LMFDB - Embedded newform 507.4.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.a (trivial)

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:	yes
Analytic conductor:	$29.9139683729$
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} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 $ x^9 - x^8 - 48x^7 + 29x^6 + 772x^5 - 150x^4 - 4745x^3 - 966x^2 + 9428*x + 5144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.23649$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23649 q^{2} -3.00000 q^{3} +2.47490 q^{4} -13.5815 q^{5} +9.70948 q^{6} +1.42933 q^{7} +17.8820 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-3.23649 q^{2} -3.00000 q^{3} +2.47490 q^{4} -13.5815 q^{5} +9.70948 q^{6} +1.42933 q^{7} +17.8820 q^{8} +9.00000 q^{9} +43.9566 q^{10} +54.5673 q^{11} -7.42469 q^{12} -4.62601 q^{14} +40.7446 q^{15} -77.6741 q^{16} -114.413 q^{17} -29.1285 q^{18} -104.933 q^{19} -33.6129 q^{20} -4.28798 q^{21} -176.607 q^{22} -64.5438 q^{23} -53.6459 q^{24} +59.4583 q^{25} -27.0000 q^{27} +3.53743 q^{28} -60.8037 q^{29} -131.870 q^{30} -148.902 q^{31} +108.336 q^{32} -163.702 q^{33} +370.296 q^{34} -19.4125 q^{35} +22.2741 q^{36} -20.9326 q^{37} +339.615 q^{38} -242.865 q^{40} +371.761 q^{41} +13.8780 q^{42} +40.2951 q^{43} +135.048 q^{44} -122.234 q^{45} +208.896 q^{46} +639.802 q^{47} +233.022 q^{48} -340.957 q^{49} -192.436 q^{50} +343.238 q^{51} +102.124 q^{53} +87.3854 q^{54} -741.108 q^{55} +25.5592 q^{56} +314.799 q^{57} +196.791 q^{58} -704.586 q^{59} +100.839 q^{60} -819.087 q^{61} +481.919 q^{62} +12.8639 q^{63} +270.764 q^{64} +529.820 q^{66} +574.918 q^{67} -283.159 q^{68} +193.631 q^{69} +62.8283 q^{70} +365.790 q^{71} +160.938 q^{72} -965.233 q^{73} +67.7482 q^{74} -178.375 q^{75} -259.698 q^{76} +77.9945 q^{77} +580.173 q^{79} +1054.93 q^{80} +81.0000 q^{81} -1203.20 q^{82} +175.372 q^{83} -10.6123 q^{84} +1553.90 q^{85} -130.415 q^{86} +182.411 q^{87} +975.770 q^{88} -20.0351 q^{89} +395.609 q^{90} -159.739 q^{92} +446.705 q^{93} -2070.72 q^{94} +1425.15 q^{95} -325.008 q^{96} +1226.72 q^{97} +1103.51 q^{98} +491.106 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9}+O(q^{10}) $ 9 * q + 8 * q^2 - 27 * q^3 + 32 * q^4 + 41 * q^5 - 24 * q^6 + q^7 + 111 * q^8 + 81 * q^9	\( 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9} + 198 q^{10} + 37 q^{11} - 96 q^{12} + 98 q^{14} - 123 q^{15} + 32 q^{16} - 134 q^{17} + 72 q^{18} - 72 q^{19} + 356 q^{20} - 3 q^{21} + 274 q^{22} + 226 q^{23} - 333 q^{24} + 612 q^{25} - 243 q^{27} + 132 q^{28} - 547 q^{29} - 594 q^{30} - 521 q^{31} + 721 q^{32} - 111 q^{33} - 100 q^{34} + 138 q^{35} + 288 q^{36} + 584 q^{37} - 416 q^{38} + 1342 q^{40} + 482 q^{41} - 294 q^{42} + 158 q^{43} + 1453 q^{44} + 369 q^{45} + 1537 q^{46} + 1500 q^{47} - 96 q^{48} + 642 q^{49} + 2777 q^{50} + 402 q^{51} + 1399 q^{53} - 216 q^{54} - 1408 q^{55} - 616 q^{56} + 216 q^{57} + 1455 q^{58} + 1541 q^{59} - 1068 q^{60} + 2092 q^{61} - 293 q^{62} + 9 q^{63} + 2481 q^{64} - 822 q^{66} + 252 q^{67} - 1579 q^{68} - 678 q^{69} + 2492 q^{70} + 2352 q^{71} + 999 q^{72} + 903 q^{73} + 1037 q^{74} - 1836 q^{75} - 485 q^{76} - 1686 q^{77} - 115 q^{79} + 5701 q^{80} + 729 q^{81} - 5147 q^{82} + 1207 q^{83} - 396 q^{84} + 4308 q^{85} + 5691 q^{86} + 1641 q^{87} - 484 q^{88} + 2336 q^{89} + 1782 q^{90} + 2087 q^{92} + 1563 q^{93} - 468 q^{94} - 222 q^{95} - 2163 q^{96} + 2155 q^{97} + 5593 q^{98} + 333 q^{99}+O(q^{100}) \) 9 * q + 8 * q^2 - 27 * q^3 + 32 * q^4 + 41 * q^5 - 24 * q^6 + q^7 + 111 * q^8 + 81 * q^9 + 198 * q^10 + 37 * q^11 - 96 * q^12 + 98 * q^14 - 123 * q^15 + 32 * q^16 - 134 * q^17 + 72 * q^18 - 72 * q^19 + 356 * q^20 - 3 * q^21 + 274 * q^22 + 226 * q^23 - 333 * q^24 + 612 * q^25 - 243 * q^27 + 132 * q^28 - 547 * q^29 - 594 * q^30 - 521 * q^31 + 721 * q^32 - 111 * q^33 - 100 * q^34 + 138 * q^35 + 288 * q^36 + 584 * q^37 - 416 * q^38 + 1342 * q^40 + 482 * q^41 - 294 * q^42 + 158 * q^43 + 1453 * q^44 + 369 * q^45 + 1537 * q^46 + 1500 * q^47 - 96 * q^48 + 642 * q^49 + 2777 * q^50 + 402 * q^51 + 1399 * q^53 - 216 * q^54 - 1408 * q^55 - 616 * q^56 + 216 * q^57 + 1455 * q^58 + 1541 * q^59 - 1068 * q^60 + 2092 * q^61 - 293 * q^62 + 9 * q^63 + 2481 * q^64 - 822 * q^66 + 252 * q^67 - 1579 * q^68 - 678 * q^69 + 2492 * q^70 + 2352 * q^71 + 999 * q^72 + 903 * q^73 + 1037 * q^74 - 1836 * q^75 - 485 * q^76 - 1686 * q^77 - 115 * q^79 + 5701 * q^80 + 729 * q^81 - 5147 * q^82 + 1207 * q^83 - 396 * q^84 + 4308 * q^85 + 5691 * q^86 + 1641 * q^87 - 484 * q^88 + 2336 * q^89 + 1782 * q^90 + 2087 * q^92 + 1563 * q^93 - 468 * q^94 - 222 * q^95 - 2163 * q^96 + 2155 * q^97 + 5593 * q^98 + 333 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−3.23649	−1.14427	−0.572137	−	0.820158i	$-0.693885\pi$
$2$	−3.23649	−1.14427	−0.572137	+	0.820158i	$0.693885\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	2.47490	0.309362
$5$	−13.5815	−1.21477	−0.607385	−	0.794408i	$-0.707781\pi$
$5$	−13.5815	−1.21477	−0.607385	+	0.794408i	$0.707781\pi$
$6$	9.70948	0.660647
$7$	1.42933	0.0771763	0.0385882	−	0.999255i	$-0.487714\pi$
$7$	1.42933	0.0771763	0.0385882	+	0.999255i	$0.487714\pi$
$8$	17.8820	0.790279
$9$	9.00000	0.333333
$10$	43.9566	1.39003
$11$	54.5673	1.49570	0.747848	−	0.663870i	$-0.231087\pi$
$11$	54.5673	1.49570	0.747848	+	0.663870i	$0.231087\pi$
$12$	−7.42469	−0.178610
$13$	0	0
$13$	0	0
$14$	−4.62601	−0.0883109
$15$	40.7446	0.701348
$16$	−77.6741	−1.21366
$17$	−114.413	−1.63230	−0.816151	−	0.577839i	$-0.803896\pi$
$17$	−114.413	−1.63230	−0.816151	+	0.577839i	$0.803896\pi$
$18$	−29.1285	−0.381425
$19$	−104.933	−1.26701	−0.633507	−	0.773737i	$-0.718385\pi$
$19$	−104.933	−1.26701	−0.633507	+	0.773737i	$0.718385\pi$
$20$	−33.6129	−0.375804
$21$	−4.28798	−0.0445578
$22$	−176.607	−1.71149
$23$	−64.5438	−0.585144	−0.292572	−	0.956244i	$-0.594511\pi$
$23$	−64.5438	−0.585144	−0.292572	+	0.956244i	$0.594511\pi$
$24$	−53.6459	−0.456268
$25$	59.4583	0.475666
$26$	0	0
$27$	−27.0000	−0.192450
$28$	3.53743	0.0238754
$29$	−60.8037	−0.389343	−0.194672	−	0.980868i	$-0.562364\pi$
$29$	−60.8037	−0.389343	−0.194672	+	0.980868i	$0.562364\pi$
$30$	−131.870	−0.802534
$31$	−148.902	−0.862694	−0.431347	−	0.902186i	$-0.641961\pi$
$31$	−148.902	−0.862694	−0.431347	+	0.902186i	$0.641961\pi$
$32$	108.336	0.598477
$33$	−163.702	−0.863541
$34$	370.296	1.86780
$35$	−19.4125	−0.0937515
$36$	22.2741	0.103121
$37$	−20.9326	−0.0930079	−0.0465040	−	0.998918i	$-0.514808\pi$
$37$	−20.9326	−0.0930079	−0.0465040	+	0.998918i	$0.514808\pi$
$38$	339.615	1.44981
$39$	0	0
$40$	−242.865	−0.960007
$41$	371.761	1.41608	0.708040	−	0.706173i	$-0.249580\pi$
$41$	371.761	1.41608	0.708040	+	0.706173i	$0.249580\pi$
$42$	13.8780	0.0509863
$43$	40.2951	0.142906	0.0714528	−	0.997444i	$-0.477236\pi$
$43$	40.2951	0.142906	0.0714528	+	0.997444i	$0.477236\pi$
$44$	135.048	0.462712
$45$	−122.234	−0.404923
$46$	208.896	0.669565
$47$	639.802	1.98563	0.992816	−	0.119652i	$-0.0381778\pi$
$47$	639.802	1.98563	0.992816	+	0.119652i	$0.0381778\pi$
$48$	233.022	0.700705
$49$	−340.957	−0.994044
$50$	−192.436	−0.544292
$51$	343.238	0.942410
$52$	0	0
$53$	102.124	0.264676	0.132338	−	0.991205i	$-0.457752\pi$
$53$	102.124	0.264676	0.132338	+	0.991205i	$0.457752\pi$
$54$	87.3854	0.220216
$55$	−741.108	−1.81693
$56$	25.5592	0.0609908
$57$	314.799	0.731511
$58$	196.791	0.445515
$59$	−704.586	−1.55473	−0.777367	−	0.629048i	$-0.783445\pi$
$59$	−704.586	−1.55473	−0.777367	+	0.629048i	$0.783445\pi$
$60$	100.839	0.216970
$61$	−819.087	−1.71924	−0.859618	−	0.510938i	$-0.829298\pi$
$61$	−819.087	−1.71924	−0.859618	+	0.510938i	$0.829298\pi$
$62$	481.919	0.987158
$63$	12.8639	0.0257254
$64$	270.764	0.528835
$65$	0	0
$66$	529.820	0.988127
$67$	574.918	1.04832	0.524159	−	0.851620i	$-0.324380\pi$
$67$	574.918	1.04832	0.524159	+	0.851620i	$0.324380\pi$
$68$	−283.159	−0.504972
$69$	193.631	0.337833
$70$	62.8283	0.107277
$71$	365.790	0.611427	0.305714	−	0.952124i	$-0.401105\pi$
$71$	365.790	0.611427	0.305714	+	0.952124i	$0.401105\pi$
$72$	160.938	0.263426
$73$	−965.233	−1.54756	−0.773780	−	0.633454i	$-0.781637\pi$
$73$	−965.233	−1.54756	−0.773780	+	0.633454i	$0.781637\pi$
$74$	67.7482	0.106427
$75$	−178.375	−0.274626
$76$	−259.698	−0.391966
$77$	77.9945	0.115432
$78$	0	0
$79$	580.173	0.826261	0.413130	−	0.910672i	$-0.364435\pi$
$79$	580.173	0.826261	0.413130	+	0.910672i	$0.364435\pi$
$80$	1054.93	1.47431
$81$	81.0000	0.111111
$82$	−1203.20	−1.62038
$83$	175.372	0.231923	0.115962	−	0.993254i	$-0.463005\pi$
$83$	175.372	0.231923	0.115962	+	0.993254i	$0.463005\pi$
$84$	−10.6123	−0.0137845
$85$	1553.90	1.98287
$86$	−130.415	−0.163523
$87$	182.411	0.224787
$88$	975.770	1.18202
$89$	−20.0351	−0.0238620	−0.0119310	−	0.999929i	$-0.503798\pi$
$89$	−20.0351	−0.0238620	−0.0119310	+	0.999929i	$0.503798\pi$
$90$	395.609	0.463343
$91$	0	0
$92$	−159.739	−0.181021
$93$	446.705	0.498076
$94$	−2070.72	−2.27211
$95$	1425.15	1.53913
$96$	−325.008	−0.345531
$97$	1226.72	1.28406	0.642031	−	0.766678i	$-0.278092\pi$
$97$	1226.72	1.28406	0.642031	+	0.766678i	$0.278092\pi$
$98$	1103.51	1.13746
$99$	491.106	0.498565
$100$	147.153	0.147153
$101$	−57.9799	−0.0571209	−0.0285605	−	0.999592i	$-0.509092\pi$
$101$	−57.9799	−0.0571209	−0.0285605	+	0.999592i	$0.509092\pi$
$102$	−1110.89	−1.07837
$103$	−954.989	−0.913571	−0.456786	−	0.889577i	$-0.650999\pi$
$103$	−954.989	−0.913571	−0.456786	+	0.889577i	$0.650999\pi$
$104$	0	0
$105$	58.2374	0.0541275
$106$	−330.524	−0.302861
$107$	1604.68	1.44982	0.724908	−	0.688846i	$-0.241882\pi$
$107$	1604.68	1.44982	0.724908	+	0.688846i	$0.241882\pi$
$108$	−66.8222	−0.0595368
$109$	−1534.93	−1.34880	−0.674401	−	0.738365i	$-0.735598\pi$
$109$	−1534.93	−1.34880	−0.674401	+	0.738365i	$0.735598\pi$
$110$	2398.59	2.07906
$111$	62.7977	0.0536982
$112$	−111.022	−0.0936656
$113$	−1789.20	−1.48951	−0.744753	−	0.667341i	$-0.767433\pi$
$113$	−1789.20	−1.48951	−0.744753	+	0.667341i	$0.767433\pi$
$114$	−1018.84	−0.837049
$115$	876.604	0.710815
$116$	−150.483	−0.120448
$117$	0	0
$118$	2280.39	1.77904
$119$	−163.533	−0.125975
$120$	728.594	0.554260
$121$	1646.59	1.23711
$122$	2650.97	1.96728
$123$	−1115.28	−0.817574
$124$	−368.516	−0.266885
$125$	890.158	0.636945
$126$	−41.6341	−0.0294370
$127$	1648.30	1.15168	0.575838	−	0.817564i	$-0.304676\pi$
$127$	1648.30	1.15168	0.575838	+	0.817564i	$0.304676\pi$
$128$	−1743.01	−1.20361
$129$	−120.885	−0.0825066
$130$	0	0
$131$	2278.88	1.51990	0.759948	−	0.649983i	$-0.225224\pi$
$131$	2278.88	1.51990	0.759948	+	0.649983i	$0.225224\pi$
$132$	−405.145	−0.267147
$133$	−149.983	−0.0977835
$134$	−1860.72	−1.19956
$135$	366.702	0.233783
$136$	−2045.92	−1.28997
$137$	−719.324	−0.448584	−0.224292	−	0.974522i	$-0.572007\pi$
$137$	−719.324	−0.448584	−0.224292	+	0.974522i	$0.572007\pi$
$138$	−626.687	−0.386573
$139$	1777.65	1.08474	0.542369	−	0.840140i	$-0.317527\pi$
$139$	1777.65	1.08474	0.542369	+	0.840140i	$0.317527\pi$
$140$	−48.0438	−0.0290032
$141$	−1919.41	−1.14641
$142$	−1183.88	−0.699640
$143$	0	0
$144$	−699.067	−0.404552
$145$	825.807	0.472963
$146$	3123.97	1.77083
$147$	1022.87	0.573911
$148$	−51.8060	−0.0287731
$149$	266.172	0.146347	0.0731734	−	0.997319i	$-0.476687\pi$
$149$	266.172	0.146347	0.0731734	+	0.997319i	$0.476687\pi$
$150$	577.309	0.314247
$151$	−1417.27	−0.763812	−0.381906	−	0.924201i	$-0.624732\pi$
$151$	−1417.27	−0.763812	−0.381906	+	0.924201i	$0.624732\pi$
$152$	−1876.41	−1.00129
$153$	−1029.71	−0.544101
$154$	−252.429	−0.132086
$155$	2022.31	1.04797
$156$	0	0
$157$	35.6073	0.0181005	0.00905023	−	0.999959i	$-0.497119\pi$
$157$	35.6073	0.0181005	0.00905023	+	0.999959i	$0.497119\pi$
$158$	−1877.73	−0.945468
$159$	−306.372	−0.152810
$160$	−1471.37	−0.727012
$161$	−92.2541	−0.0451593
$162$	−262.156	−0.127142
$163$	97.7228	0.0469585	0.0234793	−	0.999724i	$-0.492526\pi$
$163$	97.7228	0.0469585	0.0234793	+	0.999724i	$0.492526\pi$
$164$	920.069	0.438081
$165$	2223.32	1.04900
$166$	−567.592	−0.265383
$167$	2715.91	1.25846	0.629231	−	0.777218i	$-0.283370\pi$
$167$	2715.91	1.25846	0.629231	+	0.777218i	$0.283370\pi$
$168$	−76.6775	−0.0352131
$169$	0	0
$170$	−5029.19	−2.26895
$171$	−944.397	−0.422338
$172$	99.7262	0.0442096
$173$	2074.88	0.911852	0.455926	−	0.890018i	$-0.349308\pi$
$173$	2074.88	0.911852	0.455926	+	0.890018i	$0.349308\pi$
$174$	−590.372	−0.257218
$175$	84.9853	0.0367102
$176$	−4238.46	−1.81526
$177$	2113.76	0.897626
$178$	64.8435	0.0273046
$179$	1023.50	0.427372	0.213686	−	0.976902i	$-0.431453\pi$
$179$	1023.50	0.427372	0.213686	+	0.976902i	$0.431453\pi$
$180$	−302.516	−0.125268
$181$	1773.72	0.728395	0.364197	−	0.931322i	$-0.381343\pi$
$181$	1773.72	0.728395	0.364197	+	0.931322i	$0.381343\pi$
$182$	0	0
$183$	2457.26	0.992601
$184$	−1154.17	−0.462427
$185$	284.297	0.112983
$186$	−1445.76	−0.569936
$187$	−6243.18	−2.44143
$188$	1583.44	0.614279
$189$	−38.5918	−0.0148526
$190$	−4612.49	−1.76119
$191$	2251.37	0.852896	0.426448	−	0.904512i	$-0.359765\pi$
$191$	2251.37	0.852896	0.426448	+	0.904512i	$0.359765\pi$
$192$	−812.291	−0.305323
$193$	3876.48	1.44578	0.722889	−	0.690964i	$-0.242814\pi$
$193$	3876.48	1.44578	0.722889	+	0.690964i	$0.242814\pi$
$194$	−3970.26	−1.46932
$195$	0	0
$196$	−843.834	−0.307520
$197$	−756.708	−0.273671	−0.136835	−	0.990594i	$-0.543693\pi$
$197$	−756.708	−0.273671	−0.136835	+	0.990594i	$0.543693\pi$
$198$	−1589.46	−0.570495
$199$	−1986.32	−0.707572	−0.353786	−	0.935326i	$-0.615106\pi$
$199$	−1986.32	−0.707572	−0.353786	+	0.935326i	$0.615106\pi$
$200$	1063.23	0.375909
$201$	−1724.75	−0.605247
$202$	187.652	0.0653620
$203$	−86.9082	−0.0300481
$204$	849.478	0.291546
$205$	−5049.08	−1.72021
$206$	3090.82	1.04538
$207$	−580.894	−0.195048
$208$	0	0
$209$	−5725.91	−1.89507
$210$	−188.485	−0.0619366
$211$	−390.462	−0.127396	−0.0636980	−	0.997969i	$-0.520289\pi$
$211$	−390.462	−0.127396	−0.0636980	+	0.997969i	$0.520289\pi$
$212$	252.746	0.0818806
$213$	−1097.37	−0.353008
$214$	−5193.54	−1.65899
$215$	−547.269	−0.173597
$216$	−482.813	−0.152089
$217$	−212.829	−0.0665795
$218$	4967.79	1.54340
$219$	2895.70	0.893485
$220$	−1834.17	−0.562088
$221$	0	0
$222$	−203.244	−0.0614454
$223$	−1814.65	−0.544922	−0.272461	−	0.962167i	$-0.587838\pi$
$223$	−1814.65	−0.544922	−0.272461	+	0.962167i	$0.587838\pi$
$224$	154.847	0.0461883
$225$	535.124	0.158555
$226$	5790.75	1.70440
$227$	−1046.66	−0.306032	−0.153016	−	0.988224i	$-0.548899\pi$
$227$	−1046.66	−0.306032	−0.153016	+	0.988224i	$0.548899\pi$
$228$	779.095	0.226302
$229$	6018.10	1.73662	0.868312	−	0.496018i	$-0.165205\pi$
$229$	6018.10	1.73662	0.868312	+	0.496018i	$0.165205\pi$
$230$	−2837.12	−0.813367
$231$	−233.983	−0.0666449
$232$	−1087.29	−0.307690
$233$	−4312.29	−1.21248	−0.606239	−	0.795282i	$-0.707323\pi$
$233$	−4312.29	−1.21248	−0.606239	+	0.795282i	$0.707323\pi$
$234$	0	0
$235$	−8689.50	−2.41209
$236$	−1743.78	−0.480976
$237$	−1740.52	−0.477042
$238$	529.273	0.144150
$239$	3341.91	0.904478	0.452239	−	0.891897i	$-0.350625\pi$
$239$	3341.91	0.904478	0.452239	+	0.891897i	$0.350625\pi$
$240$	−3164.80	−0.851196
$241$	1981.50	0.529625	0.264812	−	0.964300i	$-0.414690\pi$
$241$	1981.50	0.529625	0.264812	+	0.964300i	$0.414690\pi$
$242$	−5329.18	−1.41559
$243$	−243.000	−0.0641500
$244$	−2027.16	−0.531866
$245$	4630.72	1.20753
$246$	3609.60	0.935528
$247$	0	0
$248$	−2662.65	−0.681768
$249$	−526.117	−0.133901
$250$	−2880.99	−0.728839
$251$	484.149	0.121750	0.0608749	−	0.998145i	$-0.480611\pi$
$251$	484.149	0.121750	0.0608749	+	0.998145i	$0.480611\pi$
$252$	31.8369	0.00795848
$253$	−3521.98	−0.875197
$254$	−5334.71	−1.31783
$255$	−4661.70	−1.14481
$256$	3475.14	0.848423
$257$	1964.83	0.476897	0.238448	−	0.971155i	$-0.423361\pi$
$257$	1964.83	0.476897	0.238448	+	0.971155i	$0.423361\pi$
$258$	391.244	0.0944101
$259$	−29.9195	−0.00717801
$260$	0	0
$261$	−547.233	−0.129781
$262$	−7375.58	−1.73918
$263$	4886.27	1.14563	0.572814	−	0.819685i	$-0.305852\pi$
$263$	4886.27	1.14563	0.572814	+	0.819685i	$0.305852\pi$
$264$	−2927.31	−0.682438
$265$	−1387.00	−0.321520
$266$	485.420	0.111891
$267$	60.1053	0.0137767
$268$	1422.86	0.324310
$269$	−2379.98	−0.539443	−0.269722	−	0.962938i	$-0.586932\pi$
$269$	−2379.98	−0.539443	−0.269722	+	0.962938i	$0.586932\pi$
$270$	−1186.83	−0.267511
$271$	−5468.73	−1.22584	−0.612919	−	0.790146i	$-0.710005\pi$
$271$	−5468.73	−1.22584	−0.612919	+	0.790146i	$0.710005\pi$
$272$	8886.89	1.98105
$273$	0	0
$274$	2328.09	0.513303
$275$	3244.48	0.711452
$276$	479.218	0.104513
$277$	5298.94	1.14940	0.574698	−	0.818366i	$-0.305120\pi$
$277$	5298.94	1.14940	0.574698	+	0.818366i	$0.305120\pi$
$278$	−5753.37	−1.24124
$279$	−1340.11	−0.287565
$280$	−347.133	−0.0740898
$281$	6632.52	1.40805	0.704027	−	0.710173i	$-0.251383\pi$
$281$	6632.52	1.40805	0.704027	+	0.710173i	$0.251383\pi$
$282$	6212.15	1.31180
$283$	−3693.82	−0.775882	−0.387941	−	0.921684i	$-0.626814\pi$
$283$	−3693.82	−0.775882	−0.387941	+	0.921684i	$0.626814\pi$
$284$	905.294	0.189152
$285$	−4275.45	−0.888618
$286$	0	0
$287$	531.367	0.109288
$288$	975.024	0.199492
$289$	8177.24	1.66441
$290$	−2672.72	−0.541199
$291$	−3680.15	−0.741354
$292$	−2388.85	−0.478757
$293$	3051.86	0.608503	0.304252	−	0.952592i	$-0.401594\pi$
$293$	3051.86	0.608503	0.304252	+	0.952592i	$0.401594\pi$
$294$	−3310.52	−0.656712
$295$	9569.36	1.88864
$296$	−374.316	−0.0735022
$297$	−1473.32	−0.287847
$298$	−861.465	−0.167461
$299$	0	0
$300$	−441.459	−0.0849589
$301$	57.5948	0.0110289
$302$	4586.98	0.874010
$303$	173.940	0.0329788
$304$	8150.57	1.53772
$305$	11124.5	2.08848
$306$	3332.66	0.622600
$307$	4737.05	0.880643	0.440322	−	0.897840i	$-0.354864\pi$
$307$	4737.05	0.880643	0.440322	+	0.897840i	$0.354864\pi$
$308$	193.028	0.0357104
$309$	2864.97	0.527451
$310$	−6545.20	−1.19917
$311$	−10095.5	−1.84072	−0.920359	−	0.391076i	$-0.872103\pi$
$311$	−10095.5	−1.84072	−0.920359	+	0.391076i	$0.872103\pi$
$312$	0	0
$313$	5235.36	0.945432	0.472716	−	0.881215i	$-0.343274\pi$
$313$	5235.36	0.945432	0.472716	+	0.881215i	$0.343274\pi$
$314$	−115.243	−0.0207119
$315$	−174.712	−0.0312505
$316$	1435.87	0.255614
$317$	6701.12	1.18730	0.593648	−	0.804725i	$-0.297687\pi$
$317$	6701.12	1.18730	0.593648	+	0.804725i	$0.297687\pi$
$318$	991.571	0.174857
$319$	−3317.89	−0.582339
$320$	−3677.39	−0.642413
$321$	−4814.04	−0.837052
$322$	298.580	0.0516746
$323$	12005.7	2.06815
$324$	200.467	0.0343736
$325$	0	0
$326$	−316.279	−0.0537334
$327$	4604.78	0.778731
$328$	6647.81	1.11910
$329$	914.485	0.153244
$330$	−7195.78	−1.20035
$331$	764.044	0.126875	0.0634376	−	0.997986i	$-0.479794\pi$
$331$	764.044	0.126875	0.0634376	+	0.997986i	$0.479794\pi$
$332$	434.029	0.0717482
$333$	−188.393	−0.0310026
$334$	−8790.02	−1.44002
$335$	−7808.27	−1.27347
$336$	333.065	0.0540779
$337$	3310.33	0.535089	0.267545	−	0.963545i	$-0.413788\pi$
$337$	3310.33	0.535089	0.267545	+	0.963545i	$0.413788\pi$
$338$	0	0
$339$	5367.61	0.859966
$340$	3845.74	0.613425
$341$	−8125.15	−1.29033
$342$	3056.53	0.483270
$343$	−977.598	−0.153893
$344$	720.555	0.112935
$345$	−2629.81	−0.410389
$346$	−6715.35	−1.04341
$347$	869.809	0.134564	0.0672821	−	0.997734i	$-0.478567\pi$
$347$	869.809	0.134564	0.0672821	+	0.997734i	$0.478567\pi$
$348$	451.448	0.0695407
$349$	−10694.9	−1.64036	−0.820181	−	0.572104i	$-0.806127\pi$
$349$	−10694.9	−1.64036	−0.820181	+	0.572104i	$0.806127\pi$
$350$	−275.054	−0.0420065
$351$	0	0
$352$	5911.60	0.895140
$353$	−1514.92	−0.228417	−0.114208	−	0.993457i	$-0.536433\pi$
$353$	−1514.92	−0.228417	−0.114208	+	0.993457i	$0.536433\pi$
$354$	−6841.16	−1.02713
$355$	−4968.00	−0.742744
$356$	−49.5848	−0.00738200
$357$	490.599	0.0727317
$358$	−3312.54	−0.489031
$359$	−7006.81	−1.03010	−0.515049	−	0.857160i	$-0.672226\pi$
$359$	−7006.81	−1.03010	−0.515049	+	0.857160i	$0.672226\pi$
$360$	−2185.78	−0.320002
$361$	4151.92	0.605325
$362$	−5740.63	−0.833483
$363$	−4939.77	−0.714244
$364$	0	0
$365$	13109.3	1.87993
$366$	−7952.91	−1.13581
$367$	−572.331	−0.0814045	−0.0407022	−	0.999171i	$-0.512960\pi$
$367$	−572.331	−0.0814045	−0.0407022	+	0.999171i	$0.512960\pi$
$368$	5013.38	0.710164
$369$	3345.85	0.472027
$370$	−920.125	−0.129284
$371$	145.968	0.0204267
$372$	1105.55	0.154086
$373$	2405.41	0.333907	0.166954	−	0.985965i	$-0.446607\pi$
$373$	2405.41	0.333907	0.166954	+	0.985965i	$0.446607\pi$
$374$	20206.0	2.79366
$375$	−2670.47	−0.367740
$376$	11440.9	1.56920
$377$	0	0
$378$	124.902	0.0169954
$379$	3161.53	0.428488	0.214244	−	0.976780i	$-0.431271\pi$
$379$	3161.53	0.428488	0.214244	+	0.976780i	$0.431271\pi$
$380$	3527.10	0.476149
$381$	−4944.89	−0.664920
$382$	−7286.53	−0.975946
$383$	7568.71	1.00977	0.504887	−	0.863186i	$-0.331534\pi$
$383$	7568.71	1.00977	0.504887	+	0.863186i	$0.331534\pi$
$384$	5229.04	0.694904
$385$	−1059.28	−0.140224
$386$	−12546.2	−1.65437
$387$	362.656	0.0476352
$388$	3036.00	0.397240
$389$	−6757.17	−0.880726	−0.440363	−	0.897820i	$-0.645150\pi$
$389$	−6757.17	−0.880726	−0.440363	+	0.897820i	$0.645150\pi$
$390$	0	0
$391$	7384.62	0.955131
$392$	−6096.98	−0.785572
$393$	−6836.63	−0.877513
$394$	2449.08	0.313154
$395$	−7879.65	−1.00372
$396$	1215.44	0.154237
$397$	7313.92	0.924623	0.462312	−	0.886718i	$-0.347020\pi$
$397$	7313.92	0.924623	0.462312	+	0.886718i	$0.347020\pi$
$398$	6428.73	0.809656
$399$	449.950	0.0564553
$400$	−4618.37	−0.577296
$401$	−5349.32	−0.666165	−0.333082	−	0.942898i	$-0.608089\pi$
$401$	−5349.32	−0.666165	−0.333082	+	0.942898i	$0.608089\pi$
$402$	5582.15	0.692568
$403$	0	0
$404$	−143.494	−0.0176711
$405$	−1100.10	−0.134974
$406$	281.278	0.0343832
$407$	−1142.23	−0.139112
$408$	6137.77	0.744766
$409$	13444.2	1.62537	0.812683	−	0.582706i	$-0.198006\pi$
$409$	13444.2	1.62537	0.812683	+	0.582706i	$0.198006\pi$
$410$	16341.3	1.96839
$411$	2157.97	0.258990
$412$	−2363.50	−0.282624
$413$	−1007.08	−0.119989
$414$	1880.06	0.223188
$415$	−2381.83	−0.281733
$416$	0	0
$417$	−5332.96	−0.626274
$418$	18531.9	2.16848
$419$	−8313.65	−0.969328	−0.484664	−	0.874700i	$-0.661058\pi$
$419$	−8313.65	−0.969328	−0.484664	+	0.874700i	$0.661058\pi$
$420$	144.131	0.0167450
$421$	3491.02	0.404138	0.202069	−	0.979371i	$-0.435234\pi$
$421$	3491.02	0.404138	0.202069	+	0.979371i	$0.435234\pi$
$422$	1263.73	0.145776
$423$	5758.22	0.661877
$424$	1826.18	0.209167
$425$	−6802.77	−0.776431
$426$	3551.64	0.403937
$427$	−1170.74	−0.132684
$428$	3971.42	0.448518
$429$	0	0
$430$	1771.23	0.198643
$431$	1269.80	0.141912	0.0709560	−	0.997479i	$-0.477395\pi$
$431$	1269.80	0.141912	0.0709560	+	0.997479i	$0.477395\pi$
$432$	2097.20	0.233568
$433$	6157.71	0.683419	0.341710	−	0.939806i	$-0.388994\pi$
$433$	6157.71	0.683419	0.341710	+	0.939806i	$0.388994\pi$
$434$	688.819	0.0761852
$435$	−2477.42	−0.273065
$436$	−3798.79	−0.417268
$437$	6772.77	0.741386
$438$	−9371.91	−1.02239
$439$	−2466.57	−0.268162	−0.134081	−	0.990970i	$-0.542808\pi$
$439$	−2466.57	−0.268162	−0.134081	+	0.990970i	$0.542808\pi$
$440$	−13252.5	−1.43588
$441$	−3068.61	−0.331348
$442$	0	0
$443$	−1858.98	−0.199374	−0.0996871	−	0.995019i	$-0.531784\pi$
$443$	−1858.98	−0.199374	−0.0996871	+	0.995019i	$0.531784\pi$
$444$	155.418	0.0166122
$445$	272.108	0.0289868
$446$	5873.09	0.623540
$447$	−798.516	−0.0844934
$448$	387.010	0.0408136
$449$	−5176.22	−0.544055	−0.272028	−	0.962289i	$-0.587694\pi$
$449$	−5176.22	−0.544055	−0.272028	+	0.962289i	$0.587694\pi$
$450$	−1731.93	−0.181431
$451$	20286.0	2.11802
$452$	−4428.09	−0.460797
$453$	4251.80	0.440987
$454$	3387.51	0.350184
$455$	0	0
$456$	5629.22	0.578098
$457$	3327.37	0.340586	0.170293	−	0.985393i	$-0.445529\pi$
$457$	3327.37	0.340586	0.170293	+	0.985393i	$0.445529\pi$
$458$	−19477.5	−1.98717
$459$	3089.14	0.314137
$460$	2169.50	0.219899
$461$	−8992.19	−0.908477	−0.454238	−	0.890880i	$-0.650089\pi$
$461$	−8992.19	−0.908477	−0.454238	+	0.890880i	$0.650089\pi$
$462$	757.286	0.0762600
$463$	13143.4	1.31928	0.659640	−	0.751582i	$-0.270709\pi$
$463$	13143.4	1.31928	0.659640	+	0.751582i	$0.270709\pi$
$464$	4722.87	0.472529
$465$	−6066.94	−0.605048
$466$	13956.7	1.38741
$467$	7798.23	0.772718	0.386359	−	0.922349i	$-0.373733\pi$
$467$	7798.23	0.772718	0.386359	+	0.922349i	$0.373733\pi$
$468$	0	0
$469$	821.745	0.0809054
$470$	28123.5	2.76009
$471$	−106.822	−0.0104503
$472$	−12599.4	−1.22867
$473$	2198.79	0.213743
$474$	5633.18	0.545866
$475$	−6239.13	−0.602676
$476$	−404.727	−0.0389719
$477$	919.115	0.0882252
$478$	−10816.1	−1.03497
$479$	−7341.94	−0.700338	−0.350169	−	0.936687i	$-0.613876\pi$
$479$	−7341.94	−0.700338	−0.350169	+	0.936687i	$0.613876\pi$
$480$	4414.11	0.419741
$481$	0	0
$482$	−6413.11	−0.606036
$483$	276.762	0.0260727
$484$	4075.14	0.382714
$485$	−16660.7	−1.55984
$486$	786.468	0.0734052
$487$	2684.77	0.249812	0.124906	−	0.992169i	$-0.460137\pi$
$487$	2684.77	0.249812	0.124906	+	0.992169i	$0.460137\pi$
$488$	−14646.9	−1.35867
$489$	−293.168	−0.0271115
$490$	−14987.3	−1.38175
$491$	−9212.74	−0.846772	−0.423386	−	0.905949i	$-0.639159\pi$
$491$	−9212.74	−0.846772	−0.423386	+	0.905949i	$0.639159\pi$
$492$	−2760.21	−0.252926
$493$	6956.70	0.635526
$494$	0	0
$495$	−6669.97	−0.605642
$496$	11565.8	1.04701
$497$	522.834	0.0471877
$498$	1702.77	0.153219
$499$	11401.9	1.02289	0.511443	−	0.859317i	$-0.329111\pi$
$499$	11401.9	1.02289	0.511443	+	0.859317i	$0.329111\pi$
$500$	2203.05	0.197047
$501$	−8147.72	−0.726573
$502$	−1566.94	−0.139315
$503$	1603.20	0.142114	0.0710570	−	0.997472i	$-0.477363\pi$
$503$	1603.20	0.142114	0.0710570	+	0.997472i	$0.477363\pi$
$504$	230.032	0.0203303
$505$	787.456	0.0693888
$506$	11398.9	1.00147
$507$	0	0
$508$	4079.37	0.356285
$509$	3427.55	0.298475	0.149237	−	0.988801i	$-0.452318\pi$
$509$	3427.55	0.298475	0.149237	+	0.988801i	$0.452318\pi$
$510$	15087.6	1.30998
$511$	−1379.63	−0.119435
$512$	2696.83	0.232781
$513$	2833.19	0.243837
$514$	−6359.15	−0.545701
$515$	12970.2	1.10978
$516$	−299.178	−0.0255244
$517$	34912.3	2.96990
$518$	96.8342	0.00821361
$519$	−6224.65	−0.526458
$520$	0	0
$521$	−5244.77	−0.441032	−0.220516	−	0.975383i	$-0.570774\pi$
$521$	−5244.77	−0.441032	−0.220516	+	0.975383i	$0.570774\pi$
$522$	1771.12	0.148505
$523$	−15817.1	−1.32243	−0.661216	−	0.750196i	$-0.729959\pi$
$523$	−15817.1	−1.32243	−0.661216	+	0.750196i	$0.729959\pi$
$524$	5639.99	0.470198
$525$	−254.956	−0.0211946
$526$	−15814.4	−1.31091
$527$	17036.2	1.40818
$528$	12715.4	1.04804
$529$	−8001.10	−0.657607
$530$	4489.02	0.367907
$531$	−6341.27	−0.518244
$532$	−371.193	−0.0302505
$533$	0	0
$534$	−194.531	−0.0157643
$535$	−21794.0	−1.76119
$536$	10280.7	0.828464
$537$	−3070.49	−0.246743
$538$	7702.81	0.617271
$539$	−18605.1	−1.48679
$540$	907.549	0.0723235
$541$	5694.69	0.452558	0.226279	−	0.974063i	$-0.427344\pi$
$541$	5694.69	0.452558	0.226279	+	0.974063i	$0.427344\pi$
$542$	17699.5	1.40269
$543$	−5321.15	−0.420539
$544$	−12395.0	−0.976895
$545$	20846.7	1.63848
$546$	0	0
$547$	−20598.9	−1.61014	−0.805070	−	0.593180i	$-0.797872\pi$
$547$	−20598.9	−1.61014	−0.805070	+	0.593180i	$0.797872\pi$
$548$	−1780.25	−0.138775
$549$	−7371.78	−0.573078
$550$	−10500.7	−0.814096
$551$	6380.31	0.493304
$552$	3462.51	0.266982
$553$	829.257	0.0637678
$554$	−17150.0	−1.31522
$555$	−852.890	−0.0652309
$556$	4399.51	0.335577
$557$	18421.7	1.40135	0.700674	−	0.713482i	$-0.252883\pi$
$557$	18421.7	1.40135	0.700674	+	0.713482i	$0.252883\pi$
$558$	4337.27	0.329053
$559$	0	0
$560$	1507.84	0.113782
$561$	18729.6	1.40956
$562$	−21466.1	−1.61120
$563$	−6516.79	−0.487833	−0.243916	−	0.969796i	$-0.578432\pi$
$563$	−6516.79	−0.487833	−0.243916	+	0.969796i	$0.578432\pi$
$564$	−4750.33	−0.354654
$565$	24300.1	1.80941
$566$	11955.0	0.887822
$567$	115.775	0.00857515
$568$	6541.05	0.483198
$569$	−24233.0	−1.78541	−0.892706	−	0.450640i	$-0.851196\pi$
$569$	−24233.0	−1.78541	−0.892706	+	0.450640i	$0.851196\pi$
$570$	13837.5	1.01682
$571$	11060.7	0.810640	0.405320	−	0.914175i	$-0.367160\pi$
$571$	11060.7	0.810640	0.405320	+	0.914175i	$0.367160\pi$
$572$	0	0
$573$	−6754.10	−0.492420
$574$	−1719.77	−0.125055
$575$	−3837.66	−0.278333
$576$	2436.87	0.176278
$577$	−19118.6	−1.37941	−0.689703	−	0.724092i	$-0.742259\pi$
$577$	−19118.6	−1.37941	−0.689703	+	0.724092i	$0.742259\pi$
$578$	−26465.6	−1.90454
$579$	−11629.4	−0.834720
$580$	2043.79	0.146317
$581$	250.664	0.0178990
$582$	11910.8	0.848312
$583$	5572.63	0.395874
$584$	−17260.3	−1.22300
$585$	0	0
$586$	−9877.32	−0.696294
$587$	−21636.2	−1.52133	−0.760667	−	0.649142i	$-0.775128\pi$
$587$	−21636.2	−1.52133	−0.760667	+	0.649142i	$0.775128\pi$
$588$	2531.50	0.177546
$589$	15624.7	1.09305
$590$	−30971.2	−2.16112
$591$	2270.12	0.158004
$592$	1625.92	0.112880
$593$	2148.58	0.148789	0.0743943	−	0.997229i	$-0.476298\pi$
$593$	2148.58	0.148789	0.0743943	+	0.997229i	$0.476298\pi$
$594$	4768.38	0.329376
$595$	2221.03	0.153031
$596$	658.749	0.0452742
$597$	5958.97	0.408517
$598$	0	0
$599$	27957.0	1.90700	0.953499	−	0.301397i	$-0.0974529\pi$
$599$	27957.0	1.90700	0.953499	+	0.301397i	$0.0974529\pi$
$600$	−3189.69	−0.217031
$601$	−13217.8	−0.897111	−0.448555	−	0.893755i	$-0.648061\pi$
$601$	−13217.8	−0.897111	−0.448555	+	0.893755i	$0.648061\pi$
$602$	−186.405	−0.0126201
$603$	5174.26	0.349440
$604$	−3507.59	−0.236295
$605$	−22363.2	−1.50280
$606$	−562.955	−0.0377368
$607$	−23642.4	−1.58091	−0.790457	−	0.612518i	$-0.790157\pi$
$607$	−23642.4	−1.58091	−0.790457	+	0.612518i	$0.790157\pi$
$608$	−11368.0	−0.758279
$609$	260.725	0.0173483
$610$	−36004.3	−2.38979
$611$	0	0
$612$	−2548.43	−0.168324
$613$	−8117.53	−0.534851	−0.267426	−	0.963578i	$-0.586173\pi$
$613$	−8117.53	−0.534851	−0.267426	+	0.963578i	$0.586173\pi$
$614$	−15331.4	−1.00770
$615$	15147.2	0.993164
$616$	1394.69	0.0912237
$617$	20928.0	1.36553	0.682764	−	0.730639i	$-0.260778\pi$
$617$	20928.0	1.36553	0.682764	+	0.730639i	$0.260778\pi$
$618$	−9272.45	−0.603548
$619$	−3529.43	−0.229175	−0.114588	−	0.993413i	$-0.536555\pi$
$619$	−3529.43	−0.229175	−0.114588	+	0.993413i	$0.536555\pi$
$620$	5005.01	0.324204
$621$	1742.68	0.112611
$622$	32674.0	2.10628
$623$	−28.6367	−0.00184158
$624$	0	0
$625$	−19522.0	−1.24941
$626$	−16944.2	−1.08183
$627$	17177.7	1.09412
$628$	88.1244	0.00559959
$629$	2394.95	0.151817
$630$	565.455	0.0357591
$631$	10466.2	0.660307	0.330154	−	0.943927i	$-0.392899\pi$
$631$	10466.2	0.660307	0.330154	+	0.943927i	$0.392899\pi$
$632$	10374.6	0.652976
$633$	1171.39	0.0735521
$634$	−21688.1	−1.35859
$635$	−22386.4	−1.39902
$636$	−758.239	−0.0472738
$637$	0	0
$638$	10738.3	0.666356
$639$	3292.11	0.203809
$640$	23672.8	1.46211
$641$	−26594.1	−1.63870	−0.819349	−	0.573295i	$-0.805665\pi$
$641$	−26594.1	−1.63870	−0.819349	+	0.573295i	$0.805665\pi$
$642$	15580.6	0.957816
$643$	−6965.66	−0.427214	−0.213607	−	0.976920i	$-0.568521\pi$
$643$	−6965.66	−0.427214	−0.213607	+	0.976920i	$0.568521\pi$
$644$	−228.319	−0.0139706
$645$	1641.81	0.100227
$646$	−38856.2	−2.36653
$647$	3956.40	0.240405	0.120203	−	0.992749i	$-0.461646\pi$
$647$	3956.40	0.240405	0.120203	+	0.992749i	$0.461646\pi$
$648$	1448.44	0.0878087
$649$	−38447.3	−2.32541
$650$	0	0
$651$	638.486	0.0384397
$652$	241.854	0.0145272
$653$	24311.1	1.45691	0.728457	−	0.685091i	$-0.240238\pi$
$653$	24311.1	1.45691	0.728457	+	0.685091i	$0.240238\pi$
$654$	−14903.4	−0.891082
$655$	−30950.7	−1.84632
$656$	−28876.2	−1.71864
$657$	−8687.09	−0.515854
$658$	−2959.73	−0.175353
$659$	8891.33	0.525580	0.262790	−	0.964853i	$-0.415357\pi$
$659$	8891.33	0.525580	0.262790	+	0.964853i	$0.415357\pi$
$660$	5502.50	0.324522
$661$	994.333	0.0585099	0.0292550	−	0.999572i	$-0.490687\pi$
$661$	994.333	0.0585099	0.0292550	+	0.999572i	$0.490687\pi$
$662$	−2472.82	−0.145180
$663$	0	0
$664$	3136.00	0.183284
$665$	2037.01	0.118785
$666$	609.733	0.0354755
$667$	3924.50	0.227822
$668$	6721.59	0.389321
$669$	5443.94	0.314611
$670$	25271.4	1.45719
$671$	−44695.4	−2.57145
$672$	−464.542	−0.0266668
$673$	5228.19	0.299453	0.149726	−	0.988727i	$-0.452161\pi$
$673$	5228.19	0.299453	0.149726	+	0.988727i	$0.452161\pi$
$674$	−10713.9	−0.612288
$675$	−1605.37	−0.0915420
$676$	0	0
$677$	−10208.1	−0.579512	−0.289756	−	0.957101i	$-0.593574\pi$
$677$	−10208.1	−0.579512	−0.289756	+	0.957101i	$0.593574\pi$
$678$	−17372.2	−0.984037
$679$	1753.38	0.0990993
$680$	27786.8	1.56702
$681$	3139.98	0.176688
$682$	26297.0	1.47649
$683$	15514.5	0.869172	0.434586	−	0.900630i	$-0.356895\pi$
$683$	15514.5	0.869172	0.434586	+	0.900630i	$0.356895\pi$
$684$	−2337.28	−0.130655
$685$	9769.53	0.544927
$686$	3163.99	0.176096
$687$	−18054.3	−1.00264
$688$	−3129.88	−0.173438
$689$	0	0
$690$	8511.37	0.469598
$691$	−17826.4	−0.981403	−0.490701	−	0.871328i	$-0.663259\pi$
$691$	−17826.4	−0.981403	−0.490701	+	0.871328i	$0.663259\pi$
$692$	5135.12	0.282092
$693$	701.950	0.0384775
$694$	−2815.13	−0.153978
$695$	−24143.3	−1.31771
$696$	3261.87	0.177645
$697$	−42534.1	−2.31147
$698$	34614.1	1.87702
$699$	12936.9	0.700025
$700$	210.330	0.0113567
$701$	9266.46	0.499271	0.249636	−	0.968340i	$-0.419689\pi$
$701$	9266.46	0.499271	0.249636	+	0.968340i	$0.419689\pi$
$702$	0	0
$703$	2196.52	0.117842
$704$	14774.8	0.790977
$705$	26068.5	1.39262
$706$	4903.04	0.261371
$707$	−82.8722	−0.00440839
$708$	5231.33	0.277691
$709$	−5844.49	−0.309583	−0.154792	−	0.987947i	$-0.549471\pi$
$709$	−5844.49	−0.309583	−0.154792	+	0.987947i	$0.549471\pi$
$710$	16078.9	0.849902
$711$	5221.56	0.275420
$712$	−358.267	−0.0188576
$713$	9610.67	0.504800
$714$	−1587.82	−0.0832250
$715$	0	0
$716$	2533.04	0.132213
$717$	−10025.7	−0.522200
$718$	22677.5	1.17872
$719$	11776.3	0.610825	0.305412	−	0.952220i	$-0.401206\pi$
$719$	11776.3	0.610825	0.305412	+	0.952220i	$0.401206\pi$
$720$	9494.40	0.491438
$721$	−1364.99	−0.0705061
$722$	−13437.7	−0.692658
$723$	−5944.50	−0.305779
$724$	4389.77	0.225338
$725$	−3615.28	−0.185197
$726$	15987.5	0.817291
$727$	26770.6	1.36570	0.682851	−	0.730558i	$-0.260740\pi$
$727$	26770.6	1.36570	0.682851	+	0.730558i	$0.260740\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−42428.3	−2.15115
$731$	−4610.26	−0.233265
$732$	6081.47	0.307073
$733$	21653.8	1.09113	0.545567	−	0.838067i	$-0.316314\pi$
$733$	21653.8	1.09113	0.545567	+	0.838067i	$0.316314\pi$
$734$	1852.35	0.0931490
$735$	−13892.2	−0.697170
$736$	−6992.41	−0.350195
$737$	31371.7	1.56797
$738$	−10828.8	−0.540128
$739$	9986.74	0.497115	0.248558	−	0.968617i	$-0.420043\pi$
$739$	9986.74	0.497115	0.248558	+	0.968617i	$0.420043\pi$
$740$	703.605	0.0349527
$741$	0	0
$742$	−472.426	−0.0233737
$743$	−7505.34	−0.370584	−0.185292	−	0.982683i	$-0.559323\pi$
$743$	−7505.34	−0.370584	−0.185292	+	0.982683i	$0.559323\pi$
$744$	7987.96	0.393619
$745$	−3615.03	−0.177778
$746$	−7785.10	−0.382081
$747$	1578.35	0.0773077
$748$	−15451.2	−0.755285
$749$	2293.61	0.111891
$750$	8642.97	0.420796
$751$	13441.9	0.653134	0.326567	−	0.945174i	$-0.394108\pi$
$751$	13441.9	0.653134	0.326567	+	0.945174i	$0.394108\pi$
$752$	−49696.0	−2.40988
$753$	−1452.45	−0.0702922
$754$	0	0
$755$	19248.7	0.927856
$756$	−95.5107	−0.00459483
$757$	32951.1	1.58207	0.791035	−	0.611771i	$-0.209543\pi$
$757$	32951.1	1.58207	0.791035	+	0.611771i	$0.209543\pi$
$758$	−10232.3	−0.490308
$759$	10565.9	0.505295
$760$	25484.5	1.21634
$761$	9866.20	0.469973	0.234986	−	0.971999i	$-0.424495\pi$
$761$	9866.20	0.469973	0.234986	+	0.971999i	$0.424495\pi$
$762$	16004.1	0.760851
$763$	−2193.91	−0.104096
$764$	5571.90	0.263854
$765$	13985.1	0.660957
$766$	−24496.1	−1.15546
$767$	0	0
$768$	−10425.4	−0.489837
$769$	31948.6	1.49818	0.749088	−	0.662471i	$-0.230492\pi$
$769$	31948.6	1.49818	0.749088	+	0.662471i	$0.230492\pi$
$770$	3428.37	0.160454
$771$	−5894.48	−0.275337
$772$	9593.89	0.447269
$773$	−10227.7	−0.475893	−0.237946	−	0.971278i	$-0.576474\pi$
$773$	−10227.7	−0.475893	−0.237946	+	0.971278i	$0.576474\pi$
$774$	−1173.73	−0.0545077
$775$	−8853.43	−0.410354
$776$	21936.1	1.01477
$777$	89.7584	0.00414423
$778$	21869.6	1.00779
$779$	−39009.9	−1.79419
$780$	0	0
$781$	19960.2	0.914510
$782$	−23900.3	−1.09293
$783$	1641.70	0.0749292
$784$	26483.5	1.20643
$785$	−483.602	−0.0219879
$786$	22126.7	1.00411
$787$	18651.6	0.844802	0.422401	−	0.906409i	$-0.361187\pi$
$787$	18651.6	0.844802	0.422401	+	0.906409i	$0.361187\pi$
$788$	−1872.77	−0.0846634
$789$	−14658.8	−0.661429
$790$	25502.4	1.14853
$791$	−2557.35	−0.114955
$792$	8781.93	0.394006
$793$	0	0
$794$	−23671.5	−1.05802
$795$	4161.00	0.185630
$796$	−4915.95	−0.218896
$797$	32566.8	1.44740	0.723698	−	0.690117i	$-0.242441\pi$
$797$	32566.8	1.44740	0.723698	+	0.690117i	$0.242441\pi$
$798$	−1456.26	−0.0646004
$799$	−73201.4	−3.24115
$800$	6441.47	0.284675
$801$	−180.316	−0.00795400
$802$	17313.0	0.762275
$803$	−52670.1	−2.31468
$804$	−4268.59	−0.187241
$805$	1252.95	0.0548581
$806$	0	0
$807$	7139.95	0.311448
$808$	−1036.79	−0.0451415
$809$	23895.3	1.03846	0.519231	−	0.854634i	$-0.326219\pi$
$809$	23895.3	1.03846	0.519231	+	0.854634i	$0.326219\pi$
$810$	3560.48	0.154448
$811$	−219.216	−0.00949165	−0.00474582	−	0.999989i	$-0.501511\pi$
$811$	−219.216	−0.00949165	−0.00474582	+	0.999989i	$0.501511\pi$
$812$	−215.089	−0.00929574
$813$	16406.2	0.707738
$814$	3696.83	0.159182
$815$	−1327.23	−0.0570438
$816$	−26660.7	−1.14376
$817$	−4228.28	−0.181063
$818$	−43512.2	−1.85986
$819$	0	0
$820$	−12496.0	−0.532168
$821$	−7550.84	−0.320982	−0.160491	−	0.987037i	$-0.551308\pi$
$821$	−7550.84	−0.320982	−0.160491	+	0.987037i	$0.551308\pi$
$822$	−6984.27	−0.296356
$823$	−14082.8	−0.596471	−0.298235	−	0.954492i	$-0.596398\pi$
$823$	−14082.8	−0.596471	−0.298235	+	0.954492i	$0.596398\pi$
$824$	−17077.1	−0.721976
$825$	−9733.43	−0.410757
$826$	3259.42	0.137300
$827$	−5424.18	−0.228074	−0.114037	−	0.993477i	$-0.536378\pi$
$827$	−5424.18	−0.228074	−0.114037	+	0.993477i	$0.536378\pi$
$828$	−1437.65	−0.0603404
$829$	26888.3	1.12650	0.563251	−	0.826286i	$-0.309550\pi$
$829$	26888.3	1.12650	0.563251	+	0.826286i	$0.309550\pi$
$830$	7708.77	0.322380
$831$	−15896.8	−0.663604
$832$	0	0
$833$	39009.8	1.62258
$834$	17260.1	0.716629
$835$	−36886.2	−1.52874
$836$	−14171.0	−0.586262
$837$	4020.34	0.166025
$838$	26907.1	1.10918
$839$	23680.9	0.974441	0.487220	−	0.873279i	$-0.338011\pi$
$839$	23680.9	0.974441	0.487220	+	0.873279i	$0.338011\pi$
$840$	1041.40	0.0427758
$841$	−20691.9	−0.848412
$842$	−11298.7	−0.462444
$843$	−19897.6	−0.812941
$844$	−966.354	−0.0394115
$845$	0	0
$846$	−18636.4	−0.757369
$847$	2353.51	0.0954754
$848$	−7932.38	−0.321225
$849$	11081.5	0.447956
$850$	22017.1	0.888449
$851$	1351.07	0.0544230
$852$	−2715.88	−0.109207
$853$	−9653.82	−0.387504	−0.193752	−	0.981051i	$-0.562066\pi$
$853$	−9653.82	−0.387504	−0.193752	+	0.981051i	$0.562066\pi$
$854$	3789.10	0.151827
$855$	12826.4	0.513044
$856$	28694.8	1.14576
$857$	41438.8	1.65172	0.825859	−	0.563877i	$-0.190691\pi$
$857$	41438.8	1.65172	0.825859	+	0.563877i	$0.190691\pi$
$858$	0	0
$859$	−20696.9	−0.822085	−0.411042	−	0.911616i	$-0.634835\pi$
$859$	−20696.9	−0.822085	−0.411042	+	0.911616i	$0.634835\pi$
$860$	−1354.43	−0.0537045
$861$	−1594.10	−0.0630974
$862$	−4109.70	−0.162386
$863$	19513.8	0.769707	0.384854	−	0.922978i	$-0.374252\pi$
$863$	19513.8	0.769707	0.384854	+	0.922978i	$0.374252\pi$
$864$	−2925.07	−0.115177
$865$	−28180.1	−1.10769
$866$	−19929.4	−0.782018
$867$	−24531.7	−0.960947
$868$	−526.729	−0.0205972
$869$	31658.5	1.23583
$870$	8018.16	0.312461
$871$	0	0
$872$	−27447.5	−1.06593
$873$	11040.4	0.428021
$874$	−21920.0	−0.848348
$875$	1272.33	0.0491571
$876$	7166.55	0.276410
$877$	35595.5	1.37055	0.685277	−	0.728282i	$-0.259681\pi$
$877$	35595.5	1.37055	0.685277	+	0.728282i	$0.259681\pi$
$878$	7983.05	0.306851
$879$	−9155.57	−0.351319
$880$	57564.9	2.20513
$881$	4277.22	0.163568	0.0817838	−	0.996650i	$-0.473938\pi$
$881$	4277.22	0.163568	0.0817838	+	0.996650i	$0.473938\pi$
$882$	9931.55	0.379153
$883$	355.428	0.0135460	0.00677300	−	0.999977i	$-0.497844\pi$
$883$	355.428	0.0135460	0.00677300	+	0.999977i	$0.497844\pi$
$884$	0	0
$885$	−28708.1	−1.09041
$886$	6016.58	0.228139
$887$	−7133.87	−0.270047	−0.135024	−	0.990842i	$-0.543111\pi$
$887$	−7133.87	−0.270047	−0.135024	+	0.990842i	$0.543111\pi$
$888$	1122.95	0.0424365
$889$	2355.96	0.0888821
$890$	−880.675	−0.0331689
$891$	4419.95	0.166188
$892$	−4491.06	−0.168578
$893$	−67136.3	−2.51582
$894$	2584.39	0.0966835
$895$	−13900.6	−0.519159
$896$	−2491.33	−0.0928902
$897$	0	0
$898$	16752.8	0.622548
$899$	9053.76	0.335884
$900$	1324.38	0.0490510
$901$	−11684.3	−0.432030
$902$	−65655.4	−2.42360
$903$	−172.784	−0.00636756
$904$	−31994.5	−1.17712
$905$	−24089.8	−0.884832
$906$	−13760.9	−0.504610
$907$	−21719.2	−0.795122	−0.397561	−	0.917576i	$-0.630143\pi$
$907$	−21719.2	−0.795122	−0.397561	+	0.917576i	$0.630143\pi$
$908$	−2590.38	−0.0946747
$909$	−521.819	−0.0190403
$910$	0	0
$911$	−39331.5	−1.43042	−0.715209	−	0.698911i	$-0.753669\pi$
$911$	−39331.5	−1.43042	−0.715209	+	0.698911i	$0.753669\pi$
$912$	−24451.7	−0.887804
$913$	9569.59	0.346886
$914$	−10769.0	−0.389723
$915$	−33373.4	−1.20578
$916$	14894.2	0.537246
$917$	3257.26	0.117300
$918$	−9997.98	−0.359458
$919$	−33117.0	−1.18871	−0.594357	−	0.804202i	$-0.702593\pi$
$919$	−33117.0	−1.18871	−0.594357	+	0.804202i	$0.702593\pi$
$920$	15675.4	0.561742
$921$	−14211.1	−0.508440
$922$	29103.2	1.03955
$923$	0	0
$924$	−579.085	−0.0206174
$925$	−1244.61	−0.0442407
$926$	−42538.6	−1.50962
$927$	−8594.90	−0.304524
$928$	−6587.22	−0.233013
$929$	−39917.7	−1.40975	−0.704874	−	0.709332i	$-0.748997\pi$
$929$	−39917.7	−1.40975	−0.704874	+	0.709332i	$0.748997\pi$
$930$	19635.6	0.692341
$931$	35777.6	1.25947
$932$	−10672.5	−0.375095
$933$	30286.5	1.06274
$934$	−25238.9	−0.884201
$935$	84792.1	2.96577
$936$	0	0
$937$	−37219.5	−1.29766	−0.648830	−	0.760934i	$-0.724741\pi$
$937$	−37219.5	−1.29766	−0.648830	+	0.760934i	$0.724741\pi$
$938$	−2659.57	−0.0925779
$939$	−15706.1	−0.545845
$940$	−21505.6	−0.746208
$941$	53360.4	1.84856	0.924282	−	0.381710i	$-0.124665\pi$
$941$	53360.4	1.84856	0.924282	+	0.381710i	$0.124665\pi$
$942$	345.728	0.0119580
$943$	−23994.8	−0.828610
$944$	54728.0	1.88691
$945$	524.136	0.0180425
$946$	−7116.38	−0.244581
$947$	29333.8	1.00657	0.503284	−	0.864121i	$-0.332125\pi$
$947$	29333.8	1.00657	0.503284	+	0.864121i	$0.332125\pi$
$948$	−4307.61	−0.147579
$949$	0	0
$950$	20192.9	0.689626
$951$	−20103.4	−0.685485
$952$	−2924.29	−0.0995554
$953$	−18645.0	−0.633759	−0.316879	−	0.948466i	$-0.602635\pi$
$953$	−18645.0	−0.633759	−0.316879	+	0.948466i	$0.602635\pi$
$954$	−2974.71	−0.100954
$955$	−30577.0	−1.03607
$956$	8270.88	0.279811
$957$	9953.67	0.336214
$958$	23762.2	0.801378
$959$	−1028.15	−0.0346201
$960$	11032.2	0.370898
$961$	−7619.34	−0.255760
$962$	0	0
$963$	14442.1	0.483272
$964$	4904.01	0.163846
$965$	−52648.6	−1.75629
$966$	−895.740	−0.0298343
$967$	−43847.6	−1.45816	−0.729081	−	0.684427i	$-0.760052\pi$
$967$	−43847.6	−1.45816	−0.729081	+	0.684427i	$0.760052\pi$
$968$	29444.3	0.977659
$969$	−36017.0	−1.19405
$970$	53922.2	1.78489
$971$	11602.4	0.383459	0.191729	−	0.981448i	$-0.438590\pi$
$971$	11602.4	0.383459	0.191729	+	0.981448i	$0.438590\pi$
$972$	−601.400	−0.0198456
$973$	2540.85	0.0837161
$974$	−8689.24	−0.285853
$975$	0	0
$976$	63621.8	2.08656
$977$	11324.2	0.370823	0.185411	−	0.982661i	$-0.440638\pi$
$977$	11324.2	0.370823	0.185411	+	0.982661i	$0.440638\pi$
$978$	948.837	0.0310230
$979$	−1093.26	−0.0356903
$980$	11460.6	0.373565
$981$	−13814.4	−0.449601
$982$	29817.0	0.968939
$983$	47443.8	1.53939	0.769696	−	0.638410i	$-0.220408\pi$
$983$	47443.8	1.53939	0.769696	+	0.638410i	$0.220408\pi$
$984$	−19943.4	−0.646111
$985$	10277.3	0.332447
$986$	−22515.3	−0.727215
$987$	−2743.46	−0.0884754
$988$	0	0
$989$	−2600.80	−0.0836203
$990$	21587.3	0.693021
$991$	51752.6	1.65891	0.829453	−	0.558577i	$-0.188653\pi$
$991$	51752.6	1.65891	0.829453	+	0.558577i	$0.188653\pi$
$992$	−16131.4	−0.516303
$993$	−2292.13	−0.0732514
$994$	−1692.15	−0.0539957
$995$	26977.3	0.859537
$996$	−1302.09	−0.0414239
$997$	16949.8	0.538420	0.269210	−	0.963082i	$-0.413237\pi$
$997$	16949.8	0.538420	0.269210	+	0.963082i	$0.413237\pi$
$998$	−36902.3	−1.17046
$999$	565.180	0.0178994

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.q.1.2	yes	9
3.2	odd	2		1521.4.a.be.1.8		9
13.5	odd	4		507.4.b.j.337.14		18
13.8	odd	4		507.4.b.j.337.5		18
13.12	even	2		507.4.a.n.1.8	✓	9
39.38	odd	2		1521.4.a.bj.1.2		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.8	✓	9	13.12	even	2
507.4.a.q.1.2	yes	9	1.1	even	1	trivial
507.4.b.j.337.5		18	13.8	odd	4
507.4.b.j.337.14		18	13.5	odd	4
1521.4.a.be.1.8		9	3.2	odd	2
1521.4.a.bj.1.2		9	39.38	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q-3.23649 q^{2} -3.00000 q^{3} +2.47490 q^{4} -13.5815 q^{5} +9.70948 q^{6} +1.42933 q^{7} +17.8820 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-3.23649 q^{2} -3.00000 q^{3} +2.47490 q^{4} -13.5815 q^{5} +9.70948 q^{6} +1.42933 q^{7} +17.8820 q^{8} +9.00000 q^{9} +43.9566 q^{10} +54.5673 q^{11} -7.42469 q^{12} -4.62601 q^{14} +40.7446 q^{15} -77.6741 q^{16} -114.413 q^{17} -29.1285 q^{18} -104.933 q^{19} -33.6129 q^{20} -4.28798 q^{21} -176.607 q^{22} -64.5438 q^{23} -53.6459 q^{24} +59.4583 q^{25} -27.0000 q^{27} +3.53743 q^{28} -60.8037 q^{29} -131.870 q^{30} -148.902 q^{31} +108.336 q^{32} -163.702 q^{33} +370.296 q^{34} -19.4125 q^{35} +22.2741 q^{36} -20.9326 q^{37} +339.615 q^{38} -242.865 q^{40} +371.761 q^{41} +13.8780 q^{42} +40.2951 q^{43} +135.048 q^{44} -122.234 q^{45} +208.896 q^{46} +639.802 q^{47} +233.022 q^{48} -340.957 q^{49} -192.436 q^{50} +343.238 q^{51} +102.124 q^{53} +87.3854 q^{54} -741.108 q^{55} +25.5592 q^{56} +314.799 q^{57} +196.791 q^{58} -704.586 q^{59} +100.839 q^{60} -819.087 q^{61} +481.919 q^{62} +12.8639 q^{63} +270.764 q^{64} +529.820 q^{66} +574.918 q^{67} -283.159 q^{68} +193.631 q^{69} +62.8283 q^{70} +365.790 q^{71} +160.938 q^{72} -965.233 q^{73} +67.7482 q^{74} -178.375 q^{75} -259.698 q^{76} +77.9945 q^{77} +580.173 q^{79} +1054.93 q^{80} +81.0000 q^{81} -1203.20 q^{82} +175.372 q^{83} -10.6123 q^{84} +1553.90 q^{85} -130.415 q^{86} +182.411 q^{87} +975.770 q^{88} -20.0351 q^{89} +395.609 q^{90} -159.739 q^{92} +446.705 q^{93} -2070.72 q^{94} +1425.15 q^{95} -325.008 q^{96} +1226.72 q^{97} +1103.51 q^{98} +491.106 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9}+O(q^{10}) \) 9 * q + 8 * q^2 - 27 * q^3 + 32 * q^4 + 41 * q^5 - 24 * q^6 + q^7 + 111 * q^8 + 81 * q^9	\( 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9} + 198 q^{10} + 37 q^{11} - 96 q^{12} + 98 q^{14} - 123 q^{15} + 32 q^{16} - 134 q^{17} + 72 q^{18} - 72 q^{19} + 356 q^{20} - 3 q^{21} + 274 q^{22} + 226 q^{23} - 333 q^{24} + 612 q^{25} - 243 q^{27} + 132 q^{28} - 547 q^{29} - 594 q^{30} - 521 q^{31} + 721 q^{32} - 111 q^{33} - 100 q^{34} + 138 q^{35} + 288 q^{36} + 584 q^{37} - 416 q^{38} + 1342 q^{40} + 482 q^{41} - 294 q^{42} + 158 q^{43} + 1453 q^{44} + 369 q^{45} + 1537 q^{46} + 1500 q^{47} - 96 q^{48} + 642 q^{49} + 2777 q^{50} + 402 q^{51} + 1399 q^{53} - 216 q^{54} - 1408 q^{55} - 616 q^{56} + 216 q^{57} + 1455 q^{58} + 1541 q^{59} - 1068 q^{60} + 2092 q^{61} - 293 q^{62} + 9 q^{63} + 2481 q^{64} - 822 q^{66} + 252 q^{67} - 1579 q^{68} - 678 q^{69} + 2492 q^{70} + 2352 q^{71} + 999 q^{72} + 903 q^{73} + 1037 q^{74} - 1836 q^{75} - 485 q^{76} - 1686 q^{77} - 115 q^{79} + 5701 q^{80} + 729 q^{81} - 5147 q^{82} + 1207 q^{83} - 396 q^{84} + 4308 q^{85} + 5691 q^{86} + 1641 q^{87} - 484 q^{88} + 2336 q^{89} + 1782 q^{90} + 2087 q^{92} + 1563 q^{93} - 468 q^{94} - 222 q^{95} - 2163 q^{96} + 2155 q^{97} + 5593 q^{98} + 333 q^{99}+O(q^{100}) \) 9 * q + 8 * q^2 - 27 * q^3 + 32 * q^4 + 41 * q^5 - 24 * q^6 + q^7 + 111 * q^8 + 81 * q^9 + 198 * q^10 + 37 * q^11 - 96 * q^12 + 98 * q^14 - 123 * q^15 + 32 * q^16 - 134 * q^17 + 72 * q^18 - 72 * q^19 + 356 * q^20 - 3 * q^21 + 274 * q^22 + 226 * q^23 - 333 * q^24 + 612 * q^25 - 243 * q^27 + 132 * q^28 - 547 * q^29 - 594 * q^30 - 521 * q^31 + 721 * q^32 - 111 * q^33 - 100 * q^34 + 138 * q^35 + 288 * q^36 + 584 * q^37 - 416 * q^38 + 1342 * q^40 + 482 * q^41 - 294 * q^42 + 158 * q^43 + 1453 * q^44 + 369 * q^45 + 1537 * q^46 + 1500 * q^47 - 96 * q^48 + 642 * q^49 + 2777 * q^50 + 402 * q^51 + 1399 * q^53 - 216 * q^54 - 1408 * q^55 - 616 * q^56 + 216 * q^57 + 1455 * q^58 + 1541 * q^59 - 1068 * q^60 + 2092 * q^61 - 293 * q^62 + 9 * q^63 + 2481 * q^64 - 822 * q^66 + 252 * q^67 - 1579 * q^68 - 678 * q^69 + 2492 * q^70 + 2352 * q^71 + 999 * q^72 + 903 * q^73 + 1037 * q^74 - 1836 * q^75 - 485 * q^76 - 1686 * q^77 - 115 * q^79 + 5701 * q^80 + 729 * q^81 - 5147 * q^82 + 1207 * q^83 - 396 * q^84 + 4308 * q^85 + 5691 * q^86 + 1641 * q^87 - 484 * q^88 + 2336 * q^89 + 1782 * q^90 + 2087 * q^92 + 1563 * q^93 - 468 * q^94 - 222 * q^95 - 2163 * q^96 + 2155 * q^97 + 5593 * q^98 + 333 * q^99

Introduction

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