LMFDB - Embedded newform 89.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [89,2,Mod(1,89)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(89, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("89.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	89.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:	$0.710668577989$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.535120.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 4x^{2} + 4x - 2 $ x^5 - 2x^4 - 6x^3 + 4x^2 + 4x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.34526$ of defining polynomial
Character	$\chi$	$=$	89.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.09813 q^{2} +2.34526 q^{3} +2.40214 q^{4} -0.402140 q^{5} -4.92067 q^{6} +0.678986 q^{7} -0.843741 q^{8} +2.50027 q^{9} +O(q^{10})$	\(q-2.09813 q^{2} +2.34526 q^{3} +2.40214 q^{4} -0.402140 q^{5} -4.92067 q^{6} +0.678986 q^{7} -0.843741 q^{8} +2.50027 q^{9} +0.843741 q^{10} +0.745613 q^{11} +5.63365 q^{12} +6.09267 q^{13} -1.42460 q^{14} -0.943125 q^{15} -3.03400 q^{16} -6.59240 q^{17} -5.24588 q^{18} +0.827996 q^{19} -0.965997 q^{20} +1.59240 q^{21} -1.56439 q^{22} -7.71478 q^{23} -1.97880 q^{24} -4.83828 q^{25} -12.7832 q^{26} -1.17200 q^{27} +1.63102 q^{28} +6.24042 q^{29} +1.97880 q^{30} +0.400348 q^{31} +8.05321 q^{32} +1.74866 q^{33} +13.8317 q^{34} -0.273047 q^{35} +6.00599 q^{36} -9.34706 q^{37} -1.73724 q^{38} +14.2889 q^{39} +0.339302 q^{40} +1.89641 q^{41} -3.34106 q^{42} -2.27864 q^{43} +1.79107 q^{44} -1.00546 q^{45} +16.1866 q^{46} +3.45064 q^{47} -7.11554 q^{48} -6.53898 q^{49} +10.1513 q^{50} -15.4609 q^{51} +14.6354 q^{52} +1.33802 q^{53} +2.45902 q^{54} -0.299841 q^{55} -0.572888 q^{56} +1.94187 q^{57} -13.0932 q^{58} +3.53177 q^{59} -2.26552 q^{60} +0.597860 q^{61} -0.839982 q^{62} +1.69765 q^{63} -10.8287 q^{64} -2.45011 q^{65} -3.66891 q^{66} +10.4367 q^{67} -15.8359 q^{68} -18.0932 q^{69} +0.572888 q^{70} +12.6941 q^{71} -2.10958 q^{72} -10.4361 q^{73} +19.6113 q^{74} -11.3471 q^{75} +1.98896 q^{76} +0.506261 q^{77} -29.9800 q^{78} +17.3471 q^{79} +1.22009 q^{80} -10.2495 q^{81} -3.97892 q^{82} -1.12476 q^{83} +3.82517 q^{84} +2.65107 q^{85} +4.78087 q^{86} +14.6354 q^{87} -0.629105 q^{88} +1.00000 q^{89} +2.10958 q^{90} +4.13684 q^{91} -18.5320 q^{92} +0.938923 q^{93} -7.23989 q^{94} -0.332970 q^{95} +18.8869 q^{96} +7.63786 q^{97} +13.7196 q^{98} +1.86423 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} - 3 q^{3} + 11 q^{4} - q^{5} - 3 q^{6} + 8 q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) $ 5 * q - q^2 - 3 * q^3 + 11 * q^4 - q^5 - 3 * q^6 + 8 * q^7 + 3 * q^8 + 2 * q^9	\( 5 q - q^{2} - 3 q^{3} + 11 q^{4} - q^{5} - 3 q^{6} + 8 q^{7} + 3 q^{8} + 2 q^{9} - 3 q^{10} + 6 q^{11} - 15 q^{12} - 14 q^{14} + 9 q^{15} + 11 q^{16} - 13 q^{17} - 18 q^{18} + 13 q^{19} - 31 q^{20} - 12 q^{21} - 14 q^{22} + q^{23} - 5 q^{24} + 4 q^{25} - 4 q^{26} + 3 q^{27} + 12 q^{28} + 2 q^{29} + 5 q^{30} + 19 q^{31} + 15 q^{32} - 4 q^{33} - 11 q^{34} + 4 q^{35} + 30 q^{36} - 14 q^{37} + 23 q^{38} + 22 q^{39} - 11 q^{40} - 2 q^{41} + 16 q^{42} + q^{43} + 6 q^{44} - 26 q^{45} + 23 q^{46} - 4 q^{47} - 45 q^{48} + 9 q^{49} + 16 q^{50} - 19 q^{51} + 12 q^{52} - 11 q^{53} + 25 q^{54} + 6 q^{55} + 10 q^{56} - 7 q^{57} - 14 q^{58} + 57 q^{60} + 4 q^{61} - 13 q^{62} + 10 q^{63} + 11 q^{64} - 12 q^{65} + 48 q^{66} + 4 q^{67} - 15 q^{68} - 39 q^{69} - 10 q^{70} - 2 q^{71} - 22 q^{72} - 25 q^{73} - 6 q^{74} - 24 q^{75} + 21 q^{76} - 8 q^{77} - 50 q^{78} + 54 q^{79} - 55 q^{80} - 7 q^{81} + 38 q^{82} - 20 q^{83} - 40 q^{84} - 11 q^{85} + 3 q^{86} + 12 q^{87} - 78 q^{88} + 5 q^{89} + 22 q^{90} - 20 q^{91} + 33 q^{92} - 27 q^{93} - 28 q^{94} + 5 q^{95} - 7 q^{96} + 13 q^{97} - 29 q^{98} + 8 q^{99}+O(q^{100}) \) 5 * q - q^2 - 3 * q^3 + 11 * q^4 - q^5 - 3 * q^6 + 8 * q^7 + 3 * q^8 + 2 * q^9 - 3 * q^10 + 6 * q^11 - 15 * q^12 - 14 * q^14 + 9 * q^15 + 11 * q^16 - 13 * q^17 - 18 * q^18 + 13 * q^19 - 31 * q^20 - 12 * q^21 - 14 * q^22 + q^23 - 5 * q^24 + 4 * q^25 - 4 * q^26 + 3 * q^27 + 12 * q^28 + 2 * q^29 + 5 * q^30 + 19 * q^31 + 15 * q^32 - 4 * q^33 - 11 * q^34 + 4 * q^35 + 30 * q^36 - 14 * q^37 + 23 * q^38 + 22 * q^39 - 11 * q^40 - 2 * q^41 + 16 * q^42 + q^43 + 6 * q^44 - 26 * q^45 + 23 * q^46 - 4 * q^47 - 45 * q^48 + 9 * q^49 + 16 * q^50 - 19 * q^51 + 12 * q^52 - 11 * q^53 + 25 * q^54 + 6 * q^55 + 10 * q^56 - 7 * q^57 - 14 * q^58 + 57 * q^60 + 4 * q^61 - 13 * q^62 + 10 * q^63 + 11 * q^64 - 12 * q^65 + 48 * q^66 + 4 * q^67 - 15 * q^68 - 39 * q^69 - 10 * q^70 - 2 * q^71 - 22 * q^72 - 25 * q^73 - 6 * q^74 - 24 * q^75 + 21 * q^76 - 8 * q^77 - 50 * q^78 + 54 * q^79 - 55 * q^80 - 7 * q^81 + 38 * q^82 - 20 * q^83 - 40 * q^84 - 11 * q^85 + 3 * q^86 + 12 * q^87 - 78 * q^88 + 5 * q^89 + 22 * q^90 - 20 * q^91 + 33 * q^92 - 27 * q^93 - 28 * q^94 + 5 * q^95 - 7 * q^96 + 13 * q^97 - 29 * q^98 + 8 * 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 $

$2$	−2.09813	−1.48360	−0.741800	−	0.670621i	$-0.766028\pi$
$2$	−2.09813	−1.48360	−0.741800	+	0.670621i	$0.766028\pi$
$3$	2.34526	1.35404	0.677020	−	0.735965i	$-0.263271\pi$
$3$	2.34526	1.35404	0.677020	+	0.735965i	$0.263271\pi$
$4$	2.40214	1.20107
$5$	−0.402140	−0.179842	−0.0899212	−	0.995949i	$-0.528662\pi$
$5$	−0.402140	−0.179842	−0.0899212	+	0.995949i	$0.528662\pi$
$6$	−4.92067	−2.00885
$7$	0.678986	0.256633	0.128316	−	0.991733i	$-0.459043\pi$
$7$	0.678986	0.256633	0.128316	+	0.991733i	$0.459043\pi$
$8$	−0.843741	−0.298308
$9$	2.50027	0.833423
$10$	0.843741	0.266814
$11$	0.745613	0.224811	0.112405	−	0.993662i	$-0.464144\pi$
$11$	0.745613	0.224811	0.112405	+	0.993662i	$0.464144\pi$
$12$	5.63365	1.62630
$13$	6.09267	1.68980	0.844901	−	0.534922i	$-0.179659\pi$
$13$	6.09267	1.68980	0.844901	+	0.534922i	$0.179659\pi$
$14$	−1.42460	−0.380740
$15$	−0.943125	−0.243514
$16$	−3.03400	−0.758501
$17$	−6.59240	−1.59889	−0.799446	−	0.600738i	$-0.794874\pi$
$17$	−6.59240	−1.59889	−0.799446	+	0.600738i	$0.794874\pi$
$18$	−5.24588	−1.23647
$19$	0.827996	0.189955	0.0949776	−	0.995479i	$-0.469722\pi$
$19$	0.827996	0.189955	0.0949776	+	0.995479i	$0.469722\pi$
$20$	−0.965997	−0.216003
$21$	1.59240	0.347491
$22$	−1.56439	−0.333530
$23$	−7.71478	−1.60864	−0.804322	−	0.594194i	$-0.797471\pi$
$23$	−7.71478	−1.60864	−0.804322	+	0.594194i	$0.797471\pi$
$24$	−1.97880	−0.403920
$25$	−4.83828	−0.967657
$26$	−12.7832	−2.50699
$27$	−1.17200	−0.225552
$28$	1.63102	0.308234
$29$	6.24042	1.15882	0.579409	−	0.815037i	$-0.303284\pi$
$29$	6.24042	1.15882	0.579409	+	0.815037i	$0.303284\pi$
$30$	1.97880	0.361277
$31$	0.400348	0.0719047	0.0359524	−	0.999354i	$-0.488554\pi$
$31$	0.400348	0.0719047	0.0359524	+	0.999354i	$0.488554\pi$
$32$	8.05321	1.42362
$33$	1.74866	0.304403
$34$	13.8317	2.37212
$35$	−0.273047	−0.0461534
$36$	6.00599	1.00100
$37$	−9.34706	−1.53665	−0.768323	−	0.640062i	$-0.778909\pi$
$37$	−9.34706	−1.53665	−0.768323	+	0.640062i	$0.778909\pi$
$38$	−1.73724	−0.281818
$39$	14.2889	2.28806
$40$	0.339302	0.0536484
$41$	1.89641	0.296170	0.148085	−	0.988975i	$-0.452689\pi$
$41$	1.89641	0.296170	0.148085	+	0.988975i	$0.452689\pi$
$42$	−3.34106	−0.515537
$43$	−2.27864	−0.347489	−0.173744	−	0.984791i	$-0.555587\pi$
$43$	−2.27864	−0.347489	−0.173744	+	0.984791i	$0.555587\pi$
$44$	1.79107	0.270014
$45$	−1.00546	−0.149885
$46$	16.1866	2.38658
$47$	3.45064	0.503328	0.251664	−	0.967815i	$-0.419022\pi$
$47$	3.45064	0.503328	0.251664	+	0.967815i	$0.419022\pi$
$48$	−7.11554	−1.02704
$49$	−6.53898	−0.934140
$50$	10.1513	1.43562
$51$	−15.4609	−2.16496
$52$	14.6354	2.02957
$53$	1.33802	0.183791	0.0918953	−	0.995769i	$-0.470707\pi$
$53$	1.33802	0.183791	0.0918953	+	0.995769i	$0.470707\pi$
$54$	2.45902	0.334630
$55$	−0.299841	−0.0404306
$56$	−0.572888	−0.0765554
$57$	1.94187	0.257207
$58$	−13.0932	−1.71922
$59$	3.53177	0.459797	0.229899	−	0.973215i	$-0.426161\pi$
$59$	3.53177	0.459797	0.229899	+	0.973215i	$0.426161\pi$
$60$	−2.26552	−0.292477
$61$	0.597860	0.0765481	0.0382741	−	0.999267i	$-0.487814\pi$
$61$	0.597860	0.0765481	0.0382741	+	0.999267i	$0.487814\pi$
$62$	−0.839982	−0.106678
$63$	1.69765	0.213883
$64$	−10.8287	−1.35358
$65$	−2.45011	−0.303898
$66$	−3.66891	−0.451612
$67$	10.4367	1.27504	0.637522	−	0.770432i	$-0.279960\pi$
$67$	10.4367	1.27504	0.637522	+	0.770432i	$0.279960\pi$
$68$	−15.8359	−1.92038
$69$	−18.0932	−2.17817
$70$	0.572888	0.0684733
$71$	12.6941	1.50651	0.753257	−	0.657726i	$-0.228482\pi$
$71$	12.6941	1.50651	0.753257	+	0.657726i	$0.228482\pi$
$72$	−2.10958	−0.248616
$73$	−10.4361	−1.22146	−0.610729	−	0.791840i	$-0.709124\pi$
$73$	−10.4361	−1.22146	−0.610729	+	0.791840i	$0.709124\pi$
$74$	19.6113	2.27977
$75$	−11.3471	−1.31025
$76$	1.98896	0.228149
$77$	0.506261	0.0576938
$78$	−29.9800	−3.39457
$79$	17.3471	1.95170	0.975848	−	0.218450i	$-0.0701000\pi$
$79$	17.3471	1.95170	0.975848	+	0.218450i	$0.0701000\pi$
$80$	1.22009	0.136411
$81$	−10.2495	−1.13883
$82$	−3.97892	−0.439398
$83$	−1.12476	−0.123458	−0.0617291	−	0.998093i	$-0.519661\pi$
$83$	−1.12476	−0.123458	−0.0617291	+	0.998093i	$0.519661\pi$
$84$	3.82517	0.417361
$85$	2.65107	0.287549
$86$	4.78087	0.515535
$87$	14.6354	1.56908
$88$	−0.629105	−0.0670628
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	2.10958	0.222369
$91$	4.13684	0.433658
$92$	−18.5320	−1.93209
$93$	0.938923	0.0973618
$94$	−7.23989	−0.746737
$95$	−0.332970	−0.0341620
$96$	18.8869	1.92764
$97$	7.63786	0.775507	0.387753	−	0.921763i	$-0.373251\pi$
$97$	7.63786	0.775507	0.387753	+	0.921763i	$0.373251\pi$
$98$	13.7196	1.38589
$99$	1.86423	0.187362
$100$	−11.6222	−1.16222
$101$	9.00487	0.896018	0.448009	−	0.894029i	$-0.352133\pi$
$101$	9.00487	0.896018	0.448009	+	0.894029i	$0.352133\pi$
$102$	32.4390	3.21194
$103$	3.56269	0.351043	0.175521	−	0.984476i	$-0.443839\pi$
$103$	3.56269	0.351043	0.175521	+	0.984476i	$0.443839\pi$
$104$	−5.14064	−0.504081
$105$	−0.640369	−0.0624936
$106$	−2.80733	−0.272672
$107$	−16.5438	−1.59935	−0.799677	−	0.600430i	$-0.794996\pi$
$107$	−16.5438	−1.59935	−0.799677	+	0.600430i	$0.794996\pi$
$108$	−2.81532	−0.270904
$109$	−14.0770	−1.34834	−0.674168	−	0.738578i	$-0.735498\pi$
$109$	−14.0770	−1.34834	−0.674168	+	0.738578i	$0.735498\pi$
$110$	0.629105	0.0599828
$111$	−21.9213	−2.08068
$112$	−2.06005	−0.194656
$113$	−5.27389	−0.496126	−0.248063	−	0.968744i	$-0.579794\pi$
$113$	−5.27389	−0.496126	−0.248063	+	0.968744i	$0.579794\pi$
$114$	−4.07429	−0.381592
$115$	3.10242	0.289302
$116$	14.9904	1.39182
$117$	15.2333	1.40832
$118$	−7.41010	−0.682155
$119$	−4.47615	−0.410328
$120$	0.795753	0.0726420
$121$	−10.4441	−0.949460
$122$	−1.25439	−0.113567
$123$	4.44759	0.401026
$124$	0.961693	0.0863626
$125$	3.95637	0.353868
$126$	−3.56188	−0.317318
$127$	0.193928	0.0172084	0.00860418	−	0.999963i	$-0.497261\pi$
$127$	0.193928	0.0172084	0.00860418	+	0.999963i	$0.497261\pi$
$128$	6.61348	0.584555
$129$	−5.34401	−0.470514
$130$	5.14064	0.450864
$131$	16.6694	1.45642	0.728208	−	0.685356i	$-0.240353\pi$
$131$	16.6694	1.45642	0.728208	+	0.685356i	$0.240353\pi$
$132$	4.20053	0.365609
$133$	0.562197	0.0487487
$134$	−21.8975	−1.89165
$135$	0.471310	0.0405639
$136$	5.56228	0.476962
$137$	−13.6911	−1.16971	−0.584853	−	0.811139i	$-0.698848\pi$
$137$	−13.6911	−1.16971	−0.584853	+	0.811139i	$0.698848\pi$
$138$	37.9619	3.23153
$139$	−2.19626	−0.186284	−0.0931420	−	0.995653i	$-0.529691\pi$
$139$	−2.19626	−0.186284	−0.0931420	+	0.995653i	$0.529691\pi$
$140$	−0.655898	−0.0554335
$141$	8.09267	0.681526
$142$	−26.6339	−2.23506
$143$	4.54278	0.379886
$144$	−7.58582	−0.632152
$145$	−2.50952	−0.208405
$146$	21.8964	1.81216
$147$	−15.3356	−1.26486
$148$	−22.4529	−1.84562
$149$	6.59353	0.540163	0.270081	−	0.962838i	$-0.412949\pi$
$149$	6.59353	0.540163	0.270081	+	0.962838i	$0.412949\pi$
$150$	23.8076	1.94388
$151$	3.18418	0.259125	0.129562	−	0.991571i	$-0.458643\pi$
$151$	3.18418	0.259125	0.129562	+	0.991571i	$0.458643\pi$
$152$	−0.698614	−0.0566651
$153$	−16.4828	−1.33255
$154$	−1.06220	−0.0855946
$155$	−0.160996	−0.0129315
$156$	34.3240	2.74812
$157$	10.4063	0.830510	0.415255	−	0.909705i	$-0.363692\pi$
$157$	10.4063	0.830510	0.415255	+	0.909705i	$0.363692\pi$
$158$	−36.3963	−2.89554
$159$	3.13800	0.248860
$160$	−3.23852	−0.256027
$161$	−5.23823	−0.412830
$162$	21.5047	1.68957
$163$	10.6512	0.834266	0.417133	−	0.908845i	$-0.363035\pi$
$163$	10.6512	0.834266	0.417133	+	0.908845i	$0.363035\pi$
$164$	4.55545	0.355721
$165$	−0.703207	−0.0547446
$166$	2.35989	0.183163
$167$	−1.85225	−0.143331	−0.0716656	−	0.997429i	$-0.522831\pi$
$167$	−1.85225	−0.143331	−0.0716656	+	0.997429i	$0.522831\pi$
$168$	−1.34358	−0.103659
$169$	24.1206	1.85543
$170$	−5.56228	−0.426607
$171$	2.07021	0.158313
$172$	−5.47361	−0.417359
$173$	15.1853	1.15452	0.577260	−	0.816560i	$-0.304122\pi$
$173$	15.1853	1.15452	0.577260	+	0.816560i	$0.304122\pi$
$174$	−30.7070	−2.32789
$175$	−3.28513	−0.248332
$176$	−2.26219	−0.170519
$177$	8.28293	0.622583
$178$	−2.09813	−0.157261
$179$	17.9782	1.34375	0.671876	−	0.740664i	$-0.265489\pi$
$179$	17.9782	1.34375	0.671876	+	0.740664i	$0.265489\pi$
$180$	−2.41525	−0.180022
$181$	−18.3834	−1.36643	−0.683214	−	0.730218i	$-0.739419\pi$
$181$	−18.3834	−1.36643	−0.683214	+	0.730218i	$0.739419\pi$
$182$	−8.67961	−0.643376
$183$	1.40214	0.103649
$184$	6.50928	0.479870
$185$	3.75883	0.276354
$186$	−1.96998	−0.144446
$187$	−4.91538	−0.359448
$188$	8.28893	0.604532
$189$	−0.795775	−0.0578841
$190$	0.698614	0.0506828
$191$	6.13884	0.444191	0.222096	−	0.975025i	$-0.428710\pi$
$191$	6.13884	0.444191	0.222096	+	0.975025i	$0.428710\pi$
$192$	−25.3961	−1.83280
$193$	8.69486	0.625870	0.312935	−	0.949775i	$-0.398688\pi$
$193$	8.69486	0.625870	0.312935	+	0.949775i	$0.398688\pi$
$194$	−16.0252	−1.15054
$195$	−5.74615	−0.411490
$196$	−15.7075	−1.12197
$197$	−23.6990	−1.68848	−0.844241	−	0.535963i	$-0.819949\pi$
$197$	−23.6990	−1.68848	−0.844241	+	0.535963i	$0.819949\pi$
$198$	−3.91140	−0.277971
$199$	−4.36867	−0.309687	−0.154843	−	0.987939i	$-0.549487\pi$
$199$	−4.36867	−0.309687	−0.154843	+	0.987939i	$0.549487\pi$
$200$	4.08226	0.288659
$201$	24.4768	1.72646
$202$	−18.8934	−1.32933
$203$	4.23716	0.297390
$204$	−37.1393	−2.60027
$205$	−0.762624	−0.0532640
$206$	−7.47499	−0.520807
$207$	−19.2890	−1.34068
$208$	−18.4852	−1.28172
$209$	0.617365	0.0427040
$210$	1.34358	0.0927155
$211$	−16.1958	−1.11497	−0.557483	−	0.830188i	$-0.688233\pi$
$211$	−16.1958	−1.11497	−0.557483	+	0.830188i	$0.688233\pi$
$212$	3.21410	0.220745
$213$	29.7711	2.03988
$214$	34.7111	2.37280
$215$	0.916331	0.0624933
$216$	0.988868	0.0672840
$217$	0.271831	0.0184531
$218$	29.5354	2.00039
$219$	−24.4755	−1.65390
$220$	−0.720260	−0.0485599
$221$	−40.1653	−2.70181
$222$	45.9937	3.08690
$223$	16.0452	1.07447	0.537234	−	0.843433i	$-0.319469\pi$
$223$	16.0452	1.07447	0.537234	+	0.843433i	$0.319469\pi$
$224$	5.46802	0.365347
$225$	−12.0970	−0.806467
$226$	11.0653	0.736053
$227$	0.986788	0.0654954	0.0327477	−	0.999464i	$-0.489574\pi$
$227$	0.986788	0.0654954	0.0327477	+	0.999464i	$0.489574\pi$
$228$	4.66464	0.308923
$229$	1.64149	0.108473	0.0542364	−	0.998528i	$-0.482728\pi$
$229$	1.64149	0.108473	0.0542364	+	0.998528i	$0.482728\pi$
$230$	−6.50928	−0.429209
$231$	1.18732	0.0781197
$232$	−5.26530	−0.345684
$233$	18.6930	1.22462	0.612311	−	0.790617i	$-0.290240\pi$
$233$	18.6930	1.22462	0.612311	+	0.790617i	$0.290240\pi$
$234$	−31.9614	−2.08938
$235$	−1.38764	−0.0905197
$236$	8.48380	0.552248
$237$	40.6834	2.64267
$238$	9.39153	0.608763
$239$	−10.2061	−0.660178	−0.330089	−	0.943950i	$-0.607079\pi$
$239$	−10.2061	−0.660178	−0.330089	+	0.943950i	$0.607079\pi$
$240$	2.86144	0.184705
$241$	14.0238	0.903355	0.451677	−	0.892181i	$-0.350826\pi$
$241$	14.0238	0.903355	0.451677	+	0.892181i	$0.350826\pi$
$242$	21.9130	1.40862
$243$	−20.5217	−1.31647
$244$	1.43614	0.0919397
$245$	2.62958	0.167998
$246$	−9.33162	−0.594962
$247$	5.04470	0.320987
$248$	−0.337790	−0.0214497
$249$	−2.63786	−0.167167
$250$	−8.30096	−0.524999
$251$	−18.7723	−1.18490	−0.592448	−	0.805609i	$-0.701838\pi$
$251$	−18.7723	−1.18490	−0.592448	+	0.805609i	$0.701838\pi$
$252$	4.07799	0.256889
$253$	−5.75224	−0.361640
$254$	−0.406886	−0.0255303
$255$	6.21746	0.389352
$256$	7.78138	0.486336
$257$	−3.94845	−0.246298	−0.123149	−	0.992388i	$-0.539299\pi$
$257$	−3.94845	−0.246298	−0.123149	+	0.992388i	$0.539299\pi$
$258$	11.2124	0.698054
$259$	−6.34652	−0.394354
$260$	−5.88550	−0.365003
$261$	15.6027	0.965785
$262$	−34.9746	−2.16074
$263$	−6.91803	−0.426584	−0.213292	−	0.976988i	$-0.568419\pi$
$263$	−6.91803	−0.426584	−0.213292	+	0.976988i	$0.568419\pi$
$264$	−1.47542	−0.0908057
$265$	−0.538070	−0.0330534
$266$	−1.17956	−0.0723236
$267$	2.34526	0.143528
$268$	25.0704	1.53142
$269$	−21.1692	−1.29071	−0.645354	−	0.763884i	$-0.723290\pi$
$269$	−21.1692	−1.29071	−0.645354	+	0.763884i	$0.723290\pi$
$270$	−0.988868	−0.0601806
$271$	2.36402	0.143604	0.0718019	−	0.997419i	$-0.477125\pi$
$271$	2.36402	0.143604	0.0718019	+	0.997419i	$0.477125\pi$
$272$	20.0014	1.21276
$273$	9.70198	0.587191
$274$	28.7256	1.73538
$275$	−3.60749	−0.217540
$276$	−43.4624	−2.61613
$277$	26.2166	1.57520	0.787600	−	0.616187i	$-0.211323\pi$
$277$	26.2166	1.57520	0.787600	+	0.616187i	$0.211323\pi$
$278$	4.60802	0.276371
$279$	1.00098	0.0599270
$280$	0.230381	0.0137679
$281$	13.3303	0.795220	0.397610	−	0.917555i	$-0.369840\pi$
$281$	13.3303	0.795220	0.397610	+	0.917555i	$0.369840\pi$
$282$	−16.9795	−1.01111
$283$	−7.39449	−0.439557	−0.219778	−	0.975550i	$-0.570533\pi$
$283$	−7.39449	−0.439557	−0.219778	+	0.975550i	$0.570533\pi$
$284$	30.4930	1.80943
$285$	−0.780903	−0.0462567
$286$	−9.53133	−0.563599
$287$	1.28764	0.0760069
$288$	20.1352	1.18648
$289$	26.4598	1.55646
$290$	5.26530	0.309189
$291$	17.9128	1.05007
$292$	−25.0691	−1.46706
$293$	19.3032	1.12771	0.563853	−	0.825875i	$-0.309319\pi$
$293$	19.3032	1.12771	0.563853	+	0.825875i	$0.309319\pi$
$294$	32.1761	1.87655
$295$	−1.42027	−0.0826910
$296$	7.88650	0.458393
$297$	−0.873862	−0.0507066
$298$	−13.8341	−0.801386
$299$	−47.0036	−2.71829
$300$	−27.2572	−1.57370
$301$	−1.54716	−0.0891770
$302$	−6.68081	−0.384437
$303$	21.1188	1.21324
$304$	−2.51214	−0.144081
$305$	−0.240423	−0.0137666
$306$	34.5830	1.97698
$307$	24.8020	1.41552	0.707761	−	0.706452i	$-0.249705\pi$
$307$	24.8020	1.41552	0.707761	+	0.706452i	$0.249705\pi$
$308$	1.21611	0.0692943
$309$	8.35546	0.475326
$310$	0.337790	0.0191852
$311$	13.5042	0.765750	0.382875	−	0.923800i	$-0.374934\pi$
$311$	13.5042	0.765750	0.382875	+	0.923800i	$0.374934\pi$
$312$	−12.0562	−0.682545
$313$	−8.16150	−0.461315	−0.230658	−	0.973035i	$-0.574088\pi$
$313$	−8.16150	−0.461315	−0.230658	+	0.973035i	$0.574088\pi$
$314$	−21.8337	−1.23214
$315$	−0.682692	−0.0384653
$316$	41.6701	2.34412
$317$	−11.9353	−0.670353	−0.335176	−	0.942155i	$-0.608796\pi$
$317$	−11.9353	−0.670353	−0.335176	+	0.942155i	$0.608796\pi$
$318$	−6.58393	−0.369208
$319$	4.65294	0.260515
$320$	4.35464	0.243432
$321$	−38.7997	−2.16559
$322$	10.9905	0.612475
$323$	−5.45848	−0.303718
$324$	−24.6206	−1.36781
$325$	−29.4781	−1.63515
$326$	−22.3476	−1.23772
$327$	−33.0144	−1.82570
$328$	−1.60008	−0.0883498
$329$	2.34294	0.129170
$330$	1.47542	0.0812190
$331$	22.0853	1.21392	0.606960	−	0.794732i	$-0.292389\pi$
$331$	22.0853	1.21392	0.606960	+	0.794732i	$0.292389\pi$
$332$	−2.70183	−0.148282
$333$	−23.3701	−1.28068
$334$	3.88625	0.212646
$335$	−4.19701	−0.229307
$336$	−4.83135	−0.263572
$337$	11.1800	0.609016	0.304508	−	0.952510i	$-0.401508\pi$
$337$	11.1800	0.609016	0.304508	+	0.952510i	$0.401508\pi$
$338$	−50.6082	−2.75272
$339$	−12.3687	−0.671774
$340$	6.36824	0.345366
$341$	0.298505	0.0161650
$342$	−4.34357	−0.234873
$343$	−9.19278	−0.496363
$344$	1.92258	0.103659
$345$	7.27600	0.391727
$346$	−31.8608	−1.71285
$347$	−9.49198	−0.509556	−0.254778	−	0.967000i	$-0.582002\pi$
$347$	−9.49198	−0.509556	−0.254778	+	0.967000i	$0.582002\pi$
$348$	35.1564	1.88458
$349$	−24.2009	−1.29545	−0.647723	−	0.761876i	$-0.724278\pi$
$349$	−24.2009	−1.29545	−0.647723	+	0.761876i	$0.724278\pi$
$350$	6.89262	0.368426
$351$	−7.14064	−0.381139
$352$	6.00458	0.320045
$353$	−1.04797	−0.0557776	−0.0278888	−	0.999611i	$-0.508878\pi$
$353$	−1.04797	−0.0557776	−0.0278888	+	0.999611i	$0.508878\pi$
$354$	−17.3786	−0.923665
$355$	−5.10481	−0.270935
$356$	2.40214	0.127313
$357$	−10.4978	−0.555600
$358$	−37.7205	−1.99359
$359$	30.8654	1.62901	0.814507	−	0.580154i	$-0.197008\pi$
$359$	30.8654	1.62901	0.814507	+	0.580154i	$0.197008\pi$
$360$	0.848346	0.0447118
$361$	−18.3144	−0.963917
$362$	38.5708	2.02723
$363$	−24.4941	−1.28561
$364$	9.93726	0.520854
$365$	4.19679	0.219670
$366$	−2.94187	−0.153774
$367$	−29.8753	−1.55948	−0.779740	−	0.626104i	$-0.784649\pi$
$367$	−29.8753	−1.55948	−0.779740	+	0.626104i	$0.784649\pi$
$368$	23.4067	1.22016
$369$	4.74154	0.246835
$370$	−7.88650	−0.409999
$371$	0.908494	0.0471667
$372$	2.25542	0.116938
$373$	16.8287	0.871355	0.435677	−	0.900103i	$-0.356509\pi$
$373$	16.8287	0.871355	0.435677	+	0.900103i	$0.356509\pi$
$374$	10.3131	0.533278
$375$	9.27873	0.479152
$376$	−2.91145	−0.150147
$377$	38.0208	1.95817
$378$	1.66964	0.0858769
$379$	−13.6608	−0.701707	−0.350853	−	0.936430i	$-0.614108\pi$
$379$	−13.6608	−0.701707	−0.350853	+	0.936430i	$0.614108\pi$
$380$	−0.799841	−0.0410310
$381$	0.454813	0.0233008
$382$	−12.8801	−0.659002
$383$	−25.6318	−1.30972	−0.654861	−	0.755749i	$-0.727273\pi$
$383$	−25.6318	−1.30972	−0.654861	+	0.755749i	$0.727273\pi$
$384$	15.5104	0.791510
$385$	−0.203588	−0.0103758
$386$	−18.2429	−0.928541
$387$	−5.69720	−0.289605
$388$	18.3472	0.931438
$389$	14.5801	0.739240	0.369620	−	0.929183i	$-0.379488\pi$
$389$	14.5801	0.739240	0.369620	+	0.929183i	$0.379488\pi$
$390$	12.0562	0.610487
$391$	50.8589	2.57205
$392$	5.51720	0.278661
$393$	39.0943	1.97205
$394$	49.7235	2.50503
$395$	−6.97595	−0.350998
$396$	4.47815	0.225035
$397$	−27.9706	−1.40380	−0.701902	−	0.712274i	$-0.747665\pi$
$397$	−27.9706	−1.40380	−0.701902	+	0.712274i	$0.747665\pi$
$398$	9.16603	0.459452
$399$	1.31850	0.0660077
$400$	14.6794	0.733968
$401$	−17.9848	−0.898116	−0.449058	−	0.893503i	$-0.648240\pi$
$401$	−17.9848	−0.898116	−0.449058	+	0.893503i	$0.648240\pi$
$402$	−51.3554	−2.56137
$403$	2.43919	0.121505
$404$	21.6310	1.07618
$405$	4.12172	0.204810
$406$	−8.89010	−0.441208
$407$	−6.96929	−0.345455
$408$	13.0450	0.645825
$409$	16.9540	0.838323	0.419162	−	0.907912i	$-0.362324\pi$
$409$	16.9540	0.838323	0.419162	+	0.907912i	$0.362324\pi$
$410$	1.60008	0.0790225
$411$	−32.1092	−1.58383
$412$	8.55809	0.421627
$413$	2.39802	0.117999
$414$	40.4708	1.98903
$415$	0.452310	0.0222030
$416$	49.0655	2.40564
$417$	−5.15080	−0.252236
$418$	−1.29531	−0.0633557
$419$	−15.4330	−0.753952	−0.376976	−	0.926223i	$-0.623036\pi$
$419$	−15.4330	−0.753952	−0.376976	+	0.926223i	$0.623036\pi$
$420$	−1.53826	−0.0750592
$421$	−9.47448	−0.461758	−0.230879	−	0.972982i	$-0.574160\pi$
$421$	−9.47448	−0.461758	−0.230879	+	0.972982i	$0.574160\pi$
$422$	33.9809	1.65416
$423$	8.62753	0.419485
$424$	−1.12894	−0.0548261
$425$	31.8959	1.54718
$426$	−62.4635	−3.02637
$427$	0.405939	0.0196447
$428$	−39.7406	−1.92094
$429$	10.6540	0.514381
$430$	−1.92258	−0.0927151
$431$	19.7151	0.949643	0.474822	−	0.880082i	$-0.342513\pi$
$431$	19.7151	0.949643	0.474822	+	0.880082i	$0.342513\pi$
$432$	3.55587	0.171082
$433$	−20.9780	−1.00814	−0.504070	−	0.863663i	$-0.668164\pi$
$433$	−20.9780	−1.00814	−0.504070	+	0.863663i	$0.668164\pi$
$434$	−0.570336	−0.0273770
$435$	−5.88550	−0.282188
$436$	−33.8150	−1.61945
$437$	−6.38780	−0.305570
$438$	51.3528	2.45373
$439$	23.2780	1.11100	0.555500	−	0.831516i	$-0.312527\pi$
$439$	23.2780	1.11100	0.555500	+	0.831516i	$0.312527\pi$
$440$	0.252988	0.0120607
$441$	−16.3492	−0.778533
$442$	84.2720	4.00841
$443$	−8.53700	−0.405605	−0.202803	−	0.979220i	$-0.565005\pi$
$443$	−8.53700	−0.405605	−0.202803	+	0.979220i	$0.565005\pi$
$444$	−52.6581	−2.49904
$445$	−0.402140	−0.0190633
$446$	−33.6650	−1.59408
$447$	15.4636	0.731402
$448$	−7.35251	−0.347373
$449$	10.7725	0.508386	0.254193	−	0.967154i	$-0.418190\pi$
$449$	10.7725	0.508386	0.254193	+	0.967154i	$0.418190\pi$
$450$	25.3811	1.19647
$451$	1.41399	0.0665823
$452$	−12.6686	−0.595882
$453$	7.46774	0.350865
$454$	−2.07041	−0.0971690
$455$	−1.66359	−0.0779902
$456$	−1.63843	−0.0767267
$457$	−31.0724	−1.45351	−0.726753	−	0.686899i	$-0.758971\pi$
$457$	−31.0724	−1.45351	−0.726753	+	0.686899i	$0.758971\pi$
$458$	−3.44406	−0.160930
$459$	7.72632	0.360634
$460$	7.45245	0.347472
$461$	−23.8796	−1.11218	−0.556092	−	0.831121i	$-0.687700\pi$
$461$	−23.8796	−1.11218	−0.556092	+	0.831121i	$0.687700\pi$
$462$	−2.49114	−0.115898
$463$	12.8814	0.598649	0.299324	−	0.954151i	$-0.403239\pi$
$463$	12.8814	0.598649	0.299324	+	0.954151i	$0.403239\pi$
$464$	−18.9335	−0.878964
$465$	−0.377579	−0.0175098
$466$	−39.2204	−1.81685
$467$	4.98245	0.230560	0.115280	−	0.993333i	$-0.463223\pi$
$467$	4.98245	0.230560	0.115280	+	0.993333i	$0.463223\pi$
$468$	36.5925	1.69149
$469$	7.08636	0.327218
$470$	2.91145	0.134295
$471$	24.4054	1.12454
$472$	−2.97990	−0.137161
$473$	−1.69898	−0.0781193
$474$	−85.3591	−3.92067
$475$	−4.00608	−0.183811
$476$	−10.7523	−0.492833
$477$	3.34540	0.153175
$478$	21.4137	0.979440
$479$	−29.9376	−1.36788	−0.683941	−	0.729537i	$-0.739736\pi$
$479$	−29.9376	−1.36788	−0.683941	+	0.729537i	$0.739736\pi$
$480$	−7.59518	−0.346671
$481$	−56.9485	−2.59663
$482$	−29.4238	−1.34022
$483$	−12.2850	−0.558988
$484$	−25.0881	−1.14037
$485$	−3.07149	−0.139469
$486$	43.0571	1.95311
$487$	21.5179	0.975069	0.487535	−	0.873104i	$-0.337896\pi$
$487$	21.5179	0.975069	0.487535	+	0.873104i	$0.337896\pi$
$488$	−0.504439	−0.0228349
$489$	24.9799	1.12963
$490$	−5.51720	−0.249242
$491$	22.2920	1.00602	0.503012	−	0.864279i	$-0.332225\pi$
$491$	22.2920	1.00602	0.503012	+	0.864279i	$0.332225\pi$
$492$	10.6837	0.481660
$493$	−41.1394	−1.85282
$494$	−10.5844	−0.476216
$495$	−0.749683	−0.0336957
$496$	−1.21466	−0.0545398
$497$	8.61913	0.386621
$498$	5.53456	0.248010
$499$	35.9789	1.61063	0.805317	−	0.592844i	$-0.201995\pi$
$499$	35.9789	1.61063	0.805317	+	0.592844i	$0.201995\pi$
$500$	9.50375	0.425021
$501$	−4.34401	−0.194076
$502$	39.3867	1.75791
$503$	−12.7057	−0.566517	−0.283259	−	0.959044i	$-0.591415\pi$
$503$	−12.7057	−0.566517	−0.283259	+	0.959044i	$0.591415\pi$
$504$	−1.43237	−0.0638030
$505$	−3.62122	−0.161142
$506$	12.0689	0.536530
$507$	56.5693	2.51233
$508$	0.465843	0.0206684
$509$	−22.2317	−0.985403	−0.492701	−	0.870198i	$-0.663991\pi$
$509$	−22.2317	−0.985403	−0.492701	+	0.870198i	$0.663991\pi$
$510$	−13.0450	−0.577643
$511$	−7.08600	−0.313466
$512$	−29.5533	−1.30608
$513$	−0.970414	−0.0428448
$514$	8.28435	0.365407
$515$	−1.43270	−0.0631324
$516$	−12.8371	−0.565120
$517$	2.57285	0.113154
$518$	13.3158	0.585063
$519$	35.6136	1.56327
$520$	2.06726	0.0906552
$521$	15.2116	0.666433	0.333216	−	0.942850i	$-0.391866\pi$
$521$	15.2116	0.666433	0.333216	+	0.942850i	$0.391866\pi$
$522$	−32.7365	−1.43284
$523$	23.2905	1.01842	0.509212	−	0.860641i	$-0.329937\pi$
$523$	23.2905	1.01842	0.509212	+	0.860641i	$0.329937\pi$
$524$	40.0424	1.74926
$525$	−7.70449	−0.336252
$526$	14.5149	0.632880
$527$	−2.63926	−0.114968
$528$	−5.30544	−0.230890
$529$	36.5178	1.58773
$530$	1.12894	0.0490380
$531$	8.83037	0.383205
$532$	1.35048	0.0585506
$533$	11.5542	0.500469
$534$	−4.92067	−0.212938
$535$	6.65294	0.287632
$536$	−8.80586	−0.380355
$537$	42.1636	1.81949
$538$	44.4156	1.91489
$539$	−4.87555	−0.210005
$540$	1.13215	0.0487201
$541$	−9.20513	−0.395760	−0.197880	−	0.980226i	$-0.563406\pi$
$541$	−9.20513	−0.395760	−0.197880	+	0.980226i	$0.563406\pi$
$542$	−4.96001	−0.213051
$543$	−43.1140	−1.85020
$544$	−53.0900	−2.27621
$545$	5.66094	0.242488
$546$	−20.3560	−0.871156
$547$	−25.6044	−1.09477	−0.547383	−	0.836882i	$-0.684376\pi$
$547$	−25.6044	−1.09477	−0.547383	+	0.836882i	$0.684376\pi$
$548$	−32.8879	−1.40490
$549$	1.49481	0.0637969
$550$	7.56897	0.322742
$551$	5.16704	0.220123
$552$	15.2660	0.649763
$553$	11.7784	0.500869
$554$	−55.0057	−2.33697
$555$	8.81544	0.374195
$556$	−5.27571	−0.223740
$557$	−13.0483	−0.552874	−0.276437	−	0.961032i	$-0.589154\pi$
$557$	−13.0483	−0.552874	−0.276437	+	0.961032i	$0.589154\pi$
$558$	−2.10018	−0.0889077
$559$	−13.8830	−0.587188
$560$	0.828427	0.0350074
$561$	−11.5279	−0.486707
$562$	−27.9687	−1.17979
$563$	38.2429	1.61175	0.805874	−	0.592087i	$-0.201696\pi$
$563$	38.2429	1.61175	0.805874	+	0.592087i	$0.201696\pi$
$564$	19.4397	0.818560
$565$	2.12084	0.0892245
$566$	15.5146	0.652126
$567$	−6.95924	−0.292261
$568$	−10.7105	−0.449405
$569$	−23.2615	−0.975174	−0.487587	−	0.873074i	$-0.662123\pi$
$569$	−23.2615	−0.975174	−0.487587	+	0.873074i	$0.662123\pi$
$570$	1.63843	0.0686265
$571$	−1.38815	−0.0580921	−0.0290461	−	0.999578i	$-0.509247\pi$
$571$	−1.38815	−0.0580921	−0.0290461	+	0.999578i	$0.509247\pi$
$572$	10.9124	0.456270
$573$	14.3972	0.601452
$574$	−2.70163	−0.112764
$575$	37.3263	1.55661
$576$	−27.0745	−1.12811
$577$	32.3652	1.34738	0.673690	−	0.739014i	$-0.264708\pi$
$577$	32.3652	1.34738	0.673690	+	0.739014i	$0.264708\pi$
$578$	−55.5160	−2.30916
$579$	20.3918	0.847453
$580$	−6.02823	−0.250309
$581$	−0.763695	−0.0316834
$582$	−37.5833	−1.55788
$583$	0.997642	0.0413181
$584$	8.80540	0.364370
$585$	−6.12592	−0.253276
$586$	−40.5006	−1.67306
$587$	−40.5897	−1.67532	−0.837658	−	0.546195i	$-0.816076\pi$
$587$	−40.5897	−1.67532	−0.837658	+	0.546195i	$0.816076\pi$
$588$	−36.8383	−1.51919
$589$	0.331487	0.0136587
$590$	2.97990	0.122680
$591$	−55.5804	−2.28627
$592$	28.3590	1.16555
$593$	5.60774	0.230282	0.115141	−	0.993349i	$-0.463268\pi$
$593$	5.60774	0.230282	0.115141	+	0.993349i	$0.463268\pi$
$594$	1.83347	0.0752284
$595$	1.80004	0.0737944
$596$	15.8386	0.648773
$597$	−10.2457	−0.419328
$598$	98.6196	4.03285
$599$	−15.2969	−0.625015	−0.312508	−	0.949915i	$-0.601169\pi$
$599$	−15.2969	−0.625015	−0.312508	+	0.949915i	$0.601169\pi$
$600$	9.57398	0.390856
$601$	−29.4967	−1.20319	−0.601597	−	0.798799i	$-0.705469\pi$
$601$	−29.4967	−1.20319	−0.601597	+	0.798799i	$0.705469\pi$
$602$	3.24615	0.132303
$603$	26.0945	1.06265
$604$	7.64884	0.311227
$605$	4.19997	0.170753
$606$	−44.3100	−1.79997
$607$	−2.66519	−0.108177	−0.0540885	−	0.998536i	$-0.517225\pi$
$607$	−2.66519	−0.108177	−0.0540885	+	0.998536i	$0.517225\pi$
$608$	6.66802	0.270424
$609$	9.93726	0.402678
$610$	0.504439	0.0204241
$611$	21.0236	0.850525
$612$	−39.5939	−1.60049
$613$	17.2285	0.695853	0.347927	−	0.937522i	$-0.386886\pi$
$613$	17.2285	0.695853	0.347927	+	0.937522i	$0.386886\pi$
$614$	−52.0377	−2.10007
$615$	−1.78856	−0.0721215
$616$	−0.427153	−0.0172105
$617$	18.2907	0.736355	0.368177	−	0.929756i	$-0.379982\pi$
$617$	18.2907	0.736355	0.368177	+	0.929756i	$0.379982\pi$
$618$	−17.5308	−0.705193
$619$	41.6253	1.67306	0.836532	−	0.547919i	$-0.184580\pi$
$619$	41.6253	1.67306	0.836532	+	0.547919i	$0.184580\pi$
$620$	−0.386735	−0.0155317
$621$	9.04176	0.362833
$622$	−28.3334	−1.13607
$623$	0.678986	0.0272030
$624$	−43.3526	−1.73549
$625$	22.6004	0.904016
$626$	17.1239	0.684408
$627$	1.44788	0.0578229
$628$	24.9973	0.997500
$629$	61.6196	2.45693
$630$	1.43237	0.0570672
$631$	35.5147	1.41382	0.706909	−	0.707304i	$-0.250089\pi$
$631$	35.5147	1.41382	0.706909	+	0.707304i	$0.250089\pi$
$632$	−14.6364	−0.582206
$633$	−37.9835	−1.50971
$634$	25.0418	0.994535
$635$	−0.0779864	−0.00309479
$636$	7.53792	0.298898
$637$	−39.8398	−1.57851
$638$	−9.76247	−0.386500
$639$	31.7387	1.25556
$640$	−2.65955	−0.105128
$641$	−6.95752	−0.274805	−0.137403	−	0.990515i	$-0.543875\pi$
$641$	−6.95752	−0.274805	−0.137403	+	0.990515i	$0.543875\pi$
$642$	81.4067	3.21287
$643$	−27.1280	−1.06982	−0.534911	−	0.844908i	$-0.679655\pi$
$643$	−27.1280	−1.06982	−0.534911	+	0.844908i	$0.679655\pi$
$644$	−12.5830	−0.495838
$645$	2.14904	0.0846184
$646$	11.4526	0.450596
$647$	−42.4343	−1.66826	−0.834132	−	0.551564i	$-0.814031\pi$
$647$	−42.4343	−1.66826	−0.834132	+	0.551564i	$0.814031\pi$
$648$	8.64789	0.339721
$649$	2.63333	0.103367
$650$	61.8487	2.42591
$651$	0.637516	0.0249862
$652$	25.5857	1.00201
$653$	5.57983	0.218356	0.109178	−	0.994022i	$-0.465178\pi$
$653$	5.57983	0.218356	0.109178	+	0.994022i	$0.465178\pi$
$654$	69.2684	2.70861
$655$	−6.70345	−0.261926
$656$	−5.75373	−0.224645
$657$	−26.0932	−1.01799
$658$	−4.91578	−0.191637
$659$	−19.1324	−0.745291	−0.372646	−	0.927974i	$-0.621549\pi$
$659$	−19.1324	−0.745291	−0.372646	+	0.927974i	$0.621549\pi$
$660$	−1.68920	−0.0657520
$661$	10.5218	0.409252	0.204626	−	0.978840i	$-0.434402\pi$
$661$	10.5218	0.409252	0.204626	+	0.978840i	$0.434402\pi$
$662$	−46.3379	−1.80097
$663$	−94.1983	−3.65836
$664$	0.949005	0.0368285
$665$	−0.226082	−0.00876709
$666$	49.0335	1.90001
$667$	−48.1435	−1.86412
$668$	−4.44936	−0.172151
$669$	37.6303	1.45487
$670$	8.80586	0.340200
$671$	0.445772	0.0172089
$672$	12.8239	0.494695
$673$	−34.9240	−1.34622	−0.673111	−	0.739542i	$-0.735042\pi$
$673$	−34.9240	−1.34622	−0.673111	+	0.739542i	$0.735042\pi$
$674$	−23.4572	−0.903536
$675$	5.67049	0.218257
$676$	57.9411	2.22850
$677$	−19.8302	−0.762135	−0.381067	−	0.924547i	$-0.624443\pi$
$677$	−19.8302	−0.762135	−0.381067	+	0.924547i	$0.624443\pi$
$678$	25.9511	0.996644
$679$	5.18600	0.199020
$680$	−2.23682	−0.0857780
$681$	2.31428	0.0886834
$682$	−0.626302	−0.0239823
$683$	5.91731	0.226420	0.113210	−	0.993571i	$-0.463887\pi$
$683$	5.91731	0.226420	0.113210	+	0.993571i	$0.463887\pi$
$684$	4.97294	0.190145
$685$	5.50573	0.210363
$686$	19.2876	0.736405
$687$	3.84973	0.146877
$688$	6.91339	0.263571
$689$	8.15209	0.310570
$690$	−15.2660	−0.581166
$691$	−34.9975	−1.33137	−0.665685	−	0.746233i	$-0.731860\pi$
$691$	−34.9975	−1.33137	−0.665685	+	0.746233i	$0.731860\pi$
$692$	36.4773	1.38666
$693$	1.26579	0.0480833
$694$	19.9154	0.755978
$695$	0.883202	0.0335018
$696$	−12.3485	−0.468070
$697$	−12.5019	−0.473544
$698$	50.7766	1.92192
$699$	43.8401	1.65819
$700$	−7.89133	−0.298264
$701$	2.61006	0.0985807	0.0492903	−	0.998784i	$-0.484304\pi$
$701$	2.61006	0.0985807	0.0492903	+	0.998784i	$0.484304\pi$
$702$	14.9820	0.565458
$703$	−7.73932	−0.291894
$704$	−8.07399	−0.304300
$705$	−3.25439	−0.122567
$706$	2.19877	0.0827517
$707$	6.11418	0.229947
$708$	19.8968	0.747766
$709$	−5.18047	−0.194557	−0.0972783	−	0.995257i	$-0.531014\pi$
$709$	−5.18047	−0.194557	−0.0972783	+	0.995257i	$0.531014\pi$
$710$	10.7105	0.401960
$711$	43.3723	1.62659
$712$	−0.843741	−0.0316205
$713$	−3.08860	−0.115669
$714$	22.0256	0.824289
$715$	−1.82683	−0.0683197
$716$	43.1861	1.61394
$717$	−23.9360	−0.893907
$718$	−64.7595	−2.41680
$719$	−22.7700	−0.849177	−0.424588	−	0.905386i	$-0.639581\pi$
$719$	−22.7700	−0.849177	−0.424588	+	0.905386i	$0.639581\pi$
$720$	3.05056	0.113688
$721$	2.41902	0.0900890
$722$	38.4260	1.43007
$723$	32.8896	1.22318
$724$	−44.1595	−1.64118
$725$	−30.1929	−1.12134
$726$	51.3917	1.90733
$727$	−21.9526	−0.814175	−0.407088	−	0.913389i	$-0.633456\pi$
$727$	−21.9526	−0.814175	−0.407088	+	0.913389i	$0.633456\pi$
$728$	−3.49042	−0.129364
$729$	−17.3804	−0.643719
$730$	−8.80540	−0.325903
$731$	15.0217	0.555597
$732$	3.36814	0.124490
$733$	−14.0420	−0.518652	−0.259326	−	0.965790i	$-0.583500\pi$
$733$	−14.0420	−0.518652	−0.259326	+	0.965790i	$0.583500\pi$
$734$	62.6823	2.31364
$735$	6.16707	0.227476
$736$	−62.1287	−2.29010
$737$	7.78173	0.286644
$738$	−9.94837	−0.366204
$739$	−32.5728	−1.19821	−0.599106	−	0.800670i	$-0.704477\pi$
$739$	−32.5728	−1.19821	−0.599106	+	0.800670i	$0.704477\pi$
$740$	9.02923	0.331921
$741$	11.8312	0.434629
$742$	−1.90614	−0.0699765
$743$	29.7292	1.09066	0.545330	−	0.838221i	$-0.316404\pi$
$743$	29.7292	1.09066	0.545330	+	0.838221i	$0.316404\pi$
$744$	−0.792208	−0.0290438
$745$	−2.65152	−0.0971442
$746$	−35.3087	−1.29274
$747$	−2.81220	−0.102893
$748$	−11.8074	−0.431723
$749$	−11.2330	−0.410446
$750$	−19.4680	−0.710869
$751$	38.1761	1.39307	0.696533	−	0.717525i	$-0.254725\pi$
$751$	38.1761	1.39307	0.696533	+	0.717525i	$0.254725\pi$
$752$	−10.4693	−0.381775
$753$	−44.0260	−1.60440
$754$	−79.7726	−2.90515
$755$	−1.28048	−0.0466016
$756$	−1.91156	−0.0695228
$757$	−29.0793	−1.05691	−0.528453	−	0.848962i	$-0.677228\pi$
$757$	−29.0793	−1.05691	−0.528453	+	0.848962i	$0.677228\pi$
$758$	28.6620	1.04105
$759$	−13.4905	−0.489675
$760$	0.280941	0.0101908
$761$	−33.0769	−1.19904	−0.599519	−	0.800361i	$-0.704641\pi$
$761$	−33.0769	−1.19904	−0.599519	+	0.800361i	$0.704641\pi$
$762$	−0.954257	−0.0345691
$763$	−9.55812	−0.346027
$764$	14.7464	0.533505
$765$	6.62838	0.239650
$766$	53.7787	1.94310
$767$	21.5179	0.776966
$768$	18.2494	0.658518
$769$	−12.4613	−0.449367	−0.224683	−	0.974432i	$-0.572135\pi$
$769$	−12.4613	−0.449367	−0.224683	+	0.974432i	$0.572135\pi$
$770$	0.427153	0.0153935
$771$	−9.26016	−0.333497
$772$	20.8863	0.751714
$773$	3.92483	0.141166	0.0705831	−	0.997506i	$-0.477514\pi$
$773$	3.92483	0.141166	0.0705831	+	0.997506i	$0.477514\pi$
$774$	11.9535	0.429658
$775$	−1.93700	−0.0695791
$776$	−6.44437	−0.231340
$777$	−14.8843	−0.533970
$778$	−30.5909	−1.09674
$779$	1.57022	0.0562591
$780$	−13.8031	−0.494229
$781$	9.46490	0.338681
$782$	−106.709	−3.81589
$783$	−7.31380	−0.261374
$784$	19.8393	0.708546
$785$	−4.18477	−0.149361
$786$	−82.0248	−2.92573
$787$	−22.9047	−0.816465	−0.408233	−	0.912878i	$-0.633855\pi$
$787$	−22.9047	−0.816465	−0.408233	+	0.912878i	$0.633855\pi$
$788$	−56.9283	−2.02799
$789$	−16.2246	−0.577612
$790$	14.6364	0.520741
$791$	−3.58090	−0.127322
$792$	−1.57293	−0.0558916
$793$	3.64256	0.129351
$794$	58.6858	2.08268
$795$	−1.26192	−0.0447555
$796$	−10.4942	−0.371956
$797$	11.5045	0.407509	0.203755	−	0.979022i	$-0.434686\pi$
$797$	11.5045	0.407509	0.203755	+	0.979022i	$0.434686\pi$
$798$	−2.76639	−0.0979290
$799$	−22.7480	−0.804767
$800$	−38.9637	−1.37758
$801$	2.50027	0.0883426
$802$	37.7343	1.33244
$803$	−7.78133	−0.274597
$804$	58.7966	2.07360
$805$	2.10650	0.0742444
$806$	−5.11773	−0.180265
$807$	−49.6473	−1.74767
$808$	−7.59778	−0.267289
$809$	−11.6004	−0.407850	−0.203925	−	0.978987i	$-0.565370\pi$
$809$	−11.6004	−0.407850	−0.203925	+	0.978987i	$0.565370\pi$
$810$	−8.64789	−0.303856
$811$	−45.9245	−1.61263	−0.806313	−	0.591489i	$-0.798540\pi$
$811$	−45.9245	−1.61263	−0.806313	+	0.591489i	$0.798540\pi$
$812$	10.1783	0.357187
$813$	5.54425	0.194445
$814$	14.6225	0.512517
$815$	−4.28327	−0.150037
$816$	46.9085	1.64213
$817$	−1.88670	−0.0660073
$818$	−35.5717	−1.24374
$819$	10.3432	0.361421
$820$	−1.83193	−0.0639738
$821$	29.3695	1.02500	0.512501	−	0.858687i	$-0.328719\pi$
$821$	29.3695	1.02500	0.512501	+	0.858687i	$0.328719\pi$
$822$	67.3692	2.34977
$823$	−7.09029	−0.247152	−0.123576	−	0.992335i	$-0.539436\pi$
$823$	−7.09029	−0.247152	−0.123576	+	0.992335i	$0.539436\pi$
$824$	−3.00599	−0.104719
$825$	−8.46052	−0.294557
$826$	−5.03136	−0.175063
$827$	5.07055	0.176320	0.0881601	−	0.996106i	$-0.471901\pi$
$827$	5.07055	0.176320	0.0881601	+	0.996106i	$0.471901\pi$
$828$	−46.3349	−1.61025
$829$	−5.34123	−0.185508	−0.0927542	−	0.995689i	$-0.529567\pi$
$829$	−5.34123	−0.185508	−0.0927542	+	0.995689i	$0.529567\pi$
$830$	−0.949005	−0.0329404
$831$	61.4848	2.13288
$832$	−65.9754	−2.28729
$833$	43.1076	1.49359
$834$	10.8070	0.374217
$835$	0.744862	0.0257770
$836$	1.48300	0.0512905
$837$	−0.469210	−0.0162183
$838$	32.3804	1.11856
$839$	30.9654	1.06904	0.534522	−	0.845154i	$-0.320492\pi$
$839$	30.9654	1.06904	0.534522	+	0.845154i	$0.320492\pi$
$840$	0.540305	0.0186423
$841$	9.94288	0.342858
$842$	19.8787	0.685064
$843$	31.2631	1.07676
$844$	−38.9046	−1.33915
$845$	−9.69987	−0.333686
$846$	−18.1017	−0.622348
$847$	−7.09137	−0.243662
$848$	−4.05954	−0.139405
$849$	−17.3420	−0.595177
$850$	−66.9217	−2.29540
$851$	72.1105	2.47192
$852$	71.5143	2.45004
$853$	21.7707	0.745414	0.372707	−	0.927949i	$-0.378430\pi$
$853$	21.7707	0.745414	0.372707	+	0.927949i	$0.378430\pi$
$854$	−0.851711	−0.0291450
$855$	−0.832515	−0.0284714
$856$	13.9587	0.477099
$857$	43.2636	1.47786	0.738929	−	0.673784i	$-0.235332\pi$
$857$	43.2636	1.47786	0.738929	+	0.673784i	$0.235332\pi$
$858$	−22.3535	−0.763135
$859$	−25.8029	−0.880385	−0.440192	−	0.897904i	$-0.645090\pi$
$859$	−25.8029	−0.880385	−0.440192	+	0.897904i	$0.645090\pi$
$860$	2.20116	0.0750588
$861$	3.01985	0.102916
$862$	−41.3648	−1.40889
$863$	−7.49398	−0.255098	−0.127549	−	0.991832i	$-0.540711\pi$
$863$	−7.49398	−0.255098	−0.127549	+	0.991832i	$0.540711\pi$
$864$	−9.43840	−0.321101
$865$	−6.10663	−0.207632
$866$	44.0146	1.49568
$867$	62.0552	2.10750
$868$	0.652976	0.0221635
$869$	12.9342	0.438763
$870$	12.3485	0.418654
$871$	63.5872	2.15457
$872$	11.8774	0.402219
$873$	19.0967	0.646325
$874$	13.4024	0.453344
$875$	2.68632	0.0908141
$876$	−58.7936	−1.98645
$877$	27.2811	0.921216	0.460608	−	0.887604i	$-0.347631\pi$
$877$	27.2811	0.921216	0.460608	+	0.887604i	$0.347631\pi$
$878$	−48.8403	−1.64828
$879$	45.2711	1.52696
$880$	0.909719	0.0306666
$881$	−43.5512	−1.46728	−0.733638	−	0.679541i	$-0.762179\pi$
$881$	−43.5512	−1.46728	−0.733638	+	0.679541i	$0.762179\pi$
$882$	34.3027	1.15503
$883$	3.51915	0.118429	0.0592145	−	0.998245i	$-0.481140\pi$
$883$	3.51915	0.118429	0.0592145	+	0.998245i	$0.481140\pi$
$884$	−96.4827	−3.24507
$885$	−3.33090	−0.111967
$886$	17.9117	0.601756
$887$	−21.8519	−0.733716	−0.366858	−	0.930277i	$-0.619567\pi$
$887$	−21.8519	−0.733716	−0.366858	+	0.930277i	$0.619567\pi$
$888$	18.4959	0.620683
$889$	0.131675	0.00441623
$890$	0.843741	0.0282823
$891$	−7.64214	−0.256021
$892$	38.5429	1.29051
$893$	2.85712	0.0956097
$894$	−32.4445	−1.08511
$895$	−7.22974	−0.241664
$896$	4.49046	0.150016
$897$	−110.236	−3.68067
$898$	−22.6021	−0.754241
$899$	2.49834	0.0833244
$900$	−29.0587	−0.968623
$901$	−8.82074	−0.293861
$902$	−2.96674	−0.0987815
$903$	−3.62851	−0.120749
$904$	4.44980	0.147998
$905$	7.39271	0.245742
$906$	−15.6683	−0.520543
$907$	−28.7325	−0.954045	−0.477023	−	0.878891i	$-0.658284\pi$
$907$	−28.7325	−0.954045	−0.477023	+	0.878891i	$0.658284\pi$
$908$	2.37040	0.0786646
$909$	22.5146	0.746762
$910$	3.49042	0.115706
$911$	21.8988	0.725540	0.362770	−	0.931879i	$-0.381831\pi$
$911$	21.8988	0.725540	0.362770	+	0.931879i	$0.381831\pi$
$912$	−5.89164	−0.195092
$913$	−0.838635	−0.0277548
$914$	65.1938	2.15642
$915$	−0.563857	−0.0186405
$916$	3.94309	0.130284
$917$	11.3183	0.373764
$918$	−16.2108	−0.535037
$919$	−17.1740	−0.566517	−0.283258	−	0.959044i	$-0.591415\pi$
$919$	−17.1740	−0.566517	−0.283258	+	0.959044i	$0.591415\pi$
$920$	−2.61764	−0.0863011
$921$	58.1672	1.91667
$922$	50.1024	1.65004
$923$	77.3410	2.54571
$924$	2.85210	0.0938272
$925$	45.2237	1.48695
$926$	−27.0268	−0.888155
$927$	8.90769	0.292567
$928$	50.2554	1.64972
$929$	−1.95712	−0.0642110	−0.0321055	−	0.999484i	$-0.510221\pi$
$929$	−1.95712	−0.0642110	−0.0321055	+	0.999484i	$0.510221\pi$
$930$	0.792208	0.0259775
$931$	−5.41424	−0.177445
$932$	44.9033	1.47086
$933$	31.6708	1.03686
$934$	−10.4538	−0.342060
$935$	1.97667	0.0646441
$936$	−12.8530	−0.420112
$937$	−32.6584	−1.06690	−0.533451	−	0.845831i	$-0.679105\pi$
$937$	−32.6584	−1.06690	−0.533451	+	0.845831i	$0.679105\pi$
$938$	−14.8681	−0.485460
$939$	−19.1409	−0.624639
$940$	−3.33331	−0.108721
$941$	16.2204	0.528770	0.264385	−	0.964417i	$-0.414831\pi$
$941$	16.2204	0.528770	0.264385	+	0.964417i	$0.414831\pi$
$942$	−51.2057	−1.66837
$943$	−14.6304	−0.476432
$944$	−10.7154	−0.348756
$945$	0.320013	0.0104100
$946$	3.56468	0.115898
$947$	11.5048	0.373857	0.186929	−	0.982374i	$-0.440147\pi$
$947$	11.5048	0.373857	0.186929	+	0.982374i	$0.440147\pi$
$948$	97.7273	3.17404
$949$	−63.5840	−2.06402
$950$	8.40526	0.272703
$951$	−27.9914	−0.907684
$952$	3.77671	0.122404
$953$	39.8366	1.29043	0.645217	−	0.764000i	$-0.276767\pi$
$953$	39.8366	1.29043	0.645217	+	0.764000i	$0.276767\pi$
$954$	−7.01907	−0.227251
$955$	−2.46868	−0.0798845
$956$	−24.5165	−0.792920
$957$	10.9124	0.352747
$958$	62.8128	2.02939
$959$	−9.29604	−0.300185
$960$	10.2128	0.329616
$961$	−30.8397	−0.994830
$962$	119.485	3.85236
$963$	−41.3640	−1.33294
$964$	33.6872	1.08499
$965$	−3.49655	−0.112558
$966$	25.7756	0.829315
$967$	32.4694	1.04414	0.522072	−	0.852901i	$-0.325159\pi$
$967$	32.4694	1.04414	0.522072	+	0.852901i	$0.325159\pi$
$968$	8.81208	0.283231
$969$	−12.8016	−0.411246
$970$	6.44437	0.206916
$971$	−1.97866	−0.0634981	−0.0317490	−	0.999496i	$-0.510108\pi$
$971$	−1.97866	−0.0634981	−0.0317490	+	0.999496i	$0.510108\pi$
$972$	−49.2960	−1.58117
$973$	−1.49123	−0.0478065
$974$	−45.1473	−1.44661
$975$	−69.1339	−2.21406
$976$	−1.81391	−0.0580618
$977$	−51.3643	−1.64329	−0.821645	−	0.569999i	$-0.806944\pi$
$977$	−51.3643	−1.64329	−0.821645	+	0.569999i	$0.806944\pi$
$978$	−52.4110	−1.67592
$979$	0.745613	0.0238299
$980$	6.31663	0.201777
$981$	−35.1964	−1.12373
$982$	−46.7715	−1.49254
$983$	−48.3938	−1.54352	−0.771761	−	0.635913i	$-0.780624\pi$
$983$	−48.3938	−1.54352	−0.771761	+	0.635913i	$0.780624\pi$
$984$	−3.75262	−0.119629
$985$	9.53031	0.303661
$986$	86.3157	2.74885
$987$	5.49481	0.174902
$988$	12.1181	0.385528
$989$	17.5792	0.558986
$990$	1.57293	0.0499910
$991$	−0.116520	−0.00370138	−0.00185069	−	0.999998i	$-0.500589\pi$
$991$	−0.116520	−0.00370138	−0.00185069	+	0.999998i	$0.500589\pi$
$992$	3.22409	0.102365
$993$	51.7960	1.64370
$994$	−18.0840	−0.573590
$995$	1.75682	0.0556949
$996$	−6.33650	−0.200780
$997$	−13.0576	−0.413539	−0.206770	−	0.978390i	$-0.566295\pi$
$997$	−13.0576	−0.413539	−0.206770	+	0.978390i	$0.566295\pi$
$998$	−75.4882	−2.38954
$999$	10.9548	0.346594

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	89.2.a.c.1.2	✓	5
3.2	odd	2		801.2.a.i.1.4		5
4.3	odd	2		1424.2.a.k.1.1		5
5.4	even	2		2225.2.a.j.1.4		5
7.6	odd	2		4361.2.a.i.1.2		5
8.3	odd	2		5696.2.a.bc.1.5		5
8.5	even	2		5696.2.a.be.1.1		5
89.88	even	2		7921.2.a.g.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
89.2.a.c.1.2	✓	5	1.1	even	1	trivial
801.2.a.i.1.4		5	3.2	odd	2
1424.2.a.k.1.1		5	4.3	odd	2
2225.2.a.j.1.4		5	5.4	even	2
4361.2.a.i.1.2		5	7.6	odd	2
5696.2.a.bc.1.5		5	8.3	odd	2
5696.2.a.be.1.1		5	8.5	even	2
7921.2.a.g.1.2		5	89.88	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	89.a (trivial)

\(f(q)\)	\(=\)	\(q-2.09813 q^{2} +2.34526 q^{3} +2.40214 q^{4} -0.402140 q^{5} -4.92067 q^{6} +0.678986 q^{7} -0.843741 q^{8} +2.50027 q^{9} +O(q^{10})\)	\(q-2.09813 q^{2} +2.34526 q^{3} +2.40214 q^{4} -0.402140 q^{5} -4.92067 q^{6} +0.678986 q^{7} -0.843741 q^{8} +2.50027 q^{9} +0.843741 q^{10} +0.745613 q^{11} +5.63365 q^{12} +6.09267 q^{13} -1.42460 q^{14} -0.943125 q^{15} -3.03400 q^{16} -6.59240 q^{17} -5.24588 q^{18} +0.827996 q^{19} -0.965997 q^{20} +1.59240 q^{21} -1.56439 q^{22} -7.71478 q^{23} -1.97880 q^{24} -4.83828 q^{25} -12.7832 q^{26} -1.17200 q^{27} +1.63102 q^{28} +6.24042 q^{29} +1.97880 q^{30} +0.400348 q^{31} +8.05321 q^{32} +1.74866 q^{33} +13.8317 q^{34} -0.273047 q^{35} +6.00599 q^{36} -9.34706 q^{37} -1.73724 q^{38} +14.2889 q^{39} +0.339302 q^{40} +1.89641 q^{41} -3.34106 q^{42} -2.27864 q^{43} +1.79107 q^{44} -1.00546 q^{45} +16.1866 q^{46} +3.45064 q^{47} -7.11554 q^{48} -6.53898 q^{49} +10.1513 q^{50} -15.4609 q^{51} +14.6354 q^{52} +1.33802 q^{53} +2.45902 q^{54} -0.299841 q^{55} -0.572888 q^{56} +1.94187 q^{57} -13.0932 q^{58} +3.53177 q^{59} -2.26552 q^{60} +0.597860 q^{61} -0.839982 q^{62} +1.69765 q^{63} -10.8287 q^{64} -2.45011 q^{65} -3.66891 q^{66} +10.4367 q^{67} -15.8359 q^{68} -18.0932 q^{69} +0.572888 q^{70} +12.6941 q^{71} -2.10958 q^{72} -10.4361 q^{73} +19.6113 q^{74} -11.3471 q^{75} +1.98896 q^{76} +0.506261 q^{77} -29.9800 q^{78} +17.3471 q^{79} +1.22009 q^{80} -10.2495 q^{81} -3.97892 q^{82} -1.12476 q^{83} +3.82517 q^{84} +2.65107 q^{85} +4.78087 q^{86} +14.6354 q^{87} -0.629105 q^{88} +1.00000 q^{89} +2.10958 q^{90} +4.13684 q^{91} -18.5320 q^{92} +0.938923 q^{93} -7.23989 q^{94} -0.332970 q^{95} +18.8869 q^{96} +7.63786 q^{97} +13.7196 q^{98} +1.86423 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} - 3 q^{3} + 11 q^{4} - q^{5} - 3 q^{6} + 8 q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10}) \) 5 * q - q^2 - 3 * q^3 + 11 * q^4 - q^5 - 3 * q^6 + 8 * q^7 + 3 * q^8 + 2 * q^9	\( 5 q - q^{2} - 3 q^{3} + 11 q^{4} - q^{5} - 3 q^{6} + 8 q^{7} + 3 q^{8} + 2 q^{9} - 3 q^{10} + 6 q^{11} - 15 q^{12} - 14 q^{14} + 9 q^{15} + 11 q^{16} - 13 q^{17} - 18 q^{18} + 13 q^{19} - 31 q^{20} - 12 q^{21} - 14 q^{22} + q^{23} - 5 q^{24} + 4 q^{25} - 4 q^{26} + 3 q^{27} + 12 q^{28} + 2 q^{29} + 5 q^{30} + 19 q^{31} + 15 q^{32} - 4 q^{33} - 11 q^{34} + 4 q^{35} + 30 q^{36} - 14 q^{37} + 23 q^{38} + 22 q^{39} - 11 q^{40} - 2 q^{41} + 16 q^{42} + q^{43} + 6 q^{44} - 26 q^{45} + 23 q^{46} - 4 q^{47} - 45 q^{48} + 9 q^{49} + 16 q^{50} - 19 q^{51} + 12 q^{52} - 11 q^{53} + 25 q^{54} + 6 q^{55} + 10 q^{56} - 7 q^{57} - 14 q^{58} + 57 q^{60} + 4 q^{61} - 13 q^{62} + 10 q^{63} + 11 q^{64} - 12 q^{65} + 48 q^{66} + 4 q^{67} - 15 q^{68} - 39 q^{69} - 10 q^{70} - 2 q^{71} - 22 q^{72} - 25 q^{73} - 6 q^{74} - 24 q^{75} + 21 q^{76} - 8 q^{77} - 50 q^{78} + 54 q^{79} - 55 q^{80} - 7 q^{81} + 38 q^{82} - 20 q^{83} - 40 q^{84} - 11 q^{85} + 3 q^{86} + 12 q^{87} - 78 q^{88} + 5 q^{89} + 22 q^{90} - 20 q^{91} + 33 q^{92} - 27 q^{93} - 28 q^{94} + 5 q^{95} - 7 q^{96} + 13 q^{97} - 29 q^{98} + 8 q^{99}+O(q^{100}) \) 5 * q - q^2 - 3 * q^3 + 11 * q^4 - q^5 - 3 * q^6 + 8 * q^7 + 3 * q^8 + 2 * q^9 - 3 * q^10 + 6 * q^11 - 15 * q^12 - 14 * q^14 + 9 * q^15 + 11 * q^16 - 13 * q^17 - 18 * q^18 + 13 * q^19 - 31 * q^20 - 12 * q^21 - 14 * q^22 + q^23 - 5 * q^24 + 4 * q^25 - 4 * q^26 + 3 * q^27 + 12 * q^28 + 2 * q^29 + 5 * q^30 + 19 * q^31 + 15 * q^32 - 4 * q^33 - 11 * q^34 + 4 * q^35 + 30 * q^36 - 14 * q^37 + 23 * q^38 + 22 * q^39 - 11 * q^40 - 2 * q^41 + 16 * q^42 + q^43 + 6 * q^44 - 26 * q^45 + 23 * q^46 - 4 * q^47 - 45 * q^48 + 9 * q^49 + 16 * q^50 - 19 * q^51 + 12 * q^52 - 11 * q^53 + 25 * q^54 + 6 * q^55 + 10 * q^56 - 7 * q^57 - 14 * q^58 + 57 * q^60 + 4 * q^61 - 13 * q^62 + 10 * q^63 + 11 * q^64 - 12 * q^65 + 48 * q^66 + 4 * q^67 - 15 * q^68 - 39 * q^69 - 10 * q^70 - 2 * q^71 - 22 * q^72 - 25 * q^73 - 6 * q^74 - 24 * q^75 + 21 * q^76 - 8 * q^77 - 50 * q^78 + 54 * q^79 - 55 * q^80 - 7 * q^81 + 38 * q^82 - 20 * q^83 - 40 * q^84 - 11 * q^85 + 3 * q^86 + 12 * q^87 - 78 * q^88 + 5 * q^89 + 22 * q^90 - 20 * q^91 + 33 * q^92 - 27 * q^93 - 28 * q^94 + 5 * q^95 - 7 * q^96 + 13 * q^97 - 29 * q^98 + 8 * q^99

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