LMFDB - Embedded newform 4009.2.a.e.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.71228 q^{2} -0.799829 q^{3} +5.35647 q^{4} +2.45685 q^{5} +2.16936 q^{6} -1.47623 q^{7} -9.10370 q^{8} -2.36027 q^{9} +O(q^{10})$	\(q-2.71228 q^{2} -0.799829 q^{3} +5.35647 q^{4} +2.45685 q^{5} +2.16936 q^{6} -1.47623 q^{7} -9.10370 q^{8} -2.36027 q^{9} -6.66368 q^{10} -3.67530 q^{11} -4.28426 q^{12} -0.840855 q^{13} +4.00396 q^{14} -1.96506 q^{15} +13.9789 q^{16} -2.83991 q^{17} +6.40173 q^{18} +1.00000 q^{19} +13.1601 q^{20} +1.18073 q^{21} +9.96846 q^{22} +1.26199 q^{23} +7.28140 q^{24} +1.03613 q^{25} +2.28064 q^{26} +4.28730 q^{27} -7.90740 q^{28} +1.56159 q^{29} +5.32980 q^{30} -5.90829 q^{31} -19.7072 q^{32} +2.93961 q^{33} +7.70264 q^{34} -3.62689 q^{35} -12.6427 q^{36} -9.30121 q^{37} -2.71228 q^{38} +0.672540 q^{39} -22.3665 q^{40} +5.99792 q^{41} -3.20248 q^{42} -9.15103 q^{43} -19.6867 q^{44} -5.79885 q^{45} -3.42287 q^{46} -9.32102 q^{47} -11.1807 q^{48} -4.82074 q^{49} -2.81027 q^{50} +2.27144 q^{51} -4.50402 q^{52} +6.66269 q^{53} -11.6284 q^{54} -9.02968 q^{55} +13.4392 q^{56} -0.799829 q^{57} -4.23547 q^{58} +11.7356 q^{59} -10.5258 q^{60} -1.48915 q^{61} +16.0249 q^{62} +3.48431 q^{63} +25.4938 q^{64} -2.06586 q^{65} -7.97306 q^{66} +7.67269 q^{67} -15.2119 q^{68} -1.00938 q^{69} +9.83714 q^{70} +13.8293 q^{71} +21.4872 q^{72} +6.37179 q^{73} +25.2275 q^{74} -0.828725 q^{75} +5.35647 q^{76} +5.42561 q^{77} -1.82412 q^{78} -1.59842 q^{79} +34.3440 q^{80} +3.65172 q^{81} -16.2681 q^{82} +6.54855 q^{83} +6.32457 q^{84} -6.97724 q^{85} +24.8202 q^{86} -1.24900 q^{87} +33.4589 q^{88} -12.9084 q^{89} +15.7281 q^{90} +1.24130 q^{91} +6.75982 q^{92} +4.72562 q^{93} +25.2812 q^{94} +2.45685 q^{95} +15.7624 q^{96} -12.3544 q^{97} +13.0752 q^{98} +8.67473 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.71228	−1.91787	−0.958937	−	0.283621i	$-0.908464\pi$
$2$	−2.71228	−1.91787	−0.958937	+	0.283621i	$0.908464\pi$
$3$	−0.799829	−0.461781	−0.230891	−	0.972980i	$-0.574164\pi$
$3$	−0.799829	−0.461781	−0.230891	+	0.972980i	$0.574164\pi$
$4$	5.35647	2.67824
$5$	2.45685	1.09874	0.549369	−	0.835580i	$-0.314868\pi$
$5$	2.45685	1.09874	0.549369	+	0.835580i	$0.314868\pi$
$6$	2.16936	0.885638
$7$	−1.47623	−0.557964	−0.278982	−	0.960296i	$-0.589997\pi$
$7$	−1.47623	−0.557964	−0.278982	+	0.960296i	$0.589997\pi$
$8$	−9.10370	−3.21865
$9$	−2.36027	−0.786758
$10$	−6.66368	−2.10724
$11$	−3.67530	−1.10815	−0.554073	−	0.832468i	$-0.686927\pi$
$11$	−3.67530	−1.10815	−0.554073	+	0.832468i	$0.686927\pi$
$12$	−4.28426	−1.23676
$13$	−0.840855	−0.233211	−0.116606	−	0.993178i	$-0.537201\pi$
$13$	−0.840855	−0.233211	−0.116606	+	0.993178i	$0.537201\pi$
$14$	4.00396	1.07010
$15$	−1.96506	−0.507377
$16$	13.9789	3.49472
$17$	−2.83991	−0.688779	−0.344390	−	0.938827i	$-0.611914\pi$
$17$	−2.83991	−0.688779	−0.344390	+	0.938827i	$0.611914\pi$
$18$	6.40173	1.50890
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	13.1601	2.94268
$21$	1.18073	0.257657
$22$	9.96846	2.12528
$23$	1.26199	0.263143	0.131572	−	0.991307i	$-0.457998\pi$
$23$	1.26199	0.263143	0.131572	+	0.991307i	$0.457998\pi$
$24$	7.28140	1.48631
$25$	1.03613	0.207226
$26$	2.28064	0.447270
$27$	4.28730	0.825091
$28$	−7.90740	−1.49436
$29$	1.56159	0.289980	0.144990	−	0.989433i	$-0.453685\pi$
$29$	1.56159	0.289980	0.144990	+	0.989433i	$0.453685\pi$
$30$	5.32980	0.973084
$31$	−5.90829	−1.06116	−0.530580	−	0.847635i	$-0.678026\pi$
$31$	−5.90829	−1.06116	−0.530580	+	0.847635i	$0.678026\pi$
$32$	−19.7072	−3.48378
$33$	2.93961	0.511721
$34$	7.70264	1.32099
$35$	−3.62689	−0.613056
$36$	−12.6427	−2.10712
$37$	−9.30121	−1.52911	−0.764555	−	0.644559i	$-0.777041\pi$
$37$	−9.30121	−1.52911	−0.764555	+	0.644559i	$0.777041\pi$
$38$	−2.71228	−0.439990
$39$	0.672540	0.107693
$40$	−22.3665	−3.53645
$41$	5.99792	0.936718	0.468359	−	0.883538i	$-0.344845\pi$
$41$	5.99792	0.936718	0.468359	+	0.883538i	$0.344845\pi$
$42$	−3.20248	−0.494154
$43$	−9.15103	−1.39552	−0.697760	−	0.716332i	$-0.745820\pi$
$43$	−9.15103	−1.39552	−0.697760	+	0.716332i	$0.745820\pi$
$44$	−19.6867	−2.96788
$45$	−5.79885	−0.864441
$46$	−3.42287	−0.504675
$47$	−9.32102	−1.35961	−0.679805	−	0.733393i	$-0.737936\pi$
$47$	−9.32102	−1.35961	−0.679805	+	0.733393i	$0.737936\pi$
$48$	−11.1807	−1.61379
$49$	−4.82074	−0.688677
$50$	−2.81027	−0.397432
$51$	2.27144	0.318065
$52$	−4.50402	−0.624595
$53$	6.66269	0.915191	0.457595	−	0.889161i	$-0.348711\pi$
$53$	6.66269	0.915191	0.457595	+	0.889161i	$0.348711\pi$
$54$	−11.6284	−1.58242
$55$	−9.02968	−1.21756
$56$	13.4392	1.79589
$57$	−0.799829	−0.105940
$58$	−4.23547	−0.556145
$59$	11.7356	1.52784	0.763921	−	0.645310i	$-0.223272\pi$
$59$	11.7356	1.52784	0.763921	+	0.645310i	$0.223272\pi$
$60$	−10.5258	−1.35888
$61$	−1.48915	−0.190667	−0.0953334	−	0.995445i	$-0.530392\pi$
$61$	−1.48915	−0.190667	−0.0953334	+	0.995445i	$0.530392\pi$
$62$	16.0249	2.03517
$63$	3.48431	0.438982
$64$	25.4938	3.18673
$65$	−2.06586	−0.256238
$66$	−7.97306	−0.981416
$67$	7.67269	0.937368	0.468684	−	0.883366i	$-0.344728\pi$
$67$	7.67269	0.937368	0.468684	+	0.883366i	$0.344728\pi$
$68$	−15.2119	−1.84471
$69$	−1.00938	−0.121515
$70$	9.83714	1.17576
$71$	13.8293	1.64123	0.820616	−	0.571480i	$-0.193631\pi$
$71$	13.8293	1.64123	0.820616	+	0.571480i	$0.193631\pi$
$72$	21.4872	2.53230
$73$	6.37179	0.745762	0.372881	−	0.927879i	$-0.378370\pi$
$73$	6.37179	0.745762	0.372881	+	0.927879i	$0.378370\pi$
$74$	25.2275	2.93264
$75$	−0.828725	−0.0956929
$76$	5.35647	0.614430
$77$	5.42561	0.618305
$78$	−1.82412	−0.206541
$79$	−1.59842	−0.179836	−0.0899182	−	0.995949i	$-0.528661\pi$
$79$	−1.59842	−0.179836	−0.0899182	+	0.995949i	$0.528661\pi$
$80$	34.3440	3.83978
$81$	3.65172	0.405746
$82$	−16.2681	−1.79651
$83$	6.54855	0.718797	0.359399	−	0.933184i	$-0.382982\pi$
$83$	6.54855	0.718797	0.359399	+	0.933184i	$0.382982\pi$
$84$	6.32457	0.690067
$85$	−6.97724	−0.756788
$86$	24.8202	2.67643
$87$	−1.24900	−0.133907
$88$	33.4589	3.56673
$89$	−12.9084	−1.36829	−0.684146	−	0.729345i	$-0.739825\pi$
$89$	−12.9084	−1.36829	−0.684146	+	0.729345i	$0.739825\pi$
$90$	15.7281	1.65789
$91$	1.24130	0.130123
$92$	6.75982	0.704760
$93$	4.72562	0.490024
$94$	25.2812	2.60756
$95$	2.45685	0.252068
$96$	15.7624	1.60874
$97$	−12.3544	−1.25440	−0.627198	−	0.778859i	$-0.715799\pi$
$97$	−12.3544	−1.25440	−0.627198	+	0.778859i	$0.715799\pi$
$98$	13.0752	1.32079
$99$	8.67473	0.871843
$100$	5.54999	0.554999
$101$	−10.5379	−1.04856	−0.524279	−	0.851547i	$-0.675665\pi$
$101$	−10.5379	−1.04856	−0.524279	+	0.851547i	$0.675665\pi$
$102$	−6.16079	−0.610009
$103$	−10.5142	−1.03599	−0.517995	−	0.855383i	$-0.673322\pi$
$103$	−10.5142	−1.03599	−0.517995	+	0.855383i	$0.673322\pi$
$104$	7.65490	0.750624
$105$	2.90089	0.283098
$106$	−18.0711	−1.75522
$107$	9.67234	0.935060	0.467530	−	0.883977i	$-0.345144\pi$
$107$	9.67234	0.935060	0.467530	+	0.883977i	$0.345144\pi$
$108$	22.9648	2.20979
$109$	6.15033	0.589095	0.294547	−	0.955637i	$-0.404831\pi$
$109$	6.15033	0.589095	0.294547	+	0.955637i	$0.404831\pi$
$110$	24.4911	2.33513
$111$	7.43937	0.706114
$112$	−20.6361	−1.94992
$113$	−3.07916	−0.289663	−0.144832	−	0.989456i	$-0.546264\pi$
$113$	−3.07916	−0.289663	−0.144832	+	0.989456i	$0.546264\pi$
$114$	2.16936	0.203179
$115$	3.10053	0.289125
$116$	8.36462	0.776635
$117$	1.98465	0.183481
$118$	−31.8302	−2.93021
$119$	4.19237	0.384314
$120$	17.8893	1.63307
$121$	2.50786	0.227988
$122$	4.03901	0.365675
$123$	−4.79731	−0.432559
$124$	−31.6476	−2.84204
$125$	−9.73865	−0.871051
$126$	−9.45044	−0.841912
$127$	12.3594	1.09672	0.548362	−	0.836241i	$-0.315252\pi$
$127$	12.3594	1.09672	0.548362	+	0.836241i	$0.315252\pi$
$128$	−29.7320	−2.62796
$129$	7.31926	0.644425
$130$	5.60319	0.491432
$131$	6.51070	0.568842	0.284421	−	0.958699i	$-0.408199\pi$
$131$	6.51070	0.568842	0.284421	+	0.958699i	$0.408199\pi$
$132$	15.7460	1.37051
$133$	−1.47623	−0.128006
$134$	−20.8105	−1.79775
$135$	10.5333	0.906559
$136$	25.8537	2.21694
$137$	−17.9528	−1.53381	−0.766906	−	0.641759i	$-0.778205\pi$
$137$	−17.9528	−1.53381	−0.766906	+	0.641759i	$0.778205\pi$
$138$	2.73771	0.233050
$139$	15.1380	1.28399	0.641996	−	0.766708i	$-0.278106\pi$
$139$	15.1380	1.28399	0.641996	+	0.766708i	$0.278106\pi$
$140$	−19.4273	−1.64191
$141$	7.45522	0.627842
$142$	−37.5089	−3.14767
$143$	3.09040	0.258432
$144$	−32.9940	−2.74950
$145$	3.83660	0.318612
$146$	−17.2821	−1.43028
$147$	3.85576	0.318018
$148$	−49.8217	−4.09532
$149$	10.1271	0.829641	0.414820	−	0.909903i	$-0.363844\pi$
$149$	10.1271	0.829641	0.414820	+	0.909903i	$0.363844\pi$
$150$	2.24774	0.183527
$151$	19.6867	1.60208	0.801039	−	0.598612i	$-0.204281\pi$
$151$	19.6867	1.60208	0.801039	+	0.598612i	$0.204281\pi$
$152$	−9.10370	−0.738408
$153$	6.70297	0.541903
$154$	−14.7158	−1.18583
$155$	−14.5158	−1.16594
$156$	3.60244	0.288426
$157$	15.7539	1.25729	0.628647	−	0.777690i	$-0.283609\pi$
$157$	15.7539	1.25729	0.628647	+	0.777690i	$0.283609\pi$
$158$	4.33537	0.344903
$159$	−5.32901	−0.422618
$160$	−48.4178	−3.82776
$161$	−1.86299	−0.146824
$162$	−9.90449	−0.778170
$163$	1.98798	0.155711	0.0778554	−	0.996965i	$-0.475193\pi$
$163$	1.98798	0.155711	0.0778554	+	0.996965i	$0.475193\pi$
$164$	32.1277	2.50875
$165$	7.22220	0.562247
$166$	−17.7615	−1.37856
$167$	4.97328	0.384844	0.192422	−	0.981312i	$-0.438366\pi$
$167$	4.97328	0.384844	0.192422	+	0.981312i	$0.438366\pi$
$168$	−10.7490	−0.829307
$169$	−12.2930	−0.945613
$170$	18.9242	1.45142
$171$	−2.36027	−0.180495
$172$	−49.0173	−3.73753
$173$	14.4357	1.09752	0.548762	−	0.835979i	$-0.315099\pi$
$173$	14.4357	1.09752	0.548762	+	0.835979i	$0.315099\pi$
$174$	3.38765	0.256817
$175$	−1.52957	−0.115624
$176$	−51.3766	−3.87266
$177$	−9.38644	−0.705528
$178$	35.0113	2.62421
$179$	14.3611	1.07340	0.536701	−	0.843773i	$-0.319670\pi$
$179$	14.3611	1.07340	0.536701	+	0.843773i	$0.319670\pi$
$180$	−31.0614	−2.31518
$181$	−14.9618	−1.11210	−0.556050	−	0.831149i	$-0.687684\pi$
$181$	−14.9618	−1.11210	−0.556050	+	0.831149i	$0.687684\pi$
$182$	−3.36675	−0.249560
$183$	1.19107	0.0880463
$184$	−11.4888	−0.846965
$185$	−22.8517	−1.68009
$186$	−12.8172	−0.939804
$187$	10.4375	0.763268
$188$	−49.9278	−3.64136
$189$	−6.32905	−0.460371
$190$	−6.66368	−0.483434
$191$	22.9058	1.65740	0.828702	−	0.559690i	$-0.189080\pi$
$191$	22.9058	1.65740	0.828702	+	0.559690i	$0.189080\pi$
$192$	−20.3907	−1.47157
$193$	9.77008	0.703266	0.351633	−	0.936138i	$-0.385627\pi$
$193$	9.77008	0.703266	0.351633	+	0.936138i	$0.385627\pi$
$194$	33.5086	2.40577
$195$	1.65233	0.118326
$196$	−25.8221	−1.84444
$197$	3.30108	0.235192	0.117596	−	0.993062i	$-0.462481\pi$
$197$	3.30108	0.235192	0.117596	+	0.993062i	$0.462481\pi$
$198$	−23.5283	−1.67208
$199$	4.77708	0.338638	0.169319	−	0.985561i	$-0.445843\pi$
$199$	4.77708	0.338638	0.169319	+	0.985561i	$0.445843\pi$
$200$	−9.43260	−0.666986
$201$	−6.13684	−0.432859
$202$	28.5817	2.01100
$203$	−2.30527	−0.161798
$204$	12.1669	0.851854
$205$	14.7360	1.02921
$206$	28.5174	1.98690
$207$	−2.97864	−0.207030
$208$	−11.7542	−0.815007
$209$	−3.67530	−0.254226
$210$	−7.86803	−0.542946
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	35.6885	2.45110
$213$	−11.0610	−0.757890
$214$	−26.2341	−1.79333
$215$	−22.4827	−1.53331
$216$	−39.0303	−2.65568
$217$	8.72201	0.592089
$218$	−16.6814	−1.12981
$219$	−5.09634	−0.344379
$220$	−48.3673	−3.26092
$221$	2.38795	0.160631
$222$	−20.1777	−1.35424
$223$	−1.56503	−0.104802	−0.0524012	−	0.998626i	$-0.516687\pi$
$223$	−1.56503	−0.104802	−0.0524012	+	0.998626i	$0.516687\pi$
$224$	29.0924	1.94382
$225$	−2.44555	−0.163036
$226$	8.35156	0.555537
$227$	−25.5881	−1.69834	−0.849171	−	0.528118i	$-0.822898\pi$
$227$	−25.5881	−1.69834	−0.849171	+	0.528118i	$0.822898\pi$
$228$	−4.28426	−0.283732
$229$	10.2025	0.674200	0.337100	−	0.941469i	$-0.390554\pi$
$229$	10.2025	0.674200	0.337100	+	0.941469i	$0.390554\pi$
$230$	−8.40950	−0.554506
$231$	−4.33955	−0.285522
$232$	−14.2163	−0.933343
$233$	2.03950	0.133612	0.0668060	−	0.997766i	$-0.478719\pi$
$233$	2.03950	0.133612	0.0668060	+	0.997766i	$0.478719\pi$
$234$	−5.38293	−0.351893
$235$	−22.9004	−1.49386
$236$	62.8613	4.09192
$237$	1.27846	0.0830451
$238$	−11.3709	−0.737065
$239$	14.6027	0.944567	0.472284	−	0.881447i	$-0.343430\pi$
$239$	14.6027	0.944567	0.472284	+	0.881447i	$0.343430\pi$
$240$	−27.4693	−1.77314
$241$	10.4529	0.673329	0.336665	−	0.941625i	$-0.390701\pi$
$241$	10.4529	0.673329	0.336665	+	0.941625i	$0.390701\pi$
$242$	−6.80203	−0.437251
$243$	−15.7826	−1.01246
$244$	−7.97662	−0.510651
$245$	−11.8438	−0.756675
$246$	13.0117	0.829593
$247$	−0.840855	−0.0535023
$248$	53.7873	3.41550
$249$	−5.23772	−0.331927
$250$	26.4140	1.67057
$251$	−1.05763	−0.0667570	−0.0333785	−	0.999443i	$-0.510627\pi$
$251$	−1.05763	−0.0667570	−0.0333785	+	0.999443i	$0.510627\pi$
$252$	18.6636	1.17570
$253$	−4.63820	−0.291601
$254$	−33.5223	−2.10338
$255$	5.58060	0.349471
$256$	29.6538	1.85336
$257$	−22.3386	−1.39344	−0.696721	−	0.717342i	$-0.745359\pi$
$257$	−22.3386	−1.39344	−0.696721	+	0.717342i	$0.745359\pi$
$258$	−19.8519	−1.23592
$259$	13.7308	0.853187
$260$	−11.0657	−0.686266
$261$	−3.68578	−0.228144
$262$	−17.6589	−1.09097
$263$	−9.96838	−0.614676	−0.307338	−	0.951600i	$-0.599438\pi$
$263$	−9.96838	−0.614676	−0.307338	+	0.951600i	$0.599438\pi$
$264$	−26.7614	−1.64705
$265$	16.3692	1.00555
$266$	4.00396	0.245499
$267$	10.3245	0.631852
$268$	41.0986	2.51049
$269$	2.48959	0.151793	0.0758966	−	0.997116i	$-0.475818\pi$
$269$	2.48959	0.151793	0.0758966	+	0.997116i	$0.475818\pi$
$270$	−28.5692	−1.73867
$271$	−17.6039	−1.06936	−0.534680	−	0.845054i	$-0.679568\pi$
$271$	−17.6039	−1.06936	−0.534680	+	0.845054i	$0.679568\pi$
$272$	−39.6987	−2.40709
$273$	−0.992826	−0.0600885
$274$	48.6931	2.94166
$275$	−3.80809	−0.229636
$276$	−5.40670	−0.325445
$277$	−29.7323	−1.78644	−0.893220	−	0.449620i	$-0.851560\pi$
$277$	−29.7323	−1.78644	−0.893220	+	0.449620i	$0.851560\pi$
$278$	−41.0587	−2.46253
$279$	13.9452	0.834876
$280$	33.0181	1.97321
$281$	24.9925	1.49092	0.745462	−	0.666548i	$-0.232229\pi$
$281$	24.9925	1.49092	0.745462	+	0.666548i	$0.232229\pi$
$282$	−20.2206	−1.20412
$283$	−10.8391	−0.644319	−0.322160	−	0.946685i	$-0.604409\pi$
$283$	−10.8391	−0.644319	−0.322160	+	0.946685i	$0.604409\pi$
$284$	74.0761	4.39561
$285$	−1.96506	−0.116400
$286$	−8.38203	−0.495640
$287$	−8.85433	−0.522654
$288$	46.5144	2.74089
$289$	−8.93491	−0.525583
$290$	−10.4059	−0.611058
$291$	9.88138	0.579257
$292$	34.1304	1.99733
$293$	12.5011	0.730321	0.365161	−	0.930945i	$-0.381014\pi$
$293$	12.5011	0.730321	0.365161	+	0.930945i	$0.381014\pi$
$294$	−10.4579	−0.609918
$295$	28.8326	1.67870
$296$	84.6755	4.92166
$297$	−15.7571	−0.914322
$298$	−27.4674	−1.59115
$299$	−1.06115	−0.0613679
$300$	−4.43904	−0.256288
$301$	13.5091	0.778649
$302$	−53.3958	−3.07258
$303$	8.42850	0.484204
$304$	13.9789	0.801743
$305$	−3.65863	−0.209493
$306$	−18.1803	−1.03930
$307$	−5.90705	−0.337133	−0.168567	−	0.985690i	$-0.553914\pi$
$307$	−5.90705	−0.337133	−0.168567	+	0.985690i	$0.553914\pi$
$308$	29.0621	1.65597
$309$	8.40952	0.478401
$310$	39.3709	2.23612
$311$	−7.38585	−0.418813	−0.209406	−	0.977829i	$-0.567153\pi$
$311$	−7.38585	−0.418813	−0.209406	+	0.977829i	$0.567153\pi$
$312$	−6.12260	−0.346624
$313$	9.50913	0.537488	0.268744	−	0.963212i	$-0.413391\pi$
$313$	9.50913	0.537488	0.268744	+	0.963212i	$0.413391\pi$
$314$	−42.7289	−2.41133
$315$	8.56045	0.482327
$316$	−8.56190	−0.481644
$317$	22.6673	1.27312	0.636562	−	0.771225i	$-0.280356\pi$
$317$	22.6673	1.27312	0.636562	+	0.771225i	$0.280356\pi$
$318$	14.4538	0.810527
$319$	−5.73932	−0.321340
$320$	62.6345	3.50138
$321$	−7.73621	−0.431793
$322$	5.05296	0.281590
$323$	−2.83991	−0.158017
$324$	19.5603	1.08668
$325$	−0.871234	−0.0483273
$326$	−5.39197	−0.298633
$327$	−4.91921	−0.272033
$328$	−54.6033	−3.01496
$329$	13.7600	0.758613
$330$	−19.5886	−1.07832
$331$	4.83227	0.265605	0.132803	−	0.991142i	$-0.457602\pi$
$331$	4.83227	0.265605	0.132803	+	0.991142i	$0.457602\pi$
$332$	35.0772	1.92511
$333$	21.9534	1.20304
$334$	−13.4889	−0.738082
$335$	18.8507	1.02992
$336$	16.5053	0.900439
$337$	−14.7426	−0.803082	−0.401541	−	0.915841i	$-0.631525\pi$
$337$	−14.7426	−0.803082	−0.401541	+	0.915841i	$0.631525\pi$
$338$	33.3420	1.81356
$339$	2.46280	0.133761
$340$	−37.3734	−2.02686
$341$	21.7148	1.17592
$342$	6.40173	0.346166
$343$	17.4502	0.942220
$344$	83.3083	4.49168
$345$	−2.47989	−0.133513
$346$	−39.1536	−2.10491
$347$	26.6494	1.43062	0.715308	−	0.698810i	$-0.246287\pi$
$347$	26.6494	1.43062	0.715308	+	0.698810i	$0.246287\pi$
$348$	−6.69026	−0.358636
$349$	11.3757	0.608930	0.304465	−	0.952524i	$-0.401522\pi$
$349$	11.3757	0.608930	0.304465	+	0.952524i	$0.401522\pi$
$350$	4.14862	0.221753
$351$	−3.60500	−0.192421
$352$	72.4300	3.86053
$353$	−12.1480	−0.646571	−0.323286	−	0.946301i	$-0.604787\pi$
$353$	−12.1480	−0.646571	−0.323286	+	0.946301i	$0.604787\pi$
$354$	25.4587	1.35311
$355$	33.9765	1.80328
$356$	−69.1437	−3.66461
$357$	−3.35318	−0.177469
$358$	−38.9514	−2.05865
$359$	−17.9251	−0.946049	−0.473025	−	0.881049i	$-0.656838\pi$
$359$	−17.9251	−0.946049	−0.473025	+	0.881049i	$0.656838\pi$
$360$	52.7910	2.78233
$361$	1.00000	0.0526316
$362$	40.5806	2.13287
$363$	−2.00586	−0.105280
$364$	6.64898	0.348501
$365$	15.6546	0.819397
$366$	−3.23051	−0.168862
$367$	0.144009	0.00751719	0.00375860	−	0.999993i	$-0.498804\pi$
$367$	0.144009	0.00751719	0.00375860	+	0.999993i	$0.498804\pi$
$368$	17.6412	0.919611
$369$	−14.1567	−0.736970
$370$	61.9803	3.22220
$371$	−9.83568	−0.510643
$372$	25.3127	1.31240
$373$	9.72792	0.503693	0.251846	−	0.967767i	$-0.418962\pi$
$373$	9.72792	0.503693	0.251846	+	0.967767i	$0.418962\pi$
$374$	−28.3095	−1.46385
$375$	7.78925	0.402235
$376$	84.8558	4.37610
$377$	−1.31307	−0.0676266
$378$	17.1662	0.882933
$379$	−25.2656	−1.29781	−0.648903	−	0.760871i	$-0.724772\pi$
$379$	−25.2656	−1.29781	−0.648903	+	0.760871i	$0.724772\pi$
$380$	13.1601	0.675097
$381$	−9.88544	−0.506446
$382$	−62.1269	−3.17869
$383$	20.0775	1.02591	0.512957	−	0.858414i	$-0.328550\pi$
$383$	20.0775	1.02591	0.512957	+	0.858414i	$0.328550\pi$
$384$	23.7805	1.21354
$385$	13.3299	0.679356
$386$	−26.4992	−1.34877
$387$	21.5989	1.09794
$388$	−66.1759	−3.35957
$389$	4.75914	0.241298	0.120649	−	0.992695i	$-0.461502\pi$
$389$	4.75914	0.241298	0.120649	+	0.992695i	$0.461502\pi$
$390$	−4.48159	−0.226934
$391$	−3.58394	−0.181248
$392$	43.8866	2.21661
$393$	−5.20744	−0.262681
$394$	−8.95345	−0.451069
$395$	−3.92708	−0.197593
$396$	46.4659	2.33500
$397$	−21.0468	−1.05631	−0.528153	−	0.849149i	$-0.677115\pi$
$397$	−21.0468	−1.05631	−0.528153	+	0.849149i	$0.677115\pi$
$398$	−12.9568	−0.649465
$399$	1.18073	0.0591106
$400$	14.4839	0.724195
$401$	25.1820	1.25753	0.628763	−	0.777597i	$-0.283561\pi$
$401$	25.1820	1.25753	0.628763	+	0.777597i	$0.283561\pi$
$402$	16.6448	0.830169
$403$	4.96802	0.247474
$404$	−56.4459	−2.80829
$405$	8.97173	0.445809
$406$	6.25255	0.310309
$407$	34.1848	1.69448
$408$	−20.6785	−1.02374
$409$	19.8929	0.983639	0.491820	−	0.870697i	$-0.336332\pi$
$409$	19.8929	0.983639	0.491820	+	0.870697i	$0.336332\pi$
$410$	−39.9682	−1.97389
$411$	14.3592	0.708286
$412$	−56.3188	−2.77463
$413$	−17.3244	−0.852480
$414$	8.07892	0.397057
$415$	16.0888	0.789770
$416$	16.5709	0.812456
$417$	−12.1078	−0.592924
$418$	9.96846	0.487573
$419$	−7.81583	−0.381828	−0.190914	−	0.981607i	$-0.561145\pi$
$419$	−7.81583	−0.381828	−0.190914	+	0.981607i	$0.561145\pi$
$420$	15.5385	0.758203
$421$	15.1876	0.740199	0.370100	−	0.928992i	$-0.379324\pi$
$421$	15.1876	0.740199	0.370100	+	0.928992i	$0.379324\pi$
$422$	2.71228	0.132032
$423$	22.0002	1.06968
$424$	−60.6551	−2.94567
$425$	−2.94251	−0.142733
$426$	30.0007	1.45354
$427$	2.19834	0.106385
$428$	51.8096	2.50431
$429$	−2.47179	−0.119339
$430$	60.9795	2.94069
$431$	−10.0020	−0.481778	−0.240889	−	0.970553i	$-0.577439\pi$
$431$	−10.0020	−0.481778	−0.240889	+	0.970553i	$0.577439\pi$
$432$	59.9316	2.88346
$433$	6.48340	0.311572	0.155786	−	0.987791i	$-0.450209\pi$
$433$	6.48340	0.311572	0.155786	+	0.987791i	$0.450209\pi$
$434$	−23.6566	−1.13555
$435$	−3.06862	−0.147129
$436$	32.9441	1.57774
$437$	1.26199	0.0603692
$438$	13.8227	0.660475
$439$	8.62375	0.411589	0.205795	−	0.978595i	$-0.434022\pi$
$439$	8.62375	0.411589	0.205795	+	0.978595i	$0.434022\pi$
$440$	82.2036	3.91890
$441$	11.3783	0.541822
$442$	−6.47680	−0.308070
$443$	5.30205	0.251908	0.125954	−	0.992036i	$-0.459801\pi$
$443$	5.30205	0.251908	0.125954	+	0.992036i	$0.459801\pi$
$444$	39.8488	1.89114
$445$	−31.7141	−1.50339
$446$	4.24481	0.200998
$447$	−8.09991	−0.383112
$448$	−37.6348	−1.77808
$449$	−9.04103	−0.426672	−0.213336	−	0.976979i	$-0.568433\pi$
$449$	−9.04103	−0.426672	−0.213336	+	0.976979i	$0.568433\pi$
$450$	6.63301	0.312683
$451$	−22.0442	−1.03802
$452$	−16.4935	−0.775787
$453$	−15.7460	−0.739810
$454$	69.4021	3.25720
$455$	3.04969	0.142972
$456$	7.28140	0.340983
$457$	12.7837	0.597998	0.298999	−	0.954253i	$-0.403347\pi$
$457$	12.7837	0.597998	0.298999	+	0.954253i	$0.403347\pi$
$458$	−27.6721	−1.29303
$459$	−12.1755	−0.568306
$460$	16.6079	0.774347
$461$	11.1578	0.519671	0.259836	−	0.965653i	$-0.416332\pi$
$461$	11.1578	0.519671	0.259836	+	0.965653i	$0.416332\pi$
$462$	11.7701	0.547594
$463$	−40.5434	−1.88421	−0.942106	−	0.335316i	$-0.891157\pi$
$463$	−40.5434	−1.88421	−0.942106	+	0.335316i	$0.891157\pi$
$464$	21.8293	1.01340
$465$	11.6102	0.538408
$466$	−5.53170	−0.256251
$467$	30.3092	1.40254	0.701271	−	0.712895i	$-0.252616\pi$
$467$	30.3092	1.40254	0.701271	+	0.712895i	$0.252616\pi$
$468$	10.6307	0.491405
$469$	−11.3267	−0.523017
$470$	62.1123	2.86503
$471$	−12.6004	−0.580595
$472$	−106.837	−4.91758
$473$	33.6328	1.54644
$474$	−3.46755	−0.159270
$475$	1.03613	0.0475408
$476$	22.4563	1.02928
$477$	−15.7258	−0.720034
$478$	−39.6065	−1.81156
$479$	31.9210	1.45851	0.729255	−	0.684242i	$-0.239867\pi$
$479$	31.9210	1.45851	0.729255	+	0.684242i	$0.239867\pi$
$480$	38.7259	1.76759
$481$	7.82097	0.356605
$482$	−28.3512	−1.29136
$483$	1.49007	0.0678007
$484$	13.4333	0.610605
$485$	−30.3529	−1.37825
$486$	42.8070	1.94176
$487$	−32.9994	−1.49535	−0.747673	−	0.664067i	$-0.768829\pi$
$487$	−32.9994	−1.49535	−0.747673	+	0.664067i	$0.768829\pi$
$488$	13.5568	0.613689
$489$	−1.59004	−0.0719043
$490$	32.1238	1.45121
$491$	−15.1884	−0.685443	−0.342721	−	0.939437i	$-0.611349\pi$
$491$	−15.1884	−0.685443	−0.342721	+	0.939437i	$0.611349\pi$
$492$	−25.6967	−1.15849
$493$	−4.43477	−0.199732
$494$	2.28064	0.102611
$495$	21.3125	0.957927
$496$	−82.5912	−3.70845
$497$	−20.4152	−0.915748
$498$	14.2062	0.636594
$499$	19.5782	0.876441	0.438221	−	0.898867i	$-0.355609\pi$
$499$	19.5782	0.876441	0.438221	+	0.898867i	$0.355609\pi$
$500$	−52.1648	−2.33288
$501$	−3.97777	−0.177714
$502$	2.86859	0.128031
$503$	34.8193	1.55252	0.776259	−	0.630414i	$-0.217115\pi$
$503$	34.8193	1.55252	0.776259	+	0.630414i	$0.217115\pi$
$504$	−31.7202	−1.41293
$505$	−25.8900	−1.15209
$506$	12.5801	0.559254
$507$	9.83226	0.436666
$508$	66.2030	2.93728
$509$	−20.5164	−0.909375	−0.454688	−	0.890651i	$-0.650249\pi$
$509$	−20.5164	−0.909375	−0.454688	+	0.890651i	$0.650249\pi$
$510$	−15.1362	−0.670240
$511$	−9.40625	−0.416108
$512$	−20.9657	−0.926560
$513$	4.28730	0.189289
$514$	60.5885	2.67245
$515$	−25.8317	−1.13828
$516$	39.2054	1.72592
$517$	34.2576	1.50665
$518$	−37.2417	−1.63630
$519$	−11.5461	−0.506816
$520$	18.8070	0.824740
$521$	23.5097	1.02998	0.514990	−	0.857196i	$-0.327796\pi$
$521$	23.5097	1.02998	0.514990	+	0.857196i	$0.327796\pi$
$522$	9.99688	0.437551
$523$	2.74665	0.120102	0.0600512	−	0.998195i	$-0.480874\pi$
$523$	2.74665	0.120102	0.0600512	+	0.998195i	$0.480874\pi$
$524$	34.8744	1.52349
$525$	1.22339	0.0533932
$526$	27.0370	1.17887
$527$	16.7790	0.730905
$528$	41.0925	1.78832
$529$	−21.4074	−0.930756
$530$	−44.3980	−1.92853
$531$	−27.6992	−1.20204
$532$	−7.90740	−0.342829
$533$	−5.04338	−0.218453
$534$	−28.0031	−1.21181
$535$	23.7635	1.02739
$536$	−69.8499	−3.01706
$537$	−11.4864	−0.495677
$538$	−6.75248	−0.291120
$539$	17.7177	0.763154
$540$	56.4212	2.42798
$541$	39.6647	1.70532	0.852659	−	0.522468i	$-0.174989\pi$
$541$	39.6647	1.70532	0.852659	+	0.522468i	$0.174989\pi$
$542$	47.7467	2.05090
$543$	11.9669	0.513547
$544$	55.9667	2.39955
$545$	15.1105	0.647261
$546$	2.69282	0.115242
$547$	21.0644	0.900649	0.450325	−	0.892865i	$-0.351308\pi$
$547$	21.0644	0.900649	0.450325	+	0.892865i	$0.351308\pi$
$548$	−96.1638	−4.10791
$549$	3.51481	0.150009
$550$	10.3286	0.440413
$551$	1.56159	0.0665260
$552$	9.18906	0.391112
$553$	2.35964	0.100342
$554$	80.6424	3.42617
$555$	18.2774	0.775834
$556$	81.0866	3.43884
$557$	17.0092	0.720704	0.360352	−	0.932816i	$-0.382657\pi$
$557$	17.0092	0.720704	0.360352	+	0.932816i	$0.382657\pi$
$558$	−37.8233	−1.60119
$559$	7.69469	0.325451
$560$	−50.6998	−2.14246
$561$	−8.34824	−0.352463
$562$	−67.7866	−2.85940
$563$	20.8209	0.877497	0.438748	−	0.898610i	$-0.355422\pi$
$563$	20.8209	0.877497	0.438748	+	0.898610i	$0.355422\pi$
$564$	39.9337	1.68151
$565$	−7.56505	−0.318264
$566$	29.3988	1.23572
$567$	−5.39079	−0.226392
$568$	−125.898	−5.28254
$569$	11.2418	0.471279	0.235639	−	0.971841i	$-0.424282\pi$
$569$	11.2418	0.471279	0.235639	+	0.971841i	$0.424282\pi$
$570$	5.32980	0.223241
$571$	4.51880	0.189106	0.0945529	−	0.995520i	$-0.469858\pi$
$571$	4.51880	0.189106	0.0945529	+	0.995520i	$0.469858\pi$
$572$	16.5536	0.692142
$573$	−18.3207	−0.765358
$574$	24.0154	1.00238
$575$	1.30758	0.0545300
$576$	−60.1724	−2.50718
$577$	−0.364448	−0.0151722	−0.00758608	−	0.999971i	$-0.502415\pi$
$577$	−0.364448	−0.0151722	−0.00758608	+	0.999971i	$0.502415\pi$
$578$	24.2340	1.00800
$579$	−7.81439	−0.324755
$580$	20.5506	0.853319
$581$	−9.66719	−0.401063
$582$	−26.8011	−1.11094
$583$	−24.4874	−1.01416
$584$	−58.0069	−2.40034
$585$	4.87599	0.201597
$586$	−33.9065	−1.40066
$587$	−3.41227	−0.140839	−0.0704197	−	0.997517i	$-0.522434\pi$
$587$	−3.41227	−0.140839	−0.0704197	+	0.997517i	$0.522434\pi$
$588$	20.6533	0.851727
$589$	−5.90829	−0.243447
$590$	−78.2021	−3.21953
$591$	−2.64030	−0.108607
$592$	−130.020	−5.34380
$593$	−30.6507	−1.25867	−0.629336	−	0.777133i	$-0.716673\pi$
$593$	−30.6507	−1.25867	−0.629336	+	0.777133i	$0.716673\pi$
$594$	42.7378	1.75355
$595$	10.3000	0.422260
$596$	54.2453	2.22197
$597$	−3.82085	−0.156377
$598$	2.87814	0.117696
$599$	−0.285237	−0.0116545	−0.00582723	−	0.999983i	$-0.501855\pi$
$599$	−0.285237	−0.0116545	−0.00582723	+	0.999983i	$0.501855\pi$
$600$	7.54447	0.308002
$601$	−7.08967	−0.289194	−0.144597	−	0.989491i	$-0.546189\pi$
$601$	−7.08967	−0.289194	−0.144597	+	0.989491i	$0.546189\pi$
$602$	−36.6404	−1.49335
$603$	−18.1097	−0.737482
$604$	105.451	4.29075
$605$	6.16145	0.250499
$606$	−22.8605	−0.928643
$607$	−41.2577	−1.67460	−0.837299	−	0.546745i	$-0.815867\pi$
$607$	−41.2577	−1.67460	−0.837299	+	0.546745i	$0.815867\pi$
$608$	−19.7072	−0.799233
$609$	1.84382	0.0747154
$610$	9.92325	0.401781
$611$	7.83762	0.317076
$612$	35.9043	1.45134
$613$	−15.3921	−0.621682	−0.310841	−	0.950462i	$-0.600611\pi$
$613$	−15.3921	−0.621682	−0.310841	+	0.950462i	$0.600611\pi$
$614$	16.0216	0.646579
$615$	−11.7863	−0.475269
$616$	−49.3931	−1.99011
$617$	6.30336	0.253764	0.126882	−	0.991918i	$-0.459503\pi$
$617$	6.30336	0.253764	0.126882	+	0.991918i	$0.459503\pi$
$618$	−22.8090	−0.917512
$619$	47.2358	1.89857	0.949283	−	0.314423i	$-0.101811\pi$
$619$	47.2358	1.89857	0.949283	+	0.314423i	$0.101811\pi$
$620$	−77.7535	−3.12266
$621$	5.41053	0.217117
$622$	20.0325	0.803230
$623$	19.0559	0.763457
$624$	9.40135	0.376355
$625$	−29.1071	−1.16428
$626$	−25.7914	−1.03083
$627$	2.93961	0.117397
$628$	84.3851	3.36733
$629$	26.4146	1.05322
$630$	−23.2184	−0.925041
$631$	−10.9022	−0.434009	−0.217005	−	0.976171i	$-0.569629\pi$
$631$	−10.9022	−0.434009	−0.217005	+	0.976171i	$0.569629\pi$
$632$	14.5515	0.578830
$633$	0.799829	0.0317903
$634$	−61.4802	−2.44169
$635$	30.3653	1.20501
$636$	−28.5447	−1.13187
$637$	4.05354	0.160607
$638$	15.5667	0.616290
$639$	−32.6409	−1.29125
$640$	−73.0471	−2.88744
$641$	42.2976	1.67065	0.835327	−	0.549753i	$-0.185278\pi$
$641$	42.2976	1.67065	0.835327	+	0.549753i	$0.185278\pi$
$642$	20.9828	0.828125
$643$	40.0207	1.57826	0.789131	−	0.614224i	$-0.210531\pi$
$643$	40.0207	1.57826	0.789131	+	0.614224i	$0.210531\pi$
$644$	−9.97907	−0.393230
$645$	17.9823	0.708054
$646$	7.70264	0.303056
$647$	39.5600	1.55527	0.777633	−	0.628719i	$-0.216420\pi$
$647$	39.5600	1.55527	0.777633	+	0.628719i	$0.216420\pi$
$648$	−33.2442	−1.30595
$649$	−43.1318	−1.69307
$650$	2.36303	0.0926857
$651$	−6.97611	−0.273415
$652$	10.6486	0.417030
$653$	−42.0626	−1.64604	−0.823019	−	0.568014i	$-0.807712\pi$
$653$	−42.0626	−1.64604	−0.823019	+	0.568014i	$0.807712\pi$
$654$	13.3423	0.521725
$655$	15.9958	0.625009
$656$	83.8441	3.27356
$657$	−15.0392	−0.586734
$658$	−37.3210	−1.45492
$659$	−30.2148	−1.17700	−0.588501	−	0.808497i	$-0.700282\pi$
$659$	−30.2148	−1.17700	−0.588501	+	0.808497i	$0.700282\pi$
$660$	38.6855	1.50583
$661$	27.5504	1.07158	0.535792	−	0.844350i	$-0.320013\pi$
$661$	27.5504	1.07158	0.535792	+	0.844350i	$0.320013\pi$
$662$	−13.1065	−0.509397
$663$	−1.90995	−0.0741764
$664$	−59.6161	−2.31355
$665$	−3.62689	−0.140645
$666$	−59.5438	−2.30728
$667$	1.97071	0.0763063
$668$	26.6393	1.03070
$669$	1.25176	0.0483958
$670$	−51.1283	−1.97526
$671$	5.47310	0.211287
$672$	−23.2690	−0.897620
$673$	6.75868	0.260528	0.130264	−	0.991479i	$-0.458418\pi$
$673$	6.75868	0.260528	0.130264	+	0.991479i	$0.458418\pi$
$674$	39.9862	1.54021
$675$	4.44219	0.170980
$676$	−65.8469	−2.53257
$677$	−11.0544	−0.424853	−0.212427	−	0.977177i	$-0.568137\pi$
$677$	−11.0544	−0.424853	−0.212427	+	0.977177i	$0.568137\pi$
$678$	−6.67982	−0.256537
$679$	18.2379	0.699908
$680$	63.5187	2.43583
$681$	20.4661	0.784262
$682$	−58.8966	−2.25527
$683$	−2.08758	−0.0798791	−0.0399396	−	0.999202i	$-0.512717\pi$
$683$	−2.08758	−0.0798791	−0.0399396	+	0.999202i	$0.512717\pi$
$684$	−12.6427	−0.483408
$685$	−44.1074	−1.68526
$686$	−47.3298	−1.80706
$687$	−8.16025	−0.311333
$688$	−127.921	−4.87694
$689$	−5.60235	−0.213433
$690$	6.72616	0.256060
$691$	8.14697	0.309926	0.154963	−	0.987920i	$-0.450474\pi$
$691$	8.14697	0.309926	0.154963	+	0.987920i	$0.450474\pi$
$692$	77.3243	2.93943
$693$	−12.8059	−0.486457
$694$	−72.2807	−2.74374
$695$	37.1920	1.41077
$696$	11.3706	0.431000
$697$	−17.0336	−0.645192
$698$	−30.8542	−1.16785
$699$	−1.63125	−0.0616995
$700$	−8.19308	−0.309669
$701$	33.2624	1.25630	0.628152	−	0.778091i	$-0.283812\pi$
$701$	33.2624	1.25630	0.628152	+	0.778091i	$0.283812\pi$
$702$	9.77777	0.369038
$703$	−9.30121	−0.350802
$704$	−93.6975	−3.53136
$705$	18.3164	0.689835
$706$	32.9487	1.24004
$707$	15.5564	0.585057
$708$	−50.2782	−1.88957
$709$	9.89398	0.371576	0.185788	−	0.982590i	$-0.440516\pi$
$709$	9.89398	0.371576	0.185788	+	0.982590i	$0.440516\pi$
$710$	−92.1538	−3.45847
$711$	3.77271	0.141488
$712$	117.515	4.40405
$713$	−7.45621	−0.279237
$714$	9.09476	0.340363
$715$	7.59266	0.283949
$716$	76.9250	2.87482
$717$	−11.6796	−0.436183
$718$	48.6179	1.81440
$719$	−42.2049	−1.57398	−0.786988	−	0.616968i	$-0.788361\pi$
$719$	−42.2049	−1.57398	−0.786988	+	0.616968i	$0.788361\pi$
$720$	−81.0613	−3.02098
$721$	15.5213	0.578045
$722$	−2.71228	−0.100941
$723$	−8.36051	−0.310931
$724$	−80.1424	−2.97847
$725$	1.61801	0.0600913
$726$	5.44046	0.201914
$727$	−44.4926	−1.65014	−0.825069	−	0.565031i	$-0.808864\pi$
$727$	−44.4926	−1.65014	−0.825069	+	0.565031i	$0.808864\pi$
$728$	−11.3004	−0.418821
$729$	1.66826	0.0617875
$730$	−42.4596	−1.57150
$731$	25.9881	0.961205
$732$	6.37993	0.235809
$733$	7.53910	0.278463	0.139232	−	0.990260i	$-0.455537\pi$
$733$	7.53910	0.278463	0.139232	+	0.990260i	$0.455537\pi$
$734$	−0.390592	−0.0144170
$735$	9.47304	0.349418
$736$	−24.8703	−0.916732
$737$	−28.1995	−1.03874
$738$	38.3971	1.41342
$739$	36.9798	1.36032	0.680162	−	0.733062i	$-0.261909\pi$
$739$	36.9798	1.36032	0.680162	+	0.733062i	$0.261909\pi$
$740$	−122.405	−4.49968
$741$	0.672540	0.0247064
$742$	26.6771	0.979349
$743$	45.9901	1.68721	0.843607	−	0.536961i	$-0.180428\pi$
$743$	45.9901	1.68721	0.843607	+	0.536961i	$0.180428\pi$
$744$	−43.0206	−1.57721
$745$	24.8807	0.911558
$746$	−26.3849	−0.966019
$747$	−15.4564	−0.565519
$748$	55.9084	2.04421
$749$	−14.2786	−0.521730
$750$	−21.1266	−0.771436
$751$	22.9971	0.839178	0.419589	−	0.907714i	$-0.362174\pi$
$751$	22.9971	0.839178	0.419589	+	0.907714i	$0.362174\pi$
$752$	−130.297	−4.75145
$753$	0.845923	0.0308271
$754$	3.56142	0.129699
$755$	48.3673	1.76027
$756$	−33.9014	−1.23298
$757$	−32.2850	−1.17342	−0.586709	−	0.809798i	$-0.699577\pi$
$757$	−32.2850	−1.17342	−0.586709	+	0.809798i	$0.699577\pi$
$758$	68.5274	2.48903
$759$	3.70976	0.134656
$760$	−22.3665	−0.811317
$761$	0.697468	0.0252832	0.0126416	−	0.999920i	$-0.495976\pi$
$761$	0.697468	0.0252832	0.0126416	+	0.999920i	$0.495976\pi$
$762$	26.8121	0.971300
$763$	−9.07932	−0.328694
$764$	122.694	4.43892
$765$	16.4682	0.595409
$766$	−54.4560	−1.96757
$767$	−9.86791	−0.356310
$768$	−23.7180	−0.855849
$769$	−0.370782	−0.0133707	−0.00668537	−	0.999978i	$-0.502128\pi$
$769$	−0.370782	−0.0133707	−0.00668537	+	0.999978i	$0.502128\pi$
$770$	−36.1545	−1.30292
$771$	17.8670	0.643466
$772$	52.3332	1.88351
$773$	−12.6975	−0.456696	−0.228348	−	0.973580i	$-0.573332\pi$
$773$	−12.6975	−0.456696	−0.228348	+	0.973580i	$0.573332\pi$
$774$	−58.5824	−2.10570
$775$	−6.12175	−0.219900
$776$	112.471	4.03746
$777$	−10.9822	−0.393986
$778$	−12.9081	−0.462779
$779$	5.99792	0.214898
$780$	8.85067	0.316905
$781$	−50.8268	−1.81872
$782$	9.72065	0.347610
$783$	6.69501	0.239260
$784$	−67.3884	−2.40673
$785$	38.7049	1.38144
$786$	14.1241	0.503788
$787$	48.6276	1.73339	0.866693	−	0.498841i	$-0.166241\pi$
$787$	48.6276	1.73339	0.866693	+	0.498841i	$0.166241\pi$
$788$	17.6821	0.629900
$789$	7.97299	0.283846
$790$	10.6514	0.378958
$791$	4.54556	0.161622
$792$	−78.9722	−2.80615
$793$	1.25216	0.0444656
$794$	57.0847	2.02586
$795$	−13.0926	−0.464346
$796$	25.5883	0.906954
$797$	−19.7509	−0.699614	−0.349807	−	0.936822i	$-0.613753\pi$
$797$	−19.7509	−0.699614	−0.349807	+	0.936822i	$0.613753\pi$
$798$	−3.20248	−0.113367
$799$	26.4708	0.936471
$800$	−20.4192	−0.721928
$801$	30.4675	1.07651
$802$	−68.3006	−2.41178
$803$	−23.4183	−0.826413
$804$	−32.8718	−1.15930
$805$	−4.57710	−0.161321
$806$	−13.4747	−0.474625
$807$	−1.99125	−0.0700953
$808$	95.9337	3.37494
$809$	−22.6012	−0.794617	−0.397309	−	0.917685i	$-0.630056\pi$
$809$	−22.6012	−0.794617	−0.397309	+	0.917685i	$0.630056\pi$
$810$	−24.3339	−0.855005
$811$	−40.1013	−1.40815	−0.704073	−	0.710128i	$-0.748637\pi$
$811$	−40.1013	−1.40815	−0.704073	+	0.710128i	$0.748637\pi$
$812$	−12.3481	−0.433334
$813$	14.0801	0.493811
$814$	−92.7187	−3.24979
$815$	4.88418	0.171085
$816$	31.7522	1.11155
$817$	−9.15103	−0.320154
$818$	−53.9551	−1.88650
$819$	−2.92980	−0.102376
$820$	78.9331	2.75646
$821$	−25.1845	−0.878945	−0.439473	−	0.898256i	$-0.644835\pi$
$821$	−25.1845	−0.878945	−0.439473	+	0.898256i	$0.644835\pi$
$822$	−38.9461	−1.35840
$823$	54.4555	1.89820	0.949099	−	0.314978i	$-0.101997\pi$
$823$	54.4555	1.89820	0.949099	+	0.314978i	$0.101997\pi$
$824$	95.7178	3.33449
$825$	3.04582	0.106042
$826$	46.9888	1.63495
$827$	−17.1538	−0.596498	−0.298249	−	0.954488i	$-0.596403\pi$
$827$	−17.1538	−0.596498	−0.298249	+	0.954488i	$0.596403\pi$
$828$	−15.9550	−0.554475
$829$	4.18673	0.145411	0.0727055	−	0.997353i	$-0.476837\pi$
$829$	4.18673	0.145411	0.0727055	+	0.997353i	$0.476837\pi$
$830$	−43.6375	−1.51468
$831$	23.7807	0.824945
$832$	−21.4366	−0.743180
$833$	13.6905	0.474346
$834$	32.8399	1.13715
$835$	12.2186	0.422843
$836$	−19.6867	−0.680878
$837$	−25.3306	−0.875554
$838$	21.1987	0.732298
$839$	−15.8364	−0.546732	−0.273366	−	0.961910i	$-0.588137\pi$
$839$	−15.8364	−0.546732	−0.273366	+	0.961910i	$0.588137\pi$
$840$	−26.4088	−0.911191
$841$	−26.5614	−0.915912
$842$	−41.1931	−1.41961
$843$	−19.9897	−0.688481
$844$	−5.35647	−0.184377
$845$	−30.2020	−1.03898
$846$	−59.6706	−2.05152
$847$	−3.70219	−0.127209
$848$	93.1368	3.19833
$849$	8.66944	0.297535
$850$	7.98092	0.273743
$851$	−11.7380	−0.402375
$852$	−59.2482	−2.02981
$853$	25.5819	0.875908	0.437954	−	0.898997i	$-0.355703\pi$
$853$	25.5819	0.875908	0.437954	+	0.898997i	$0.355703\pi$
$854$	−5.96252	−0.204033
$855$	−5.79885	−0.198316
$856$	−88.0541	−3.00963
$857$	36.4245	1.24424	0.622118	−	0.782924i	$-0.286272\pi$
$857$	36.4245	1.24424	0.622118	+	0.782924i	$0.286272\pi$
$858$	6.70419	0.228877
$859$	33.6887	1.14944	0.574722	−	0.818349i	$-0.305110\pi$
$859$	33.6887	1.14944	0.574722	+	0.818349i	$0.305110\pi$
$860$	−120.428	−4.10657
$861$	7.08194	0.241352
$862$	27.1282	0.923989
$863$	17.5732	0.598197	0.299099	−	0.954222i	$-0.403314\pi$
$863$	17.5732	0.598197	0.299099	+	0.954222i	$0.403314\pi$
$864$	−84.4908	−2.87443
$865$	35.4663	1.20589
$866$	−17.5848	−0.597556
$867$	7.14640	0.242704
$868$	46.7192	1.58575
$869$	5.87468	0.199285
$870$	8.32297	0.282175
$871$	−6.45162	−0.218605
$872$	−55.9908	−1.89609
$873$	29.1597	0.986907
$874$	−3.42287	−0.115780
$875$	14.3765	0.486015
$876$	−27.2984	−0.922329
$877$	−37.4732	−1.26538	−0.632690	−	0.774405i	$-0.718049\pi$
$877$	−37.4732	−1.26538	−0.632690	+	0.774405i	$0.718049\pi$
$878$	−23.3900	−0.789376
$879$	−9.99872	−0.337249
$880$	−126.225	−4.25504
$881$	27.5202	0.927179	0.463589	−	0.886050i	$-0.346561\pi$
$881$	27.5202	0.927179	0.463589	+	0.886050i	$0.346561\pi$
$882$	−30.8610	−1.03915
$883$	20.5747	0.692393	0.346196	−	0.938162i	$-0.387473\pi$
$883$	20.5747	0.692393	0.346196	+	0.938162i	$0.387473\pi$
$884$	12.7910	0.430208
$885$	−23.0611	−0.775191
$886$	−14.3807	−0.483128
$887$	−29.4657	−0.989361	−0.494680	−	0.869075i	$-0.664715\pi$
$887$	−29.4657	−0.989361	−0.494680	+	0.869075i	$0.664715\pi$
$888$	−67.7258	−2.27273
$889$	−18.2454	−0.611932
$890$	86.0177	2.88332
$891$	−13.4212	−0.449626
$892$	−8.38306	−0.280686
$893$	−9.32102	−0.311916
$894$	21.9692	0.734761
$895$	35.2832	1.17939
$896$	43.8913	1.46631
$897$	0.848739	0.0283386
$898$	24.5218	0.818304
$899$	−9.22633	−0.307715
$900$	−13.0995	−0.436650
$901$	−18.9214	−0.630364
$902$	59.7900	1.99079
$903$	−10.8049	−0.359566
$904$	28.0318	0.932323
$905$	−36.7589	−1.22191
$906$	42.7075	1.41886
$907$	6.74109	0.223834	0.111917	−	0.993718i	$-0.464301\pi$
$907$	6.74109	0.223834	0.111917	+	0.993718i	$0.464301\pi$
$908$	−137.062	−4.54856
$909$	24.8723	0.824961
$910$	−8.27161	−0.274201
$911$	8.41905	0.278936	0.139468	−	0.990227i	$-0.455461\pi$
$911$	8.41905	0.278936	0.139468	+	0.990227i	$0.455461\pi$
$912$	−11.1807	−0.370230
$913$	−24.0679	−0.796532
$914$	−34.6731	−1.14688
$915$	2.92628	0.0967398
$916$	54.6494	1.80567
$917$	−9.61131	−0.317393
$918$	33.0235	1.08994
$919$	−26.9291	−0.888310	−0.444155	−	0.895950i	$-0.646496\pi$
$919$	−26.9291	−0.888310	−0.444155	+	0.895950i	$0.646496\pi$
$920$	−28.2263	−0.930592
$921$	4.72463	0.155682
$922$	−30.2632	−0.996664
$923$	−11.6284	−0.382754
$924$	−23.2447	−0.764695
$925$	−9.63724	−0.316871
$926$	109.965	3.61368
$927$	24.8163	0.815074
$928$	−30.7746	−1.01023
$929$	−6.09447	−0.199953	−0.0999765	−	0.994990i	$-0.531877\pi$
$929$	−6.09447	−0.199953	−0.0999765	+	0.994990i	$0.531877\pi$
$930$	−31.4900	−1.03260
$931$	−4.82074	−0.157993
$932$	10.9245	0.357845
$933$	5.90741	0.193400
$934$	−82.2071	−2.68990
$935$	25.6435	0.838632
$936$	−18.0677	−0.590560
$937$	−39.3441	−1.28532	−0.642658	−	0.766153i	$-0.722168\pi$
$937$	−39.3441	−1.28532	−0.642658	+	0.766153i	$0.722168\pi$
$938$	30.7211	1.00308
$939$	−7.60567	−0.248202
$940$	−122.665	−4.00090
$941$	7.71358	0.251455	0.125728	−	0.992065i	$-0.459873\pi$
$941$	7.71358	0.251455	0.125728	+	0.992065i	$0.459873\pi$
$942$	34.1758	1.11351
$943$	7.56932	0.246491
$944$	164.050	5.33937
$945$	−15.5496	−0.505827
$946$	−91.2217	−2.96587
$947$	18.3728	0.597037	0.298518	−	0.954404i	$-0.403508\pi$
$947$	18.3728	0.597037	0.298518	+	0.954404i	$0.403508\pi$
$948$	6.84805	0.222414
$949$	−5.35776	−0.173920
$950$	−2.81027	−0.0911773
$951$	−18.1300	−0.587905
$952$	−38.1661	−1.23697
$953$	−31.7844	−1.02960	−0.514799	−	0.857311i	$-0.672133\pi$
$953$	−31.7844	−1.02960	−0.514799	+	0.857311i	$0.672133\pi$
$954$	42.6527	1.38093
$955$	56.2761	1.82105
$956$	78.2187	2.52978
$957$	4.59047	0.148389
$958$	−86.5789	−2.79724
$959$	26.5025	0.855812
$960$	−50.0969	−1.61687
$961$	3.90789	0.126061
$962$	−21.2127	−0.683924
$963$	−22.8294	−0.735666
$964$	55.9906	1.80334
$965$	24.0036	0.772705
$966$	−4.04150	−0.130033
$967$	−4.79849	−0.154309	−0.0771546	−	0.997019i	$-0.524583\pi$
$967$	−4.79849	−0.154309	−0.0771546	+	0.997019i	$0.524583\pi$
$968$	−22.8309	−0.733811
$969$	2.27144	0.0729692
$970$	82.3256	2.64332
$971$	42.3773	1.35995	0.679976	−	0.733234i	$-0.261990\pi$
$971$	42.3773	1.35995	0.679976	+	0.733234i	$0.261990\pi$
$972$	−84.5394	−2.71160
$973$	−22.3473	−0.716421
$974$	89.5037	2.86788
$975$	0.696838	0.0223167
$976$	−20.8167	−0.666326
$977$	44.7260	1.43091	0.715456	−	0.698658i	$-0.246219\pi$
$977$	44.7260	1.43091	0.715456	+	0.698658i	$0.246219\pi$
$978$	4.31265	0.137903
$979$	47.4425	1.51627
$980$	−63.4412	−2.02656
$981$	−14.5165	−0.463475
$982$	41.1952	1.31459
$983$	−44.3591	−1.41484	−0.707418	−	0.706795i	$-0.750140\pi$
$983$	−44.3591	−1.41484	−0.707418	+	0.706795i	$0.750140\pi$
$984$	43.6733	1.39225
$985$	8.11026	0.258415
$986$	12.0284	0.383061
$987$	−11.0056	−0.350313
$988$	−4.50402	−0.143292
$989$	−11.5485	−0.367221
$990$	−57.8056	−1.83718
$991$	−46.5096	−1.47743	−0.738713	−	0.674020i	$-0.764566\pi$
$991$	−46.5096	−1.47743	−0.738713	+	0.674020i	$0.764566\pi$
$992$	116.436	3.69685
$993$	−3.86499	−0.122652
$994$	55.3718	1.75629
$995$	11.7366	0.372075
$996$	−28.0557	−0.888979
$997$	30.3045	0.959754	0.479877	−	0.877336i	$-0.340681\pi$
$997$	30.3045	0.959754	0.479877	+	0.877336i	$0.340681\pi$
$998$	−53.1017	−1.68090
$999$	−39.8771	−1.26165

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.e.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-2.71228 q^{2} -0.799829 q^{3} +5.35647 q^{4} +2.45685 q^{5} +2.16936 q^{6} -1.47623 q^{7} -9.10370 q^{8} -2.36027 q^{9} +O(q^{10})\)	\(q-2.71228 q^{2} -0.799829 q^{3} +5.35647 q^{4} +2.45685 q^{5} +2.16936 q^{6} -1.47623 q^{7} -9.10370 q^{8} -2.36027 q^{9} -6.66368 q^{10} -3.67530 q^{11} -4.28426 q^{12} -0.840855 q^{13} +4.00396 q^{14} -1.96506 q^{15} +13.9789 q^{16} -2.83991 q^{17} +6.40173 q^{18} +1.00000 q^{19} +13.1601 q^{20} +1.18073 q^{21} +9.96846 q^{22} +1.26199 q^{23} +7.28140 q^{24} +1.03613 q^{25} +2.28064 q^{26} +4.28730 q^{27} -7.90740 q^{28} +1.56159 q^{29} +5.32980 q^{30} -5.90829 q^{31} -19.7072 q^{32} +2.93961 q^{33} +7.70264 q^{34} -3.62689 q^{35} -12.6427 q^{36} -9.30121 q^{37} -2.71228 q^{38} +0.672540 q^{39} -22.3665 q^{40} +5.99792 q^{41} -3.20248 q^{42} -9.15103 q^{43} -19.6867 q^{44} -5.79885 q^{45} -3.42287 q^{46} -9.32102 q^{47} -11.1807 q^{48} -4.82074 q^{49} -2.81027 q^{50} +2.27144 q^{51} -4.50402 q^{52} +6.66269 q^{53} -11.6284 q^{54} -9.02968 q^{55} +13.4392 q^{56} -0.799829 q^{57} -4.23547 q^{58} +11.7356 q^{59} -10.5258 q^{60} -1.48915 q^{61} +16.0249 q^{62} +3.48431 q^{63} +25.4938 q^{64} -2.06586 q^{65} -7.97306 q^{66} +7.67269 q^{67} -15.2119 q^{68} -1.00938 q^{69} +9.83714 q^{70} +13.8293 q^{71} +21.4872 q^{72} +6.37179 q^{73} +25.2275 q^{74} -0.828725 q^{75} +5.35647 q^{76} +5.42561 q^{77} -1.82412 q^{78} -1.59842 q^{79} +34.3440 q^{80} +3.65172 q^{81} -16.2681 q^{82} +6.54855 q^{83} +6.32457 q^{84} -6.97724 q^{85} +24.8202 q^{86} -1.24900 q^{87} +33.4589 q^{88} -12.9084 q^{89} +15.7281 q^{90} +1.24130 q^{91} +6.75982 q^{92} +4.72562 q^{93} +25.2812 q^{94} +2.45685 q^{95} +15.7624 q^{96} -12.3544 q^{97} +13.0752 q^{98} +8.67473 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) 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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