LMFDB - Embedded newform 338.4.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	338.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:	$19.9426455819$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.66487 q^{3} +4.00000 q^{4} -8.53079 q^{5} +7.32975 q^{6} +4.20105 q^{7} -8.00000 q^{8} -13.5687 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.66487 q^{3} +4.00000 q^{4} -8.53079 q^{5} +7.32975 q^{6} +4.20105 q^{7} -8.00000 q^{8} -13.5687 q^{9} +17.0616 q^{10} +65.3726 q^{11} -14.6595 q^{12} -8.40209 q^{14} +31.2643 q^{15} +16.0000 q^{16} -26.9019 q^{17} +27.1374 q^{18} +13.3730 q^{19} -34.1232 q^{20} -15.3963 q^{21} -130.745 q^{22} +159.929 q^{23} +29.3190 q^{24} -52.2255 q^{25} +148.679 q^{27} +16.8042 q^{28} -301.288 q^{29} -62.5286 q^{30} +73.0232 q^{31} -32.0000 q^{32} -239.582 q^{33} +53.8038 q^{34} -35.8383 q^{35} -54.2748 q^{36} +118.781 q^{37} -26.7459 q^{38} +68.2464 q^{40} +432.901 q^{41} +30.7926 q^{42} -356.508 q^{43} +261.490 q^{44} +115.752 q^{45} -319.858 q^{46} -588.614 q^{47} -58.6380 q^{48} -325.351 q^{49} +104.451 q^{50} +98.5921 q^{51} -269.462 q^{53} -297.358 q^{54} -557.680 q^{55} -33.6084 q^{56} -49.0102 q^{57} +602.576 q^{58} -230.340 q^{59} +125.057 q^{60} -380.816 q^{61} -146.046 q^{62} -57.0027 q^{63} +64.0000 q^{64} +479.164 q^{66} -435.848 q^{67} -107.608 q^{68} -586.120 q^{69} +71.6765 q^{70} -65.9622 q^{71} +108.550 q^{72} -885.517 q^{73} -237.562 q^{74} +191.400 q^{75} +53.4918 q^{76} +274.633 q^{77} -385.463 q^{79} -136.493 q^{80} -178.536 q^{81} -865.803 q^{82} +254.207 q^{83} -61.5852 q^{84} +229.495 q^{85} +713.016 q^{86} +1104.18 q^{87} -522.980 q^{88} +372.612 q^{89} -231.504 q^{90} +639.716 q^{92} -267.621 q^{93} +1177.23 q^{94} -114.082 q^{95} +117.276 q^{96} -1313.88 q^{97} +650.702 q^{98} -887.020 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9	\( 3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9} - 24 q^{10} + 82 q^{11} - 48 q^{12} + 54 q^{14} - 90 q^{15} + 48 q^{16} - 90 q^{17} - 18 q^{18} + 130 q^{19} + 48 q^{20} + 234 q^{21} - 164 q^{22} + 19 q^{23} + 96 q^{24} - 61 q^{25} - 69 q^{27} - 108 q^{28} - 101 q^{29} + 180 q^{30} + 519 q^{31} - 96 q^{32} - 146 q^{33} + 180 q^{34} - 458 q^{35} + 36 q^{36} + 84 q^{37} - 260 q^{38} - 96 q^{40} + 187 q^{41} - 468 q^{42} - 1205 q^{43} + 328 q^{44} + 645 q^{45} - 38 q^{46} - 536 q^{47} - 192 q^{48} - 184 q^{49} + 122 q^{50} - 207 q^{51} - 1095 q^{53} + 138 q^{54} - 526 q^{55} + 216 q^{56} - 1409 q^{57} + 202 q^{58} - 1413 q^{59} - 360 q^{60} - 1108 q^{61} - 1038 q^{62} - 1404 q^{63} + 192 q^{64} + 292 q^{66} - 1605 q^{67} - 360 q^{68} - 314 q^{69} + 916 q^{70} - 909 q^{71} - 72 q^{72} + 287 q^{73} - 168 q^{74} - 505 q^{75} + 520 q^{76} + 480 q^{77} - 1961 q^{79} + 192 q^{80} + 915 q^{81} - 374 q^{82} + 191 q^{83} + 936 q^{84} + 67 q^{85} + 2410 q^{86} + 1636 q^{87} - 656 q^{88} + 1091 q^{89} - 1290 q^{90} + 76 q^{92} - 1614 q^{93} + 1072 q^{94} + 1829 q^{95} + 384 q^{96} - 947 q^{97} + 368 q^{98} - 2057 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9 - 24 * q^10 + 82 * q^11 - 48 * q^12 + 54 * q^14 - 90 * q^15 + 48 * q^16 - 90 * q^17 - 18 * q^18 + 130 * q^19 + 48 * q^20 + 234 * q^21 - 164 * q^22 + 19 * q^23 + 96 * q^24 - 61 * q^25 - 69 * q^27 - 108 * q^28 - 101 * q^29 + 180 * q^30 + 519 * q^31 - 96 * q^32 - 146 * q^33 + 180 * q^34 - 458 * q^35 + 36 * q^36 + 84 * q^37 - 260 * q^38 - 96 * q^40 + 187 * q^41 - 468 * q^42 - 1205 * q^43 + 328 * q^44 + 645 * q^45 - 38 * q^46 - 536 * q^47 - 192 * q^48 - 184 * q^49 + 122 * q^50 - 207 * q^51 - 1095 * q^53 + 138 * q^54 - 526 * q^55 + 216 * q^56 - 1409 * q^57 + 202 * q^58 - 1413 * q^59 - 360 * q^60 - 1108 * q^61 - 1038 * q^62 - 1404 * q^63 + 192 * q^64 + 292 * q^66 - 1605 * q^67 - 360 * q^68 - 314 * q^69 + 916 * q^70 - 909 * q^71 - 72 * q^72 + 287 * q^73 - 168 * q^74 - 505 * q^75 + 520 * q^76 + 480 * q^77 - 1961 * q^79 + 192 * q^80 + 915 * q^81 - 374 * q^82 + 191 * q^83 + 936 * q^84 + 67 * q^85 + 2410 * q^86 + 1636 * q^87 - 656 * q^88 + 1091 * q^89 - 1290 * q^90 + 76 * q^92 - 1614 * q^93 + 1072 * q^94 + 1829 * q^95 + 384 * q^96 - 947 * q^97 + 368 * q^98 - 2057 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−3.66487	−0.705305	−0.352653	−	0.935754i	$-0.614720\pi$
$3$	−3.66487	−0.705305	−0.352653	+	0.935754i	$0.614720\pi$
$4$	4.00000	0.500000
$5$	−8.53079	−0.763017	−0.381509	−	0.924365i	$-0.624595\pi$
$5$	−8.53079	−0.763017	−0.381509	+	0.924365i	$0.624595\pi$
$6$	7.32975	0.498726
$7$	4.20105	0.226835	0.113418	−	0.993547i	$-0.463820\pi$
$7$	4.20105	0.226835	0.113418	+	0.993547i	$0.463820\pi$
$8$	−8.00000	−0.353553
$9$	−13.5687	−0.502544
$10$	17.0616	0.539535
$11$	65.3726	1.79187	0.895935	−	0.444186i	$-0.146507\pi$
$11$	65.3726	1.79187	0.895935	+	0.444186i	$0.146507\pi$
$12$	−14.6595	−0.352653
$13$	0	0
$13$	0	0
$14$	−8.40209	−0.160397
$15$	31.2643	0.538160
$16$	16.0000	0.250000
$17$	−26.9019	−0.383804	−0.191902	−	0.981414i	$-0.561466\pi$
$17$	−26.9019	−0.383804	−0.191902	+	0.981414i	$0.561466\pi$
$18$	27.1374	0.355352
$19$	13.3730	0.161472	0.0807360	−	0.996736i	$-0.474273\pi$
$19$	13.3730	0.161472	0.0807360	+	0.996736i	$0.474273\pi$
$20$	−34.1232	−0.381509
$21$	−15.3963	−0.159988
$22$	−130.745	−1.26704
$23$	159.929	1.44989	0.724946	−	0.688806i	$-0.241865\pi$
$23$	159.929	1.44989	0.724946	+	0.688806i	$0.241865\pi$
$24$	29.3190	0.249363
$25$	−52.2255	−0.417804
$26$	0	0
$27$	148.679	1.05975
$28$	16.8042	0.113418
$29$	−301.288	−1.92923	−0.964616	−	0.263658i	$-0.915071\pi$
$29$	−301.288	−1.92923	−0.964616	+	0.263658i	$0.915071\pi$
$30$	−62.5286	−0.380537
$31$	73.0232	0.423076	0.211538	−	0.977370i	$-0.432153\pi$
$31$	73.0232	0.423076	0.211538	+	0.977370i	$0.432153\pi$
$32$	−32.0000	−0.176777
$33$	−239.582	−1.26382
$34$	53.8038	0.271390
$35$	−35.8383	−0.173079
$36$	−54.2748	−0.251272
$37$	118.781	0.527769	0.263885	−	0.964554i	$-0.414996\pi$
$37$	118.781	0.527769	0.263885	+	0.964554i	$0.414996\pi$
$38$	−26.7459	−0.114178
$39$	0	0
$40$	68.2464	0.269767
$41$	432.901	1.64897	0.824486	−	0.565883i	$-0.191465\pi$
$41$	432.901	1.64897	0.824486	+	0.565883i	$0.191465\pi$
$42$	30.7926	0.113129
$43$	−356.508	−1.26435	−0.632174	−	0.774827i	$-0.717837\pi$
$43$	−356.508	−1.26435	−0.632174	+	0.774827i	$0.717837\pi$
$44$	261.490	0.895935
$45$	115.752	0.383450
$46$	−319.858	−1.02523
$47$	−588.614	−1.82677	−0.913385	−	0.407096i	$-0.866541\pi$
$47$	−588.614	−1.82677	−0.913385	+	0.407096i	$0.866541\pi$
$48$	−58.6380	−0.176326
$49$	−325.351	−0.948546
$50$	104.451	0.295432
$51$	98.5921	0.270699
$52$	0	0
$53$	−269.462	−0.698366	−0.349183	−	0.937054i	$-0.613541\pi$
$53$	−269.462	−0.698366	−0.349183	+	0.937054i	$0.613541\pi$
$54$	−297.358	−0.749358
$55$	−557.680	−1.36723
$56$	−33.6084	−0.0801983
$57$	−49.0102	−0.113887
$58$	602.576	1.36417
$59$	−230.340	−0.508265	−0.254133	−	0.967169i	$-0.581790\pi$
$59$	−230.340	−0.508265	−0.254133	+	0.967169i	$0.581790\pi$
$60$	125.057	0.269080
$61$	−380.816	−0.799319	−0.399659	−	0.916664i	$-0.630872\pi$
$61$	−380.816	−0.799319	−0.399659	+	0.916664i	$0.630872\pi$
$62$	−146.046	−0.299160
$63$	−57.0027	−0.113995
$64$	64.0000	0.125000
$65$	0	0
$66$	479.164	0.893652
$67$	−435.848	−0.794736	−0.397368	−	0.917659i	$-0.630076\pi$
$67$	−435.848	−0.794736	−0.397368	+	0.917659i	$0.630076\pi$
$68$	−107.608	−0.191902
$69$	−586.120	−1.02262
$70$	71.6765	0.122385
$71$	−65.9622	−0.110257	−0.0551287	−	0.998479i	$-0.517557\pi$
$71$	−65.9622	−0.110257	−0.0551287	+	0.998479i	$0.517557\pi$
$72$	108.550	0.177676
$73$	−885.517	−1.41975	−0.709876	−	0.704326i	$-0.751249\pi$
$73$	−885.517	−1.41975	−0.709876	+	0.704326i	$0.751249\pi$
$74$	−237.562	−0.373189
$75$	191.400	0.294680
$76$	53.4918	0.0807360
$77$	274.633	0.406459
$78$	0	0
$79$	−385.463	−0.548962	−0.274481	−	0.961592i	$-0.588506\pi$
$79$	−385.463	−0.548962	−0.274481	+	0.961592i	$0.588506\pi$
$80$	−136.493	−0.190754
$81$	−178.536	−0.244905
$82$	−865.803	−1.16600
$83$	254.207	0.336179	0.168089	−	0.985772i	$-0.446240\pi$
$83$	254.207	0.336179	0.168089	+	0.985772i	$0.446240\pi$
$84$	−61.5852	−0.0799940
$85$	229.495	0.292849
$86$	713.016	0.894029
$87$	1104.18	1.36070
$88$	−522.980	−0.633522
$89$	372.612	0.443785	0.221892	−	0.975071i	$-0.428777\pi$
$89$	372.612	0.443785	0.221892	+	0.975071i	$0.428777\pi$
$90$	−231.504	−0.271140
$91$	0	0
$92$	639.716	0.724946
$93$	−267.621	−0.298398
$94$	1177.23	1.29172
$95$	−114.082	−0.123206
$96$	117.276	0.124682
$97$	−1313.88	−1.37531	−0.687653	−	0.726039i	$-0.741359\pi$
$97$	−1313.88	−1.37531	−0.687653	+	0.726039i	$0.741359\pi$
$98$	650.702	0.670723
$99$	−887.020	−0.900494
$100$	−208.902	−0.208902
$101$	1463.72	1.44204	0.721019	−	0.692915i	$-0.243674\pi$
$101$	1463.72	1.44204	0.721019	+	0.692915i	$0.243674\pi$
$102$	−197.184	−0.191413
$103$	210.886	0.201740	0.100870	−	0.994900i	$-0.467837\pi$
$103$	210.886	0.201740	0.100870	+	0.994900i	$0.467837\pi$
$104$	0	0
$105$	131.343	0.122074
$106$	538.924	0.493820
$107$	391.061	0.353321	0.176660	−	0.984272i	$-0.443471\pi$
$107$	391.061	0.353321	0.176660	+	0.984272i	$0.443471\pi$
$108$	594.717	0.529876
$109$	−1331.40	−1.16996	−0.584978	−	0.811049i	$-0.698897\pi$
$109$	−1331.40	−1.16996	−0.584978	+	0.811049i	$0.698897\pi$
$110$	1115.36	0.966776
$111$	−435.317	−0.372238
$112$	67.2167	0.0567088
$113$	−711.912	−0.592665	−0.296332	−	0.955085i	$-0.595764\pi$
$113$	−711.912	−0.592665	−0.296332	+	0.955085i	$0.595764\pi$
$114$	98.0204	0.0805303
$115$	−1364.32	−1.10629
$116$	−1205.15	−0.964616
$117$	0	0
$118$	460.679	0.359398
$119$	−113.016	−0.0870603
$120$	−250.114	−0.190268
$121$	2942.57	2.21080
$122$	761.631	0.565204
$123$	−1586.53	−1.16303
$124$	292.093	0.211538
$125$	1511.87	1.08181
$126$	114.005	0.0806064
$127$	−1171.70	−0.818677	−0.409338	−	0.912383i	$-0.634240\pi$
$127$	−1171.70	−0.818677	−0.409338	+	0.912383i	$0.634240\pi$
$128$	−128.000	−0.0883883
$129$	1306.56	0.891751
$130$	0	0
$131$	−1128.61	−0.752726	−0.376363	−	0.926472i	$-0.622825\pi$
$131$	−1128.61	−0.752726	−0.376363	+	0.926472i	$0.622825\pi$
$132$	−958.329	−0.631908
$133$	56.1804	0.0366275
$134$	871.696	0.561963
$135$	−1268.35	−0.808610
$136$	215.215	0.135695
$137$	1747.93	1.09004	0.545022	−	0.838422i	$-0.316521\pi$
$137$	1747.93	1.09004	0.545022	+	0.838422i	$0.316521\pi$
$138$	1172.24	0.723099
$139$	1676.54	1.02304	0.511518	−	0.859273i	$-0.329083\pi$
$139$	1676.54	1.02304	0.511518	+	0.859273i	$0.329083\pi$
$140$	−143.353	−0.0865396
$141$	2157.20	1.28843
$142$	131.924	0.0779637
$143$	0	0
$144$	−217.099	−0.125636
$145$	2570.22	1.47204
$146$	1771.03	1.00392
$147$	1192.37	0.669014
$148$	475.124	0.263885
$149$	896.961	0.493167	0.246584	−	0.969122i	$-0.420692\pi$
$149$	896.961	0.493167	0.246584	+	0.969122i	$0.420692\pi$
$150$	−382.800	−0.208370
$151$	2078.28	1.12005	0.560027	−	0.828475i	$-0.310791\pi$
$151$	2078.28	1.12005	0.560027	+	0.828475i	$0.310791\pi$
$152$	−106.984	−0.0570890
$153$	365.024	0.192879
$154$	−549.266	−0.287410
$155$	−622.946	−0.322814
$156$	0	0
$157$	−3494.75	−1.77650	−0.888252	−	0.459357i	$-0.848080\pi$
$157$	−3494.75	−1.77650	−0.888252	+	0.459357i	$0.848080\pi$
$158$	770.927	0.388175
$159$	987.544	0.492562
$160$	272.985	0.134884
$161$	671.869	0.328887
$162$	357.071	0.173174
$163$	451.254	0.216840	0.108420	−	0.994105i	$-0.465421\pi$
$163$	451.254	0.216840	0.108420	+	0.994105i	$0.465421\pi$
$164$	1731.61	0.824486
$165$	2043.83	0.964313
$166$	−508.414	−0.237714
$167$	−3203.35	−1.48433	−0.742164	−	0.670218i	$-0.766200\pi$
$167$	−3203.35	−1.48433	−0.742164	+	0.670218i	$0.766200\pi$
$168$	123.170	0.0565643
$169$	0	0
$170$	−458.989	−0.207076
$171$	−181.454	−0.0811468
$172$	−1426.03	−0.632174
$173$	1899.38	0.834725	0.417362	−	0.908740i	$-0.362955\pi$
$173$	1899.38	0.834725	0.417362	+	0.908740i	$0.362955\pi$
$174$	−2208.36	−0.962159
$175$	−219.402	−0.0947727
$176$	1045.96	0.447967
$177$	844.165	0.358482
$178$	−745.225	−0.313803
$179$	2650.89	1.10691	0.553455	−	0.832879i	$-0.313309\pi$
$179$	2650.89	1.10691	0.553455	+	0.832879i	$0.313309\pi$
$180$	463.007	0.191725
$181$	−2289.94	−0.940387	−0.470194	−	0.882563i	$-0.655816\pi$
$181$	−2289.94	−0.940387	−0.470194	+	0.882563i	$0.655816\pi$
$182$	0	0
$183$	1395.64	0.563764
$184$	−1279.43	−0.512614
$185$	−1013.30	−0.402697
$186$	535.242	0.210999
$187$	−1758.65	−0.687727
$188$	−2354.46	−0.913385
$189$	624.608	0.240389
$190$	228.164	0.0871198
$191$	−338.313	−0.128165	−0.0640825	−	0.997945i	$-0.520412\pi$
$191$	−338.313	−0.128165	−0.0640825	+	0.997945i	$0.520412\pi$
$192$	−234.552	−0.0881632
$193$	−2684.05	−1.00105	−0.500524	−	0.865723i	$-0.666859\pi$
$193$	−2684.05	−1.00105	−0.500524	+	0.865723i	$0.666859\pi$
$194$	2627.77	0.972489
$195$	0	0
$196$	−1301.40	−0.474273
$197$	−1897.24	−0.686156	−0.343078	−	0.939307i	$-0.611470\pi$
$197$	−1897.24	−0.686156	−0.343078	+	0.939307i	$0.611470\pi$
$198$	1774.04	0.636745
$199$	−2415.60	−0.860491	−0.430245	−	0.902712i	$-0.641573\pi$
$199$	−2415.60	−0.860491	−0.430245	+	0.902712i	$0.641573\pi$
$200$	417.804	0.147716
$201$	1597.33	0.560532
$202$	−2927.44	−1.01967
$203$	−1265.72	−0.437618
$204$	394.368	0.135350
$205$	−3692.99	−1.25819
$206$	−421.772	−0.142652
$207$	−2170.03	−0.728635
$208$	0	0
$209$	874.225	0.289337
$210$	−262.685	−0.0863191
$211$	2668.42	0.870623	0.435312	−	0.900280i	$-0.356638\pi$
$211$	2668.42	0.870623	0.435312	+	0.900280i	$0.356638\pi$
$212$	−1077.85	−0.349183
$213$	241.743	0.0777651
$214$	−782.122	−0.249835
$215$	3041.30	0.964719
$216$	−1189.43	−0.374679
$217$	306.774	0.0959685
$218$	2662.81	0.827284
$219$	3245.31	1.00136
$220$	−2230.72	−0.683614
$221$	0	0
$222$	870.634	0.263212
$223$	286.358	0.0859907	0.0429953	−	0.999075i	$-0.486310\pi$
$223$	286.358	0.0859907	0.0429953	+	0.999075i	$0.486310\pi$
$224$	−134.433	−0.0400992
$225$	708.632	0.209965
$226$	1423.82	0.419077
$227$	−5201.24	−1.52079	−0.760393	−	0.649463i	$-0.774994\pi$
$227$	−5201.24	−1.52079	−0.760393	+	0.649463i	$0.774994\pi$
$228$	−196.041	−0.0569435
$229$	−890.458	−0.256957	−0.128478	−	0.991712i	$-0.541009\pi$
$229$	−890.458	−0.256957	−0.128478	+	0.991712i	$0.541009\pi$
$230$	2728.64	0.782267
$231$	−1006.50	−0.286678
$232$	2410.30	0.682087
$233$	4753.11	1.33642	0.668212	−	0.743971i	$-0.267060\pi$
$233$	4753.11	1.33642	0.668212	+	0.743971i	$0.267060\pi$
$234$	0	0
$235$	5021.35	1.39386
$236$	−921.358	−0.254133
$237$	1412.67	0.387186
$238$	226.032	0.0615609
$239$	2292.62	0.620491	0.310245	−	0.950656i	$-0.399589\pi$
$239$	2292.62	0.620491	0.310245	+	0.950656i	$0.399589\pi$
$240$	500.229	0.134540
$241$	−1975.21	−0.527943	−0.263972	−	0.964530i	$-0.585033\pi$
$241$	−1975.21	−0.527943	−0.263972	+	0.964530i	$0.585033\pi$
$242$	−5885.14	−1.56327
$243$	−3360.03	−0.887020
$244$	−1523.26	−0.399659
$245$	2775.50	0.723757
$246$	3173.06	0.822385
$247$	0	0
$248$	−584.186	−0.149580
$249$	−931.637	−0.237109
$250$	−3023.75	−0.764955
$251$	−7465.74	−1.87742	−0.938711	−	0.344705i	$-0.887979\pi$
$251$	−7465.74	−1.87742	−0.938711	+	0.344705i	$0.887979\pi$
$252$	−228.011	−0.0569974
$253$	10455.0	2.59802
$254$	2343.41	0.578892
$255$	−841.069	−0.206548
$256$	256.000	0.0625000
$257$	−554.966	−0.134700	−0.0673499	−	0.997729i	$-0.521454\pi$
$257$	−554.966	−0.134700	−0.0673499	+	0.997729i	$0.521454\pi$
$258$	−2613.11	−0.630563
$259$	499.004	0.119717
$260$	0	0
$261$	4088.08	0.969525
$262$	2257.22	0.532258
$263$	−1993.07	−0.467292	−0.233646	−	0.972322i	$-0.575066\pi$
$263$	−1993.07	−0.467292	−0.233646	+	0.972322i	$0.575066\pi$
$264$	1916.66	0.446826
$265$	2298.72	0.532866
$266$	−112.361	−0.0258996
$267$	−1365.58	−0.313004
$268$	−1743.39	−0.397368
$269$	−5207.76	−1.18038	−0.590191	−	0.807263i	$-0.700948\pi$
$269$	−5207.76	−1.18038	−0.590191	+	0.807263i	$0.700948\pi$
$270$	2536.70	0.571773
$271$	−4084.90	−0.915646	−0.457823	−	0.889043i	$-0.651371\pi$
$271$	−4084.90	−0.915646	−0.457823	+	0.889043i	$0.651371\pi$
$272$	−430.430	−0.0959510
$273$	0	0
$274$	−3495.87	−0.770778
$275$	−3414.12	−0.748651
$276$	−2344.48	−0.511308
$277$	437.993	0.0950051	0.0475026	−	0.998871i	$-0.484874\pi$
$277$	437.993	0.0950051	0.0475026	+	0.998871i	$0.484874\pi$
$278$	−3353.07	−0.723395
$279$	−990.829	−0.212614
$280$	286.706	0.0611927
$281$	−5462.34	−1.15963	−0.579815	−	0.814748i	$-0.696875\pi$
$281$	−5462.34	−1.15963	−0.579815	+	0.814748i	$0.696875\pi$
$282$	−4314.40	−0.911058
$283$	−796.676	−0.167341	−0.0836704	−	0.996493i	$-0.526664\pi$
$283$	−796.676	−0.167341	−0.0836704	+	0.996493i	$0.526664\pi$
$284$	−263.849	−0.0551287
$285$	418.096	0.0868978
$286$	0	0
$287$	1818.64	0.374045
$288$	434.198	0.0888381
$289$	−4189.29	−0.852694
$290$	−5140.45	−1.04089
$291$	4815.22	0.970011
$292$	−3542.07	−0.709876
$293$	−1697.32	−0.338426	−0.169213	−	0.985580i	$-0.554123\pi$
$293$	−1697.32	−0.338426	−0.169213	+	0.985580i	$0.554123\pi$
$294$	−2384.74	−0.473065
$295$	1964.98	0.387815
$296$	−950.247	−0.186595
$297$	9719.54	1.89894
$298$	−1793.92	−0.348722
$299$	0	0
$300$	765.600	0.147340
$301$	−1497.71	−0.286798
$302$	−4156.56	−0.791997
$303$	−5364.36	−1.01708
$304$	213.967	0.0403680
$305$	3248.66	0.609894
$306$	−730.047	−0.136386
$307$	9385.86	1.74488	0.872442	−	0.488718i	$-0.162535\pi$
$307$	9385.86	1.74488	0.872442	+	0.488718i	$0.162535\pi$
$308$	1098.53	0.203230
$309$	−772.870	−0.142288
$310$	1245.89	0.228264
$311$	−3282.65	−0.598527	−0.299264	−	0.954170i	$-0.596741\pi$
$311$	−3282.65	−0.598527	−0.299264	+	0.954170i	$0.596741\pi$
$312$	0	0
$313$	−5924.67	−1.06991	−0.534955	−	0.844880i	$-0.679672\pi$
$313$	−5924.67	−1.06991	−0.534955	+	0.844880i	$0.679672\pi$
$314$	6989.49	1.25618
$315$	486.278	0.0869800
$316$	−1541.85	−0.274481
$317$	6467.88	1.14597	0.572985	−	0.819566i	$-0.305785\pi$
$317$	6467.88	1.14597	0.572985	+	0.819566i	$0.305785\pi$
$318$	−1975.09	−0.348294
$319$	−19696.0	−3.45693
$320$	−545.971	−0.0953772
$321$	−1433.19	−0.249199
$322$	−1343.74	−0.232558
$323$	−359.758	−0.0619736
$324$	−714.143	−0.122452
$325$	0	0
$326$	−902.508	−0.153329
$327$	4879.42	0.825177
$328$	−3463.21	−0.582999
$329$	−2472.80	−0.414376
$330$	−4087.65	−0.681872
$331$	3509.24	0.582735	0.291368	−	0.956611i	$-0.405890\pi$
$331$	3509.24	0.582735	0.291368	+	0.956611i	$0.405890\pi$
$332$	1016.83	0.168089
$333$	−1611.70	−0.265227
$334$	6406.70	1.04958
$335$	3718.13	0.606398
$336$	−246.341	−0.0399970
$337$	−1834.82	−0.296584	−0.148292	−	0.988944i	$-0.547378\pi$
$337$	−1834.82	−0.296584	−0.148292	+	0.988944i	$0.547378\pi$
$338$	0	0
$339$	2609.07	0.418010
$340$	917.978	0.146425
$341$	4773.71	0.758097
$342$	362.907	0.0573795
$343$	−2807.77	−0.441999
$344$	2852.06	0.447014
$345$	5000.07	0.780274
$346$	−3798.76	−0.590239
$347$	−7259.30	−1.12305	−0.561527	−	0.827459i	$-0.689786\pi$
$347$	−7259.30	−1.12305	−0.561527	+	0.827459i	$0.689786\pi$
$348$	4416.73	0.680349
$349$	−1795.24	−0.275349	−0.137674	−	0.990478i	$-0.543963\pi$
$349$	−1795.24	−0.275349	−0.137674	+	0.990478i	$0.543963\pi$
$350$	438.804	0.0670144
$351$	0	0
$352$	−2091.92	−0.316761
$353$	−4552.38	−0.686398	−0.343199	−	0.939263i	$-0.611511\pi$
$353$	−4552.38	−0.686398	−0.343199	+	0.939263i	$0.611511\pi$
$354$	−1688.33	−0.253485
$355$	562.710	0.0841283
$356$	1490.45	0.221892
$357$	414.190	0.0614041
$358$	−5301.78	−0.782703
$359$	10165.4	1.49445	0.747227	−	0.664569i	$-0.231385\pi$
$359$	10165.4	1.49445	0.747227	+	0.664569i	$0.231385\pi$
$360$	−926.014	−0.135570
$361$	−6680.16	−0.973927
$362$	4579.88	0.664954
$363$	−10784.2	−1.55929
$364$	0	0
$365$	7554.16	1.08330
$366$	−2791.28	−0.398641
$367$	−10401.6	−1.47945	−0.739723	−	0.672911i	$-0.765044\pi$
$367$	−10401.6	−1.47945	−0.739723	+	0.672911i	$0.765044\pi$
$368$	2558.87	0.362473
$369$	−5873.91	−0.828681
$370$	2026.59	0.284750
$371$	−1132.02	−0.158414
$372$	−1070.48	−0.149199
$373$	−7953.41	−1.10405	−0.552027	−	0.833826i	$-0.686145\pi$
$373$	−7953.41	−1.10405	−0.552027	+	0.833826i	$0.686145\pi$
$374$	3517.29	0.486296
$375$	−5540.83	−0.763006
$376$	4708.91	0.645861
$377$	0	0
$378$	−1249.22	−0.169981
$379$	9014.81	1.22179	0.610897	−	0.791710i	$-0.290809\pi$
$379$	9014.81	1.22179	0.610897	+	0.791710i	$0.290809\pi$
$380$	−456.328	−0.0616030
$381$	4294.15	0.577417
$382$	676.627	0.0906263
$383$	−2496.24	−0.333034	−0.166517	−	0.986039i	$-0.553252\pi$
$383$	−2496.24	−0.333034	−0.166517	+	0.986039i	$0.553252\pi$
$384$	469.104	0.0623408
$385$	−2342.84	−0.310135
$386$	5368.10	0.707848
$387$	4837.35	0.635391
$388$	−5255.54	−0.687653
$389$	10896.9	1.42029	0.710145	−	0.704056i	$-0.248629\pi$
$389$	10896.9	1.42029	0.710145	+	0.704056i	$0.248629\pi$
$390$	0	0
$391$	−4302.40	−0.556475
$392$	2602.81	0.335362
$393$	4136.21	0.530902
$394$	3794.48	0.485186
$395$	3288.31	0.418868
$396$	−3548.08	−0.450247
$397$	10887.1	1.37635	0.688174	−	0.725546i	$-0.258413\pi$
$397$	10887.1	1.37635	0.688174	+	0.725546i	$0.258413\pi$
$398$	4831.21	0.608459
$399$	−205.894	−0.0258336
$400$	−835.609	−0.104451
$401$	9968.76	1.24144	0.620718	−	0.784034i	$-0.286841\pi$
$401$	9968.76	1.24144	0.620718	+	0.784034i	$0.286841\pi$
$402$	−3194.66	−0.396356
$403$	0	0
$404$	5854.89	0.721019
$405$	1523.05	0.186867
$406$	2531.45	0.309442
$407$	7765.01	0.945693
$408$	−788.737	−0.0957066
$409$	4036.32	0.487979	0.243989	−	0.969778i	$-0.421544\pi$
$409$	4036.32	0.487979	0.243989	+	0.969778i	$0.421544\pi$
$410$	7385.98	0.889677
$411$	−6405.96	−0.768814
$412$	843.543	0.100870
$413$	−967.667	−0.115292
$414$	4340.06	0.515223
$415$	−2168.59	−0.256510
$416$	0	0
$417$	−6144.29	−0.721552
$418$	−1748.45	−0.204592
$419$	15194.9	1.77164	0.885821	−	0.464028i	$-0.153596\pi$
$419$	15194.9	1.77164	0.885821	+	0.464028i	$0.153596\pi$
$420$	525.371	0.0610368
$421$	10154.8	1.17556	0.587782	−	0.809019i	$-0.300001\pi$
$421$	10154.8	1.17556	0.587782	+	0.809019i	$0.300001\pi$
$422$	−5336.84	−0.615624
$423$	7986.73	0.918033
$424$	2155.69	0.246910
$425$	1404.97	0.160355
$426$	−483.486	−0.0549882
$427$	−1599.82	−0.181314
$428$	1564.24	0.176660
$429$	0	0
$430$	−6082.59	−0.682159
$431$	−6616.84	−0.739494	−0.369747	−	0.929133i	$-0.620556\pi$
$431$	−6616.84	−0.739494	−0.369747	+	0.929133i	$0.620556\pi$
$432$	2378.87	0.264938
$433$	5697.60	0.632354	0.316177	−	0.948700i	$-0.397601\pi$
$433$	5697.60	0.632354	0.316177	+	0.948700i	$0.397601\pi$
$434$	−613.548	−0.0678600
$435$	−9419.55	−1.03824
$436$	−5325.61	−0.584978
$437$	2138.73	0.234117
$438$	−6490.62	−0.708068
$439$	436.007	0.0474020	0.0237010	−	0.999719i	$-0.492455\pi$
$439$	436.007	0.0474020	0.0237010	+	0.999719i	$0.492455\pi$
$440$	4461.44	0.483388
$441$	4414.59	0.476686
$442$	0	0
$443$	6609.58	0.708873	0.354436	−	0.935080i	$-0.384673\pi$
$443$	6609.58	0.708873	0.354436	+	0.935080i	$0.384673\pi$
$444$	−1741.27	−0.186119
$445$	−3178.68	−0.338616
$446$	−572.715	−0.0608046
$447$	−3287.25	−0.347833
$448$	268.867	0.0283544
$449$	8015.31	0.842463	0.421231	−	0.906953i	$-0.361598\pi$
$449$	8015.31	0.842463	0.421231	+	0.906953i	$0.361598\pi$
$450$	−1417.26	−0.148468
$451$	28299.9	2.95474
$452$	−2847.65	−0.296332
$453$	−7616.64	−0.789980
$454$	10402.5	1.07536
$455$	0	0
$456$	392.082	0.0402652
$457$	−4507.03	−0.461335	−0.230668	−	0.973033i	$-0.574091\pi$
$457$	−4507.03	−0.461335	−0.230668	+	0.973033i	$0.574091\pi$
$458$	1780.92	0.181696
$459$	−3999.75	−0.406737
$460$	−5457.29	−0.553146
$461$	2103.55	0.212521	0.106260	−	0.994338i	$-0.466112\pi$
$461$	2103.55	0.212521	0.106260	+	0.994338i	$0.466112\pi$
$462$	2012.99	0.202712
$463$	5468.28	0.548883	0.274441	−	0.961604i	$-0.411507\pi$
$463$	5468.28	0.548883	0.274441	+	0.961604i	$0.411507\pi$
$464$	−4820.60	−0.482308
$465$	2283.02	0.227683
$466$	−9506.22	−0.944994
$467$	−6043.79	−0.598872	−0.299436	−	0.954116i	$-0.596799\pi$
$467$	−6043.79	−0.598872	−0.299436	+	0.954116i	$0.596799\pi$
$468$	0	0
$469$	−1831.02	−0.180274
$470$	−10042.7	−0.985606
$471$	12807.8	1.25298
$472$	1842.72	0.179699
$473$	−23305.8	−2.26555
$474$	−2825.35	−0.273782
$475$	−698.410	−0.0674637
$476$	−452.065	−0.0435301
$477$	3656.25	0.350960
$478$	−4585.24	−0.438753
$479$	−8481.30	−0.809020	−0.404510	−	0.914534i	$-0.632558\pi$
$479$	−8481.30	−0.809020	−0.404510	+	0.914534i	$0.632558\pi$
$480$	−1000.46	−0.0951342
$481$	0	0
$482$	3950.42	0.373312
$483$	−2462.32	−0.231965
$484$	11770.3	1.10540
$485$	11208.5	1.04938
$486$	6720.05	0.627218
$487$	−6623.36	−0.616289	−0.308145	−	0.951340i	$-0.599708\pi$
$487$	−6623.36	−0.616289	−0.308145	+	0.951340i	$0.599708\pi$
$488$	3046.53	0.282602
$489$	−1653.79	−0.152939
$490$	−5551.01	−0.511774
$491$	16379.4	1.50548	0.752739	−	0.658319i	$-0.228732\pi$
$491$	16379.4	1.50548	0.752739	+	0.658319i	$0.228732\pi$
$492$	−6346.12	−0.581514
$493$	8105.21	0.740447
$494$	0	0
$495$	7566.99	0.687093
$496$	1168.37	0.105769
$497$	−277.110	−0.0250102
$498$	1863.27	0.167661
$499$	−7915.70	−0.710131	−0.355065	−	0.934841i	$-0.615541\pi$
$499$	−7915.70	−0.710131	−0.355065	+	0.934841i	$0.615541\pi$
$500$	6047.50	0.540905
$501$	11739.9	1.04690
$502$	14931.5	1.32754
$503$	−479.511	−0.0425056	−0.0212528	−	0.999774i	$-0.506765\pi$
$503$	−479.511	−0.0425056	−0.0212528	+	0.999774i	$0.506765\pi$
$504$	456.022	0.0403032
$505$	−12486.7	−1.10030
$506$	−20909.9	−1.83708
$507$	0	0
$508$	−4686.82	−0.409338
$509$	1192.33	0.103829	0.0519145	−	0.998652i	$-0.483468\pi$
$509$	1192.33	0.103829	0.0519145	+	0.998652i	$0.483468\pi$
$510$	1682.14	0.146052
$511$	−3720.10	−0.322050
$512$	−512.000	−0.0441942
$513$	1988.28	0.171120
$514$	1109.93	0.0952471
$515$	−1799.02	−0.153931
$516$	5226.23	0.445876
$517$	−38479.2	−3.27333
$518$	−998.008	−0.0846524
$519$	−6961.00	−0.588736
$520$	0	0
$521$	−23238.5	−1.95412	−0.977062	−	0.212955i	$-0.931691\pi$
$521$	−23238.5	−1.95412	−0.977062	+	0.212955i	$0.931691\pi$
$522$	−8176.16	−0.685557
$523$	−11026.3	−0.921884	−0.460942	−	0.887430i	$-0.652488\pi$
$523$	−11026.3	−0.921884	−0.460942	+	0.887430i	$0.652488\pi$
$524$	−4514.44	−0.376363
$525$	804.080	0.0668437
$526$	3986.14	0.330425
$527$	−1964.46	−0.162378
$528$	−3833.32	−0.315954
$529$	13410.3	1.10219
$530$	−4597.45	−0.376793
$531$	3125.41	0.255426
$532$	224.722	0.0183138
$533$	0	0
$534$	2731.16	0.221327
$535$	−3336.06	−0.269590
$536$	3486.79	0.280982
$537$	−9715.18	−0.780709
$538$	10415.5	0.834657
$539$	−21269.0	−1.69967
$540$	−5073.41	−0.404305
$541$	8987.70	0.714254	0.357127	−	0.934056i	$-0.383756\pi$
$541$	8987.70	0.714254	0.357127	+	0.934056i	$0.383756\pi$
$542$	8169.80	0.647459
$543$	8392.35	0.663260
$544$	860.861	0.0678476
$545$	11357.9	0.892697
$546$	0	0
$547$	−10734.8	−0.839101	−0.419550	−	0.907732i	$-0.637812\pi$
$547$	−10734.8	−0.839101	−0.419550	+	0.907732i	$0.637812\pi$
$548$	6991.74	0.545022
$549$	5167.17	0.401693
$550$	6828.23	0.529376
$551$	−4029.11	−0.311517
$552$	4688.96	0.361550
$553$	−1619.35	−0.124524
$554$	−875.985	−0.0671788
$555$	3713.60	0.284024
$556$	6706.14	0.511518
$557$	13228.3	1.00628	0.503141	−	0.864204i	$-0.332178\pi$
$557$	13228.3	1.00628	0.503141	+	0.864204i	$0.332178\pi$
$558$	1981.66	0.150341
$559$	0	0
$560$	−573.412	−0.0432698
$561$	6445.22	0.485058
$562$	10924.7	0.819983
$563$	10734.0	0.803522	0.401761	−	0.915745i	$-0.368398\pi$
$563$	10734.0	0.803522	0.401761	+	0.915745i	$0.368398\pi$
$564$	8628.79	0.644216
$565$	6073.18	0.452213
$566$	1593.35	0.118328
$567$	−750.037	−0.0555531
$568$	527.698	0.0389819
$569$	5313.45	0.391479	0.195739	−	0.980656i	$-0.437289\pi$
$569$	5313.45	0.391479	0.195739	+	0.980656i	$0.437289\pi$
$570$	−836.192	−0.0614460
$571$	4629.37	0.339287	0.169644	−	0.985505i	$-0.445738\pi$
$571$	4629.37	0.339287	0.169644	+	0.985505i	$0.445738\pi$
$572$	0	0
$573$	1239.88	0.0903954
$574$	−3637.28	−0.264490
$575$	−8352.38	−0.605771
$576$	−868.397	−0.0628180
$577$	−504.750	−0.0364177	−0.0182088	−	0.999834i	$-0.505796\pi$
$577$	−504.750	−0.0364177	−0.0182088	+	0.999834i	$0.505796\pi$
$578$	8378.58	0.602946
$579$	9836.71	0.706045
$580$	10280.9	0.736019
$581$	1067.94	0.0762572
$582$	−9630.44	−0.685901
$583$	−17615.4	−1.25138
$584$	7084.14	0.501958
$585$	0	0
$586$	3394.65	0.239303
$587$	5532.93	0.389044	0.194522	−	0.980898i	$-0.437684\pi$
$587$	5532.93	0.389044	0.194522	+	0.980898i	$0.437684\pi$
$588$	4769.49	0.334507
$589$	976.536	0.0683149
$590$	−3929.96	−0.274227
$591$	6953.15	0.483950
$592$	1900.49	0.131942
$593$	−18079.8	−1.25202	−0.626009	−	0.779816i	$-0.715313\pi$
$593$	−18079.8	−1.25202	−0.626009	+	0.779816i	$0.715313\pi$
$594$	−19439.1	−1.34275
$595$	964.117	0.0664285
$596$	3587.84	0.246584
$597$	8852.88	0.606909
$598$	0	0
$599$	−1837.55	−0.125342	−0.0626712	−	0.998034i	$-0.519962\pi$
$599$	−1837.55	−0.125342	−0.0626712	+	0.998034i	$0.519962\pi$
$600$	−1531.20	−0.104185
$601$	−23487.3	−1.59412	−0.797060	−	0.603901i	$-0.793612\pi$
$601$	−23487.3	−1.59412	−0.797060	+	0.603901i	$0.793612\pi$
$602$	2995.41	0.202797
$603$	5913.89	0.399390
$604$	8313.12	0.560027
$605$	−25102.5	−1.68688
$606$	10728.7	0.719182
$607$	−16310.8	−1.09067	−0.545335	−	0.838218i	$-0.683598\pi$
$607$	−16310.8	−1.09067	−0.545335	+	0.838218i	$0.683598\pi$
$608$	−427.935	−0.0285445
$609$	4638.72	0.308654
$610$	−6497.32	−0.431260
$611$	0	0
$612$	1460.09	0.0964393
$613$	8453.11	0.556962	0.278481	−	0.960442i	$-0.410169\pi$
$613$	8453.11	0.556962	0.278481	+	0.960442i	$0.410169\pi$
$614$	−18771.7	−1.23382
$615$	13534.4	0.887411
$616$	−2197.06	−0.143705
$617$	−13410.3	−0.875008	−0.437504	−	0.899216i	$-0.644137\pi$
$617$	−13410.3	−0.875008	−0.437504	+	0.899216i	$0.644137\pi$
$618$	1545.74	0.100613
$619$	−890.135	−0.0577990	−0.0288995	−	0.999582i	$-0.509200\pi$
$619$	−890.135	−0.0577990	−0.0288995	+	0.999582i	$0.509200\pi$
$620$	−2491.78	−0.161407
$621$	23778.1	1.53653
$622$	6565.30	0.423223
$623$	1565.36	0.100666
$624$	0	0
$625$	−6369.30	−0.407635
$626$	11849.3	0.756541
$627$	−3203.92	−0.204071
$628$	−13979.0	−0.888252
$629$	−3195.43	−0.202560
$630$	−972.557	−0.0615041
$631$	−10582.4	−0.667639	−0.333819	−	0.942637i	$-0.608338\pi$
$631$	−10582.4	−0.667639	−0.333819	+	0.942637i	$0.608338\pi$
$632$	3083.71	0.194087
$633$	−9779.42	−0.614055
$634$	−12935.8	−0.810323
$635$	9995.57	0.624665
$636$	3950.17	0.246281
$637$	0	0
$638$	39391.9	2.44442
$639$	895.021	0.0554092
$640$	1091.94	0.0674419
$641$	26884.2	1.65657	0.828286	−	0.560305i	$-0.189316\pi$
$641$	26884.2	1.65657	0.828286	+	0.560305i	$0.189316\pi$
$642$	2866.38	0.176210
$643$	5691.12	0.349045	0.174523	−	0.984653i	$-0.444162\pi$
$643$	5691.12	0.349045	0.174523	+	0.984653i	$0.444162\pi$
$644$	2687.48	0.164443
$645$	−11146.0	−0.680422
$646$	719.516	0.0438220
$647$	1809.79	0.109969	0.0549847	−	0.998487i	$-0.482489\pi$
$647$	1809.79	0.109969	0.0549847	+	0.998487i	$0.482489\pi$
$648$	1428.29	0.0865870
$649$	−15057.9	−0.910745
$650$	0	0
$651$	−1124.29	−0.0676871
$652$	1805.02	0.108420
$653$	9459.16	0.566869	0.283435	−	0.958992i	$-0.408526\pi$
$653$	9459.16	0.566869	0.283435	+	0.958992i	$0.408526\pi$
$654$	−9758.85	−0.583488
$655$	9627.94	0.574343
$656$	6926.42	0.412243
$657$	12015.3	0.713488
$658$	4945.59	0.293008
$659$	1015.32	0.0600171	0.0300085	−	0.999550i	$-0.490447\pi$
$659$	1015.32	0.0600171	0.0300085	+	0.999550i	$0.490447\pi$
$660$	8175.31	0.482157
$661$	23648.4	1.39156	0.695778	−	0.718257i	$-0.255060\pi$
$661$	23648.4	1.39156	0.695778	+	0.718257i	$0.255060\pi$
$662$	−7018.49	−0.412056
$663$	0	0
$664$	−2033.66	−0.118857
$665$	−479.264	−0.0279474
$666$	3223.40	0.187544
$667$	−48184.7	−2.79718
$668$	−12813.4	−0.742164
$669$	−1049.46	−0.0606497
$670$	−7436.26	−0.428788
$671$	−24894.9	−1.43228
$672$	492.682	0.0282822
$673$	17747.0	1.01649	0.508243	−	0.861214i	$-0.330295\pi$
$673$	17747.0	1.01649	0.508243	+	0.861214i	$0.330295\pi$
$674$	3669.63	0.209717
$675$	−7764.85	−0.442769
$676$	0	0
$677$	10754.2	0.610511	0.305256	−	0.952270i	$-0.401258\pi$
$677$	10754.2	0.610511	0.305256	+	0.952270i	$0.401258\pi$
$678$	−5218.14	−0.295577
$679$	−5519.69	−0.311968
$680$	−1835.96	−0.103538
$681$	19061.9	1.07262
$682$	−9547.42	−0.536055
$683$	−23981.7	−1.34354	−0.671768	−	0.740762i	$-0.734465\pi$
$683$	−23981.7	−1.34354	−0.671768	+	0.740762i	$0.734465\pi$
$684$	−725.815	−0.0405734
$685$	−14911.3	−0.831723
$686$	5615.55	0.312540
$687$	3263.42	0.181233
$688$	−5704.13	−0.316087
$689$	0	0
$690$	−10000.1	−0.551737
$691$	29001.4	1.59662	0.798312	−	0.602244i	$-0.205727\pi$
$691$	29001.4	1.59662	0.798312	+	0.602244i	$0.205727\pi$
$692$	7597.53	0.417362
$693$	−3726.41	−0.204264
$694$	14518.6	0.794119
$695$	−14302.2	−0.780594
$696$	−8833.46	−0.481079
$697$	−11645.9	−0.632882
$698$	3590.47	0.194701
$699$	−17419.6	−0.942587
$700$	−877.608	−0.0473864
$701$	−31031.5	−1.67196	−0.835979	−	0.548761i	$-0.815100\pi$
$701$	−31031.5	−1.67196	−0.835979	+	0.548761i	$0.815100\pi$
$702$	0	0
$703$	1588.45	0.0852199
$704$	4183.84	0.223984
$705$	−18402.6	−0.983096
$706$	9104.75	0.485357
$707$	6149.16	0.327105
$708$	3376.66	0.179241
$709$	−4382.59	−0.232146	−0.116073	−	0.993241i	$-0.537031\pi$
$709$	−4382.59	−0.232146	−0.116073	+	0.993241i	$0.537031\pi$
$710$	−1125.42	−0.0594877
$711$	5230.23	0.275878
$712$	−2980.90	−0.156902
$713$	11678.5	0.613414
$714$	−828.380	−0.0434192
$715$	0	0
$716$	10603.6	0.553455
$717$	−8402.17	−0.437636
$718$	−20330.8	−1.05674
$719$	−28764.6	−1.49199	−0.745994	−	0.665953i	$-0.768025\pi$
$719$	−28764.6	−1.49199	−0.745994	+	0.665953i	$0.768025\pi$
$720$	1852.03	0.0958625
$721$	885.941	0.0457617
$722$	13360.3	0.688670
$723$	7238.89	0.372361
$724$	−9159.77	−0.470194
$725$	15734.9	0.806042
$726$	21568.3	1.10258
$727$	25408.5	1.29621	0.648107	−	0.761549i	$-0.275561\pi$
$727$	25408.5	1.29621	0.648107	+	0.761549i	$0.275561\pi$
$728$	0	0
$729$	17134.5	0.870525
$730$	−15108.3	−0.766006
$731$	9590.74	0.485262
$732$	5582.57	0.281882
$733$	−22341.8	−1.12580	−0.562902	−	0.826524i	$-0.690315\pi$
$733$	−22341.8	−1.12580	−0.562902	+	0.826524i	$0.690315\pi$
$734$	20803.1	1.04613
$735$	−10171.9	−0.510470
$736$	−5117.73	−0.256307
$737$	−28492.5	−1.42406
$738$	11747.8	0.585966
$739$	−8210.90	−0.408718	−0.204359	−	0.978896i	$-0.565511\pi$
$739$	−8210.90	−0.408718	−0.204359	+	0.978896i	$0.565511\pi$
$740$	−4053.18	−0.201349
$741$	0	0
$742$	2264.04	0.112016
$743$	−32752.3	−1.61718	−0.808590	−	0.588373i	$-0.799769\pi$
$743$	−32752.3	−1.61718	−0.808590	+	0.588373i	$0.799769\pi$
$744$	2140.97	0.105500
$745$	−7651.79	−0.376295
$746$	15906.8	0.780684
$747$	−3449.26	−0.168945
$748$	−7034.58	−0.343863
$749$	1642.87	0.0801455
$750$	11081.7	0.539527
$751$	8907.41	0.432804	0.216402	−	0.976304i	$-0.430568\pi$
$751$	8907.41	0.432804	0.216402	+	0.976304i	$0.430568\pi$
$752$	−9417.83	−0.456693
$753$	27361.0	1.32416
$754$	0	0
$755$	−17729.4	−0.854620
$756$	2498.43	0.120195
$757$	−10744.6	−0.515879	−0.257940	−	0.966161i	$-0.583044\pi$
$757$	−10744.6	−0.515879	−0.257940	+	0.966161i	$0.583044\pi$
$758$	−18029.6	−0.863938
$759$	−38316.2	−1.83240
$760$	912.656	0.0435599
$761$	−16997.8	−0.809683	−0.404841	−	0.914387i	$-0.632673\pi$
$761$	−16997.8	−0.809683	−0.404841	+	0.914387i	$0.632673\pi$
$762$	−8588.30	−0.408296
$763$	−5593.28	−0.265387
$764$	−1353.25	−0.0640825
$765$	−3113.94	−0.147170
$766$	4992.48	0.235490
$767$	0	0
$768$	−938.208	−0.0440816
$769$	19032.4	0.892492	0.446246	−	0.894910i	$-0.352761\pi$
$769$	19032.4	0.892492	0.446246	+	0.894910i	$0.352761\pi$
$770$	4685.68	0.219299
$771$	2033.88	0.0950045
$772$	−10736.2	−0.500524
$773$	23750.6	1.10511	0.552555	−	0.833477i	$-0.313653\pi$
$773$	23750.6	1.10511	0.552555	+	0.833477i	$0.313653\pi$
$774$	−9674.69	−0.449289
$775$	−3813.68	−0.176763
$776$	10511.1	0.486244
$777$	−1828.79	−0.0844368
$778$	−21793.7	−1.00430
$779$	5789.17	0.266263
$780$	0	0
$781$	−4312.12	−0.197567
$782$	8604.79	0.393487
$783$	−44795.2	−2.04451
$784$	−5205.62	−0.237136
$785$	29813.0	1.35550
$786$	−8272.42	−0.375404
$787$	17825.9	0.807403	0.403702	−	0.914891i	$-0.367723\pi$
$787$	17825.9	0.807403	0.403702	+	0.914891i	$0.367723\pi$
$788$	−7588.96	−0.343078
$789$	7304.34	0.329584
$790$	−6576.62	−0.296184
$791$	−2990.78	−0.134437
$792$	7096.16	0.318373
$793$	0	0
$794$	−21774.3	−0.973225
$795$	−8424.53	−0.375833
$796$	−9662.42	−0.430245
$797$	−26461.2	−1.17604	−0.588020	−	0.808846i	$-0.700092\pi$
$797$	−26461.2	−1.17604	−0.588020	+	0.808846i	$0.700092\pi$
$798$	411.788	0.0182671
$799$	15834.8	0.701122
$800$	1671.22	0.0738581
$801$	−5055.87	−0.223021
$802$	−19937.5	−0.877828
$803$	−57888.5	−2.54401
$804$	6389.32	0.280266
$805$	−5731.58	−0.250946
$806$	0	0
$807$	19085.8	0.832530
$808$	−11709.8	−0.509837
$809$	21873.3	0.950588	0.475294	−	0.879827i	$-0.342342\pi$
$809$	21873.3	0.950588	0.475294	+	0.879827i	$0.342342\pi$
$810$	−3046.10	−0.132135
$811$	39113.8	1.69355	0.846776	−	0.531949i	$-0.178540\pi$
$811$	39113.8	1.69355	0.846776	+	0.531949i	$0.178540\pi$
$812$	−5062.90	−0.218809
$813$	14970.6	0.645810
$814$	−15530.0	−0.668706
$815$	−3849.56	−0.165453
$816$	1577.47	0.0676748
$817$	−4767.57	−0.204157
$818$	−8072.65	−0.345053
$819$	0	0
$820$	−14772.0	−0.629097
$821$	−26296.8	−1.11786	−0.558931	−	0.829214i	$-0.688788\pi$
$821$	−26296.8	−1.11786	−0.558931	+	0.829214i	$0.688788\pi$
$822$	12811.9	0.543634
$823$	24681.1	1.04536	0.522678	−	0.852530i	$-0.324933\pi$
$823$	24681.1	1.04536	0.522678	+	0.852530i	$0.324933\pi$
$824$	−1687.09	−0.0713258
$825$	12512.3	0.528027
$826$	1935.33	0.0815241
$827$	−3081.68	−0.129578	−0.0647888	−	0.997899i	$-0.520637\pi$
$827$	−3081.68	−0.129578	−0.0647888	+	0.997899i	$0.520637\pi$
$828$	−8680.12	−0.364317
$829$	−8498.53	−0.356051	−0.178026	−	0.984026i	$-0.556971\pi$
$829$	−8498.53	−0.356051	−0.178026	+	0.984026i	$0.556971\pi$
$830$	4337.18	0.181380
$831$	−1605.19	−0.0670076
$832$	0	0
$833$	8752.57	0.364056
$834$	12288.6	0.510215
$835$	27327.1	1.13257
$836$	3496.90	0.144668
$837$	10857.0	0.448356
$838$	−30389.7	−1.25274
$839$	18680.1	0.768664	0.384332	−	0.923195i	$-0.374432\pi$
$839$	18680.1	0.768664	0.384332	+	0.923195i	$0.374432\pi$
$840$	−1050.74	−0.0431596
$841$	66385.3	2.72194
$842$	−20309.5	−0.831249
$843$	20018.8	0.817894
$844$	10673.7	0.435312
$845$	0	0
$846$	−15973.5	−0.649147
$847$	12361.9	0.501486
$848$	−4311.39	−0.174592
$849$	2919.72	0.118026
$850$	−2809.93	−0.113388
$851$	18996.5	0.765208
$852$	966.973	0.0388826
$853$	24129.9	0.968574	0.484287	−	0.874909i	$-0.339079\pi$
$853$	24129.9	0.968574	0.484287	+	0.874909i	$0.339079\pi$
$854$	3199.65	0.128208
$855$	1547.94	0.0619164
$856$	−3128.49	−0.124918
$857$	−7739.30	−0.308482	−0.154241	−	0.988033i	$-0.549293\pi$
$857$	−7739.30	−0.308482	−0.154241	+	0.988033i	$0.549293\pi$
$858$	0	0
$859$	4302.59	0.170899	0.0854496	−	0.996342i	$-0.472767\pi$
$859$	4302.59	0.170899	0.0854496	+	0.996342i	$0.472767\pi$
$860$	12165.2	0.482360
$861$	−6665.08	−0.263816
$862$	13233.7	0.522901
$863$	−16253.2	−0.641096	−0.320548	−	0.947232i	$-0.603867\pi$
$863$	−16253.2	−0.641096	−0.320548	+	0.947232i	$0.603867\pi$
$864$	−4757.73	−0.187340
$865$	−16203.2	−0.636910
$866$	−11395.2	−0.447142
$867$	15353.2	0.601410
$868$	1227.10	0.0479842
$869$	−25198.7	−0.983669
$870$	18839.1	0.734144
$871$	0	0
$872$	10651.2	0.413642
$873$	17827.7	0.691152
$874$	−4277.45	−0.165546
$875$	6351.46	0.245392
$876$	12981.2	0.500680
$877$	−9441.33	−0.363525	−0.181762	−	0.983342i	$-0.558180\pi$
$877$	−9441.33	−0.363525	−0.181762	+	0.983342i	$0.558180\pi$
$878$	−872.014	−0.0335183
$879$	6220.48	0.238693
$880$	−8922.88	−0.341807
$881$	−11872.4	−0.454021	−0.227011	−	0.973892i	$-0.572895\pi$
$881$	−11872.4	−0.454021	−0.227011	+	0.973892i	$0.572895\pi$
$882$	−8829.18	−0.337068
$883$	−493.891	−0.0188231	−0.00941153	−	0.999956i	$-0.502996\pi$
$883$	−493.891	−0.0188231	−0.00941153	+	0.999956i	$0.502996\pi$
$884$	0	0
$885$	−7201.40	−0.273528
$886$	−13219.2	−0.501249
$887$	20543.6	0.777664	0.388832	−	0.921309i	$-0.372879\pi$
$887$	20543.6	0.777664	0.388832	+	0.921309i	$0.372879\pi$
$888$	3482.54	0.131606
$889$	−4922.38	−0.185705
$890$	6357.36	0.239437
$891$	−11671.3	−0.438838
$892$	1145.43	0.0429953
$893$	−7871.52	−0.294972
$894$	6574.50	0.245955
$895$	−22614.2	−0.844592
$896$	−537.734	−0.0200496
$897$	0	0
$898$	−16030.6	−0.595711
$899$	−22001.0	−0.816212
$900$	2834.53	0.104983
$901$	7249.03	0.268036
$902$	−56599.7	−2.08932
$903$	5488.90	0.202280
$904$	5695.30	0.209539
$905$	19535.0	0.717532
$906$	15233.3	0.558600
$907$	−5386.61	−0.197199	−0.0985994	−	0.995127i	$-0.531436\pi$
$907$	−5386.61	−0.197199	−0.0985994	+	0.995127i	$0.531436\pi$
$908$	−20805.0	−0.760393
$909$	−19860.8	−0.724688
$910$	0	0
$911$	31793.5	1.15627	0.578137	−	0.815940i	$-0.303780\pi$
$911$	31793.5	1.15627	0.578137	+	0.815940i	$0.303780\pi$
$912$	−784.164	−0.0284718
$913$	16618.2	0.602389
$914$	9014.07	0.326213
$915$	−11905.9	−0.430162
$916$	−3561.83	−0.128478
$917$	−4741.34	−0.170745
$918$	7999.51	0.287607
$919$	−42991.4	−1.54315	−0.771575	−	0.636139i	$-0.780531\pi$
$919$	−42991.4	−1.54315	−0.771575	+	0.636139i	$0.780531\pi$
$920$	10914.6	0.391134
$921$	−34398.0	−1.23068
$922$	−4207.10	−0.150275
$923$	0	0
$924$	−4025.98	−0.143339
$925$	−6203.40	−0.220504
$926$	−10936.6	−0.388119
$927$	−2861.45	−0.101383
$928$	9641.21	0.341043
$929$	−40676.7	−1.43655	−0.718277	−	0.695757i	$-0.755069\pi$
$929$	−40676.7	−1.43655	−0.718277	+	0.695757i	$0.755069\pi$
$930$	−4566.04	−0.160996
$931$	−4350.91	−0.153164
$932$	19012.4	0.668212
$933$	12030.5	0.422145
$934$	12087.6	0.423467
$935$	15002.6	0.524748
$936$	0	0
$937$	21008.3	0.732456	0.366228	−	0.930525i	$-0.380649\pi$
$937$	21008.3	0.732456	0.366228	+	0.930525i	$0.380649\pi$
$938$	3662.04	0.127473
$939$	21713.2	0.754614
$940$	20085.4	0.696929
$941$	4525.50	0.156777	0.0783885	−	0.996923i	$-0.475023\pi$
$941$	4525.50	0.156777	0.0783885	+	0.996923i	$0.475023\pi$
$942$	−25615.6	−0.885989
$943$	69233.5	2.39083
$944$	−3685.43	−0.127066
$945$	−5328.40	−0.183421
$946$	46611.7	1.60198
$947$	27401.6	0.940267	0.470134	−	0.882595i	$-0.344206\pi$
$947$	27401.6	0.940267	0.470134	+	0.882595i	$0.344206\pi$
$948$	5650.70	0.193593
$949$	0	0
$950$	1396.82	0.0477040
$951$	−23704.0	−0.808259
$952$	904.129	0.0307805
$953$	26226.2	0.891449	0.445725	−	0.895170i	$-0.352946\pi$
$953$	26226.2	0.891449	0.445725	+	0.895170i	$0.352946\pi$
$954$	−7312.49	−0.248166
$955$	2886.08	0.0977921
$956$	9170.48	0.310245
$957$	72183.2	2.43819
$958$	16962.6	0.572063
$959$	7343.15	0.247260
$960$	2000.91	0.0672700
$961$	−24458.6	−0.821007
$962$	0	0
$963$	−5306.19	−0.177559
$964$	−7900.83	−0.263972
$965$	22897.1	0.763817
$966$	4924.63	0.164024
$967$	−20843.6	−0.693158	−0.346579	−	0.938021i	$-0.612657\pi$
$967$	−20843.6	−0.693158	−0.346579	+	0.938021i	$0.612657\pi$
$968$	−23540.6	−0.781635
$969$	1318.47	0.0437103
$970$	−22417.0	−0.742026
$971$	32577.3	1.07668	0.538339	−	0.842729i	$-0.319052\pi$
$971$	32577.3	1.07668	0.538339	+	0.842729i	$0.319052\pi$
$972$	−13440.1	−0.443510
$973$	7043.20	0.232060
$974$	13246.7	0.435782
$975$	0	0
$976$	−6093.05	−0.199830
$977$	43864.1	1.43637	0.718186	−	0.695851i	$-0.244973\pi$
$977$	43864.1	1.43637	0.718186	+	0.695851i	$0.244973\pi$
$978$	3307.58	0.108144
$979$	24358.6	0.795204
$980$	11102.0	0.361879
$981$	18065.4	0.587955
$982$	−32758.7	−1.06453
$983$	14758.8	0.478875	0.239437	−	0.970912i	$-0.423037\pi$
$983$	14758.8	0.478875	0.239437	+	0.970912i	$0.423037\pi$
$984$	12692.2	0.411193
$985$	16185.0	0.523549
$986$	−16210.4	−0.523575
$987$	9062.49	0.292262
$988$	0	0
$989$	−57016.0	−1.83317
$990$	−15134.0	−0.485848
$991$	48622.4	1.55857	0.779284	−	0.626671i	$-0.215583\pi$
$991$	48622.4	1.55857	0.779284	+	0.626671i	$0.215583\pi$
$992$	−2336.74	−0.0747900
$993$	−12860.9	−0.411006
$994$	554.220	0.0176849
$995$	20607.0	0.656569
$996$	−3726.55	−0.118554
$997$	−18812.3	−0.597586	−0.298793	−	0.954318i	$-0.596584\pi$
$997$	−18812.3	−0.597586	−0.298793	+	0.954318i	$0.596584\pi$
$998$	15831.4	0.502138
$999$	17660.2	0.559305

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.4.a.j.1.2	✓	3
13.2	odd	12		338.4.e.h.147.2		12
13.3	even	3		338.4.c.l.191.2		6
13.4	even	6		338.4.c.k.315.2		6
13.5	odd	4		338.4.b.f.337.5		6
13.6	odd	12		338.4.e.h.23.5		12
13.7	odd	12		338.4.e.h.23.2		12
13.8	odd	4		338.4.b.f.337.2		6
13.9	even	3		338.4.c.l.315.2		6
13.10	even	6		338.4.c.k.191.2		6
13.11	odd	12		338.4.e.h.147.5		12
13.12	even	2		338.4.a.k.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.4.a.j.1.2	✓	3	1.1	even	1	trivial
338.4.a.k.1.2	yes	3	13.12	even	2
338.4.b.f.337.2		6	13.8	odd	4
338.4.b.f.337.5		6	13.5	odd	4
338.4.c.k.191.2		6	13.10	even	6
338.4.c.k.315.2		6	13.4	even	6
338.4.c.l.191.2		6	13.3	even	3
338.4.c.l.315.2		6	13.9	even	3
338.4.e.h.23.2		12	13.7	odd	12
338.4.e.h.23.5		12	13.6	odd	12
338.4.e.h.147.2		12	13.2	odd	12
338.4.e.h.147.5		12	13.11	odd	12

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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.66487 q^{3} +4.00000 q^{4} -8.53079 q^{5} +7.32975 q^{6} +4.20105 q^{7} -8.00000 q^{8} -13.5687 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.66487 q^{3} +4.00000 q^{4} -8.53079 q^{5} +7.32975 q^{6} +4.20105 q^{7} -8.00000 q^{8} -13.5687 q^{9} +17.0616 q^{10} +65.3726 q^{11} -14.6595 q^{12} -8.40209 q^{14} +31.2643 q^{15} +16.0000 q^{16} -26.9019 q^{17} +27.1374 q^{18} +13.3730 q^{19} -34.1232 q^{20} -15.3963 q^{21} -130.745 q^{22} +159.929 q^{23} +29.3190 q^{24} -52.2255 q^{25} +148.679 q^{27} +16.8042 q^{28} -301.288 q^{29} -62.5286 q^{30} +73.0232 q^{31} -32.0000 q^{32} -239.582 q^{33} +53.8038 q^{34} -35.8383 q^{35} -54.2748 q^{36} +118.781 q^{37} -26.7459 q^{38} +68.2464 q^{40} +432.901 q^{41} +30.7926 q^{42} -356.508 q^{43} +261.490 q^{44} +115.752 q^{45} -319.858 q^{46} -588.614 q^{47} -58.6380 q^{48} -325.351 q^{49} +104.451 q^{50} +98.5921 q^{51} -269.462 q^{53} -297.358 q^{54} -557.680 q^{55} -33.6084 q^{56} -49.0102 q^{57} +602.576 q^{58} -230.340 q^{59} +125.057 q^{60} -380.816 q^{61} -146.046 q^{62} -57.0027 q^{63} +64.0000 q^{64} +479.164 q^{66} -435.848 q^{67} -107.608 q^{68} -586.120 q^{69} +71.6765 q^{70} -65.9622 q^{71} +108.550 q^{72} -885.517 q^{73} -237.562 q^{74} +191.400 q^{75} +53.4918 q^{76} +274.633 q^{77} -385.463 q^{79} -136.493 q^{80} -178.536 q^{81} -865.803 q^{82} +254.207 q^{83} -61.5852 q^{84} +229.495 q^{85} +713.016 q^{86} +1104.18 q^{87} -522.980 q^{88} +372.612 q^{89} -231.504 q^{90} +639.716 q^{92} -267.621 q^{93} +1177.23 q^{94} -114.082 q^{95} +117.276 q^{96} -1313.88 q^{97} +650.702 q^{98} -887.020 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9	\( 3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9} - 24 q^{10} + 82 q^{11} - 48 q^{12} + 54 q^{14} - 90 q^{15} + 48 q^{16} - 90 q^{17} - 18 q^{18} + 130 q^{19} + 48 q^{20} + 234 q^{21} - 164 q^{22} + 19 q^{23} + 96 q^{24} - 61 q^{25} - 69 q^{27} - 108 q^{28} - 101 q^{29} + 180 q^{30} + 519 q^{31} - 96 q^{32} - 146 q^{33} + 180 q^{34} - 458 q^{35} + 36 q^{36} + 84 q^{37} - 260 q^{38} - 96 q^{40} + 187 q^{41} - 468 q^{42} - 1205 q^{43} + 328 q^{44} + 645 q^{45} - 38 q^{46} - 536 q^{47} - 192 q^{48} - 184 q^{49} + 122 q^{50} - 207 q^{51} - 1095 q^{53} + 138 q^{54} - 526 q^{55} + 216 q^{56} - 1409 q^{57} + 202 q^{58} - 1413 q^{59} - 360 q^{60} - 1108 q^{61} - 1038 q^{62} - 1404 q^{63} + 192 q^{64} + 292 q^{66} - 1605 q^{67} - 360 q^{68} - 314 q^{69} + 916 q^{70} - 909 q^{71} - 72 q^{72} + 287 q^{73} - 168 q^{74} - 505 q^{75} + 520 q^{76} + 480 q^{77} - 1961 q^{79} + 192 q^{80} + 915 q^{81} - 374 q^{82} + 191 q^{83} + 936 q^{84} + 67 q^{85} + 2410 q^{86} + 1636 q^{87} - 656 q^{88} + 1091 q^{89} - 1290 q^{90} + 76 q^{92} - 1614 q^{93} + 1072 q^{94} + 1829 q^{95} + 384 q^{96} - 947 q^{97} + 368 q^{98} - 2057 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9 - 24 * q^10 + 82 * q^11 - 48 * q^12 + 54 * q^14 - 90 * q^15 + 48 * q^16 - 90 * q^17 - 18 * q^18 + 130 * q^19 + 48 * q^20 + 234 * q^21 - 164 * q^22 + 19 * q^23 + 96 * q^24 - 61 * q^25 - 69 * q^27 - 108 * q^28 - 101 * q^29 + 180 * q^30 + 519 * q^31 - 96 * q^32 - 146 * q^33 + 180 * q^34 - 458 * q^35 + 36 * q^36 + 84 * q^37 - 260 * q^38 - 96 * q^40 + 187 * q^41 - 468 * q^42 - 1205 * q^43 + 328 * q^44 + 645 * q^45 - 38 * q^46 - 536 * q^47 - 192 * q^48 - 184 * q^49 + 122 * q^50 - 207 * q^51 - 1095 * q^53 + 138 * q^54 - 526 * q^55 + 216 * q^56 - 1409 * q^57 + 202 * q^58 - 1413 * q^59 - 360 * q^60 - 1108 * q^61 - 1038 * q^62 - 1404 * q^63 + 192 * q^64 + 292 * q^66 - 1605 * q^67 - 360 * q^68 - 314 * q^69 + 916 * q^70 - 909 * q^71 - 72 * q^72 + 287 * q^73 - 168 * q^74 - 505 * q^75 + 520 * q^76 + 480 * q^77 - 1961 * q^79 + 192 * q^80 + 915 * q^81 - 374 * q^82 + 191 * q^83 + 936 * q^84 + 67 * q^85 + 2410 * q^86 + 1636 * q^87 - 656 * q^88 + 1091 * q^89 - 1290 * q^90 + 76 * q^92 - 1614 * q^93 + 1072 * q^94 + 1829 * q^95 + 384 * q^96 - 947 * q^97 + 368 * q^98 - 2057 * q^99

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