LMFDB - Embedded newform 4031.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$1$
Dimension:	$61$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62306 q^{2} +1.48834 q^{3} +4.88043 q^{4} +0.915377 q^{5} -3.90400 q^{6} -2.34390 q^{7} -7.55555 q^{8} -0.784847 q^{9} +O(q^{10})$	\(q-2.62306 q^{2} +1.48834 q^{3} +4.88043 q^{4} +0.915377 q^{5} -3.90400 q^{6} -2.34390 q^{7} -7.55555 q^{8} -0.784847 q^{9} -2.40109 q^{10} -2.86549 q^{11} +7.26374 q^{12} +4.85001 q^{13} +6.14819 q^{14} +1.36239 q^{15} +10.0578 q^{16} +2.11615 q^{17} +2.05870 q^{18} -1.70430 q^{19} +4.46744 q^{20} -3.48852 q^{21} +7.51634 q^{22} +2.87531 q^{23} -11.2452 q^{24} -4.16208 q^{25} -12.7219 q^{26} -5.63314 q^{27} -11.4393 q^{28} -1.00000 q^{29} -3.57363 q^{30} +4.12414 q^{31} -11.2710 q^{32} -4.26482 q^{33} -5.55078 q^{34} -2.14555 q^{35} -3.83039 q^{36} -9.79358 q^{37} +4.47047 q^{38} +7.21846 q^{39} -6.91618 q^{40} +10.6933 q^{41} +9.15059 q^{42} -4.66701 q^{43} -13.9848 q^{44} -0.718431 q^{45} -7.54210 q^{46} +10.4466 q^{47} +14.9694 q^{48} -1.50613 q^{49} +10.9174 q^{50} +3.14955 q^{51} +23.6702 q^{52} -4.88080 q^{53} +14.7760 q^{54} -2.62300 q^{55} +17.7094 q^{56} -2.53657 q^{57} +2.62306 q^{58} -2.10982 q^{59} +6.64906 q^{60} +1.94878 q^{61} -10.8178 q^{62} +1.83960 q^{63} +9.44900 q^{64} +4.43959 q^{65} +11.1869 q^{66} +2.44695 q^{67} +10.3277 q^{68} +4.27943 q^{69} +5.62791 q^{70} +6.20279 q^{71} +5.92995 q^{72} -6.92011 q^{73} +25.6891 q^{74} -6.19459 q^{75} -8.31771 q^{76} +6.71642 q^{77} -18.9344 q^{78} -3.33388 q^{79} +9.20665 q^{80} -6.02947 q^{81} -28.0490 q^{82} -4.26583 q^{83} -17.0255 q^{84} +1.93708 q^{85} +12.2418 q^{86} -1.48834 q^{87} +21.6503 q^{88} +5.49919 q^{89} +1.88449 q^{90} -11.3679 q^{91} +14.0327 q^{92} +6.13811 q^{93} -27.4021 q^{94} -1.56007 q^{95} -16.7751 q^{96} +7.50513 q^{97} +3.95067 q^{98} +2.24897 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) $ 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * 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 $

$2$	−2.62306	−1.85478	−0.927391	−	0.374093i	$-0.877954\pi$
$2$	−2.62306	−1.85478	−0.927391	+	0.374093i	$0.877954\pi$
$3$	1.48834	0.859293	0.429646	−	0.902997i	$-0.358638\pi$
$3$	1.48834	0.859293	0.429646	+	0.902997i	$0.358638\pi$
$4$	4.88043	2.44022
$5$	0.915377	0.409369	0.204685	−	0.978828i	$-0.434383\pi$
$5$	0.915377	0.409369	0.204685	+	0.978828i	$0.434383\pi$
$6$	−3.90400	−1.59380
$7$	−2.34390	−0.885911	−0.442956	−	0.896544i	$-0.646070\pi$
$7$	−2.34390	−0.885911	−0.442956	+	0.896544i	$0.646070\pi$
$8$	−7.55555	−2.67129
$9$	−0.784847	−0.261616
$10$	−2.40109	−0.759291
$11$	−2.86549	−0.863977	−0.431988	−	0.901879i	$-0.642188\pi$
$11$	−2.86549	−0.863977	−0.431988	+	0.901879i	$0.642188\pi$
$12$	7.26374	2.09686
$13$	4.85001	1.34515	0.672575	−	0.740029i	$-0.265188\pi$
$13$	4.85001	1.34515	0.672575	+	0.740029i	$0.265188\pi$
$14$	6.14819	1.64317
$15$	1.36239	0.351768
$16$	10.0578	2.51444
$17$	2.11615	0.513242	0.256621	−	0.966512i	$-0.417391\pi$
$17$	2.11615	0.513242	0.256621	+	0.966512i	$0.417391\pi$
$18$	2.05870	0.485240
$19$	−1.70430	−0.390992	−0.195496	−	0.980704i	$-0.562632\pi$
$19$	−1.70430	−0.390992	−0.195496	+	0.980704i	$0.562632\pi$
$20$	4.46744	0.998950
$21$	−3.48852	−0.761257
$22$	7.51634	1.60249
$23$	2.87531	0.599543	0.299772	−	0.954011i	$-0.403090\pi$
$23$	2.87531	0.599543	0.299772	+	0.954011i	$0.403090\pi$
$24$	−11.2452	−2.29542
$25$	−4.16208	−0.832417
$26$	−12.7219	−2.49496
$27$	−5.63314	−1.08410
$28$	−11.4393	−2.16182
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−3.57363	−0.652453
$31$	4.12414	0.740717	0.370358	−	0.928889i	$-0.379235\pi$
$31$	4.12414	0.740717	0.370358	+	0.928889i	$0.379235\pi$
$32$	−11.2710	−1.99245
$33$	−4.26482	−0.742409
$34$	−5.55078	−0.951951
$35$	−2.14555	−0.362665
$36$	−3.83039	−0.638399
$37$	−9.79358	−1.61005	−0.805027	−	0.593238i	$-0.797849\pi$
$37$	−9.79358	−1.61005	−0.805027	+	0.593238i	$0.797849\pi$
$38$	4.47047	0.725206
$39$	7.21846	1.15588
$40$	−6.91618	−1.09354
$41$	10.6933	1.67001	0.835003	−	0.550245i	$-0.185466\pi$
$41$	10.6933	1.67001	0.835003	+	0.550245i	$0.185466\pi$
$42$	9.15059	1.41197
$43$	−4.66701	−0.711712	−0.355856	−	0.934541i	$-0.615811\pi$
$43$	−4.66701	−0.711712	−0.355856	+	0.934541i	$0.615811\pi$
$44$	−13.9848	−2.10829
$45$	−0.718431	−0.107097
$46$	−7.54210	−1.11202
$47$	10.4466	1.52380	0.761899	−	0.647695i	$-0.224267\pi$
$47$	10.4466	1.52380	0.761899	+	0.647695i	$0.224267\pi$
$48$	14.9694	2.16064
$49$	−1.50613	−0.215162
$50$	10.9174	1.54395
$51$	3.14955	0.441025
$52$	23.6702	3.28246
$53$	−4.88080	−0.670429	−0.335215	−	0.942142i	$-0.608809\pi$
$53$	−4.88080	−0.670429	−0.335215	+	0.942142i	$0.608809\pi$
$54$	14.7760	2.01076
$55$	−2.62300	−0.353686
$56$	17.7094	2.36652
$57$	−2.53657	−0.335977
$58$	2.62306	0.344424
$59$	−2.10982	−0.274675	−0.137338	−	0.990524i	$-0.543855\pi$
$59$	−2.10982	−0.274675	−0.137338	+	0.990524i	$0.543855\pi$
$60$	6.64906	0.858390
$61$	1.94878	0.249516	0.124758	−	0.992187i	$-0.460185\pi$
$61$	1.94878	0.249516	0.124758	+	0.992187i	$0.460185\pi$
$62$	−10.8178	−1.37387
$63$	1.83960	0.231768
$64$	9.44900	1.18112
$65$	4.43959	0.550663
$66$	11.1869	1.37701
$67$	2.44695	0.298943	0.149471	−	0.988766i	$-0.452243\pi$
$67$	2.44695	0.298943	0.149471	+	0.988766i	$0.452243\pi$
$68$	10.3277	1.25242
$69$	4.27943	0.515183
$70$	5.62791	0.672664
$71$	6.20279	0.736135	0.368068	−	0.929799i	$-0.380019\pi$
$71$	6.20279	0.736135	0.368068	+	0.929799i	$0.380019\pi$
$72$	5.92995	0.698851
$73$	−6.92011	−0.809938	−0.404969	−	0.914330i	$-0.632718\pi$
$73$	−6.92011	−0.809938	−0.404969	+	0.914330i	$0.632718\pi$
$74$	25.6891	2.98630
$75$	−6.19459	−0.715290
$76$	−8.31771	−0.954106
$77$	6.71642	0.765407
$78$	−18.9344	−2.14390
$79$	−3.33388	−0.375091	−0.187545	−	0.982256i	$-0.560053\pi$
$79$	−3.33388	−0.375091	−0.187545	+	0.982256i	$0.560053\pi$
$80$	9.20665	1.02934
$81$	−6.02947	−0.669941
$82$	−28.0490	−3.09750
$83$	−4.26583	−0.468236	−0.234118	−	0.972208i	$-0.575220\pi$
$83$	−4.26583	−0.468236	−0.234118	+	0.972208i	$0.575220\pi$
$84$	−17.0255	−1.85763
$85$	1.93708	0.210105
$86$	12.2418	1.32007
$87$	−1.48834	−0.159567
$88$	21.6503	2.30793
$89$	5.49919	0.582913	0.291456	−	0.956584i	$-0.405860\pi$
$89$	5.49919	0.582913	0.291456	+	0.956584i	$0.405860\pi$
$90$	1.88449	0.198642
$91$	−11.3679	−1.19168
$92$	14.0327	1.46302
$93$	6.13811	0.636493
$94$	−27.4021	−2.82631
$95$	−1.56007	−0.160060
$96$	−16.7751	−1.71210
$97$	7.50513	0.762031	0.381015	−	0.924569i	$-0.375575\pi$
$97$	7.50513	0.762031	0.381015	+	0.924569i	$0.375575\pi$
$98$	3.95067	0.399078
$99$	2.24897	0.226030
$100$	−20.3128	−2.03128
$101$	−5.27766	−0.525147	−0.262574	−	0.964912i	$-0.584571\pi$
$101$	−5.27766	−0.525147	−0.262574	+	0.964912i	$0.584571\pi$
$102$	−8.26145	−0.818005
$103$	−10.8229	−1.06641	−0.533207	−	0.845985i	$-0.679013\pi$
$103$	−10.8229	−1.06641	−0.533207	+	0.845985i	$0.679013\pi$
$104$	−36.6445	−3.59329
$105$	−3.19331	−0.311635
$106$	12.8026	1.24350
$107$	−11.6751	−1.12868	−0.564339	−	0.825543i	$-0.690869\pi$
$107$	−11.6751	−1.12868	−0.564339	+	0.825543i	$0.690869\pi$
$108$	−27.4921	−2.64543
$109$	−15.1364	−1.44980	−0.724902	−	0.688852i	$-0.758115\pi$
$109$	−15.1364	−1.44980	−0.724902	+	0.688852i	$0.758115\pi$
$110$	6.88029	0.656010
$111$	−14.5762	−1.38351
$112$	−23.5744	−2.22757
$113$	−2.19835	−0.206804	−0.103402	−	0.994640i	$-0.532973\pi$
$113$	−2.19835	−0.206804	−0.103402	+	0.994640i	$0.532973\pi$
$114$	6.65357	0.623164
$115$	2.63199	0.245434
$116$	−4.88043	−0.453137
$117$	−3.80652	−0.351913
$118$	5.53419	0.509463
$119$	−4.96004	−0.454686
$120$	−10.2936	−0.939674
$121$	−2.78898	−0.253544
$122$	−5.11177	−0.462798
$123$	15.9152	1.43502
$124$	20.1276	1.80751
$125$	−8.38676	−0.750135
$126$	−4.82539	−0.429880
$127$	−4.63282	−0.411096	−0.205548	−	0.978647i	$-0.565898\pi$
$127$	−4.63282	−0.411096	−0.205548	+	0.978647i	$0.565898\pi$
$128$	−2.24324	−0.198276
$129$	−6.94609	−0.611569
$130$	−11.6453	−1.02136
$131$	−0.229703	−0.0200692	−0.0100346	−	0.999950i	$-0.503194\pi$
$131$	−0.229703	−0.0200692	−0.0100346	+	0.999950i	$0.503194\pi$
$132$	−20.8142	−1.81164
$133$	3.99470	0.346385
$134$	−6.41850	−0.554474
$135$	−5.15645	−0.443796
$136$	−15.9887	−1.37102
$137$	−2.52853	−0.216027	−0.108013	−	0.994149i	$-0.534449\pi$
$137$	−2.52853	−0.216027	−0.108013	+	0.994149i	$0.534449\pi$
$138$	−11.2252	−0.955552
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	−10.4712	−0.884981
$141$	15.5481	1.30939
$142$	−16.2703	−1.36537
$143$	−13.8976	−1.16218
$144$	−7.89381	−0.657818
$145$	−0.915377	−0.0760180
$146$	18.1519	1.50226
$147$	−2.24163	−0.184887
$148$	−47.7969	−3.92888
$149$	−18.9421	−1.55180	−0.775898	−	0.630858i	$-0.782703\pi$
$149$	−18.9421	−1.55180	−0.775898	+	0.630858i	$0.782703\pi$
$150$	16.2488	1.32671
$151$	−21.1079	−1.71774	−0.858870	−	0.512194i	$-0.828833\pi$
$151$	−21.1079	−1.71774	−0.858870	+	0.512194i	$0.828833\pi$
$152$	12.8769	1.04445
$153$	−1.66085	−0.134272
$154$	−17.6176	−1.41966
$155$	3.77514	0.303227
$156$	35.2292	2.82059
$157$	2.85108	0.227541	0.113770	−	0.993507i	$-0.463707\pi$
$157$	2.85108	0.227541	0.113770	+	0.993507i	$0.463707\pi$
$158$	8.74496	0.695712
$159$	−7.26428	−0.576095
$160$	−10.3172	−0.815649
$161$	−6.73943	−0.531142
$162$	15.8157	1.24260
$163$	21.6563	1.69625	0.848125	−	0.529796i	$-0.177732\pi$
$163$	21.6563	1.69625	0.848125	+	0.529796i	$0.177732\pi$
$164$	52.1877	4.07518
$165$	−3.90392	−0.303919
$166$	11.1895	0.868475
$167$	4.31008	0.333524	0.166762	−	0.985997i	$-0.446669\pi$
$167$	4.31008	0.333524	0.166762	+	0.985997i	$0.446669\pi$
$168$	26.3577	2.03354
$169$	10.5226	0.809431
$170$	−5.08106	−0.389700
$171$	1.33761	0.102290
$172$	−22.7770	−1.73673
$173$	−4.02159	−0.305756	−0.152878	−	0.988245i	$-0.548854\pi$
$173$	−4.02159	−0.305756	−0.152878	+	0.988245i	$0.548854\pi$
$174$	3.90400	0.295961
$175$	9.75551	0.737447
$176$	−28.8204	−2.17242
$177$	−3.14013	−0.236027
$178$	−14.4247	−1.08118
$179$	11.8116	0.882844	0.441422	−	0.897300i	$-0.354474\pi$
$179$	11.8116	0.882844	0.441422	+	0.897300i	$0.354474\pi$
$180$	−3.50626	−0.261341
$181$	6.72152	0.499606	0.249803	−	0.968297i	$-0.419634\pi$
$181$	6.72152	0.499606	0.249803	+	0.968297i	$0.419634\pi$
$182$	29.8188	2.21031
$183$	2.90045	0.214407
$184$	−21.7245	−1.60155
$185$	−8.96482	−0.659107
$186$	−16.1006	−1.18056
$187$	−6.06380	−0.443429
$188$	50.9841	3.71840
$189$	13.2035	0.960414
$190$	4.09217	0.296877
$191$	−11.3992	−0.824815	−0.412408	−	0.910999i	$-0.635312\pi$
$191$	−11.3992	−0.824815	−0.412408	+	0.910999i	$0.635312\pi$
$192$	14.0633	1.01493
$193$	−20.5310	−1.47786	−0.738928	−	0.673784i	$-0.764668\pi$
$193$	−20.5310	−1.47786	−0.738928	+	0.673784i	$0.764668\pi$
$194$	−19.6864	−1.41340
$195$	6.60761	0.473181
$196$	−7.35057	−0.525041
$197$	−7.02343	−0.500399	−0.250200	−	0.968194i	$-0.580496\pi$
$197$	−7.02343	−0.500399	−0.250200	+	0.968194i	$0.580496\pi$
$198$	−5.89918	−0.419236
$199$	−2.17144	−0.153929	−0.0769647	−	0.997034i	$-0.524523\pi$
$199$	−2.17144	−0.153929	−0.0769647	+	0.997034i	$0.524523\pi$
$200$	31.4468	2.22363
$201$	3.64189	0.256879
$202$	13.8436	0.974034
$203$	2.34390	0.164510
$204$	15.3712	1.07620
$205$	9.78837	0.683649
$206$	28.3892	1.97797
$207$	−2.25668	−0.156850
$208$	48.7803	3.38230
$209$	4.88364	0.337808
$210$	8.37624	0.578015
$211$	−0.727573	−0.0500882	−0.0250441	−	0.999686i	$-0.507973\pi$
$211$	−0.727573	−0.0500882	−0.0250441	+	0.999686i	$0.507973\pi$
$212$	−23.8204	−1.63599
$213$	9.23185	0.632556
$214$	30.6246	2.09345
$215$	−4.27208	−0.291353
$216$	42.5614	2.89594
$217$	−9.66656	−0.656209
$218$	39.7036	2.68907
$219$	−10.2995	−0.695974
$220$	−12.8014	−0.863070
$221$	10.2633	0.690387
$222$	38.2341	2.56611
$223$	−14.7702	−0.989083	−0.494542	−	0.869154i	$-0.664664\pi$
$223$	−14.7702	−0.989083	−0.494542	+	0.869154i	$0.664664\pi$
$224$	26.4181	1.76514
$225$	3.26660	0.217773
$226$	5.76641	0.383576
$227$	24.4236	1.62105	0.810525	−	0.585705i	$-0.199182\pi$
$227$	24.4236	1.62105	0.810525	+	0.585705i	$0.199182\pi$
$228$	−12.3796	−0.819857
$229$	−22.7679	−1.50455	−0.752273	−	0.658852i	$-0.771043\pi$
$229$	−22.7679	−1.50455	−0.752273	+	0.658852i	$0.771043\pi$
$230$	−6.90387	−0.455228
$231$	9.99630	0.657709
$232$	7.55555	0.496046
$233$	6.41975	0.420572	0.210286	−	0.977640i	$-0.432560\pi$
$233$	6.41975	0.420572	0.210286	+	0.977640i	$0.432560\pi$
$234$	9.98471	0.652721
$235$	9.56262	0.623796
$236$	−10.2968	−0.670268
$237$	−4.96195	−0.322313
$238$	13.0105	0.843344
$239$	−16.0960	−1.04116	−0.520582	−	0.853812i	$-0.674285\pi$
$239$	−16.0960	−1.04116	−0.520582	+	0.853812i	$0.674285\pi$
$240$	13.7026	0.884500
$241$	−5.06697	−0.326392	−0.163196	−	0.986594i	$-0.552180\pi$
$241$	−5.06697	−0.326392	−0.163196	+	0.986594i	$0.552180\pi$
$242$	7.31566	0.470269
$243$	7.92551	0.508421
$244$	9.51090	0.608873
$245$	−1.37868	−0.0880805
$246$	−41.7465	−2.66166
$247$	−8.26586	−0.525944
$248$	−31.1601	−1.97867
$249$	−6.34900	−0.402352
$250$	21.9990	1.39134
$251$	14.9398	0.942995	0.471497	−	0.881868i	$-0.343714\pi$
$251$	14.9398	0.942995	0.471497	+	0.881868i	$0.343714\pi$
$252$	8.97806	0.565565
$253$	−8.23916	−0.517991
$254$	12.1522	0.762494
$255$	2.88302	0.180542
$256$	−13.0139	−0.813366
$257$	−15.2828	−0.953318	−0.476659	−	0.879088i	$-0.658152\pi$
$257$	−15.2828	−0.953318	−0.476659	+	0.879088i	$0.658152\pi$
$258$	18.2200	1.13433
$259$	22.9552	1.42637
$260$	21.6671	1.34374
$261$	0.784847	0.0485808
$262$	0.602524	0.0372240
$263$	−6.21287	−0.383102	−0.191551	−	0.981483i	$-0.561352\pi$
$263$	−6.21287	−0.383102	−0.191551	+	0.981483i	$0.561352\pi$
$264$	32.2230	1.98319
$265$	−4.46777	−0.274453
$266$	−10.4783	−0.642468
$267$	8.18465	0.500893
$268$	11.9422	0.729485
$269$	−14.6538	−0.893456	−0.446728	−	0.894670i	$-0.647411\pi$
$269$	−14.6538	−0.893456	−0.446728	+	0.894670i	$0.647411\pi$
$270$	13.5257	0.823145
$271$	1.04771	0.0636438	0.0318219	−	0.999494i	$-0.489869\pi$
$271$	1.04771	0.0636438	0.0318219	+	0.999494i	$0.489869\pi$
$272$	21.2837	1.29052
$273$	−16.9193	−1.02401
$274$	6.63247	0.400682
$275$	11.9264	0.719189
$276$	20.8855	1.25716
$277$	−7.66740	−0.460690	−0.230345	−	0.973109i	$-0.573985\pi$
$277$	−7.66740	−0.460690	−0.230345	+	0.973109i	$0.573985\pi$
$278$	2.62306	0.157321
$279$	−3.23682	−0.193783
$280$	16.2108	0.968782
$281$	−23.1368	−1.38022	−0.690112	−	0.723703i	$-0.742439\pi$
$281$	−23.1368	−1.38022	−0.690112	+	0.723703i	$0.742439\pi$
$282$	−40.7837	−2.42863
$283$	6.53094	0.388224	0.194112	−	0.980979i	$-0.437817\pi$
$283$	6.53094	0.388224	0.194112	+	0.980979i	$0.437817\pi$
$284$	30.2723	1.79633
$285$	−2.32192	−0.137539
$286$	36.4543	2.15559
$287$	−25.0639	−1.47948
$288$	8.84603	0.521257
$289$	−12.5219	−0.736583
$290$	2.40109	0.140997
$291$	11.1702	0.654808
$292$	−33.7731	−1.97642
$293$	−7.03906	−0.411226	−0.205613	−	0.978633i	$-0.565919\pi$
$293$	−7.03906	−0.411226	−0.205613	+	0.978633i	$0.565919\pi$
$294$	5.87994	0.342925
$295$	−1.93128	−0.112444
$296$	73.9958	4.30092
$297$	16.1417	0.936635
$298$	49.6862	2.87824
$299$	13.9453	0.806476
$300$	−30.2323	−1.74546
$301$	10.9390	0.630514
$302$	55.3673	3.18603
$303$	−7.85495	−0.451255
$304$	−17.1414	−0.983128
$305$	1.78387	0.102144
$306$	4.35652	0.249045
$307$	8.21476	0.468841	0.234421	−	0.972135i	$-0.424681\pi$
$307$	8.21476	0.468841	0.234421	+	0.972135i	$0.424681\pi$
$308$	32.7790	1.86776
$309$	−16.1082	−0.916363
$310$	−9.90241	−0.562419
$311$	−13.9130	−0.788935	−0.394468	−	0.918910i	$-0.629071\pi$
$311$	−13.9130	−0.788935	−0.394468	+	0.918910i	$0.629071\pi$
$312$	−54.5394	−3.08769
$313$	28.4028	1.60542	0.802712	−	0.596367i	$-0.203390\pi$
$313$	28.4028	1.60542	0.802712	+	0.596367i	$0.203390\pi$
$314$	−7.47854	−0.422039
$315$	1.68393	0.0948788
$316$	−16.2708	−0.915303
$317$	4.92573	0.276656	0.138328	−	0.990386i	$-0.455827\pi$
$317$	4.92573	0.276656	0.138328	+	0.990386i	$0.455827\pi$
$318$	19.0546	1.06853
$319$	2.86549	0.160436
$320$	8.64940	0.483516
$321$	−17.3766	−0.969865
$322$	17.6779	0.985152
$323$	−3.60655	−0.200674
$324$	−29.4264	−1.63480
$325$	−20.1862	−1.11973
$326$	−56.8056	−3.14617
$327$	−22.5281	−1.24581
$328$	−80.7934	−4.46107
$329$	−24.4859	−1.34995
$330$	10.2402	0.563704
$331$	29.9712	1.64737	0.823683	−	0.567051i	$-0.191916\pi$
$331$	29.9712	1.64737	0.823683	+	0.567051i	$0.191916\pi$
$332$	−20.8191	−1.14260
$333$	7.68646	0.421216
$334$	−11.3056	−0.618614
$335$	2.23988	0.122378
$336$	−35.0867	−1.91414
$337$	−5.77862	−0.314781	−0.157391	−	0.987536i	$-0.550308\pi$
$337$	−5.77862	−0.314781	−0.157391	+	0.987536i	$0.550308\pi$
$338$	−27.6014	−1.50132
$339$	−3.27190	−0.177705
$340$	9.45377	0.512703
$341$	−11.8177	−0.639962
$342$	−3.50864	−0.189725
$343$	19.9375	1.07653
$344$	35.2618	1.90119
$345$	3.91730	0.210900
$346$	10.5489	0.567110
$347$	−34.2569	−1.83900	−0.919502	−	0.393085i	$-0.871408\pi$
$347$	−34.2569	−1.83900	−0.919502	+	0.393085i	$0.871408\pi$
$348$	−7.26374	−0.389377
$349$	−3.58106	−0.191690	−0.0958450	−	0.995396i	$-0.530555\pi$
$349$	−3.58106	−0.191690	−0.0958450	+	0.995396i	$0.530555\pi$
$350$	−25.5893	−1.36780
$351$	−27.3208	−1.45827
$352$	32.2970	1.72143
$353$	−0.368654	−0.0196215	−0.00981074	−	0.999952i	$-0.503123\pi$
$353$	−0.368654	−0.0196215	−0.00981074	+	0.999952i	$0.503123\pi$
$354$	8.23674	0.437778
$355$	5.67789	0.301351
$356$	26.8384	1.42243
$357$	−7.38223	−0.390709
$358$	−30.9826	−1.63748
$359$	13.8984	0.733531	0.366766	−	0.930313i	$-0.380465\pi$
$359$	13.8984	0.733531	0.366766	+	0.930313i	$0.380465\pi$
$360$	5.42814	0.286088
$361$	−16.0954	−0.847125
$362$	−17.6309	−0.926661
$363$	−4.15095	−0.217868
$364$	−55.4805	−2.90797
$365$	−6.33451	−0.331564
$366$	−7.60804	−0.397679
$367$	−20.3413	−1.06181	−0.530904	−	0.847432i	$-0.678148\pi$
$367$	−20.3413	−1.06181	−0.530904	+	0.847432i	$0.678148\pi$
$368$	28.9192	1.50752
$369$	−8.39257	−0.436900
$370$	23.5152	1.22250
$371$	11.4401	0.593941
$372$	29.9567	1.55318
$373$	−14.8871	−0.770824	−0.385412	−	0.922745i	$-0.625941\pi$
$373$	−14.8871	−0.770824	−0.385412	+	0.922745i	$0.625941\pi$
$374$	15.9057	0.822464
$375$	−12.4823	−0.644586
$376$	−78.9300	−4.07051
$377$	−4.85001	−0.249788
$378$	−34.6336	−1.78136
$379$	−5.94730	−0.305493	−0.152746	−	0.988265i	$-0.548812\pi$
$379$	−5.94730	−0.305493	−0.152746	+	0.988265i	$0.548812\pi$
$380$	−7.61384	−0.390582
$381$	−6.89521	−0.353252
$382$	29.9007	1.52985
$383$	25.1310	1.28414	0.642068	−	0.766647i	$-0.278077\pi$
$383$	25.1310	1.28414	0.642068	+	0.766647i	$0.278077\pi$
$384$	−3.33870	−0.170377
$385$	6.14806	0.313334
$386$	53.8541	2.74110
$387$	3.66289	0.186195
$388$	36.6283	1.85952
$389$	29.3804	1.48965	0.744824	−	0.667261i	$-0.232534\pi$
$389$	29.3804	1.48965	0.744824	+	0.667261i	$0.232534\pi$
$390$	−17.3322	−0.877648
$391$	6.08458	0.307710
$392$	11.3796	0.574759
$393$	−0.341876	−0.0172453
$394$	18.4229	0.928131
$395$	−3.05176	−0.153551
$396$	10.9759	0.551562
$397$	5.34593	0.268304	0.134152	−	0.990961i	$-0.457169\pi$
$397$	5.34593	0.268304	0.134152	+	0.990961i	$0.457169\pi$
$398$	5.69582	0.285506
$399$	5.94547	0.297646
$400$	−41.8613	−2.09306
$401$	31.7714	1.58659	0.793294	−	0.608838i	$-0.208364\pi$
$401$	31.7714	1.58659	0.793294	+	0.608838i	$0.208364\pi$
$402$	−9.55290	−0.476455
$403$	20.0021	0.996376
$404$	−25.7573	−1.28147
$405$	−5.51924	−0.274253
$406$	−6.14819	−0.305129
$407$	28.0634	1.39105
$408$	−23.7966	−1.17810
$409$	−14.3410	−0.709117	−0.354558	−	0.935034i	$-0.615369\pi$
$409$	−14.3410	−0.709117	−0.354558	+	0.935034i	$0.615369\pi$
$410$	−25.6755	−1.26802
$411$	−3.76330	−0.185630
$412$	−52.8206	−2.60228
$413$	4.94521	0.243338
$414$	5.91940	0.290922
$415$	−3.90485	−0.191681
$416$	−54.6645	−2.68015
$417$	−1.48834	−0.0728843
$418$	−12.8101	−0.626561
$419$	0.753347	0.0368034	0.0184017	−	0.999831i	$-0.494142\pi$
$419$	0.753347	0.0368034	0.0184017	+	0.999831i	$0.494142\pi$
$420$	−15.5847	−0.760458
$421$	7.09072	0.345580	0.172790	−	0.984959i	$-0.444722\pi$
$421$	7.09072	0.345580	0.172790	+	0.984959i	$0.444722\pi$
$422$	1.90847	0.0929027
$423$	−8.19901	−0.398650
$424$	36.8771	1.79091
$425$	−8.80759	−0.427231
$426$	−24.2157	−1.17325
$427$	−4.56775	−0.221049
$428$	−56.9797	−2.75422
$429$	−20.6844	−0.998652
$430$	11.2059	0.540397
$431$	−13.9586	−0.672361	−0.336180	−	0.941798i	$-0.609135\pi$
$431$	−13.9586	−0.672361	−0.336180	+	0.941798i	$0.609135\pi$
$432$	−56.6568	−2.72590
$433$	−32.0601	−1.54071	−0.770356	−	0.637615i	$-0.779921\pi$
$433$	−32.0601	−1.54071	−0.770356	+	0.637615i	$0.779921\pi$
$434$	25.3560	1.21712
$435$	−1.36239	−0.0653217
$436$	−73.8722	−3.53783
$437$	−4.90038	−0.234417
$438$	27.0161	1.29088
$439$	22.2163	1.06033	0.530164	−	0.847895i	$-0.322131\pi$
$439$	22.2163	1.06033	0.530164	+	0.847895i	$0.322131\pi$
$440$	19.8182	0.944796
$441$	1.18208	0.0562897
$442$	−26.9214	−1.28052
$443$	39.4190	1.87285	0.936427	−	0.350863i	$-0.114112\pi$
$443$	39.4190	1.87285	0.936427	+	0.350863i	$0.114112\pi$
$444$	−71.1380	−3.37606
$445$	5.03383	0.238627
$446$	38.7430	1.83453
$447$	−28.1922	−1.33345
$448$	−22.1475	−1.04637
$449$	−25.6917	−1.21247	−0.606234	−	0.795286i	$-0.707321\pi$
$449$	−25.6917	−1.21247	−0.606234	+	0.795286i	$0.707321\pi$
$450$	−8.56848	−0.403922
$451$	−30.6414	−1.44285
$452$	−10.7289	−0.504646
$453$	−31.4158	−1.47604
$454$	−64.0644	−3.00669
$455$	−10.4060	−0.487839
$456$	19.1652	0.897492
$457$	16.1178	0.753959	0.376979	−	0.926222i	$-0.376963\pi$
$457$	16.1178	0.753959	0.376979	+	0.926222i	$0.376963\pi$
$458$	59.7216	2.79060
$459$	−11.9206	−0.556404
$460$	12.8453	0.598913
$461$	−22.4907	−1.04749	−0.523747	−	0.851874i	$-0.675466\pi$
$461$	−22.4907	−1.04749	−0.523747	+	0.851874i	$0.675466\pi$
$462$	−26.2209	−1.21991
$463$	−0.544971	−0.0253270	−0.0126635	−	0.999920i	$-0.504031\pi$
$463$	−0.544971	−0.0253270	−0.0126635	+	0.999920i	$0.504031\pi$
$464$	−10.0578	−0.466920
$465$	5.61869	0.260560
$466$	−16.8394	−0.780069
$467$	17.0702	0.789915	0.394958	−	0.918699i	$-0.370759\pi$
$467$	17.0702	0.789915	0.394958	+	0.918699i	$0.370759\pi$
$468$	−18.5775	−0.858743
$469$	−5.73541	−0.264837
$470$	−25.0833	−1.15701
$471$	4.24337	0.195524
$472$	15.9409	0.733737
$473$	13.3733	0.614903
$474$	13.0155	0.597820
$475$	7.09343	0.325469
$476$	−24.2072	−1.10953
$477$	3.83068	0.175395
$478$	42.2207	1.93113
$479$	1.64739	0.0752710	0.0376355	−	0.999292i	$-0.488017\pi$
$479$	1.64739	0.0752710	0.0376355	+	0.999292i	$0.488017\pi$
$480$	−15.3555	−0.700881
$481$	−47.4990	−2.16577
$482$	13.2909	0.605386
$483$	−10.0306	−0.456406
$484$	−13.6114	−0.618702
$485$	6.87003	0.311952
$486$	−20.7891	−0.943011
$487$	18.3500	0.831518	0.415759	−	0.909475i	$-0.363516\pi$
$487$	18.3500	0.831518	0.415759	+	0.909475i	$0.363516\pi$
$488$	−14.7241	−0.666529
$489$	32.2319	1.45758
$490$	3.61635	0.163370
$491$	−1.79664	−0.0810812	−0.0405406	−	0.999178i	$-0.512908\pi$
$491$	−1.79664	−0.0810812	−0.0405406	+	0.999178i	$0.512908\pi$
$492$	77.6730	3.50177
$493$	−2.11615	−0.0953066
$494$	21.6818	0.975511
$495$	2.05866	0.0925297
$496$	41.4796	1.86249
$497$	−14.5387	−0.652150
$498$	16.6538	0.746275
$499$	−6.50403	−0.291161	−0.145580	−	0.989346i	$-0.546505\pi$
$499$	−6.50403	−0.291161	−0.145580	+	0.989346i	$0.546505\pi$
$500$	−40.9311	−1.83049
$501$	6.41485	0.286595
$502$	−39.1881	−1.74905
$503$	6.20492	0.276664	0.138332	−	0.990386i	$-0.455826\pi$
$503$	6.20492	0.276664	0.138332	+	0.990386i	$0.455826\pi$
$504$	−13.8992	−0.619120
$505$	−4.83105	−0.214979
$506$	21.6118	0.960761
$507$	15.6612	0.695538
$508$	−22.6102	−1.00316
$509$	3.74128	0.165829	0.0829147	−	0.996557i	$-0.473577\pi$
$509$	3.74128	0.165829	0.0829147	+	0.996557i	$0.473577\pi$
$510$	−7.56234	−0.334866
$511$	16.2201	0.717533
$512$	38.6226	1.70689
$513$	9.60053	0.423874
$514$	40.0878	1.76820
$515$	−9.90707	−0.436557
$516$	−33.8999	−1.49236
$517$	−29.9347	−1.31653
$518$	−60.2128	−2.64560
$519$	−5.98548	−0.262734
$520$	−33.5435	−1.47098
$521$	−20.8778	−0.914673	−0.457337	−	0.889294i	$-0.651197\pi$
$521$	−20.8778	−0.914673	−0.457337	+	0.889294i	$0.651197\pi$
$522$	−2.05870	−0.0901068
$523$	1.23932	0.0541916	0.0270958	−	0.999633i	$-0.491374\pi$
$523$	1.23932	0.0541916	0.0270958	+	0.999633i	$0.491374\pi$
$524$	−1.12105	−0.0489733
$525$	14.5195	0.633683
$526$	16.2967	0.710571
$527$	8.72729	0.380167
$528$	−42.8945	−1.86674
$529$	−14.7326	−0.640548
$530$	11.7192	0.509051
$531$	1.65589	0.0718594
$532$	19.4959	0.845253
$533$	51.8624	2.24641
$534$	−21.4688	−0.929047
$535$	−10.6872	−0.462046
$536$	−18.4881	−0.798562
$537$	17.5797	0.758621
$538$	38.4377	1.65717
$539$	4.31580	0.185895
$540$	−25.1657	−1.08296
$541$	−31.7533	−1.36518	−0.682591	−	0.730801i	$-0.739147\pi$
$541$	−31.7533	−1.36518	−0.682591	+	0.730801i	$0.739147\pi$
$542$	−2.74820	−0.118045
$543$	10.0039	0.429308
$544$	−23.8512	−1.02261
$545$	−13.8555	−0.593505
$546$	44.3804	1.89931
$547$	−33.9854	−1.45311	−0.726556	−	0.687107i	$-0.758880\pi$
$547$	−33.9854	−1.45311	−0.726556	+	0.687107i	$0.758880\pi$
$548$	−12.3403	−0.527152
$549$	−1.52950	−0.0652773
$550$	−31.2836	−1.33394
$551$	1.70430	0.0726055
$552$	−32.3334	−1.37620
$553$	7.81429	0.332297
$554$	20.1120	0.854479
$555$	−13.3427	−0.566366
$556$	−4.88043	−0.206977
$557$	−11.5342	−0.488719	−0.244360	−	0.969685i	$-0.578578\pi$
$557$	−11.5342	−0.488719	−0.244360	+	0.969685i	$0.578578\pi$
$558$	8.49036	0.359426
$559$	−22.6350	−0.957360
$560$	−21.5795	−0.911899
$561$	−9.02499	−0.381035
$562$	60.6891	2.56001
$563$	2.14007	0.0901931	0.0450966	−	0.998983i	$-0.485640\pi$
$563$	2.14007	0.0901931	0.0450966	+	0.998983i	$0.485640\pi$
$564$	75.8817	3.19519
$565$	−2.01232	−0.0846591
$566$	−17.1310	−0.720071
$567$	14.1325	0.593509
$568$	−46.8654	−1.96643
$569$	−8.51745	−0.357070	−0.178535	−	0.983934i	$-0.557136\pi$
$569$	−8.51745	−0.357070	−0.178535	+	0.983934i	$0.557136\pi$
$570$	6.09053	0.255104
$571$	−32.3728	−1.35476	−0.677381	−	0.735633i	$-0.736885\pi$
$571$	−32.3728	−1.35476	−0.677381	+	0.735633i	$0.736885\pi$
$572$	−67.8265	−2.83597
$573$	−16.9658	−0.708758
$574$	65.7441	2.74411
$575$	−11.9673	−0.499070
$576$	−7.41602	−0.309001
$577$	3.38068	0.140739	0.0703697	−	0.997521i	$-0.477582\pi$
$577$	3.38068	0.140739	0.0703697	+	0.997521i	$0.477582\pi$
$578$	32.8457	1.36620
$579$	−30.5571	−1.26991
$580$	−4.46744	−0.185500
$581$	9.99868	0.414815
$582$	−29.3000	−1.21453
$583$	13.9859	0.579235
$584$	52.2852	2.16358
$585$	−3.48440	−0.144062
$586$	18.4639	0.762735
$587$	0.405404	0.0167328	0.00836642	−	0.999965i	$-0.497337\pi$
$587$	0.405404	0.0167328	0.00836642	+	0.999965i	$0.497337\pi$
$588$	−10.9401	−0.451164
$589$	−7.02875	−0.289615
$590$	5.06587	0.208558
$591$	−10.4533	−0.429989
$592$	−98.5015	−4.04839
$593$	−16.8671	−0.692649	−0.346325	−	0.938115i	$-0.612570\pi$
$593$	−16.8671	−0.692649	−0.346325	+	0.938115i	$0.612570\pi$
$594$	−42.3406	−1.73725
$595$	−4.54031	−0.186135
$596$	−92.4456	−3.78672
$597$	−3.23184	−0.132271
$598$	−36.5793	−1.49584
$599$	−16.9366	−0.692013	−0.346006	−	0.938232i	$-0.612462\pi$
$599$	−16.9366	−0.692013	−0.346006	+	0.938232i	$0.612462\pi$
$600$	46.8035	1.91075
$601$	6.15835	0.251204	0.125602	−	0.992081i	$-0.459914\pi$
$601$	6.15835	0.251204	0.125602	+	0.992081i	$0.459914\pi$
$602$	−28.6937	−1.16947
$603$	−1.92048	−0.0782081
$604$	−103.016	−4.19166
$605$	−2.55297	−0.103793
$606$	20.6040	0.836980
$607$	−12.3246	−0.500242	−0.250121	−	0.968215i	$-0.580470\pi$
$607$	−12.3246	−0.500242	−0.250121	+	0.968215i	$0.580470\pi$
$608$	19.2092	0.779034
$609$	3.48852	0.141362
$610$	−4.67920	−0.189455
$611$	50.6663	2.04974
$612$	−8.10569	−0.327653
$613$	30.0916	1.21539	0.607693	−	0.794172i	$-0.292095\pi$
$613$	30.0916	1.21539	0.607693	+	0.794172i	$0.292095\pi$
$614$	−21.5478	−0.869598
$615$	14.5684	0.587455
$616$	−50.7462	−2.04462
$617$	−12.7070	−0.511563	−0.255781	−	0.966735i	$-0.582333\pi$
$617$	−12.7070	−0.511563	−0.255781	+	0.966735i	$0.582333\pi$
$618$	42.2527	1.69965
$619$	−9.32837	−0.374939	−0.187470	−	0.982270i	$-0.560029\pi$
$619$	−9.32837	−0.374939	−0.187470	+	0.982270i	$0.560029\pi$
$620$	18.4243	0.739939
$621$	−16.1970	−0.649963
$622$	36.4947	1.46330
$623$	−12.8895	−0.516409
$624$	72.6016	2.90639
$625$	13.1334	0.525335
$626$	−74.5023	−2.97771
$627$	7.26851	0.290276
$628$	13.9145	0.555249
$629$	−20.7247	−0.826347
$630$	−4.41705	−0.175979
$631$	10.7719	0.428821	0.214410	−	0.976744i	$-0.431217\pi$
$631$	10.7719	0.428821	0.214410	+	0.976744i	$0.431217\pi$
$632$	25.1893	1.00198
$633$	−1.08288	−0.0430404
$634$	−12.9205	−0.513137
$635$	−4.24078	−0.168290
$636$	−35.4528	−1.40580
$637$	−7.30475	−0.289425
$638$	−7.51634	−0.297575
$639$	−4.86824	−0.192585
$640$	−2.05341	−0.0811681
$641$	−18.1401	−0.716489	−0.358245	−	0.933628i	$-0.616625\pi$
$641$	−18.1401	−0.716489	−0.358245	+	0.933628i	$0.616625\pi$
$642$	45.5797	1.79889
$643$	24.8009	0.978052	0.489026	−	0.872269i	$-0.337352\pi$
$643$	24.8009	0.978052	0.489026	+	0.872269i	$0.337352\pi$
$644$	−32.8914	−1.29610
$645$	−6.35830	−0.250358
$646$	9.46018	0.372206
$647$	−23.4154	−0.920556	−0.460278	−	0.887775i	$-0.652250\pi$
$647$	−23.4154	−0.920556	−0.460278	+	0.887775i	$0.652250\pi$
$648$	45.5560	1.78961
$649$	6.04567	0.237313
$650$	52.9494	2.07685
$651$	−14.3871	−0.563876
$652$	105.692	4.13922
$653$	30.5678	1.19621	0.598105	−	0.801418i	$-0.295921\pi$
$653$	30.5678	1.19621	0.598105	+	0.801418i	$0.295921\pi$
$654$	59.0925	2.31070
$655$	−0.210265	−0.00821572
$656$	107.550	4.19913
$657$	5.43123	0.211892
$658$	64.2279	2.50386
$659$	−0.707816	−0.0275726	−0.0137863	−	0.999905i	$-0.504388\pi$
$659$	−0.707816	−0.0275726	−0.0137863	+	0.999905i	$0.504388\pi$
$660$	−19.0528	−0.741630
$661$	9.90137	0.385119	0.192559	−	0.981285i	$-0.438321\pi$
$661$	9.90137	0.385119	0.192559	+	0.981285i	$0.438321\pi$
$662$	−78.6162	−3.05550
$663$	15.2753	0.593245
$664$	32.2307	1.25079
$665$	3.65666	0.141799
$666$	−20.1620	−0.781263
$667$	−2.87531	−0.111332
$668$	21.0350	0.813870
$669$	−21.9830	−0.849912
$670$	−5.87535	−0.226984
$671$	−5.58421	−0.215576
$672$	39.3191	1.51677
$673$	5.85484	0.225688	0.112844	−	0.993613i	$-0.464004\pi$
$673$	5.85484	0.225688	0.112844	+	0.993613i	$0.464004\pi$
$674$	15.1576	0.583851
$675$	23.4456	0.902421
$676$	51.3548	1.97519
$677$	−19.2989	−0.741718	−0.370859	−	0.928689i	$-0.620937\pi$
$677$	−19.2989	−0.741718	−0.370859	+	0.928689i	$0.620937\pi$
$678$	8.58237	0.329604
$679$	−17.5913	−0.675091
$680$	−14.6357	−0.561252
$681$	36.3505	1.39296
$682$	30.9984	1.18699
$683$	−16.9564	−0.648820	−0.324410	−	0.945917i	$-0.605166\pi$
$683$	−16.9564	−0.648820	−0.324410	+	0.945917i	$0.605166\pi$
$684$	6.52813	0.249609
$685$	−2.31456	−0.0884346
$686$	−52.2973	−1.99672
$687$	−33.8864	−1.29285
$688$	−46.9397	−1.78956
$689$	−23.6719	−0.901828
$690$	−10.2753	−0.391174
$691$	−43.1034	−1.63973	−0.819865	−	0.572558i	$-0.805951\pi$
$691$	−43.1034	−1.63973	−0.819865	+	0.572558i	$0.805951\pi$
$692$	−19.6271	−0.746110
$693$	−5.27136	−0.200242
$694$	89.8577	3.41095
$695$	−0.915377	−0.0347222
$696$	11.2452	0.426249
$697$	22.6285	0.857117
$698$	9.39334	0.355543
$699$	9.55477	0.361394
$700$	47.6111	1.79953
$701$	32.9783	1.24557	0.622786	−	0.782392i	$-0.286001\pi$
$701$	32.9783	1.24557	0.622786	+	0.782392i	$0.286001\pi$
$702$	71.6640	2.70478
$703$	16.6912	0.629519
$704$	−27.0760	−1.02046
$705$	14.2324	0.536024
$706$	0.967001	0.0363936
$707$	12.3703	0.465234
$708$	−15.3252	−0.575956
$709$	−0.0267051	−0.00100293	−0.000501465	−	1.00000i	$-0.500160\pi$
$709$	−0.0267051	−0.00100293	−0.000501465		1.00000i	$0.500160\pi$
$710$	−14.8934	−0.558941
$711$	2.61659	0.0981297
$712$	−41.5494	−1.55713
$713$	11.8582	0.444092
$714$	19.3640	0.724680
$715$	−12.7216	−0.475760
$716$	57.6459	2.15433
$717$	−23.9563	−0.894665
$718$	−36.4564	−1.36054
$719$	25.0500	0.934206	0.467103	−	0.884203i	$-0.345298\pi$
$719$	25.0500	0.934206	0.467103	+	0.884203i	$0.345298\pi$
$720$	−7.22582	−0.269290
$721$	25.3679	0.944749
$722$	42.2191	1.57123
$723$	−7.54136	−0.280466
$724$	32.8039	1.21915
$725$	4.16208	0.154576
$726$	10.8882	0.404098
$727$	−16.3071	−0.604797	−0.302399	−	0.953182i	$-0.597787\pi$
$727$	−16.3071	−0.604797	−0.302399	+	0.953182i	$0.597787\pi$
$728$	85.8910	3.18333
$729$	29.8843	1.10682
$730$	16.6158	0.614978
$731$	−9.87609	−0.365280
$732$	14.1554	0.523200
$733$	43.4785	1.60592	0.802958	−	0.596036i	$-0.203258\pi$
$733$	43.4785	1.60592	0.802958	+	0.596036i	$0.203258\pi$
$734$	53.3565	1.96942
$735$	−2.05194	−0.0756870
$736$	−32.4076	−1.19456
$737$	−7.01171	−0.258280
$738$	22.0142	0.810354
$739$	−7.19993	−0.264854	−0.132427	−	0.991193i	$-0.542277\pi$
$739$	−7.19993	−0.264854	−0.132427	+	0.991193i	$0.542277\pi$
$740$	−43.7522	−1.60836
$741$	−12.3024	−0.451940
$742$	−30.0081	−1.10163
$743$	42.7231	1.56736	0.783679	−	0.621166i	$-0.213341\pi$
$743$	42.7231	1.56736	0.783679	+	0.621166i	$0.213341\pi$
$744$	−46.3768	−1.70026
$745$	−17.3392	−0.635258
$746$	39.0497	1.42971
$747$	3.34803	0.122498
$748$	−29.5940	−1.08206
$749$	27.3654	0.999908
$750$	32.7419	1.19557
$751$	1.07082	0.0390748	0.0195374	−	0.999809i	$-0.493781\pi$
$751$	1.07082	0.0390748	0.0195374	+	0.999809i	$0.493781\pi$
$752$	105.070	3.83150
$753$	22.2356	0.810309
$754$	12.7219	0.463303
$755$	−19.3217	−0.703190
$756$	64.4388	2.34362
$757$	−5.17697	−0.188160	−0.0940800	−	0.995565i	$-0.529991\pi$
$757$	−5.17697	−0.188160	−0.0940800	+	0.995565i	$0.529991\pi$
$758$	15.6001	0.566622
$759$	−12.2627	−0.445106
$760$	11.7872	0.427567
$761$	31.3871	1.13778	0.568890	−	0.822414i	$-0.307373\pi$
$761$	31.3871	1.13778	0.568890	+	0.822414i	$0.307373\pi$
$762$	18.0865	0.655206
$763$	35.4782	1.28440
$764$	−55.6329	−2.01273
$765$	−1.52031	−0.0549669
$766$	−65.9202	−2.38179
$767$	−10.2327	−0.369480
$768$	−19.3690	−0.698919
$769$	29.2998	1.05658	0.528289	−	0.849065i	$-0.322834\pi$
$769$	29.2998	1.05658	0.528289	+	0.849065i	$0.322834\pi$
$770$	−16.1267	−0.581166
$771$	−22.7461	−0.819179
$772$	−100.200	−3.60629
$773$	35.5761	1.27958	0.639792	−	0.768548i	$-0.279020\pi$
$773$	35.5761	1.27958	0.639792	+	0.768548i	$0.279020\pi$
$774$	−9.60797	−0.345351
$775$	−17.1650	−0.616585
$776$	−56.7054	−2.03560
$777$	34.1651	1.22567
$778$	−77.0666	−2.76297
$779$	−18.2245	−0.652960
$780$	32.2480	1.15466
$781$	−17.7740	−0.636004
$782$	−15.9602	−0.570736
$783$	5.63314	0.201312
$784$	−15.1483	−0.541011
$785$	2.60981	0.0931482
$786$	0.896759	0.0319863
$787$	2.22740	0.0793981	0.0396990	−	0.999212i	$-0.487360\pi$
$787$	2.22740	0.0793981	0.0396990	+	0.999212i	$0.487360\pi$
$788$	−34.2774	−1.22108
$789$	−9.24686	−0.329197
$790$	8.00494	0.284803
$791$	5.15272	0.183210
$792$	−16.9922	−0.603791
$793$	9.45161	0.335636
$794$	−14.0227	−0.497646
$795$	−6.64956	−0.235836
$796$	−10.5976	−0.375621
$797$	13.4700	0.477133	0.238567	−	0.971126i	$-0.423322\pi$
$797$	13.4700	0.477133	0.238567	+	0.971126i	$0.423322\pi$
$798$	−15.5953	−0.552068
$799$	22.1066	0.782077
$800$	46.9109	1.65855
$801$	−4.31602	−0.152499
$802$	−83.3383	−2.94278
$803$	19.8295	0.699768
$804$	17.7740	0.626841
$805$	−6.16913	−0.217433
$806$	−52.4667	−1.84806
$807$	−21.8098	−0.767740
$808$	39.8756	1.40282
$809$	−41.5677	−1.46144	−0.730722	−	0.682676i	$-0.760816\pi$
$809$	−41.5677	−1.46144	−0.730722	+	0.682676i	$0.760816\pi$
$810$	14.4773	0.508680
$811$	−32.2720	−1.13322	−0.566612	−	0.823984i	$-0.691746\pi$
$811$	−32.2720	−1.13322	−0.566612	+	0.823984i	$0.691746\pi$
$812$	11.4393	0.401439
$813$	1.55935	0.0546886
$814$	−73.6119	−2.58009
$815$	19.8237	0.694392
$816$	31.6774	1.10893
$817$	7.95397	0.278274
$818$	37.6173	1.31526
$819$	8.92210	0.311763
$820$	47.7715	1.66825
$821$	−26.5092	−0.925178	−0.462589	−	0.886573i	$-0.653079\pi$
$821$	−26.5092	−0.925178	−0.462589	+	0.886573i	$0.653079\pi$
$822$	9.87136	0.344303
$823$	38.1358	1.32933	0.664664	−	0.747142i	$-0.268575\pi$
$823$	38.1358	1.32933	0.664664	+	0.747142i	$0.268575\pi$
$824$	81.7731	2.84870
$825$	17.7505	0.617994
$826$	−12.9716	−0.451339
$827$	36.0350	1.25306	0.626530	−	0.779397i	$-0.284475\pi$
$827$	36.0350	1.25306	0.626530	+	0.779397i	$0.284475\pi$
$828$	−11.0136	−0.382748
$829$	−28.7597	−0.998865	−0.499432	−	0.866353i	$-0.666458\pi$
$829$	−28.7597	−0.998865	−0.499432	+	0.866353i	$0.666458\pi$
$830$	10.2426	0.355527
$831$	−11.4117	−0.395867
$832$	45.8277	1.58879
$833$	−3.18720	−0.110430
$834$	3.90400	0.135184
$835$	3.94535	0.136534
$836$	23.8343	0.824326
$837$	−23.2318	−0.803009
$838$	−1.97607	−0.0682623
$839$	−24.4262	−0.843286	−0.421643	−	0.906762i	$-0.638546\pi$
$839$	−24.4262	−0.843286	−0.421643	+	0.906762i	$0.638546\pi$
$840$	24.1272	0.832468
$841$	1.00000	0.0344828
$842$	−18.5994	−0.640977
$843$	−34.4353	−1.18602
$844$	−3.55087	−0.122226
$845$	9.63215	0.331356
$846$	21.5065	0.739408
$847$	6.53710	0.224617
$848$	−49.0899	−1.68575
$849$	9.72025	0.333598
$850$	23.1028	0.792420
$851$	−28.1596	−0.965297
$852$	45.0554	1.54357
$853$	48.8276	1.67183	0.835913	−	0.548862i	$-0.184939\pi$
$853$	48.8276	1.67183	0.835913	+	0.548862i	$0.184939\pi$
$854$	11.9815	0.409997
$855$	1.22442	0.0418743
$856$	88.2120	3.01503
$857$	−55.9134	−1.90997	−0.954983	−	0.296659i	$-0.904127\pi$
$857$	−55.9134	−1.90997	−0.954983	+	0.296659i	$0.904127\pi$
$858$	54.2564	1.85228
$859$	36.3786	1.24122	0.620611	−	0.784118i	$-0.286885\pi$
$859$	36.3786	1.24122	0.620611	+	0.784118i	$0.286885\pi$
$860$	−20.8496	−0.710965
$861$	−37.3036	−1.27130
$862$	36.6141	1.24708
$863$	51.8008	1.76332	0.881660	−	0.471885i	$-0.156426\pi$
$863$	51.8008	1.76332	0.881660	+	0.471885i	$0.156426\pi$
$864$	63.4912	2.16001
$865$	−3.68127	−0.125167
$866$	84.0956	2.85768
$867$	−18.6368	−0.632941
$868$	−47.1770	−1.60129
$869$	9.55319	0.324070
$870$	3.57363	0.121158
$871$	11.8677	0.402123
$872$	114.364	3.87284
$873$	−5.89038	−0.199359
$874$	12.8540	0.434792
$875$	19.6577	0.664553
$876$	−50.2659	−1.69833
$877$	−18.9152	−0.638721	−0.319361	−	0.947633i	$-0.603468\pi$
$877$	−18.9152	−0.638721	−0.319361	+	0.947633i	$0.603468\pi$
$878$	−58.2747	−1.96668
$879$	−10.4765	−0.353364
$880$	−26.3815	−0.889322
$881$	−24.5575	−0.827363	−0.413681	−	0.910422i	$-0.635757\pi$
$881$	−24.5575	−0.827363	−0.413681	+	0.910422i	$0.635757\pi$
$882$	−3.10067	−0.104405
$883$	−54.1537	−1.82242	−0.911209	−	0.411944i	$-0.864850\pi$
$883$	−54.1537	−1.82242	−0.911209	+	0.411944i	$0.864850\pi$
$884$	50.0896	1.68469
$885$	−2.87440	−0.0966220
$886$	−103.398	−3.47374
$887$	−47.6484	−1.59988	−0.799938	−	0.600082i	$-0.795135\pi$
$887$	−47.6484	−1.59988	−0.799938	+	0.600082i	$0.795135\pi$
$888$	110.131	3.69575
$889$	10.8589	0.364195
$890$	−13.2040	−0.442600
$891$	17.2774	0.578814
$892$	−72.0848	−2.41358
$893$	−17.8042	−0.595794
$894$	73.9499	2.47325
$895$	10.8121	0.361409
$896$	5.25793	0.175655
$897$	20.7553	0.692999
$898$	67.3909	2.24886
$899$	−4.12414	−0.137548
$900$	15.9424	0.531414
$901$	−10.3285	−0.344092
$902$	80.3741	2.67617
$903$	16.2810	0.541796
$904$	16.6098	0.552432
$905$	6.15272	0.204523
$906$	82.4054	2.73773
$907$	−47.5465	−1.57876	−0.789379	−	0.613907i	$-0.789597\pi$
$907$	−47.5465	−1.57876	−0.789379	+	0.613907i	$0.789597\pi$
$908$	119.198	3.95571
$909$	4.14216	0.137387
$910$	27.2954	0.904835
$911$	−10.4122	−0.344972	−0.172486	−	0.985012i	$-0.555180\pi$
$911$	−10.4122	−0.344972	−0.172486	+	0.985012i	$0.555180\pi$
$912$	−25.5122	−0.844795
$913$	12.2237	0.404545
$914$	−42.2779	−1.39843
$915$	2.65500	0.0877717
$916$	−111.117	−3.67142
$917$	0.538400	0.0177795
$918$	31.2683	1.03201
$919$	−45.3542	−1.49610	−0.748049	−	0.663644i	$-0.769009\pi$
$919$	−45.3542	−1.49610	−0.748049	+	0.663644i	$0.769009\pi$
$920$	−19.8861	−0.655626
$921$	12.2263	0.402872
$922$	58.9943	1.94287
$923$	30.0836	0.990213
$924$	48.7863	1.60495
$925$	40.7617	1.34024
$926$	1.42949	0.0469760
$927$	8.49435	0.278991
$928$	11.2710	0.369989
$929$	−18.2898	−0.600070	−0.300035	−	0.953928i	$-0.596998\pi$
$929$	−18.2898	−0.600070	−0.300035	+	0.953928i	$0.596998\pi$
$930$	−14.7381	−0.483283
$931$	2.56689	0.0841266
$932$	31.3312	1.02629
$933$	−20.7073	−0.677926
$934$	−44.7762	−1.46512
$935$	−5.55067	−0.181526
$936$	28.7603	0.940060
$937$	−54.3968	−1.77707	−0.888533	−	0.458812i	$-0.848275\pi$
$937$	−54.3968	−1.77707	−0.888533	+	0.458812i	$0.848275\pi$
$938$	15.0443	0.491214
$939$	42.2731	1.37953
$940$	46.6697	1.52220
$941$	45.5995	1.48650	0.743251	−	0.669013i	$-0.233283\pi$
$941$	45.5995	1.48650	0.743251	+	0.669013i	$0.233283\pi$
$942$	−11.1306	−0.362655
$943$	30.7464	1.00124
$944$	−21.2201	−0.690655
$945$	12.0862	0.393164
$946$	−35.0788	−1.14051
$947$	59.3017	1.92705	0.963524	−	0.267623i	$-0.0862381\pi$
$947$	59.3017	1.92705	0.963524	+	0.267623i	$0.0862381\pi$
$948$	−24.2164	−0.786514
$949$	−33.5626	−1.08949
$950$	−18.6065	−0.603674
$951$	7.33115	0.237729
$952$	37.4758	1.21460
$953$	31.2359	1.01183	0.505915	−	0.862583i	$-0.331155\pi$
$953$	31.2359	1.01183	0.505915	+	0.862583i	$0.331155\pi$
$954$	−10.0481	−0.325319
$955$	−10.4345	−0.337654
$956$	−78.5555	−2.54067
$957$	4.26482	0.137862
$958$	−4.32119	−0.139611
$959$	5.92661	0.191380
$960$	12.8732	0.415482
$961$	−13.9915	−0.451339
$962$	124.593	4.01702
$963$	9.16320	0.295280
$964$	−24.7290	−0.796467
$965$	−18.7936	−0.604989
$966$	26.3107	0.846534
$967$	27.1569	0.873308	0.436654	−	0.899629i	$-0.356163\pi$
$967$	27.1569	0.873308	0.436654	+	0.899629i	$0.356163\pi$
$968$	21.0723	0.677289
$969$	−5.36776	−0.172437
$970$	−18.0205	−0.578603
$971$	−26.1244	−0.838372	−0.419186	−	0.907900i	$-0.637684\pi$
$971$	−26.1244	−0.838372	−0.419186	+	0.907900i	$0.637684\pi$
$972$	38.6799	1.24066
$973$	2.34390	0.0751420
$974$	−48.1331	−1.54228
$975$	−30.0438	−0.962173
$976$	19.6004	0.627393
$977$	37.4883	1.19936	0.599678	−	0.800241i	$-0.295295\pi$
$977$	37.4883	1.19936	0.599678	+	0.800241i	$0.295295\pi$
$978$	−84.5461	−2.70348
$979$	−15.7579	−0.503623
$980$	−6.72855	−0.214936
$981$	11.8798	0.379291
$982$	4.71269	0.150388
$983$	38.0740	1.21437	0.607186	−	0.794560i	$-0.292298\pi$
$983$	38.0740	1.21437	0.607186	+	0.794560i	$0.292298\pi$
$984$	−120.248	−3.83336
$985$	−6.42909	−0.204848
$986$	5.55078	0.176773
$987$	−36.4433	−1.16000
$988$	−40.3410	−1.28342
$989$	−13.4191	−0.426702
$990$	−5.39997	−0.171622
$991$	−55.9117	−1.77609	−0.888047	−	0.459752i	$-0.847938\pi$
$991$	−55.9117	−1.77609	−0.888047	+	0.459752i	$0.847938\pi$
$992$	−46.4832	−1.47584
$993$	44.6073	1.41557
$994$	38.1359	1.20960
$995$	−1.98769	−0.0630140
$996$	−30.9859	−0.981825
$997$	−4.07868	−0.129173	−0.0645865	−	0.997912i	$-0.520573\pi$
$997$	−4.07868	−0.129173	−0.0645865	+	0.997912i	$0.520573\pi$
$998$	17.0605	0.540039
$999$	55.1686	1.74546

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.2	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.2	✓	61	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.62306 q^{2} +1.48834 q^{3} +4.88043 q^{4} +0.915377 q^{5} -3.90400 q^{6} -2.34390 q^{7} -7.55555 q^{8} -0.784847 q^{9} +O(q^{10})\)	\(q-2.62306 q^{2} +1.48834 q^{3} +4.88043 q^{4} +0.915377 q^{5} -3.90400 q^{6} -2.34390 q^{7} -7.55555 q^{8} -0.784847 q^{9} -2.40109 q^{10} -2.86549 q^{11} +7.26374 q^{12} +4.85001 q^{13} +6.14819 q^{14} +1.36239 q^{15} +10.0578 q^{16} +2.11615 q^{17} +2.05870 q^{18} -1.70430 q^{19} +4.46744 q^{20} -3.48852 q^{21} +7.51634 q^{22} +2.87531 q^{23} -11.2452 q^{24} -4.16208 q^{25} -12.7219 q^{26} -5.63314 q^{27} -11.4393 q^{28} -1.00000 q^{29} -3.57363 q^{30} +4.12414 q^{31} -11.2710 q^{32} -4.26482 q^{33} -5.55078 q^{34} -2.14555 q^{35} -3.83039 q^{36} -9.79358 q^{37} +4.47047 q^{38} +7.21846 q^{39} -6.91618 q^{40} +10.6933 q^{41} +9.15059 q^{42} -4.66701 q^{43} -13.9848 q^{44} -0.718431 q^{45} -7.54210 q^{46} +10.4466 q^{47} +14.9694 q^{48} -1.50613 q^{49} +10.9174 q^{50} +3.14955 q^{51} +23.6702 q^{52} -4.88080 q^{53} +14.7760 q^{54} -2.62300 q^{55} +17.7094 q^{56} -2.53657 q^{57} +2.62306 q^{58} -2.10982 q^{59} +6.64906 q^{60} +1.94878 q^{61} -10.8178 q^{62} +1.83960 q^{63} +9.44900 q^{64} +4.43959 q^{65} +11.1869 q^{66} +2.44695 q^{67} +10.3277 q^{68} +4.27943 q^{69} +5.62791 q^{70} +6.20279 q^{71} +5.92995 q^{72} -6.92011 q^{73} +25.6891 q^{74} -6.19459 q^{75} -8.31771 q^{76} +6.71642 q^{77} -18.9344 q^{78} -3.33388 q^{79} +9.20665 q^{80} -6.02947 q^{81} -28.0490 q^{82} -4.26583 q^{83} -17.0255 q^{84} +1.93708 q^{85} +12.2418 q^{86} -1.48834 q^{87} +21.6503 q^{88} +5.49919 q^{89} +1.88449 q^{90} -11.3679 q^{91} +14.0327 q^{92} +6.13811 q^{93} -27.4021 q^{94} -1.56007 q^{95} -16.7751 q^{96} +7.50513 q^{97} +3.95067 q^{98} +2.24897 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

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