LMFDB - Embedded newform 786.4.a.g.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [786,4,Mod(1,786)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(786, 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("786.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 786 = 2 \cdot 3 \cdot 131 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	786.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:	$46.3755012645$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 436x^{6} + 1403x^{5} + 41156x^{4} - 104947x^{3} - 993314x^{2} + 1535040x + 1863168 $ x^8 - x^7 - 436x^6 + 1403x^5 + 41156x^4 - 104947x^3 - 993314x^2 + 1535040x + 1863168
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.824306$ of defining polynomial
Character	$\chi$	$=$	786.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +0.109133 q^{5} +6.00000 q^{6} +11.7436 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +0.109133 q^{5} +6.00000 q^{6} +11.7436 q^{7} -8.00000 q^{8} +9.00000 q^{9} -0.218267 q^{10} -21.1379 q^{11} -12.0000 q^{12} -4.17345 q^{13} -23.4872 q^{14} -0.327400 q^{15} +16.0000 q^{16} +15.0724 q^{17} -18.0000 q^{18} +44.1214 q^{19} +0.436534 q^{20} -35.2308 q^{21} +42.2757 q^{22} -86.0400 q^{23} +24.0000 q^{24} -124.988 q^{25} +8.34691 q^{26} -27.0000 q^{27} +46.9744 q^{28} -115.944 q^{29} +0.654800 q^{30} +253.860 q^{31} -32.0000 q^{32} +63.4136 q^{33} -30.1448 q^{34} +1.28162 q^{35} +36.0000 q^{36} +113.118 q^{37} -88.2429 q^{38} +12.5204 q^{39} -0.873067 q^{40} -133.278 q^{41} +70.4616 q^{42} -336.592 q^{43} -84.5514 q^{44} +0.982201 q^{45} +172.080 q^{46} +584.133 q^{47} -48.0000 q^{48} -205.088 q^{49} +249.976 q^{50} -45.2172 q^{51} -16.6938 q^{52} +163.782 q^{53} +54.0000 q^{54} -2.30685 q^{55} -93.9488 q^{56} -132.364 q^{57} +231.888 q^{58} -338.423 q^{59} -1.30960 q^{60} +858.720 q^{61} -507.720 q^{62} +105.692 q^{63} +64.0000 q^{64} -0.455463 q^{65} -126.827 q^{66} +598.030 q^{67} +60.2896 q^{68} +258.120 q^{69} -2.56324 q^{70} -204.646 q^{71} -72.0000 q^{72} +468.328 q^{73} -226.235 q^{74} +374.964 q^{75} +176.486 q^{76} -248.235 q^{77} -25.0407 q^{78} -845.879 q^{79} +1.74613 q^{80} +81.0000 q^{81} +266.556 q^{82} -1386.09 q^{83} -140.923 q^{84} +1.64490 q^{85} +673.185 q^{86} +347.832 q^{87} +169.103 q^{88} -932.433 q^{89} -1.96440 q^{90} -49.0114 q^{91} -344.160 q^{92} -761.580 q^{93} -1168.27 q^{94} +4.81512 q^{95} +96.0000 q^{96} -1523.84 q^{97} +410.176 q^{98} -190.241 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{2} - 24 q^{3} + 32 q^{4} + q^{5} + 48 q^{6} + 22 q^{7} - 64 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q - 16 * q^2 - 24 * q^3 + 32 * q^4 + q^5 + 48 * q^6 + 22 * q^7 - 64 * q^8 + 72 * q^9	\( 8 q - 16 q^{2} - 24 q^{3} + 32 q^{4} + q^{5} + 48 q^{6} + 22 q^{7} - 64 q^{8} + 72 q^{9} - 2 q^{10} - 77 q^{11} - 96 q^{12} + 29 q^{13} - 44 q^{14} - 3 q^{15} + 128 q^{16} + 59 q^{17} - 144 q^{18} - 150 q^{19} + 4 q^{20} - 66 q^{21} + 154 q^{22} - 269 q^{23} + 192 q^{24} + 273 q^{25} - 58 q^{26} - 216 q^{27} + 88 q^{28} - 123 q^{29} + 6 q^{30} - 86 q^{31} - 256 q^{32} + 231 q^{33} - 118 q^{34} - 292 q^{35} + 288 q^{36} + 412 q^{37} + 300 q^{38} - 87 q^{39} - 8 q^{40} - 114 q^{41} + 132 q^{42} + 1087 q^{43} - 308 q^{44} + 9 q^{45} + 538 q^{46} - 442 q^{47} - 384 q^{48} + 1292 q^{49} - 546 q^{50} - 177 q^{51} + 116 q^{52} + 5 q^{53} + 432 q^{54} + 456 q^{55} - 176 q^{56} + 450 q^{57} + 246 q^{58} + 252 q^{59} - 12 q^{60} + 1482 q^{61} + 172 q^{62} + 198 q^{63} + 512 q^{64} + 475 q^{65} - 462 q^{66} + 330 q^{67} + 236 q^{68} + 807 q^{69} + 584 q^{70} - 2946 q^{71} - 576 q^{72} - 214 q^{73} - 824 q^{74} - 819 q^{75} - 600 q^{76} - 960 q^{77} + 174 q^{78} - 64 q^{79} + 16 q^{80} + 648 q^{81} + 228 q^{82} - 276 q^{83} - 264 q^{84} + 80 q^{85} - 2174 q^{86} + 369 q^{87} + 616 q^{88} - 3177 q^{89} - 18 q^{90} - 781 q^{91} - 1076 q^{92} + 258 q^{93} + 884 q^{94} - 2700 q^{95} + 768 q^{96} + 200 q^{97} - 2584 q^{98} - 693 q^{99}+O(q^{100}) \) 8 * q - 16 * q^2 - 24 * q^3 + 32 * q^4 + q^5 + 48 * q^6 + 22 * q^7 - 64 * q^8 + 72 * q^9 - 2 * q^10 - 77 * q^11 - 96 * q^12 + 29 * q^13 - 44 * q^14 - 3 * q^15 + 128 * q^16 + 59 * q^17 - 144 * q^18 - 150 * q^19 + 4 * q^20 - 66 * q^21 + 154 * q^22 - 269 * q^23 + 192 * q^24 + 273 * q^25 - 58 * q^26 - 216 * q^27 + 88 * q^28 - 123 * q^29 + 6 * q^30 - 86 * q^31 - 256 * q^32 + 231 * q^33 - 118 * q^34 - 292 * q^35 + 288 * q^36 + 412 * q^37 + 300 * q^38 - 87 * q^39 - 8 * q^40 - 114 * q^41 + 132 * q^42 + 1087 * q^43 - 308 * q^44 + 9 * q^45 + 538 * q^46 - 442 * q^47 - 384 * q^48 + 1292 * q^49 - 546 * q^50 - 177 * q^51 + 116 * q^52 + 5 * q^53 + 432 * q^54 + 456 * q^55 - 176 * q^56 + 450 * q^57 + 246 * q^58 + 252 * q^59 - 12 * q^60 + 1482 * q^61 + 172 * q^62 + 198 * q^63 + 512 * q^64 + 475 * q^65 - 462 * q^66 + 330 * q^67 + 236 * q^68 + 807 * q^69 + 584 * q^70 - 2946 * q^71 - 576 * q^72 - 214 * q^73 - 824 * q^74 - 819 * q^75 - 600 * q^76 - 960 * q^77 + 174 * q^78 - 64 * q^79 + 16 * q^80 + 648 * q^81 + 228 * q^82 - 276 * q^83 - 264 * q^84 + 80 * q^85 - 2174 * q^86 + 369 * q^87 + 616 * q^88 - 3177 * q^89 - 18 * q^90 - 781 * q^91 - 1076 * q^92 + 258 * q^93 + 884 * q^94 - 2700 * q^95 + 768 * q^96 + 200 * q^97 - 2584 * q^98 - 693 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	0.109133	0.00976119	0.00488059	−	0.999988i	$-0.498446\pi$
$5$	0.109133	0.00976119	0.00488059	+	0.999988i	$0.498446\pi$
$6$	6.00000	0.408248
$7$	11.7436	0.634095	0.317047	−	0.948410i	$-0.397309\pi$
$7$	11.7436	0.634095	0.317047	+	0.948410i	$0.397309\pi$
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−0.218267	−0.00690220
$11$	−21.1379	−0.579391	−0.289696	−	0.957119i	$-0.593554\pi$
$11$	−21.1379	−0.579391	−0.289696	+	0.957119i	$0.593554\pi$
$12$	−12.0000	−0.288675
$13$	−4.17345	−0.0890391	−0.0445195	−	0.999009i	$-0.514176\pi$
$13$	−4.17345	−0.0890391	−0.0445195	+	0.999009i	$0.514176\pi$
$14$	−23.4872	−0.448373
$15$	−0.327400	−0.00563562
$16$	16.0000	0.250000
$17$	15.0724	0.215035	0.107517	−	0.994203i	$-0.465710\pi$
$17$	15.0724	0.215035	0.107517	+	0.994203i	$0.465710\pi$
$18$	−18.0000	−0.235702
$19$	44.1214	0.532745	0.266372	−	0.963870i	$-0.414175\pi$
$19$	44.1214	0.532745	0.266372	+	0.963870i	$0.414175\pi$
$20$	0.436534	0.00488059
$21$	−35.2308	−0.366095
$22$	42.2757	0.409691
$23$	−86.0400	−0.780025	−0.390013	−	0.920809i	$-0.627529\pi$
$23$	−86.0400	−0.780025	−0.390013	+	0.920809i	$0.627529\pi$
$24$	24.0000	0.204124
$25$	−124.988	−0.999905
$26$	8.34691	0.0629601
$27$	−27.0000	−0.192450
$28$	46.9744	0.317047
$29$	−115.944	−0.742422	−0.371211	−	0.928549i	$-0.621057\pi$
$29$	−115.944	−0.742422	−0.371211	+	0.928549i	$0.621057\pi$
$30$	0.654800	0.00398499
$31$	253.860	1.47079	0.735397	−	0.677637i	$-0.236996\pi$
$31$	253.860	1.47079	0.735397	+	0.677637i	$0.236996\pi$
$32$	−32.0000	−0.176777
$33$	63.4136	0.334512
$34$	−30.1448	−0.152053
$35$	1.28162	0.00618952
$36$	36.0000	0.166667
$37$	113.118	0.502607	0.251303	−	0.967908i	$-0.419141\pi$
$37$	113.118	0.502607	0.251303	+	0.967908i	$0.419141\pi$
$38$	−88.2429	−0.376708
$39$	12.5204	0.0514067
$40$	−0.873067	−0.00345110
$41$	−133.278	−0.507671	−0.253835	−	0.967247i	$-0.581692\pi$
$41$	−133.278	−0.507671	−0.253835	+	0.967247i	$0.581692\pi$
$42$	70.4616	0.258868
$43$	−336.592	−1.19372	−0.596859	−	0.802346i	$-0.703585\pi$
$43$	−336.592	−1.19372	−0.596859	+	0.802346i	$0.703585\pi$
$44$	−84.5514	−0.289696
$45$	0.982201	0.00325373
$46$	172.080	0.551561
$47$	584.133	1.81286	0.906431	−	0.422354i	$-0.138796\pi$
$47$	584.133	1.81286	0.906431	+	0.422354i	$0.138796\pi$
$48$	−48.0000	−0.144338
$49$	−205.088	−0.597924
$50$	249.976	0.707039
$51$	−45.2172	−0.124151
$52$	−16.6938	−0.0445195
$53$	163.782	0.424477	0.212238	−	0.977218i	$-0.431925\pi$
$53$	163.782	0.424477	0.212238	+	0.977218i	$0.431925\pi$
$54$	54.0000	0.136083
$55$	−2.30685	−0.00565555
$56$	−93.9488	−0.224186
$57$	−132.364	−0.307580
$58$	231.888	0.524971
$59$	−338.423	−0.746762	−0.373381	−	0.927678i	$-0.621802\pi$
$59$	−338.423	−0.746762	−0.373381	+	0.927678i	$0.621802\pi$
$60$	−1.30960	−0.00281781
$61$	858.720	1.80242	0.901212	−	0.433379i	$-0.142679\pi$
$61$	858.720	1.80242	0.901212	+	0.433379i	$0.142679\pi$
$62$	−507.720	−1.04001
$63$	105.692	0.211365
$64$	64.0000	0.125000
$65$	−0.455463	−0.000869127 0
$66$	−126.827	−0.236535
$67$	598.030	1.09046	0.545231	−	0.838286i	$-0.316442\pi$
$67$	598.030	1.09046	0.545231	+	0.838286i	$0.316442\pi$
$68$	60.2896	0.107517
$69$	258.120	0.450348
$70$	−2.56324	−0.00437665
$71$	−204.646	−0.342071	−0.171035	−	0.985265i	$-0.554711\pi$
$71$	−204.646	−0.342071	−0.171035	+	0.985265i	$0.554711\pi$
$72$	−72.0000	−0.117851
$73$	468.328	0.750872	0.375436	−	0.926848i	$-0.377493\pi$
$73$	468.328	0.750872	0.375436	+	0.926848i	$0.377493\pi$
$74$	−226.235	−0.355396
$75$	374.964	0.577295
$76$	176.486	0.266372
$77$	−248.235	−0.367389
$78$	−25.0407	−0.0363500
$79$	−845.879	−1.20467	−0.602334	−	0.798244i	$-0.705763\pi$
$79$	−845.879	−1.20467	−0.602334	+	0.798244i	$0.705763\pi$
$80$	1.74613	0.00244030
$81$	81.0000	0.111111
$82$	266.556	0.358977
$83$	−1386.09	−1.83305	−0.916525	−	0.399978i	$-0.869018\pi$
$83$	−1386.09	−1.83305	−0.916525	+	0.399978i	$0.869018\pi$
$84$	−140.923	−0.183047
$85$	1.64490	0.00209900
$86$	673.185	0.844086
$87$	347.832	0.428637
$88$	169.103	0.204846
$89$	−932.433	−1.11054	−0.555268	−	0.831672i	$-0.687384\pi$
$89$	−932.433	−1.11054	−0.555268	+	0.831672i	$0.687384\pi$
$90$	−1.96440	−0.00230073
$91$	−49.0114	−0.0564592
$92$	−344.160	−0.390013
$93$	−761.580	−0.849163
$94$	−1168.27	−1.28189
$95$	4.81512	0.00520022
$96$	96.0000	0.102062
$97$	−1523.84	−1.59508	−0.797538	−	0.603269i	$-0.793865\pi$
$97$	−1523.84	−1.59508	−0.797538	+	0.603269i	$0.793865\pi$
$98$	410.176	0.422796
$99$	−190.241	−0.193130
$100$	−499.952	−0.499952
$101$	447.507	0.440877	0.220439	−	0.975401i	$-0.429251\pi$
$101$	447.507	0.440877	0.220439	+	0.975401i	$0.429251\pi$
$102$	90.4344	0.0877877
$103$	−1305.06	−1.24846	−0.624232	−	0.781239i	$-0.714588\pi$
$103$	−1305.06	−1.24846	−0.624232	+	0.781239i	$0.714588\pi$
$104$	33.3876	0.0314801
$105$	−3.84486	−0.00357352
$106$	−327.565	−0.300150
$107$	−1291.14	−1.16654	−0.583268	−	0.812280i	$-0.698226\pi$
$107$	−1291.14	−1.16654	−0.583268	+	0.812280i	$0.698226\pi$
$108$	−108.000	−0.0962250
$109$	1398.85	1.22922	0.614611	−	0.788830i	$-0.289313\pi$
$109$	1398.85	1.22922	0.614611	+	0.788830i	$0.289313\pi$
$110$	4.61369	0.00399907
$111$	−339.353	−0.290180
$112$	187.898	0.158524
$113$	662.379	0.551428	0.275714	−	0.961240i	$-0.411086\pi$
$113$	662.379	0.551428	0.275714	+	0.961240i	$0.411086\pi$
$114$	264.729	0.217492
$115$	−9.38984	−0.00761397
$116$	−463.775	−0.371211
$117$	−37.5611	−0.0296797
$118$	676.846	0.528040
$119$	177.004	0.136353
$120$	2.61920	0.00199249
$121$	−884.191	−0.664306
$122$	−1717.44	−1.27451
$123$	399.833	0.293104
$124$	1015.44	0.735397
$125$	−27.2821	−0.0195214
$126$	−211.385	−0.149458
$127$	−390.369	−0.272753	−0.136377	−	0.990657i	$-0.543546\pi$
$127$	−390.369	−0.272753	−0.136377	+	0.990657i	$0.543546\pi$
$128$	−128.000	−0.0883883
$129$	1009.78	0.689193
$130$	0.910926	0.000614566 0
$131$	131.000	0.0873704
$131$	131.000	0.0873704
$132$	253.654	0.167256
$133$	518.145	0.337811
$134$	−1196.06	−0.771073
$135$	−2.94660	−0.00187854
$136$	−120.579	−0.0760264
$137$	−2224.74	−1.38739	−0.693696	−	0.720268i	$-0.744019\pi$
$137$	−2224.74	−1.38739	−0.693696	+	0.720268i	$0.744019\pi$
$138$	−516.240	−0.318444
$139$	−720.007	−0.439354	−0.219677	−	0.975573i	$-0.570500\pi$
$139$	−720.007	−0.439354	−0.219677	+	0.975573i	$0.570500\pi$
$140$	5.12648	0.00309476
$141$	−1752.40	−1.04666
$142$	409.292	0.241881
$143$	88.2178	0.0515884
$144$	144.000	0.0833333
$145$	−12.6533	−0.00724692
$146$	−936.656	−0.530946
$147$	615.263	0.345211
$148$	452.471	0.251303
$149$	−3304.46	−1.81686	−0.908428	−	0.418041i	$-0.862717\pi$
$149$	−3304.46	−1.81686	−0.908428	+	0.418041i	$0.862717\pi$
$150$	−749.929	−0.408209
$151$	−3295.23	−1.77591	−0.887955	−	0.459931i	$-0.847874\pi$
$151$	−3295.23	−1.77591	−0.887955	+	0.459931i	$0.847874\pi$
$152$	−352.972	−0.188354
$153$	135.652	0.0716783
$154$	496.469	0.259783
$155$	27.7046	0.0143567
$156$	50.0814	0.0257034
$157$	3258.74	1.65653	0.828266	−	0.560335i	$-0.189328\pi$
$157$	3258.74	1.65653	0.828266	+	0.560335i	$0.189328\pi$
$158$	1691.76	0.851829
$159$	−491.347	−0.245072
$160$	−3.49227	−0.00172555
$161$	−1010.42	−0.494610
$162$	−162.000	−0.0785674
$163$	273.339	0.131347	0.0656735	−	0.997841i	$-0.479080\pi$
$163$	273.339	0.131347	0.0656735	+	0.997841i	$0.479080\pi$
$164$	−533.111	−0.253835
$165$	6.92054	0.00326523
$166$	2772.18	1.29616
$167$	598.861	0.277493	0.138746	−	0.990328i	$-0.455693\pi$
$167$	598.861	0.277493	0.138746	+	0.990328i	$0.455693\pi$
$168$	281.846	0.129434
$169$	−2179.58	−0.992072
$170$	−3.28981	−0.00148422
$171$	397.093	0.177582
$172$	−1346.37	−0.596859
$173$	−2753.55	−1.21011	−0.605053	−	0.796185i	$-0.706848\pi$
$173$	−2753.55	−1.21011	−0.605053	+	0.796185i	$0.706848\pi$
$174$	−695.663	−0.303092
$175$	−1467.81	−0.634034
$176$	−338.206	−0.144848
$177$	1015.27	0.431143
$178$	1864.87	0.785267
$179$	−1712.89	−0.715238	−0.357619	−	0.933868i	$-0.616411\pi$
$179$	−1712.89	−0.715238	−0.357619	+	0.933868i	$0.616411\pi$
$180$	3.92880	0.00162686
$181$	−1559.91	−0.640593	−0.320297	−	0.947317i	$-0.603783\pi$
$181$	−1559.91	−0.640593	−0.320297	+	0.947317i	$0.603783\pi$
$182$	98.0228	0.0399227
$183$	−2576.16	−1.04063
$184$	688.320	0.275781
$185$	12.3449	0.00490604
$186$	1523.16	0.600449
$187$	−318.598	−0.124589
$188$	2336.53	0.906431
$189$	−317.077	−0.122032
$190$	−9.63025	−0.00367711
$191$	−66.3782	−0.0251464	−0.0125732	−	0.999921i	$-0.504002\pi$
$191$	−66.3782	−0.0251464	−0.0125732	+	0.999921i	$0.504002\pi$
$192$	−192.000	−0.0721688
$193$	1261.78	0.470595	0.235298	−	0.971923i	$-0.424394\pi$
$193$	1261.78	0.470595	0.235298	+	0.971923i	$0.424394\pi$
$194$	3047.68	1.12789
$195$	1.36639	0.000501791 0
$196$	−820.351	−0.298962
$197$	−2516.64	−0.910168	−0.455084	−	0.890448i	$-0.650391\pi$
$197$	−2516.64	−0.910168	−0.455084	+	0.890448i	$0.650391\pi$
$198$	380.481	0.136564
$199$	−2196.07	−0.782288	−0.391144	−	0.920330i	$-0.627920\pi$
$199$	−2196.07	−0.782288	−0.391144	+	0.920330i	$0.627920\pi$
$200$	999.905	0.353520
$201$	−1794.09	−0.629579
$202$	−895.013	−0.311747
$203$	−1361.60	−0.470766
$204$	−180.869	−0.0620753
$205$	−14.5451	−0.00495547
$206$	2610.13	0.882798
$207$	−774.360	−0.260008
$208$	−66.7753	−0.0222598
$209$	−932.633	−0.308668
$210$	7.68972	0.00252686
$211$	−1624.89	−0.530153	−0.265077	−	0.964227i	$-0.585397\pi$
$211$	−1624.89	−0.530153	−0.265077	+	0.964227i	$0.585397\pi$
$212$	655.130	0.212238
$213$	613.939	0.197495
$214$	2582.28	0.824865
$215$	−36.7335	−0.0116521
$216$	216.000	0.0680414
$217$	2981.23	0.932623
$218$	−2797.70	−0.869192
$219$	−1404.98	−0.433516
$220$	−9.22738	−0.00282777
$221$	−62.9040	−0.0191465
$222$	678.706	0.205188
$223$	6198.94	1.86149	0.930743	−	0.365673i	$-0.119161\pi$
$223$	6198.94	1.86149	0.930743	+	0.365673i	$0.119161\pi$
$224$	−375.795	−0.112093
$225$	−1124.89	−0.333302
$226$	−1324.76	−0.389918
$227$	−483.179	−0.141276	−0.0706381	−	0.997502i	$-0.522504\pi$
$227$	−483.179	−0.141276	−0.0706381	+	0.997502i	$0.522504\pi$
$228$	−529.457	−0.153790
$229$	838.827	0.242058	0.121029	−	0.992649i	$-0.461381\pi$
$229$	838.827	0.242058	0.121029	+	0.992649i	$0.461381\pi$
$230$	18.7797	0.00538389
$231$	744.704	0.212112
$232$	927.551	0.262486
$233$	−1338.17	−0.376251	−0.188126	−	0.982145i	$-0.560241\pi$
$233$	−1338.17	−0.376251	−0.188126	+	0.982145i	$0.560241\pi$
$234$	75.1222	0.0209867
$235$	63.7484	0.0176957
$236$	−1353.69	−0.373381
$237$	2537.64	0.695516
$238$	−354.009	−0.0964158
$239$	−4329.90	−1.17187	−0.585937	−	0.810357i	$-0.699273\pi$
$239$	−4329.90	−1.17187	−0.585937	+	0.810357i	$0.699273\pi$
$240$	−5.23840	−0.00140891
$241$	3027.31	0.809154	0.404577	−	0.914504i	$-0.367419\pi$
$241$	3027.31	0.809154	0.404577	+	0.914504i	$0.367419\pi$
$242$	1768.38	0.469735
$243$	−243.000	−0.0641500
$244$	3434.88	0.901212
$245$	−22.3819	−0.00583645
$246$	−799.667	−0.207256
$247$	−184.139	−0.0474351
$248$	−2030.88	−0.520004
$249$	4158.27	1.05831
$250$	54.5641	0.0138037
$251$	5605.94	1.40974	0.704868	−	0.709339i	$-0.251006\pi$
$251$	5605.94	1.40974	0.704868	+	0.709339i	$0.251006\pi$
$252$	422.770	0.105682
$253$	1818.70	0.451940
$254$	780.739	0.192866
$255$	−4.93471	−0.00121186
$256$	256.000	0.0625000
$257$	−7170.97	−1.74052	−0.870258	−	0.492596i	$-0.836048\pi$
$257$	−7170.97	−1.74052	−0.870258	+	0.492596i	$0.836048\pi$
$258$	−2019.55	−0.487333
$259$	1328.41	0.318700
$260$	−1.82185	−0.000434564 0
$261$	−1043.49	−0.247474
$262$	−262.000	−0.0617802
$263$	−5059.42	−1.18623	−0.593113	−	0.805119i	$-0.702101\pi$
$263$	−5059.42	−1.18623	−0.593113	+	0.805119i	$0.702101\pi$
$264$	−507.308	−0.118268
$265$	17.8741	0.00414340
$266$	−1036.29	−0.238868
$267$	2797.30	0.641168
$268$	2392.12	0.545231
$269$	274.958	0.0623216	0.0311608	−	0.999514i	$-0.490080\pi$
$269$	274.958	0.0623216	0.0311608	+	0.999514i	$0.490080\pi$
$270$	5.89320	0.00132833
$271$	5202.35	1.16613	0.583063	−	0.812427i	$-0.301854\pi$
$271$	5202.35	1.16613	0.583063	+	0.812427i	$0.301854\pi$
$272$	241.158	0.0537587
$273$	147.034	0.0325967
$274$	4449.49	0.981034
$275$	2641.98	0.579336
$276$	1032.48	0.225174
$277$	−6119.82	−1.32745	−0.663726	−	0.747976i	$-0.731026\pi$
$277$	−6119.82	−1.32745	−0.663726	+	0.747976i	$0.731026\pi$
$278$	1440.01	0.310670
$279$	2284.74	0.490265
$280$	−10.2530	−0.00218833
$281$	5239.59	1.11234	0.556171	−	0.831068i	$-0.312270\pi$
$281$	5239.59	1.11234	0.556171	+	0.831068i	$0.312270\pi$
$282$	3504.80	0.740098
$283$	−5313.01	−1.11599	−0.557996	−	0.829844i	$-0.688429\pi$
$283$	−5313.01	−1.11599	−0.557996	+	0.829844i	$0.688429\pi$
$284$	−818.585	−0.171035
$285$	−14.4454	−0.00300235
$286$	−176.436	−0.0364785
$287$	−1565.16	−0.321911
$288$	−288.000	−0.0589256
$289$	−4685.82	−0.953760
$290$	25.3067	0.00512435
$291$	4571.51	0.920917
$292$	1873.31	0.375436
$293$	9033.32	1.80113	0.900567	−	0.434717i	$-0.143152\pi$
$293$	9033.32	1.80113	0.900567	+	0.434717i	$0.143152\pi$
$294$	−1230.53	−0.244101
$295$	−36.9333	−0.00728928
$296$	−904.942	−0.177698
$297$	570.722	0.111504
$298$	6608.91	1.28471
$299$	359.084	0.0694527
$300$	1499.86	0.288648
$301$	−3952.81	−0.756930
$302$	6590.47	1.25576
$303$	−1342.52	−0.254541
$304$	705.943	0.133186
$305$	93.7151	0.0175938
$306$	−271.303	−0.0506842
$307$	439.299	0.0816681	0.0408340	−	0.999166i	$-0.486999\pi$
$307$	439.299	0.0816681	0.0408340	+	0.999166i	$0.486999\pi$
$308$	−992.938	−0.183694
$309$	3915.19	0.720801
$310$	−55.4092	−0.0101517
$311$	−8362.15	−1.52468	−0.762338	−	0.647179i	$-0.775948\pi$
$311$	−8362.15	−1.52468	−0.762338	+	0.647179i	$0.775948\pi$
$312$	−100.163	−0.0181750
$313$	10006.6	1.80706	0.903528	−	0.428528i	$-0.140968\pi$
$313$	10006.6	1.80706	0.903528	+	0.428528i	$0.140968\pi$
$314$	−6517.48	−1.17135
$315$	11.5346	0.00206317
$316$	−3383.52	−0.602334
$317$	−4607.82	−0.816407	−0.408204	−	0.912891i	$-0.633845\pi$
$317$	−4607.82	−0.816407	−0.408204	+	0.912891i	$0.633845\pi$
$318$	982.695	0.173292
$319$	2450.80	0.430153
$320$	6.98454	0.00122015
$321$	3873.42	0.673500
$322$	2020.84	0.349742
$323$	665.016	0.114559
$324$	324.000	0.0555556
$325$	521.632	0.0890306
$326$	−546.678	−0.0928764
$327$	−4196.54	−0.709692
$328$	1066.22	0.179489
$329$	6859.82	1.14953
$330$	−13.8411	−0.00230887
$331$	3542.93	0.588329	0.294165	−	0.955755i	$-0.404959\pi$
$331$	3542.93	0.588329	0.294165	+	0.955755i	$0.404959\pi$
$332$	−5544.36	−0.916525
$333$	1018.06	0.167536
$334$	−1197.72	−0.196217
$335$	65.2650	0.0106442
$336$	−563.693	−0.0915237
$337$	4042.00	0.653359	0.326679	−	0.945135i	$-0.394070\pi$
$337$	4042.00	0.653359	0.326679	+	0.945135i	$0.394070\pi$
$338$	4359.16	0.701501
$339$	−1987.14	−0.318367
$340$	6.57961	0.00104950
$341$	−5366.05	−0.852165
$342$	−794.186	−0.125569
$343$	−6436.53	−1.01324
$344$	2692.74	0.422043
$345$	28.1695	0.00439593
$346$	5507.10	0.855675
$347$	−7191.33	−1.11254	−0.556269	−	0.831002i	$-0.687768\pi$
$347$	−7191.33	−1.11254	−0.556269	+	0.831002i	$0.687768\pi$
$348$	1391.33	0.214319
$349$	9984.32	1.53137	0.765685	−	0.643216i	$-0.222400\pi$
$349$	9984.32	1.53137	0.765685	+	0.643216i	$0.222400\pi$
$350$	2935.62	0.448330
$351$	112.683	0.0171356
$352$	676.411	0.102423
$353$	8733.26	1.31678	0.658392	−	0.752675i	$-0.271237\pi$
$353$	8733.26	1.31678	0.658392	+	0.752675i	$0.271237\pi$
$354$	−2030.54	−0.304864
$355$	−22.3337	−0.00333902
$356$	−3729.73	−0.555268
$357$	−531.013	−0.0787232
$358$	3425.79	0.505750
$359$	8366.30	1.22996	0.614981	−	0.788542i	$-0.289164\pi$
$359$	8366.30	1.22996	0.614981	+	0.788542i	$0.289164\pi$
$360$	−7.85761	−0.00115037
$361$	−4912.30	−0.716183
$362$	3119.82	0.452968
$363$	2652.57	0.383537
$364$	−196.046	−0.0282296
$365$	51.1102	0.00732940
$366$	5152.32	0.735836
$367$	10479.8	1.49057	0.745284	−	0.666747i	$-0.232314\pi$
$367$	10479.8	1.49057	0.745284	+	0.666747i	$0.232314\pi$
$368$	−1376.64	−0.195006
$369$	−1199.50	−0.169224
$370$	−24.6898	−0.00346909
$371$	1923.40	0.269158
$372$	−3046.32	−0.424582
$373$	7882.13	1.09416	0.547080	−	0.837081i	$-0.315739\pi$
$373$	7882.13	1.09416	0.547080	+	0.837081i	$0.315739\pi$
$374$	637.196	0.0880980
$375$	81.8462	0.0112707
$376$	−4673.06	−0.640943
$377$	483.886	0.0661045
$378$	634.154	0.0862894
$379$	−11147.5	−1.51084	−0.755419	−	0.655242i	$-0.772567\pi$
$379$	−11147.5	−1.51084	−0.755419	+	0.655242i	$0.772567\pi$
$380$	19.2605	0.00260011
$381$	1171.11	0.157474
$382$	132.756	0.0177812
$383$	−3374.76	−0.450241	−0.225120	−	0.974331i	$-0.572278\pi$
$383$	−3374.76	−0.450241	−0.225120	+	0.974331i	$0.572278\pi$
$384$	384.000	0.0510310
$385$	−27.0907	−0.00358615
$386$	−2523.56	−0.332761
$387$	−3029.33	−0.397906
$388$	−6095.35	−0.797538
$389$	5057.19	0.659151	0.329576	−	0.944129i	$-0.393094\pi$
$389$	5057.19	0.659151	0.329576	+	0.944129i	$0.393094\pi$
$390$	−2.73278	−0.000354820 0
$391$	−1296.83	−0.167733
$392$	1640.70	0.211398
$393$	−393.000	−0.0504433
$394$	5033.28	0.643586
$395$	−92.3136	−0.0117590
$396$	−760.963	−0.0965652
$397$	7177.48	0.907374	0.453687	−	0.891161i	$-0.350108\pi$
$397$	7177.48	0.907374	0.453687	+	0.891161i	$0.350108\pi$
$398$	4392.14	0.553161
$399$	−1554.43	−0.195035
$400$	−1999.81	−0.249976
$401$	3231.50	0.402427	0.201213	−	0.979547i	$-0.435512\pi$
$401$	3231.50	0.402427	0.201213	+	0.979547i	$0.435512\pi$
$402$	3588.18	0.445179
$403$	−1059.47	−0.130958
$404$	1790.03	0.220439
$405$	8.83981	0.00108458
$406$	2723.20	0.332882
$407$	−2391.07	−0.291206
$408$	361.738	0.0438938
$409$	−13359.5	−1.61512	−0.807559	−	0.589787i	$-0.799212\pi$
$409$	−13359.5	−1.61512	−0.807559	+	0.589787i	$0.799212\pi$
$410$	29.0901	0.00350405
$411$	6674.23	0.801011
$412$	−5220.26	−0.624232
$413$	−3974.31	−0.473518
$414$	1548.72	0.183854
$415$	−151.269	−0.0178927
$416$	133.551	0.0157400
$417$	2160.02	0.253661
$418$	1865.27	0.218261
$419$	−9223.51	−1.07541	−0.537706	−	0.843132i	$-0.680709\pi$
$419$	−9223.51	−1.07541	−0.537706	+	0.843132i	$0.680709\pi$
$420$	−15.3794	−0.00178676
$421$	9932.01	1.14978	0.574889	−	0.818231i	$-0.305045\pi$
$421$	9932.01	1.14978	0.574889	+	0.818231i	$0.305045\pi$
$422$	3249.79	0.374875
$423$	5257.19	0.604287
$424$	−1310.26	−0.150075
$425$	−1883.87	−0.215015
$426$	−1227.88	−0.139650
$427$	10084.5	1.14291
$428$	−5164.56	−0.583268
$429$	−264.654	−0.0297846
$430$	73.4669	0.00823928
$431$	5551.94	0.620481	0.310240	−	0.950658i	$-0.399590\pi$
$431$	5551.94	0.620481	0.310240	+	0.950658i	$0.399590\pi$
$432$	−432.000	−0.0481125
$433$	−4941.40	−0.548426	−0.274213	−	0.961669i	$-0.588417\pi$
$433$	−4941.40	−0.548426	−0.274213	+	0.961669i	$0.588417\pi$
$434$	−5962.46	−0.659464
$435$	37.9600	0.00418401
$436$	5595.39	0.614611
$437$	−3796.21	−0.415555
$438$	2809.97	0.306542
$439$	−13763.2	−1.49632	−0.748158	−	0.663521i	$-0.769062\pi$
$439$	−13763.2	−1.49632	−0.748158	+	0.663521i	$0.769062\pi$
$440$	18.4548	0.00199954
$441$	−1845.79	−0.199308
$442$	125.808	0.0135386
$443$	−12514.3	−1.34215	−0.671073	−	0.741391i	$-0.734166\pi$
$443$	−12514.3	−1.34215	−0.671073	+	0.741391i	$0.734166\pi$
$444$	−1357.41	−0.145090
$445$	−101.760	−0.0108401
$446$	−12397.9	−1.31627
$447$	9913.37	1.04896
$448$	751.591	0.0792619
$449$	−4972.29	−0.522622	−0.261311	−	0.965255i	$-0.584155\pi$
$449$	−4972.29	−0.522622	−0.261311	+	0.965255i	$0.584155\pi$
$450$	2249.79	0.235680
$451$	2817.21	0.294140
$452$	2649.51	0.275714
$453$	9885.70	1.02532
$454$	966.358	0.0998974
$455$	−5.34878	−0.000551109 0
$456$	1058.91	0.108746
$457$	−11770.4	−1.20481	−0.602405	−	0.798191i	$-0.705791\pi$
$457$	−11770.4	−1.20481	−0.602405	+	0.798191i	$0.705791\pi$
$458$	−1677.65	−0.171161
$459$	−406.955	−0.0413835
$460$	−37.5594	−0.00380699
$461$	19190.3	1.93878	0.969392	−	0.245518i	$-0.0789581\pi$
$461$	19190.3	1.93878	0.969392	+	0.245518i	$0.0789581\pi$
$462$	−1489.41	−0.149986
$463$	9808.05	0.984489	0.492245	−	0.870457i	$-0.336177\pi$
$463$	9808.05	0.984489	0.492245	+	0.870457i	$0.336177\pi$
$464$	−1855.10	−0.185605
$465$	−83.1138	−0.00828884
$466$	2676.34	0.266050
$467$	15597.6	1.54554	0.772772	−	0.634683i	$-0.218869\pi$
$467$	15597.6	1.54554	0.772772	+	0.634683i	$0.218869\pi$
$468$	−150.244	−0.0148398
$469$	7023.02	0.691456
$470$	−127.497	−0.0125127
$471$	−9776.21	−0.956399
$472$	2707.38	0.264020
$473$	7114.84	0.691629
$474$	−5075.27	−0.491804
$475$	−5514.65	−0.532694
$476$	708.017	0.0681763
$477$	1474.04	0.141492
$478$	8659.79	0.828639
$479$	−15928.9	−1.51944	−0.759718	−	0.650253i	$-0.774663\pi$
$479$	−15928.9	−1.51944	−0.759718	+	0.650253i	$0.774663\pi$
$480$	10.4768	0.000996247 0
$481$	−472.092	−0.0447516
$482$	−6054.61	−0.572158
$483$	3031.26	0.285563
$484$	−3536.76	−0.332153
$485$	−166.302	−0.0155698
$486$	486.000	0.0453609
$487$	−3002.85	−0.279409	−0.139704	−	0.990193i	$-0.544615\pi$
$487$	−3002.85	−0.279409	−0.139704	+	0.990193i	$0.544615\pi$
$488$	−6869.76	−0.637253
$489$	−820.017	−0.0758332
$490$	44.7639	0.00412699
$491$	−6848.31	−0.629450	−0.314725	−	0.949183i	$-0.601912\pi$
$491$	−6848.31	−0.629450	−0.314725	+	0.949183i	$0.601912\pi$
$492$	1599.33	0.146552
$493$	−1747.55	−0.159647
$494$	368.278	0.0335417
$495$	−20.7616	−0.00188518
$496$	4061.76	0.367698
$497$	−2403.28	−0.216905
$498$	−8316.54	−0.748339
$499$	−17239.8	−1.54661	−0.773306	−	0.634033i	$-0.781398\pi$
$499$	−17239.8	−1.54661	−0.773306	+	0.634033i	$0.781398\pi$
$500$	−109.128	−0.00976072
$501$	−1796.58	−0.160210
$502$	−11211.9	−0.996833
$503$	−12651.2	−1.12145	−0.560726	−	0.828001i	$-0.689478\pi$
$503$	−12651.2	−1.12145	−0.560726	+	0.828001i	$0.689478\pi$
$504$	−845.539	−0.0747288
$505$	48.8379	0.00430348
$506$	−3637.40	−0.319570
$507$	6538.75	0.572773
$508$	−1561.48	−0.136377
$509$	13211.8	1.15049	0.575247	−	0.817980i	$-0.304906\pi$
$509$	13211.8	1.15049	0.575247	+	0.817980i	$0.304906\pi$
$510$	9.86942	0.000856912 0
$511$	5499.86	0.476124
$512$	−512.000	−0.0441942
$513$	−1191.28	−0.102527
$514$	14341.9	1.23073
$515$	−142.426	−0.0121865
$516$	4039.11	0.344597
$517$	−12347.3	−1.05036
$518$	−2656.82	−0.225355
$519$	8260.65	0.698656
$520$	3.64371	0.000307283 0
$521$	−1382.22	−0.116230	−0.0581152	−	0.998310i	$-0.518509\pi$
$521$	−1382.22	−0.116230	−0.0581152	+	0.998310i	$0.518509\pi$
$522$	2086.99	0.174990
$523$	−12601.6	−1.05360	−0.526799	−	0.849990i	$-0.676608\pi$
$523$	−12601.6	−1.05360	−0.526799	+	0.849990i	$0.676608\pi$
$524$	524.000	0.0436852
$525$	4403.43	0.366060
$526$	10118.8	0.838788
$527$	3826.28	0.316272
$528$	1014.62	0.0836279
$529$	−4764.12	−0.391560
$530$	−35.7483	−0.00292982
$531$	−3045.81	−0.248921
$532$	2072.58	0.168905
$533$	556.229	0.0452025
$534$	−5594.60	−0.453374
$535$	−140.907	−0.0113868
$536$	−4784.24	−0.385537
$537$	5138.68	0.412943
$538$	−549.917	−0.0440680
$539$	4335.12	0.346432
$540$	−11.7864	−0.000939271 0
$541$	19175.8	1.52391	0.761953	−	0.647633i	$-0.224241\pi$
$541$	19175.8	1.52391	0.761953	+	0.647633i	$0.224241\pi$
$542$	−10404.7	−0.824576
$543$	4679.74	0.369847
$544$	−482.317	−0.0380132
$545$	152.661	0.0119987
$546$	−294.068	−0.0230494
$547$	−403.751	−0.0315597	−0.0157798	−	0.999875i	$-0.505023\pi$
$547$	−403.751	−0.0315597	−0.0157798	+	0.999875i	$0.505023\pi$
$548$	−8898.98	−0.693696
$549$	7728.48	0.600808
$550$	−5283.96	−0.409652
$551$	−5115.61	−0.395521
$552$	−2064.96	−0.159222
$553$	−9933.66	−0.763874
$554$	12239.6	0.938650
$555$	−37.0348	−0.00283250
$556$	−2880.03	−0.219677
$557$	17208.9	1.30909	0.654547	−	0.756022i	$-0.272860\pi$
$557$	17208.9	1.30909	0.654547	+	0.756022i	$0.272860\pi$
$558$	−4569.48	−0.346669
$559$	1404.75	0.106287
$560$	20.5059	0.00154738
$561$	955.795	0.0719317
$562$	−10479.2	−0.786544
$563$	18475.3	1.38302	0.691510	−	0.722367i	$-0.256946\pi$
$563$	18475.3	1.38302	0.691510	+	0.722367i	$0.256946\pi$
$564$	−7009.59	−0.523328
$565$	72.2876	0.00538259
$566$	10626.0	0.789125
$567$	951.232	0.0704550
$568$	1637.17	0.120940
$569$	−18122.8	−1.33523	−0.667617	−	0.744504i	$-0.732686\pi$
$569$	−18122.8	−1.33523	−0.667617	+	0.744504i	$0.732686\pi$
$570$	28.8907	0.00212298
$571$	−15034.6	−1.10189	−0.550946	−	0.834541i	$-0.685733\pi$
$571$	−15034.6	−1.10189	−0.550946	+	0.834541i	$0.685733\pi$
$572$	352.871	0.0257942
$573$	199.135	0.0145183
$574$	3130.32	0.227626
$575$	10754.0	0.779951
$576$	576.000	0.0416667
$577$	11583.7	0.835763	0.417882	−	0.908501i	$-0.362773\pi$
$577$	11583.7	0.835763	0.417882	+	0.908501i	$0.362773\pi$
$578$	9371.65	0.674410
$579$	−3785.34	−0.271698
$580$	−50.6134	−0.00362346
$581$	−16277.7	−1.16233
$582$	−9143.03	−0.651187
$583$	−3462.01	−0.245938
$584$	−3746.62	−0.265473
$585$	−4.09917	−0.000289709 0
$586$	−18066.6	−1.27359
$587$	10186.0	0.716221	0.358111	−	0.933679i	$-0.383421\pi$
$587$	10186.0	0.716221	0.358111	+	0.933679i	$0.383421\pi$
$588$	2461.05	0.172606
$589$	11200.7	0.783558
$590$	73.8665	0.00515430
$591$	7549.92	0.525486
$592$	1809.88	0.125652
$593$	7897.09	0.546871	0.273435	−	0.961890i	$-0.411840\pi$
$593$	7897.09	0.546871	0.273435	+	0.961890i	$0.411840\pi$
$594$	−1141.44	−0.0788451
$595$	19.3171	0.00133096
$596$	−13217.8	−0.908428
$597$	6588.21	0.451654
$598$	−718.168	−0.0491105
$599$	−6899.55	−0.470631	−0.235315	−	0.971919i	$-0.575612\pi$
$599$	−6899.55	−0.470631	−0.235315	+	0.971919i	$0.575612\pi$
$600$	−2999.71	−0.204105
$601$	−22810.7	−1.54820	−0.774100	−	0.633064i	$-0.781797\pi$
$601$	−22810.7	−1.54820	−0.774100	+	0.633064i	$0.781797\pi$
$602$	7905.61	0.535230
$603$	5382.27	0.363487
$604$	−13180.9	−0.887955
$605$	−96.4948	−0.00648442
$606$	2685.04	0.179987
$607$	−17477.7	−1.16869	−0.584347	−	0.811504i	$-0.698649\pi$
$607$	−17477.7	−1.16869	−0.584347	+	0.811504i	$0.698649\pi$
$608$	−1411.89	−0.0941769
$609$	4084.79	0.271797
$610$	−187.430	−0.0124407
$611$	−2437.85	−0.161416
$612$	542.607	0.0358392
$613$	3591.21	0.236619	0.118310	−	0.992977i	$-0.462252\pi$
$613$	3591.21	0.236619	0.118310	+	0.992977i	$0.462252\pi$
$614$	−878.597	−0.0577480
$615$	43.6352	0.00286104
$616$	1985.88	0.129892
$617$	−596.501	−0.0389209	−0.0194605	−	0.999811i	$-0.506195\pi$
$617$	−596.501	−0.0389209	−0.0194605	+	0.999811i	$0.506195\pi$
$618$	−7830.39	−0.509683
$619$	3623.20	0.235265	0.117632	−	0.993057i	$-0.462470\pi$
$619$	3623.20	0.235265	0.117632	+	0.993057i	$0.462470\pi$
$620$	110.818	0.00717835
$621$	2323.08	0.150116
$622$	16724.3	1.07811
$623$	−10950.1	−0.704185
$624$	200.326	0.0128517
$625$	15620.5	0.999714
$626$	−20013.3	−1.27778
$627$	2797.90	0.178209
$628$	13035.0	0.828266
$629$	1704.96	0.108078
$630$	−23.0691	−0.00145888
$631$	−13279.6	−0.837804	−0.418902	−	0.908032i	$-0.637585\pi$
$631$	−13279.6	−0.837804	−0.418902	+	0.908032i	$0.637585\pi$
$632$	6767.03	0.425915
$633$	4874.68	0.306084
$634$	9215.64	0.577287
$635$	−42.6023	−0.00266240
$636$	−1965.39	−0.122536
$637$	855.925	0.0532386
$638$	−4901.61	−0.304164
$639$	−1841.82	−0.114024
$640$	−13.9691	−0.000862775 0
$641$	−16193.4	−0.997819	−0.498910	−	0.866654i	$-0.666266\pi$
$641$	−16193.4	−0.997819	−0.498910	+	0.866654i	$0.666266\pi$
$642$	−7746.84	−0.476236
$643$	−17975.3	−1.10245	−0.551225	−	0.834356i	$-0.685840\pi$
$643$	−17975.3	−1.10245	−0.551225	+	0.834356i	$0.685840\pi$
$644$	−4041.68	−0.247305
$645$	110.200	0.00672734
$646$	−1330.03	−0.0810053
$647$	−5243.03	−0.318586	−0.159293	−	0.987231i	$-0.550921\pi$
$647$	−5243.03	−0.318586	−0.159293	+	0.987231i	$0.550921\pi$
$648$	−648.000	−0.0392837
$649$	7153.54	0.432667
$650$	−1043.26	−0.0629541
$651$	−8943.69	−0.538450
$652$	1093.36	0.0656735
$653$	26418.5	1.58321	0.791604	−	0.611035i	$-0.209247\pi$
$653$	26418.5	1.58321	0.791604	+	0.611035i	$0.209247\pi$
$654$	8393.09	0.501828
$655$	14.2965	0.000852839 0
$656$	−2132.44	−0.126918
$657$	4214.95	0.250291
$658$	−13719.6	−0.812838
$659$	−13941.7	−0.824116	−0.412058	−	0.911158i	$-0.635190\pi$
$659$	−13941.7	−0.824116	−0.412058	+	0.911158i	$0.635190\pi$
$660$	27.6821	0.00163262
$661$	5227.12	0.307581	0.153791	−	0.988103i	$-0.450852\pi$
$661$	5227.12	0.307581	0.153791	+	0.988103i	$0.450852\pi$
$662$	−7085.86	−0.416012
$663$	188.712	0.0110542
$664$	11088.7	0.648081
$665$	56.5469	0.00329744
$666$	−2036.12	−0.118465
$667$	9975.81	0.579108
$668$	2395.44	0.138746
$669$	−18596.8	−1.07473
$670$	−130.530	−0.00752659
$671$	−18151.5	−1.04431
$672$	1127.39	0.0647170
$673$	26036.0	1.49125	0.745626	−	0.666364i	$-0.232150\pi$
$673$	26036.0	1.49125	0.745626	+	0.666364i	$0.232150\pi$
$674$	−8084.01	−0.461995
$675$	3374.68	0.192432
$676$	−8718.33	−0.496036
$677$	8553.19	0.485562	0.242781	−	0.970081i	$-0.421940\pi$
$677$	8553.19	0.485562	0.242781	+	0.970081i	$0.421940\pi$
$678$	3974.27	0.225119
$679$	−17895.3	−1.01143
$680$	−13.1592	−0.000742108 0
$681$	1449.54	0.0815659
$682$	10732.1	0.602571
$683$	6280.37	0.351847	0.175924	−	0.984404i	$-0.443709\pi$
$683$	6280.37	0.351847	0.175924	+	0.984404i	$0.443709\pi$
$684$	1588.37	0.0887908
$685$	−242.794	−0.0135426
$686$	12873.1	0.716466
$687$	−2516.48	−0.139752
$688$	−5385.48	−0.298429
$689$	−683.539	−0.0377950
$690$	−56.3390	−0.00310839
$691$	−8424.08	−0.463773	−0.231887	−	0.972743i	$-0.574490\pi$
$691$	−8424.08	−0.463773	−0.231887	+	0.972743i	$0.574490\pi$
$692$	−11014.2	−0.605053
$693$	−2234.11	−0.122463
$694$	14382.7	0.786684
$695$	−78.5768	−0.00428861
$696$	−2782.65	−0.151546
$697$	−2008.82	−0.109167
$698$	−19968.6	−1.08284
$699$	4014.51	0.217229
$700$	−5871.24	−0.317017
$701$	−4442.79	−0.239375	−0.119687	−	0.992812i	$-0.538189\pi$
$701$	−4442.79	−0.239375	−0.119687	+	0.992812i	$0.538189\pi$
$702$	−225.367	−0.0121167
$703$	4990.92	0.267761
$704$	−1352.82	−0.0724239
$705$	−191.245	−0.0102166
$706$	−17466.5	−0.931107
$707$	5255.34	0.279558
$708$	4061.08	0.215572
$709$	−29813.9	−1.57924	−0.789622	−	0.613593i	$-0.789724\pi$
$709$	−29813.9	−1.57924	−0.789622	+	0.613593i	$0.789724\pi$
$710$	44.6675	0.00236104
$711$	−7612.91	−0.401556
$712$	7459.46	0.392634
$713$	−21842.1	−1.14726
$714$	1062.03	0.0556657
$715$	9.62751	0.000503565 0
$716$	−6851.57	−0.357619
$717$	12989.7	0.676581
$718$	−16732.6	−0.869715
$719$	−8547.14	−0.443331	−0.221665	−	0.975123i	$-0.571149\pi$
$719$	−8547.14	−0.443331	−0.221665	+	0.975123i	$0.571149\pi$
$720$	15.7152	0.000813432 0
$721$	−15326.2	−0.791645
$722$	9824.60	0.506418
$723$	−9081.92	−0.467165
$724$	−6239.65	−0.320297
$725$	14491.6	0.742351
$726$	−5305.15	−0.271202
$727$	−29379.1	−1.49878	−0.749389	−	0.662130i	$-0.769653\pi$
$727$	−29379.1	−1.49878	−0.749389	+	0.662130i	$0.769653\pi$
$728$	392.091	0.0199613
$729$	729.000	0.0370370
$730$	−102.220	−0.00518267
$731$	−5073.26	−0.256691
$732$	−10304.6	−0.520315
$733$	−9594.43	−0.483463	−0.241731	−	0.970343i	$-0.577715\pi$
$733$	−9594.43	−0.483463	−0.241731	+	0.970343i	$0.577715\pi$
$734$	−20959.5	−1.05399
$735$	67.1458	0.00336967
$736$	2753.28	0.137890
$737$	−12641.1	−0.631804
$738$	2399.00	0.119659
$739$	10369.1	0.516149	0.258075	−	0.966125i	$-0.416912\pi$
$739$	10369.1	0.516149	0.258075	+	0.966125i	$0.416912\pi$
$740$	49.3797	0.00245302
$741$	552.416	0.0273867
$742$	−3846.79	−0.190324
$743$	24168.6	1.19335	0.596675	−	0.802483i	$-0.296488\pi$
$743$	24168.6	1.19335	0.596675	+	0.802483i	$0.296488\pi$
$744$	6092.64	0.300224
$745$	−360.627	−0.0177347
$746$	−15764.3	−0.773687
$747$	−12474.8	−0.611016
$748$	−1274.39	−0.0622947
$749$	−15162.6	−0.739694
$750$	−163.692	−0.00796960
$751$	9389.07	0.456208	0.228104	−	0.973637i	$-0.426747\pi$
$751$	9389.07	0.456208	0.228104	+	0.973637i	$0.426747\pi$
$752$	9346.12	0.453215
$753$	−16817.8	−0.813911
$754$	−967.773	−0.0467430
$755$	−359.620	−0.0173350
$756$	−1268.31	−0.0610158
$757$	8984.35	0.431363	0.215681	−	0.976464i	$-0.430803\pi$
$757$	8984.35	0.431363	0.215681	+	0.976464i	$0.430803\pi$
$758$	22295.0	1.06832
$759$	−5456.10	−0.260928
$760$	−38.5210	−0.00183856
$761$	32561.4	1.55105	0.775526	−	0.631316i	$-0.217485\pi$
$761$	32561.4	1.55105	0.775526	+	0.631316i	$0.217485\pi$
$762$	−2342.22	−0.111351
$763$	16427.5	0.779444
$764$	−265.513	−0.0125732
$765$	14.8041	0.000699666 0
$766$	6749.52	0.318368
$767$	1412.39	0.0664910
$768$	−768.000	−0.0360844
$769$	−31392.2	−1.47208	−0.736042	−	0.676936i	$-0.763307\pi$
$769$	−31392.2	−1.47208	−0.736042	+	0.676936i	$0.763307\pi$
$770$	54.1814	0.00253579
$771$	21512.9	1.00489
$772$	5047.12	0.235298
$773$	−32842.2	−1.52814	−0.764070	−	0.645134i	$-0.776802\pi$
$773$	−32842.2	−1.52814	−0.764070	+	0.645134i	$0.776802\pi$
$774$	6058.66	0.281362
$775$	−31729.5	−1.47065
$776$	12190.7	0.563944
$777$	−3985.23	−0.184002
$778$	−10114.4	−0.466090
$779$	−5880.41	−0.270459
$780$	5.46556	0.000250895 0
$781$	4325.78	0.198193
$782$	2593.66	0.118605
$783$	3130.48	0.142879
$784$	−3281.41	−0.149481
$785$	355.637	0.0161697
$786$	786.000	0.0356688
$787$	19969.7	0.904504	0.452252	−	0.891890i	$-0.350621\pi$
$787$	19969.7	0.904504	0.452252	+	0.891890i	$0.350621\pi$
$788$	−10066.6	−0.455084
$789$	15178.3	0.684868
$790$	184.627	0.00831487
$791$	7778.71	0.349658
$792$	1521.93	0.0682819
$793$	−3583.83	−0.160486
$794$	−14355.0	−0.641610
$795$	−53.6224	−0.00239219
$796$	−8784.28	−0.391144
$797$	−21555.8	−0.958026	−0.479013	−	0.877808i	$-0.659005\pi$
$797$	−21555.8	−0.958026	−0.479013	+	0.877808i	$0.659005\pi$
$798$	3108.87	0.137911
$799$	8804.28	0.389829
$800$	3999.62	0.176760
$801$	−8391.90	−0.370179
$802$	−6462.99	−0.284559
$803$	−9899.44	−0.435048
$804$	−7176.36	−0.314789
$805$	−110.271	−0.00482798
$806$	2118.95	0.0926013
$807$	−824.875	−0.0359814
$808$	−3580.05	−0.155874
$809$	35351.7	1.53634	0.768171	−	0.640245i	$-0.221167\pi$
$809$	35351.7	1.53634	0.768171	+	0.640245i	$0.221167\pi$
$810$	−17.6796	−0.000766911 0
$811$	5132.96	0.222247	0.111124	−	0.993807i	$-0.464555\pi$
$811$	5132.96	0.222247	0.111124	+	0.993807i	$0.464555\pi$
$812$	−5446.39	−0.235383
$813$	−15607.0	−0.673263
$814$	4782.13	0.205914
$815$	29.8304	0.00128210
$816$	−723.475	−0.0310376
$817$	−14850.9	−0.635947
$818$	26718.9	1.14206
$819$	−441.102	−0.0188197
$820$	−58.1802	−0.00247773
$821$	16423.7	0.698162	0.349081	−	0.937093i	$-0.386494\pi$
$821$	16423.7	0.698162	0.349081	+	0.937093i	$0.386494\pi$
$822$	−13348.5	−0.566400
$823$	29397.7	1.24513	0.622564	−	0.782569i	$-0.286091\pi$
$823$	29397.7	1.24513	0.622564	+	0.782569i	$0.286091\pi$
$824$	10440.5	0.441399
$825$	−7925.94	−0.334480
$826$	7948.61	0.334828
$827$	32269.5	1.35686	0.678429	−	0.734666i	$-0.262661\pi$
$827$	32269.5	1.35686	0.678429	+	0.734666i	$0.262661\pi$
$828$	−3097.44	−0.130004
$829$	−33129.3	−1.38797	−0.693986	−	0.719988i	$-0.744147\pi$
$829$	−33129.3	−1.38797	−0.693986	+	0.719988i	$0.744147\pi$
$830$	302.537	0.0126521
$831$	18359.5	0.766405
$832$	−267.101	−0.0111299
$833$	−3091.17	−0.128575
$834$	−4320.04	−0.179365
$835$	65.3557	0.00270866
$836$	−3730.53	−0.154334
$837$	−6854.22	−0.283054
$838$	18447.0	0.760432
$839$	25063.8	1.03134	0.515672	−	0.856786i	$-0.327542\pi$
$839$	25063.8	1.03134	0.515672	+	0.856786i	$0.327542\pi$
$840$	30.7589	0.00126343
$841$	−10946.0	−0.448810
$842$	−19864.0	−0.813016
$843$	−15718.8	−0.642210
$844$	−6499.58	−0.265077
$845$	−237.865	−0.00968380
$846$	−10514.4	−0.427296
$847$	−10383.6	−0.421233
$848$	2620.52	0.106119
$849$	15939.0	0.644318
$850$	3767.74	0.152038
$851$	−9732.65	−0.392046
$852$	2455.75	0.0987474
$853$	−35574.8	−1.42797	−0.713986	−	0.700160i	$-0.753112\pi$
$853$	−35574.8	−1.42797	−0.713986	+	0.700160i	$0.753112\pi$
$854$	−20168.9	−0.808158
$855$	43.3361	0.00173341
$856$	10329.1	0.412433
$857$	37464.4	1.49330	0.746651	−	0.665216i	$-0.231660\pi$
$857$	37464.4	1.49330	0.746651	+	0.665216i	$0.231660\pi$
$858$	529.307	0.0210609
$859$	−691.587	−0.0274699	−0.0137350	−	0.999906i	$-0.504372\pi$
$859$	−691.587	−0.0274699	−0.0137350	+	0.999906i	$0.504372\pi$
$860$	−146.934	−0.00582605
$861$	4695.48	0.185856
$862$	−11103.9	−0.438746
$863$	41812.2	1.64925	0.824625	−	0.565680i	$-0.191386\pi$
$863$	41812.2	1.64925	0.824625	+	0.565680i	$0.191386\pi$
$864$	864.000	0.0340207
$865$	−300.504	−0.0118121
$866$	9882.80	0.387796
$867$	14057.5	0.550654
$868$	11924.9	0.466311
$869$	17880.1	0.697974
$870$	−75.9201	−0.00295854
$871$	−2495.85	−0.0970937
$872$	−11190.8	−0.434596
$873$	−13714.5	−0.531692
$874$	7592.42	0.293841
$875$	−320.390	−0.0123784
$876$	−5619.93	−0.216758
$877$	34444.0	1.32622	0.663108	−	0.748524i	$-0.269237\pi$
$877$	34444.0	1.32622	0.663108	+	0.748524i	$0.269237\pi$
$878$	27526.4	1.05806
$879$	−27100.0	−1.03989
$880$	−36.9095	−0.00141389
$881$	38152.0	1.45899	0.729497	−	0.683984i	$-0.239754\pi$
$881$	38152.0	1.45899	0.729497	+	0.683984i	$0.239754\pi$
$882$	3691.58	0.140932
$883$	−118.733	−0.00452512	−0.00226256	−	0.999997i	$-0.500720\pi$
$883$	−118.733	−0.00452512	−0.00226256	+	0.999997i	$0.500720\pi$
$884$	−251.616	−0.00957326
$885$	110.800	0.00420847
$886$	25028.5	0.949040
$887$	30052.7	1.13762	0.568811	−	0.822468i	$-0.307404\pi$
$887$	30052.7	1.13762	0.568811	+	0.822468i	$0.307404\pi$
$888$	2714.83	0.102594
$889$	−4584.34	−0.172952
$890$	203.519	0.00766514
$891$	−1712.17	−0.0643768
$892$	24795.7	0.930743
$893$	25772.8	0.965793
$894$	−19826.7	−0.741728
$895$	−186.934	−0.00698158
$896$	−1503.18	−0.0560466
$897$	−1077.25	−0.0400986
$898$	9944.59	0.369549
$899$	−29433.5	−1.09195
$900$	−4499.57	−0.166651
$901$	2468.60	0.0912773
$902$	−5634.41	−0.207988
$903$	11858.4	0.437014
$904$	−5299.03	−0.194959
$905$	−170.239	−0.00625295
$906$	−19771.4	−0.725012
$907$	24168.5	0.884789	0.442394	−	0.896821i	$-0.354129\pi$
$907$	24168.5	0.884789	0.442394	+	0.896821i	$0.354129\pi$
$908$	−1932.72	−0.0706381
$909$	4027.56	0.146959
$910$	10.6976	0.000389693 0
$911$	−40430.8	−1.47040	−0.735199	−	0.677851i	$-0.762911\pi$
$911$	−40430.8	−1.47040	−0.735199	+	0.677851i	$0.762911\pi$
$912$	−2117.83	−0.0768951
$913$	29299.0	1.06205
$914$	23540.9	0.851929
$915$	−281.145	−0.0101578
$916$	3355.31	0.121029
$917$	1538.41	0.0554011
$918$	813.910	0.0292626
$919$	−8581.72	−0.308036	−0.154018	−	0.988068i	$-0.549221\pi$
$919$	−8581.72	−0.308036	−0.154018	+	0.988068i	$0.549221\pi$
$920$	75.1187	0.00269195
$921$	−1317.90	−0.0471511
$922$	−38380.5	−1.37093
$923$	854.081	0.0304577
$924$	2978.81	0.106056
$925$	−14138.4	−0.502559
$926$	−19616.1	−0.696139
$927$	−11745.6	−0.416155
$928$	3710.20	0.131243
$929$	9113.23	0.321846	0.160923	−	0.986967i	$-0.448553\pi$
$929$	9113.23	0.321846	0.160923	+	0.986967i	$0.448553\pi$
$930$	166.228	0.00586110
$931$	−9048.77	−0.318541
$932$	−5352.69	−0.188126
$933$	25086.5	0.880272
$934$	−31195.1	−1.09287
$935$	−34.7697	−0.00121614
$936$	300.489	0.0104934
$937$	51522.7	1.79634	0.898171	−	0.439645i	$-0.144896\pi$
$937$	51522.7	1.79634	0.898171	+	0.439645i	$0.144896\pi$
$938$	−14046.0	−0.488934
$939$	−30019.9	−1.04330
$940$	254.994	0.00884784
$941$	49037.9	1.69882	0.849411	−	0.527732i	$-0.176957\pi$
$941$	49037.9	1.69882	0.849411	+	0.527732i	$0.176957\pi$
$942$	19552.4	0.676276
$943$	11467.2	0.395996
$944$	−5414.77	−0.186690
$945$	−34.6037	−0.00119117
$946$	−14229.7	−0.489056
$947$	−5903.34	−0.202569	−0.101285	−	0.994857i	$-0.532295\pi$
$947$	−5903.34	−0.202569	−0.101285	+	0.994857i	$0.532295\pi$
$948$	10150.5	0.347758
$949$	−1954.54	−0.0668569
$950$	11029.3	0.376672
$951$	13823.5	0.471353
$952$	−1416.03	−0.0482079
$953$	6006.28	0.204158	0.102079	−	0.994776i	$-0.467451\pi$
$953$	6006.28	0.204158	0.102079	+	0.994776i	$0.467451\pi$
$954$	−2948.08	−0.100050
$955$	−7.24408	−0.000245459 0
$956$	−17319.6	−0.585937
$957$	−7352.41	−0.248349
$958$	31857.8	1.07440
$959$	−26126.5	−0.879738
$960$	−20.9536	−0.000704453 0
$961$	34653.9	1.16323
$962$	944.183	0.0316442
$963$	−11620.3	−0.388845
$964$	12109.2	0.404577
$965$	137.702	0.00459357
$966$	−6062.52	−0.201924
$967$	33730.4	1.12171	0.560857	−	0.827913i	$-0.310472\pi$
$967$	33730.4	1.12171	0.560857	+	0.827913i	$0.310472\pi$
$968$	7073.53	0.234868
$969$	−1995.05	−0.0661406
$970$	332.603	0.0110095
$971$	−13563.2	−0.448265	−0.224132	−	0.974559i	$-0.571955\pi$
$971$	−13563.2	−0.448265	−0.224132	+	0.974559i	$0.571955\pi$
$972$	−972.000	−0.0320750
$973$	−8455.47	−0.278592
$974$	6005.70	0.197572
$975$	−1564.90	−0.0514018
$976$	13739.5	0.450606
$977$	−37485.8	−1.22751	−0.613755	−	0.789497i	$-0.710342\pi$
$977$	−37485.8	−1.22751	−0.613755	+	0.789497i	$0.710342\pi$
$978$	1640.03	0.0536222
$979$	19709.6	0.643435
$980$	−89.5277	−0.00291822
$981$	12589.6	0.409741
$982$	13696.6	0.445088
$983$	38059.8	1.23491	0.617457	−	0.786605i	$-0.288163\pi$
$983$	38059.8	1.23491	0.617457	+	0.786605i	$0.288163\pi$
$984$	−3198.67	−0.103628
$985$	−274.649	−0.00888432
$986$	3495.10	0.112887
$987$	−20579.5	−0.663679
$988$	−736.555	−0.0237176
$989$	28960.4	0.931130
$990$	41.5232	0.00133302
$991$	−53695.2	−1.72117	−0.860587	−	0.509303i	$-0.829903\pi$
$991$	−53695.2	−1.72117	−0.860587	+	0.509303i	$0.829903\pi$
$992$	−8123.52	−0.260002
$993$	−10628.8	−0.339672
$994$	4806.57	0.153375
$995$	−239.665	−0.00763606
$996$	16633.1	0.529156
$997$	−42982.5	−1.36537	−0.682683	−	0.730714i	$-0.739187\pi$
$997$	−42982.5	−1.36537	−0.682683	+	0.730714i	$0.739187\pi$
$998$	34479.6	1.09362
$999$	−3054.18	−0.0967267

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	786.4.a.g.1.4	✓	8
3.2	odd	2		2358.4.a.i.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
786.4.a.g.1.4	✓	8	1.1	even	1	trivial
2358.4.a.i.1.5		8	3.2	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:	\( N \)	\(=\)	\( 786 = 2 \cdot 3 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	786.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +0.109133 q^{5} +6.00000 q^{6} +11.7436 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +0.109133 q^{5} +6.00000 q^{6} +11.7436 q^{7} -8.00000 q^{8} +9.00000 q^{9} -0.218267 q^{10} -21.1379 q^{11} -12.0000 q^{12} -4.17345 q^{13} -23.4872 q^{14} -0.327400 q^{15} +16.0000 q^{16} +15.0724 q^{17} -18.0000 q^{18} +44.1214 q^{19} +0.436534 q^{20} -35.2308 q^{21} +42.2757 q^{22} -86.0400 q^{23} +24.0000 q^{24} -124.988 q^{25} +8.34691 q^{26} -27.0000 q^{27} +46.9744 q^{28} -115.944 q^{29} +0.654800 q^{30} +253.860 q^{31} -32.0000 q^{32} +63.4136 q^{33} -30.1448 q^{34} +1.28162 q^{35} +36.0000 q^{36} +113.118 q^{37} -88.2429 q^{38} +12.5204 q^{39} -0.873067 q^{40} -133.278 q^{41} +70.4616 q^{42} -336.592 q^{43} -84.5514 q^{44} +0.982201 q^{45} +172.080 q^{46} +584.133 q^{47} -48.0000 q^{48} -205.088 q^{49} +249.976 q^{50} -45.2172 q^{51} -16.6938 q^{52} +163.782 q^{53} +54.0000 q^{54} -2.30685 q^{55} -93.9488 q^{56} -132.364 q^{57} +231.888 q^{58} -338.423 q^{59} -1.30960 q^{60} +858.720 q^{61} -507.720 q^{62} +105.692 q^{63} +64.0000 q^{64} -0.455463 q^{65} -126.827 q^{66} +598.030 q^{67} +60.2896 q^{68} +258.120 q^{69} -2.56324 q^{70} -204.646 q^{71} -72.0000 q^{72} +468.328 q^{73} -226.235 q^{74} +374.964 q^{75} +176.486 q^{76} -248.235 q^{77} -25.0407 q^{78} -845.879 q^{79} +1.74613 q^{80} +81.0000 q^{81} +266.556 q^{82} -1386.09 q^{83} -140.923 q^{84} +1.64490 q^{85} +673.185 q^{86} +347.832 q^{87} +169.103 q^{88} -932.433 q^{89} -1.96440 q^{90} -49.0114 q^{91} -344.160 q^{92} -761.580 q^{93} -1168.27 q^{94} +4.81512 q^{95} +96.0000 q^{96} -1523.84 q^{97} +410.176 q^{98} -190.241 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{2} - 24 q^{3} + 32 q^{4} + q^{5} + 48 q^{6} + 22 q^{7} - 64 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q - 16 * q^2 - 24 * q^3 + 32 * q^4 + q^5 + 48 * q^6 + 22 * q^7 - 64 * q^8 + 72 * q^9	\( 8 q - 16 q^{2} - 24 q^{3} + 32 q^{4} + q^{5} + 48 q^{6} + 22 q^{7} - 64 q^{8} + 72 q^{9} - 2 q^{10} - 77 q^{11} - 96 q^{12} + 29 q^{13} - 44 q^{14} - 3 q^{15} + 128 q^{16} + 59 q^{17} - 144 q^{18} - 150 q^{19} + 4 q^{20} - 66 q^{21} + 154 q^{22} - 269 q^{23} + 192 q^{24} + 273 q^{25} - 58 q^{26} - 216 q^{27} + 88 q^{28} - 123 q^{29} + 6 q^{30} - 86 q^{31} - 256 q^{32} + 231 q^{33} - 118 q^{34} - 292 q^{35} + 288 q^{36} + 412 q^{37} + 300 q^{38} - 87 q^{39} - 8 q^{40} - 114 q^{41} + 132 q^{42} + 1087 q^{43} - 308 q^{44} + 9 q^{45} + 538 q^{46} - 442 q^{47} - 384 q^{48} + 1292 q^{49} - 546 q^{50} - 177 q^{51} + 116 q^{52} + 5 q^{53} + 432 q^{54} + 456 q^{55} - 176 q^{56} + 450 q^{57} + 246 q^{58} + 252 q^{59} - 12 q^{60} + 1482 q^{61} + 172 q^{62} + 198 q^{63} + 512 q^{64} + 475 q^{65} - 462 q^{66} + 330 q^{67} + 236 q^{68} + 807 q^{69} + 584 q^{70} - 2946 q^{71} - 576 q^{72} - 214 q^{73} - 824 q^{74} - 819 q^{75} - 600 q^{76} - 960 q^{77} + 174 q^{78} - 64 q^{79} + 16 q^{80} + 648 q^{81} + 228 q^{82} - 276 q^{83} - 264 q^{84} + 80 q^{85} - 2174 q^{86} + 369 q^{87} + 616 q^{88} - 3177 q^{89} - 18 q^{90} - 781 q^{91} - 1076 q^{92} + 258 q^{93} + 884 q^{94} - 2700 q^{95} + 768 q^{96} + 200 q^{97} - 2584 q^{98} - 693 q^{99}+O(q^{100}) \) 8 * q - 16 * q^2 - 24 * q^3 + 32 * q^4 + q^5 + 48 * q^6 + 22 * q^7 - 64 * q^8 + 72 * q^9 - 2 * q^10 - 77 * q^11 - 96 * q^12 + 29 * q^13 - 44 * q^14 - 3 * q^15 + 128 * q^16 + 59 * q^17 - 144 * q^18 - 150 * q^19 + 4 * q^20 - 66 * q^21 + 154 * q^22 - 269 * q^23 + 192 * q^24 + 273 * q^25 - 58 * q^26 - 216 * q^27 + 88 * q^28 - 123 * q^29 + 6 * q^30 - 86 * q^31 - 256 * q^32 + 231 * q^33 - 118 * q^34 - 292 * q^35 + 288 * q^36 + 412 * q^37 + 300 * q^38 - 87 * q^39 - 8 * q^40 - 114 * q^41 + 132 * q^42 + 1087 * q^43 - 308 * q^44 + 9 * q^45 + 538 * q^46 - 442 * q^47 - 384 * q^48 + 1292 * q^49 - 546 * q^50 - 177 * q^51 + 116 * q^52 + 5 * q^53 + 432 * q^54 + 456 * q^55 - 176 * q^56 + 450 * q^57 + 246 * q^58 + 252 * q^59 - 12 * q^60 + 1482 * q^61 + 172 * q^62 + 198 * q^63 + 512 * q^64 + 475 * q^65 - 462 * q^66 + 330 * q^67 + 236 * q^68 + 807 * q^69 + 584 * q^70 - 2946 * q^71 - 576 * q^72 - 214 * q^73 - 824 * q^74 - 819 * q^75 - 600 * q^76 - 960 * q^77 + 174 * q^78 - 64 * q^79 + 16 * q^80 + 648 * q^81 + 228 * q^82 - 276 * q^83 - 264 * q^84 + 80 * q^85 - 2174 * q^86 + 369 * q^87 + 616 * q^88 - 3177 * q^89 - 18 * q^90 - 781 * q^91 - 1076 * q^92 + 258 * q^93 + 884 * q^94 - 2700 * q^95 + 768 * q^96 + 200 * q^97 - 2584 * q^98 - 693 * q^99

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