LMFDB - Embedded newform 445.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [445,2,Mod(1,445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(445, 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("445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 445 = 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	445.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:	$3.55334288995$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.09529$ of defining polynomial
Character	$\chi$	$=$	445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.39026 q^{2} +0.294963 q^{3} +3.71333 q^{4} -1.00000 q^{5} -0.705037 q^{6} +1.95452 q^{7} -4.09529 q^{8} -2.91300 q^{9} +O(q^{10})$	\(q-2.39026 q^{2} +0.294963 q^{3} +3.71333 q^{4} -1.00000 q^{5} -0.705037 q^{6} +1.95452 q^{7} -4.09529 q^{8} -2.91300 q^{9} +2.39026 q^{10} -4.29496 q^{11} +1.09529 q^{12} +1.05377 q^{13} -4.67180 q^{14} -0.294963 q^{15} +2.36215 q^{16} +3.65443 q^{17} +6.96281 q^{18} -3.07792 q^{19} -3.71333 q^{20} +0.576511 q^{21} +10.2661 q^{22} -1.46385 q^{23} -1.20796 q^{24} +1.00000 q^{25} -2.51878 q^{26} -1.74411 q^{27} +7.25777 q^{28} -8.44007 q^{29} +0.705037 q^{30} +4.24040 q^{31} +2.54445 q^{32} -1.26685 q^{33} -8.73503 q^{34} -1.95452 q^{35} -10.8169 q^{36} -4.61163 q^{37} +7.35702 q^{38} +0.310823 q^{39} +4.09529 q^{40} +0.785638 q^{41} -1.37801 q^{42} -3.60895 q^{43} -15.9486 q^{44} +2.91300 q^{45} +3.49897 q^{46} -10.8220 q^{47} +0.696746 q^{48} -3.17985 q^{49} -2.39026 q^{50} +1.07792 q^{51} +3.91300 q^{52} -9.07035 q^{53} +4.16888 q^{54} +4.29496 q^{55} -8.00433 q^{56} -0.907873 q^{57} +20.1739 q^{58} -5.36459 q^{59} -1.09529 q^{60} -5.59950 q^{61} -10.1356 q^{62} -5.69351 q^{63} -10.8062 q^{64} -1.05377 q^{65} +3.02811 q^{66} -7.92129 q^{67} +13.5701 q^{68} -0.431780 q^{69} +4.67180 q^{70} +10.0621 q^{71} +11.9296 q^{72} +2.47330 q^{73} +11.0230 q^{74} +0.294963 q^{75} -11.4293 q^{76} -8.39459 q^{77} -0.742948 q^{78} -11.6532 q^{79} -2.36215 q^{80} +8.22454 q^{81} -1.87788 q^{82} +12.5329 q^{83} +2.14077 q^{84} -3.65443 q^{85} +8.62632 q^{86} -2.48951 q^{87} +17.5891 q^{88} -1.00000 q^{89} -6.96281 q^{90} +2.05962 q^{91} -5.43574 q^{92} +1.25076 q^{93} +25.8674 q^{94} +3.07792 q^{95} +0.750517 q^{96} +5.96756 q^{97} +7.60066 q^{98} +12.5112 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} - 6 q^{6} + 2 q^{7} - 9 q^{8} - 4 q^{9}+O(q^{10}) $ 4 * q + q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 - 6 * q^6 + 2 * q^7 - 9 * q^8 - 4 * q^9	\( 4 q + q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} - 6 q^{6} + 2 q^{7} - 9 q^{8} - 4 q^{9} - q^{10} - 14 q^{11} - 3 q^{12} - 5 q^{13} - 5 q^{14} + 2 q^{15} - 3 q^{16} - 3 q^{17} + 7 q^{18} - q^{19} - 3 q^{20} - 4 q^{21} + 2 q^{22} - 3 q^{23} + 6 q^{24} + 4 q^{25} - 9 q^{26} + q^{27} + 5 q^{28} - 10 q^{29} + 6 q^{30} - 11 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{35} - 9 q^{36} + 3 q^{37} + 2 q^{38} + 11 q^{39} + 9 q^{40} - 3 q^{41} - 6 q^{42} + 9 q^{43} - 9 q^{44} + 4 q^{45} - 4 q^{46} - 24 q^{47} + 21 q^{48} + q^{50} - 7 q^{51} + 8 q^{52} + 3 q^{53} + 17 q^{54} + 14 q^{55} - 13 q^{56} + 19 q^{57} + 19 q^{58} - 22 q^{59} + 3 q^{60} - 3 q^{61} - 24 q^{62} + 6 q^{63} - 11 q^{64} + 5 q^{65} + 14 q^{66} - 9 q^{67} + 31 q^{68} + 7 q^{69} + 5 q^{70} + 16 q^{71} + 10 q^{72} + 3 q^{73} + 31 q^{74} - 2 q^{75} - 24 q^{76} - 4 q^{77} + 16 q^{78} - 27 q^{79} + 3 q^{80} - 8 q^{81} - 15 q^{82} + 6 q^{83} + 7 q^{84} + 3 q^{85} + 15 q^{86} + 4 q^{87} + 30 q^{88} - 4 q^{89} - 7 q^{90} - 29 q^{91} - 17 q^{92} - 2 q^{93} + 13 q^{94} + q^{95} + 12 q^{96} + 41 q^{97} + 22 q^{98} + 21 q^{99}+O(q^{100}) \) 4 * q + q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 - 6 * q^6 + 2 * q^7 - 9 * q^8 - 4 * q^9 - q^10 - 14 * q^11 - 3 * q^12 - 5 * q^13 - 5 * q^14 + 2 * q^15 - 3 * q^16 - 3 * q^17 + 7 * q^18 - q^19 - 3 * q^20 - 4 * q^21 + 2 * q^22 - 3 * q^23 + 6 * q^24 + 4 * q^25 - 9 * q^26 + q^27 + 5 * q^28 - 10 * q^29 + 6 * q^30 - 11 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^35 - 9 * q^36 + 3 * q^37 + 2 * q^38 + 11 * q^39 + 9 * q^40 - 3 * q^41 - 6 * q^42 + 9 * q^43 - 9 * q^44 + 4 * q^45 - 4 * q^46 - 24 * q^47 + 21 * q^48 + q^50 - 7 * q^51 + 8 * q^52 + 3 * q^53 + 17 * q^54 + 14 * q^55 - 13 * q^56 + 19 * q^57 + 19 * q^58 - 22 * q^59 + 3 * q^60 - 3 * q^61 - 24 * q^62 + 6 * q^63 - 11 * q^64 + 5 * q^65 + 14 * q^66 - 9 * q^67 + 31 * q^68 + 7 * q^69 + 5 * q^70 + 16 * q^71 + 10 * q^72 + 3 * q^73 + 31 * q^74 - 2 * q^75 - 24 * q^76 - 4 * q^77 + 16 * q^78 - 27 * q^79 + 3 * q^80 - 8 * q^81 - 15 * q^82 + 6 * q^83 + 7 * q^84 + 3 * q^85 + 15 * q^86 + 4 * q^87 + 30 * q^88 - 4 * q^89 - 7 * q^90 - 29 * q^91 - 17 * q^92 - 2 * q^93 + 13 * q^94 + q^95 + 12 * q^96 + 41 * q^97 + 22 * q^98 + 21 * 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 $

$2$	−2.39026	−1.69017	−0.845083	−	0.534634i	$-0.820449\pi$
$2$	−2.39026	−1.69017	−0.845083	+	0.534634i	$0.820449\pi$
$3$	0.294963	0.170297	0.0851485	−	0.996368i	$-0.472864\pi$
$3$	0.294963	0.170297	0.0851485	+	0.996368i	$0.472864\pi$
$4$	3.71333	1.85666
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.705037	−0.287830
$7$	1.95452	0.738739	0.369370	−	0.929283i	$-0.379574\pi$
$7$	1.95452	0.738739	0.369370	+	0.929283i	$0.379574\pi$
$8$	−4.09529	−1.44791
$9$	−2.91300	−0.970999
$10$	2.39026	0.755866
$11$	−4.29496	−1.29498	−0.647490	−	0.762074i	$-0.724181\pi$
$11$	−4.29496	−1.29498	−0.647490	+	0.762074i	$0.724181\pi$
$12$	1.09529	0.316184
$13$	1.05377	0.292263	0.146132	−	0.989265i	$-0.453318\pi$
$13$	1.05377	0.292263	0.146132	+	0.989265i	$0.453318\pi$
$14$	−4.67180	−1.24859
$15$	−0.294963	−0.0761591
$16$	2.36215	0.590537
$17$	3.65443	0.886330	0.443165	−	0.896440i	$-0.353856\pi$
$17$	3.65443	0.886330	0.443165	+	0.896440i	$0.353856\pi$
$18$	6.96281	1.64115
$19$	−3.07792	−0.706124	−0.353062	−	0.935600i	$-0.614859\pi$
$19$	−3.07792	−0.706124	−0.353062	+	0.935600i	$0.614859\pi$
$20$	−3.71333	−0.830325
$21$	0.576511	0.125805
$22$	10.2661	2.18873
$23$	−1.46385	−0.305233	−0.152616	−	0.988286i	$-0.548770\pi$
$23$	−1.46385	−0.305233	−0.152616	+	0.988286i	$0.548770\pi$
$24$	−1.20796	−0.246574
$25$	1.00000	0.200000
$26$	−2.51878	−0.493974
$27$	−1.74411	−0.335655
$28$	7.25777	1.37159
$29$	−8.44007	−1.56728	−0.783641	−	0.621214i	$-0.786640\pi$
$29$	−8.44007	−1.56728	−0.783641	+	0.621214i	$0.786640\pi$
$30$	0.705037	0.128722
$31$	4.24040	0.761599	0.380799	−	0.924658i	$-0.375649\pi$
$31$	4.24040	0.761599	0.380799	+	0.924658i	$0.375649\pi$
$32$	2.54445	0.449799
$33$	−1.26685	−0.220531
$34$	−8.73503	−1.49805
$35$	−1.95452	−0.330374
$36$	−10.8169	−1.80282
$37$	−4.61163	−0.758148	−0.379074	−	0.925366i	$-0.623757\pi$
$37$	−4.61163	−0.758148	−0.379074	+	0.925366i	$0.623757\pi$
$38$	7.35702	1.19347
$39$	0.310823	0.0497716
$40$	4.09529	0.647523
$41$	0.785638	0.122696	0.0613480	−	0.998116i	$-0.480460\pi$
$41$	0.785638	0.122696	0.0613480	+	0.998116i	$0.480460\pi$
$42$	−1.37801	−0.212631
$43$	−3.60895	−0.550360	−0.275180	−	0.961393i	$-0.588737\pi$
$43$	−3.60895	−0.550360	−0.275180	+	0.961393i	$0.588737\pi$
$44$	−15.9486	−2.40434
$45$	2.91300	0.434244
$46$	3.49897	0.515894
$47$	−10.8220	−1.57856	−0.789278	−	0.614036i	$-0.789545\pi$
$47$	−10.8220	−1.57856	−0.789278	+	0.614036i	$0.789545\pi$
$48$	0.696746	0.100567
$49$	−3.17985	−0.454265
$50$	−2.39026	−0.338033
$51$	1.07792	0.150939
$52$	3.91300	0.542635
$53$	−9.07035	−1.24591	−0.622954	−	0.782258i	$-0.714068\pi$
$53$	−9.07035	−1.24591	−0.622954	+	0.782258i	$0.714068\pi$
$54$	4.16888	0.567313
$55$	4.29496	0.579133
$56$	−8.00433	−1.06962
$57$	−0.907873	−0.120251
$58$	20.1739	2.64897
$59$	−5.36459	−0.698411	−0.349205	−	0.937046i	$-0.613548\pi$
$59$	−5.36459	−0.698411	−0.349205	+	0.937046i	$0.613548\pi$
$60$	−1.09529	−0.141402
$61$	−5.59950	−0.716942	−0.358471	−	0.933541i	$-0.616702\pi$
$61$	−5.59950	−0.716942	−0.358471	+	0.933541i	$0.616702\pi$
$62$	−10.1356	−1.28723
$63$	−5.69351	−0.717315
$64$	−10.8062	−1.35077
$65$	−1.05377	−0.130704
$66$	3.02811	0.372734
$67$	−7.92129	−0.967739	−0.483870	−	0.875140i	$-0.660769\pi$
$67$	−7.92129	−0.967739	−0.483870	+	0.875140i	$0.660769\pi$
$68$	13.5701	1.64562
$69$	−0.431780	−0.0519802
$70$	4.67180	0.558387
$71$	10.0621	1.19415	0.597074	−	0.802187i	$-0.296330\pi$
$71$	10.0621	1.19415	0.597074	+	0.802187i	$0.296330\pi$
$72$	11.9296	1.40591
$73$	2.47330	0.289478	0.144739	−	0.989470i	$-0.453766\pi$
$73$	2.47330	0.289478	0.144739	+	0.989470i	$0.453766\pi$
$74$	11.0230	1.28140
$75$	0.294963	0.0340594
$76$	−11.4293	−1.31103
$77$	−8.39459	−0.956652
$78$	−0.742948	−0.0841222
$79$	−11.6532	−1.31108	−0.655541	−	0.755159i	$-0.727559\pi$
$79$	−11.6532	−1.31108	−0.655541	+	0.755159i	$0.727559\pi$
$80$	−2.36215	−0.264096
$81$	8.22454	0.913838
$82$	−1.87788	−0.207377
$83$	12.5329	1.37567	0.687833	−	0.725869i	$-0.258562\pi$
$83$	12.5329	1.37567	0.687833	+	0.725869i	$0.258562\pi$
$84$	2.14077	0.233578
$85$	−3.65443	−0.396379
$86$	8.62632	0.930201
$87$	−2.48951	−0.266903
$88$	17.5891	1.87501
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	−6.96281	−0.733945
$91$	2.05962	0.215906
$92$	−5.43574	−0.566715
$93$	1.25076	0.129698
$94$	25.8674	2.66802
$95$	3.07792	0.315788
$96$	0.750517	0.0765993
$97$	5.96756	0.605914	0.302957	−	0.953004i	$-0.402026\pi$
$97$	5.96756	0.605914	0.302957	+	0.953004i	$0.402026\pi$
$98$	7.60066	0.767783
$99$	12.5112	1.25742
$100$	3.71333	0.371333
$101$	6.15863	0.612807	0.306403	−	0.951902i	$-0.400874\pi$
$101$	6.15863	0.612807	0.306403	+	0.951902i	$0.400874\pi$
$102$	−2.57651	−0.255113
$103$	15.0166	1.47963	0.739814	−	0.672812i	$-0.234914\pi$
$103$	15.0166	1.47963	0.739814	+	0.672812i	$0.234914\pi$
$104$	−4.31550	−0.423170
$105$	−0.576511	−0.0562617
$106$	21.6805	2.10579
$107$	2.68955	0.260009	0.130004	−	0.991513i	$-0.458501\pi$
$107$	2.68955	0.260009	0.130004	+	0.991513i	$0.458501\pi$
$108$	−6.47647	−0.623199
$109$	−0.671805	−0.0643472	−0.0321736	−	0.999482i	$-0.510243\pi$
$109$	−0.671805	−0.0643472	−0.0321736	+	0.999482i	$0.510243\pi$
$110$	−10.2661	−0.978831
$111$	−1.36026	−0.129110
$112$	4.61687	0.436253
$113$	−8.72825	−0.821085	−0.410543	−	0.911841i	$-0.634661\pi$
$113$	−8.72825	−0.821085	−0.410543	+	0.911841i	$0.634661\pi$
$114$	2.17005	0.203244
$115$	1.46385	0.136504
$116$	−31.3408	−2.90992
$117$	−3.06963	−0.283788
$118$	12.8228	1.18043
$119$	7.14266	0.654767
$120$	1.20796	0.110271
$121$	7.44671	0.676973
$122$	13.3842	1.21175
$123$	0.231734	0.0208948
$124$	15.7460	1.41403
$125$	−1.00000	−0.0894427
$126$	13.6090	1.21238
$127$	−4.55006	−0.403752	−0.201876	−	0.979411i	$-0.564704\pi$
$127$	−4.55006	−0.403752	−0.201876	+	0.979411i	$0.564704\pi$
$128$	20.7406	1.83323
$129$	−1.06451	−0.0937246
$130$	2.51878	0.220912
$131$	−9.12736	−0.797461	−0.398731	−	0.917068i	$-0.630549\pi$
$131$	−9.12736	−0.797461	−0.398731	+	0.917068i	$0.630549\pi$
$132$	−4.70425	−0.409452
$133$	−6.01586	−0.521641
$134$	18.9339	1.63564
$135$	1.74411	0.150109
$136$	−14.9660	−1.28332
$137$	16.8521	1.43978	0.719888	−	0.694090i	$-0.244193\pi$
$137$	16.8521	1.43978	0.719888	+	0.694090i	$0.244193\pi$
$138$	1.03207	0.0878552
$139$	3.20205	0.271594	0.135797	−	0.990737i	$-0.456641\pi$
$139$	3.20205	0.271594	0.135797	+	0.990737i	$0.456641\pi$
$140$	−7.25777	−0.613394
$141$	−3.19210	−0.268823
$142$	−24.0509	−2.01831
$143$	−4.52591	−0.378475
$144$	−6.88093	−0.573411
$145$	8.44007	0.700910
$146$	−5.91183	−0.489267
$147$	−0.937938	−0.0773598
$148$	−17.1245	−1.40763
$149$	14.6448	1.19975	0.599874	−	0.800094i	$-0.295217\pi$
$149$	14.6448	1.19975	0.599874	+	0.800094i	$0.295217\pi$
$150$	−0.705037	−0.0575660
$151$	−17.8411	−1.45189	−0.725943	−	0.687755i	$-0.758596\pi$
$151$	−17.8411	−1.45189	−0.725943	+	0.687755i	$0.758596\pi$
$152$	12.6050	1.02240
$153$	−10.6454	−0.860626
$154$	20.0652	1.61690
$155$	−4.24040	−0.340597
$156$	1.15419	0.0924091
$157$	15.4767	1.23518	0.617588	−	0.786502i	$-0.288110\pi$
$157$	15.4767	1.23518	0.617588	+	0.786502i	$0.288110\pi$
$158$	27.8540	2.21595
$159$	−2.67542	−0.212174
$160$	−2.54445	−0.201156
$161$	−2.86111	−0.225487
$162$	−19.6588	−1.54454
$163$	−2.39177	−0.187338	−0.0936689	−	0.995603i	$-0.529860\pi$
$163$	−2.39177	−0.187338	−0.0936689	+	0.995603i	$0.529860\pi$
$164$	2.91733	0.227805
$165$	1.26685	0.0986245
$166$	−29.9569	−2.32511
$167$	0.644864	0.0499011	0.0249505	−	0.999689i	$-0.492057\pi$
$167$	0.644864	0.0499011	0.0249505	+	0.999689i	$0.492057\pi$
$168$	−2.36098	−0.182154
$169$	−11.8896	−0.914582
$170$	8.73503	0.669947
$171$	8.96598	0.685645
$172$	−13.4012	−1.02183
$173$	−19.2152	−1.46091	−0.730453	−	0.682963i	$-0.760691\pi$
$173$	−19.2152	−1.46091	−0.730453	+	0.682963i	$0.760691\pi$
$174$	5.95056	0.451111
$175$	1.95452	0.147748
$176$	−10.1453	−0.764734
$177$	−1.58236	−0.118937
$178$	2.39026	0.179157
$179$	−14.9898	−1.12039	−0.560193	−	0.828362i	$-0.689273\pi$
$179$	−14.9898	−1.12039	−0.560193	+	0.828362i	$0.689273\pi$
$180$	10.8169	0.806245
$181$	23.1716	1.72233	0.861167	−	0.508322i	$-0.169734\pi$
$181$	23.1716	1.72233	0.861167	+	0.508322i	$0.169734\pi$
$182$	−4.92301	−0.364918
$183$	−1.65164	−0.122093
$184$	5.99488	0.441948
$185$	4.61163	0.339054
$186$	−2.98964	−0.219211
$187$	−15.6957	−1.14778
$188$	−40.1858	−2.93085
$189$	−3.40891	−0.247962
$190$	−7.35702	−0.533735
$191$	0.390024	0.0282211	0.0141106	−	0.999900i	$-0.495508\pi$
$191$	0.390024	0.0282211	0.0141106	+	0.999900i	$0.495508\pi$
$192$	−3.18742	−0.230032
$193$	9.91428	0.713645	0.356823	−	0.934172i	$-0.383860\pi$
$193$	9.91428	0.713645	0.356823	+	0.934172i	$0.383860\pi$
$194$	−14.2640	−1.02410
$195$	−0.310823	−0.0222585
$196$	−11.8078	−0.843417
$197$	−1.01097	−0.0720286	−0.0360143	−	0.999351i	$-0.511466\pi$
$197$	−1.01097	−0.0720286	−0.0360143	+	0.999351i	$0.511466\pi$
$198$	−29.9050	−2.12526
$199$	−13.3504	−0.946384	−0.473192	−	0.880959i	$-0.656898\pi$
$199$	−13.3504	−0.946384	−0.473192	+	0.880959i	$0.656898\pi$
$200$	−4.09529	−0.289581
$201$	−2.33649	−0.164803
$202$	−14.7207	−1.03575
$203$	−16.4963	−1.15781
$204$	4.00268	0.280244
$205$	−0.785638	−0.0548713
$206$	−35.8935	−2.50082
$207$	4.26418	0.296381
$208$	2.48916	0.172592
$209$	13.2196	0.914416
$210$	1.37801	0.0950917
$211$	−25.4818	−1.75424	−0.877118	−	0.480274i	$-0.840537\pi$
$211$	−25.4818	−1.75424	−0.877118	+	0.480274i	$0.840537\pi$
$212$	−33.6812	−2.31323
$213$	2.96793	0.203360
$214$	−6.42872	−0.439459
$215$	3.60895	0.246129
$216$	7.14266	0.485997
$217$	8.28795	0.562623
$218$	1.60579	0.108758
$219$	0.729533	0.0492972
$220$	15.9486	1.07525
$221$	3.85094	0.259042
$222$	3.25137	0.218218
$223$	20.3456	1.36244	0.681222	−	0.732076i	$-0.261449\pi$
$223$	20.3456	1.36244	0.681222	+	0.732076i	$0.261449\pi$
$224$	4.97317	0.332284
$225$	−2.91300	−0.194200
$226$	20.8628	1.38777
$227$	16.0605	1.06597	0.532986	−	0.846124i	$-0.321070\pi$
$227$	16.0605	1.06597	0.532986	+	0.846124i	$0.321070\pi$
$228$	−3.37123	−0.223265
$229$	15.2661	1.00881	0.504405	−	0.863467i	$-0.331712\pi$
$229$	15.2661	1.00881	0.504405	+	0.863467i	$0.331712\pi$
$230$	−3.49897	−0.230715
$231$	−2.47609	−0.162915
$232$	34.5646	2.26928
$233$	18.5117	1.21274	0.606371	−	0.795182i	$-0.292625\pi$
$233$	18.5117	1.21274	0.606371	+	0.795182i	$0.292625\pi$
$234$	7.33721	0.479648
$235$	10.8220	0.705952
$236$	−19.9205	−1.29671
$237$	−3.43725	−0.223273
$238$	−17.0728	−1.10667
$239$	−27.6511	−1.78860	−0.894300	−	0.447468i	$-0.852326\pi$
$239$	−27.6511	−1.78860	−0.894300	+	0.447468i	$0.852326\pi$
$240$	−0.696746	−0.0449748
$241$	27.2269	1.75384	0.876920	−	0.480636i	$-0.159594\pi$
$241$	27.2269	1.75384	0.876920	+	0.480636i	$0.159594\pi$
$242$	−17.7995	−1.14420
$243$	7.65828	0.491279
$244$	−20.7928	−1.33112
$245$	3.17985	0.203153
$246$	−0.553904	−0.0353156
$247$	−3.24342	−0.206374
$248$	−17.3657	−1.10272
$249$	3.69675	0.234272
$250$	2.39026	0.151173
$251$	12.9645	0.818310	0.409155	−	0.912465i	$-0.365824\pi$
$251$	12.9645	0.818310	0.409155	+	0.912465i	$0.365824\pi$
$252$	−21.1419	−1.33181
$253$	6.28716	0.395270
$254$	10.8758	0.682409
$255$	−1.07792	−0.0675021
$256$	−27.9631	−1.74769
$257$	18.9552	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$257$	18.9552	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$258$	2.54445	0.158410
$259$	−9.01353	−0.560073
$260$	−3.91300	−0.242674
$261$	24.5859	1.52183
$262$	21.8167	1.34784
$263$	−16.4109	−1.01194	−0.505971	−	0.862551i	$-0.668866\pi$
$263$	−16.4109	−1.01194	−0.505971	+	0.862551i	$0.668866\pi$
$264$	5.18814	0.319308
$265$	9.07035	0.557187
$266$	14.3795	0.881661
$267$	−0.294963	−0.0180514
$268$	−29.4143	−1.79677
$269$	28.9069	1.76248	0.881242	−	0.472664i	$-0.156708\pi$
$269$	28.9069	1.76248	0.881242	+	0.472664i	$0.156708\pi$
$270$	−4.16888	−0.253710
$271$	6.39573	0.388513	0.194256	−	0.980951i	$-0.437771\pi$
$271$	6.39573	0.388513	0.194256	+	0.980951i	$0.437771\pi$
$272$	8.63231	0.523411
$273$	0.607510	0.0367682
$274$	−40.2810	−2.43346
$275$	−4.29496	−0.258996
$276$	−1.60334	−0.0965098
$277$	−2.90811	−0.174731	−0.0873656	−	0.996176i	$-0.527845\pi$
$277$	−2.90811	−0.174731	−0.0873656	+	0.996176i	$0.527845\pi$
$278$	−7.65371	−0.459039
$279$	−12.3523	−0.739512
$280$	8.00433	0.478350
$281$	−6.27691	−0.374449	−0.187225	−	0.982317i	$-0.559949\pi$
$281$	−6.27691	−0.374449	−0.187225	+	0.982317i	$0.559949\pi$
$282$	7.62994	0.454356
$283$	11.8814	0.706277	0.353139	−	0.935571i	$-0.385114\pi$
$283$	11.8814	0.706277	0.353139	+	0.935571i	$0.385114\pi$
$284$	37.3637	2.21713
$285$	0.907873	0.0537777
$286$	10.8181	0.639686
$287$	1.53554	0.0906403
$288$	−7.41196	−0.436754
$289$	−3.64512	−0.214419
$290$	−20.1739	−1.18465
$291$	1.76021	0.103185
$292$	9.18419	0.537464
$293$	−15.4163	−0.900630	−0.450315	−	0.892870i	$-0.648688\pi$
$293$	−15.4163	−0.900630	−0.450315	+	0.892870i	$0.648688\pi$
$294$	2.24191	0.130751
$295$	5.36459	0.312339
$296$	18.8860	1.09773
$297$	7.49091	0.434667
$298$	−35.0048	−2.02777
$299$	−1.54256	−0.0892084
$300$	1.09529	0.0632368
$301$	−7.05377	−0.406573
$302$	42.6447	2.45393
$303$	1.81657	0.104359
$304$	−7.27051	−0.416992
$305$	5.59950	0.320626
$306$	25.4451	1.45460
$307$	−13.1695	−0.751623	−0.375811	−	0.926696i	$-0.622636\pi$
$307$	−13.1695	−0.751623	−0.375811	+	0.926696i	$0.622636\pi$
$308$	−31.1719	−1.77618
$309$	4.42933	0.251976
$310$	10.1356	0.575666
$311$	13.6116	0.771841	0.385920	−	0.922532i	$-0.373884\pi$
$311$	13.6116	0.771841	0.385920	+	0.922532i	$0.373884\pi$
$312$	−1.27291	−0.0720645
$313$	−20.0023	−1.13059	−0.565297	−	0.824887i	$-0.691239\pi$
$313$	−20.0023	−1.13059	−0.565297	+	0.824887i	$0.691239\pi$
$314$	−36.9933	−2.08765
$315$	5.69351	0.320793
$316$	−43.2720	−2.43424
$317$	17.9215	1.00657	0.503286	−	0.864120i	$-0.332124\pi$
$317$	17.9215	1.00657	0.503286	+	0.864120i	$0.332124\pi$
$318$	6.39494	0.358610
$319$	36.2498	2.02960
$320$	10.8062	0.604084
$321$	0.793319	0.0442787
$322$	6.83880	0.381111
$323$	−11.2481	−0.625859
$324$	30.5404	1.69669
$325$	1.05377	0.0584527
$326$	5.71694	0.316632
$327$	−0.198158	−0.0109581
$328$	−3.21742	−0.177652
$329$	−21.1519	−1.16614
$330$	−3.02811	−0.166692
$331$	−20.2646	−1.11384	−0.556920	−	0.830566i	$-0.688017\pi$
$331$	−20.2646	−1.11384	−0.556920	+	0.830566i	$0.688017\pi$
$332$	46.5388	2.55415
$333$	13.4337	0.736160
$334$	−1.54139	−0.0843411
$335$	7.92129	0.432786
$336$	1.36180	0.0742925
$337$	−16.1392	−0.879158	−0.439579	−	0.898204i	$-0.644872\pi$
$337$	−16.1392	−0.879158	−0.439579	+	0.898204i	$0.644872\pi$
$338$	28.4191	1.54580
$339$	−2.57451	−0.139828
$340$	−13.5701	−0.735942
$341$	−18.2124	−0.986255
$342$	−21.4310	−1.15886
$343$	−19.8967	−1.07432
$344$	14.7797	0.796869
$345$	0.431780	0.0232463
$346$	45.9293	2.46918
$347$	14.3396	0.769789	0.384895	−	0.922961i	$-0.374238\pi$
$347$	14.3396	0.769789	0.384895	+	0.922961i	$0.374238\pi$
$348$	−9.24436	−0.495550
$349$	−31.8954	−1.70732	−0.853660	−	0.520830i	$-0.825622\pi$
$349$	−31.8954	−1.70732	−0.853660	+	0.520830i	$0.825622\pi$
$350$	−4.67180	−0.249718
$351$	−1.83790	−0.0980997
$352$	−10.9283	−0.582480
$353$	2.46050	0.130959	0.0654796	−	0.997854i	$-0.479142\pi$
$353$	2.46050	0.130959	0.0654796	+	0.997854i	$0.479142\pi$
$354$	3.78224	0.201024
$355$	−10.0621	−0.534039
$356$	−3.71333	−0.196806
$357$	2.10682	0.111505
$358$	35.8294	1.89364
$359$	4.79276	0.252952	0.126476	−	0.991970i	$-0.459633\pi$
$359$	4.79276	0.252952	0.126476	+	0.991970i	$0.459633\pi$
$360$	−11.9296	−0.628744
$361$	−9.52640	−0.501389
$362$	−55.3862	−2.91103
$363$	2.19650	0.115286
$364$	7.64803	0.400866
$365$	−2.47330	−0.129459
$366$	3.94785	0.206358
$367$	−23.3647	−1.21963	−0.609813	−	0.792545i	$-0.708755\pi$
$367$	−23.3647	−1.21963	−0.609813	+	0.792545i	$0.708755\pi$
$368$	−3.45782	−0.180251
$369$	−2.28856	−0.119138
$370$	−11.0230	−0.573058
$371$	−17.7282	−0.920402
$372$	4.64449	0.240805
$373$	23.7500	1.22973	0.614863	−	0.788634i	$-0.289211\pi$
$373$	23.7500	1.22973	0.614863	+	0.788634i	$0.289211\pi$
$374$	37.5166	1.93994
$375$	−0.294963	−0.0152318
$376$	44.3194	2.28560
$377$	−8.89390	−0.458059
$378$	8.14816	0.419096
$379$	−27.7001	−1.42286	−0.711429	−	0.702758i	$-0.751952\pi$
$379$	−27.7001	−1.42286	−0.711429	+	0.702758i	$0.751952\pi$
$380$	11.4293	0.586312
$381$	−1.34210	−0.0687578
$382$	−0.932257	−0.0476984
$383$	6.03526	0.308388	0.154194	−	0.988041i	$-0.450722\pi$
$383$	6.03526	0.308388	0.154194	+	0.988041i	$0.450722\pi$
$384$	6.11772	0.312194
$385$	8.39459	0.427828
$386$	−23.6977	−1.20618
$387$	10.5129	0.534399
$388$	22.1595	1.12498
$389$	8.53031	0.432504	0.216252	−	0.976338i	$-0.430617\pi$
$389$	8.53031	0.432504	0.216252	+	0.976338i	$0.430617\pi$
$390$	0.742948	0.0376206
$391$	−5.34952	−0.270537
$392$	13.0224	0.657732
$393$	−2.69223	−0.135805
$394$	2.41648	0.121740
$395$	11.6532	0.586334
$396$	46.4582	2.33461
$397$	−14.6163	−0.733572	−0.366786	−	0.930305i	$-0.619542\pi$
$397$	−14.6163	−0.733572	−0.366786	+	0.930305i	$0.619542\pi$
$398$	31.9109	1.59955
$399$	−1.77446	−0.0888339
$400$	2.36215	0.118107
$401$	17.2987	0.863858	0.431929	−	0.901908i	$-0.357833\pi$
$401$	17.2987	0.863858	0.431929	+	0.901908i	$0.357833\pi$
$402$	5.58480	0.278545
$403$	4.46841	0.222587
$404$	22.8690	1.13778
$405$	−8.22454	−0.408681
$406$	39.4304	1.95690
$407$	19.8068	0.981786
$408$	−4.41441	−0.218546
$409$	−2.78264	−0.137592	−0.0687962	−	0.997631i	$-0.521916\pi$
$409$	−2.78264	−0.137592	−0.0687962	+	0.997631i	$0.521916\pi$
$410$	1.87788	0.0927417
$411$	4.97076	0.245189
$412$	55.7615	2.74717
$413$	−10.4852	−0.515943
$414$	−10.1925	−0.500933
$415$	−12.5329	−0.615217
$416$	2.68126	0.131460
$417$	0.944485	0.0462516
$418$	−31.5981	−1.54552
$419$	−4.20691	−0.205521	−0.102761	−	0.994706i	$-0.532768\pi$
$419$	−4.20691	−0.205521	−0.102761	+	0.994706i	$0.532768\pi$
$420$	−2.14077	−0.104459
$421$	19.0766	0.929737	0.464869	−	0.885380i	$-0.346102\pi$
$421$	19.0766	0.929737	0.464869	+	0.885380i	$0.346102\pi$
$422$	60.9079	2.96495
$423$	31.5246	1.53278
$424$	37.1458	1.80396
$425$	3.65443	0.177266
$426$	−7.09413	−0.343712
$427$	−10.9443	−0.529633
$428$	9.98720	0.482749
$429$	−1.33497	−0.0644532
$430$	−8.62632	−0.415998
$431$	−4.11651	−0.198285	−0.0991427	−	0.995073i	$-0.531610\pi$
$431$	−4.11651	−0.198285	−0.0991427	+	0.995073i	$0.531610\pi$
$432$	−4.11986	−0.198217
$433$	−34.4248	−1.65435	−0.827176	−	0.561943i	$-0.810054\pi$
$433$	−34.4248	−1.65435	−0.827176	+	0.561943i	$0.810054\pi$
$434$	−19.8103	−0.950926
$435$	2.48951	0.119363
$436$	−2.49463	−0.119471
$437$	4.50560	0.215532
$438$	−1.74377	−0.0833206
$439$	15.1883	0.724896	0.362448	−	0.932004i	$-0.381941\pi$
$439$	15.1883	0.724896	0.362448	+	0.932004i	$0.381941\pi$
$440$	−17.5891	−0.838529
$441$	9.26290	0.441090
$442$	−9.20472	−0.437824
$443$	2.14752	0.102032	0.0510159	−	0.998698i	$-0.483754\pi$
$443$	2.14752	0.102032	0.0510159	+	0.998698i	$0.483754\pi$
$444$	−5.05109	−0.239714
$445$	1.00000	0.0474045
$446$	−48.6313	−2.30276
$447$	4.31967	0.204313
$448$	−21.1209	−0.997868
$449$	32.5132	1.53439	0.767197	−	0.641411i	$-0.221651\pi$
$449$	32.5132	1.53439	0.767197	+	0.641411i	$0.221651\pi$
$450$	6.96281	0.328230
$451$	−3.37429	−0.158889
$452$	−32.4109	−1.52448
$453$	−5.26245	−0.247252
$454$	−38.3887	−1.80167
$455$	−2.05962	−0.0965563
$456$	3.71801	0.174112
$457$	−15.9654	−0.746831	−0.373415	−	0.927664i	$-0.621813\pi$
$457$	−15.9654	−0.746831	−0.373415	+	0.927664i	$0.621813\pi$
$458$	−36.4898	−1.70506
$459$	−6.37375	−0.297501
$460$	5.43574	0.253443
$461$	−0.406983	−0.0189551	−0.00947754	−	0.999955i	$-0.503017\pi$
$461$	−0.406983	−0.0189551	−0.00947754	+	0.999955i	$0.503017\pi$
$462$	5.91850	0.275353
$463$	28.6219	1.33017	0.665087	−	0.746766i	$-0.268395\pi$
$463$	28.6219	1.33017	0.665087	+	0.746766i	$0.268395\pi$
$464$	−19.9367	−0.925538
$465$	−1.25076	−0.0580027
$466$	−44.2477	−2.04974
$467$	−38.5772	−1.78514	−0.892571	−	0.450907i	$-0.851100\pi$
$467$	−38.5772	−1.78514	−0.892571	+	0.450907i	$0.851100\pi$
$468$	−11.3985	−0.526898
$469$	−15.4823	−0.714907
$470$	−25.8674	−1.19318
$471$	4.56505	0.210347
$472$	21.9696	1.01123
$473$	15.5003	0.712705
$474$	8.21591	0.377369
$475$	−3.07792	−0.141225
$476$	26.5230	1.21568
$477$	26.4219	1.20978
$478$	66.0932	3.02303
$479$	−14.1350	−0.645843	−0.322922	−	0.946426i	$-0.604665\pi$
$479$	−14.1350	−0.645843	−0.322922	+	0.946426i	$0.604665\pi$
$480$	−0.750517	−0.0342563
$481$	−4.85960	−0.221579
$482$	−65.0793	−2.96428
$483$	−0.843923	−0.0383998
$484$	27.6521	1.25691
$485$	−5.96756	−0.270973
$486$	−18.3053	−0.830343
$487$	−12.5869	−0.570369	−0.285184	−	0.958473i	$-0.592055\pi$
$487$	−12.5869	−0.570369	−0.285184	+	0.958473i	$0.592055\pi$
$488$	22.9316	1.03806
$489$	−0.705483	−0.0319030
$490$	−7.60066	−0.343363
$491$	−13.4541	−0.607174	−0.303587	−	0.952804i	$-0.598184\pi$
$491$	−13.4541	−0.607174	−0.303587	+	0.952804i	$0.598184\pi$
$492$	0.860504	0.0387945
$493$	−30.8437	−1.38913
$494$	7.75262	0.348807
$495$	−12.5112	−0.562337
$496$	10.0165	0.449752
$497$	19.6665	0.882163
$498$	−8.83617	−0.395958
$499$	17.5343	0.784945	0.392472	−	0.919764i	$-0.371620\pi$
$499$	17.5343	0.784945	0.392472	+	0.919764i	$0.371620\pi$
$500$	−3.71333	−0.166065
$501$	0.190211	0.00849800
$502$	−30.9884	−1.38308
$503$	−32.4398	−1.44642	−0.723209	−	0.690629i	$-0.757334\pi$
$503$	−32.4398	−1.44642	−0.723209	+	0.690629i	$0.757334\pi$
$504$	23.3166	1.03860
$505$	−6.15863	−0.274056
$506$	−15.0279	−0.668073
$507$	−3.50698	−0.155751
$508$	−16.8959	−0.749632
$509$	37.9507	1.68214	0.841068	−	0.540929i	$-0.181927\pi$
$509$	37.9507	1.68214	0.841068	+	0.540929i	$0.181927\pi$
$510$	2.57651	0.114090
$511$	4.83412	0.213849
$512$	25.3577	1.12066
$513$	5.36825	0.237014
$514$	−45.3079	−1.99845
$515$	−15.0166	−0.661710
$516$	−3.95286	−0.174015
$517$	46.4802	2.04420
$518$	21.5446	0.946617
$519$	−5.66778	−0.248788
$520$	4.31550	0.189247
$521$	−12.0211	−0.526656	−0.263328	−	0.964706i	$-0.584820\pi$
$521$	−12.0211	−0.526656	−0.263328	+	0.964706i	$0.584820\pi$
$522$	−58.7666	−2.57214
$523$	−23.7443	−1.03826	−0.519132	−	0.854694i	$-0.673745\pi$
$523$	−23.7443	−1.03826	−0.519132	+	0.854694i	$0.673745\pi$
$524$	−33.8929	−1.48062
$525$	0.576511	0.0251610
$526$	39.2264	1.71035
$527$	15.4963	0.675028
$528$	−2.99250	−0.130232
$529$	−20.8572	−0.906833
$530$	−21.6805	−0.941740
$531$	15.6270	0.678156
$532$	−22.3389	−0.968513
$533$	0.827882	0.0358596
$534$	0.705037	0.0305099
$535$	−2.68955	−0.116280
$536$	32.4400	1.40119
$537$	−4.42142	−0.190798
$538$	−69.0949	−2.97889
$539$	13.6573	0.588263
$540$	6.47647	0.278703
$541$	3.22247	0.138545	0.0692725	−	0.997598i	$-0.477932\pi$
$541$	3.22247	0.138545	0.0692725	+	0.997598i	$0.477932\pi$
$542$	−15.2874	−0.656651
$543$	6.83477	0.293308
$544$	9.29851	0.398670
$545$	0.671805	0.0287770
$546$	−1.45211	−0.0621444
$547$	−0.363571	−0.0155452	−0.00777259	−	0.999970i	$-0.502474\pi$
$547$	−0.363571	−0.0155452	−0.00777259	+	0.999970i	$0.502474\pi$
$548$	62.5775	2.67318
$549$	16.3113	0.696150
$550$	10.2661	0.437746
$551$	25.9779	1.10669
$552$	1.76827	0.0752624
$553$	−22.7763	−0.968548
$554$	6.95112	0.295325
$555$	1.36026	0.0577398
$556$	11.8902	0.504259
$557$	−10.6808	−0.452561	−0.226280	−	0.974062i	$-0.572657\pi$
$557$	−10.6808	−0.452561	−0.226280	+	0.974062i	$0.572657\pi$
$558$	29.5251	1.24990
$559$	−3.80301	−0.160850
$560$	−4.61687	−0.195098
$561$	−4.62964	−0.195463
$562$	15.0034	0.632882
$563$	−16.3376	−0.688549	−0.344274	−	0.938869i	$-0.611875\pi$
$563$	−16.3376	−0.688549	−0.344274	+	0.938869i	$0.611875\pi$
$564$	−11.8533	−0.499114
$565$	8.72825	0.367200
$566$	−28.3996	−1.19373
$567$	16.0750	0.675088
$568$	−41.2071	−1.72901
$569$	−40.6357	−1.70354	−0.851768	−	0.523919i	$-0.824470\pi$
$569$	−40.6357	−1.70354	−0.851768	+	0.523919i	$0.824470\pi$
$570$	−2.17005	−0.0908934
$571$	−38.8677	−1.62656	−0.813281	−	0.581871i	$-0.802321\pi$
$571$	−38.8677	−1.62656	−0.813281	+	0.581871i	$0.802321\pi$
$572$	−16.8062	−0.702702
$573$	0.115043	0.00480597
$574$	−3.67035	−0.153197
$575$	−1.46385	−0.0610466
$576$	31.4784	1.31160
$577$	−29.1209	−1.21232	−0.606158	−	0.795344i	$-0.707290\pi$
$577$	−29.1209	−1.21232	−0.606158	+	0.795344i	$0.707290\pi$
$578$	8.71277	0.362403
$579$	2.92434	0.121532
$580$	31.3408	1.30135
$581$	24.4958	1.01626
$582$	−4.20735	−0.174400
$583$	38.9568	1.61343
$584$	−10.1289	−0.419137
$585$	3.06963	0.126914
$586$	36.8489	1.52221
$587$	−11.3820	−0.469787	−0.234894	−	0.972021i	$-0.575474\pi$
$587$	−11.3820	−0.469787	−0.234894	+	0.972021i	$0.575474\pi$
$588$	−3.48287	−0.143631
$589$	−13.0516	−0.537783
$590$	−12.8228	−0.527905
$591$	−0.298198	−0.0122662
$592$	−10.8934	−0.447714
$593$	−5.10274	−0.209544	−0.104772	−	0.994496i	$-0.533411\pi$
$593$	−5.10274	−0.209544	−0.104772	+	0.994496i	$0.533411\pi$
$594$	−17.9052	−0.734659
$595$	−7.14266	−0.292821
$596$	54.3809	2.22753
$597$	−3.93787	−0.161166
$598$	3.68711	0.150777
$599$	22.6299	0.924631	0.462316	−	0.886715i	$-0.347019\pi$
$599$	22.6299	0.924631	0.462316	+	0.886715i	$0.347019\pi$
$600$	−1.20796	−0.0493148
$601$	27.2940	1.11335	0.556673	−	0.830732i	$-0.312078\pi$
$601$	27.2940	1.11335	0.556673	+	0.830732i	$0.312078\pi$
$602$	16.8603	0.687176
$603$	23.0747	0.939674
$604$	−66.2497	−2.69566
$605$	−7.44671	−0.302752
$606$	−4.34207	−0.176384
$607$	34.1969	1.38801	0.694004	−	0.719971i	$-0.255845\pi$
$607$	34.1969	1.38801	0.694004	+	0.719971i	$0.255845\pi$
$608$	−7.83161	−0.317614
$609$	−4.86579	−0.197172
$610$	−13.3842	−0.541912
$611$	−11.4039	−0.461354
$612$	−39.5297	−1.59789
$613$	24.2886	0.981009	0.490504	−	0.871439i	$-0.336813\pi$
$613$	24.2886	0.981009	0.490504	+	0.871439i	$0.336813\pi$
$614$	31.4785	1.27037
$615$	−0.231734	−0.00934442
$616$	34.3783	1.38514
$617$	31.9292	1.28542	0.642711	−	0.766109i	$-0.277810\pi$
$617$	31.9292	1.28542	0.642711	+	0.766109i	$0.277810\pi$
$618$	−10.5872	−0.425882
$619$	−29.1111	−1.17007	−0.585037	−	0.811007i	$-0.698920\pi$
$619$	−29.1111	−1.17007	−0.585037	+	0.811007i	$0.698920\pi$
$620$	−15.7460	−0.632375
$621$	2.55311	0.102453
$622$	−32.5351	−1.30454
$623$	−1.95452	−0.0783062
$624$	0.734211	0.0293920
$625$	1.00000	0.0400000
$626$	47.8106	1.91089
$627$	3.89928	0.155722
$628$	57.4701	2.29331
$629$	−16.8529	−0.671969
$630$	−13.6090	−0.542194
$631$	26.7634	1.06543	0.532717	−	0.846293i	$-0.321171\pi$
$631$	26.7634	1.06543	0.532717	+	0.846293i	$0.321171\pi$
$632$	47.7231	1.89832
$633$	−7.51617	−0.298741
$634$	−42.8370	−1.70128
$635$	4.55006	0.180564
$636$	−9.93470	−0.393937
$637$	−3.35084	−0.132765
$638$	−86.6463	−3.43036
$639$	−29.3108	−1.15952
$640$	−20.7406	−0.819846
$641$	−16.4357	−0.649173	−0.324586	−	0.945856i	$-0.605225\pi$
$641$	−16.4357	−0.649173	−0.324586	+	0.945856i	$0.605225\pi$
$642$	−1.89624	−0.0748384
$643$	−28.2642	−1.11463	−0.557316	−	0.830300i	$-0.688169\pi$
$643$	−28.2642	−1.11463	−0.557316	+	0.830300i	$0.688169\pi$
$644$	−10.6243	−0.418654
$645$	1.06451	0.0419149
$646$	26.8858	1.05781
$647$	16.9725	0.667258	0.333629	−	0.942704i	$-0.391727\pi$
$647$	16.9725	0.667258	0.333629	+	0.942704i	$0.391727\pi$
$648$	−33.6819	−1.32315
$649$	23.0407	0.904428
$650$	−2.51878	−0.0987948
$651$	2.44464	0.0958129
$652$	−8.88142	−0.347823
$653$	−39.2631	−1.53648	−0.768242	−	0.640160i	$-0.778868\pi$
$653$	−39.2631	−1.53648	−0.768242	+	0.640160i	$0.778868\pi$
$654$	0.473647	0.0185211
$655$	9.12736	0.356635
$656$	1.85579	0.0724566
$657$	−7.20472	−0.281083
$658$	50.5584	1.97097
$659$	3.77282	0.146968	0.0734841	−	0.997296i	$-0.476588\pi$
$659$	3.77282	0.146968	0.0734841	+	0.997296i	$0.476588\pi$
$660$	4.70425	0.183113
$661$	1.17732	0.0457923	0.0228962	−	0.999738i	$-0.492711\pi$
$661$	1.17732	0.0457923	0.0228962	+	0.999738i	$0.492711\pi$
$662$	48.4375	1.88258
$663$	1.13588	0.0441140
$664$	−51.3260	−1.99183
$665$	6.01586	0.233285
$666$	−32.1099	−1.24423
$667$	12.3550	0.478386
$668$	2.39459	0.0926495
$669$	6.00121	0.232020
$670$	−18.9339	−0.731481
$671$	24.0496	0.928425
$672$	1.46690	0.0565869
$673$	45.6233	1.75865	0.879324	−	0.476224i	$-0.157995\pi$
$673$	45.6233	1.75865	0.879324	+	0.476224i	$0.157995\pi$
$674$	38.5768	1.48592
$675$	−1.74411	−0.0671310
$676$	−44.1499	−1.69807
$677$	−43.1019	−1.65654	−0.828270	−	0.560329i	$-0.810675\pi$
$677$	−43.1019	−1.65654	−0.828270	+	0.560329i	$0.810675\pi$
$678$	6.15374	0.236333
$679$	11.6637	0.447612
$680$	14.9660	0.573919
$681$	4.73725	0.181532
$682$	43.5322	1.66694
$683$	24.1509	0.924108	0.462054	−	0.886852i	$-0.347113\pi$
$683$	24.1509	0.924108	0.462054	+	0.886852i	$0.347113\pi$
$684$	33.2936	1.27301
$685$	−16.8521	−0.643887
$686$	47.5583	1.81578
$687$	4.50292	0.171797
$688$	−8.52488	−0.325008
$689$	−9.55807	−0.364134
$690$	−1.03207	−0.0392900
$691$	−31.4456	−1.19625	−0.598124	−	0.801404i	$-0.704087\pi$
$691$	−31.4456	−1.19625	−0.598124	+	0.801404i	$0.704087\pi$
$692$	−71.3524	−2.71241
$693$	24.4534	0.928909
$694$	−34.2753	−1.30107
$695$	−3.20205	−0.121461
$696$	10.1953	0.386451
$697$	2.87106	0.108749
$698$	76.2381	2.88566
$699$	5.46027	0.206526
$700$	7.25777	0.274318
$701$	−45.2299	−1.70831	−0.854155	−	0.520019i	$-0.825925\pi$
$701$	−45.2299	−1.70831	−0.854155	+	0.520019i	$0.825925\pi$
$702$	4.39305	0.165805
$703$	14.1942	0.535346
$704$	46.4121	1.74922
$705$	3.19210	0.120221
$706$	−5.88122	−0.221343
$707$	12.0372	0.452705
$708$	−5.87581	−0.220826
$709$	11.5007	0.431919	0.215960	−	0.976402i	$-0.430712\pi$
$709$	11.5007	0.431919	0.215960	+	0.976402i	$0.430712\pi$
$710$	24.0509	0.902615
$711$	33.9456	1.27306
$712$	4.09529	0.153478
$713$	−6.20729	−0.232465
$714$	−5.03584	−0.188462
$715$	4.52591	0.169259
$716$	−55.6619	−2.08018
$717$	−8.15604	−0.304593
$718$	−11.4559	−0.427532
$719$	−51.2805	−1.91244	−0.956220	−	0.292648i	$-0.905463\pi$
$719$	−51.2805	−1.91244	−0.956220	+	0.292648i	$0.905463\pi$
$720$	6.88093	0.256437
$721$	29.3502	1.09306
$722$	22.7705	0.847431
$723$	8.03093	0.298674
$724$	86.0439	3.19780
$725$	−8.44007	−0.313456
$726$	−5.25020	−0.194853
$727$	36.9416	1.37009	0.685044	−	0.728502i	$-0.259783\pi$
$727$	36.9416	1.37009	0.685044	+	0.728502i	$0.259783\pi$
$728$	−8.43473	−0.312612
$729$	−22.4147	−0.830175
$730$	5.91183	0.218807
$731$	−13.1887	−0.487801
$732$	−6.13309	−0.226686
$733$	14.0425	0.518673	0.259337	−	0.965787i	$-0.416496\pi$
$733$	14.0425	0.518673	0.259337	+	0.965787i	$0.416496\pi$
$734$	55.8476	2.06137
$735$	0.937938	0.0345964
$736$	−3.72467	−0.137293
$737$	34.0216	1.25320
$738$	5.47025	0.201363
$739$	−18.4116	−0.677282	−0.338641	−	0.940916i	$-0.609967\pi$
$739$	−18.4116	−0.677282	−0.338641	+	0.940916i	$0.609967\pi$
$740$	17.1245	0.629509
$741$	−0.956690	−0.0351449
$742$	42.3749	1.55563
$743$	−20.0671	−0.736191	−0.368096	−	0.929788i	$-0.619990\pi$
$743$	−20.0671	−0.736191	−0.368096	+	0.929788i	$0.619990\pi$
$744$	−5.12224	−0.187790
$745$	−14.6448	−0.536544
$746$	−56.7685	−2.07844
$747$	−36.5084	−1.33577
$748$	−58.2831	−2.13104
$749$	5.25679	0.192079
$750$	0.705037	0.0257443
$751$	−17.5280	−0.639607	−0.319803	−	0.947484i	$-0.603617\pi$
$751$	−17.5280	−0.639607	−0.319803	+	0.947484i	$0.603617\pi$
$752$	−25.5633	−0.932196
$753$	3.82404	0.139356
$754$	21.2587	0.774196
$755$	17.8411	0.649303
$756$	−12.6584	−0.460381
$757$	21.0316	0.764406	0.382203	−	0.924078i	$-0.375166\pi$
$757$	21.0316	0.764406	0.382203	+	0.924078i	$0.375166\pi$
$758$	66.2103	2.40487
$759$	1.85448	0.0673133
$760$	−12.6050	−0.457231
$761$	−37.2035	−1.34863	−0.674313	−	0.738445i	$-0.735560\pi$
$761$	−37.2035	−1.34863	−0.674313	+	0.738445i	$0.735560\pi$
$762$	3.20796	0.116212
$763$	−1.31306	−0.0475358
$764$	1.44829	0.0523972
$765$	10.6454	0.384884
$766$	−14.4258	−0.521227
$767$	−5.65305	−0.204120
$768$	−8.24808	−0.297627
$769$	−15.2246	−0.549013	−0.274507	−	0.961585i	$-0.588515\pi$
$769$	−15.2246	−0.549013	−0.274507	+	0.961585i	$0.588515\pi$
$770$	−20.0652	−0.723101
$771$	5.59109	0.201358
$772$	36.8150	1.32500
$773$	26.7520	0.962202	0.481101	−	0.876665i	$-0.340237\pi$
$773$	26.7520	0.962202	0.481101	+	0.876665i	$0.340237\pi$
$774$	−25.1285	−0.903224
$775$	4.24040	0.152320
$776$	−24.4389	−0.877306
$777$	−2.65866	−0.0953787
$778$	−20.3896	−0.731004
$779$	−2.41813	−0.0866386
$780$	−1.15419	−0.0413266
$781$	−43.2162	−1.54640
$782$	12.7867	0.457253
$783$	14.7205	0.526066
$784$	−7.51128	−0.268260
$785$	−15.4767	−0.552387
$786$	6.43513	0.229533
$787$	28.4808	1.01523	0.507615	−	0.861584i	$-0.330527\pi$
$787$	28.4808	1.01523	0.507615	+	0.861584i	$0.330527\pi$
$788$	−3.75406	−0.133733
$789$	−4.84062	−0.172331
$790$	−27.8540	−0.991002
$791$	−17.0595	−0.606568
$792$	−51.2371	−1.82063
$793$	−5.90058	−0.209536
$794$	34.9367	1.23986
$795$	2.67542	0.0948873
$796$	−49.5744	−1.75712
$797$	44.0413	1.56002	0.780012	−	0.625764i	$-0.215213\pi$
$797$	44.0413	1.56002	0.780012	+	0.625764i	$0.215213\pi$
$798$	4.24140	0.150144
$799$	−39.5484	−1.39912
$800$	2.54445	0.0899597
$801$	2.91300	0.102926
$802$	−41.3484	−1.46006
$803$	−10.6227	−0.374869
$804$	−8.67614	−0.305984
$805$	2.86111	0.100841
$806$	−10.6807	−0.376210
$807$	8.52646	0.300146
$808$	−25.2214	−0.887286
$809$	−30.2316	−1.06289	−0.531444	−	0.847094i	$-0.678350\pi$
$809$	−30.2316	−1.06289	−0.531444	+	0.847094i	$0.678350\pi$
$810$	19.6588	0.690739
$811$	−34.5210	−1.21220	−0.606099	−	0.795389i	$-0.707266\pi$
$811$	−34.5210	−1.21220	−0.606099	+	0.795389i	$0.707266\pi$
$812$	−61.2561	−2.14967
$813$	1.88650	0.0661625
$814$	−47.3433	−1.65938
$815$	2.39177	0.0837800
$816$	2.54621	0.0891353
$817$	11.1081	0.388622
$818$	6.65121	0.232554
$819$	−5.99966	−0.209645
$820$	−2.91733	−0.101878
$821$	35.2307	1.22956	0.614780	−	0.788698i	$-0.289245\pi$
$821$	35.2307	1.22956	0.614780	+	0.788698i	$0.289245\pi$
$822$	−11.8814	−0.414411
$823$	−40.5379	−1.41306	−0.706532	−	0.707681i	$-0.749741\pi$
$823$	−40.5379	−1.41306	−0.706532	+	0.707681i	$0.749741\pi$
$824$	−61.4973	−2.14236
$825$	−1.26685	−0.0441062
$826$	25.0623	0.872030
$827$	−30.0036	−1.04333	−0.521664	−	0.853151i	$-0.674688\pi$
$827$	−30.0036	−1.04333	−0.521664	+	0.853151i	$0.674688\pi$
$828$	15.8343	0.550279
$829$	11.3949	0.395761	0.197881	−	0.980226i	$-0.436594\pi$
$829$	11.3949	0.395761	0.197881	+	0.980226i	$0.436594\pi$
$830$	29.9569	1.03982
$831$	−0.857783	−0.0297562
$832$	−11.3872	−0.394781
$833$	−11.6206	−0.402628
$834$	−2.25756	−0.0781729
$835$	−0.644864	−0.0223164
$836$	49.0886	1.69776
$837$	−7.39575	−0.255634
$838$	10.0556	0.347365
$839$	38.3055	1.32245	0.661226	−	0.750187i	$-0.270037\pi$
$839$	38.3055	1.32245	0.661226	+	0.750187i	$0.270037\pi$
$840$	2.36098	0.0814616
$841$	42.2348	1.45637
$842$	−45.5980	−1.57141
$843$	−1.85146	−0.0637675
$844$	−94.6221	−3.25703
$845$	11.8896	0.409014
$846$	−75.3518	−2.59065
$847$	14.5547	0.500107
$848$	−21.4255	−0.735755
$849$	3.50458	0.120277
$850$	−8.73503	−0.299609
$851$	6.75071	0.231412
$852$	11.0209	0.377570
$853$	−5.70914	−0.195477	−0.0977386	−	0.995212i	$-0.531161\pi$
$853$	−5.70914	−0.195477	−0.0977386	+	0.995212i	$0.531161\pi$
$854$	26.1597	0.895168
$855$	−8.96598	−0.306630
$856$	−11.0145	−0.376468
$857$	−18.5138	−0.632419	−0.316210	−	0.948689i	$-0.602410\pi$
$857$	−18.5138	−0.632419	−0.316210	+	0.948689i	$0.602410\pi$
$858$	3.19093	0.108937
$859$	36.1682	1.23404	0.617021	−	0.786947i	$-0.288339\pi$
$859$	36.1682	1.23404	0.617021	+	0.786947i	$0.288339\pi$
$860$	13.4012	0.456978
$861$	0.452929	0.0154358
$862$	9.83952	0.335135
$863$	−48.1408	−1.63873	−0.819366	−	0.573271i	$-0.805674\pi$
$863$	−48.1408	−1.63873	−0.819366	+	0.573271i	$0.805674\pi$
$864$	−4.43781	−0.150977
$865$	19.2152	0.653337
$866$	82.2842	2.79613
$867$	−1.07517	−0.0365149
$868$	30.7759	1.04460
$869$	50.0499	1.69783
$870$	−5.95056	−0.201743
$871$	−8.34722	−0.282835
$872$	2.75124	0.0931687
$873$	−17.3835	−0.588342
$874$	−10.7695	−0.364285
$875$	−1.95452	−0.0660748
$876$	2.70899	0.0915284
$877$	16.7949	0.567122	0.283561	−	0.958954i	$-0.408484\pi$
$877$	16.7949	0.567122	0.283561	+	0.958954i	$0.408484\pi$
$878$	−36.3038	−1.22519
$879$	−4.54724	−0.153374
$880$	10.1453	0.341999
$881$	10.4309	0.351425	0.175713	−	0.984442i	$-0.443777\pi$
$881$	10.4309	0.351425	0.175713	+	0.984442i	$0.443777\pi$
$882$	−22.1407	−0.745516
$883$	29.5575	0.994690	0.497345	−	0.867553i	$-0.334308\pi$
$883$	29.5575	0.994690	0.497345	+	0.867553i	$0.334308\pi$
$884$	14.2998	0.480954
$885$	1.58236	0.0531903
$886$	−5.13313	−0.172451
$887$	30.1297	1.01166	0.505828	−	0.862634i	$-0.331187\pi$
$887$	30.1297	1.01166	0.505828	+	0.862634i	$0.331187\pi$
$888$	5.57067	0.186939
$889$	−8.89318	−0.298268
$890$	−2.39026	−0.0801216
$891$	−35.3241	−1.18340
$892$	75.5500	2.52960
$893$	33.3094	1.11466
$894$	−10.3251	−0.345324
$895$	14.9898	0.501052
$896$	40.5380	1.35428
$897$	−0.454997	−0.0151919
$898$	−77.7150	−2.59338
$899$	−35.7893	−1.19364
$900$	−10.8169	−0.360564
$901$	−33.1470	−1.10429
$902$	8.06541	0.268549
$903$	−2.08060	−0.0692381
$904$	35.7448	1.18885
$905$	−23.1716	−0.770251
$906$	12.5786	0.417896
$907$	34.4041	1.14237	0.571185	−	0.820821i	$-0.306484\pi$
$907$	34.4041	1.14237	0.571185	+	0.820821i	$0.306484\pi$
$908$	59.6378	1.97915
$909$	−17.9401	−0.595035
$910$	4.92301	0.163196
$911$	−48.0626	−1.59239	−0.796193	−	0.605043i	$-0.793156\pi$
$911$	−48.0626	−1.59239	−0.796193	+	0.605043i	$0.793156\pi$
$912$	−2.14453	−0.0710125
$913$	−53.8284	−1.78146
$914$	38.1615	1.26227
$915$	1.65164	0.0546016
$916$	56.6879	1.87302
$917$	−17.8396	−0.589116
$918$	15.2349	0.502827
$919$	9.64985	0.318319	0.159160	−	0.987253i	$-0.449122\pi$
$919$	9.64985	0.318319	0.159160	+	0.987253i	$0.449122\pi$
$920$	−5.99488	−0.197645
$921$	−3.88451	−0.127999
$922$	0.972793	0.0320372
$923$	10.6031	0.349006
$924$	−9.19454	−0.302478
$925$	−4.61163	−0.151630
$926$	−68.4137	−2.24822
$927$	−43.7433	−1.43672
$928$	−21.4753	−0.704961
$929$	−4.87453	−0.159928	−0.0799641	−	0.996798i	$-0.525481\pi$
$929$	−4.87453	−0.159928	−0.0799641	+	0.996798i	$0.525481\pi$
$930$	2.98964	0.0980342
$931$	9.78734	0.320767
$932$	68.7400	2.25165
$933$	4.01491	0.131442
$934$	92.2095	3.01719
$935$	15.6957	0.513303
$936$	12.5710	0.410897
$937$	6.79035	0.221831	0.110916	−	0.993830i	$-0.464622\pi$
$937$	6.79035	0.221831	0.110916	+	0.993830i	$0.464622\pi$
$938$	37.0067	1.20831
$939$	−5.89993	−0.192537
$940$	40.1858	1.31072
$941$	−26.9378	−0.878147	−0.439074	−	0.898451i	$-0.644693\pi$
$941$	−26.9378	−0.878147	−0.439074	+	0.898451i	$0.644693\pi$
$942$	−10.9116	−0.355521
$943$	−1.15005	−0.0374508
$944$	−12.6720	−0.412437
$945$	3.40891	0.110892
$946$	−37.0497	−1.20459
$947$	0.0394219	0.00128104	0.000640519	−	1.00000i	$-0.499796\pi$
$947$	0.0394219	0.00128104	0.000640519		1.00000i	$0.499796\pi$
$948$	−12.7636	−0.414543
$949$	2.60629	0.0846039
$950$	7.35702	0.238693
$951$	5.28618	0.171416
$952$	−29.2513	−0.948040
$953$	−3.27013	−0.105930	−0.0529650	−	0.998596i	$-0.516867\pi$
$953$	−3.27013	−0.105930	−0.0529650	+	0.998596i	$0.516867\pi$
$954$	−63.1551	−2.04472
$955$	−0.390024	−0.0126209
$956$	−102.678	−3.32083
$957$	10.6923	0.345634
$958$	33.7862	1.09158
$959$	32.9379	1.06362
$960$	3.18742	0.102874
$961$	−13.0190	−0.419967
$962$	11.6157	0.374505
$963$	−7.83466	−0.252468
$964$	101.102	3.25629
$965$	−9.91428	−0.319152
$966$	2.01719	0.0649021
$967$	47.5329	1.52855	0.764277	−	0.644888i	$-0.223096\pi$
$967$	47.5329	1.52855	0.764277	+	0.644888i	$0.223096\pi$
$968$	−30.4965	−0.980193
$969$	−3.31776	−0.106582
$970$	14.2640	0.457989
$971$	−25.1811	−0.808101	−0.404050	−	0.914737i	$-0.632398\pi$
$971$	−25.1811	−0.808101	−0.404050	+	0.914737i	$0.632398\pi$
$972$	28.4377	0.912140
$973$	6.25846	0.200637
$974$	30.0860	0.964019
$975$	0.310823	0.00995431
$976$	−13.2268	−0.423381
$977$	−33.1228	−1.05969	−0.529846	−	0.848094i	$-0.677750\pi$
$977$	−33.1228	−1.05969	−0.529846	+	0.848094i	$0.677750\pi$
$978$	1.68629	0.0539215
$979$	4.29496	0.137268
$980$	11.8078	0.377187
$981$	1.95697	0.0624811
$982$	32.1587	1.02623
$983$	−22.3428	−0.712625	−0.356313	−	0.934367i	$-0.615966\pi$
$983$	−22.3428	−0.712625	−0.356313	+	0.934367i	$0.615966\pi$
$984$	−0.949019	−0.0302536
$985$	1.01097	0.0322122
$986$	73.7243	2.34786
$987$	−6.23902	−0.198590
$988$	−12.0439	−0.383167
$989$	5.28295	0.167988
$990$	29.9050	0.950444
$991$	37.5767	1.19366	0.596832	−	0.802366i	$-0.296426\pi$
$991$	37.5767	1.19366	0.596832	+	0.802366i	$0.296426\pi$
$992$	10.7895	0.342566
$993$	−5.97729	−0.189684
$994$	−47.0080	−1.49100
$995$	13.3504	0.423236
$996$	13.7272	0.434964
$997$	21.4843	0.680416	0.340208	−	0.940350i	$-0.389502\pi$
$997$	21.4843	0.680416	0.340208	+	0.940350i	$0.389502\pi$
$998$	−41.9116	−1.32669
$999$	8.04321	0.254476

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	445.2.a.d.1.1	✓	4
3.2	odd	2		4005.2.a.l.1.4		4
4.3	odd	2		7120.2.a.bc.1.2		4
5.4	even	2		2225.2.a.i.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.d.1.1	✓	4	1.1	even	1	trivial
2225.2.a.i.1.4		4	5.4	even	2
4005.2.a.l.1.4		4	3.2	odd	2
7120.2.a.bc.1.2		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 445 = 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	445.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39026 q^{2} +0.294963 q^{3} +3.71333 q^{4} -1.00000 q^{5} -0.705037 q^{6} +1.95452 q^{7} -4.09529 q^{8} -2.91300 q^{9} +O(q^{10})\)	\(q-2.39026 q^{2} +0.294963 q^{3} +3.71333 q^{4} -1.00000 q^{5} -0.705037 q^{6} +1.95452 q^{7} -4.09529 q^{8} -2.91300 q^{9} +2.39026 q^{10} -4.29496 q^{11} +1.09529 q^{12} +1.05377 q^{13} -4.67180 q^{14} -0.294963 q^{15} +2.36215 q^{16} +3.65443 q^{17} +6.96281 q^{18} -3.07792 q^{19} -3.71333 q^{20} +0.576511 q^{21} +10.2661 q^{22} -1.46385 q^{23} -1.20796 q^{24} +1.00000 q^{25} -2.51878 q^{26} -1.74411 q^{27} +7.25777 q^{28} -8.44007 q^{29} +0.705037 q^{30} +4.24040 q^{31} +2.54445 q^{32} -1.26685 q^{33} -8.73503 q^{34} -1.95452 q^{35} -10.8169 q^{36} -4.61163 q^{37} +7.35702 q^{38} +0.310823 q^{39} +4.09529 q^{40} +0.785638 q^{41} -1.37801 q^{42} -3.60895 q^{43} -15.9486 q^{44} +2.91300 q^{45} +3.49897 q^{46} -10.8220 q^{47} +0.696746 q^{48} -3.17985 q^{49} -2.39026 q^{50} +1.07792 q^{51} +3.91300 q^{52} -9.07035 q^{53} +4.16888 q^{54} +4.29496 q^{55} -8.00433 q^{56} -0.907873 q^{57} +20.1739 q^{58} -5.36459 q^{59} -1.09529 q^{60} -5.59950 q^{61} -10.1356 q^{62} -5.69351 q^{63} -10.8062 q^{64} -1.05377 q^{65} +3.02811 q^{66} -7.92129 q^{67} +13.5701 q^{68} -0.431780 q^{69} +4.67180 q^{70} +10.0621 q^{71} +11.9296 q^{72} +2.47330 q^{73} +11.0230 q^{74} +0.294963 q^{75} -11.4293 q^{76} -8.39459 q^{77} -0.742948 q^{78} -11.6532 q^{79} -2.36215 q^{80} +8.22454 q^{81} -1.87788 q^{82} +12.5329 q^{83} +2.14077 q^{84} -3.65443 q^{85} +8.62632 q^{86} -2.48951 q^{87} +17.5891 q^{88} -1.00000 q^{89} -6.96281 q^{90} +2.05962 q^{91} -5.43574 q^{92} +1.25076 q^{93} +25.8674 q^{94} +3.07792 q^{95} +0.750517 q^{96} +5.96756 q^{97} +7.60066 q^{98} +12.5112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} - 6 q^{6} + 2 q^{7} - 9 q^{8} - 4 q^{9}+O(q^{10}) \) 4 * q + q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 - 6 * q^6 + 2 * q^7 - 9 * q^8 - 4 * q^9	\( 4 q + q^{2} - 2 q^{3} + 3 q^{4} - 4 q^{5} - 6 q^{6} + 2 q^{7} - 9 q^{8} - 4 q^{9} - q^{10} - 14 q^{11} - 3 q^{12} - 5 q^{13} - 5 q^{14} + 2 q^{15} - 3 q^{16} - 3 q^{17} + 7 q^{18} - q^{19} - 3 q^{20} - 4 q^{21} + 2 q^{22} - 3 q^{23} + 6 q^{24} + 4 q^{25} - 9 q^{26} + q^{27} + 5 q^{28} - 10 q^{29} + 6 q^{30} - 11 q^{31} - 2 q^{32} - 8 q^{34} - 2 q^{35} - 9 q^{36} + 3 q^{37} + 2 q^{38} + 11 q^{39} + 9 q^{40} - 3 q^{41} - 6 q^{42} + 9 q^{43} - 9 q^{44} + 4 q^{45} - 4 q^{46} - 24 q^{47} + 21 q^{48} + q^{50} - 7 q^{51} + 8 q^{52} + 3 q^{53} + 17 q^{54} + 14 q^{55} - 13 q^{56} + 19 q^{57} + 19 q^{58} - 22 q^{59} + 3 q^{60} - 3 q^{61} - 24 q^{62} + 6 q^{63} - 11 q^{64} + 5 q^{65} + 14 q^{66} - 9 q^{67} + 31 q^{68} + 7 q^{69} + 5 q^{70} + 16 q^{71} + 10 q^{72} + 3 q^{73} + 31 q^{74} - 2 q^{75} - 24 q^{76} - 4 q^{77} + 16 q^{78} - 27 q^{79} + 3 q^{80} - 8 q^{81} - 15 q^{82} + 6 q^{83} + 7 q^{84} + 3 q^{85} + 15 q^{86} + 4 q^{87} + 30 q^{88} - 4 q^{89} - 7 q^{90} - 29 q^{91} - 17 q^{92} - 2 q^{93} + 13 q^{94} + q^{95} + 12 q^{96} + 41 q^{97} + 22 q^{98} + 21 q^{99}+O(q^{100}) \) 4 * q + q^2 - 2 * q^3 + 3 * q^4 - 4 * q^5 - 6 * q^6 + 2 * q^7 - 9 * q^8 - 4 * q^9 - q^10 - 14 * q^11 - 3 * q^12 - 5 * q^13 - 5 * q^14 + 2 * q^15 - 3 * q^16 - 3 * q^17 + 7 * q^18 - q^19 - 3 * q^20 - 4 * q^21 + 2 * q^22 - 3 * q^23 + 6 * q^24 + 4 * q^25 - 9 * q^26 + q^27 + 5 * q^28 - 10 * q^29 + 6 * q^30 - 11 * q^31 - 2 * q^32 - 8 * q^34 - 2 * q^35 - 9 * q^36 + 3 * q^37 + 2 * q^38 + 11 * q^39 + 9 * q^40 - 3 * q^41 - 6 * q^42 + 9 * q^43 - 9 * q^44 + 4 * q^45 - 4 * q^46 - 24 * q^47 + 21 * q^48 + q^50 - 7 * q^51 + 8 * q^52 + 3 * q^53 + 17 * q^54 + 14 * q^55 - 13 * q^56 + 19 * q^57 + 19 * q^58 - 22 * q^59 + 3 * q^60 - 3 * q^61 - 24 * q^62 + 6 * q^63 - 11 * q^64 + 5 * q^65 + 14 * q^66 - 9 * q^67 + 31 * q^68 + 7 * q^69 + 5 * q^70 + 16 * q^71 + 10 * q^72 + 3 * q^73 + 31 * q^74 - 2 * q^75 - 24 * q^76 - 4 * q^77 + 16 * q^78 - 27 * q^79 + 3 * q^80 - 8 * q^81 - 15 * q^82 + 6 * q^83 + 7 * q^84 + 3 * q^85 + 15 * q^86 + 4 * q^87 + 30 * q^88 - 4 * q^89 - 7 * q^90 - 29 * q^91 - 17 * q^92 - 2 * q^93 + 13 * q^94 + q^95 + 12 * q^96 + 41 * q^97 + 22 * q^98 + 21 * q^99

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