LMFDB - Embedded newform 4009.2.a.e.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.93317 q^{2} -1.52394 q^{3} +1.73714 q^{4} -2.99074 q^{5} +2.94604 q^{6} -0.667472 q^{7} +0.508147 q^{8} -0.677595 q^{9} +O(q^{10})$	\(q-1.93317 q^{2} -1.52394 q^{3} +1.73714 q^{4} -2.99074 q^{5} +2.94604 q^{6} -0.667472 q^{7} +0.508147 q^{8} -0.677595 q^{9} +5.78161 q^{10} -3.08456 q^{11} -2.64731 q^{12} -1.24975 q^{13} +1.29034 q^{14} +4.55772 q^{15} -4.45662 q^{16} +5.21873 q^{17} +1.30991 q^{18} +1.00000 q^{19} -5.19535 q^{20} +1.01719 q^{21} +5.96299 q^{22} -3.84991 q^{23} -0.774387 q^{24} +3.94453 q^{25} +2.41598 q^{26} +5.60445 q^{27} -1.15949 q^{28} -3.08226 q^{29} -8.81085 q^{30} -4.24675 q^{31} +7.59911 q^{32} +4.70070 q^{33} -10.0887 q^{34} +1.99624 q^{35} -1.17708 q^{36} -5.48384 q^{37} -1.93317 q^{38} +1.90455 q^{39} -1.51973 q^{40} +0.516072 q^{41} -1.96640 q^{42} -8.47193 q^{43} -5.35833 q^{44} +2.02651 q^{45} +7.44253 q^{46} +4.41516 q^{47} +6.79164 q^{48} -6.55448 q^{49} -7.62544 q^{50} -7.95305 q^{51} -2.17099 q^{52} -10.8873 q^{53} -10.8343 q^{54} +9.22513 q^{55} -0.339173 q^{56} -1.52394 q^{57} +5.95854 q^{58} -2.59163 q^{59} +7.91741 q^{60} -2.95475 q^{61} +8.20968 q^{62} +0.452276 q^{63} -5.77712 q^{64} +3.73767 q^{65} -9.08726 q^{66} -1.33132 q^{67} +9.06568 q^{68} +5.86705 q^{69} -3.85906 q^{70} -6.50886 q^{71} -0.344317 q^{72} +3.32910 q^{73} +10.6012 q^{74} -6.01124 q^{75} +1.73714 q^{76} +2.05886 q^{77} -3.68181 q^{78} -13.4558 q^{79} +13.3286 q^{80} -6.50808 q^{81} -0.997655 q^{82} +4.16758 q^{83} +1.76700 q^{84} -15.6079 q^{85} +16.3777 q^{86} +4.69720 q^{87} -1.56741 q^{88} -0.335254 q^{89} -3.91759 q^{90} +0.834172 q^{91} -6.68784 q^{92} +6.47181 q^{93} -8.53524 q^{94} -2.99074 q^{95} -11.5806 q^{96} -16.6971 q^{97} +12.6709 q^{98} +2.09009 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) $ 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * 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 $

$2$	−1.93317	−1.36696	−0.683479	−	0.729971i	$-0.739534\pi$
$2$	−1.93317	−1.36696	−0.683479	+	0.729971i	$0.739534\pi$
$3$	−1.52394	−0.879849	−0.439925	−	0.898035i	$-0.644995\pi$
$3$	−1.52394	−0.879849	−0.439925	+	0.898035i	$0.644995\pi$
$4$	1.73714	0.868572
$5$	−2.99074	−1.33750	−0.668750	−	0.743487i	$-0.733170\pi$
$5$	−2.99074	−1.33750	−0.668750	+	0.743487i	$0.733170\pi$
$6$	2.94604	1.20272
$7$	−0.667472	−0.252281	−0.126140	−	0.992012i	$-0.540259\pi$
$7$	−0.667472	−0.252281	−0.126140	+	0.992012i	$0.540259\pi$
$8$	0.508147	0.179657
$9$	−0.677595	−0.225865
$10$	5.78161	1.82830
$11$	−3.08456	−0.930031	−0.465016	−	0.885302i	$-0.653951\pi$
$11$	−3.08456	−0.930031	−0.465016	+	0.885302i	$0.653951\pi$
$12$	−2.64731	−0.764212
$13$	−1.24975	−0.346618	−0.173309	−	0.984868i	$-0.555446\pi$
$13$	−1.24975	−0.346618	−0.173309	+	0.984868i	$0.555446\pi$
$14$	1.29034	0.344857
$15$	4.55772	1.17680
$16$	−4.45662	−1.11415
$17$	5.21873	1.26573	0.632864	−	0.774263i	$-0.281879\pi$
$17$	5.21873	1.26573	0.632864	+	0.774263i	$0.281879\pi$
$18$	1.30991	0.308748
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−5.19535	−1.16171
$21$	1.01719	0.221969
$22$	5.96299	1.27131
$23$	−3.84991	−0.802762	−0.401381	−	0.915911i	$-0.631470\pi$
$23$	−3.84991	−0.802762	−0.401381	+	0.915911i	$0.631470\pi$
$24$	−0.774387	−0.158071
$25$	3.94453	0.788906
$26$	2.41598	0.473812
$27$	5.60445	1.07858
$28$	−1.15949	−0.219124
$29$	−3.08226	−0.572362	−0.286181	−	0.958176i	$-0.592386\pi$
$29$	−3.08226	−0.572362	−0.286181	+	0.958176i	$0.592386\pi$
$30$	−8.81085	−1.60863
$31$	−4.24675	−0.762739	−0.381369	−	0.924423i	$-0.624547\pi$
$31$	−4.24675	−0.762739	−0.381369	+	0.924423i	$0.624547\pi$
$32$	7.59911	1.34335
$33$	4.70070	0.818287
$34$	−10.0887	−1.73019
$35$	1.99624	0.337425
$36$	−1.17708	−0.196180
$37$	−5.48384	−0.901538	−0.450769	−	0.892641i	$-0.648850\pi$
$37$	−5.48384	−0.901538	−0.450769	+	0.892641i	$0.648850\pi$
$38$	−1.93317	−0.313601
$39$	1.90455	0.304972
$40$	−1.51973	−0.240291
$41$	0.516072	0.0805970	0.0402985	−	0.999188i	$-0.487169\pi$
$41$	0.516072	0.0805970	0.0402985	+	0.999188i	$0.487169\pi$
$42$	−1.96640	−0.303422
$43$	−8.47193	−1.29196	−0.645979	−	0.763355i	$-0.723551\pi$
$43$	−8.47193	−1.29196	−0.645979	+	0.763355i	$0.723551\pi$
$44$	−5.35833	−0.807799
$45$	2.02651	0.302094
$46$	7.44253	1.09734
$47$	4.41516	0.644017	0.322008	−	0.946737i	$-0.395642\pi$
$47$	4.41516	0.644017	0.322008	+	0.946737i	$0.395642\pi$
$48$	6.79164	0.980289
$49$	−6.55448	−0.936354
$50$	−7.62544	−1.07840
$51$	−7.95305	−1.11365
$52$	−2.17099	−0.301063
$53$	−10.8873	−1.49549	−0.747744	−	0.663987i	$-0.768863\pi$
$53$	−10.8873	−1.49549	−0.747744	+	0.663987i	$0.768863\pi$
$54$	−10.8343	−1.47437
$55$	9.22513	1.24392
$56$	−0.339173	−0.0453240
$57$	−1.52394	−0.201851
$58$	5.95854	0.782394
$59$	−2.59163	−0.337402	−0.168701	−	0.985667i	$-0.553957\pi$
$59$	−2.59163	−0.337402	−0.168701	+	0.985667i	$0.553957\pi$
$60$	7.91741	1.02213
$61$	−2.95475	−0.378317	−0.189158	−	0.981947i	$-0.560576\pi$
$61$	−2.95475	−0.378317	−0.189158	+	0.981947i	$0.560576\pi$
$62$	8.20968	1.04263
$63$	0.452276	0.0569814
$64$	−5.77712	−0.722140
$65$	3.73767	0.463601
$66$	−9.08726	−1.11856
$67$	−1.33132	−0.162647	−0.0813235	−	0.996688i	$-0.525915\pi$
$67$	−1.33132	−0.162647	−0.0813235	+	0.996688i	$0.525915\pi$
$68$	9.06568	1.09937
$69$	5.86705	0.706309
$70$	−3.85906	−0.461246
$71$	−6.50886	−0.772459	−0.386230	−	0.922403i	$-0.626223\pi$
$71$	−6.50886	−0.772459	−0.386230	+	0.922403i	$0.626223\pi$
$72$	−0.344317	−0.0405782
$73$	3.32910	0.389642	0.194821	−	0.980839i	$-0.437587\pi$
$73$	3.32910	0.389642	0.194821	+	0.980839i	$0.437587\pi$
$74$	10.6012	1.23236
$75$	−6.01124	−0.694118
$76$	1.73714	0.199264
$77$	2.05886	0.234629
$78$	−3.68181	−0.416883
$79$	−13.4558	−1.51389	−0.756947	−	0.653476i	$-0.773310\pi$
$79$	−13.4558	−1.51389	−0.756947	+	0.653476i	$0.773310\pi$
$80$	13.3286	1.49018
$81$	−6.50808	−0.723120
$82$	−0.997655	−0.110173
$83$	4.16758	0.457451	0.228726	−	0.973491i	$-0.426544\pi$
$83$	4.16758	0.457451	0.228726	+	0.973491i	$0.426544\pi$
$84$	1.76700	0.192796
$85$	−15.6079	−1.69291
$86$	16.3777	1.76605
$87$	4.69720	0.503592
$88$	−1.56741	−0.167087
$89$	−0.335254	−0.0355369	−0.0177684	−	0.999842i	$-0.505656\pi$
$89$	−0.335254	−0.0355369	−0.0177684	+	0.999842i	$0.505656\pi$
$90$	−3.91759	−0.412950
$91$	0.834172	0.0874450
$92$	−6.68784	−0.697256
$93$	6.47181	0.671095
$94$	−8.53524	−0.880343
$95$	−2.99074	−0.306844
$96$	−11.5806	−1.18194
$97$	−16.6971	−1.69534	−0.847668	−	0.530526i	$-0.821994\pi$
$97$	−16.6971	−1.69534	−0.847668	+	0.530526i	$0.821994\pi$
$98$	12.6709	1.27996
$99$	2.09009	0.210061
$100$	6.85221	0.685221
$101$	11.8797	1.18207	0.591037	−	0.806645i	$-0.298719\pi$
$101$	11.8797	1.18207	0.591037	+	0.806645i	$0.298719\pi$
$102$	15.3746	1.52231
$103$	−4.15080	−0.408991	−0.204495	−	0.978868i	$-0.565555\pi$
$103$	−4.15080	−0.408991	−0.204495	+	0.978868i	$0.565555\pi$
$104$	−0.635055	−0.0622723
$105$	−3.04215	−0.296883
$106$	21.0470	2.04427
$107$	1.14702	0.110886	0.0554431	−	0.998462i	$-0.482343\pi$
$107$	1.14702	0.110886	0.0554431	+	0.998462i	$0.482343\pi$
$108$	9.73573	0.936821
$109$	−13.7823	−1.32011	−0.660053	−	0.751219i	$-0.729466\pi$
$109$	−13.7823	−1.32011	−0.660053	+	0.751219i	$0.729466\pi$
$110$	−17.8337	−1.70038
$111$	8.35707	0.793218
$112$	2.97467	0.281080
$113$	−1.98141	−0.186396	−0.0931979	−	0.995648i	$-0.529709\pi$
$113$	−1.98141	−0.186396	−0.0931979	+	0.995648i	$0.529709\pi$
$114$	2.94604	0.275922
$115$	11.5141	1.07369
$116$	−5.35433	−0.497137
$117$	0.846823	0.0782888
$118$	5.01006	0.461214
$119$	−3.48335	−0.319318
$120$	2.31599	0.211420
$121$	−1.48546	−0.135042
$122$	5.71203	0.517143
$123$	−0.786465	−0.0709132
$124$	−7.37721	−0.662493
$125$	3.15664	0.282338
$126$	−0.874325	−0.0778911
$127$	−9.98894	−0.886376	−0.443188	−	0.896429i	$-0.646153\pi$
$127$	−9.98894	−0.886376	−0.443188	+	0.896429i	$0.646153\pi$
$128$	−4.03006	−0.356211
$129$	12.9107	1.13673
$130$	−7.22556	−0.633723
$131$	−17.1107	−1.49497	−0.747486	−	0.664278i	$-0.768739\pi$
$131$	−17.1107	−1.49497	−0.747486	+	0.664278i	$0.768739\pi$
$132$	8.16580	0.710741
$133$	−0.667472	−0.0578771
$134$	2.57367	0.222331
$135$	−16.7615	−1.44260
$136$	2.65188	0.227397
$137$	−19.6797	−1.68135	−0.840674	−	0.541541i	$-0.817841\pi$
$137$	−19.6797	−1.68135	−0.840674	+	0.541541i	$0.817841\pi$
$138$	−11.3420	−0.965494
$139$	−2.40956	−0.204376	−0.102188	−	0.994765i	$-0.532584\pi$
$139$	−2.40956	−0.204376	−0.102188	+	0.994765i	$0.532584\pi$
$140$	3.46775	0.293078
$141$	−6.72845	−0.566638
$142$	12.5827	1.05592
$143$	3.85493	0.322366
$144$	3.01978	0.251649
$145$	9.21825	0.765534
$146$	−6.43571	−0.532623
$147$	9.98866	0.823851
$148$	−9.52622	−0.783050
$149$	−21.7616	−1.78278	−0.891391	−	0.453236i	$-0.850270\pi$
$149$	−21.7616	−1.78278	−0.891391	+	0.453236i	$0.850270\pi$
$150$	11.6207	0.948830
$151$	14.5908	1.18738	0.593692	−	0.804693i	$-0.297670\pi$
$151$	14.5908	1.18738	0.593692	+	0.804693i	$0.297670\pi$
$152$	0.508147	0.0412161
$153$	−3.53618	−0.285883
$154$	−3.98012	−0.320728
$155$	12.7009	1.02016
$156$	3.30847	0.264890
$157$	−6.02143	−0.480563	−0.240281	−	0.970703i	$-0.577240\pi$
$157$	−6.02143	−0.480563	−0.240281	+	0.970703i	$0.577240\pi$
$158$	26.0123	2.06943
$159$	16.5917	1.31580
$160$	−22.7270	−1.79672
$161$	2.56971	0.202521
$162$	12.5812	0.988474
$163$	−24.6376	−1.92977	−0.964885	−	0.262674i	$-0.915396\pi$
$163$	−24.6376	−1.92977	−0.964885	+	0.262674i	$0.915396\pi$
$164$	0.896492	0.0700042
$165$	−14.0586	−1.09446
$166$	−8.05663	−0.625316
$167$	−0.955624	−0.0739484	−0.0369742	−	0.999316i	$-0.511772\pi$
$167$	−0.955624	−0.0739484	−0.0369742	+	0.999316i	$0.511772\pi$
$168$	0.516881	0.0398783
$169$	−11.4381	−0.879856
$170$	30.1726	2.31414
$171$	−0.677595	−0.0518170
$172$	−14.7170	−1.12216
$173$	11.5780	0.880257	0.440128	−	0.897935i	$-0.354933\pi$
$173$	11.5780	0.880257	0.440128	+	0.897935i	$0.354933\pi$
$174$	−9.08048	−0.688389
$175$	−2.63286	−0.199026
$176$	13.7467	1.03620
$177$	3.94950	0.296863
$178$	0.648103	0.0485774
$179$	−1.51008	−0.112869	−0.0564343	−	0.998406i	$-0.517973\pi$
$179$	−1.51008	−0.112869	−0.0564343	+	0.998406i	$0.517973\pi$
$180$	3.52034	0.262391
$181$	19.0238	1.41403	0.707016	−	0.707198i	$-0.250041\pi$
$181$	19.0238	1.41403	0.707016	+	0.707198i	$0.250041\pi$
$182$	−1.61260	−0.119534
$183$	4.50287	0.332862
$184$	−1.95632	−0.144222
$185$	16.4007	1.20581
$186$	−12.5111	−0.917358
$187$	−16.0975	−1.17717
$188$	7.66976	0.559375
$189$	−3.74081	−0.272104
$190$	5.78161	0.419442
$191$	13.4032	0.969823	0.484911	−	0.874563i	$-0.338852\pi$
$191$	13.4032	0.969823	0.484911	+	0.874563i	$0.338852\pi$
$192$	8.80401	0.635375
$193$	−3.64715	−0.262527	−0.131264	−	0.991347i	$-0.541903\pi$
$193$	−3.64715	−0.262527	−0.131264	+	0.991347i	$0.541903\pi$
$194$	32.2784	2.31745
$195$	−5.69601	−0.407899
$196$	−11.3861	−0.813291
$197$	−6.51937	−0.464486	−0.232243	−	0.972658i	$-0.574606\pi$
$197$	−6.51937	−0.464486	−0.232243	+	0.972658i	$0.574606\pi$
$198$	−4.04049	−0.287145
$199$	10.0993	0.715918	0.357959	−	0.933737i	$-0.383473\pi$
$199$	10.0993	0.715918	0.357959	+	0.933737i	$0.383473\pi$
$200$	2.00440	0.141732
$201$	2.02886	0.143105
$202$	−22.9655	−1.61584
$203$	2.05732	0.144396
$204$	−13.8156	−0.967284
$205$	−1.54344	−0.107798
$206$	8.02420	0.559073
$207$	2.60868	0.181316
$208$	5.56966	0.386186
$209$	−3.08456	−0.213364
$210$	5.88099	0.405827
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	−18.9128	−1.29894
$213$	9.91913	0.679648
$214$	−2.21738	−0.151577
$215$	25.3373	1.72799
$216$	2.84788	0.193774
$217$	2.83459	0.192424
$218$	26.6435	1.80453
$219$	−5.07336	−0.342826
$220$	16.0254	1.08043
$221$	−6.52210	−0.438724
$222$	−16.1556	−1.08429
$223$	−9.63658	−0.645313	−0.322656	−	0.946516i	$-0.604576\pi$
$223$	−9.63658	−0.645313	−0.322656	+	0.946516i	$0.604576\pi$
$224$	−5.07219	−0.338900
$225$	−2.67279	−0.178186
$226$	3.83041	0.254795
$227$	20.1051	1.33442	0.667211	−	0.744869i	$-0.267488\pi$
$227$	20.1051	1.33442	0.667211	+	0.744869i	$0.267488\pi$
$228$	−2.64731	−0.175322
$229$	2.95634	0.195360	0.0976802	−	0.995218i	$-0.468858\pi$
$229$	2.95634	0.195360	0.0976802	+	0.995218i	$0.468858\pi$
$230$	−22.2587	−1.46769
$231$	−3.13759	−0.206438
$232$	−1.56624	−0.102829
$233$	−3.17405	−0.207939	−0.103970	−	0.994580i	$-0.533154\pi$
$233$	−3.17405	−0.207939	−0.103970	+	0.994580i	$0.533154\pi$
$234$	−1.63705	−0.107017
$235$	−13.2046	−0.861372
$236$	−4.50204	−0.293058
$237$	20.5059	1.33200
$238$	6.73391	0.436495
$239$	3.62553	0.234516	0.117258	−	0.993101i	$-0.462590\pi$
$239$	3.62553	0.234516	0.117258	+	0.993101i	$0.462590\pi$
$240$	−20.3120	−1.31114
$241$	−8.31799	−0.535809	−0.267904	−	0.963446i	$-0.586331\pi$
$241$	−8.31799	−0.535809	−0.267904	+	0.963446i	$0.586331\pi$
$242$	2.87165	0.184597
$243$	−6.89540	−0.442340
$244$	−5.13282	−0.328595
$245$	19.6028	1.25237
$246$	1.52037	0.0969353
$247$	−1.24975	−0.0795196
$248$	−2.15797	−0.137031
$249$	−6.35116	−0.402488
$250$	−6.10231	−0.385944
$251$	−16.3631	−1.03283	−0.516414	−	0.856339i	$-0.672733\pi$
$251$	−16.3631	−1.03283	−0.516414	+	0.856339i	$0.672733\pi$
$252$	0.785667	0.0494924
$253$	11.8753	0.746593
$254$	19.3103	1.21164
$255$	23.7855	1.48951
$256$	19.3450	1.20906
$257$	−14.3830	−0.897184	−0.448592	−	0.893737i	$-0.648074\pi$
$257$	−14.3830	−0.897184	−0.448592	+	0.893737i	$0.648074\pi$
$258$	−24.9587	−1.55386
$259$	3.66031	0.227441
$260$	6.49288	0.402671
$261$	2.08853	0.129277
$262$	33.0779	2.04356
$263$	−1.51617	−0.0934913	−0.0467457	−	0.998907i	$-0.514885\pi$
$263$	−1.51617	−0.0934913	−0.0467457	+	0.998907i	$0.514885\pi$
$264$	2.38865	0.147011
$265$	32.5611	2.00022
$266$	1.29034	0.0791156
$267$	0.510909	0.0312671
$268$	−2.31270	−0.141271
$269$	5.10189	0.311068	0.155534	−	0.987831i	$-0.450290\pi$
$269$	5.10189	0.311068	0.155534	+	0.987831i	$0.450290\pi$
$270$	32.4027	1.97197
$271$	17.0840	1.03778	0.518888	−	0.854842i	$-0.326346\pi$
$271$	17.0840	1.03778	0.518888	+	0.854842i	$0.326346\pi$
$272$	−23.2579	−1.41022
$273$	−1.27123	−0.0769384
$274$	38.0442	2.29833
$275$	−12.1672	−0.733707
$276$	10.1919	0.613480
$277$	0.266293	0.0160000	0.00799999	−	0.999968i	$-0.497453\pi$
$277$	0.266293	0.0160000	0.00799999	+	0.999968i	$0.497453\pi$
$278$	4.65809	0.279374
$279$	2.87758	0.172276
$280$	1.01438	0.0606208
$281$	29.4646	1.75771	0.878855	−	0.477088i	$-0.158308\pi$
$281$	29.4646	1.75771	0.878855	+	0.477088i	$0.158308\pi$
$282$	13.0072	0.774569
$283$	−17.5453	−1.04296	−0.521480	−	0.853263i	$-0.674620\pi$
$283$	−17.5453	−1.04296	−0.521480	+	0.853263i	$0.674620\pi$
$284$	−11.3068	−0.670936
$285$	4.55772	0.269976
$286$	−7.45223	−0.440660
$287$	−0.344464	−0.0203331
$288$	−5.14912	−0.303415
$289$	10.2351	0.602065
$290$	−17.8204	−1.04645
$291$	25.4455	1.49164
$292$	5.78312	0.338432
$293$	−2.69968	−0.157717	−0.0788584	−	0.996886i	$-0.525127\pi$
$293$	−2.69968	−0.157717	−0.0788584	+	0.996886i	$0.525127\pi$
$294$	−19.3098	−1.12617
$295$	7.75090	0.451275
$296$	−2.78659	−0.161968
$297$	−17.2873	−1.00311
$298$	42.0689	2.43699
$299$	4.81142	0.278252
$300$	−10.4424	−0.602892
$301$	5.65478	0.325936
$302$	−28.2065	−1.62310
$303$	−18.1040	−1.04005
$304$	−4.45662	−0.255605
$305$	8.83689	0.505999
$306$	6.83604	0.390790
$307$	−2.32930	−0.132940	−0.0664700	−	0.997788i	$-0.521174\pi$
$307$	−2.32930	−0.132940	−0.0664700	+	0.997788i	$0.521174\pi$
$308$	3.57653	0.203792
$309$	6.32559	0.359850
$310$	−24.5530	−1.39452
$311$	13.5922	0.770744	0.385372	−	0.922761i	$-0.374073\pi$
$311$	13.5922	0.770744	0.385372	+	0.922761i	$0.374073\pi$
$312$	0.967789	0.0547903
$313$	8.81208	0.498088	0.249044	−	0.968492i	$-0.419884\pi$
$313$	8.81208	0.498088	0.249044	+	0.968492i	$0.419884\pi$
$314$	11.6404	0.656908
$315$	−1.35264	−0.0762126
$316$	−23.3746	−1.31493
$317$	−1.60610	−0.0902074	−0.0451037	−	0.998982i	$-0.514362\pi$
$317$	−1.60610	−0.0902074	−0.0451037	+	0.998982i	$0.514362\pi$
$318$	−32.0745	−1.79865
$319$	9.50744	0.532314
$320$	17.2779	0.965862
$321$	−1.74799	−0.0975632
$322$	−4.96768	−0.276838
$323$	5.21873	0.290378
$324$	−11.3055	−0.628082
$325$	−4.92967	−0.273449
$326$	47.6287	2.63791
$327$	21.0035	1.16149
$328$	0.262240	0.0144798
$329$	−2.94699	−0.162473
$330$	27.1776	1.49608
$331$	−8.77527	−0.482332	−0.241166	−	0.970484i	$-0.577530\pi$
$331$	−8.77527	−0.482332	−0.241166	+	0.970484i	$0.577530\pi$
$332$	7.23968	0.397329
$333$	3.71582	0.203626
$334$	1.84738	0.101084
$335$	3.98164	0.217540
$336$	−4.53323	−0.247308
$337$	12.9580	0.705869	0.352934	−	0.935648i	$-0.385184\pi$
$337$	12.9580	0.705869	0.352934	+	0.935648i	$0.385184\pi$
$338$	22.1118	1.20273
$339$	3.01956	0.164000
$340$	−27.1131	−1.47041
$341$	13.0994	0.709371
$342$	1.30991	0.0708316
$343$	9.04723	0.488505
$344$	−4.30498	−0.232109
$345$	−17.5468	−0.944689
$346$	−22.3822	−1.20327
$347$	3.73981	0.200763	0.100382	−	0.994949i	$-0.467994\pi$
$347$	3.73981	0.200763	0.100382	+	0.994949i	$0.467994\pi$
$348$	8.15970	0.437406
$349$	−18.6277	−0.997119	−0.498559	−	0.866856i	$-0.666137\pi$
$349$	−18.6277	−0.997119	−0.498559	+	0.866856i	$0.666137\pi$
$350$	5.08977	0.272060
$351$	−7.00415	−0.373854
$352$	−23.4399	−1.24935
$353$	−2.62764	−0.139855	−0.0699277	−	0.997552i	$-0.522277\pi$
$353$	−2.62764	−0.139855	−0.0699277	+	0.997552i	$0.522277\pi$
$354$	−7.63506	−0.405799
$355$	19.4663	1.03316
$356$	−0.582385	−0.0308663
$357$	5.30843	0.280952
$358$	2.91924	0.154286
$359$	23.8842	1.26056	0.630279	−	0.776368i	$-0.282940\pi$
$359$	23.8842	1.26056	0.630279	+	0.776368i	$0.282940\pi$
$360$	1.02976	0.0542733
$361$	1.00000	0.0526316
$362$	−36.7763	−1.93292
$363$	2.26376	0.118817
$364$	1.44908	0.0759522
$365$	−9.95647	−0.521145
$366$	−8.70481	−0.455008
$367$	17.4253	0.909593	0.454797	−	0.890595i	$-0.349712\pi$
$367$	17.4253	0.909593	0.454797	+	0.890595i	$0.349712\pi$
$368$	17.1576	0.894401
$369$	−0.349688	−0.0182040
$370$	−31.7054	−1.64829
$371$	7.26698	0.377283
$372$	11.2425	0.582894
$373$	−20.0585	−1.03859	−0.519294	−	0.854596i	$-0.673805\pi$
$373$	−20.0585	−1.03859	−0.519294	+	0.854596i	$0.673805\pi$
$374$	31.1192	1.60914
$375$	−4.81054	−0.248415
$376$	2.24355	0.115702
$377$	3.85206	0.198391
$378$	7.23162	0.371954
$379$	−6.30801	−0.324021	−0.162010	−	0.986789i	$-0.551798\pi$
$379$	−6.30801	−0.324021	−0.162010	+	0.986789i	$0.551798\pi$
$380$	−5.19535	−0.266516
$381$	15.2226	0.779877
$382$	−25.9107	−1.32571
$383$	16.4924	0.842722	0.421361	−	0.906893i	$-0.361553\pi$
$383$	16.4924	0.842722	0.421361	+	0.906893i	$0.361553\pi$
$384$	6.14159	0.313412
$385$	−6.15752	−0.313816
$386$	7.05056	0.358864
$387$	5.74054	0.291808
$388$	−29.0053	−1.47252
$389$	20.6693	1.04798	0.523988	−	0.851726i	$-0.324444\pi$
$389$	20.6693	1.04798	0.523988	+	0.851726i	$0.324444\pi$
$390$	11.0113	0.557581
$391$	−20.0916	−1.01608
$392$	−3.33064	−0.168223
$393$	26.0758	1.31535
$394$	12.6031	0.634933
$395$	40.2428	2.02483
$396$	3.63078	0.182453
$397$	−6.14904	−0.308611	−0.154306	−	0.988023i	$-0.549314\pi$
$397$	−6.14904	−0.308611	−0.154306	+	0.988023i	$0.549314\pi$
$398$	−19.5236	−0.978629
$399$	1.01719	0.0509232
$400$	−17.5793	−0.878963
$401$	−12.3067	−0.614567	−0.307283	−	0.951618i	$-0.599420\pi$
$401$	−12.3067	−0.614567	−0.307283	+	0.951618i	$0.599420\pi$
$402$	−3.92213	−0.195618
$403$	5.30737	0.264379
$404$	20.6367	1.02672
$405$	19.4640	0.967173
$406$	−3.97716	−0.197383
$407$	16.9153	0.838458
$408$	−4.04131	−0.200075
$409$	−8.80689	−0.435473	−0.217736	−	0.976008i	$-0.569867\pi$
$409$	−8.80689	−0.435473	−0.217736	+	0.976008i	$0.569867\pi$
$410$	2.98373	0.147356
$411$	29.9907	1.47933
$412$	−7.21054	−0.355238
$413$	1.72984	0.0851199
$414$	−5.04302	−0.247851
$415$	−12.4641	−0.611841
$416$	−9.49697	−0.465628
$417$	3.67204	0.179821
$418$	5.96299	0.291659
$419$	7.66547	0.374483	0.187241	−	0.982314i	$-0.440045\pi$
$419$	7.66547	0.374483	0.187241	+	0.982314i	$0.440045\pi$
$420$	−5.28465	−0.257865
$421$	−11.3391	−0.552634	−0.276317	−	0.961067i	$-0.589114\pi$
$421$	−11.3391	−0.552634	−0.276317	+	0.961067i	$0.589114\pi$
$422$	1.93317	0.0941052
$423$	−2.99169	−0.145461
$424$	−5.53235	−0.268675
$425$	20.5854	0.998540
$426$	−19.1754	−0.929049
$427$	1.97221	0.0954420
$428$	1.99253	0.0963127
$429$	−5.87470	−0.283633
$430$	−48.9814	−2.36209
$431$	11.4201	0.550085	0.275042	−	0.961432i	$-0.411308\pi$
$431$	11.4201	0.550085	0.275042	+	0.961432i	$0.411308\pi$
$432$	−24.9769	−1.20170
$433$	11.7742	0.565831	0.282916	−	0.959145i	$-0.408698\pi$
$433$	11.7742	0.565831	0.282916	+	0.959145i	$0.408698\pi$
$434$	−5.47973	−0.263036
$435$	−14.0481	−0.673555
$436$	−23.9418	−1.14661
$437$	−3.84991	−0.184166
$438$	9.80766	0.468628
$439$	15.9872	0.763028	0.381514	−	0.924363i	$-0.375403\pi$
$439$	15.9872	0.763028	0.381514	+	0.924363i	$0.375403\pi$
$440$	4.68772	0.223478
$441$	4.44128	0.211490
$442$	12.6083	0.599717
$443$	−27.3811	−1.30091	−0.650457	−	0.759543i	$-0.725423\pi$
$443$	−27.3811	−1.30091	−0.650457	+	0.759543i	$0.725423\pi$
$444$	14.5174	0.688966
$445$	1.00266	0.0475306
$446$	18.6291	0.882115
$447$	33.1635	1.56858
$448$	3.85607	0.182182
$449$	27.2875	1.28778	0.643889	−	0.765119i	$-0.277320\pi$
$449$	27.2875	1.28778	0.643889	+	0.765119i	$0.277320\pi$
$450$	5.16696	0.243573
$451$	−1.59186	−0.0749577
$452$	−3.44200	−0.161898
$453$	−22.2356	−1.04472
$454$	−38.8666	−1.82410
$455$	−2.49479	−0.116958
$456$	−0.774387	−0.0362640
$457$	−22.4717	−1.05118	−0.525591	−	0.850737i	$-0.676156\pi$
$457$	−22.4717	−1.05118	−0.525591	+	0.850737i	$0.676156\pi$
$458$	−5.71510	−0.267049
$459$	29.2481	1.36518
$460$	20.0016	0.932580
$461$	1.90199	0.0885847	0.0442923	−	0.999019i	$-0.485897\pi$
$461$	1.90199	0.0885847	0.0442923	+	0.999019i	$0.485897\pi$
$462$	6.06549	0.282192
$463$	−7.23635	−0.336302	−0.168151	−	0.985761i	$-0.553780\pi$
$463$	−7.23635	−0.336302	−0.168151	+	0.985761i	$0.553780\pi$
$464$	13.7365	0.637700
$465$	−19.3555	−0.897590
$466$	6.13598	0.284244
$467$	−42.5786	−1.97030	−0.985152	−	0.171682i	$-0.945080\pi$
$467$	−42.5786	−1.97030	−0.985152	+	0.171682i	$0.945080\pi$
$468$	1.47105	0.0679995
$469$	0.888621	0.0410327
$470$	25.5267	1.17746
$471$	9.17633	0.422823
$472$	−1.31693	−0.0606166
$473$	26.1322	1.20156
$474$	−39.6413	−1.82079
$475$	3.94453	0.180987
$476$	−6.05108	−0.277351
$477$	7.37719	0.337778
$478$	−7.00876	−0.320573
$479$	10.8134	0.494079	0.247039	−	0.969005i	$-0.420542\pi$
$479$	10.8134	0.494079	0.247039	+	0.969005i	$0.420542\pi$
$480$	34.6346	1.58085
$481$	6.85342	0.312489
$482$	16.0801	0.732427
$483$	−3.91609	−0.178188
$484$	−2.58046	−0.117294
$485$	49.9368	2.26751
$486$	13.3300	0.604660
$487$	42.8823	1.94318	0.971591	−	0.236666i	$-0.0760548\pi$
$487$	42.8823	1.94318	0.971591	+	0.236666i	$0.0760548\pi$
$488$	−1.50144	−0.0679672
$489$	37.5464	1.69791
$490$	−37.8954	−1.71194
$491$	−37.9353	−1.71200	−0.855998	−	0.516980i	$-0.827056\pi$
$491$	−37.9353	−1.71200	−0.855998	+	0.516980i	$0.827056\pi$
$492$	−1.36620	−0.0615932
$493$	−16.0855	−0.724454
$494$	2.41598	0.108700
$495$	−6.25090	−0.280957
$496$	18.9261	0.849809
$497$	4.34448	0.194877
$498$	12.2779	0.550184
$499$	−12.2763	−0.549561	−0.274780	−	0.961507i	$-0.588605\pi$
$499$	−12.2763	−0.549561	−0.274780	+	0.961507i	$0.588605\pi$
$500$	5.48353	0.245231
$501$	1.45632	0.0650635
$502$	31.6326	1.41183
$503$	11.8685	0.529190	0.264595	−	0.964360i	$-0.414762\pi$
$503$	11.8685	0.529190	0.264595	+	0.964360i	$0.414762\pi$
$504$	0.229822	0.0102371
$505$	−35.5291	−1.58102
$506$	−22.9570	−1.02056
$507$	17.4311	0.774141
$508$	−17.3522	−0.769881
$509$	1.74148	0.0771895	0.0385948	−	0.999255i	$-0.487712\pi$
$509$	1.74148	0.0771895	0.0385948	+	0.999255i	$0.487712\pi$
$510$	−45.9814	−2.03609
$511$	−2.22208	−0.0982990
$512$	−29.3371	−1.29653
$513$	5.60445	0.247442
$514$	27.8047	1.22641
$515$	12.4140	0.547025
$516$	22.4278	0.987330
$517$	−13.6188	−0.598956
$518$	−7.07600	−0.310901
$519$	−17.6442	−0.774493
$520$	1.89929	0.0832892
$521$	42.4838	1.86125	0.930624	−	0.365977i	$-0.119265\pi$
$521$	42.4838	1.86125	0.930624	+	0.365977i	$0.119265\pi$
$522$	−4.03747	−0.176715
$523$	6.21164	0.271616	0.135808	−	0.990735i	$-0.456637\pi$
$523$	6.21164	0.271616	0.135808	+	0.990735i	$0.456637\pi$
$524$	−29.7238	−1.29849
$525$	4.01233	0.175113
$526$	2.93102	0.127799
$527$	−22.1626	−0.965419
$528$	−20.9492	−0.911699
$529$	−8.17820	−0.355574
$530$	−62.9462	−2.73421
$531$	1.75608	0.0762072
$532$	−1.15949	−0.0502705
$533$	−0.644961	−0.0279364
$534$	−0.987673	−0.0427408
$535$	−3.43043	−0.148310
$536$	−0.676507	−0.0292207
$537$	2.30127	0.0993073
$538$	−9.86282	−0.425216
$539$	20.2177	0.870839
$540$	−29.1170	−1.25300
$541$	17.5586	0.754905	0.377452	−	0.926029i	$-0.376800\pi$
$541$	17.5586	0.754905	0.377452	+	0.926029i	$0.376800\pi$
$542$	−33.0262	−1.41860
$543$	−28.9913	−1.24413
$544$	39.6577	1.70031
$545$	41.2193	1.76564
$546$	2.45751	0.105172
$547$	23.6315	1.01041	0.505206	−	0.862999i	$-0.331417\pi$
$547$	23.6315	1.01041	0.505206	+	0.862999i	$0.331417\pi$
$548$	−34.1864	−1.46037
$549$	2.00212	0.0854485
$550$	23.5212	1.00295
$551$	−3.08226	−0.131309
$552$	2.98132	0.126893
$553$	8.98136	0.381926
$554$	−0.514789	−0.0218713
$555$	−24.9938	−1.06093
$556$	−4.18576	−0.177516
$557$	−28.0234	−1.18739	−0.593695	−	0.804690i	$-0.702332\pi$
$557$	−28.0234	−1.18739	−0.593695	+	0.804690i	$0.702332\pi$
$558$	−5.56284	−0.235494
$559$	10.5878	0.447816
$560$	−8.89646	−0.375944
$561$	24.5317	1.03573
$562$	−56.9601	−2.40271
$563$	−38.8777	−1.63850	−0.819250	−	0.573437i	$-0.805610\pi$
$563$	−38.8777	−1.63850	−0.819250	+	0.573437i	$0.805610\pi$
$564$	−11.6883	−0.492165
$565$	5.92590	0.249304
$566$	33.9181	1.42568
$567$	4.34396	0.182429
$568$	−3.30745	−0.138778
$569$	32.2615	1.35247	0.676236	−	0.736685i	$-0.263610\pi$
$569$	32.2615	1.35247	0.676236	+	0.736685i	$0.263610\pi$
$570$	−8.81085	−0.369046
$571$	−5.55302	−0.232386	−0.116193	−	0.993227i	$-0.537069\pi$
$571$	−5.55302	−0.232386	−0.116193	+	0.993227i	$0.537069\pi$
$572$	6.69657	0.279998
$573$	−20.4258	−0.853298
$574$	0.665907	0.0277944
$575$	−15.1861	−0.633303
$576$	3.91455	0.163106
$577$	24.7085	1.02863	0.514313	−	0.857602i	$-0.328047\pi$
$577$	24.7085	1.02863	0.514313	+	0.857602i	$0.328047\pi$
$578$	−19.7862	−0.822997
$579$	5.55805	0.230985
$580$	16.0134	0.664921
$581$	−2.78174	−0.115406
$582$	−49.1904	−2.03901
$583$	33.5826	1.39085
$584$	1.69167	0.0700018
$585$	−2.53263	−0.104711
$586$	5.21893	0.215592
$587$	−30.4073	−1.25504	−0.627521	−	0.778599i	$-0.715930\pi$
$587$	−30.4073	−1.25504	−0.627521	+	0.778599i	$0.715930\pi$
$588$	17.3517	0.715574
$589$	−4.24675	−0.174984
$590$	−14.9838	−0.616873
$591$	9.93516	0.408678
$592$	24.4394	1.00445
$593$	−39.3107	−1.61430	−0.807149	−	0.590348i	$-0.798991\pi$
$593$	−39.3107	−1.61430	−0.807149	+	0.590348i	$0.798991\pi$
$594$	33.4192	1.37121
$595$	10.4178	0.427088
$596$	−37.8030	−1.54847
$597$	−15.3907	−0.629900
$598$	−9.30129	−0.380358
$599$	11.4033	0.465925	0.232962	−	0.972486i	$-0.425158\pi$
$599$	11.4033	0.465925	0.232962	+	0.972486i	$0.425158\pi$
$600$	−3.05459	−0.124703
$601$	−22.9681	−0.936889	−0.468445	−	0.883493i	$-0.655185\pi$
$601$	−22.9681	−0.936889	−0.468445	+	0.883493i	$0.655185\pi$
$602$	−10.9316	−0.445540
$603$	0.902098	0.0367363
$604$	25.3463	1.03133
$605$	4.44263	0.180619
$606$	34.9981	1.42170
$607$	19.3914	0.787071	0.393536	−	0.919309i	$-0.371252\pi$
$607$	19.3914	0.787071	0.393536	+	0.919309i	$0.371252\pi$
$608$	7.59911	0.308184
$609$	−3.13525	−0.127047
$610$	−17.0832	−0.691678
$611$	−5.51784	−0.223228
$612$	−6.14286	−0.248310
$613$	15.8973	0.642084	0.321042	−	0.947065i	$-0.395967\pi$
$613$	15.8973	0.642084	0.321042	+	0.947065i	$0.395967\pi$
$614$	4.50293	0.181723
$615$	2.35211	0.0948464
$616$	1.04620	0.0421527
$617$	29.2068	1.17582	0.587910	−	0.808926i	$-0.299951\pi$
$617$	29.2068	1.17582	0.587910	+	0.808926i	$0.299951\pi$
$618$	−12.2284	−0.491900
$619$	20.6302	0.829199	0.414599	−	0.910004i	$-0.363922\pi$
$619$	20.6302	0.829199	0.414599	+	0.910004i	$0.363922\pi$
$620$	22.0633	0.886085
$621$	−21.5766	−0.865840
$622$	−26.2761	−1.05357
$623$	0.223773	0.00896527
$624$	−8.48784	−0.339786
$625$	−29.1633	−1.16653
$626$	−17.0352	−0.680865
$627$	4.70070	0.187728
$628$	−10.4601	−0.417403
$629$	−28.6187	−1.14110
$630$	2.61488	0.104179
$631$	−8.65055	−0.344373	−0.172187	−	0.985064i	$-0.555083\pi$
$631$	−8.65055	−0.344373	−0.172187	+	0.985064i	$0.555083\pi$
$632$	−6.83751	−0.271982
$633$	1.52394	0.0605713
$634$	3.10486	0.123310
$635$	29.8743	1.18553
$636$	28.8221	1.14287
$637$	8.19146	0.324557
$638$	−18.3795	−0.727651
$639$	4.41037	0.174471
$640$	12.0529	0.476432
$641$	−2.70849	−0.106979	−0.0534895	−	0.998568i	$-0.517034\pi$
$641$	−2.70849	−0.106979	−0.0534895	+	0.998568i	$0.517034\pi$
$642$	3.37916	0.133365
$643$	−19.0789	−0.752397	−0.376198	−	0.926539i	$-0.622769\pi$
$643$	−19.0789	−0.752397	−0.376198	+	0.926539i	$0.622769\pi$
$644$	4.46395	0.175904
$645$	−38.6127	−1.52037
$646$	−10.0887	−0.396934
$647$	17.4328	0.685354	0.342677	−	0.939453i	$-0.388666\pi$
$647$	17.4328	0.685354	0.342677	+	0.939453i	$0.388666\pi$
$648$	−3.30706	−0.129914
$649$	7.99406	0.313794
$650$	9.52989	0.373793
$651$	−4.31975	−0.169304
$652$	−42.7991	−1.67614
$653$	25.2960	0.989909	0.494955	−	0.868919i	$-0.335185\pi$
$653$	25.2960	0.989909	0.494955	+	0.868919i	$0.335185\pi$
$654$	−40.6032	−1.58771
$655$	51.1738	1.99952
$656$	−2.29994	−0.0897975
$657$	−2.25578	−0.0880064
$658$	5.69703	0.222094
$659$	−1.33630	−0.0520549	−0.0260275	−	0.999661i	$-0.508286\pi$
$659$	−1.33630	−0.0520549	−0.0260275	+	0.999661i	$0.508286\pi$
$660$	−24.4218	−0.950616
$661$	−18.3138	−0.712324	−0.356162	−	0.934424i	$-0.615915\pi$
$661$	−18.3138	−0.712324	−0.356162	+	0.934424i	$0.615915\pi$
$662$	16.9641	0.659327
$663$	9.93931	0.386011
$664$	2.11774	0.0821843
$665$	1.99624	0.0774107
$666$	−7.18331	−0.278348
$667$	11.8664	0.459470
$668$	−1.66006	−0.0642295
$669$	14.6856	0.567778
$670$	−7.69719	−0.297368
$671$	9.11411	0.351846
$672$	7.72973	0.298181
$673$	−37.9390	−1.46244	−0.731221	−	0.682141i	$-0.761049\pi$
$673$	−37.9390	−1.46244	−0.731221	+	0.682141i	$0.761049\pi$
$674$	−25.0501	−0.964892
$675$	22.1069	0.850896
$676$	−19.8697	−0.764218
$677$	−9.90545	−0.380697	−0.190349	−	0.981717i	$-0.560962\pi$
$677$	−9.90545	−0.380697	−0.190349	+	0.981717i	$0.560962\pi$
$678$	−5.83733	−0.224181
$679$	11.1449	0.427701
$680$	−7.93108	−0.304143
$681$	−30.6390	−1.17409
$682$	−25.3233	−0.969679
$683$	41.0277	1.56988	0.784942	−	0.619570i	$-0.212693\pi$
$683$	41.0277	1.56988	0.784942	+	0.619570i	$0.212693\pi$
$684$	−1.17708	−0.0450068
$685$	58.8568	2.24880
$686$	−17.4898	−0.667765
$687$	−4.50530	−0.171888
$688$	37.7562	1.43944
$689$	13.6064	0.518363
$690$	33.9210	1.29135
$691$	−40.4997	−1.54068	−0.770341	−	0.637633i	$-0.779914\pi$
$691$	−40.4997	−1.54068	−0.770341	+	0.637633i	$0.779914\pi$
$692$	20.1126	0.764566
$693$	−1.39507	−0.0529944
$694$	−7.22968	−0.274435
$695$	7.20638	0.273354
$696$	2.38686	0.0904738
$697$	2.69324	0.102014
$698$	36.0105	1.36302
$699$	4.83708	0.182955
$700$	−4.57366	−0.172868
$701$	7.92864	0.299461	0.149730	−	0.988727i	$-0.452159\pi$
$701$	7.92864	0.299461	0.149730	+	0.988727i	$0.452159\pi$
$702$	13.5402	0.511042
$703$	−5.48384	−0.206827
$704$	17.8199	0.671613
$705$	20.1230	0.757878
$706$	5.07968	0.191176
$707$	−7.92936	−0.298214
$708$	6.86085	0.257847
$709$	26.8426	1.00809	0.504047	−	0.863676i	$-0.331844\pi$
$709$	26.8426	1.00809	0.504047	+	0.863676i	$0.331844\pi$
$710$	−37.6317	−1.41229
$711$	9.11758	0.341936
$712$	−0.170358	−0.00638445
$713$	16.3496	0.612297
$714$	−10.2621	−0.384050
$715$	−11.5291	−0.431164
$716$	−2.62322	−0.0980344
$717$	−5.52510	−0.206339
$718$	−46.1722	−1.72313
$719$	20.5822	0.767585	0.383793	−	0.923419i	$-0.374618\pi$
$719$	20.5822	0.767585	0.383793	+	0.923419i	$0.374618\pi$
$720$	−9.03139	−0.336580
$721$	2.77054	0.103180
$722$	−1.93317	−0.0719451
$723$	12.6761	0.471431
$724$	33.0471	1.22819
$725$	−12.1581	−0.451540
$726$	−4.37623	−0.162417
$727$	25.5964	0.949319	0.474660	−	0.880169i	$-0.342571\pi$
$727$	25.5964	0.949319	0.474660	+	0.880169i	$0.342571\pi$
$728$	0.423882	0.0157101
$729$	30.0324	1.11231
$730$	19.2475	0.712384
$731$	−44.2127	−1.63527
$732$	7.82213	0.289114
$733$	−20.1226	−0.743245	−0.371623	−	0.928384i	$-0.621198\pi$
$733$	−20.1226	−0.743245	−0.371623	+	0.928384i	$0.621198\pi$
$734$	−33.6861	−1.24338
$735$	−29.8735	−1.10190
$736$	−29.2559	−1.07839
$737$	4.10655	0.151267
$738$	0.676006	0.0248841
$739$	15.2676	0.561627	0.280814	−	0.959762i	$-0.409396\pi$
$739$	15.2676	0.561627	0.280814	+	0.959762i	$0.409396\pi$
$740$	28.4904	1.04733
$741$	1.90455	0.0699653
$742$	−14.0483	−0.515729
$743$	−29.1920	−1.07095	−0.535475	−	0.844551i	$-0.679867\pi$
$743$	−29.1920	−1.07095	−0.535475	+	0.844551i	$0.679867\pi$
$744$	3.28863	0.120567
$745$	65.0834	2.38447
$746$	38.7764	1.41970
$747$	−2.82393	−0.103322
$748$	−27.9637	−1.02245
$749$	−0.765601	−0.0279745
$750$	9.29959	0.339573
$751$	−26.4630	−0.965649	−0.482824	−	0.875717i	$-0.660389\pi$
$751$	−26.4630	−0.965649	−0.482824	+	0.875717i	$0.660389\pi$
$752$	−19.6767	−0.717534
$753$	24.9364	0.908733
$754$	−7.44667	−0.271192
$755$	−43.6373	−1.58813
$756$	−6.49833	−0.236342
$757$	25.6106	0.930834	0.465417	−	0.885092i	$-0.345904\pi$
$757$	25.6106	0.930834	0.465417	+	0.885092i	$0.345904\pi$
$758$	12.1944	0.442922
$759$	−18.0973	−0.656890
$760$	−1.51973	−0.0551266
$761$	−28.0677	−1.01745	−0.508727	−	0.860928i	$-0.669884\pi$
$761$	−28.0677	−1.01745	−0.508727	+	0.860928i	$0.669884\pi$
$762$	−29.4278	−1.06606
$763$	9.19930	0.333037
$764$	23.2833	0.842361
$765$	10.5758	0.382369
$766$	−31.8826	−1.15196
$767$	3.23889	0.116950
$768$	−29.4807	−1.06379
$769$	−22.4378	−0.809126	−0.404563	−	0.914510i	$-0.632576\pi$
$769$	−22.4378	−0.809126	−0.404563	+	0.914510i	$0.632576\pi$
$770$	11.9035	0.428973
$771$	21.9188	0.789387
$772$	−6.33562	−0.228024
$773$	−7.97622	−0.286885	−0.143442	−	0.989659i	$-0.545817\pi$
$773$	−7.97622	−0.286885	−0.143442	+	0.989659i	$0.545817\pi$
$774$	−11.0974	−0.398889
$775$	−16.7514	−0.601729
$776$	−8.48459	−0.304579
$777$	−5.57811	−0.200113
$778$	−39.9573	−1.43254
$779$	0.516072	0.0184902
$780$	−9.89478	−0.354290
$781$	20.0770	0.718411
$782$	38.8405	1.38893
$783$	−17.2744	−0.617336
$784$	29.2108	1.04324
$785$	18.0085	0.642752
$786$	−50.4089	−1.79803
$787$	−32.8599	−1.17133	−0.585664	−	0.810554i	$-0.699166\pi$
$787$	−32.8599	−1.17133	−0.585664	+	0.810554i	$0.699166\pi$
$788$	−11.3251	−0.403440
$789$	2.31057	0.0822583
$790$	−77.7961	−2.76786
$791$	1.32254	0.0470241
$792$	1.06207	0.0377390
$793$	3.69269	0.131131
$794$	11.8871	0.421858
$795$	−49.6214	−1.75989
$796$	17.5439	0.621826
$797$	−35.4110	−1.25432	−0.627162	−	0.778889i	$-0.715783\pi$
$797$	−35.4110	−1.25432	−0.627162	+	0.778889i	$0.715783\pi$
$798$	−1.96640	−0.0696098
$799$	23.0415	0.815149
$800$	29.9749	1.05977
$801$	0.227167	0.00802654
$802$	23.7909	0.840086
$803$	−10.2688	−0.362379
$804$	3.52442	0.124297
$805$	−7.68532	−0.270872
$806$	−10.2600	−0.361395
$807$	−7.77500	−0.273693
$808$	6.03662	0.212368
$809$	−7.08062	−0.248941	−0.124471	−	0.992223i	$-0.539723\pi$
$809$	−7.08062	−0.248941	−0.124471	+	0.992223i	$0.539723\pi$
$810$	−37.6272	−1.32208
$811$	−1.50598	−0.0528822	−0.0264411	−	0.999650i	$-0.508417\pi$
$811$	−1.50598	−0.0528822	−0.0264411	+	0.999650i	$0.508417\pi$
$812$	3.57387	0.125418
$813$	−26.0350	−0.913087
$814$	−32.7001	−1.14614
$815$	73.6848	2.58107
$816$	35.4437	1.24078
$817$	−8.47193	−0.296395
$818$	17.0252	0.595272
$819$	−0.565231	−0.0197508
$820$	−2.68117	−0.0936307
$821$	−19.5007	−0.680580	−0.340290	−	0.940321i	$-0.610525\pi$
$821$	−19.5007	−0.680580	−0.340290	+	0.940321i	$0.610525\pi$
$822$	−57.9772	−2.02219
$823$	17.0809	0.595402	0.297701	−	0.954659i	$-0.403780\pi$
$823$	17.0809	0.595402	0.297701	+	0.954659i	$0.403780\pi$
$824$	−2.10922	−0.0734780
$825$	18.5421	0.645552
$826$	−3.34408	−0.116355
$827$	6.84157	0.237905	0.118952	−	0.992900i	$-0.462046\pi$
$827$	6.84157	0.237905	0.118952	+	0.992900i	$0.462046\pi$
$828$	4.53165	0.157486
$829$	3.23649	0.112408	0.0562039	−	0.998419i	$-0.482100\pi$
$829$	3.23649	0.112408	0.0562039	+	0.998419i	$0.482100\pi$
$830$	24.0953	0.836360
$831$	−0.405815	−0.0140776
$832$	7.21995	0.250307
$833$	−34.2060	−1.18517
$834$	−7.09867	−0.245807
$835$	2.85802	0.0989060
$836$	−5.35833	−0.185322
$837$	−23.8007	−0.822672
$838$	−14.8187	−0.511902
$839$	23.9425	0.826585	0.413293	−	0.910598i	$-0.364379\pi$
$839$	23.9425	0.826585	0.413293	+	0.910598i	$0.364379\pi$
$840$	−1.54586	−0.0533372
$841$	−19.4997	−0.672402
$842$	21.9204	0.755426
$843$	−44.9024	−1.54652
$844$	−1.73714	−0.0597949
$845$	34.2085	1.17681
$846$	5.78344	0.198839
$847$	0.991504	0.0340685
$848$	48.5206	1.66621
$849$	26.7381	0.917649
$850$	−39.7951	−1.36496
$851$	21.1123	0.723720
$852$	17.2310	0.590323
$853$	−51.8246	−1.77444	−0.887220	−	0.461346i	$-0.847367\pi$
$853$	−51.8246	−1.77444	−0.887220	+	0.461346i	$0.847367\pi$
$854$	−3.81262	−0.130465
$855$	2.02651	0.0693052
$856$	0.582853	0.0199215
$857$	−8.19711	−0.280008	−0.140004	−	0.990151i	$-0.544712\pi$
$857$	−8.19711	−0.280008	−0.140004	+	0.990151i	$0.544712\pi$
$858$	11.3568	0.387714
$859$	38.1697	1.30233	0.651167	−	0.758934i	$-0.274280\pi$
$859$	38.1697	1.30233	0.651167	+	0.758934i	$0.274280\pi$
$860$	44.0146	1.50089
$861$	0.524943	0.0178900
$862$	−22.0769	−0.751942
$863$	−47.1078	−1.60357	−0.801784	−	0.597613i	$-0.796116\pi$
$863$	−47.1078	−1.60357	−0.801784	+	0.597613i	$0.796116\pi$
$864$	42.5888	1.44890
$865$	−34.6267	−1.17734
$866$	−22.7615	−0.773467
$867$	−15.5977	−0.529727
$868$	4.92408	0.167134
$869$	41.5053	1.40797
$870$	27.1573	0.920720
$871$	1.66382	0.0563764
$872$	−7.00343	−0.237166
$873$	11.3139	0.382917
$874$	7.44253	0.251747
$875$	−2.10697	−0.0712285
$876$	−8.81315	−0.297769
$877$	33.2792	1.12376	0.561880	−	0.827219i	$-0.310078\pi$
$877$	33.2792	1.12376	0.561880	+	0.827219i	$0.310078\pi$
$878$	−30.9060	−1.04303
$879$	4.11416	0.138767
$880$	−41.1129	−1.38592
$881$	−0.695860	−0.0234441	−0.0117221	−	0.999931i	$-0.503731\pi$
$881$	−0.695860	−0.0234441	−0.0117221	+	0.999931i	$0.503731\pi$
$882$	−8.58575	−0.289097
$883$	−22.7435	−0.765379	−0.382689	−	0.923877i	$-0.625002\pi$
$883$	−22.7435	−0.765379	−0.382689	+	0.923877i	$0.625002\pi$
$884$	−11.3298	−0.381063
$885$	−11.8119	−0.397054
$886$	52.9322	1.77829
$887$	3.31223	0.111214	0.0556068	−	0.998453i	$-0.482291\pi$
$887$	3.31223	0.111214	0.0556068	+	0.998453i	$0.482291\pi$
$888$	4.24661	0.142507
$889$	6.66734	0.223615
$890$	−1.93831	−0.0649723
$891$	20.0746	0.672524
$892$	−16.7401	−0.560500
$893$	4.41516	0.147748
$894$	−64.1106	−2.14418
$895$	4.51625	0.150962
$896$	2.68995	0.0898650
$897$	−7.33233	−0.244819
$898$	−52.7514	−1.76034
$899$	13.0896	0.436563
$900$	−4.64302	−0.154767
$901$	−56.8179	−1.89288
$902$	3.07733	0.102464
$903$	−8.61756	−0.286774
$904$	−1.00685	−0.0334873
$905$	−56.8954	−1.89127
$906$	42.9851	1.42809
$907$	−51.8499	−1.72165	−0.860824	−	0.508903i	$-0.830051\pi$
$907$	−51.8499	−1.72165	−0.860824	+	0.508903i	$0.830051\pi$
$908$	34.9254	1.15904
$909$	−8.04962	−0.266989
$910$	4.82286	0.159876
$911$	51.0439	1.69116	0.845580	−	0.533848i	$-0.179255\pi$
$911$	51.0439	1.69116	0.845580	+	0.533848i	$0.179255\pi$
$912$	6.79164	0.224894
$913$	−12.8552	−0.425444
$914$	43.4416	1.43692
$915$	−13.4669	−0.445203
$916$	5.13558	0.169685
$917$	11.4209	0.377152
$918$	−56.5415	−1.86615
$919$	−34.9757	−1.15374	−0.576871	−	0.816835i	$-0.695727\pi$
$919$	−34.9757	−1.15374	−0.576871	+	0.816835i	$0.695727\pi$
$920$	5.85084	0.192896
$921$	3.54972	0.116967
$922$	−3.67688	−0.121091
$923$	8.13444	0.267748
$924$	−5.45044	−0.179306
$925$	−21.6312	−0.711229
$926$	13.9891	0.459710
$927$	2.81256	0.0923767
$928$	−23.4225	−0.768880
$929$	−29.6764	−0.973650	−0.486825	−	0.873499i	$-0.661845\pi$
$929$	−29.6764	−0.973650	−0.486825	+	0.873499i	$0.661845\pi$
$930$	37.4175	1.22697
$931$	−6.55448	−0.214814
$932$	−5.51379	−0.180610
$933$	−20.7138	−0.678139
$934$	82.3117	2.69332
$935$	48.1434	1.57446
$936$	0.430310	0.0140651
$937$	8.38723	0.273999	0.136999	−	0.990571i	$-0.456254\pi$
$937$	8.38723	0.273999	0.136999	+	0.990571i	$0.456254\pi$
$938$	−1.71785	−0.0560899
$939$	−13.4291	−0.438242
$940$	−22.9383	−0.748164
$941$	0.182692	0.00595559	0.00297780	−	0.999996i	$-0.499052\pi$
$941$	0.182692	0.00595559	0.00297780	+	0.999996i	$0.499052\pi$
$942$	−17.7394	−0.577981
$943$	−1.98683	−0.0647001
$944$	11.5499	0.375918
$945$	11.1878	0.363939
$946$	−50.5180	−1.64248
$947$	17.5100	0.569000	0.284500	−	0.958676i	$-0.408172\pi$
$947$	17.5100	0.569000	0.284500	+	0.958676i	$0.408172\pi$
$948$	35.6216	1.15694
$949$	−4.16054	−0.135057
$950$	−7.62544	−0.247402
$951$	2.44760	0.0793689
$952$	−1.77005	−0.0573678
$953$	−10.6413	−0.344707	−0.172353	−	0.985035i	$-0.555137\pi$
$953$	−10.6413	−0.344707	−0.172353	+	0.985035i	$0.555137\pi$
$954$	−14.2614	−0.461728
$955$	−40.0855	−1.29714
$956$	6.29806	0.203694
$957$	−14.4888	−0.468357
$958$	−20.9042	−0.675385
$959$	13.1356	0.424172
$960$	−26.3305	−0.849813
$961$	−12.9651	−0.418230
$962$	−13.2488	−0.427159
$963$	−0.777213	−0.0250453
$964$	−14.4495	−0.465388
$965$	10.9077	0.351130
$966$	7.57046	0.243576
$967$	−44.5552	−1.43280	−0.716400	−	0.697690i	$-0.754211\pi$
$967$	−44.5552	−1.43280	−0.716400	+	0.697690i	$0.754211\pi$
$968$	−0.754832	−0.0242612
$969$	−7.95305	−0.255489
$970$	−96.5363	−3.09959
$971$	−23.1427	−0.742684	−0.371342	−	0.928496i	$-0.621102\pi$
$971$	−23.1427	−0.742684	−0.371342	+	0.928496i	$0.621102\pi$
$972$	−11.9783	−0.384204
$973$	1.60832	0.0515602
$974$	−82.8987	−2.65625
$975$	7.51254	0.240594
$976$	13.1682	0.421503
$977$	−52.7828	−1.68867	−0.844336	−	0.535814i	$-0.820005\pi$
$977$	−52.7828	−1.68867	−0.844336	+	0.535814i	$0.820005\pi$
$978$	−72.5835	−2.32097
$979$	1.03411	0.0330504
$980$	34.0528	1.08778
$981$	9.33882	0.298166
$982$	73.3353	2.34022
$983$	−49.6613	−1.58395	−0.791975	−	0.610554i	$-0.790947\pi$
$983$	−49.6613	−1.58395	−0.791975	+	0.610554i	$0.790947\pi$
$984$	−0.399640	−0.0127400
$985$	19.4978	0.621250
$986$	31.0960	0.990298
$987$	4.49105	0.142952
$988$	−2.17099	−0.0690685
$989$	32.6162	1.03713
$990$	12.0841	0.384056
$991$	34.3406	1.09087	0.545433	−	0.838154i	$-0.316365\pi$
$991$	34.3406	1.09087	0.545433	+	0.838154i	$0.316365\pi$
$992$	−32.2715	−1.02462
$993$	13.3730	0.424380
$994$	−8.39861	−0.266388
$995$	−30.2043	−0.957540
$996$	−11.0329	−0.349590
$997$	30.8696	0.977650	0.488825	−	0.872382i	$-0.337426\pi$
$997$	30.8696	0.977650	0.488825	+	0.872382i	$0.337426\pi$
$998$	23.7321	0.751226
$999$	−30.7339	−0.972378

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.e.1.13	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.e.1.13	✓	82	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-1.93317 q^{2} -1.52394 q^{3} +1.73714 q^{4} -2.99074 q^{5} +2.94604 q^{6} -0.667472 q^{7} +0.508147 q^{8} -0.677595 q^{9} +O(q^{10})\)	\(q-1.93317 q^{2} -1.52394 q^{3} +1.73714 q^{4} -2.99074 q^{5} +2.94604 q^{6} -0.667472 q^{7} +0.508147 q^{8} -0.677595 q^{9} +5.78161 q^{10} -3.08456 q^{11} -2.64731 q^{12} -1.24975 q^{13} +1.29034 q^{14} +4.55772 q^{15} -4.45662 q^{16} +5.21873 q^{17} +1.30991 q^{18} +1.00000 q^{19} -5.19535 q^{20} +1.01719 q^{21} +5.96299 q^{22} -3.84991 q^{23} -0.774387 q^{24} +3.94453 q^{25} +2.41598 q^{26} +5.60445 q^{27} -1.15949 q^{28} -3.08226 q^{29} -8.81085 q^{30} -4.24675 q^{31} +7.59911 q^{32} +4.70070 q^{33} -10.0887 q^{34} +1.99624 q^{35} -1.17708 q^{36} -5.48384 q^{37} -1.93317 q^{38} +1.90455 q^{39} -1.51973 q^{40} +0.516072 q^{41} -1.96640 q^{42} -8.47193 q^{43} -5.35833 q^{44} +2.02651 q^{45} +7.44253 q^{46} +4.41516 q^{47} +6.79164 q^{48} -6.55448 q^{49} -7.62544 q^{50} -7.95305 q^{51} -2.17099 q^{52} -10.8873 q^{53} -10.8343 q^{54} +9.22513 q^{55} -0.339173 q^{56} -1.52394 q^{57} +5.95854 q^{58} -2.59163 q^{59} +7.91741 q^{60} -2.95475 q^{61} +8.20968 q^{62} +0.452276 q^{63} -5.77712 q^{64} +3.73767 q^{65} -9.08726 q^{66} -1.33132 q^{67} +9.06568 q^{68} +5.86705 q^{69} -3.85906 q^{70} -6.50886 q^{71} -0.344317 q^{72} +3.32910 q^{73} +10.6012 q^{74} -6.01124 q^{75} +1.73714 q^{76} +2.05886 q^{77} -3.68181 q^{78} -13.4558 q^{79} +13.3286 q^{80} -6.50808 q^{81} -0.997655 q^{82} +4.16758 q^{83} +1.76700 q^{84} -15.6079 q^{85} +16.3777 q^{86} +4.69720 q^{87} -1.56741 q^{88} -0.335254 q^{89} -3.91759 q^{90} +0.834172 q^{91} -6.68784 q^{92} +6.47181 q^{93} -8.53524 q^{94} -2.99074 q^{95} -11.5806 q^{96} -16.6971 q^{97} +12.6709 q^{98} +2.09009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 82 * q + 15 * q^2 + 12 * q^3 + 89 * q^4 + 9 * q^5 + 9 * q^6 + 14 * q^7 + 42 * q^8 + 92 * q^9 + 4 * q^10 + 41 * q^11 + 26 * q^12 + 13 * q^13 + 22 * q^14 + 41 * q^15 + 87 * q^16 + 12 * q^17 + 24 * q^18 + 82 * q^19 + 26 * q^20 + 29 * q^21 + 2 * q^22 + 59 * q^23 + 16 * q^24 + 67 * q^25 + 24 * q^26 + 42 * q^27 - 2 * q^28 + 101 * q^29 - 22 * q^30 + 48 * q^31 + 69 * q^32 + 3 * q^33 + q^34 + 38 * q^35 + 82 * q^36 + 16 * q^37 + 15 * q^38 + 82 * q^39 + 20 * q^40 + 86 * q^41 - q^42 + 9 * q^43 + 82 * q^44 - 8 * q^45 + 43 * q^46 + 24 * q^47 + 34 * q^48 + 76 * q^49 + 82 * q^50 + 57 * q^51 - 22 * q^52 + 39 * q^53 + 17 * q^54 - 21 * q^55 + 50 * q^56 + 12 * q^57 + 33 * q^58 + 79 * q^59 + 87 * q^60 + 4 * q^61 + 40 * q^62 + 44 * q^63 + 90 * q^64 + 66 * q^65 - 39 * q^66 + 33 * q^67 - 9 * q^68 + 60 * q^69 + 30 * q^70 + 168 * q^71 + 15 * q^72 - 28 * q^73 + 35 * q^74 + 55 * q^75 + 89 * q^76 + 19 * q^77 - 41 * q^78 + 121 * q^79 + 64 * q^80 + 110 * q^81 + 41 * q^82 + 28 * q^84 + 17 * q^85 + 80 * q^86 + 29 * q^87 + 49 * q^88 + 83 * q^89 - 42 * q^90 + 38 * q^91 + 71 * q^92 - q^93 + 89 * q^94 + 9 * q^95 + 35 * q^96 - 23 * q^97 + 135 * q^98 + 93 * q^99

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