931339i	$0.381358\pi$
$908$	−19759.8	−0.722195
$909$	75730.7	2.76329
$910$	129.058	0.00470136
$911$	−10997.6	−0.399962	−0.199981	−	0.979800i	$-0.564088\pi$
$911$	−10997.6	−0.399962	−0.199981	+	0.979800i	$0.564088\pi$
$912$	23771.3	0.863100
$913$	30818.5	1.11713
$914$	−62.5889	−0.00226505
$915$	33903.3	1.22493
$916$	−41470.9	−1.49589
$917$	−2232.80	−0.0804072
$918$	45.2477	0.00162679
$919$	−11458.0	−0.411277	−0.205639	−	0.978628i	$-0.565927\pi$
$919$	−11458.0	−0.411277	−0.205639	+	0.978628i	$0.565927\pi$
$920$	201.076	0.00720575
$921$	64999.3	2.32552
$922$	−29.4051	−0.00105033
$923$	12130.5	0.432591
$924$	−32346.2	−1.15163
$925$	−5159.67	−0.183404
$926$	−105.360	−0.00373903
$927$	−2878.98	−0.102004
$928$	308.227	0.0109031
$929$	−966.228	−0.0341237	−0.0170619	−	0.999854i	$-0.505431\pi$
$929$	−966.228	−0.0341237	−0.0170619	+	0.999854i	$0.505431\pi$
$930$	−390.845	−0.0137810
$931$	−2181.85	−0.0768068
$932$	8302.73	0.291808
$933$	−37477.6	−1.31507
$934$	−364.594	−0.0127729
$935$	14086.5	0.492703
$936$	1050.08	0.0366699
$937$	−2117.39	−0.0738231	−0.0369116	−	0.999319i	$-0.511752\pi$
$937$	−2117.39	−0.0738231	−0.0369116	+	0.999319i	$0.511752\pi$
$938$	8.08738	0.000281516 0
$939$	−55315.3	−1.92241
$940$	17737.6	0.615465
$941$	−3067.14	−0.106255	−0.0531274	−	0.998588i	$-0.516919\pi$
$941$	−3067.14	−0.106255	−0.0531274	+	0.998588i	$0.516919\pi$
$942$	348.933	0.0120689
$943$	−9970.66	−0.344315
$944$	−39111.5	−1.34849
$945$	10904.9	0.375382
$946$	161.240	0.00554161
$947$	20989.5	0.720240	0.360120	−	0.932906i	$-0.382736\pi$
$947$	20989.5	0.720240	0.360120	+	0.932906i	$0.382736\pi$
$948$	82975.7	2.84275
$949$	16913.8	0.578550
$950$	16.5932	0.000566688 0
$951$	−58892.1	−2.00810
$952$	38.9305	0.00132536
$953$	40255.6	1.36832	0.684158	−	0.729334i	$-0.260170\pi$
$953$	40255.6	1.36832	0.684158	+	0.729334i	$0.260170\pi$
$954$	−603.819	−0.0204920
$955$	10956.4	0.371246
$956$	−36833.0	−1.24609
$957$	−45354.7	−1.53199
$958$	120.684	0.00407007
$959$	2614.81	0.0880464
$960$	−51104.2	−1.71810
$961$	6860.87	0.230300
$962$	436.147	0.0146174
$963$	7429.86	0.248623
$964$	42092.6	1.40634
$965$	−33041.1	−1.10221
$966$	−61.3273	−0.00204262
$967$	−27866.3	−0.926702	−0.463351	−	0.886175i	$-0.653353\pi$
$967$	−27866.3	−0.926702	−0.463351	+	0.886175i	$0.653353\pi$
$968$	1132.88	0.0376157
$969$	6315.25	0.209365
$970$	−36.3452	−0.00120307
$971$	30301.7	1.00147	0.500735	−	0.865601i	$-0.333063\pi$
$971$	30301.7	1.00147	0.500735	+	0.865601i	$0.333063\pi$
$972$	−32405.5	−1.06935
$973$	−8233.66	−0.271284
$974$	3.99115	0.000131299 0
$975$	−11455.7	−0.376285
$976$	21728.6	0.712618
$977$	−5422.77	−0.177574	−0.0887869	−	0.996051i	$-0.528299\pi$
$977$	−5422.77	−0.177574	−0.0887869	+	0.996051i	$0.528299\pi$
$978$	−574.548	−0.0187853
$979$	−15286.6	−0.499041
$980$	4691.08	0.152909
$981$	72451.7	2.35801
$982$	−24.0463	−0.000781414 0
$983$	51694.2	1.67730	0.838652	−	0.544668i	$-0.183344\pi$
$983$	51694.2	1.67730	0.838652	+	0.544668i	$0.183344\pi$
$984$	−529.872	−0.0171664
$985$	−2519.00	−0.0814842
$986$	27.2928	0.000881522 0
$987$	−10820.1	−0.348943
$988$	26837.3	0.864179
$989$	−5849.28	−0.188065
$990$	−721.814	−0.0231725
$991$	−16723.3	−0.536057	−0.268028	−	0.963411i	$-0.586372\pi$
$991$	−16723.3	−0.536057	−0.268028	+	0.963411i	$0.586372\pi$
$992$	−751.542	−0.0240539
$993$	−57035.3	−1.82272
$994$	−23.0446	−0.000735341 0
$995$	19507.0	0.621520
$996$	−29706.2	−0.945058
$997$	−9223.60	−0.292993	−0.146497	−	0.989211i	$-0.546800\pi$
$997$	−9223.60	−0.292993	−0.146497	+	0.989211i	$0.546800\pi$
$998$	191.781	0.00608290
$999$	36852.6	1.16713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	119.4.a.e.1.5	✓	9
3.2	odd	2		1071.4.a.r.1.5		9
4.3	odd	2		1904.4.a.s.1.9		9
7.6	odd	2		833.4.a.g.1.5		9
17.16	even	2		2023.4.a.h.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.4.a.e.1.5	✓	9	1.1	even	1	trivial
833.4.a.g.1.5		9	7.6	odd	2
1071.4.a.r.1.5		9	3.2	odd	2
1904.4.a.s.1.9		9	4.3	odd	2
2023.4.a.h.1.5		9	17.16	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 119 = 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	119.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0204472 q^{2} -8.34282 q^{3} -7.99958 q^{4} -11.9677 q^{5} +0.170588 q^{6} +7.00000 q^{7} +0.327147 q^{8} +42.6027 q^{9} +0.244705 q^{10} -69.2380 q^{11} +66.7391 q^{12} +75.3431 q^{13} -0.143131 q^{14} +99.8441 q^{15} +63.9900 q^{16} +17.0000 q^{17} -0.871107 q^{18} -44.5275 q^{19} +95.7363 q^{20} -58.3998 q^{21} +1.41572 q^{22} -51.3581 q^{23} -2.72933 q^{24} +18.2250 q^{25} -1.54056 q^{26} -130.171 q^{27} -55.9971 q^{28} -78.5173 q^{29} -2.04153 q^{30} +191.447 q^{31} -3.92559 q^{32} +577.640 q^{33} -0.347603 q^{34} -83.7736 q^{35} -340.804 q^{36} -283.110 q^{37} +0.910463 q^{38} -628.574 q^{39} -3.91518 q^{40} +194.140 q^{41} +1.19411 q^{42} +113.892 q^{43} +553.875 q^{44} -509.855 q^{45} +1.05013 q^{46} +185.276 q^{47} -533.857 q^{48} +49.0000 q^{49} -0.372650 q^{50} -141.828 q^{51} -602.713 q^{52} +693.163 q^{53} +2.66163 q^{54} +828.617 q^{55} +2.29003 q^{56} +371.485 q^{57} +1.60546 q^{58} -611.213 q^{59} -798.711 q^{60} +339.563 q^{61} -3.91455 q^{62} +298.219 q^{63} -511.839 q^{64} -901.681 q^{65} -11.8111 q^{66} -56.5036 q^{67} -135.993 q^{68} +428.471 q^{69} +1.71294 q^{70} +161.004 q^{71} +13.9373 q^{72} +224.490 q^{73} +5.78881 q^{74} -152.048 q^{75} +356.201 q^{76} -484.666 q^{77} +12.8526 q^{78} +1243.28 q^{79} -765.810 q^{80} -64.2819 q^{81} -3.96962 q^{82} -445.110 q^{83} +467.174 q^{84} -203.450 q^{85} -2.32878 q^{86} +655.056 q^{87} -22.6510 q^{88} +220.783 q^{89} +10.4251 q^{90} +527.402 q^{91} +410.843 q^{92} -1597.21 q^{93} -3.78837 q^{94} +532.890 q^{95} +32.7505 q^{96} -148.526 q^{97} -1.00191 q^{98} -2949.73 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 2 q^{2} + 11 q^{3} + 38 q^{4} - 3 q^{5} + 9 q^{6} + 63 q^{7} + 24 q^{8} + 74 q^{9} + 134 q^{10} - 8 q^{11} + 56 q^{12} + 164 q^{13} + 14 q^{14} + 34 q^{15} + 178 q^{16} + 153 q^{17} + 98 q^{18} + 244 q^{19}+ \cdots - 2396 q^{99}+O(q^{100}) \) 9 * q + 2 * q^2 + 11 * q^3 + 38 * q^4 - 3 * q^5 + 9 * q^6 + 63 * q^7 + 24 * q^8 + 74 * q^9 + 134 * q^10 - 8 * q^11 + 56 * q^12 + 164 * q^13 + 14 * q^14 + 34 * q^15 + 178 * q^16 + 153 * q^17 + 98 * q^18 + 244 * q^19 - 41 * q^20 + 77 * q^21 - 80 * q^22 - 14 * q^23 + 298 * q^24 + 684 * q^25 + 326 * q^26 + 218 * q^27 + 266 * q^28 - 234 * q^29 - 335 * q^30 + 555 * q^31 - 181 * q^32 + 458 * q^33 + 34 * q^34 - 21 * q^35 - 1221 * q^36 - 364 * q^37 - 714 * q^38 - 52 * q^39 + 123 * q^40 - 45 * q^41 + 63 * q^42 - 135 * q^43 - 748 * q^44 - 844 * q^45 - 1576 * q^46 - 172 * q^47 - 949 * q^48 + 441 * q^49 - 2901 * q^50 + 187 * q^51 - 1596 * q^52 + 101 * q^53 - 1163 * q^54 + 1260 * q^55 + 168 * q^56 - 602 * q^57 + 1062 * q^58 + 280 * q^59 - 1727 * q^60 + 639 * q^61 - 1708 * q^62 + 518 * q^63 - 2390 * q^64 + 638 * q^65 - 2476 * q^66 + 35 * q^67 + 646 * q^68 + 1288 * q^69 + 938 * q^70 - 1616 * q^71 + 1335 * q^72 + 1049 * q^73 - 370 * q^74 + 1260 * q^75 + 4964 * q^76 - 56 * q^77 - 4714 * q^78 + 2304 * q^79 - 3996 * q^80 - 791 * q^81 - 215 * q^82 + 2508 * q^83 + 392 * q^84 - 51 * q^85 + 623 * q^86 + 166 * q^87 - 416 * q^88 + 2762 * q^89 + 2935 * q^90 + 1148 * q^91 - 2392 * q^92 + 2784 * q^93 - 862 * q^94 - 3462 * q^95 + 2928 * q^96 + 3107 * q^97 + 98 * q^98 - 2396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more